Abstract

Seismoelectric methods measure electrical fields generated when seismic deformation drives relative motion of electrolyte in porous media. The resulting observations include coseismic fields that travel with mechanical wave modes and electromagnetic interface conversions generated when the seismic wave reaches contrasts in elastic, hydraulic, electrical or electrokinetic properties. This review combines the supplied theory and application documents into one continuous framework, beginning with electrical-double-layer physics and the coupled Biot-Maxwell-Pride formulation, then developing electrical petrophysics, dynamic permeability, effective excess charge, saturation effects, field acquisition, signal processing and inversion.

The same framework is then applied to grounded-dipole geometry, lateral and vertical resolution, reflected-wave contamination and reflection-assisted calibration, fracture detection, elastic/geotechnical stiffness, thermal sensitivity, and groundwater quality including salinity and NAPL/DNAPL multiphase effects. Throughout, measured quantities, derived diagnostics and model-dependent inversions are kept distinct. The review therefore emphasizes not only what seismoelectric data are sensitive to, but also which independent velocity, density, fluid, electrical, petrophysical or chemical constraints are required before a formation property can be reported quantitatively.

Keywords: seismoelectric, electrokinetic coupling, Biot poroelasticity, Maxwell equations, streaming potential, interface conversion, grounded dipole, reflection, permeability, vertical resolution, fracture, elastic modulus, thermal monitoring, salinity, NAPL, DNAPL.

Source and synthesis policy. The governing scientific content, equations, illustrations and application discussions below are carried forward from the supplied documents and their embedded bibliographies. Repeated introductory material and duplicate reference lists are consolidated for continuity. A small number of transition paragraphs and the final cross-application decision table are editorial synthesis only; they do not introduce a new constitutive law.

Part I

Fundamental seismoelectric theory

The physical chain from pore-scale charge separation through coupled Biot-Maxwell-electrokinetic equations to observable coseismic and interface-conversion signals.

Integrated source module

Core Theory: From Pore-Scale Physics to Field Inversion

Primary synthesis backbone. It defines terminology, governing equations, field observables, acquisition, processing, inversion and the main identifiability limits used by every later application.

1.Scope, terminology, and the complete physical chain

Seismoelectric conversion is an electrokinetic coupling phenomenon in a fluid-bearing porous medium. A mechanical disturbance produces relative motion between pore fluid and the solid frame; that motion transports the mobile part of the electrical double layer, creates streaming current and charge separation, and therefore produces an electric field. In the reciprocal direction, an applied electric field can force the mobile charge and generate relative fluid motion. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016)

seismic forcing→frame / pore-fluid relative motion→mobile excess charge transport→streaming current→electric field / voltage→processing & inversion

Source policy. Scientific statements, equations and interpretation rules in this synthesis are tied to papers physically present in the supplied ZIP archive. Equation provenance distinguishes: direct source relations, rearrangements/derived estimators, and general measurement identities. The bibliography lists the exact archive PDF used for each cited paper. No scientific claim is added from the public website.

Generation

Seismic strain and acceleration create pore-pressure gradients and fluid/frame slip. The moving diffuse-layer charge produces streaming current. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012)

Observation

The electrical response appears either as a field travelling with the mechanical wave or as a fast converted response triggered at a material contrast. (Haartsen & Pride, 1997; Haines et al., 2007; Schakel et al., 2011)

Inference

Amplitude, phase, moveout, conductivity, saturation and frequency dependence can constrain coupling and hydraulic properties, but unique permeability requires supporting information and an appropriate forward model. (Jouniaux & Zyserman, 2016; Bordes et al., 2015; Bonnetier et al., 2019)

2.Pore-scale origin: the electrical double layer

A seismic wave squeezes and accelerates the rock skeleton. In a water-bearing porous rock, the pore water does not move perfectly with the grains. Because the grain-water boundary carries a microscopic electrical charge structure, that relative fluid motion transports charge and creates a measurable voltage. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012)

The crucial word is relative. If solid grains and pore fluid moved together with no relative motion, there would be little electrokinetic charge transport. Permeability, pore geometry, frequency, and saturation matter because they control how easily fluid can move relative to the frame. (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016)

Figure 1. A water-filled pore is electrically neutral overall, but charge is separated over a very thin zone at the grain-water boundary. Pressure-driven fluid motion preferentially drags mobile ions. The animation in the HTML version represents this relative motion.

Mineral surfaces in water acquire charge through surface chemistry. Oppositely charged ions in the water gather close to that surface. Some are strongly associated with the surface; others remain mobile in a diffuse layer. The result is called the electrical double layer. (Jouniaux & Ishido, 2012; Beamish, 1999)

Imagine a crowded sidewalk next to a wall. The people pressed directly against the wall barely move, while people one or two steps away can be carried along by a passing crowd. In a pore, the mobile ions are the part of the charge cloud that can be dragged by flowing water. The electrical potential at the effective slipping surface is summarized by a parameter called the zeta potential. (Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016)

Surface charge

Mineral-water chemistry establishes charge at the pore wall.

Mobile counter-charge

The pore water contains ions that balance the surface charge.

Relative flow

Seismic pressure and acceleration shift water relative to grains and carry some of those ions.

3.What is observed: coseismic fields, interface conversion, and reciprocity

Figure 2. Two observable signatures. A small electric field can travel with the seismic wave inside a porous layer (coseismic response). When the seismic wave crosses a strong contrast, the disturbed charge distribution can create an electromagnetic interface response. (Haartsen & Pride, 1997; Haines et al., 2007; Schakel et al., 2011)

Coseismic field

This electric field exists in the neighborhood of the seismic disturbance. On a shot gather it tends to show seismic-like moveout because it travels with the P, S, or surface-wave motion that drives the pore-fluid displacement. (Jouniaux & Zyserman, 2016; Haines et al., 2003; Haines et al., 2007)

Think: electricity riding with the mechanical wave.

Interface response

When the seismic wave reaches a boundary where electrical, hydraulic, or mechanical properties change, the balanced charge distribution is disturbed. That disturbance can launch an electromagnetic response. Across a surface array it may appear nearly flat in time compared with seismic moveout. (Haartsen & Pride, 1997; Schakel et al., 2011; Dupuis et al., 2009)

Think: a fast electrical flash triggered when the slower seismic wave reaches a boundary.

Not every flat electrical event is a buried interface. Source electronics, metal moving in the Earth's magnetic field, power-line interference, and other cultural noise can mimic useful signals. Field geometry and source-control tests are part of the physics, not an optional cleanup step. (Haines et al., 2003; Haines et al., 2007; Warden et al., 2012)

The coupling works both ways. A pressure gradient can move charge and create voltage (the streaming-potential direction). An applied electric field can act on mobile charge in the pore water and create relative fluid motion, which launches a seismic disturbance. The latter is commonly called electroseismic conversion. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012)

seismic force→relative pore-water motion→charge transport→voltage

electric field→force on mobile charge→relative pore-water motion→seismic wave

Property	Why it matters	What the field team might measure separately
Water conductivity / salinity	Changes electrical current paths and how streaming voltage is expressed.	Water sample conductivity, temperature, electrical resistivity.
pH and mineral surface chemistry	Controls surface charge and zeta potential.	Water pH, lithology/mineralogy, laboratory streaming-potential tests.
Porosity and tortuosity	Control connected pore volume and how winding the flow/electrical paths are.	Core density, logs, formation factor, calibrated petrophysical models.
Permeability	Controls how easily fluid can move relative to the frame and sets a characteristic transition frequency.	Slug/pump tests, core permeameter, or calibrated dynamic-coupling analysis.
Water saturation	Changes conductive pathways, capillary flow geometry, effective fluid properties, and coupling strength.	Moisture probes, resistivity, cores, time-lapse hydrologic data.
Seismic stiffness and density	Control wave speeds, acceleration, impedance contrasts, and travel times.	Geophones/accelerometers, refraction/reflection velocity analysis, density data.
Frequency	At high enough frequency, pore-fluid inertia changes permeability and electrokinetic coupling.	Source spectrum and calibrated receiver responses.

Sources for the parameter controls. (Jouniaux & Zyserman, 2016; Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015; Jougnot & Solazzi, 2021)

4.Coupled continuum framework and sign conventions

The most useful way to understand seismoelectric theory is as a three-block coupled system: poroelastic mechanics determines how the frame and pore fluid move; electrokinetic transport converts relative fluid motion to electrical current and electric force back to fluid motion; Maxwell equations propagate the electromagnetic response. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012)

Figure 3. The coupled model is not a single empirical formula. It is the combination of poroelastic mechanics, Maxwell electromagnetics, and reciprocal electrokinetic transport. Turning the coupling coefficient L to zero mathematically decouples the mechanical and electromagnetic subsystems.

Sign conventions matter. Papers differ in the sign of pressure, the direction assigned to an electrode dipole, the definition of filtration displacement, and the Fourier convention. Here, harmonic fields use exp(-iωt), electric field is E = -∇V, and the streaming-potential coefficient is C_s = ΔV/ΔP. If another paper defines either difference in the opposite direction, the polarity flips while the amplitude physics remains the same.

Equation 1 - Equation 1 - Electric field from a finite dipole

E_∥ ≈ - ΔV / ℓ

Meaning. A pair of electrodes measures a voltage difference. If the field does not vary strongly over the dipole length, dividing by the known separation estimates the field component parallel to the dipole.

ΔV measured voltage difference [V]

ℓ electrode separation [m]

E_∥ electric-field component [V m^-1]

Derivation. The exact definition is the line integral ΔV = -∫ E · dl. Approximating E as constant along a short straight dipole gives the formula above.

Field use. Calibrate channel gain and polarity, survey electrode positions, and use the actual dipole length. Bordes et al. use this conversion explicitly in laboratory measurements.

Source provenance. General electric-potential measurement identity; seismoelectric laboratory application is documented in (Bordes et al., 2015)

5.Governing electromagnetic and poroelastic laws

2.1 Maxwell equations in the frequency domain

Equation 2 - Faraday induction

∇ × E = iω B

Meaning. Time-varying magnetic flux is tied to circulation of the electric field under the exp(-iωt) convention.

E electric field [V m^-1]

B magnetic flux density [T]

ω angular frequency [rad s^-1]

Derivation. It is Maxwell-Faraday in the time domain, ∇×E = -∂B/∂t. Substitution ∂/∂t → -iω gives the displayed form.

Field use. For low-frequency shallow surveys the electric response is often treated in a quasi-static or diffusive electromagnetic regime, but the coupled equations retain the full constitutive path.

Source provenance. Maxwell-Faraday relation as embedded in the coupled porous-medium system in (Pride & Haartsen, 1996; Santos, 2011)

Equation 3 - Ampere-Maxwell law

∇ × H = J - iωD + J_src

Meaning. Conduction current, displacement current and any impressed current source produce the magnetic field.

H magnetic field [A m^-1]

J conduction/current density [A m^-2]

D electric displacement [C m^-2]

J_src impressed source current [A m^-2]

Derivation. Use D = εE and B = μH to close the electromagnetic constitutive part. At sufficiently low frequencies in conductive earth, |ωε| can be much smaller than conductivity, yielding a diffusion-dominated limit.

Field use. The ratio |ωε/σ| should be checked rather than assumed. Bernardo et al. (2024) specifically examine the influence of dielectric permittivity at an interface.

Source provenance. Ampere-Maxwell relation in coupled formulations; dielectric-permittivity sensitivity at interfaces is treated in (Pride & Haartsen, 1996; Santos, 2011; Bernardo et al., 2024)

2.2 Biot variables: the fluid moves relative to the frame

Let u denote solid-frame displacement and define the filtration displacement w = φ(u_f-u). Its time derivative is the Darcy-scale relative volume flux. The macroscopic average displacement is U = u + w. This variable choice makes the source of electrokinetic coupling explicit: relative fluid motion is the quantity that transports mobile excess charge.

Equation 4 - Biot coefficient

α = 1 - K_d/K_s

Meaning. The Biot coefficient measures how strongly pore pressure contributes to bulk stress in a porous skeleton.

α Biot coefficient [-]

K_d drained bulk modulus of the porous frame [Pa]

K_s bulk modulus of the solid grains [Pa]

Derivation. If the drained frame is much softer than the grains, Kd/Ks is small and α approaches one. A frame as stiff as the grains has little pore volume compliance and α approaches zero.

Field use. Kd can be obtained from drained laboratory tests or a calibrated rock-physics model; Ks comes from mineral composition or grain measurements.

Source provenance. Standard poroelastic coefficient used within the Biot-based seismoelectric framework summarized or implemented in (Pride & Haartsen, 1996; Revil et al., 2014)

Equation 5 - Biot storage modulus

M = [(α-φ)/K_s + φ/K_f]^-1

Meaning. M converts a change in fluid content at fixed strain into pore pressure. It combines compressibility of the solid grains and pore fluid.

M Biot modulus [Pa]

φ porosity [-]

K_f pore-fluid bulk modulus [Pa]

Derivation. The reciprocal modulus is a weighted storage compliance. The first term accounts for the part of pore-volume change associated with grain compressibility; the second is fluid compressibility.

Field use. For a clean single-fluid saturated material, Kf may be calculated from fluid composition, pressure and temperature. Gas or multiphase saturation requires an effective-fluid model and makes M strongly saturation dependent.

Source provenance. Standard Biot storage relation used within the poroelastic component of (Pride & Haartsen, 1996; Revil et al., 2014)

Equation 6 - Undrained bulk modulus

K_u = K_d + α²M

Meaning. When pore fluid cannot drain during a rapid compression, fluid pressure stiffens the composite relative to the drained frame.

K_u undrained bulk modulus [Pa]

Derivation. The pressure-strain coupling contributes an added stiffness α²M in linear isotropic Biot theory.

Field use. Compare frequencies and hydraulic diffusion length to sample or field scale before treating a measured seismic modulus as fully drained or undrained.

Source provenance. Standard undrained-modulus relation consistent with the poroelastic models used in (Pride & Haartsen, 1996; Revil et al., 2014)

Equation 7 - Fast-P and shear moduli in the low-loss effective limit

H = K_u + 4G/3, V_P ≈ √(H/ρ_b), V_S ≈ √(G/ρ_b)

Meaning. H is the P-wave modulus. These expressions are the familiar effective-medium limits for the fast body waves, not the full two-compressional-wave Biot dispersion relation.

G shear modulus [Pa]

ρ_b effective bulk density [kg m^-3]

V_P, V_S body-wave speeds [m s^-1]

Derivation. For isotropic elasticity the compressional modulus is bulk plus 4G/3. In a saturated porous medium the appropriate bulk modulus becomes frequency- and drainage-dependent; the full Biot eigenproblem also contains a slow compressional mode.

Field use. Given density and picked Vp/Vs, use G ≈ ρ_b Vs² and Ku ≈ ρ_b(Vp²-4Vs²/3) only when the effective-limit assumptions are appropriate.

Source provenance. Low-loss effective-wave-speed limit; the archive discusses the full Biot framework and low-frequency transfer limits in (Pride & Haartsen, 1996; Bordes et al., 2015; Jouniaux & Zyserman, 2016)

6.Electrokinetic transport, streaming potential, and Onsager reciprocity

The Pride formulation uses a reciprocal transport matrix. In isotropic notation, the same coupling coefficient appears in the mechanically driven electrical current and the electrically driven fluid flux. This symmetry is the theoretical link between seismoelectric and electroseismic conversion. (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012; Schoemaker et al., 2012)

Equation 8 - Isotropic coupled transport pair

J = σ(ω)E + L(ω)X_p
q = L(ω)E + k(ω)X_p/η

Meaning. The first line is electrical current: Ohmic conduction plus streaming current. The second is Darcy-scale relative fluid flux: electro-osmotic flow plus pressure/inertial flow.

J electric current density [A m^-2]

q Darcy/filtration velocity [m s^-1]

σ(ω) bulk electrical conductivity [S m^-1]

L(ω) electrokinetic coupling coefficient in this transport convention

k(ω) dynamic permeability [m²]

η dynamic fluid viscosity [Pa s]

X_p mechanical driving-force density per volume, commonly -∇p plus inertial term

Derivation. Linear irreversible thermodynamics writes fluxes as a matrix times generalized forces. Onsager reciprocity gives the same cross coefficient L for both directions under the usual microscopic reversibility conditions. Pride supplies the porous-medium dynamic form and embeds it in the full Maxwell-Biot wave equations.

Field use. At low frequency and in a static core, Xp reduces to -∇p. In a seismic wave it includes the inertial forcing associated with acceleration of the porous frame and fluid.

Source provenance. Reciprocal electrokinetic transport pair is the core Pride coupling, reproduced and discussed in (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012; Schoemaker et al., 2012)

3.1 Streaming potential and open-circuit condition

Equation 9 - Streaming-potential coefficient

C_s = ΔV/ΔP

Meaning. C_s is the directly measurable voltage-per-pressure coupling in a steady or quasi-static laboratory experiment.

C_s streaming-potential coefficient [V Pa^-1]

ΔV open-circuit voltage [V]

ΔP applied pressure difference [Pa]

Derivation. Under open circuit, net electric current is zero. From J = σE + L(-∇p), setting J=0 and using E=-∇V gives, for the sign convention used here, L = -σC_s. Other electrode and pressure conventions can reverse the displayed sign.

Field use. Laboratory core holders can impose ΔP and record ΔV. In the field, the low-frequency seismic transfer relation can provide an effective C_s estimate if acceleration and fluid density are known.

Source provenance. Streaming-potential coefficient is directly defined and measured in the electrokinetic literature, including (Jouniaux & Ishido, 2012; Jougnot & Solazzi, 2021)

Equation 10 - Helmholtz-Smoluchowski low-frequency limit

C_s ≈ ε_fζ / (η_fσ_w)

Meaning. For a thin electrical double layer and negligible surface conductivity, the streaming voltage is set by fluid permittivity, zeta potential, viscosity and pore-water conductivity.

ε_f fluid dielectric permittivity [F m^-1]

ζ zeta potential at the slipping surface [V]

η_f fluid viscosity [Pa s]

σ_w pore-water conductivity [S m^-1]

Derivation. Pressure-driven Poiseuille flow weights the mobile charge in the diffuse layer. Integrating that advective charge flux across a capillary and balancing it by Ohmic conduction under open circuit yields the Helmholtz-Smoluchowski relation. The derivation assumes the double layer is thin relative to pore size and ignores important surface-conduction corrections.

Field use. Measure fluid conductivity and temperature; obtain viscosity and permittivity at that temperature; then C_s gives an effective zeta potential if the model assumptions are met. Surface chemistry and pH can strongly alter ζ.

Source provenance. Helmholtz-Smoluchowski low-frequency limit and its assumptions are discussed in (Beamish, 1999; Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016)

Equation 11 - Coupling coefficient from streaming potential

L = -σ₀C_s, |L| ≈ ε_f|ζ|/(η_fF)

Meaning. The first equality follows from the open-circuit transport equations. The magnitude form on the right follows for a clean saturated medium when bulk conduction is pore-water conduction divided by formation factor.

σ₀ bulk low-frequency conductivity [S m^-1]

F electrical formation factor [-]

Derivation. Insert C_s from Equation 9 and σ₀ ≈ σ_w/F. Polarity depends on the chosen signs of ζ, ΔP and electrode order, so the magnitude form is safest when comparing conventions across the archive.

Field use. Electrical resistivity and pore-water conductivity constrain F; laboratory streaming-potential measurements constrain C_s; the two together determine the transport coupling L under the adopted convention.

Source provenance. The low-frequency relation between electrokinetic coupling, streaming potential, conductivity and formation factor is given in (Jouniaux & Zyserman, 2016)

7.Formation factor, bulk conductivity, and surface conduction

Equation 12 - Bulk conductivity with surface contribution

σ₀ = σ_w/F + σ_s

Meaning. Bulk electrical conduction is represented as pore-water conduction reduced by geometrical formation factor plus an effective surface-conduction term.

σ_s effective bulk-equivalent surface conductivity [S m^-1]

Derivation. F summarizes reduced conducting cross-section and tortuous current paths. Surface conduction represents current carried near mineral surfaces; it becomes especially important in low-salinity fluids or clay-rich material.

Field use. Measure bulk conductivity with ERT/resistivity or core impedance and pore-water conductivity from samples. If surface conduction is negligible, F ≈ σ_w/σ₀. If not, a surface-conduction model or multi-salinity laboratory calibration is needed.

Source provenance. This is a simplified bulk-plus-surface conductivity representation; surface-conduction corrections are discussed in (Jouniaux & Zyserman, 2016; Revil et al., 2014)

Equation 13 - Archie formation factor

F = φ^-m

Meaning. The formation factor is empirically related to porosity by the cementation exponent m in clean porous media.

m cementation exponent [-]

Derivation. This is a petrophysical power law, not a universal theorem. It compresses pore-connectivity and tortuosity effects into one empirical exponent.

Field use. If φ and F are measured, m = -ln(F)/ln(φ). Use the relationship cautiously in conductive minerals, clays, fractures or strongly heterogeneous media.

Source provenance. Archie formation-factor scaling is used explicitly in the supplied seismoelectric petrophysical treatments (Jouniaux & Zyserman, 2016; Revil et al., 2014; Jougnot & Solazzi, 2021)

Equation 14 - Unsaturated Archie relation

σ_b - σ_s ≈ σ_wφ^mS_wⁿ

Meaning. A saturation exponent n extends the clean-rock Archie relation to partially saturated pore space.

S_w water saturation [-]

n saturation exponent [-]

σ_b measured bulk conductivity [S m^-1]

Derivation. At S_w=1 and negligible surface conduction, the equation reduces to σ_b=σ_w/F=σ_wφ^m. The extra S_wⁿ factor represents loss of connected water-filled electrical pathways as saturation declines.

Field use. If m and n are site-calibrated, S_w = [(σ_b-σ_s)/(σ_wφ^m)]^1/n. This saturation estimate comes primarily from electrical petrophysics and should then constrain, not be confused with, the seismoelectric inversion.

Source provenance. Unsaturated Archie-type conductivity scaling and surface-conduction variants are treated in (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015)

8.Frequency dependence, dynamic permeability, and transition frequency

Figure 4. The Biot/electrokinetic transition frequency gives a first check on whether a low-frequency coupling approximation is justified. The exact transition and complex phase depend on pore geometry and the adopted dynamic-permeability model.

Equation 15 - Characteristic Biot/electrokinetic transition frequency

ω_c = φη_f/(α_∞k₀ρ_f), f_c = ω_c/(2π)

Meaning. The transition compares viscous drag to pore-fluid inertia. Below it, permeability is approximately its static value; around and above it, dynamic permeability and coupling acquire stronger frequency dependence and phase.

α_∞ high-frequency tortuosity [-]

k₀ intrinsic permeability [m²]

ρ_f fluid density [kg m^-3]

Derivation. Dimensional check: η/(ρk) has units 1/s; multiplying by porosity/tortuosity preserves frequency. The expression is a standard characteristic scale in the Pride/Biot family of models.

Field use. Compute f_c from independent permeability, or if a reliable coupling transition is resolved spectrally, invert k₀ = φη/(2πα_∞ρ_ff_c). The latter is model dependent and needs adequate bandwidth.

Source provenance. The displayed Biot transition-frequency expression is given explicitly in the laboratory transfer-function treatment of (Bordes et al., 2015)

Equation 16 - Dynamic permeability factorization

k*( ω ) = k₀ k_rel*( ω )

Meaning. Dynamic permeability is written as the static permeability times a complex, dimensionless frequency response.

k_rel* complex relative dynamic permeability [-]

Derivation. At low frequency k_rel* approaches one. Viscous boundary layers and inertia change both amplitude and phase as frequency increases. Johnson-type dynamic permeability models provide specific pore-geometry-dependent forms.

Field use. Use the source and receiver spectra to determine whether a quasi-static k₀ approximation is defensible. Bordes et al. show that using a low-frequency transfer function at kilohertz frequencies can bias amplitude in some saturation ranges.

Source provenance. Dynamic permeability is factorized into static permeability and a relative frequency-dependent term in (Jougnot & Solazzi, 2021)

9.Effective excess charge formulation

Instead of packaging electrokinetics only into L or C_s, the effective-excess-charge approach asks how much net diffuse-layer charge is actually transported by the velocity profile. This is attractive because it connects the electrical source directly to Darcy flux.

Equation 17 - Quasi-static coupling from effective excess charge

C_EK⁰ = - Q̂_v⁰ k₀ / (η_wσ₀)

Meaning. The streaming-potential coupling equals transported effective charge density times hydraulic mobility, divided by electrical conductivity. This is the REV-scale form used by Jougnot and Solazzi.

Q̂_v⁰ quasi-static effective excess charge density [C m^-3]

C_EK⁰ quasi-static electrokinetic coupling [V Pa^-1]

Derivation. Streaming current density can be represented as transported charge density times Darcy flux. Under open circuit the conduction current cancels streaming current. Substituting Darcy q = -(k/η)∇p gives the displayed pressure-to-voltage coupling.

Field use. If k, η, σ and C_EK are independently estimated, infer Q̂_v⁰ = -C_EK⁰ησ/k. Because k can span orders of magnitude, uncertainty propagation is essential.

Source provenance. Quasi-static effective-excess-charge coupling is developed in the effective-charge formulation used by (Revil & Mahardika, 2013; Revil et al., 2014; Jougnot & Solazzi, 2021)

Equation 18 - Frequency-dependent effective-charge coupling

C_EK*( ω ) = - Q̂_v*( ω ) k*( ω ) / [η_wσ*( ω )]

Meaning. Frequency dependence can be split among transported effective charge, dynamic permeability and electrical conductivity.

Q̂_v*(ω) complex frequency-dependent effective excess charge [C m^-3]

σ*(ω) complex electrical conductivity [S m^-1]

Derivation. Jougnot and Solazzi upscale the flow-weighted charge at pore scale and show that the effective charge itself can become frequency dependent. The total coupling spectrum is therefore not controlled by k(ω) alone.

Field use. Fit amplitude and phase over a calibrated bandwidth only after instrument response removal. A spectral trend can constrain dynamic parameters, but trade-offs among Q̂_v, k and σ mean that independent electrical/hydraulic data remain valuable.

Source provenance. The frequency-dependent effective-charge form is developed explicitly in (Jougnot & Solazzi, 2021)

10.From coupled equations to observable wave signatures

7.1 Low-frequency coseismic P-wave transfer

Equation 19 - Approximate P-wave electric field / acceleration transfer

E_∥ ≈ - C_sρ_fa_s,∥

Meaning. In the low-frequency fast-P limit, the coseismic electric field is proportional to solid-frame acceleration, with fluid density and streaming-potential coupling setting the scale.

a_s,∥ solid-frame acceleration along propagation [m s^-2]

Derivation. The seismic acceleration creates an inertial pressure/force term on the pore fluid. The electrokinetic constitutive law converts that force to current; the open-circuit electric field balances the current. Jouniaux and Zyserman summarize this low-frequency proportionality from the Pride framework.

Field use. Estimate C_s ≈ -E/(ρ_fa_s) after converting voltage to E, calibrating acceleration, aligning component direction, and windowing a P-dominated arrival. If the window mixes P and S energy or sits near/above f_c, use the full dynamic transfer function instead.

Source provenance. The low-frequency fast-P electric-field/acceleration proportionality is summarized and tested in (Jouniaux & Zyserman, 2016; Bordes et al., 2015)

Equation 20 - Measured transfer function

H_SE(ω) = E(ω) / A_s(ω)

Meaning. The frequency-domain ratio of calibrated electric field to solid acceleration is the most direct empirical transfer function for comparing theory and data.

E(ω) Fourier transform of electric field

A_s(ω) Fourier transform of acceleration

Derivation. If Equation 18 is valid, H_SE is approximately constant and equal to -ρ_fC_s over the low-frequency band. Departures in amplitude/phase can indicate dynamic coupling, mixed wave modes, instrument response, attenuation or other physics.

Field use. Compute cross-spectra and coherence rather than a naive pointwise division where A is small. Restrict inversion to frequencies with adequate signal-to-noise ratio and coherence.

Source provenance. This empirical transfer-function ratio follows the laboratory and spectral comparison used in (Bordes et al., 2015; Schoemaker et al., 2012)

7.2 Interface conversion

At an interface, mechanical traction/displacement, pore pressure/fluid flux, and electromagnetic boundary conditions must be satisfied simultaneously. A contrast in elastic moduli, density, permeability, porosity, electrical conductivity, dielectric permittivity or electrokinetic coupling changes the mode mixture required to satisfy those conditions. The scattered solution therefore contains both mechanical and electromagnetic converted modes. Haartsen & Pride, White & Zhou, Schakel et al., Wang et al. and Zheng et al. develop layered or interface solutions in increasing detail. (Haartsen & Pride, 1997; White & Zhou, 2006; Schakel et al., 2011; Wang et al., 2020; Zheng et al., 2021; Bernardo et al., 2024)

Equation 21 - First-order interface-response timing

t_IR = ∫_{source→interface} ds/V_P(s) + t_EM ≈ t_{P,to interface}

Meaning. The interface response is triggered when the seismic wave reaches the converter. On many shallow survey scales the electromagnetic propagation time to the receivers is small compared with the seismic travel time, so the observed time is dominated by the one-way seismic path.

t_IR interface-response time after source trigger [s]

t_EM electromagnetic propagation/diffusion delay [s]

Derivation. The first term is the seismic travel-time integral along the source-to-interface ray/path. A simple vertical constant-velocity geometry gives z ≈ V_Pt_IR. This is not a conventional two-way seismic reflection formula.

Field use. Use the seismic velocity model and source/interface geometry. The very high apparent velocity across a surface electrode array helps distinguish a conversion from a coseismic event, but source-related electrical transients can also be nearly simultaneous and must be controlled.

Source provenance. One-way seismic triggering of a fast interface conversion and its spatial signature are developed or observed in (Haartsen & Pride, 1997; Haines et al., 2007; Dupuis et al., 2009; Schakel et al., 2011)

11.Unsaturated and two-phase extensions

Partial saturation changes nearly every block in Figure 1: effective density and compressibility, electrical connectivity, permeability, capillary pressure, surface area contacted by water, zeta-potential environment, and the spatial distribution of transported excess charge. Therefore, multiplying a saturated transfer function by a single saturation factor is generally a model choice, not a universal law.

Equation 22 - Relative permeability and hydraulic conductivity

k(S_w) = k₀k_r(S_w), K_h = k(S_w)ρ_fg/η_f

Meaning. Relative permeability reduces hydraulic mobility as the connected water phase shrinks. Hydraulic conductivity converts intrinsic permeability to a gravity-driven hydraulic-flow coefficient.

k_r relative permeability [-]

K_h hydraulic conductivity [m s^-1]

g gravitational acceleration [m s^-2]

Derivation. Revil & Mahardika use constitutive saturation functions linked to capillary pressure; specific exponents depend on the selected Brooks-Corey/van Genuchten-style model. The hydraulic-conductivity expression follows Darcy law written in hydraulic-head form.

Field use. Measure k₀ by permeameter or hydraulic test and calibrate k_r(S_w) from retention/flow data where possible. Do not assume an uncalibrated relative-permeability curve is a direct seismoelectric observation.

Source provenance. Saturation-dependent hydraulic mobility and coupled unsaturated formulations are developed in (Revil & Mahardika, 2013; Revil et al., 2014; Jardani & Revil, 2015)

Laboratory evidence from the archive. Bordes et al. measured water conductivity, porosity, intrinsic permeability and water saturation independently, then compared measured seismic/electric transfer behavior with saturation-dependent models. That design is important: saturation was not inferred from one electrical amplitude alone. (Bordes et al., 2015)

12.Field acquisition: source, electrodes, mechanical channels, and geometry

Figure 5. Minimal acquisition geometry. Electrical dipoles estimate a component of E; mechanical receivers provide velocity or acceleration; geometry and moveout help distinguish signal classes. Source-control experiments remain necessary because an electrical transient generated at the source can appear with very high apparent velocity.

9.1 Electrical channels

Use differential electrode pairs with known spacing and surveyed orientation. Record contact resistance/impedance, because the source impedance of the electrode-ground interface must remain small relative to recorder input impedance. Non-polarizing electrodes can reduce electrochemical drift for low-frequency work; stainless-steel rods have also been used successfully in active shallow experiments. The selection is a bandwidth, contact and deployment trade-off.

9.2 Mechanical channels

Co-locate or geometrically relate geophones/accelerometers to the electrical dipoles. A velocity geophone does not measure acceleration directly; remove its instrument response to obtain ground velocity and differentiate only within the reliable band. An accelerometer provides a more direct path to Equation 18, again after response calibration.

9.3 Source reference and electrical hygiene

Record a clean source trigger and a pretrigger noise window. Insulate trigger circuits from energetic sources. Test source plates/materials and cable layouts. Haines et al. found that recording individual hammer impacts, rejecting electrically contaminated records, then stacking the remaining shots improved repeatability. Power-line harmonics can be estimated and subtracted, but raw-data preservation is essential. (Haines et al., 2003; Haines et al., 2007)

9.4 Geometry as a filter

Use multiple offsets and, where practical, multiple dipole orientations. Coseismic arrivals have seismic slowness; an interface response has much smaller apparent slowness across the electrical array and can show a characteristic dipolar amplitude/polarity pattern. Warden et al. exploit this separation with f-k, Radon and curvelet-domain processing. (Haines et al., 2007; Warden et al., 2012; Dupuis et al., 2009)

Practical measurement view

Figure 6. A practical survey records seismic motion and voltage differences together. The electrical receiver is a dipole: two grounded electrodes connected to one differential channel. Repeated source impacts are commonly stacked to improve signal-to-noise ratio. (Mikhailov et al., 1997; Haines et al., 2003; Haines et al., 2007)

1. GenerateA hammer, weight drop, vibrator, or other controlled source puts a seismic pulse into the ground.

2. RecordGeophones or accelerometers measure mechanical motion while electrode pairs measure voltage differences.

3. RepeatMany nominally identical shots are recorded. Noisy or contaminated shots can be rejected before stacking.

4. SeparateMoveout, polarization, geometry and filters separate coseismic, interface and source-related energy.

6.1 Why two electrodes?

A voltmeter measures a difference in potential. If the electrodes are separated by a known distance, the measured voltage difference can be converted approximately to the component of electric field along the dipole. Shorter dipoles improve spatial localization; longer dipoles can increase voltage but also spatially average the field. The papers show that electrode spacing therefore affects the recorded frequency content. (Bordes et al., 2015; Strahser et al., 2011)

6.2 Why stack many shots?

A coherent seismoelectric response repeats with the source. Uncorrelated electrical noise does not repeat with the same phase. Averaging or stacking repeated records can therefore strengthen the repeatable signal relative to random noise. Haines and co-workers also emphasize recording impacts individually so contaminated traces can be rejected before stacking. (Haines et al., 2003; Haines et al., 2007)

13.Signal processing and event-separation logic

Step	Purpose	Diagnostic / caution
Instrument-response correction	Convert digitizer counts to volts, V/m, velocity or acceleration.	Never compare theoretical SI-unit transfer functions with uncalibrated counts.
Shot QC and stacking	Increase repeatable signal relative to random noise.	Reject contaminated shots before stacking; preserve a record of rejection rules.
Mains/harmonic treatment	Suppress coherent power-line contamination.	Use narrow or model-based subtraction; avoid removing target energy near harmonics without documenting it.
Moveout / slowness analysis	Separate seismic-slowness coseismic events from near-zero-slowness conversions.	Flat source electrical transients can masquerade as interface responses.
Polarization / dipole orientation	Use vector behavior and expected radiation patterns.	Coordinate and electrode polarity conventions must be consistent.
f-k / Radon / curvelet filtering	Exploit differences in wavefront geometry.	Filtering can alter amplitudes; Warden et al. developed curvelet masking partly to preserve interface-response patterns better. (Warden et al., 2012)
Cross-spectral transfer estimation	Estimate H_SE(ω) and phase with coherence.	A ratio is unreliable where the mechanical denominator has low energy.

Shot-gather diagnostics

Ask about moveout

Does an event arrive later as receiver offset grows, with a slope comparable to seismic velocity? That behavior supports a coseismic interpretation. (Haines et al., 2003; Haines et al., 2007; Warden et al., 2012)

Ask about polarity

An interface response often has a dipolar radiation pattern. Polarity reversals across the source or interface geometry can be diagnostic. (Haartsen & Pride, 1997; Schakel et al., 2011; Dupuis et al., 2009)

Ask about simultaneity

A converted electromagnetic response can have very small apparent slowness across the array compared with a seismic wave. (Haartsen & Pride, 1997; Haines et al., 2007; Dupuis et al., 2009)

Ask what the source itself did

Trigger circuits, moving metal, cables and grounding can create electrical transients. Control experiments are essential. (Haines et al., 2003; Haines et al., 2007)

14.General parameter derivation from field data

The following chain separates measured quantities, direct calculations and model-dependent inversions. That distinction is critical for defensible reporting.

raw voltage ΔV(t)→E(t)→H_SE(ω)→C_s→L and/or ζ→joint inversion with σ, Vp, Vs, φ, k, Sw

Equation 23 - Estimate low-frequency streaming coupling from a P arrival

Ĉ_s = - E_∥ / (ρ_fa_s,∥)

Meaning. This rearranges Equation 18 to estimate the effective low-frequency coupling coefficient from a calibrated coseismic P-wave window.

Ĉ_s field estimate [V Pa^-1]

Derivation. No new physics is introduced; the estimator is simply the inverse of the low-frequency transfer relation.

Field use. Prefer a regression or complex spectral estimate over many samples/frequencies rather than one amplitude ratio. Mask frequencies with low coherence; propagate receiver-calibration uncertainty and density uncertainty.

Source provenance. Derived here by rearranging Equation 18; the parent low-frequency transfer relation is supported by (Jouniaux & Zyserman, 2016; Bordes et al., 2015)

Equation 24 - Estimate zeta potential under Helmholtz-Smoluchowski assumptions

ζ̂ = Ĉ_sη_fσ_w/ε_f

Meaning. If surface conduction is negligible and the double layer is thin, the field/lab coupling coefficient can be converted to an effective zeta potential.

ζ̂ effective zeta-potential estimate [V]

Derivation. This is Equation 9 rearranged. It is not valid merely because C_s has been estimated; its electrochemical assumptions must also hold.

Field use. Measure σ_w, pH and temperature from water samples. Use temperature-corrected η and ε. If clays or low salinity make surface conduction important, use a corrected conductivity/electrochemical model rather than this simple inversion.

Source provenance. Derived here by rearranging Equation 9; the parent Helmholtz-Smoluchowski relation is supported by (Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016)

Equation 25 - Formation factor from saturated conductivity

F̂ = σ_w/(σ₀-σ_s)

Meaning. Formation factor is obtained from pore-water conductivity divided by the bulk pore-water contribution.

F̂ formation-factor estimate [-]

Derivation. Rearrange Equation 11.

Field use. ERT/resistivity provides a spatial bulk-conductivity model; water samples provide σ_w. Surface conduction must be estimated or shown negligible. With porosity, infer m̂ = -ln(F̂)/ln(φ).

Source provenance. Derived here by rearranging the conductivity/formation-factor relation; relevant surface-conduction treatments are in (Jouniaux & Zyserman, 2016; Revil et al., 2014)

Equation 26 - Permeability from an observed transition frequency

k̂₀ = φη_f / (2πα_∞ρ_ff_c)

Meaning. If the dynamic transition is resolved and the Pride/Biot characteristic-frequency model is appropriate, its location can be inverted for permeability.

k̂₀ intrinsic-permeability estimate [m²]

Derivation. Rearrange Equation 14.

Field use. This is a model-based estimate, not a direct measurement. It is strongest when φ, α_∞, fluid properties and f_c are independently constrained and the data span both sides of the transition. Compare with slug/pump/core permeability whenever possible.

Source provenance. Derived here by rearranging Equation 14; the parent Biot-frequency expression is given in (Bordes et al., 2015)

Equation 27 - Porosity from bulk density in a simple saturated two-component mixture

φ̂ = (ρ_s-ρ_b)/(ρ_s-ρ_f)

Meaning. If a representative solid-grain density and single pore-fluid density apply, bulk density provides a porosity estimate.

ρ_s grain density [kg m^-3]

ρ_b bulk density [kg m^-3]

Derivation. Use volume averaging ρ_b=(1-φ)ρ_s+φρ_f and solve for φ.

Field use. Obtain ρ_b from cores/logs or other density constraints and ρ_s from mineralogy. Gas, multiple minerals and bound water require a multi-component mixing model.

Source provenance. General two-component density mixing identity; comparable saturation-dependent effective-fluid averaging is used in (Revil et al., 2014; Bordes et al., 2015)

Equation 28 - Water saturation from a calibrated Archie model

Ŝ_w = [(σ_b-σ_s)/(σ_wφ^m)]^1/n

Meaning. This provides a resistivity-derived saturation constraint for the coupled interpretation.

Ŝ_w water-saturation estimate [-]

Derivation. Rearrange Equation 13.

Field use. Calibrate m and n locally. Use time-lapse resistivity and moisture data where available. Feed S_w into saturation-dependent mechanical/electrokinetic models rather than treating the formula as a seismoelectric amplitude law.

Source provenance. Derived here from the Archie-type saturation relation; parent models and laboratory usage are in (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015)

15.Single-dipole near-source inversion: from ΔV(t) to coupling and permeability

A special case considered in the supplied workflow document places one short electric dipole immediately beside the seismic source. The geometry reduces long-path ambiguity and targets the source-zone formation, but it removes an independently measured mechanical channel; therefore the local acceleration must be supplied by calibrated source/elastic modeling. This is a focused inversion strategy, not a claim that one voltage trace uniquely determines formation properties. (Beamish, 1999; Haines et al., 2003; Jouniaux & Zyserman, 2016; Schoemaker et al., 2012)

Figure 7. The single-dipole measurement is intentionally local. Instead of imaging a long path or an interface conversion, it targets the source-zone formation response. That reduces geometric ambiguity, but it also means the mechanical side of the problem must be supplied by a calibrated source model rather than a colocated geophone (Beamish, 1999; Haines et al., 2003; Jouniaux & Zyserman, 2016).

Required independent inputs

Only one electrical receiver is recorded, but the inversion is not based on one number. The following assumptions and external inputs are required for the chain to close.

Item	Why it is needed	Typical source
Dipole spacing ℓ and orientation	Needed to convert voltage to electric field and to project the correct field component.	Field survey / instrument layout
Source force-time signature F(ω)	Needed to predict the local acceleration spectrum when no geophone is recorded.	Factory calibration, reference impact test, or pre-calibrated source model
Elastic properties V_P, V_S, ρ_b	Needed in the source-to-acceleration transfer operator.	Shallow seismic model, site logs, or prior survey
Bulk electrical conductivity σ and pore-water conductivity σ_w	Needed to convert C_s to L and to assess electrochemical assumptions.	Resistivity survey, water sample, lab test
Fluid density and viscosity ρ_f, η_f	Needed in the low-frequency transfer law and permeability estimate.	Water chemistry / temperature measurement
Porosity and tortuosity α_∞	Needed to convert transition frequency to intrinsic permeability.	Core/log estimate, literature prior, calibrated site model

Identifiability warning. A single near-source electric trace cannot, by itself, uniquely determine every property of the formation. What it can do is support a tightly focused inversion when the other required variables are constrained independently. Without that support, the trace is informative but not uniquely diagnostic.

Inversion chain

Figure 8. A single-trace inversion still has multiple stages. The electrical trace is measured; the coupling coefficient is estimated by comparison with a source-derived mechanical model; permeability is then obtained by fitting the dynamic spectral transition (Pride & Haartsen, 1996; Schoemaker et al., 2012; Bordes et al., 2015).

Additional equations unique to the single-dipole implementation

Equation 29 - Predict local acceleration from the source model

A_mod(ω) = G_a(r,ω; m) F(ω)

Meaning. Because only one electrical channel is recorded, the local solid acceleration must be predicted from a source-to-receiver mechanical transfer operator rather than measured directly.

F(ω) source force-time spectrum

G_a source-to-acceleration operator

r source-to-dipole offset [m]

m elastic model parameters (V_P, V_S, ρ_b, attenuation, layering)

Field use. In practice, G_a may be a simple half-space operator or a local layered model. The source spectrum F(ω) should come from a controlled calibration rather than guesswork.

Source provenance. The need to connect the electrical response to the local mechanical field follows from the coupled Pride/Biot framework (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016).

Equation 30 - Weighted estimate of the streaming-potential coefficient

Ĉ_s = - [ Σ_i w_i Re{E_iA_i^*} ] / [ ρ_f Σ_i w_i|A_i|² ]

Meaning. Rather than dividing one noisy sample by another, estimate C_s by a weighted regression in the frequency domain. Use only frequencies with adequate coherence and signal-to-noise ratio.

E_i measured electric field at frequency i

A_i modeled acceleration at frequency i

w_i spectral weight or coherence weight

* complex conjugate

Field use. This estimator is robust and explicit. It also makes uncertainty propagation easier because weights can reflect stacking variance, coherence, or estimated spectral noise.

Source provenance. Derived estimator based on the low-frequency transfer relation and the transfer-function practice used in seismoelectric spectral analysis (Bordes et al., 2015; Schoemaker et al., 2012).

Equation 31 - Practical dynamic-response model for fitting the single-trace spectrum

E(ω) = -ρ_fĈ_sA_mod(ω) G_d(ω; f_c)

Meaning. Once Ĉ_s is known, the full spectrum can be fit with a dynamic correction factor G_d whose corner or transition frequency is tied to permeability.

G_d dynamic correction / roll-off function

f_c transition frequency [Hz]

E(ω) observed electric spectrum

A_mod(ω) modeled acceleration spectrum

Field use. The exact form of G_d depends on the chosen electrokinetic dynamic model. A practical first pass is a monotonic low-pass response consistent with the known flat low-frequency regime and high-frequency roll-off.

Source provenance. Dynamic coupling and characteristic-frequency behavior are discussed within the Pride/Biot framework and laboratory transfer-function studies (Pride & Haartsen, 1996; Bordes et al., 2015).

Use the master equations already defined above for the remaining steps. Voltage-to-field uses Equation 1; the low-frequency fast-P transfer is Equation 18; streaming coupling and L use Equations 10 and 22; the transition-frequency relation is Equation 14 and its permeability rearrangement is Equation 25; hydraulic conductivity follows Equation 21. The arbitrary first-pass roll-off formula from the supplied implementation note is intentionally not treated as a governing law; the fitted dynamic factor in Equation 32 should be chosen from a justified Pride/Biot/dynamic-permeability model. (Pride & Haartsen, 1996; Bordes et al., 2015; Jougnot & Solazzi, 2021)

Figure 9. The low-frequency plateau and the dynamic roll-off play different roles. The plateau constrains coupling; the roll-off location constrains the transition frequency and therefore permeability. This is the cleanest route from a single trace to hydraulic properties when a broadband source is available (Bordes et al., 2015; Jouniaux & Zyserman, 2016).

Reproducible inversion sequence

Acquire repeated source hits. Keep the dipole fixed; record raw voltage and trigger time for every shot; perform source-control tests. (Haines et al., 2003; Haines et al., 2007)
Reject contaminated shots before stacking. De-mean, control drift and power-line contamination, and quantify shot-to-shot variance. (Haines et al., 2003; Warden et al., 2012)
Convert voltage to electric field. Apply Equation 1 using surveyed dipole length and documented electrode polarity. (Bordes et al., 2015)
Predict local mechanical acceleration. Apply Equation 30 using a calibrated source spectrum and a local elastic/poroelastic model.
Identify a low-frequency P-dominated fitting band. The approximation behind Equation 18 requires a suitable wave mode and frequencies well below the relevant transition. (Bordes et al., 2015; Jouniaux & Zyserman, 2016)
Estimate the streaming-potential coefficient. Use Equation 31 or an equivalent weighted complex regression; propagate source and spectral uncertainty. (Bordes et al., 2015; Schoemaker et al., 2012)
Convert coupling representations. Use Equation 10 (or Equation 22 where appropriate) with an independently constrained bulk conductivity. (Jouniaux & Ishido, 2012)
Fit frequency dependence. Use Equation 32 with a justified dynamic model. Resolve the transition rather than forcing a corner into data that lack bandwidth. (Pride & Haartsen, 1996; Bordes et al., 2015; Jougnot & Solazzi, 2021)
Infer intrinsic permeability only if the transition is resolved. Apply Equation 25 with independent porosity, tortuosity, fluid density and viscosity; then obtain hydraulic conductivity from Equation 21. (Bordes et al., 2015)
Validate externally. Compare coupling with laboratory streaming-potential measurements where available and compare permeability/hydraulic conductivity with permeameter, slug or pumping tests. (Bordes et al., 2015; Haines et al., 2007)

Single-dipole failure modes and validation

The single-dipole near-source approach is efficient, but it has well-defined failure modes.

Risk	Why it matters	Recommended mitigation
Electrical source transient mistaken for seismoelectric signal	A cable spark, trigger leak, or inductive pulse can mimic a near-instantaneous field.	Electrical isolation tests, dummy-source tests, repeatability checks, and alternative source wiring (Haines et al., 2003)
Poor source calibration	The method depends on modeled acceleration rather than measured acceleration.	Use a controlled source signature, pre-survey calibration, and sensitivity analysis on source amplitude and bandwidth.
No resolved transition frequency	Without a visible corner frequency, permeability from Equation 25 is weakly constrained.	Use a broader-band source, longer record, or report only coupling rather than permeability.
Uncertain σ, φ, or α_∞	These enter directly into L̂ and k̂.	Use independent conductivity, porosity, and tortuosity information; report the propagated uncertainty.
Violation of low-frequency assumption	The transfer relation used to estimate Ĉ_s can fail above the transition or if the window mixes modes.	Restrict the fit to a validated plateau band and inspect amplitude-phase behavior carefully.

Recommended validation ladder. The most defensible workflow is: (1) validate the inversion on a laboratory or test-site formation with known permeability, (2) compare Ĉ_s against independent streaming-potential measurements if available, and (3) compare k̂ or K̂_h against permeameter, slug-test, or pumping-test values wherever possible (Bordes et al., 2015; Haines et al., 2007).

16.Forward modeling, inversion, and non-uniqueness

Equation 32 - Regularized joint inverse problem

Φ(m) = ||W_d[d_obs-F(m)]||² + λ²||W_m[m-m_ref]||²

Meaning. A practical inversion balances fit to observed seismoelectric/seismic/electrical data against prior information or smoothness/structure constraints.

m model vector: e.g., σ, k, φ, ζ, elastic moduli, saturation

F(m) coupled forward-model prediction

W_d data weighting / uncertainty matrix

W_m model regularization weighting

λ regularization strength

m_ref reference/prior model

Derivation. This is a maximum-likelihood/least-squares framework when errors are Gaussian and W_d represents inverse standard deviations or covariance whitening. The second term supplies prior structure needed because multiple parameter combinations can produce similar coupled amplitudes.

Field use. Use independent resistivity, seismic velocity, hydraulic tests and fluid chemistry to constrain subsets of m. Report posterior resolution or sensitivity, not only a best-fit image.

Source provenance. A practical Tikhonov-style least-squares formulation introduced here for field inversion; the supplied archive provides electroseismic inverse-problem and layered-forward-model context in (Bonnetier et al., 2019; White, 2005; White & Zhou, 2006)

Equation 33 - First-order uncertainty propagation for a derived scalar

u_y² ≈ Σ_i (∂y/∂x_i)²u_{x_i}²

Meaning. For approximately independent small errors, uncertainty in a calculated result is the quadrature sum of input uncertainties scaled by local sensitivity.

u_y standard uncertainty of derived y

u_{x_i} standard uncertainty of input x_i

Derivation. Linearize y(x) with a first-order Taylor expansion around the estimated inputs. If inputs are correlated, covariance cross-terms must be added.

Field use. Apply to C_s, ζ, F, k or S_w estimates. A parameter that is mathematically derivable but has enormous propagated uncertainty should not be presented as practically resolved.

Source provenance. General first-order uncertainty-propagation identity introduced here rather than quoted from a seismoelectric paper; it should be applied to the strongly parameter-dependent relations discussed in (Jouniaux & Zyserman, 2016; Bordes et al., 2015; Jougnot & Solazzi, 2021)

17.Parameter-to-measurement crosswalk

Parameter	Theoretical role	Preferred practical constraint	Main caveat
ΔV, ℓ, E	Electrical observable	Calibrated differential electrodes + surveyed spacing	Contact impedance, spatial averaging, cultural noise
a_s, Vp, Vs	Mechanical forcing and elastic waves	Accelerometers/geophones + velocity analysis	Receiver response and mixed wave modes
σ₀, σ_w	Ohmic current and streaming-potential scale	ERT/core conductivity + fluid sample	Temperature, salinity and surface conduction
ζ	Electrochemical coupling at slip plane	Streaming-potential lab test, or constrained Equation 23	pH/mineralogy dependence; sign convention
φ	Storage, electrical geometry, transition frequency	Core density/logs; calibrated electrical relation	Scale and heterogeneity
F, m	Electrical tortuosity/connectivity	Bulk + fluid conductivity; porosity calibration	Archie assumptions
k₀	Hydraulic mobility and f_c	Pump/slug/core test; dynamic spectrum if calibrated	Strong scale dependence and anisotropy
α_∞	Inertial tortuosity	Lab/acoustic/electrical model calibration	Rarely measured directly in routine field work
Kd, Ks, Kf, G, M	Biot poroelastic constitutive system	Seismic velocities, density, lab moduli, mineral/fluid data	Drainage/frequency regime
S_w	Multiphase conductivity, mobility, coupling	Resistivity + moisture/core/hydrologic calibration	Patchiness and model dependence
Q̂_v^eff	Transported diffuse-layer charge	Derived from C_EK, k, η, σ or mechanistic model	Strong dependence on pore-size weighting

18.Worked end-to-end workflow: shallow saturated sand over a contrasting layer

Step A - calibrate the observables. Convert electrode counts to ΔV(t), then to E(t) with Equation 1. Convert accelerometer/geophone channels to a_s(t) in SI units. Preserve polarity and coordinate metadata.

Step B - build the seismic model. Pick P arrivals across the array to estimate Vp and, if S waves are usable, Vs. Combine with density constraints to estimate effective G and Ku using Equation 6. This gives travel-time geometry for a candidate converter.

Step C - build the electrical/fluid model. Use bulk resistivity and water conductivity to estimate F (Equation 24) subject to surface-conduction tests. Use cores/logs for porosity and independent hydraulics for k where possible.

Step D - classify electrical events. Events with seismic-like moveout are candidates for coseismic transfer; near-zero-slowness events timed to the source-to-interface seismic travel time are candidates for interface conversion. Reject interpretations inconsistent with source-control experiments.

Step E - estimate coupling. In a P-dominated, low-frequency, high-coherence window, estimate H_SE(ω) and C_s using Equations 19 and 22. Compare the inferred coupling with fluid chemistry and a laboratory streaming-potential expectation.

Step F - decide whether dynamic theory is needed. Compute f_c from the independent k estimate. If the measurement band is not safely below f_c, fit a dynamic transfer model that includes k(ω) and possibly Q̂_v(ω).

Step G - invert jointly, not in isolation. Fit the coupled data while constraining electrical conductivity from ERT, elastic structure from seismic data, and hydraulic parameters from tests. Quantify uncertainty and resolution. Only parameters that are both sensitive and independently constrained should be reported as practical results.

19.Assumption failures, identifiability limits, and defensible reporting

Surface conduction is not negligible

Then σ₀≠σ_w/F, the simple Helmholtz-Smoluchowski inversion for ζ is biased, and Archie-based formation-factor estimates can fail. (Jouniaux & Zyserman, 2016; Revil et al., 2014)

The data are not below f_c

Then a constant C_s or k₀ approximation misses amplitude and phase effects from dynamic permeability and effective charge. (Bordes et al., 2015; Jougnot & Solazzi, 2021)

Multiple wave modes occupy the window

Then E/a is not a pure P-wave transfer coefficient. Mode separation or a full vector/tensor forward model is required. (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016)

Saturation is patchy

Effective moduli, conductivity and relative permeability may be scale dependent; a homogeneous saturation parameter can become misleading. (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015)

The interface is not planar

Simple timing and dipole patterns no longer apply directly. Finite-element, spectral or layered/3-D forward models may be required. (White & Zhou, 2006; Zheng et al., 2021)

The electrical event is source-generated

No amount of inversion can turn an acquisition artifact into a subsurface property. Control shots and source/electrode geometry are decisive. (Haines et al., 2003; Haines et al., 2007)

Interpretation boundaries

Electrokinetic seismoelectric theory does not imply that every electrical anomaly before an earthquake has an electrokinetic origin, that hydrocarbons have a unique seismoelectric signature, or that permeability can always be recovered from a single field record. The archive contains both rigorous coupling theory and broader seismo-electromagnetic studies; competing mechanisms and cultural noise must be tested. (Jouniaux & Zyserman, 2016; Haines et al., 2007)

Field-scale prediction is hardest when pore water is very conductive, surface conduction is important, saturation is patchy, the source frequency approaches the dynamic transition, the ground is electrically noisy, or the assumed geometry is wrong. Those are model-selection problems, not just signal-processing problems. (Jouniaux & Zyserman, 2016; Revil & Mahardika, 2013; Bordes et al., 2015; Jougnot & Solazzi, 2021)

20.Conclusions

From beginning to end, the seismoelectric chain is a coupled transport problem: a seismic source creates poroelastic deformation; relative fluid motion transports electrical-double-layer charge; streaming current and charge separation produce a measurable electric field; material contrasts can additionally radiate an electromagnetic interface response; and the recorded voltages are interpreted only after geometry, source behavior, electrical conductivity, fluid properties, frequency regime and saturation are accounted for. (Pride & Haartsen, 1996; Haartsen & Pride, 1997; Jouniaux & Zyserman, 2016)

The strongest quantitative use is not an unconstrained conversion of voltage amplitude directly to permeability. A defensible inversion separates what is measured from what is independently supplied, uses the low-frequency transfer to estimate coupling, uses dynamic behavior only where the transition is resolved, and reports parameter trade-offs explicitly. (Bordes et al., 2015; Bonnetier et al., 2019; Jougnot & Solazzi, 2021)

Operational summary. ΔV(t) is the instrument observable. It becomes E(t) after dipole calibration; coupling is inferred by comparison with the mechanical field; permeability is only inferred through a justified frequency-dependent model plus independent petrophysical inputs. Field controls and external validation are part of the scientific method, not optional post-processing. (Haines et al., 2003; Haines et al., 2007; Bordes et al., 2015)

Part II

Acquisition geometry and resolution

How a grounded dipole samples the electric field, how near-source geometry controls lateral sensitivity, and how sampling, bandwidth, noise and filtering control vertical resolution.

Integrated source module

Grounded-Dipole Geometry and Lateral Sensitivity

Specialized acquisition section. It deepens the grounded-dipole line-integral measurement, source-adjacent geometry, baseline-length trade-off and the strongly local lateral sensitivity of a single near-source receiver.

1. Scope and practical claim

The statement “a 2 m grounded dipole with one electrode next to the seismic source is most effective” should be read as a single-dipole near-source design rule, not as a universal theorem for all seismoelectric surveys.

In a multi-channel interface-imaging survey one may want larger apertures and distributed dipoles. In contrast, the present geometry is optimized for a very specific objective: measuring the local source-zone seismoelectric response with maximum repeatability and minimum spatial smearing. That objective appears repeatedly in the literature on near-surface and laboratory seismoelectric coupling, where the electrical response is often interpreted using local transfer functions, normalized electric-to-pressure ratios, or local electric-to-motion comparisons (Beamish, 1999; Schoemaker et al., 2012; Bordes et al., 2015).

Operational conclusion. In a single-channel surface layout, the most useful signal usually comes from deliberately biasing the measurement toward the source neighborhood. Putting one electrode right next to the source and keeping the second only a short distance away is the simplest way to do that.

Figure 10. The seismoelectric response of the source zone is a symmetric two-lobe (dipolar) electric field that pulses outward with the seismic wavefront, centred on a vertical dipole axis beneath the source. A near-source grounded dipole - one pin at the source, the second a short distance away - samples the steep field of the near lobe, which is why the recorded voltage is dominated by the ground directly beneath the source point (Mikhailov et al., 1997; Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016).

Figure 11. The source-adjacent electrode samples the strongest electric-potential gradient, while the second electrode defines a short baseline over which the voltage difference is measured. The differential voltage is therefore strongly weighted toward the first part of the dipole and toward the formation directly below the source point (Pride & Haartsen, 1996; Beamish, 1999; Haines et al., 2007).

2. What the grounded dipole actually measures

A grounded dipole does not directly measure “electrical amplitude at a point.” It measures the potential difference between two electrodes. In continuum form, the recorded voltage is the line integral of the electric field along the dipole path:

Equation 34 - Voltage measured by a grounded dipole

V_AB(t) = φ(B,t) - φ(A,t) = - ∫_A^B E(r,t) · dl

Meaning. The dipole integrates the component of the electric field that lies parallel to the line joining electrodes A and B.

V_AB measured voltage [V]

φ electric potential [V]

E electric field [V m^-1]

dl differential element along the dipole

Why it matters. Because this is an integral, spacing and orientation are not incidental geometric details. They determine which part of the field is emphasized and which part is averaged out.

Source provenance. The relation between potential, electric field, and grounded-electrode measurement follows standard electromagnetic theory and the seismoelectric transport formulations summarized in (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012).

If the dipole is short enough that the field does not change dramatically over its length, then the measured voltage approximates the field component along the dipole multiplied by spacing:

Equation 35 - Short-dipole approximation

V_AB(t) ≈ - E_∥(r_eff,t) · ℓ, E_∥(r_eff,t) ≈ -V_AB(t)/ℓ

Meaning. The dipole estimates the field component parallel to its axis. The approximation improves as spacing becomes small relative to the scale over which the field changes direction and amplitude.

Practical consequence. A short dipole improves localization, whereas a long dipole increases signal amplitude but causes spatial averaging. The literature explicitly notes that electrode spacing changes the observed response and its spectral content (Strahser et al., 2011; Bordes et al., 2015).

3. Why one electrode belongs next to the source

The electrokinetic source of the seismoelectric field is mechanically driven fluid motion. In a near-source single-dipole measurement, the earliest and most repeatable electrical energy normally comes from the neighborhood where the seismic source injects the strongest stress and pressure gradient into the ground. The local potential field is therefore steepest near the source zone, which is exactly where one wants one of the electrodes to sit (Beamish, 1999; Haines et al., 2003; Haines et al., 2007).

It maximizes the potential contrast

The nearer electrode sees the largest local perturbation in potential. Moving both electrodes away from the source reduces the differential voltage because both ends then sit on weaker parts of the field.

It anchors the measurement in time and space

The early-time window is then tied to the local source zone instead of to a distributed or distant region. That simplifies interpretation and transfer-function analysis.

It biases the measurement toward the target interval

If the goal is to infer source-zone coupling or source-point permeability, placing one electrode at the source is the most direct way to weight the measurement toward the relevant volume.

There is also a purely geometric reason. The electric field of a localized electrokinetic source is not laterally uniform. To first order, the potential field generated by a small effective electric dipole moment p embedded in a conductive medium is dipolar:

Equation 36 - Quasi-static dipole field of a localized source region

φ(r) = (p·r) / (4πσr³), E(r) = [3&hatr(&hatr·p) - p] / (4πσr³)

Meaning. The field amplitude decays rapidly with distance and depends strongly on direction. The exact source mechanism in real seismoelectric coupling is more complicated than a static dipole, but this expression captures the key geometry: rapid decay and strong directionality.

Practical consequence. Because the field drops off quickly, moving the first electrode away from the source sacrifices the most energetic and information-rich part of the measurement.

Source provenance. The detailed seismoelectric field equations are developed in coupled-wave form in (Pride & Haartsen, 1996; White, 2005; White & Zhou, 2006). The expression shown here is a compact field-geometry analogue used to explain why a localized source yields a strongly localized measured voltage.

4. Why about 2 m is often the best spacing

The preference for about 2 m emerges from competing design pressures. The spacing must be large enough to produce a measurable voltage difference but short enough to preserve locality and avoid averaging over unrelated structure. In practice, around 2 m is often a useful middle ground for shallow active surveys with a single grounded dipole beside the source.

Spacing regime	Main advantage	Main penalty	Interpretive effect
Very short (< 1 m)	Excellent spatial localization	Small ΔV, lower SNR, stronger vulnerability to electrode noise and contact-impedance variability	Good locality but sometimes not enough usable signal
Moderate (~ 2 m)	Good compromise between amplitude and locality	Still some averaging, but usually limited	Often the best engineering balance for a single near-source dipole
Long (> 3-5 m)	Larger raw voltage	Broader footprint, more off-axis sensitivity, more later arrivals and lateral geology mixed in	Interpretation becomes less local and more ambiguous

This trade-off can be formalized by assuming the dipole samples a field component that decays approximately as E_∥(s) ∝ (r₀+s)^-n along the dipole, where r₀ is the offset from the source to the near electrode and n > 1 represents rapid geometric decay. Then:

Equation 37 - Along-dipole weighting for a rapidly decaying local field

V_AB ∝ ∫₀^ℓ (r₀+s)^-n ds = [(r₀)^1-n - (r₀+ℓ)^1-n] / (n-1)

Meaning. For any n > 1, the contribution from the first part of the dipole is disproportionately large. In other words, the measured voltage is not equally sensitive along the entire 2 m baseline - it is heavily weighted toward the near electrode.

Interpretive consequence. Increasing spacing from 0.5 m to 2 m usually adds useful voltage because the integral extends farther into the field. But increasing from 2 m to 5 m yields diminishing returns in useful near-source signal while increasingly admitting spatial averaging and unrelated lateral contributions.

Figure 12. Spacing trade-off. Very short dipoles preserve locality but may produce insufficient voltage; long dipoles produce more raw voltage but integrate a broader region. Around 2 m is often the best compromise in a single near-source surface setup, especially when the objective is to isolate source-point coupling rather than map lateral structure (Haines et al., 2003; Bordes et al., 2015; Strahser et al., 2011).

5. Why dipole-field geometry is the controlling factor

Geometry matters because the electric field is a vector field, while the dipole is a one-dimensional sampler. A dipole measures only the projection of the local field onto the dipole axis. If the dipole is aligned with the dominant field direction, the recorded voltage is maximized. If it is oblique, the measured voltage is reduced by the cosine of the misalignment angle:

Equation 38 - Directional projection onto the dipole axis

V_AB(t) ≈ ℓ |E(r_eff,t)| cos θ

Meaning. θ is the angle between the dipole axis and the local electric-field vector. The signal is largest when the dipole points along the dominant field direction.

Why this is central. In a near-source experiment, the source-zone field is often approximately radial or dipolar with respect to the source. The dipole geometry therefore determines not only how much signal is recorded, but also which part of the vector field is visible at all.

This is the deeper reason the geometry cannot be treated as an afterthought. In a single-dipole experiment there is no spatial redundancy: there is only one baseline, one orientation, and one projected field component. Therefore the measured trace is governed more by geometry than by any hope of broad subsurface averaging.

Geometric concentration

The differential voltage is weighted toward the near electrode because the source-zone field decays quickly with distance. That makes the first part of the dipole the most important part.

Geometric selectivity

The dipole only sees the field component parallel to itself. If the field is mostly vertical or orthogonal to the dipole axis, that part of the field is largely invisible in the recorded voltage.

6. Why the data mainly represent the geology directly below the source

In a source-adjacent single-dipole layout, the recorded trace is dominated by the local source-point sensitivity kernel. A useful conceptual expression is:

Equation 39 - Conceptual sensitivity-kernel form

V(t) = &iint_Ω K(x,z,t) · C(x,z) dΩ

Meaning. C(x,z) denotes the local electrokinetic “coupling content” of the formation (a shorthand for the elastic, hydraulic, electrical and electrochemical properties that control the response), while K(x,z,t) is the experiment-dependent weighting kernel.

Key point. For a single near-source dipole, K is sharply peaked beneath and immediately around the source point. Therefore the integral is dominated by the geology in that small neighborhood, not by a broad lateral swath.

Source provenance. The exact kernel depends on the full coupled-wave solution and acquisition geometry as developed in (Pride & Haartsen, 1996; White, 2005; Bonnetier et al., 2019). The compact form above is used here to explain the footprint of the single-dipole measurement.

Three mechanisms force this source-point dominance:

Rapid field decay

The electrokinetic field drops quickly with distance from the source zone, so the strongest contribution comes from material nearest the source and the near electrode.

Early-time gating

In practice the interpreter uses the earliest repeatable window, which isolates the local coseismic response before later reflections or distant interface conversions arrive (Haines et al., 2007; Schoemaker et al., 2012).

Lack of lateral aperture

With only one dipole there is no receiver spread to triangulate or stack lateral events. The geometry therefore cannot accumulate much lateral information.

Figure 13. Conceptual sensitivity footprint of a single grounded dipole placed beside the source. The weighting is strongest immediately below the source and decays rapidly with both depth and lateral distance. For that reason, the resulting data set is dominated by the source-point neighborhood rather than by wide lateral geology.

7. Why lateral geology contributes very little

The weak visibility of lateral geology in a single near-source dipole is not an accident; it is the natural consequence of the acquisition physics.

7.1 No lateral receiver aperture

Surface seismoelectric lateral imaging normally relies on differences in moveout, polarity, and timing across multiple receivers. A single dipole provides none of that redundancy. Without a receiver spread, there is no effective way to image lateral variation except through changes in repeated measurements at different source positions (Haines et al., 2007; Warden et al., 2012).

7.2 Lateral contributions are geometrically disadvantaged

Off-axis geology lies farther from both the source and the most sensitive electrode. Its contribution therefore suffers from more geometric decay, poorer field projection onto the dipole axis, and - if a short early window is used - later arrival time.

Equation 40 - Why laterally displaced geology is suppressed

R = (x² + z²)^1/2, K(x,z) ∝ cos θ / Rⁿ

Meaning. If a feature lies laterally displaced by x and at depth z, then its source-to-feature distance is larger, and the measurement is further reduced by the projection factor cos θ.

Implication. Even before time-gating is applied, lateral features are penalized twice: they are farther away and usually less well aligned with the dipole axis.

7.3 Early-time gating favors vertical rather than lateral sensitivity

In practical near-source seismoelectric analysis, one usually windows the earliest repeatable part of the trace to isolate the local coseismic response. That decision alone removes much of the later energy that would have carried broader or more complicated path effects. If a reflector or lateral boundary requires extra seismic travel time, it arrives after the source-zone window and is intentionally excluded (Haines et al., 2007; Schoemaker et al., 2012).

Source pulse → local pore-fluid motion→ local electric field at the source point→ near electrode sees strongest gradient→ short early-time window keeps local response→ lateral information remains weak

Important nuance. “Very little lateral geology” does not mean “no lateral geology under any circumstance.” A very shallow lateral boundary, a strong conductivity or permeability contrast close to the source, or repeated measurements made at many different source points can all introduce lateral information. The point is that the default weighting of a single near-source dipole is overwhelmingly local.

8. Limits, exceptions, and failure modes

When 2 m may be too short

If the formation is very resistive, the coupling is weak, or the source energy is small, a 2 m baseline may not produce enough measurable voltage. In that case one may need stronger stacking, improved grounding, or a somewhat longer baseline.

When 2 m may be too long

If the site is electrically noisy, strongly heterogeneous at metre scale, or crowded with cultural conductors, a 2 m dipole may integrate more unwanted variability than desired. A shorter baseline can then improve locality.

When lateral geology may appear

Shallow interfaces, lateral fractures, or strong off-axis contrasts very near the source can leak into the signal, especially if they generate strong interface responses or if the time window is too broad.

When geometry is misaligned

If the dipole is not aligned with the dominant field component, the signal can be suppressed even though strong coupling exists. Apparent “weak geology” can therefore be a geometry problem rather than a formation problem.

9. Practical design recommendations

Place electrode A immediately adjacent to the source. This captures the strongest local gradient and ties the measurement to the source zone.
Use a short baseline - often around 2 m - as the default starting geometry. It is usually the best compromise between signal level and spatial locality.
Orient the dipole deliberately. Align it with the expected strongest field component; do not treat orientation as arbitrary.
Record repeated hits and stack selectively. Near-source measurements can be small and are vulnerable to source-related electrical artefacts, so repeatability testing is essential (Haines et al., 2003).
Use a short early-time analysis window if the objective is source-point coupling or permeability, because that window emphasizes the local coseismic response and suppresses later nonlocal energy.
Do not over-interpret lateral structure from a single dipole. If lateral imaging is required, add source positions, receiver positions, or both.

Bottom line. The reason a grounded 2 m dipole beside the seismic source works so well is fundamentally geometric. The recorded voltage is a local, directionally projected, rapidly decaying field integral. That makes the source-adjacent geometry highly efficient for sensing the formation directly below the source point - but correspondingly poor for seeing broad lateral geology.

Integrated source module

Vertical Resolution, Digitization, Noise and Filtering

Specialized resolution section. It separates sample-grid spacing, wavelet/bandwidth resolving power and depth uncertainty, then traces how digitization and electrical-noise filtering affect an interface-depth estimate.

Interpretation policy. SOURCE-SUPPORTED marks relationships explicitly represented in the supplied seismoelectric material. DERIVED DSP marks standard sampling/noise/filter calculations introduced here. IMPLEMENTATION marks recommended workflow choices. A number such as 0.045 m/sample is a digital depth interval, not proof that a 4.5 cm-thick bed can be geologically resolved.

1. What "vertical resolution" means

A digital recorder, a seismic wavelet and a geological model each impose a different vertical scale. They must not be collapsed into one number.

Sample-grid increment

The depth represented by one sample of time after applying the velocity model. At 44.1 kHz this can be centimetric.

Interface-separation resolution

The minimum spacing at which two separate conversions can be distinguished. It is primarily wavelet/bandwidth controlled and is normally much coarser than one sample.

Depth accuracy

The uncertainty of the depth assigned to one picked event. Velocity error, trigger latency, filter phase, SNR and picking jitter all contribute.

The useful reporting format is therefore three numbers: (1) digital depth increment, (2) estimated vertical resolving thickness, and (3) depth uncertainty. Reporting only "resolution = V_P/44,100" overstates what the data can resolve.

2. What the grounded dipole measures

Equation 41 - Differential voltage measured by the grounded dipole

V_AB(t) = φ(B,t)-φ(A,t) = -∫_A^B E(r,t)·dl

SOURCE-SUPPORTED A two-electrode dipole measures a potential difference, not an electric field at a mathematical point.

V_AB recorded differential voltage [V]

E electric field [V/m]

dl element along electrode baseline

Source basis: differential electrode measurement and its line-integral interpretation are developed in the supplied Dipole and advanced-theory notes from the Pride/Jouniaux framework.

Equation 42 - Short-dipole field approximation

E_∥(t) ≈ - V_AB(t) / ℓ

If the field does not vary strongly along a short baseline, voltage divided by surveyed electrode spacing estimates the field component parallel to the dipole.

Source basis: the supplied material explicitly notes that electrode spacing and orientation affect amplitude, spatial averaging and observed frequency content (Bordes et al., 2015).

Figure 14. The geological interface is located from the event time relative to a known source time-zero. The audio input only digitizes the voltage waveform; it does not supply the seismic velocity model needed to turn time into metres. The source material supports the interface timing concept and differential dipole measurement; the ADC illustration is the derived implementation used here.

Instrumentation warning. A consumer computer audio input may be AC-coupled, may contain anti-alias filtering, automatic gain control, noise suppression or other "enhancements," and its input impedance may not be appropriate for direct electrode connection. Use a protected high-input-impedance differential front end and galvanic isolation where necessary, disable automatic processing, and calibrate amplitude and phase versus frequency. The supplied seismoelectric material specifically stresses electrode/contact impedance, recorder input impedance and instrument-response calibration.

3. Interface conversion time to depth

Equation 43 - One-way seismoelectric interface timing

t_IR = ∫_{source→interface} ds / V_P(s) + t_EM ≈ t_{P,to interface}

SOURCE-SUPPORTED The interface response is launched when the seismic wave reaches a contrast. On typical shallow scales the electromagnetic propagation/diffusion time is commonly small relative to the seismic travel time.

Source basis: Haartsen & Pride (1997), Haines et al. (2007), Dupuis et al. (2009), Schakel et al. (2011), as summarized in the supplied advanced-theory note.

Equation 44 - Constant-velocity vertical depth approximation

z ≈ V_P t_IR

This is a one-way depth relation for a primary interface conversion. It is not the conventional seismic-reflection formula z = V_Pt/2.

Equation 45 - Reflected-coseismic timing for comparison

z ≈ V_P t_ref / 2

A returning mechanical reflection carrying a coseismic electric field has approximately two-way seismic travel time in a simple near-zero-offset geometry. The factor of two changes the depth interval represented by each digital time sample.

4. 44.1 kHz raw sampling limits

Equation 46 - Sample interval

Δt_s = 1/f_s = 1/44,100 = 22.675736... µs

DERIVED DSP This is the exact nominal spacing of the recorded time grid.

Equation 47 - Nyquist frequency

f_N = f_s/2 = 22,050 Hz

The ADC cannot uniquely represent frequency components above Nyquist. In practice, the computer audio input's analog anti-alias filter and the seismic/electrokinetic source bandwidth usually impose a lower usable upper frequency.

Equation 48 - One-way interface depth interval per sample

Δz_s,IR = V_PΔt_s = V_P/44,100

At V_P=2000 m/s, one raw sample corresponds to 0.04535 m of one-way interface depth.

Equation 49 - Two-way reflected-coseismic depth interval per sample

Δz_s,ref = V_PΔt_s/2 = V_P/88,200

At V_P=2000 m/s, one sample corresponds to 0.02268 m of two-way reflection depth.

Figure 15. The ADC time grid is fixed, but its depth spacing is not: it scales linearly with the velocity model. A faster formation maps the same 22.676 µs sample interval to a larger depth increment.

5. Minimum and maximum vertical data resolution of the raw unfiltered samples

Without a stated velocity range there is no unique minimum or maximum value in metres. The exact raw digital increment is the function V_P/44,100 for a one-way interface conversion. The table below therefore uses a broad illustrative velocity envelope; these velocities are examples for understanding the recorder, not site-specific values inferred from the supplied papers.

Illustrative V_P (m/s)	Raw one-way interface increment (m/sample)	Raw two-way reflection increment (m/sample)	Interpretation
500	0.01134	0.00567	Very fine digital grid; true geological resolution will be wavelet/SNR limited.
1000	0.02268	0.01134	2.27 cm/sample one-way.
1500	0.03401	0.01701	3.40 cm/sample one-way.
2000	0.04535	0.02268	4.54 cm/sample one-way.
2500	0.05669	0.02834	5.67 cm/sample one-way.
3000	0.06803	0.03401	6.80 cm/sample one-way.
4000	0.09070	0.04535	9.07 cm/sample one-way.
6000	0.13605	0.06803	13.61 cm/sample one-way.

Best-case digital floor. If one sample is used as the raw timing quantum, the table gives the nominal depth-grid spacing. High-SNR correlation can estimate a peak at a fractional-sample delay, but that improves timing precision; it does not create geological bandwidth that was never recorded.

There is no finite worst-case resolution. If the event is buried in noise, clipped, removed by filtering, or the velocity is unconstrained, the interface becomes unresolved. In that case vertical resolution is undefined rather than merely "coarse."

Equation 50 - Maximum depth range is record-length controlled, not sample-rate controlled

z_max,IR ≈ V_PT_record, z_max,ref ≈ V_PT_record/2

A 44.1 kHz sample rate fixes time spacing but not recording duration. Do not confuse maximum depth range with vertical resolution.

6. Why bandwidth, not the sample interval, usually controls real vertical resolution

A geological interface response is a band-limited pulse. Two interfaces separated in depth create two converted wavelets separated in time. If their wavelets strongly overlap, the interfaces cannot be distinguished even though dozens of ADC samples lie between adjacent points on each waveform.

Equation 51 - Approximate quarter-wavelength vertical-resolution heuristic

Δz_λ/4 ≈ V_P / (4 f_dom)

DERIVED / HEURISTIC The familiar quarter-wavelength criterion is a useful order-of-magnitude check for the separation of two reflectors/converters carried by a wavelet with dominant frequency f_dom. It is not a universal seismoelectric law.

Equation 52 - Bandwidth-controlled time cell

Δt_B ~ 1/B_eff, Δz_B,IR ~ V_P/B_eff

A broad effective bandwidth produces a short correlation/impulse-response peak. A narrow band produces a long oscillatory wavelet and poor interface separation. The precise constant depends on the resolution criterion and spectral shape.

Equation 53 - Practical vertical-resolution envelope

Δz_resolve ≳ max(Δz_sample, Δz_wavelet, V_Pσ_t,pick)

DERIVED DSP This deliberately conservative expression says that the useful depth resolution cannot be finer than the largest active limitation. In most field data the wavelet width and pick uncertainty dominate the centimetric sample grid.

Figure 16. Vertical resolution is controlled by whether two converted pulses can be separated, not by how many digital samples describe one pulse. Oversampling is valuable for stable filtering and sub-sample picking, but it does not bypass the source/formation bandwidth.

7. What 16-bit amplitude resolution changes

Equation 54 - 16-bit quantization step

ΔV_q = V_FS,pp/2¹⁶ = V_FS,pp/65,536

DERIVED DSP The voltage represented by one digital code depends on the actual peak-to-peak full-scale input range of the computer audio interface. That range must be calibrated for the specific device.

Equation 55 - RMS quantization noise for an ideal uniform quantizer

σ_q ≈ ΔV_q/√12

This is an ideal model. Real audio inputs also contain analog noise, hum, distortion, clock error and front-end nonlinearities.

Equation 56 - Ideal full-scale sine quantization SNR

SNR_ideal,dB ≈ 6.02N + 1.76 = 98.08 dB for N=16

The theoretical 98 dB figure is not guaranteed in field use. If the seismoelectric signal uses only 1/10 of full scale, approximately 20 dB of available quantization SNR is forfeited before environmental noise is considered.

Figure 17. Sixteen-bit quantization determines amplitude granularity. Poor gain staging can make a nominally 16-bit recording behave like a much lower effective-resolution system, increasing event-time jitter and therefore depth uncertainty.

8. Arrival-time picking and depth uncertainty

Equation 57 - Depth estimated from a picked interface time

ẑ = V̂_P (t̂_IR - t̂₀ - t̂_sys)

The source time-zero and recorder/system delay must be calibrated. A fixed latency creates a fixed depth bias.

Equation 58 - One-way timing error to depth error

δz_time = V_Pδt

At 2000 m/s, 0.1 ms timing error is 0.20 m depth error. This is why filter phase and trigger latency can dominate the centimetric sample spacing.

Equation 59 - First-order depth uncertainty

u_z² ≈ (t u_Vp)² + (V_Pu_t)² + 2tV_PCov(V_P,t)

If velocity and time errors are treated as independent, the covariance term is zero. This shows why a 5% velocity uncertainty usually overwhelms one-sample timing quantization at meaningful depths.

Equation 60 - Effective RMS bandwidth of a picked event

β_rms² = [∫(f-f̄)²|S(f)|²df] / [∫|S(f)|²df]

A wider RMS bandwidth produces a sharper correlation peak and better delay estimation. Any filter that removes useful spectral energy can therefore worsen depth-pick precision even if it visually cleans the waveform.

Equation 61 - Timing-estimation scaling in additive noise

σ_t ∝ 1 / (β_rms√SNR)

DERIVED DSP This expresses the key delay-estimation principle without claiming a universal coefficient: timing variance improves with both bandwidth and SNR. A source-matched or physically modeled correlation picker can attain fractional-sample precision when SNR is high.

9. Noise model for the recorded dipole trace

Equation 62 - Practical single-channel observation model

x(t) = s_SE(t) + p_line(t) + n_white(t) + p_fence(t) + a_source(t) + n_instr(t)

The source material explicitly warns that cultural electrical noise and source-related electrical transients can mimic useful seismoelectric events. Power-line treatment and preservation of raw records are specifically recommended. Electric-fence pulses are added here as an application-specific impulsive-noise term.

Figure 18. Narrowband hum is primarily a spectral-line problem; white noise is an SNR problem; electric-fence pulses are an impulsive/transient problem. They should therefore not be attacked with the same filter.

10. Power-line noise: effect on vertical resolution and how to filter it

Equation 63 - Harmonic power-line model

p_line(t) = Σ_m=1^M A_m cos(2πm f₀t + φ_m), f₀=50 or 60 Hz

DERIVED DSP The fundamental depends on the local mains system. Harmonics can extend through the seismoelectric band.

Equation 64 - Notch-filter bandwidth

B_notch ≈ f₀/Q

A high-Q notch removes a narrow line and preserves most neighboring bandwidth. A low-Q notch can erase a large part of the useful wavelet and degrade vertical resolution.

Equation 65 - Filtering is convolution

y(t) = h(t) * x(t)

Every filter has an impulse response. A narrow frequency notch has a long time-domain response and can introduce ringing around a short interface event.

Equation 66 - Causal filter group-delay depth bias

δz_filter(f) = V_Pτ_g(f), τ_g(f) = -dφ(f)/dω

A causal filter with frequency-dependent phase can move or reshape a time pick. Offline zero-phase forward/reverse filtering removes net phase delay but can produce symmetric pre/post-ringing; therefore the unfiltered pick must remain a QC reference.

Preferred power-line strategy. Preserve the raw trace. Estimate and subtract stable sinusoidal/harmonic components or use the narrowest defensible notches. Avoid broad "hum removal" that unnecessarily reduces β_rms. The supplied advanced-theory material specifically recommends estimating/subtracting power-line harmonics and preserving raw data.

11. White noise: effect on vertical resolution and how stacking helps

White noise does not create a deterministic travel-time shift. Instead it broadens the statistical distribution of the picked arrival time. The source material supports recording repeated impacts, rejecting contaminated shots and stacking the accepted traces because uncorrelated noise does not repeat with the same phase.

Equation 67 - Stacking N repeatable shots

x̄(t) = (1/N)Σ_k=1^Nx_k(t)

SOURCE-SUPPORTED PRINCIPLE Repeatable seismoelectric energy adds coherently; random noise averages down.

Equation 68 - White-noise reduction under ideal stacking

σ_n,stack = σ_n/√N, SNR_amplitude ∝ √N

Twenty-five clean shots ideally reduce uncorrelated RMS noise by a factor of five. This improves pick precision without deliberately removing signal bandwidth.

Equation 69 - Matched/correlation time pick for a known event template

r_xs(τ) = Σ_n x[n] s[n-τ], t̂_IR = arg max_τ r_xs(τ)

DERIVED DSP If the source wavelet or a forward-modeled interface template is known, correlation is usually a better white-noise timing estimator than aggressive low-pass filtering. Sub-sample interpolation of the correlation peak can improve pick precision, but the physical resolving bandwidth remains unchanged.

Figure 19. Stacking improves white-noise-limited timing precision without sacrificing useful spectral bandwidth. The supplied field methodology specifically recommends individual-shot QC before stacking so contaminated impacts are not made more coherent by averaging.

12. Electric-fence pulses: effect on vertical resolution and how to filter them

Electric-fence interference is fundamentally different from 50/60 Hz hum. A short high-voltage pulse is broadband, so a conventional notch cannot remove it. It may also clip the 16-bit ADC or overload the analog input, in which case the affected samples contain irrecoverable information loss.

Equation 70 - Generic fence-pulse train

p_fence(t) = Σ_k A_k q(t-t_k)

DERIVED DSP The pulse shape q(t), repetition interval and amplitude are site dependent. The correct filter should therefore be based on observed pulse morphology, not a hard-coded repetition rate.

Equation 71 - Depth interval destroyed by pulse blanking

Δz_blank,IR ≈ V_PT_blank

If a contaminated interval of duration T_blank is excised from a one-way interface trace, any interface arrival within the corresponding depth interval is unavailable or degraded. At 2000 m/s, a 0.5 ms blank is equivalent to 1.0 m of one-way depth.

Equation 72 - Clipping test

|x[n]| ≥ x_FS ⇒ sample clipped ⇒ amplitude/phase information lost

Do not attempt to "filter back" a clipped event. Reject the shot when the overload recovery overlaps the target interface window.

Best option: reject the contaminated shot

If repeat impacts are available, discard records where a fence pulse overlaps the target window. This preserves true wavelet bandwidth in the accepted stack.

Second option: template subtraction

If a fence pulse is stable and does not saturate the input, estimate its waveform from fence-only windows and fit/subtract it. Validate that the target pick is stable before and after subtraction.

Last option: blank/interpolate

Use only when the pulse is far from the target. Blanking creates a literal missing-depth zone according to Equation 31 and can inject spectral sidelobes.

Figure 20. An electric-fence pulse is not just "extra noise" when it overlaps the target event. If it saturates or must be blanked, the corresponding depth interval is missing. Repeated-shot rejection is therefore much safer than attempting to repair a contaminated target window.

13. Combined practical vertical resolution after noise treatment

Equation 73 - Filtered effective bandwidth

S_out(f) = H(f)S_in(f), B_eff,out ≤ B_eff,in in ordinary noise-suppressing filtering

Filtering cannot create missing physical bandwidth. It can improve SNR enough to make an event pickable, but a strong low-pass or many wide notches also broaden the effective wavelet.

Equation 74 - Practical depth-resolution budget

R_z = max[R_sample, R_wavelet, R_pick, R_filter, R_gap]

DERIVED IMPLEMENTATION This is a reporting framework rather than a published seismoelectric constitutive equation. Each term should be quantified or explicitly marked "not active." A fence blank, for example, enters as R_gap; a zero-phase notch may leave R_filter near zero in timing bias but still increase R_wavelet through lost bandwidth.

Equation 75 - Depth accuracy is separate from resolution

U_z ≈ k √(u_velocity² + u_pick² + u_trigger² + u_filter-delay² + u_clock²)

The terms must be expressed in common depth units before combination. Use an appropriate coverage factor k for the reporting convention. This uncertainty describes where a single interface lies; it is not identical to the minimum spacing at which two interfaces are separable.

Disturbance / process	Main effect on vertical resolution	Best treatment for timing	When to reject
50/60 Hz power-line + harmonics	Raises coherent spectral lines; wide notches reduce effective bandwidth and can ring.	Narrow sinusoidal regression/subtraction or high-Q zero-phase notch; compare picks with raw trace.	If hum dominates the event band or causes front-end overload.
White/broadband noise	Increases timing jitter; does not inherently blur the wavelet deterministically.	Repeat-shot QC + stacking + correlation/matched filtering.	If SNR remains too low to produce a stable pick.
Electric-fence pulse	Broadband transient; may create missing depth interval or clipping.	Reject contaminated shot; otherwise validated template subtraction.	Whenever pulse or overload recovery overlaps the target window.
Low-pass filter	Improves high-frequency noise but broadens the wavelet and reduces vertical resolution.	Set cutoff from measured signal spectrum, not visual preference.	If required cutoff removes the frequency content needed for the desired layer separation.
Causal IIR filter	Can shift event time via group delay.	Calibrate delay or use offline zero-phase processing.	If phase is unknown and cannot be reconstructed.

14. Practical implementation for a 44.1 kHz, 16-bit audio recording system

Figure 21. Resolution-safe implementation. Filtering occurs only after raw acquisition, front-end calibration and shot-level contamination checks. The filtered result is always compared with the original timing pick so that a cleaner-looking trace is not mistaken for a more accurate depth estimate.

Recorder/front-end calibration

Measure actual sample clock, channel gain, frequency response, full-scale voltage, input impedance and phase response. Disable AGC, echo cancellation, noise suppression and operating-system audio "enhancements." Record in an uncompressed PCM format.

Time-zero calibration

Depth is impossible without a source reference. Measure the source trigger/time-zero and any fixed audio-path latency. A known waveform can be injected to measure recorder latency and filter phase.

Shot QC before stacking

Use a pretrigger window to estimate RMS noise, mains components and fence pulses. Reject clipped records and any record where an impulsive artifact overlaps the target conversion.

Pick stability test

Re-pick the event on raw, hum-subtracted, stacked and final-filtered traces. If depth changes materially with reasonable filter choices, the interface is not robustly resolved.

15. Interactive 44.1 kHz vertical-resolution calculator

Use the calculator as a design/QC tool. The site velocity and useful frequency must come from the survey/model; the defaults below are examples, not claimed site parameters.

P-wave velocity Vp (m/s)

Effective resolving frequency f (Hz)

Pick uncertainty (samples, 1-sigma)

Velocity uncertainty (%)

Fence/blank interval (ms)

Record length (s)

44.1 kHz sample interval22.676 µs

One-way depth / sample

Two-way depth / sample

Quarter-wave heuristic

Pick-jitter depth sigma

Fence blank depth span

Velocity-only depth uncertainty at 10 m

Max one-way depth in record

Max two-way reflection depth

16. Worked examples

Example A: Vp = 1500 m/s

Raw one-way sample grid0.0340 m

Two-way sample grid0.0170 m

If the useful event frequency is only 500 Hz, the quarter-wavelength heuristic is 0.75 m: over twenty times coarser than the ADC depth grid.

Example B: Vp = 2000 m/s

Raw one-way sample grid0.0454 m

0.1 ms timing error0.20 m

A 0.5 ms electric-fence blank removes 1.0 m of one-way depth coverage, showing why impulsive contamination is often more damaging than sampling quantization.

Example C: Vp = 4000 m/s

Raw one-way sample grid0.0907 m

Two-way sample grid0.0454 m

At 1000 Hz the quarter-wavelength heuristic is 1.0 m. Again, signal bandwidth rather than ADC sample spacing controls layer separation.

Example interpretation. Suppose a primary interface conversion is picked at 8.00 ms and V_P=2000 m/s. The one-way depth estimate is 16.0 m. One sample is 0.045 m, but a 5% velocity uncertainty alone is ±0.8 m before pick noise, trigger error or filtering are included. This is why displaying depth to the nearest centimetre would be false precision.

17. Conditions under which vertical resolution should not be quoted

No independent velocity model

The data contain time, not metres. Report interface time rather than a depth if V_P is not constrained.

Unknown source time-zero

A fixed timing offset maps directly into a fixed depth bias. Sample rate cannot correct an unknown trigger latency.

Audio processing is active

AGC, noise suppression, resampling or undocumented codec processing can alter amplitude and phase. Use raw PCM and a calibrated signal path.

Target event is clipped

ADC saturation is nonlinear information loss. A clipped interface event does not have a defensible amplitude or phase and may have a biased time pick.

Fence pulse overlaps the target

If the shot cannot be rejected and the pulse cannot be independently modeled without altering the target, resolution is undefined across that depth interval.

Depth changes with filter settings

If reasonable hum/noise filters move the inferred interface by more than the quoted uncertainty, the event is not robustly resolved.

Final resolution rule. For a 44.1 kHz, 16-bit recorder, the sampling grid can be centimetric, but the geological vertical resolution should be reported from the usable event bandwidth and pick stability, while depth accuracy should include the velocity and timing uncertainty. Power-line filtering should remove as little bandwidth as possible; white noise is best attacked by repeat-shot QC, stacking and correlation; electric-fence pulses should preferably cause shot rejection rather than broad filtering or target-window interpolation.

Part III

Reflections, signal separation, calibration and inversion

How reflected seismic waves create secondary seismoelectric energy, how those events are separated or deliberately used, and how source-normalized timing and spectra constrain depth and permeability.

Integrated source module

Reflected Coseismic Energy, Reflection Noise and Signal Separation

Specialized signal-separation section. It classifies reflected coseismic energy, secondary conversions and reverberations, and develops timing, slowness, polarity and filtering tests that protect quantitative inversion.

Interpretation policy. This note uses “does not significantly affect” to mean “normally does not dominate the selected seismoelectric observable.” It does not imply that reflected mechanical or electromagnetic modes are absent from the coupled solution.

1. Three different responses must be kept separate

Much confusion comes from treating every late electrical arrival as the same phenomenon. In coupled seismoelectric theory there are at least three physically different responses.

Primary coseismic field

An electric field travels with the incident P, S, or surface-wave disturbance because local pressure gradients and relative fluid motion drive electrokinetic current. It has seismic-like velocity and moveout (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016).

Interface response

When the incident seismic wave reaches a property contrast, the boundary conditions generate an electromagnetic conversion. Because the electromagnetic field propagates much faster than the seismic wave over typical shallow scales, the interface response appears close to the one-way seismic time and can be nearly flat across a surface receiver array (Haartsen & Pride, 1997; Schakel et al., 2011).

Reflected coseismic field

The seismic wave reflected from the boundary can also carry a coseismic electric field back toward the receivers. This is a real coupled event, but it is later and usually smaller because the mechanical reflection has already suffered partitioning, geometric spreading, and attenuation.

Figure 22. The interface-generated electrical response is launched when the incident seismic wave first reaches the contrast. A reflected coseismic field cannot be observed until the reflected mechanical wave travels back toward the receiver. Haartsen & Pride showed that converted electromagnetic arrivals can coincide approximately with one-way seismic time, whereas later coseismic arrivals follow the seismic reflection timing (Haartsen & Pride, 1997).

2. Reason 1 - the target interface response is generated before the reflection returns

For a horizontal interface at depth z and a near-vertical incident P wave, the first-order timing is especially simple. The interface response is triggered when the incident seismic wave reaches the interface. The converted electromagnetic field then reaches surface electrodes with a delay that is small compared with the seismic travel time. The reflected mechanical wave, by contrast, must complete the upward return path.

Equation 76 - first-order interface-response time

t_IR = t_down + t_EM ≈ z / V_P

Meaning. At shallow scale, t_EM is commonly small enough that the observed interface-response time is dominated by the one-way seismic path.

t_IR interface-response time [s]

V_P P-wave speed [m s^-1]

z interface depth [m]

t_EM electromagnetic propagation/diffusion time [s]

Source provenance. One-way seismic triggering of fast interface conversion is developed in layered-media theory and observed in shallow surveys (Haartsen & Pride, 1997; Haines et al., 2007).

Equation 77 - approximate reflected seismic return time

t_ref ≈ t_down + t_up ≈ 2z / V_P

Meaning. In the simplest zero-offset geometry, the reflected wave arrives roughly one additional one-way travel time after the interface response.

Survey consequence. A time gate around the interface response can be selected before the principal reflected coseismic event arrives. The separation is not perfect in every geometry, but it is a strong first-order discriminator.

Source provenance. General seismic reflection timing applied to the interface-response kinematics described by Haartsen & Pride and later layered-media studies.

Timing is the first defense. The primary interface response is not produced by the reflected seismic wave. It is produced on the first encounter of the incident wave with the contrast. The reflection is therefore a later secondary event unless the geometry is extremely shallow or strongly reverberant.

3. Reason 2 - the reflected mechanical wave has already lost amplitude

At a boundary, incident mechanical energy is partitioned among reflected, transmitted, and converted wave modes. A reflected wave therefore begins with only a fraction of the incident amplitude. It then travels an additional path and loses more amplitude through geometric spreading and material attenuation. Because the low-frequency coseismic electric field is approximately proportional to local solid acceleration, the reflected coseismic electric field inherits that mechanical amplitude reduction (Pride & Haartsen, 1996; Bordes et al., 2015).

Equation 78 - low-frequency coseismic electric field

E_c ≈ -ρ_fC_sa_s

Meaning. For a P-dominated low-frequency wave, the coseismic electric field scales with the local solid-frame acceleration.

E_c coseismic electric field [V m^-1]

ρ_f fluid density [kg m^-3]

C_s streaming-potential coefficient [V Pa^-1]

a_s solid-frame acceleration [m s^-2]

Source provenance. Low-frequency fast-P transfer summarized and tested in (Jouniaux & Zyserman, 2016; Bordes et al., 2015).

Equation 79 - practical scaling of the reflected mechanical amplitude

|a_ref| / |a_inc| ≈ |R_s| G_spread exp(-α_sΔL)

Meaning. The reflected wave is reduced by the seismic reflection coefficient, by the extra geometric spreading of the return path, and by attenuation accumulated over the additional path.

R_s seismic reflection coefficient

G_spread spreading factor, generally less than 1 for the added path

α_s seismic attenuation coefficient

ΔL additional reflected-wave path length

Important note. This is a general propagation scaling, not a special seismoelectric constitutive law.

Equation 80 - consequence for the reflected coseismic electric field

|E_ref| / |E_inc| ≈ |a_ref| / |a_inc|

Meaning. If the incident and reflected wave portions sample similar electrokinetic coupling, the coseismic electric field scales approximately in proportion to mechanical acceleration. The weaker reflected mechanical wave therefore produces a weaker reflected coseismic electric field.

Literature consistency. White & Zhou's layered calculations explicitly show primary and multiple converted events, noting that the multiples are considerably smaller than the primary events. Wang et al. likewise found reflected electromagnetic events weaker than the electromagnetic fields accompanying mechanical waves in their marine layered example (White & Zhou, 2006; Wang et al., 2020).

Figure 23. A reflected coseismic field is not an independent strong electromagnetic source; it is coupled to a mechanically reflected wave that has already lost amplitude. This is one reason late reflected coseismic energy is commonly subordinate to primary converted responses.

4. Reason 3 - reflected coseismic energy retains seismic moveout

Even when reflected coseismic energy is visible, it does not have the same kinematic signature as a fast interface-generated electrical response. The reflected coseismic field remains spatially associated with the reflected mechanical wave and therefore follows seismic slowness and reflection moveout. Interface radiation is generated at the boundary and, over common shallow layouts, can appear almost simultaneous across the electric array because the electromagnetic propagation time is short (Haartsen & Pride, 1997; Haines et al., 2007; Warden et al., 2012).

Equation 81 - conventional reflected-wave moveout as a discriminator

t_ref(x) ≈ √(t₀² + x²/V_RMS²)

Meaning. A reflected mechanical wave and its coseismic electric field generally acquire offset-dependent travel time. The target interface response often has much smaller apparent slowness across the electric array.

Field use. Plot electric traces as a shot gather. A sloping or hyperbolic event consistent with seismic velocity is classified as coseismic/reflected energy; a nearly flat event at the predicted one-way interface time is a candidate interface conversion.

Source provenance. The moveout distinction is central to experimental seismoelectric signal separation (Haines et al., 2007; Warden et al., 2012).

Figure 24. Conceptual electric shot gather. The interface response is differentiated from direct or reflected coseismic energy by apparent slowness, timing, polarity, and geometry - not merely by amplitude. This distinction is emphasized in field experiments and processing studies (Haines et al., 2007; Warden et al., 2012).

5. Reason 4 - survey geometry can be chosen to minimize reflected/coseismic contamination

Haines et al. demonstrated that seismoelectric acquisition geometry is unusually flexible because the target can be an electromagnetic conversion rather than a mechanically reflected arrival. In their shallow-target work, off-line geometries were used to isolate interface response energy from coseismic fields. The paper explicitly treats coseismic contamination as a signal-separation problem and notes that geometry can be exploited to avoid it (Haines et al., 2007).

Surface interface-response survey

Receiver geometry is chosen so that the target conversion has the expected one-way timing and low apparent slowness, while seismic-like energy has a different moveout. Reflected coseismic arrivals are then late and geometrically distinguishable.

Near-source / local-coupling survey

If a dipole is located immediately beside the source, the selected early window samples the local source-zone coseismic response. Reflections must complete a much longer down-and-up path before returning, so a short early-time window naturally excludes them.

Important consequence. The statement that reflections “do not significantly affect” results is strongest when the interpretation is based on a deliberately selected early-time or interface-response window. It is much less defensible if the entire late-time record is inverted without separating wave modes.

6. Major forms of reflection-related noise

For inversion, “reflection noise” is not one waveform. It is a family of source-coherent arrivals produced after the first local/primary response. Each class enters the electric record differently, and the filter must be chosen from its kinematics rather than from amplitude alone.

Reflected coseismic P-wave field

A P reflection carries a coseismic electric field because the reflected mechanical wave again drives relative pore-fluid motion. It has two-way reflection timing and P-wave moveout. It is usually the most obvious reflection contaminant.

Mode-converted reflected S / PS / SP energy

Elastic contrasts can partition the incident field into reflected and converted modes. Their seismoelectric fields arrive on different moveout branches and can broaden the late-time wave train. In a short-offset single-dipole record these contributions are handled by the same early-time exclusion rule: do not allow their first plausible arrival into the inversion window.

Multiples and reverberation

Repeated bounces between the free surface and shallow contrasts, or between strong layers, generate a train of coherent delayed mechanical arrivals. Each bounce can carry a coseismic field. Narrow-band sites can ring long enough for these arrivals to overlap the tail of the desired signal.

Secondary conversion caused by a reflected wave

A reflected seismic wave can re-cross another hydraulic/electrical contrast from below and generate an additional interface electromagnetic response. This is not simply the electric field riding with the reflection; it is a new conversion launched by the reflected mechanical wave. Layered models explicitly contain such events (Zheng et al., 2021; White & Zhou, 2006).

Near-offset flat top of a reflection hyperbola

Near zero offset, the apex of a reflection hyperbola can appear nearly flat. That is a known weakness of pure apparent-velocity filtering: a small part of the reflection can occupy the same low-slowness region as a desired interface response. Warden et al. note this limitation for f-k separation and cite discarding the first few near-offset traces as one mitigation.

Source-related coherent electrical transients

Trigger leakage, cable motion, inductive pickup and the Lorentz field of moving metal are not reflections, but they are coherent and can be confused with early seismoelectric energy. They must be removed by source-control tests before reflection filtering; otherwise an apparent “clean early window” can still be non-geologic (Haines et al., 2007; Haines et al., 2003).

Equation 82 - Convert the measured dipole voltage to electric field before filtering

E_obs,∥(t) ≈ -ΔV(t)/ℓ

Filtering should be performed on calibrated physical units whenever possible. ℓ is the surveyed electrode spacing and the sign depends on electrode order.

Source provenance. General finite-dipole potential identity used directly in seismoelectric laboratory and field measurements (Bordes et al., 2015).

Equation 83 - Observation model used to think about contamination

E_obs(t)=E_local(t)+Σ_mE_ref,m(t)+Σ_qE_conv,q^(ref)(t)+E_src(t)+n(t)

Meaning. The trace contains the wanted local/primary response, electric fields riding with reflected seismic modes, secondary interface conversions triggered by reflected waves, source-related electrical contamination, and incoherent noise. Stacking mainly reduces the last term; it does not remove the coherent reflection terms.

Provenance. Bookkeeping/implementation equation that separates the coupled event classes described by Haartsen & Pride (1997), Haines et al. (2007), White & Zhou (2006), Warden et al. (2012), and Zheng et al. (2021).

Equation 84 - Why a delayed reflection can create a false spectral corner

H_obs(ω)=H₀(ω)+R(ω)e^-iωτ
|H_obs|²=|H₀|²+|R|²+2 Re{H₀R^*e^iωτ}

Meaning. The wanted transfer function H₀ and a delayed reflected term R e^-iωτ interfere. Even if |R| is modest, the cross term oscillates with frequency and can produce spectral peaks and notches. Those artificial features can be mistaken for dynamic coupling roll-off if the full record is transformed and fitted.

Approximate ripple spacing. For a nearly constant delay, adjacent interference cycles are separated by Δf ≈ 1/τ.

Provenance. Derived Fourier interference identity applied here to the reflection problem; the need to preserve waveform/amplitude during seismoelectric wave separation is emphasized by Warden et al. (2012).

7. Exact filtering workflow for a single near-source dipole

A single dipole has no receiver-offset dimension, so f-k, Radon, and curvelet slowness separation cannot be performed on that one trace. The correct reflection filter is primarily a physics-defined early-time gate, supported by source-control tests, tapered edges, residual tests, and - only when necessary - model-based nuisance subtraction.

1. CalibrateΔV(t) → E(t); correct gain, polarity, trigger and electrode spacing.

2. PredictUse the seismic model to calculate the earliest plausible reflected arrival.

3. Gate + taperEnd the local window before that arrival; use soft edges to protect the spectrum.

4. Residual testCheck the late-time energy and, if overlap exists, subtract a constrained reflection template.

5. Invert clean spectrumEstimate coupling and f_c only from the gated/cleaned response.

Equation 85 - Predict the earliest physically plausible reflected arrival

t_ref,min=min_m[L_m/V_m] and for a simple near-vertical P reflection t_ref≈2z/V_P

How to use it. Build the earliest reflected path allowed by the shallow seismic model. A conservative gate uses the shallowest credible reflector and the fastest credible P velocity, because that combination produces the earliest return. The inversion window must end before this time after allowing for the source wavelet duration and timing uncertainty.

Provenance. Standard reflection kinematics applied to the seismoelectric event separation already developed in Sections 2-4.

Equation 86 - Raised-cosine gate used instead of a hard rectangular cut

w(t)=0 outside the accepted interval; w(t)=1 in its central plateau;
w(t)=½[1-cos(π(t-t_a)/T_r)] on the leading ramp, and the reversed cosine on the trailing ramp.

Meaning. A hard time cut has a sinc-like spectral response and can manufacture oscillations. A short cosine/Tukey taper reduces edge leakage while still zeroing late reflected energy. Choose the ramp length long enough to avoid a discontinuity but short relative to the usable local wavelet.

Provenance. Standard spectral-windowing implementation; not a seismoelectric constitutive equation.

Equation 87 - The actual trace sent to the transfer-function calculation

E_gate(t)=w(t)E_obs(t), E_gate(ω)=ℱ{E_gate(t)}

Critical rule. The FFT is taken after the physically justified time gate. Transforming the full record first and then fitting a frequency band does not remove reflection interference, because the reflection has already contributed its phase-delayed spectral term at every frequency.

Equation 88 - Quantify potential reflection leakage into the chosen gate

ε_ref=∫ |w(t)E_ref,mod(t)|²dt / ∫ |w(t)E_local,mod(t)|²dt

Meaning. If a forward model is available, calculate how much modeled reflected energy survives the same gate. The gate is acceptable only when this fraction is small relative to the measurement and model uncertainty. There is no universal numerical threshold; it must be reported and tested against the resulting permeability stability.

Provenance. Derived energy-leakage QC metric for this implementation.

Equation 89 - Optional nuisance-template subtraction when a reflection overlaps the local tail

β̂=arg min_β Σ_t∈C|E_obs(t)-Σ_mβ_mr_m(t)|²
E_clean(t)=E_gate(t)-w(t)Σ_mβ̂_mr_m(t)

Meaning. If shallow reverberation overlaps the gate, use predicted reflected wavelets r_m as nuisance templates. Fit their amplitudes only in a control interval C dominated by reflection energy (or from independent multi-trace/seismic data), then subtract the fitted nuisance terms. Do not estimate the template amplitude from the target portion alone, because the fit can subtract true electrokinetic signal and bias the transfer amplitude.

Provenance. General constrained least-squares nuisance subtraction proposed here as an implementation option when simple time exclusion is impossible; it should be validated by synthetic injection/recovery.

Figure 25. Reflection rejection for the single-dipole permeability workflow. The earliest physically plausible reflected arrival is predicted from the seismic model. The selected local response ends before that arrival and is tapered rather than hard-clipped. The FFT and transfer-function inversion are then performed on the cleaned gated trace, not on the entire record. The animation emphasizes that the late reflected energy is excluded rather than averaged away.

A single dipole cannot prove reflection rejection from slowness. If the earliest reflection overlaps the desired local response so strongly that a clean gate cannot be drawn, the permeability inversion is not identifiable from that trace alone. Add a geophone/accelerometer, repeat measurements at offsets, or acquire an electrical array so the reflection can be separated kinematically rather than forcing the inversion.

8. f-k, Radon and curvelet filters for multi-trace data

When several dipoles or repeated source positions create a time-offset gather, reflected/coseismic energy has a spatial slope and can be removed in transform domains. Warden et al. directly compared f-k, linear Radon, and curvelet filters. Their central result is that wave separation must preserve the wanted waveform and amplitude if the filtered signal will later be inverted; a filter that looks visually clean but distorts amplitude can still bias quantitative interpretation (Warden et al., 2012).

Equation 90 - f-k transform and high-apparent-velocity mask

D(f,k)=∫∫ d(t,x)e^-i2π(ft-kx)dtdx
D_keep(f,k)=M_fk(f,k)D(f,k), with M_fk≈1 near k=0 and 0 for seismic-like slopes.

Because apparent velocity satisfies v_app=f/k (for consistent transform units), a near-zero-slowness interface response lies close to k=0. Warden et al. used a tapered velocity boundary rather than a hard pie-slice. Their synthetic example preserved samples faster than a chosen high-velocity threshold and zeroed slower energy, with tapering between thresholds to reduce Gibbs artifacts.

Limitation. The apex of a reflection hyperbola can be locally flat, so f-k alone may fail at the first near-offset traces; Warden et al. explicitly discuss this problem.

Equation 91 - Linear Radon transform / slowness filter

R(τ,p)=∫ d[x,τ+px] dx

A linear arrival with slowness p collapses into the τ-p domain. Keep coefficients near p≈0 for a fast interface response and reject the higher-slowness reflection/coseismic region. Warden et al. found the Radon approach could preserve relative amplitude better than f-k, but absolute amplitude required correction and the output could be biased toward lower frequencies.

Equation 92 - Curvelet-domain acceleration mask (conceptual implementation)

C_E,clean(j,l,k)=M(j,l,k)C_E(j,l,k)
M=0 where the corresponding acceleration coefficient exceeds a scale/angle threshold; otherwise M=1, with the threshold weighted to favour horizontal / zero-slowness directions.

Warden et al. transform both electric data and seismic acceleration into the curvelet domain. Coefficients associated with the coseismic acceleration are zeroed, while a scale- and angle-dependent Gaussian threshold favours the horizontal directions expected for the interface response. Their paper reports better preservation of the desired radiation pattern than conventional dip-based filtering in their tests.

Source provenance. Algorithmic summary of Warden et al. (2012); use their published threshold definitions and calibration if implementing their exact curvelet filter rather than the conceptual mask shown here.

Figure 26. Transform-domain separation for multi-trace data. Warden et al. compared f-k, linear Radon, and curvelet-domain filtering. f-k and Radon separate events by apparent velocity/slowness; the curvelet method additionally uses seismic acceleration to construct a multiscale directional mask and was designed to better preserve the interface-response waveform and radiation pattern. None of these spatial filters substitutes for the early-time gate in a true single-dipole permeability experiment.

For permeability from a local single-dipole response, transform-domain filters are a diagnostic extension, not the first step. If multiple traces are available, they can confirm that late energy follows seismic moveout and can supply reflection templates. The final local coupling/permeability spectrum should still be formed from a time window that contains the intended response and excludes later modes.

9. Clean transfer function and permeability-safe inversion

Once reflection energy is excluded, the filtered electric field is allowed into the quantitative inversion. The order matters: filter first, then estimate coupling, then fit the dynamic corner, then calculate permeability.

E_clean(t)gated, calibrated electric field

H_SE(ω)electric / modeled or measured acceleration

Ĉ_s, L̂low-frequency coupling plateau

f̂_cdynamic roll-off/corner fit

k̂, K̂_hpermeability and hydraulic conductivity

Equation 93 - Clean empirical seismoelectric transfer function

H_SE,clean(ω)=E_clean(ω)/A_mod,∥(ω)

Meaning. The denominator is measured acceleration if a co-located mechanical channel exists, or the calibrated source-derived local acceleration model in the single-dipole workflow. Bordes et al. use transfer functions because electric amplitude alone is not sufficient when seismic amplitude changes.

Source provenance. Transfer-function practice and dynamic formulation in Pride & Haartsen (1996), Schoemaker et al. (2012), and Bordes et al. (2015).

Equation 94 - Magnitude-squared coherence used as a spectral QC weight when paired mechanical data exist

γ²_EA(f)=|S_EA(f)|²/[S_EE(f)S_AA(f)]

Use. Frequencies with poor coherence or poor shot-to-shot repeatability should receive little or zero weight. Coherence is a QC measure, not a reflection filter: a reflection is source-coherent and may have high coherence.

Provenance. Standard cross-spectral coherence definition applied to transfer-function quality control.

Equation 95 - Weighted low-frequency estimate of the streaming-potential coefficient

Ĉ_s=- [Σ_iw_i Re{E_clean,iA_i^*}] / [ρ_fΣ_iw_i|A_i|²]

Fit only the low-frequency plateau band for which the fast-P approximation is valid. Weights can combine coherence, stacking variance, calibration uncertainty and a mask that is zero outside the accepted band.

Source provenance. Derived weighted estimator from the low-frequency transfer relation; the same estimator is used in the supplied single-dipole permeability document and is consistent with transfer-function analysis in Bordes et al. (2015) and Schoemaker et al. (2012).

Equation 96 - Convert streaming coupling to the Pride transport coefficient

L̂=-σĈ_s

Under open-circuit conditions the electric current balance gives this direct conversion. Conductivity uncertainty therefore transfers directly into L̂.

Equation 97 - Fit the dynamic corner only on the cleaned spectrum

f̂_c=arg min_{f_c} Σ_iw_i|H_SE,clean(ω_i)-H₀G_d(ω_i;f_c)|²

Meaning. The dynamic model is fitted after time-domain reflection rejection. H₀ represents the low-frequency plateau and G_d the normalized dynamic roll-off.

Important. A detailed Pride/Biot/JKD implementation should be used when data quality supports it. A simple monotonic roll-off is acceptable only as a transparent first-pass model.

Equation 98 - Compact first-pass dynamic roll-off used for screening

G_d(ω;f_c)≈[1+iω/(2πf_c)]^-1/2

This form provides a flat low-frequency plateau and a smooth amplitude/phase transition. It is a practical fitting approximation, not a unique constitutive law.

Equation 99 - Transition frequency to intrinsic permeability

f_c=φη_f/(2πα_∞ρ_fk₀) ⇒ k̂=φη_f/(2πα_∞ρ_ff̂_c)

Meaning. Once a defensible transition frequency is resolved, the Biot characteristic-frequency relationship can be rearranged for intrinsic permeability.

Bandwidth requirement. If the cleaned source/receiver spectrum does not span both sides of the transition, report the corner/permeability as weakly constrained or do not report it.

Source provenance. Biot/electrokinetic characteristic frequency explicitly used in Bordes et al. (2015); rearranged for permeability as in the supplied permeability-inversion document.

Equation 100 - Intrinsic permeability to hydraulic conductivity

K̂_h=k̂ρ_fg/η_f

Report the fluid temperature/chemistry used for ρ_f and η_f.

Figure 27. Mechanism of permeability bias. A delayed reflected contribution adds coherently to the intended signal. In frequency space the phase delay creates alternating constructive and destructive interference. The resulting notches can mimic or move the dynamic transition. Therefore the transition frequency is fitted only after reflection rejection, and its stability is checked against reasonable changes in the gate and fitting band.

10. Verification that filtering did not manufacture permeability

Filtering becomes part of the inversion operator, so it must be validated quantitatively. A believable permeability estimate is one that survives reasonable changes in the reflection-rejection choices and can recover known synthetic or test-site parameters.

Test	What is changed	Pass criterion	Failure means
Gate-end sensitivity	Move the trailing gate edge earlier/later within the reflection-free uncertainty interval.	f̂_c and k̂ remain within the stated uncertainty.	The fitted corner is controlled by the window or by leaked reflection energy.
Taper sensitivity	Vary cosine/Tukey ramp length.	Low-frequency coupling and corner are stable.	Spectral leakage from the window edge is influencing the fit.
Late-energy null test	Process an intentionally late window containing the reflection but not the local target.	It must not reproduce the same permeability corner as the accepted local window.	The “corner” may be a source/reflection spectral feature rather than a formation parameter.
Reflection injection/recovery	Add modeled reflected arrivals of realistic amplitude/delay to a clean synthetic local response, then run the full filter.	Recovered k returns to the injected true value without systematic bias.	The filter is not sufficiently rejecting the modeled reflection class.
Fit-band sensitivity	Shift the accepted frequency band while retaining adequate SNR/coherence.	Corner remains stable when the data genuinely span it.	A narrow-band notch/ripple is being mistaken for dynamic roll-off.
Independent validation	Compare with slug/pumping/core permeability or a calibrated site.	Agreement within propagated uncertainty.	Model inputs, source calibration, saturation assumptions, or reflection rejection require revision.

Equation 101 - Simple permeability stability metric

S_k=max_j|k̂_j-median(k̂)| / median(k̂)

Here j indexes reasonable processing variants (gate end, taper, fitting band, reflection-template amplitude). S_k is not a universal acceptance standard; it is a transparent way to report how much the permeability estimate depends on filtering choices.

Provenance. Reporting/QC metric proposed here; the accepted tolerance should be set by the project's total uncertainty budget.

Permeability is accepted only after a “filter invariance” check. The filtering parameters may change the noise level, but they should not move the physical transition frequency systematically. If f̂_c walks with the gate edge or a spectral notch disappears when the gate is shortened, the dataset does not yet support a reflection-independent permeability estimate.

11. When reflected seismic waves can significantly affect the result

The source literature contains clear examples showing that reflected-wave effects are not always negligible. These cases define the boundary of the “normally secondary” assumption.

Very shallow or high-Q reverberant systems

Laboratory review literature notes that acoustic absorbers are sometimes added specifically to reduce reflected-wave effects in harmonic measurements. That is direct evidence that reflections can bias measurements when the apparatus supports reverberation (Jouniaux & Zyserman, 2016).

Strong layered reflectors and multiples

Layered-media theory produces primary and multiple conversions, P-wave reflections, and seismic reverberations. White & Zhou found the multiples smaller than primaries, but they are still present and identifiable (White & Zhou, 2006).

Reflected waves re-cross another electrokinetic interface

Zheng et al. modeled P-wave reflections that generated additional electric signals when the reflected wave impinged on a shallower interface from below. Multiple P reflections could therefore create additional electric arrivals (Zheng et al., 2021).

Marine or highly layered full-wave problems

Wang et al. explicitly modeled reflected electromagnetic waves and reflected seismic contributions in a marine layered model. Their reflected EM events were weaker than the fields accompanying mechanical waves in the illustrated case, but were still potentially detectable (Wang et al., 2020).

Do not suppress reflections by assumption when the data contradict it. If late coherent events are large, if the target is extremely shallow, if there are strong impedance contrasts, or if the source is narrow-band and the site reverberates, use a full layered or numerical coupled model that includes reflected and transmitted seismic and electromagnetic modes.

12. Practical interpretation rule

A useful field rule is to treat reflected seismic waves as a secondary modeled contaminant, never as an identically zero contribution. For a near-source permeability inversion, the result is defensible only when the local response is isolated before the FFT and the recovered corner is invariant to reasonable filtering choices:

predict earliest reflection→gate + taper before it→test residual reflection leakage→form H_SE,clean→fit f_c→check k stability

If the desired response and the earliest reflection cannot be separated in time and there is no array/mechanical channel to separate them by moveout or waveform, the correct outcome is not to force a permeability value. The reflected-wave terms must be included explicitly in the forward model or the acquisition must be expanded.

Bottom line. Reflected seismic waves usually do not dominate seismoelectric survey results because the primary electrical conversion is generated on the first arrival at the target, whereas the reflected mechanical wave must return along a longer path, carries less mechanical amplitude, and produces an electric field with seismic-like timing and moveout that can be isolated. Reflections are therefore generally a later and weaker contribution - not a nonexistent one.

Integrated source module

Reflection-Assisted Calibration for Depth and Permeability

Specialized single-dipole inversion section. It uses a known seismic impulse, primary interface conversion and returning reflected coseismic energy as mutually constraining observations for depth and permeability.

DIRECTLY SUPPORTED physical event classes, low-frequency electric/acceleration transfer, one-way interface timing, reflected seismic/coseismic events, layered-media conversions, characteristic-frequency permeability, source-control and stacking. DERIVED SYNTHESIS using the primary interface response and reflected coseismic return jointly as a two-event single-dipole calibration pair. No supplied paper was identified as validating this exact end-to-end architecture as a standalone method.

Meaning of “only one sensor.” The workflow requires no field geophone or accelerometer at the measurement station. It still requires a seismic source, a trigger/time-zero, calibrated source waveform, known dipole geometry, and fixed or prior values for the quantities needed by the equations. “One sensor” therefore means one measured subsurface response channel, not “one number with no supporting model.”

1. What the supplied source material actually supports

The method is strongest when each piece is tied to a documented physical result, and the new synthesis is clearly labelled.

Coseismic electric fields are real wave-coupled observables

Electrokinetic fields accompany seismic motion in fluid-bearing porous media. In the low-frequency fast-P regime the electric field is approximately proportional to local solid-frame acceleration (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016; Bordes et al., 2015).

Interfaces generate fast ES responses

A seismic wave incident on a contrast can generate electromagnetic conversion energy. In shallow settings, its observation time is dominated by the one-way seismic time to the converter (Haartsen & Pride, 1997; Haines et al., 2007; Dupuis et al., 2009; Schakel et al., 2011).

Reflections remain in the coupled solution

Reflected seismic waves can carry returning coseismic fields, and layered models can contain multiply reflected seismic and electromagnetic events (White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

A known source can replace a measured mechanical channel in a constrained inversion

The supplied single-dipole workflow explicitly uses a source-to-acceleration forward operator when no geophone is recorded (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016).

Permeability can be related to a dynamic transition

The archive gives a characteristic Pride/Biot frequency and its inversion to intrinsic permeability, subject to porosity, tortuosity, fluid properties, bandwidth and model assumptions (Bordes et al., 2015; Jouniaux & Zyserman, 2016; Jougnot & Solazzi, 2021).

The exact “reflection-assisted self-calibration” is a synthesis

The archive supports all component relationships, but it does not present a direct published formula saying that the ratio of one interface return and one reflected coseismic return uniquely yields aquifer permeability. That ratio is introduced here as a forward-model constraint, not a new empirical law.

Figure 28. The proposed hardware-minimal architecture. The physical event classes - coseismic fields, interface conversions, reflected seismic/coseismic returns, and additional layered-media events - are all supported by the supplied coupled-wave literature. Using the pair as an end-to-end single-dipole calibration is a synthesis described in this note, not a claim that one cited paper validates the complete workflow.

2. Minimal hardware and required priors

Live field hardware

Seismic impulse sourceknown F(t), amplitude, trigger

Grounded dipoleone differential electrical channel

Recorderknown gain, sampling, clock

Repeat shotssource repeatability / stacking

Quantities still required by the model

Dipole spacing/orientationℓ, polarity

Velocity modelV_P(z) or prior

Fluid propertiesρ_f, η_f

Petrophysical priorsφ, α_∞

Electrical properties if L is neededσ, σ_w

The requirement for source signature, elastic properties, conductivity, fluid density/viscosity, porosity and tortuosity is explicit in the supplied single-dipole permeability workflow. Removing the geophone reduces hardware but increases dependence on these calibrated inputs (Haines et al., 2003; Jouniaux & Zyserman, 2016; Bordes et al., 2015).

Equation 102 - Grounded-dipole voltage measurement

V_AB(t)=φ(B,t)-φ(A,t)=-∫_A^BE(r,t)·dl

Meaning. The receiver measures a potential difference, which is the line integral of electric field along the grounded electrode pair.

V_AB measured differential voltage [V]

E electric field [V m^-1]

ℓ electrode spacing [m]

Field use. Survey electrode order, spacing and orientation; polarity must remain fixed across repeated shots.

Source provenance. The potential/field relation and importance of dipole geometry are summarized in the supplied dipole note and electrokinetic theory (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012).

Equation 103 - Short-dipole electric-field calibration

E_∥(t)≈-ΔV(t)/ℓ

Meaning. If the field varies slowly over the electrode separation, dividing the differential voltage by spacing estimates the field component along the dipole.

E_∥ field component along dipole [V m^-1]

ΔV measured voltage difference [V]

ℓ dipole spacing [m]

Field use. This conversion must be performed before comparing the data with SI-unit transfer functions or forward-model amplitudes.

Source provenance. Short-dipole calibration is used throughout the supplied theory and single-dipole material (Bordes et al., 2015; Jouniaux & Zyserman, 2016).

3. The four return-event classes in the single trace

To use reflections as calibration rather than treating them only as noise, the trace must distinguish physically different coupled responses. The supplied reflection note stresses that these are not interchangeable (Pride & Haartsen, 1996; Haartsen & Pride, 1997; Haines et al., 2007; Jouniaux & Zyserman, 2016).

A. Source/local response

Electrical source transient plus the near-source local coseismic field. This is earliest and may contain instrumentation artifacts; control tests are essential (Haines et al., 2003; Haines et al., 2007).

B. Primary interface ES response

Generated when the incident seismic wave first reaches the aquifer boundary or other contrast. It is observed near the one-way seismic time (Haartsen & Pride, 1997; Dupuis et al., 2009; Schakel et al., 2011).

C. Reflected coseismic return

The mechanically reflected seismic wave returns toward the receiver while carrying a coseismic electric field. It is later, normally weaker, and inherits mechanical reflection/spreading/attenuation (Pride & Haartsen, 1996; Bordes et al., 2015; White & Zhou, 2006).

D. Secondary conversions / multiples

The reflected mechanical wave may re-encounter contrasts and generate additional interface responses; multiples can create further arrivals in layered media (White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Figure 29. Conceptual single-dipole record. The interface response occurs near the one-way seismic time; the reflected coseismic event is later, near a two-way path; secondary interface conversions and multiples may create additional coherent arrivals in layered media.

Equation 104 - First-order primary interface-response time

t_IR=t_down+t_EM≈∫_{source→interface}ds/V_P(s)

Meaning. The interface response is launched when the seismic wave reaches the converter; shallow electromagnetic travel/diffusion time is commonly small compared with the seismic travel time.

t_IR interface ES arrival time [s]

V_P P-wave velocity along path [m s^-1]

t_EM EM propagation/diffusion delay [s]

Field use. Use this as the first target timing marker. It is a one-way seismic time, not a conventional two-way reflection time.

Source provenance. One-way interface timing is developed and observed in (Haartsen & Pride, 1997; Haines et al., 2007; Dupuis et al., 2009; Schakel et al., 2011).

Equation 105 - First-order reflected seismic/coseismic return time

t_ref≈t_down+t_up≈2z/V_P

Meaning. In the simplest near-vertical constant-velocity geometry, the reflected mechanical wave and its coseismic field return after approximately a two-way seismic path.

t_ref returning reflected-coseismic time [s]

z reflector/interface depth [m]

V_P P-wave velocity [m s^-1]

Field use. Use as the second timing marker. The expected later arrival is one reason it can be separated from the primary interface ES response.

Source provenance. The supplied reflection note applies conventional reflection timing to the coupled layered-media problem (Haartsen & Pride, 1997; White & Zhou, 2006; Haines et al., 2007).

Equation 106 - Reflection-pair timing consistency check

q_t=t_ref/(2t_IR)≈1

Meaning. For a simple zero-offset path and negligible EM delay, the reflected-coseismic return should occur near twice the primary interface-conversion time.

q_t dimensionless timing consistency ratio

Field use. A stable value near unity across repeated shots strongly supports the event pairing. A large systematic deviation signals nonvertical path, layering, mis-picking, EM diffusion delay, or a different event class.

Source provenance. Derived here by combining Equations 3 and 4; the parent timing relations are source-supported.

Equation 107 - Time separation of the pair

Δt=t_ref-t_IR≈t_down≈z/V_P

Meaning. The gap from interface flash to reflected return is approximately one additional one-way seismic travel time in the simplest geometry.

Field use. Use the separation to construct nonoverlapping tapered windows and to detect whether the reflected event is physically plausible.

Source provenance. Derived algebraically from the source-supported one-way and two-way timing relations.

4. Forward model from known source to one electrical trace

The key hardware simplification is possible only because the source is assumed known. The supplied single-dipole workflow replaces an unmeasured acceleration channel with a mechanical source-to-receiver operator. Layered coupled theory then provides the conceptual machinery for the incident, reflected and converted fields (Pride & Haartsen, 1996; Haartsen & Pride, 1997; White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Equation 108 - Known source spectrum

F(ω)=ℱ{F(t)}

Meaning. The seismic source is treated as a calibrated deterministic force/time input with known spectral shape and amplitude.

F(t) source force/time function

F(ω) source spectrum

Field use. The source trigger defines t=0. Repeated shots should be checked for reproducibility before stacking.

Source provenance. Source control and repeat-impact QC are emphasized in (Haines et al., 2003; Haines et al., 2007).

Equation 109 - Incident mechanical field from the source model

A_inc(r,ω)=G_a,down(r,ω;m_el)F(ω)

Meaning. A mechanical Green/transfer operator predicts acceleration along the down-going path from the known source.

G_a,down source-to-acceleration operator

m_el elastic model parameters

A_inc predicted incident acceleration spectrum

Field use. This substitutes for an accelerometer. The operator must include enough elastic structure, attenuation and geometry to model the target band.

Source provenance. The source-model substitution follows the supplied single-dipole inversion logic and coupled-wave framework (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016).

Equation 110 - Reflected mechanical field

A_ref(ω)=G_a,up(ω;m_el)R_s(ω)G_a,down(ω;m_el)F(ω)

Meaning. The reflected mechanical spectrum is the source propagated to the reflector, multiplied by the mechanical reflection response, and propagated back.

R_s mechanical reflection response

G_a,up up-going propagation operator

Field use. Use a layered forward model when the reflection coefficient, attenuation or path is frequency dependent.

Source provenance. Layered reflected-wave behavior is treated in (Haartsen & Pride, 1997; White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Equation 111 - Practical reflected-amplitude scaling

|a_ref|/|a_inc|≈|R_s|G_spreadexp(-α_sΔL)

Meaning. The reflected mechanical wave is normally reduced by energy partitioning at the boundary, geometric spreading, and additional attenuation.

G_spread geometric-spreading factor

α_s seismic attenuation coefficient

ΔL additional path length

Field use. This is why the returning coseismic field is often weaker than the primary/local coseismic field; do not assume it is negligible when the measured trace shows otherwise.

Source provenance. This practical scaling is stated in the supplied reflection document and is consistent with (Pride & Haartsen, 1996; Bordes et al., 2015).

Equation 112 - Low-frequency coseismic electric field

E_c(ω)≈-ρ_fC_sA_s(ω)

Meaning. In the fast-P low-frequency regime, the coseismic electric field scales with local solid-frame acceleration.

ρ_f fluid density [kg m^-3]

C_s streaming-potential coefficient [V Pa^-1]

A_s solid acceleration spectrum

Field use. Apply separately to the mechanically modeled incident or reflected P-dominated wave only in a band where the quasi-static approximation is valid.

Source provenance. This transfer is summarized and tested in (Jouniaux & Zyserman, 2016; Bordes et al., 2015).

Equation 113 - Consequence for a reflected coseismic field

|E_ref|/|E_inc|≈|a_ref|/|a_inc|

Meaning. If the same low-frequency coupling coefficient applies, the electrical amplitude ratio inherits the mechanical reflected/incident ratio.

Field use. This ratio is useful as a calibration diagnostic, but a boundary crossing or change in coupling invalidates a simple constant-C_s interpretation.

Source provenance. The supplied reflection note derives this consequence from the low-frequency transfer (Pride & Haartsen, 1996; Bordes et al., 2015; Jouniaux & Zyserman, 2016).

Equation 114 - Interface-conversion forward operator

E_IR(ω)=G_IR(ω;m_layer)F(ω)

Meaning. The primary interface ES response is represented as a linear source-to-electrical operator for a layered coupled medium. The operator embodies boundary conditions and contrasts in mechanical, hydraulic, electrical and electrokinetic properties.

G_IR interface-conversion transfer operator

m_layer layered elastic/electrical/hydraulic/electrokinetic model

Field use. Do not reduce interface amplitude to permeability alone; fit a coupled layered model with fixed/prior parameters.

Source provenance. Layered interface-conversion operators are the natural forward-model abstraction of (Haartsen & Pride, 1997; White, 2005; White & Zhou, 2006; Schakel et al., 2011; Wang et al., 2020; Zheng et al., 2021).

Equation 115 - Single-channel observation model

E_obs(t)=E_src/local(t)+E_IR(t)+E_ref(t)+ΣE_sec(t)+n(t)

Meaning. The recorded electric field is a superposition of source/local energy, primary interface conversion, returning reflected coseismic energy, secondary conversions/multiples, and noise.

Field use. Use timing and known-source templates to isolate event windows. With one channel there is no f-k or moveout axis, so time-domain and forward-model separation carry more responsibility.

Source provenance. The event decomposition follows the supplied reflection and processing documents (Haines et al., 2007; Warden et al., 2012; White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

5. Reflection-assisted timing and aquifer depth

The interface response is generated on the first encounter with the target boundary, whereas the reflected mechanical wave must return to the surface. That separation provides an internal timing check that a single sensor can exploit even without a receiver array (Haartsen & Pride, 1997; Haines et al., 2007; Schakel et al., 2011).

Figure 30. The interface-conversion and reflected-coseismic arrivals form a powerful timing consistency pair. They do not, by themselves, solve separately for depth and seismic velocity.

Equation 116 - Constant-velocity depth from the primary interface response

z≈V_P(t_IR-t_EM)≈V_Pt_IR

Meaning. For a near-vertical path with known V_P and negligible EM delay, aquifer/interface depth is one-way P-wave velocity times interface-response time.

z depth [m]

V_P calibrated P-wave velocity [m s^-1]

Field use. This is not the seismic-reflection half-time formula. Use only after the event has been classified as an interface conversion.

Source provenance. The one-way depth interpretation is explicitly given in the supplied advanced theory and reflection notes (Haartsen & Pride, 1997; Haines et al., 2007; Dupuis et al., 2009; Schakel et al., 2011).

Equation 117 - Constant-velocity depth from the reflected-coseismic return

z≈V_Pt_ref/2

Meaning. The reflected mechanical/coseismic return gives the familiar two-way time depth when the same vertical constant-velocity path applies.

t_ref reflected-coseismic return time [s]

Field use. Compare this estimate with Equation 15. Agreement is a strong internal calibration check; disagreement is diagnostic rather than something to average blindly.

Source provenance. Reflection timing is source-supported in the supplied reflection note and layered-media context (White & Zhou, 2006; Haines et al., 2007).

Equation 118 - Depth self-consistency residual

r_z=V_Pt_IR-V_Pt_ref/2

Meaning. The two depth estimates should agree in the simple model. A nonzero residual measures departure from the assumed path/model.

Field use. Use r_z as a model-QC metric, not as random noise to be forced to zero by changing permeability.

Source provenance. Derived here from Equations 15 and 16.

Equation 119 - Layered one-way travel time

t_IR≈Σ_j=1^Nh_j/V_P,j

Meaning. For vertical propagation through known layers, the one-way interface time is the sum of layer thickness divided by layer velocity.

h_j layer thickness [m]

V_P,j layer P-wave velocity [m s^-1]

Field use. If velocities are known, solve for the unknown target thickness/depth. If both thickness and velocity are unknown in the same layer, one single path does not uniquely separate them.

Source provenance. Layered travel-time modeling is consistent with the source-to-interface integral in the supplied advanced theory and layered media papers (Haartsen & Pride, 1997; White & Zhou, 2006; Zheng et al., 2021).

Equation 120 - Why the two times do not independently solve z and VP

t_IR=z/V_P, t_ref=2z/V_P ⇒ t_ref=2t_IR

Meaning. The two equations are linearly dependent in the simple geometry; source amplitude does not add an independent travel-time relation.

Field use. Therefore report travel time or z/V_P if V_P is not independently fixed.

Source provenance. Algebraic consequence of the source-supported timing equations.

6. Source normalization and single-trace event decomposition

Because the impulse waveform is known, it supplies the reference normally provided by a mechanical measurement. Repeated shots allow selective stacking and give an empirical noise/repeatability estimate. Haines and co-workers emphasize recording impacts individually and rejecting contaminated shots before stacking (Haines et al., 2003; Haines et al., 2007).

Figure 31. A known deterministic source lets the single electrical record be converted into event-specific source-normalized transfer functions. The reflection-normalized ratio is a derived synthesis: it follows from linearity of the source-to-response operators in the supplied coupled-wave framework, but the archive does not present this exact ratio as a published permeability formula.

Equation 121 - Source-normalized electrical transfer for any isolated event

H_e(ω)=E_e(ω)/F(ω)

Meaning. For event e (interface, reflected coseismic, secondary conversion), division by the known source spectrum removes the deterministic source amplitude and phase.

e event class IR, ref, secondary…

H_e source-to-electrical transfer function

Field use. Do not divide where |F(ω)| is small; mask source spectral nulls and frequencies below acceptable repeatability. This is a transfer-operator estimate, not automatically an electrokinetic coefficient.

Source provenance. Transfer-function practice is supported by (Schoemaker et al., 2012; Bordes et al., 2015); the use of known F as the denominator follows the supplied no-geophone source-model workflow.

Equation 122 - Known-source correlation for event picking

C_EF(τ)=∫E_obs(t)F(t-τ)dt

Meaning. Cross-correlation with the known impulse shape provides a repeatable timing/pattern detector for source-related wavelets in the single electrical trace.

Field use. Use correlation peaks only as candidate arrivals; event physics still determines whether the peak is an interface conversion, reflected coseismic field, source artifact, or multiple.

Source provenance. Cross-correlation/interferometric processing has source support in (Schoemaker et al., 2012); applying the known impulse as a single-channel event template is a derived implementation.

Equation 123 - Tapered event window

E_e(t)=w_e(t)E_obs(t)

Meaning. A smooth finite window extracts the selected event while limiting spectral leakage compared with a hard rectangular gate.

Field use. Vary the gate start/end and taper; a permeability corner that moves substantially with reasonable windows is not stable.

Source provenance. Signal-windowing implementation introduced here to operationalize the source-supported timing separation.

Equation 124 - Two-template single-channel decomposition

{β̂_IR,β̂_ref}=arg min_β||E_obs-β_IRr_IR-β_refr_ref||²

Meaning. When event tails overlap, source-derived forward templates for the interface and reflected return can be fit simultaneously as nuisance/target components.

Field use. Use only when the templates are demonstrably distinguishable. If they are nearly collinear, the amplitudes are not identifiable from one trace.

Source provenance. Template separation is a practical synthesis built on known-source forward modeling; source-control and mode separation are central in (Haines et al., 2007; Warden et al., 2012; White & Zhou, 2006).

7. How the reflected return calibrates the interface ES response

The reflected return is useful because it is mechanically linked to the same source and same reflector that produced the interface event. It therefore supplies an additional source- and path-constrained observable. The reflected event should be used as a calibration constraint, not as a claim that reflection amplitude alone equals permeability.

Equation 125 - Interface-event source transfer

H_IR(ω)=E_IR(ω)/F(ω)

Meaning. This transfer contains the down-going mechanical propagation and the interface electroseismic conversion operator.

Field use. Fit its timing, phase and spectral shape with the layered model; do not interpret raw amplitude without source normalization.

Source provenance. The conceptual operator follows layered interface-conversion models (Haartsen & Pride, 1997; White & Zhou, 2006; Schakel et al., 2011; Zheng et al., 2021).

Equation 126 - Reflected-coseismic source transfer

H_ref(ω)=E_ref(ω)/F(ω)≈-ρ_fC_sG_a,ref(ω)

Meaning. The returning coseismic transfer combines mechanical reflected-path propagation with local electrokinetic conversion of acceleration to electric field.

Field use. It can calibrate reflection-path amplitude/phase and test the assumed C_s band if the returning P wave is isolated.

Source provenance. Combines reflected mechanical propagation with the low-frequency coseismic transfer supported by (Pride & Haartsen, 1996; Jouniaux & Zyserman, 2016; Bordes et al., 2015).

Equation 127 - Reflection-normalized interface ratio

Q_R(ω)=H_IR(ω)/H_ref(ω)=E_IR(ω)/E_ref(ω)

Meaning. For a linear repeatable experiment, the known source spectrum cancels. The ratio compares conversion strength/phase with the reflected-coseismic reference recorded by the same electrodes and electronics.

Field use. Use Q_R to fit the coupled layered model and reduce sensitivity to absolute source amplitude and receiver gain. It is not a published direct permeability formula in the supplied archive.

Source provenance. Derived here from source-normalized linear operators. Layered linear coupled-wave responses are supported by (Pride & Haartsen, 1996; Haartsen & Pride, 1997; White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Equation 128 - Reflection-amplitude prediction used as a calibration check

|H_ref|≈ρ_f|C_s||R_s|G_spreade^-α_sΔL|G_a,inc|

Meaning. The returning electrical amplitude should be compatible with the mechanical reflection and low-frequency coupling model.

Field use. A reflected event much larger than physically predicted is a warning for source/cultural electrical contamination, wrong event classification, or an incomplete layered model.

Source provenance. Combines the supplied reflected-amplitude scaling with the low-frequency transfer (Haines et al., 2003; Pride & Haartsen, 1996; Bordes et al., 2015).

8. Permeability calibration

Permeability is the part of the workflow where the strongest caution is required. The supplied material supports permeability through a calibrated dynamic electrokinetic transition - not a unique reflection-amplitude law.

Do not use raw ES reflection amplitude as a direct permeability equation. Interface amplitude can depend on elastic moduli, density, porosity, electrical conductivity, dielectric properties, electrokinetic coupling, geometry and hydraulic contrasts; reflected coseismic amplitude also includes the mechanical reflection coefficient, spreading and attenuation (Haartsen & Pride, 1997; White & Zhou, 2006; Schakel et al., 2011; Jouniaux & Zyserman, 2016; Wang et al., 2020; Zheng et al., 2021). The archive therefore supports a constrained forward inversion, not a one-amplitude lookup.

Figure 32. The source archive ties permeability to the characteristic dynamic transition frequency under Pride/Biot-type assumptions. Reflection-assisted calibration is used here to improve event attribution, source normalization, and model consistency before fitting that corner.

Equation 129 - Measured/calibrated electrokinetic transfer

H_SE(ω)=E(ω)/A_mod(ω)

Meaning. The physically meaningful electrokinetic transfer compares calibrated electric field with modeled solid acceleration.

A_mod modeled acceleration spectrum

Field use. With no geophone, A_mod comes from the known source and the mechanical forward model. Use the event window tied to the target and mask frequencies where the model/source has poor support.

Source provenance. The transfer-function form is explicit in the supplied advanced theory and laboratory literature (Schoemaker et al., 2012; Bordes et al., 2015; Jouniaux & Zyserman, 2016).

Equation 130 - Weighted low-frequency estimate of streaming-potential coupling

Ĉ_s=-[Σ_iw_iRe{E_iA_i^*}]/[ρ_fΣ_iw_i|A_i|²]

Meaning. A weighted regression estimates the low-frequency coupling from many spectral samples instead of one noisy ratio.

w_i spectral weights

A_i modeled acceleration sample

Field use. Weights may reflect shot variance, source spectral support and model confidence. With no measured accelerometer, “coherence with measured acceleration” is unavailable and should not be claimed.

Source provenance. This estimator is supplied in the single-dipole permeability document, based on the low-frequency transfer and transfer-function practice (Bordes et al., 2015; Schoemaker et al., 2012).

Equation 131 - Pride transport coefficient under open-circuit conditions

L̂=-σĈ_s

Meaning. The transport coupling coefficient is obtained from streaming-potential coupling and bulk electrical conductivity under the stated convention.

σ bulk electrical conductivity [S m^-1]

Field use. This step requires an independent/prescribed bulk conductivity if L rather than C_s is reported.

Source provenance. The open-circuit relation is summarized in (Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016).

Equation 132 - Dynamic response used to fit the corner

E(ω)=-ρ_fĈ_sA_mod(ω)G_d(ω;f_c)

Meaning. Once low-frequency coupling is calibrated, frequency-dependent departure from the plateau is fit with a dynamic correction having a characteristic transition frequency.

Field use. Fit the interface-associated or otherwise target-attributed spectrum after reflection/template separation and source normalization.

Source provenance. Dynamic coupling and transition behavior are supported by (Pride & Haartsen, 1996; Bordes et al., 2015; Jouniaux & Zyserman, 2016; Jougnot & Solazzi, 2021).

Equation 133 - Compact first-pass dynamic roll-off

G_d(ω;f_c)≈[1+iω/(2πf_c)]^-1/2

Meaning. This convenient screening form has a flat low-frequency plateau and a high-frequency roll-off. It is not the only valid dynamic permeability/effective-charge model.

Field use. Use as a first-pass fit only; replace with the selected Pride/Biot/JKD/effective-charge formulation when the data quality and model warrant it.

Source provenance. This first-pass approximation appears in the supplied single-dipole permeability material and is framed there as a practical rather than unique model (Pride & Haartsen, 1996; Bordes et al., 2015).

Equation 134 - Characteristic transition frequency

ω_c=φη_f/(α_∞k₀ρ_f), f_c=ω_c/(2π)

Meaning. The transition compares viscous drag and pore-fluid inertia; around/above it, dynamic permeability and coupling become frequency dependent.

φ porosity

η_f fluid dynamic viscosity [Pa s]

α_∞ high-frequency/inertial tortuosity

k₀ intrinsic permeability [m²]

ρ_f fluid density [kg m^-3]

Field use. The data need to span enough of both sides of the transition for f_c to be meaningfully resolved.

Source provenance. The characteristic-frequency expression is explicitly given in the supplied theory with provenance to (Bordes et al., 2015).

Equation 135 - Intrinsic permeability from a resolved transition

k̂₀=φη_f/(2πα_∞ρ_ff̂_c)

Meaning. Rearranging the characteristic-frequency relation converts a calibrated transition frequency to intrinsic permeability.

k̂₀ intrinsic-permeability estimate [m²]

f̂_c fitted transition frequency [Hz]

Field use. Report as model-based. If φ or α_∞ is unconstrained, report the constrained combination rather than an artificially precise k.

Source provenance. This inversion is explicitly present in the supplied advanced theory and permeability workflow (Bordes et al., 2015; Jouniaux & Zyserman, 2016).

Equation 136 - Hydraulic conductivity

K̂_h=k̂₀ρ_fg/η_f

Meaning. Hydraulic conductivity converts intrinsic permeability to a flow coefficient for the working fluid under gravity.

K̂_h hydraulic conductivity [m s^-1]

g gravitational acceleration [m s^-2]

Field use. Report fluid temperature/chemistry because viscosity is temperature dependent.

Source provenance. Standard Darcy-law conversion used in the supplied permeability document.

8.1 What exactly does the reflected event contribute to permeability?

It helps calibrate the mechanical/path side

The reflected-coseismic return is generated by a mechanical wave whose timing and approximate attenuation/reflection physics are predictable. Matching it constrains whether the source/velocity/attenuation/reflection model is plausible before the interface ES spectrum is interpreted (White & Zhou, 2006; Haines et al., 2007; Wang et al., 2020; Zheng et al., 2021).

It helps normalize the target conversion

The ratio Q_R compares two events driven by the same source and recorded by the same dipole. This can reduce source and receiver scale uncertainty in a forward inversion, but it does not remove trade-offs with conductivity, porosity, saturation, tortuosity or elastic contrast.

It helps detect false spectral corners

A delayed reflection superposed on a target event produces interference ripples/notches. Isolating and modeling the reflected event prevents those interference features from being misfit as f_c. The supplied reflection-filtering material emphasizes this risk.

It does not make k unique

The archive repeatedly states that dynamic coupling can depend on more than static permeability alone; effective charge, conductivity and saturation can be frequency dependent (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015; Jougnot & Solazzi, 2021).

9. Joint depth-permeability inversion

The most defensible implementation is a constrained inverse problem that fits timing and spectra simultaneously. Timing constrains travel-time geometry; the reflected event constrains the mechanical/reflection path; the primary interface event constrains conversion; the dynamic spectral shape constrains the transition frequency. The supplied advanced theory uses a regularized joint-inversion framework for exactly this type of non-unique coupled problem (White, 2005; White & Zhou, 2006; Bonnetier et al., 2019).

Figure 33. The reflection-assisted interpretation is best posed as a constrained forward/inverse problem. This follows the layered coupled-wave and inverse-problem logic in the supplied archive rather than relying on a unique voltage-to-permeability or time-to-depth shortcut.

Equation 137 - Joint objective function

Φ(m)=||W_t[t_obs-t_pred(m)]||²+||W_ω[d_ω,obs-d_ω,pred(m)]||²+λ²||W_m[m-m_ref]||²

Meaning. The inversion fits event times and source-normalized complex spectra while penalizing departures from prior/regularized model structure.

W_t timing uncertainty weights

W_ω spectral uncertainty weights

W_m prior/regularization weights

λ regularization strength

Field use. A possible model vector is m={z or layer thicknesses, V_P, attenuation/reflection terms, C_s, f_c/k, conductivity, porosity, tortuosity}. In a true minimal-hardware inversion, most of these must be fixed or tightly constrained rather than freely solved from one trace.

Source provenance. Adapted from the regularized inverse-problem form in the supplied advanced theory; related inverse/layered context is in (Bonnetier et al., 2019; White, 2005; White & Zhou, 2006).

Equation 138 - Reflection-normalized spectral misfit term

Φ_Q=Σ_iw_i|Q_R,obs(ω_i)-Q_R,pred(ω_i;m)|²

Meaning. The measured interface/reflected ratio is compared with the layered forward-model ratio over reliable frequencies.

Field use. This term is useful because common source and receiver scaling cancel, but the model can remain non-unique if multiple parameters alter the ratio similarly.

Source provenance. Derived synthesis for the reflection-assisted workflow; parent response operators are source-supported.

Equation 139 - Timing misfit term

Φ_t=w_IR[t_IR,obs-t_IR,pred]²+w_ref[t_ref,obs-t_ref,pred]²

Meaning. Both timing markers are fitted together, which helps reject wrong event associations and wrong path models.

Field use. Do not count them as two independent z/VP constraints in a simple one-layer geometry; their main extra value there is consistency and event identification.

Source provenance. Derived least-squares implementation of the source-supported event times.

10. Secondary conversions and multiples

Reflected waves can generate further electric events when they cross interfaces again, and layered models can contain multiple P reflections and reflected electromagnetic contributions. This means late energy can either improve calibration or destroy a simple single-event interpretation, depending on separability (White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Figure 34. Layered-media theory supports multiple reflected and converted events. They can provide extra constraints only when individually modelable; otherwise they are a source of spectral bias and must be included explicitly in the forward model.

Equation 140 - Nth path arrival time

t_n=Σ_jL_n,j/V_P,j+t_EM,n

Meaning. Each candidate multiple/secondary conversion has a calculable mechanical path length through the velocity model plus a usually small EM term for shallow targets.

Field use. Predict candidate arrival families before interpreting late peaks. Matching several arrivals can constrain a layered model; unmodeled overlap can bias the spectrum.

Source provenance. General path-sum form consistent with layered-media timing in (Haartsen & Pride, 1997; White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Equation 141 - Frequency-domain interference from an overlapping reflected event

H_obs(ω)=H_target(ω)+R(ω)e^-iωτ

Meaning. A delayed coherent reflected component adds a phase-rotating term to the target response. Its interference can create artificial peaks, notches or apparent roll-off.

Field use. If the reflection overlaps the target window, model/subtract it or fit both terms jointly before estimating f_c.

Source provenance. The supplied reflection-filtering document uses this superposition to explain false permeability corners.

11. Field-to-inversion procedure with one grounded dipole

Characterize the source before the survey. Store the force/time waveform, amplitude, bandwidth, trigger latency and repeatability.
Survey the dipole. Record electrode positions, spacing ℓ, orientation, polarity and contact condition. Convert counts to volts, then to V/m.
Acquire many individual shots. Preserve raw traces and pretrigger data. Reject source-electrical artifacts before stacking (Haines et al., 2003; Haines et al., 2007).
Construct the mechanical prior model. Specify V_P(z), density/attenuation as required, and reflector/interface geometry. If V_P is not known, the experiment yields travel time rather than unique depth.
Predict event-time families. Compute primary interface times, reflected-coseismic return times and plausible multiples.
Pick candidate events using the known source waveform. Use time prediction, repeated-shot consistency, polarity and waveform similarity. With one receiver, do not claim array moveout discrimination.
Check the reflection pair. Evaluate q_t=t_ref/(2t_IR) in the simple geometry or use the layered path model.
Estimate depth only after the velocity model is fixed. Use the one-way interface time and independently check it with the two-way reflected time.
Window the interface and reflected events with smooth tapers. Vary gates to test stability.
Source-normalize. Form H_IR=E_IR/F and H_ref=E_ref/F only where the source has spectral energy.
Fit the reflected path first. Verify that reflected timing, phase and amplitude are plausible under the mechanical reflection/attenuation model.
Form the reflection-normalized ratio Q_R. Use it as a calibration constraint on the interface conversion, not as a direct k formula.
Build the target acceleration spectrum from the known source and layered model. This replaces the absent accelerometer.
Estimate low-frequency coupling. Fit C_s over a validated low-frequency P-dominated band.
Inspect the calibrated target spectrum for a dynamic transition. Confirm the apparent corner is stable to gate position and reflection-model changes.
Fit f_c. Use a suitable dynamic coupling/permeability model, not automatically the screening approximation if the data justify more physics.
Convert f_c to k only with φ, α_∞, ρ_f, η_f fixed or constrained.
Propagate uncertainty. Include source amplitude/shape, timing, V_P, φ, α_∞, fluid properties, event window and model choice.
Perform null tests. Source-off/dummy-source electrical tests, alternative cable/source configurations, and late-window tests guard against source artifacts (Haines et al., 2003; Haines et al., 2007).
Report model dependence explicitly. Distinguish direct measurements (ΔV, times) from derived/calibrated quantities (E, H, C_s) and model-based estimates (z, f_c, k).

12. Uncertainty and identifiability

Equation 142 - First-order depth uncertainty

u_z²≈(t_IRu_VP)²+(V_Pu_tIR)²

Meaning. For z=V_Pt_IR, velocity and pick-time uncertainties both contribute. Correlation terms should be added when the inputs are correlated.

Field use. The reflected-time depth estimate provides a consistency check but does not remove V_P uncertainty.

Source provenance. Derived by applying the general first-order uncertainty relation included in the supplied advanced theory.

Equation 143 - First-order permeability relative uncertainty

(u_k/k)²≈(u_φ/φ)²+(u_η/η)²+(u_α/α_∞)²+(u_ρ/ρ_f)²+(u_fc/f_c)²

Meaning. For independent small errors, the relative uncertainty in k is the quadrature sum of the relative uncertainties of the multiplicative/divisive inputs.

Field use. This equation makes clear why a sharply picked f_c does not produce accurate permeability if porosity or tortuosity is poorly known.

Source provenance. Derived from the supplied characteristic-frequency equation and its general uncertainty-propagation equation.

Equation 144 - Window/model stability metric for permeability

S_k=max_j|k̂_j-median(k̂)|/median(k̂)

Meaning. Repeat the full fit over reasonable event gates, reflection-template variants and spectral bands. The spread measures processing/model sensitivity.

Field use. A large S_k means the permeability estimate is not robust to plausible data-treatment choices.

Source provenance. Practical stability metric introduced in the supplied reflection-filtering document.

Unknown / uncertainty	What the single dipole actually constrains	What must come from a prior or extra assumption	If absent
Depth z and V_P	Travel-time combination z/V_P; event pair consistency	V_P model or a known layer thickness/velocity	Report travel time, not unique depth
Intrinsic permeability k	Potentially f_c if target spectrum spans the dynamic transition	φ, α_∞, ρ_f, η_f; correct dynamic model	Report f_c or constrained combination, not unique k
Electrokinetic coupling C_s	E versus modeled acceleration in a validated low-f band	Source/mechanical model and fluid density	Raw amplitude is not C_s
Interface conversion amplitude	Source-normalized H_IR, reflection-normalized Q_R	Layered elastic/electrical/hydraulic model	Cannot attribute amplitude uniquely to permeability
Saturation / surface conduction	Spectral behavior may be sensitive	Appropriate multiphase/electrical model	Single saturated model can be biased (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015; Jardani & Revil, 2015)

13. Worked illustrative calculation

Illustrative only. The numbers below are not taken from a particular supplied field dataset; they demonstrate the equations and are not evidence that every site will resolve the same parameters.

Assume a repeated known impulse produces a primary interface ES peak at t_IR=25 ms and a reflected-coseismic return at t_ref=50 ms. The timing check gives q_t=50/(2×25)=1, consistent with the simple near-vertical path. If the prescribed velocity model gives V_P=800 m/s, then both paths give z≈800×0.025=20 m and z≈800×0.050/2=20 m.

Now suppose the isolated, source-normalized target spectrum has a stable fitted transition f_c=120 Hz, with prescribed φ=0.30, α_∞=2.0, ρ_f=1000 kg m^-3 and η_f=1.0×10^-3 Pa s. Equation 34 gives approximately k≈2.0×10^-10 m². The example is valid only if the measured bandwidth genuinely resolves the transition and if the assigned porosity/tortuosity/fluid values and dynamic model are defensible (Bordes et al., 2015; Jouniaux & Zyserman, 2016).

tIR = 25 ms→tref = 50 ms→VP prior = 800 m/s→z = 20 m+fc = 120 Hz→k ≈ 2×10⁻¹⁰ m²

14. Failure conditions and stop rules

Figure 35. A strict acceptance rule prevents a hardware-minimal experiment from becoming an unconstrained inversion. The source archive repeatedly emphasizes source-control tests, independent parameter constraints, and explicit failure conditions.

No independent V_P prior

The interface/reflection pair validates travel-time geometry but does not yield unique depth. Report z/V_P or travel time.

No resolved f_c

Do not convert an arbitrary spectral bend to permeability. Report coupling or a limit instead (Bordes et al., 2015; Jouniaux & Zyserman, 2016).

Interface and reflected events overlap

If one-channel templates cannot separate them stably, use a full joint forward model or do not claim event-specific permeability.

Reflection pair fails timing consistency

Revisit the velocity/path model, event labels, source trigger and possible multiples rather than forcing the simple model.

Source electrical transient mimics the target

Control/dummy-source tests are decisive; no inversion can turn a source artifact into geology (Haines et al., 2003; Haines et al., 2007).

Strong layering / reverberation

Include multiple reflected and transmitted seismic and electromagnetic modes explicitly (White & Zhou, 2006; Wang et al., 2020; Zheng et al., 2021).

Patchy or partial saturation

Static saturated coupling/permeability models can fail; use saturation-dependent electrokinetic and hydraulic formulations (Revil & Mahardika, 2013; Revil et al., 2014; Bordes et al., 2015; Jardani & Revil, 2015).

Dynamic coupling is not controlled only by k

Frequency-dependent effective charge and conductivity can alter the response, so a spectral corner must be interpreted with the selected physical model (Jougnot & Solazzi, 2021).

Bottom line. With one known seismic impulse source and one grounded dipole, the reflected-coseismic return can be deliberately retained and modeled as a calibration event rather than automatically discarded. Its most defensible roles are to validate target timing, constrain the reflection/mechanical path, normalize the primary interface ES conversion, and expose false spectral features. Absolute aquifer depth still requires a velocity model, and absolute permeability still requires a resolved dynamic transition plus petrophysical/fluid priors. Those limitations are not weaknesses of the processing - they are the identifiability conditions implied by the supplied source material.

Part IV

Formation and site applications

Application of the coupled framework to fractures, elastic/geotechnical properties, thermal state and groundwater quality or multiphase contamination.

Integrated source module

Fracture Detection and Hydraulic Characterization

Application section. It treats hydraulically active fractures as localized contrasts in elastic, hydraulic, electrical and electrokinetic properties and develops borehole, crosshole and surface detection logic.

Evidence policy. The numerical fracture examples below are reported only where they are explicitly summarized in the cited source literature. Derived equations for practical localization are labeled as such and are not presented as equations copied from the experimental papers.

1. Why fractures can generate a seismoelectric response

A fracture becomes seismoelectrically visible when the seismic disturbance encounters a sufficiently strong contrast in fluid flow, electrical, or mechanical properties to produce localized electrokinetic charge separation.

In a saturated porous medium, fluid motion relative to the mineral frame transports excess charge associated with the electrical double layer. The coupled transport description links pressure gradients, electric field, electric current, and fluid flow. A fracture changes the geometry and often the hydraulic mobility of that system. If the fracture is open and fluid-filled, it can also support strong tube/Stoneley-wave motion, which can enhance local relative fluid motion and therefore the converted electrical response (Jouniaux & Ishido, 2012; Jouniaux & Zyserman, 2016).

Figure 36. Conceptual fracture conversion. A fracture can act as a localized conversion interface because it changes hydraulic and electrokinetic properties. The literature treats the fracture response as an interface-response problem and reports Stoneley-wave-driven conversion in fluid-filled fracture models (Jouniaux & Zyserman, 2016; Schakel et al., 2012).

Equation 145 - Coupled transport relation that makes a hydraulic fracture electrically visible

J = σE + L(ω)(-∇p), q = L(ω)E + [k(ω)/η](-∇p)

Meaning. Pressure-gradient-driven flow and electric current are coupled through L(ω). A fracture can change k, σ, pore geometry, fluid chemistry, and therefore the local transfer between seismic pressure and electric response.

J electric current density

E electric field

q fluid flux / relative flow variable

p pore pressure

σ electrical conductivity

L(ω) electrokinetic coupling coefficient

k(ω) dynamic permeability

η fluid viscosity

Fracture implication. A fracture need not be electrically visible merely because it exists. Detectability requires a contrast strong enough to generate a measurable local conversion and a geometry that lets that response be separated from background coseismic fields.

Source provenance. Reciprocal electrokinetic transport is the standard framework summarized by Jouniaux & Ishido (2012) and Jouniaux & Zyserman (2016).

Equation 146 - What an electrode dipole measures

E_∥(t) ≈ -ΔV(t)/ℓ

The measured voltage difference is converted to the electric-field component along the dipole. Shorter dipoles improve spatial localization; longer dipoles increase voltage but spatially average the field. For fracture work, localization is particularly important because the interface response may be confined to a narrow depth or lateral zone.

2. Which fractures are most detectable?

Open and fluid-filled

Open fractures permit relative fluid motion and can support tube/Stoneley waves. The review literature reports strong correlations between borehole electrokinetic signals and open fractures over apertures from millimetres to centimetres (Jouniaux & Zyserman, 2016).

Hydraulically contrasting

Early field modeling explicitly identified permeability and fluid-chemistry contrasts as sites where electroseismic conversion can occur, motivating fracture-zone detection as a high-permeability target (Mikhailov et al., 1997).

Geometrically resolvable

A fracture is easier to localize when source and receiver geometry produce a clear travel-time trend that can be separated from source noise and from ordinary coseismic fields (Dupuis et al., 2009; Warden et al., 2012).

Closed or dry fractures are a weak target. A mechanically visible fracture can be seismoelectrically weak if it does not create appreciable relative fluid motion or electrokinetic contrast. Conversely, a strong electrical response is not automatically a fracture: water tables, lithologic boundaries, salinity changes, cemented layers, and other interfaces can also generate conversions.

3. Borehole fracture detection: the most direct geometry

Borehole acquisition reduces the distance between the electrical receiver and the target fracture, lowers the influence of surface electrical noise, and allows the receiver to pass directly through fracture zones. Jouniaux & Zyserman (2016) summarize a borehole electrokinetic fracture experiment in which mechanically generated pressure pulses produced signals up to 1500 mV MPa^-1 and the electrical responses correlated strongly with the locations of open fractures approximately 1 mm to 5 cm wide. The same review notes that borehole electric noise can be very low compared with surface work, which helps reveal weak interface responses.

Figure 37. Borehole fracture logging concept. A moving source/receiver pair can identify depth-localized electrokinetic responses associated with open or permeable fractures. The strongest published evidence summarized in the review literature uses mechanically generated borehole pressure pulses to avoid electromechanical source noise (Jouniaux & Zyserman, 2016; Dupuis et al., 2009).

Equation 147 - A useful borehole response normalization

H_EP(ω) = E(ω) / P(ω)

Meaning. Divide the electric response by local acoustic pressure to remove first-order source-amplitude variation. This produces an electrokinetic response metric with units of electric field per pressure.

Fracture use. In a depth log, localized changes in |H_EP| or its phase can identify zones where hydraulic/electrokinetic properties depart from the background. Jouniaux & Zyserman (2016) summarize borehole work in which normalized electric/pressure response increased with sandstone permeability, while other modeling showed that amplitude can also be porosity-sensitive and phase may carry additional permeability information. Therefore this ratio is a characterization aid, not a unique fracture-permeability equation.

Source provenance. The electric-field/pressure normalization and permeability discussion are summarized in Jouniaux & Zyserman (2016).

4. Crosshole measurements: locating the fracture and estimating dip

Crosshole geometry is particularly powerful because the source can be moved systematically in one borehole while the electrical receiver remains fixed in the other. A fracture-generated interface response then forms a coherent event whose arrival time changes predictably with source position. Jouniaux & Zyserman (2016) summarize laboratory crosshole experiments in which a vertical or dipping, water-filled fracture separated Lucite and sandstone blocks. The fracture-generated response propagated with the P-wave travel-time behavior in the source block, and the event was used to estimate both fracture position and dip.

Figure 38. Crosshole fracture localization. The fracture-generated electric event is picked at each source position. Its moveout is fitted with a seismic travel-time model to recover the fracture geometry. In the experiment summarized by Jouniaux & Zyserman (2016), the inferred distance was 4.9 cm for a true 5 cm distance, and the inferred fracture inclination was 69.2° for a true 70° inclination.

Equation 148 - Fracture-generated event travel time

t_IR,i ≈ t_seis(s_i → x_F) + t_EM(x_F → r) ≈ t_seis(s_i → x_F)

Meaning. The seismic wave needs finite time to reach the fracture. The converted electromagnetic disturbance reaches the electrode much faster than the seismic wave. Therefore the picked interface-response time is controlled mainly by seismic travel time from the source to the fracture.

Diagnostic use. If the electric event moves with source position according to the P-wave velocity in the source block, it supports a fracture-interface interpretation. The Jouniaux & Zyserman review reports a fracture response observed with the 2600 m s^-1 P-wave velocity of Lucite in the crosshole fracture model.

Equation 149 - Practical fracture-geometry inversion (derived here)

F̂ = arg min_F Σ_i w_i[t_IR,i^obs - t_seis(s_i,F)]²

Meaning. Represent the fracture F as a line in 2-D or a plane in 3-D, predict the seismic travel time from each source to that fracture, and adjust fracture position and dip until the predicted times match the picked electric event.

F̂ estimated fracture geometry

s_i source position i

t_IR,i^obs picked electric interface-response time

w_i confidence / SNR weight

Provenance. This least-squares form is a practical mathematical expression of the moveout/location method described in the fracture experiments summarized by Jouniaux & Zyserman (2016); it is not quoted as an equation from that paper.

5. Can fracture aperture or hydraulic openness be estimated?

The literature supports sensitivity to aperture, but it does not support a universal aperture-from-voltage calibration. In the laboratory fracture model summarized by Jouniaux & Zyserman (2016), increasing fracture aperture from 0.5 mm to 9 mm increased the interface-response amplitude from about 50 µV to 200 µV for tap water with conductivity of approximately 0.1 S m^-1. The same review states that the response was generated mainly by a Stoneley wave excited in the fracture.

Figure 39. End-point values reported in the fracture experiment summarized by Jouniaux & Zyserman (2016). The connecting line is only a visual guide. The source does not establish a universal linear conversion between voltage and fracture aperture.

Equation 150 - Correct way to express aperture inversion

Â_IR = g(a, σ_w, source, geometry, rock, saturation, frequency) ⇒ â = g^-1(Â_IR | calibrated conditions)

Meaning. Interface-response amplitude depends on fracture aperture a, but also on fluid conductivity, source strength, geometry, material contrast, saturation, and frequency. Aperture can therefore be estimated only through a calibrated forward/empirical model appropriate to the site or tool.

Why this matters. Treating the 50-200 µV laboratory result as a universal straight-line aperture calibration would overstate what the source literature supports.

Hydraulic characterization is stronger than “aperture from volts.” The most defensible interpretation is that a repeatable fracture-localized interface response indicates a hydraulically/electrokinetically contrasting discontinuity. Aperture and permeability should then be estimated with calibration, pressure measurements, borehole logs, or independent hydraulic data rather than from raw voltage amplitude alone.

6. Surface fracture surveys: possible, but less direct

Surface seismoelectric surveys can respond to fractured aquifers and high-permeability zones, and the 2016 review explicitly identifies fractured reservoirs and groundwater exploration in fractured rock as application targets. However, the same literature emphasizes the difficulty of separating weak interface responses from coseismic fields and electrical noise at the surface. Lateral heterogeneity also violates simple 1-D sounding assumptions. For detailed fracture mapping, surface data are therefore best treated as a screening or imaging dataset and, where possible, integrated with borehole control.

Early shallow-subsurface work by Mikhailov et al. (1997) explicitly states that conversion can occur at permeability or fluid-chemistry contrasts and identifies high-permeability fractured zones as a potential target. Haines et al. (2003, 2007) show why source-generated electrical noise and coseismic energy must be controlled before assigning an interface response to a subsurface boundary. Warden et al. (2012) demonstrates advanced seismoelectric processing designed to separate weak coherent signals from stronger contaminating energy.

Surface indicators that increase confidence

repeatabilitymoveout consistencypolarity patternindependent resistivityseismic tie

A fracture interpretation becomes stronger when the electrical event is repeatable, coherent across source positions, compatible with seismic travel times to a plausible fracture zone, and coincident with independent evidence for hydraulic or electrical contrast.

Indicators that are not sufficient alone

single spikeamplitude onlyno moveoutsource-coincident pulse

A large voltage by itself is not a fracture detector. Source wiring, electrode polarization, power-line noise, water-table conversions, lithologic interfaces, and near-surface heterogeneity can all create plausible-looking events.

7. Processing and decision workflow

repeatable shots→noise rejection→stacking→seismic/electric event separation→fracture-response picks→moveout inversion→fracture map

Step	Action	Fracture-specific purpose
1. Acquire repeats	Record many nominally identical source activations and retain individual traces.	Real fracture responses should repeat with source timing; random noise should not.
2. Reject source artefacts	Identify induction, trigger leakage, cable motion, electrode spikes, and ground-loop events.	Near-source electrical transients are a major false-positive risk (Haines et al., 2003).
3. Stack accepted records	Align on source trigger and stack only clean repeats.	Enhances coherent converted events relative to uncorrelated electrical noise.
4. Separate coseismic and interface energy	Use moveout, polarity, seismic correlation, f-k / curvelet / coherent-noise methods as appropriate.	A fracture response should be treated as a localized conversion event rather than ordinary coseismic electric motion (Warden et al., 2012).
5. Pick fracture event	Pick arrival time, amplitude, polarity, bandwidth, and uncertainty at each source/receiver position.	Provides the data used for location, orientation, and comparative hydraulic characterization.
6. Invert geometry	Fit a fracture line/plane to the picked moveout with a seismic travel-time model.	Converts electric-event timing into fracture position and dip.
7. Characterize strength	Compare amplitude/phase or E/P response to calibrated or independent hydraulic information.	Tests whether the fracture is likely open and hydraulically significant.

Equation 151 - Stacking repeated fracture-sensitive records

Ē(t) = (1/N) Σ_n=1^N E_n(t)

If the fracture-generated event repeats coherently while random electrical noise is uncorrelated, stacking improves detectability. In the ideal uncorrelated-noise limit, random-noise amplitude decreases approximately as N^-1/2. Haines et al. emphasize retaining individual impacts so contaminated records can be rejected before stacking.

8. Limitations and false positives

Issue	Effect on fracture interpretation	Mitigation
Dry / closed fracture	May be seismically visible but electrokinetically weak.	Use complementary seismic, resistivity, image-log, or hydraulic data.
High fluid salinity	Higher conductivity can reduce streaming-potential voltage for a given forcing.	Measure pore-water conductivity and model electrical attenuation/coupling.
Clay or surface conduction	Changes bulk conductivity and electrokinetic transfer behavior.	Use petrophysical/electrical constraints; avoid simple clean-sand assumptions.
Water table or lithologic interface	Can generate an interface response that resembles a fracture event.	Require geometric consistency with a fracture and independent structural evidence.
Source electrical noise	Can produce coherent, source-synchronous spikes.	Mechanical isolation, shielded wiring, dummy shots, polarity tests, and offset tests (Haines et al., 2003).
Amplitude-only inversion	Aperture/permeability estimates become non-unique.	Use timing, geometry, phase, pressure normalization, and independent hydraulic calibration.

Evidence hierarchy for reporting. High confidence: repeatable converted event + fracture-consistent moveout + independently plausible fracture geometry. Moderate confidence: depth-localized borehole electric/pressure anomaly coincident with fracture evidence. Low confidence: amplitude anomaly alone with no travel-time or structural constraint.

9. What the source literature supports most strongly

Directly supported

Open, hydraulically active fractures can produce detectable borehole electrokinetic responses.
Crosshole interface-response moveout can locate a fracture and estimate its dip.
Fracture aperture can affect interface-response amplitude in controlled laboratory models.
Stoneley/tube-wave motion can be a key mechanism in fluid-filled fractures.
Fracture detection is more reliable when the converted event is separated from coseismic and source-generated electrical energy.

Not yet a universal field calibration

Voltage amplitude does not uniquely determine aperture.
A single transfer-function amplitude does not uniquely determine permeability.
Fractured-media frequency-dependent effective-charge models remain an area of continued development (Jougnot & Solazzi, 2021).
Surface fracture imaging remains more ambiguous than controlled borehole/crosshole configurations.

Integrated source module

Formation Elastic and Geotechnical Properties

Application section. It translates seismoelectric kinematics and interface contrasts into dynamic elastic properties when velocity and density are constrained, while separating those products from static engineering strength parameters.

Source scope. Governing seismoelectric and poroelastic relations below are taken from or explicitly derived from the supplied GeoVue source material. Standard isotropic-elastic identities for Young’s modulus, Poisson’s ratio, Lamé parameter and impedance are labeled as derived consequences of the source-supported V_P, V_S, density and modulus relations rather than presented as independent seismoelectric laws.

Contents

What “geotechnical properties from seismoelectrics” means
Mechanical information carried by the ES field
Acquisition geometries
Voltage to calibrated electric field
Recovering Vp and Vs
Elastic interfaces and depth
Velocity and density to elastic moduli
Drained versus undrained poroelastic stiffness
Elastic contrast and impedance
Building a geotechnical stiffness profile
Resolution and uncertainty
Field implementation
Worked example and calculator
Unsupported or calibration-dependent claims
References

1. What “geotechnical properties from seismoelectrics” means

The seismoelectric method is mechanically informative because part of the electrical field is coseismic: it moves with the deforming porous frame and pore fluid. Another part is an interface conversion generated when the seismic wave encounters a property contrast.

Kinematic observables

Arrival time, moveout, apparent slowness and mode conversion constrain Vp, Vs, depth and layer geometry.

Dynamic elastic properties

Once density is known or estimated, velocity determines G, H, Ku, λ, ν and E under isotropic low-loss assumptions.

Poroelastic properties

Biot relations connect drained frame modulus, grain modulus, fluid modulus, storage and undrained stiffness.

Not an amplitude-only inversion. Interface-conversion amplitude also depends on porosity, permeability, conductivity, dielectric permittivity and electrokinetic coupling. The supplied theory therefore supports joint inversion and event classification, not a universal conversion such as “X microvolts = Y MPa.”

Figure 40. Seismoelectric geotechnical interpretation is a chain: mechanical-wave kinematics are observed electrically, then elastic parameters are calculated with density and poroelastic constraints. The source material repeatedly separates measured observables, direct calculations and model-dependent inversions.

2. Why the electrical record contains elastic information

Equation 152 - Low-frequency fast-P coseismic transfer

E_∥(ω) ≈ -ρ_fC_sA_s,∥(ω)

Meaning. In the low-frequency fast-P regime, the coseismic electric field follows local solid-frame acceleration. The electrical waveform can therefore inherit P-wave arrival time, phase and moveout.

Source provenance. Low-frequency fast-P transfer summarized in the supplied advanced theory from Jouniaux & Zyserman (2016) and Bordes et al. (2015).

Equation 153 - Empirical seismoelectric transfer function

H_SE(ω) = E(ω) / A_s(ω)

When a mechanical receiver is available, the calibrated electric/acceleration transfer helps separate electrokinetic coupling from the elastic waveform. When no mechanical receiver is used, the supplied single-dipole workflow requires a calibrated source and elastic forward model for A_s(ω).

Source provenance. Transfer-function treatment in Bordes et al. (2015) and Schoemaker et al. (2012), as summarized in the supplied advanced theory.

Equation 154 - General interface sensitivity

Δm = {ΔK, ΔG, Δρ, Δφ, Δk, Δσ, Δε, ΔL, …} ⇒ converted ES modes

This is a compact statement of the layered-media boundary-condition result: elastic-modulus and density contrasts are among the properties capable of generating or changing an interface conversion, but they are not the only ones.

Source provenance. Interface conversion in Haartsen & Pride (1997), White & Zhou (2006), Schakel et al. (2011), Wang et al. (2020), Zheng et al. (2021) and Bernardo et al. (2024), summarized in the supplied advanced theory.

3. Acquisition geometries for elastic-property recovery

Multi-offset electrical array

This is the strongest geometry for elastic characterization because a coseismic event shows seismic-like moveout. Fitting arrival time versus offset provides apparent slowness and therefore velocity. P-, S- and surface-wave branches can potentially be separated by timing, moveout, polarity and processing.

Single grounded dipole

A single trace can be useful when the source signature and elastic geometry are externally constrained, but it cannot independently determine both velocity and unknown depth from one arrival time. The supplied single-dipole material explicitly requires V_P,V_S,ρ_b in the source-to-acceleration model.

Figure 41. The supplied basic theory states that coseismic electrical energy shows seismic-like moveout, while interface responses can have very small apparent slowness. That makes an electrical shot gather mechanically interpretable when event classes are correctly separated.

Equation 155 - Straight-ray apparent velocity

p = dt/dx, V_app = 1/|p|

Derived implementation For a locally planar arrival in a homogeneous or weakly varying layer, the slope of the coseismic branch gives apparent velocity. Geometry/ray bending must be included for layered or refracting media.

Source basis. The supplied basic theory explicitly uses seismic-like moveout of coseismic electrical events as the diagnostic; the algebraic slope-to-velocity conversion is standard kinematics derived from that relation.

Equation 156 - Two-station velocity estimate

V_app ≈ (x₂-x₁)/(t₂-t₁)

Use regression across many stations rather than a two-point estimate whenever possible. Timing uncertainty, source jitter and misclassification of interface energy directly bias the result.

4. What the grounded dipole measures

Equation 157 - Grounded-dipole voltage

V_AB(t) = φ(B,t) - φ(A,t) = -∫_A^B E(r,t)·dl

The receiver measures a potential difference, not an elastic property directly. Dipole spacing and orientation determine which component of the ES field is sampled.

Source provenance. Supplied Dipole.html, based on Pride & Haartsen (1996) and Jouniaux & Ishido (2012).

Equation 158 - Short-dipole approximation

E_∥(t) ≈ -V_AB(t)/ℓ

Shorter dipoles improve locality; longer dipoles increase raw voltage but spatially average the field. Correct gain, polarity, spacing and electrode geometry before using arrival amplitudes or spectra.

Source provenance. Supplied Dipole.html and advanced theory; laboratory use in Bordes et al. (2015).

5. Recovering P- and S-wave velocity

The source archive gives two independent ways for elastic-wave timing to enter the electrical record: a coseismic field that travels with a mechanical mode, and an interface-generated response triggered when the incident P wave reaches a boundary.

Equation 159 - Layer travel time

t_P = ∫_ray ds/V_P(s), t_S = ∫_ray ds/V_S(s)

Derived kinematic form This is the travel-time integral underlying moveout analysis. In a constant-velocity straight path, it reduces to t=r/V.

Equation 160 - Velocity ratio

R_PS = V_P/V_S

The ratio is valuable because several isotropic elastic parameters depend on the ratio rather than absolute density. Correctly identifying an S-related coseismic or converted branch is essential; a mixed P/S window cannot be treated as a pure mode.

Practical event classification. The supplied processing guidance recommends moveout/slowness analysis, polarization/dipole orientation, source-control tests, and - in arrays - f-k, Radon or curvelet separation. A flat electrical event must not automatically be labeled an elastic interface because source electronics and cultural noise can mimic high apparent velocity.

6. Elastic interfaces and depth

Figure 42. An interface ES response is generated when the incident wave first reaches a contrast. Elastic modulus and density contrasts are part of the boundary-condition change, but amplitude also depends on hydraulic and electrical properties.

Equation 161 - First-order interface-response time

t_IR = ∫_{source→interface} ds/V_P(s) + t_EM ≈ t_{P,to interface}

At many shallow survey scales the electromagnetic propagation/diffusion delay is small relative to seismic time.

Source provenance. Supplied advanced theory; Haartsen & Pride (1997), Haines et al. (2007), Dupuis et al. (2009), Schakel et al. (2011).

Equation 162 - Simple vertical interface depth

z ≈ V_P t_IR

This is a one-way seismoelectric interface-conversion relation, not the conventional two-way seismic reflection expression.

Equation 163 - Layered depth relation

t_IR ≈ Σ_j h_j/V_P,j

Use a layered or ray-traced velocity model when velocity changes with depth. A single time pick does not by itself separate unknown thickness from unknown velocity.

7. From Vp, Vs and density to dynamic elastic moduli

Figure 43. The source material explicitly supplies the first three modulus relations. The Lamé, Poisson and Young relations are standard isotropic-elastic consequences and are labeled derived.

Equation 164 - Source-supported P and S wave moduli

H = K_u + 4G/3, V_P ≈ √(H/ρ_b), V_S ≈ √(G/ρ_b)

The supplied advanced theory calls these low-loss effective-medium limits, not the full dispersive Biot eigenproblem.

Source provenance. Pride & Haartsen (1996), Bordes et al. (2015), Jouniaux & Zyserman (2016), as summarized in advancedtheory.html.

Equation 165 - Dynamic shear modulus

G = ρ_b V_S²

This is one of the most directly useful geotechnical products. In soil dynamics it is commonly interpreted as a small-strain dynamic shear stiffness at the measurement frequency and strain level.

Equation 166 - P-wave or constrained modulus

H = ρ_b V_P²

Here H is the P-wave modulus used in the supplied theory. It includes both bulk and shear stiffness.

Equation 167 - Undrained/effective bulk modulus from Vp and Vs

K_u ≈ ρ_b(V_P² - 4V_S²/3)

This rearranges Equation 13. The label undrained is appropriate only when the frequency/drainage regime matches the Biot effective limit used in the model.

Equation 168 - Dynamic Lamé parameter derived

λ_u = K_u - 2G/3 = ρ_b(V_P² - 2V_S²)

Useful for isotropic constitutive modeling and finite-element parameterization.

Equation 169 - Dynamic Poisson’s ratio derived

ν_u = (V_P² - 2V_S²) / [2(V_P² - V_S²)]

Density cancels. This makes Poisson’s ratio particularly attractive when both P and S velocities are reliable. In saturated near-incompressible materials, dynamic undrained ν may approach 0.5.

Equation 170 - Dynamic Young’s modulus derived

E_u = 2G(1+ν_u) = 9K_uG/(3K_u+G)

This is a small-strain dynamic modulus unless a site-specific dynamic-to-static calibration has been established.

8. Drained versus undrained poroelastic stiffness

Formation stiffness in a fluid-filled porous medium is not a single number. The source material uses Biot poroelasticity to separate the solid frame, grains and pore fluid.

Figure 44. The same formation can show different stiffness depending on whether pore fluid has time to drain. Seismically inferred modulus is therefore frequency- and drainage-regime dependent.

Equation 171 - Biot coefficient

α = 1 - K_d/K_s

K_d is drained frame bulk modulus and K_s grain modulus.

Source provenance. Standard Biot coefficient as given in the supplied advanced theory; Pride & Haartsen (1996), Revil et al. (2014).

Equation 172 - Biot storage modulus

M = [(α-φ)/K_s + φ/K_f]^-1

Storage depends on porosity, grain compressibility and fluid compressibility. Multiphase or gas-bearing formations require an effective-fluid model.

Source provenance. Supplied advanced theory; Pride & Haartsen (1996), Revil et al. (2014).

Equation 173 - Undrained bulk modulus

K_u = K_d + α²M

If K_u is obtained from velocities and K_s,K_f,φ are independently constrained, the equations can be solved for a compatible drained-frame modulus. This is a poroelastic inversion, not a direct electrical measurement.

Source provenance. Supplied advanced theory; Pride & Haartsen (1996), Revil et al. (2014).

Practical drained-modulus inversion. Choose trial K_d, calculate α=1-K_d/K_s, calculate M, predict K_u=K_d+α^2M, and iterate until the predicted K_u matches the velocity-derived value. Report uncertainty from φ,K_s,K_f,ρ,V_P,V_S.

9. Elastic contrast, impedance and interface strength

Equation 174 - P and S acoustic impedances derived

Z_P = ρ_bV_P, Z_S = ρ_bV_S

Impedance is useful for describing mechanical contrast between adjacent layers. It is not by itself the seismoelectric interface-conversion coefficient because the coupled boundary problem also contains hydraulic and electromagnetic properties.

Equation 175 - Normal-incidence mechanical reflection coefficient context only

R_P = (Z_P2-Z_P1)/(Z_P2+Z_P1)

Useful as a mechanical contrast diagnostic for reflected coseismic energy. Do not equate the measured interface ES amplitude directly with R_P.

Source basis. The supplied reflection document states that reflected mechanical energy inherits reflection, spreading and attenuation losses, while the interface response is governed by the full coupled contrast.

Equation 176 - Relative stiffness contrast

C_G = (G₂-G₁)/[(G₂+G₁)/2]

Derived reporting metric This dimensionless contrast is a convenient way to compare layer stiffness after velocity-density inversion. It is not a published seismoelectric constitutive law.

10. Building a formation geotechnical stiffness profile

Figure 45. A geotechnical section can be assembled from interface depths and layer velocities, then converted to dynamic stiffness where density is known. Interface amplitude alone does not quantify stiffness contrast.

Geotechnical output	Primary seismoelectric observable	Additional constraint	Interpretation level
Vp	P-coseismic moveout	geometry / source timing	direct kinematic estimate
Vs	S-coseismic or converted-mode moveout	mode classification	direct kinematic estimate when resolved
Interface depth	interface ES time	Vp model	one-way travel-time inversion
G	Vs	bulk density	dynamic calculation
Ku / H	Vp and Vs	bulk density	dynamic/effective calculation
νu	Vp/Vs ratio	isotropic assumption	derived dynamic parameter
Eu	Vp, Vs	density + isotropic assumption	derived dynamic parameter
Kd	velocity-derived Ku	φ, Ks, Kf, Biot model	model-dependent inversion
Static E, constrained modulus for settlement	dynamic ES velocity model	local lab/CPT/pressuremeter correlation	site-calibrated engineering product
c′, φ′, su, CBR, bearing capacity	none uniquely	direct geotechnical testing/correlation	not a direct seismoelectric property

11. Resolution, uncertainty and identifiability

Equation 177 - Depth uncertainty

u_z² ≈ (t_IRu_Vp)² + (V_Pu_t)²

Derived first-order propagation Velocity uncertainty and time-pick uncertainty both control interface-depth uncertainty.

Equation 178 - Shear-modulus relative uncertainty

(u_G/G)² ≈ (u_ρ/ρ)² + (2u_Vs/V_S)²

Velocity error enters twice because G∝V_S^2. Small errors in S-wave picking can therefore dominate modulus uncertainty.

Equation 179 - P-wave-modulus relative uncertainty

(u_H/H)² ≈ (u_ρ/ρ)² + (2u_Vp/V_P)²

Use covariance terms if density and velocity estimates are correlated.

Figure 46. Elastic-property uncertainty starts before the modulus equations. Incorrect event classification or velocity picking cannot be repaired later by precise algebra.

Single-station identifiability. With one electrical trace and an unknown interface depth, a time pick constrains approximately z/V_P, not z and V_P independently. Likewise, modulus requires density. The supplied single-dipole material explicitly warns that one electric trace cannot uniquely determine every formation property.

12. Practical field implementation

Calibrate acquisition

Record source trigger, electrode spacing/orientation, channel gain and pretrigger noise. Preserve raw records.

Acquire repeat shots

Record impacts individually; reject contaminated shots before stacking, as recommended in the supplied field material.

Classify events

Separate P/S coseismic branches, interface conversions, source transients and reflections by timing/moveout/polarity.

Fit velocities

Use regression or layered travel-time inversion across receiver offsets.

Pick interfaces

Use one-way interface-conversion timing with the calibrated Vp model.

Add density

Use cores/logs or a constrained mixture model. Do not silently infer density from voltage amplitude.

Calculate moduli

Compute H, G and Ku; derive ν and E if isotropic assumptions are defensible.

Validate geotechnically

Compare against borehole velocity, downhole/CPT correlations, pressuremeter, resonant column, bender element or laboratory modulus data where available.

Equation 180 - Joint inverse objective

Φ(m) = ‖W_d[d_obs-F(m)]‖² + λ²‖W_m[m-m_ref]‖²

The supplied advanced theory recommends joint inversion when parameters trade off. For geotechnical use, m may include velocity, density, layer depth, poroelastic moduli and nuisance electrokinetic parameters.

13. Worked elastic example and interactive calculator

Consider a layer with V_P=1800 m/s, V_S=500 m/s and bulk density ρ_b=1900 kg/m³. The source-supported relations give approximately G = 0.475 GPa, H = 6.156 GPa and Ku = 5.523 GPa. The derived isotropic relations give νu ≈ 0.458 and Eu ≈ 1.385 GPa. These are dynamic/effective properties at the survey frequency and strain level.

Elastic-property calculator

P velocity Vp (m/s)

S velocity Vs (m/s)

Bulk density ρb (kg/m³)

Interface time tIR (ms)

Velocity uncertainty uV (%)

Depth z -

Shear modulus G -

Bulk modulus Ku -

P-wave modulus H -

Poisson νu -

Young Eu -

Lamé λu -

Approx. velocity-only modulus uncertainty -

Calculator uses the simple vertical one-way depth relation and isotropic low-loss formulas. It is a calculation aid, not a substitute for event classification, density calibration or a Biot drainage assessment.

Figure 47. Seismoelectric data can support elastic geotechnical characterization, but design-level properties should be validated against conventional geotechnical measurements.

14. What the method does not determine uniquely

Requested property	Can ES contribute?	Can it be obtained uniquely from ES alone?	Reason
Vp / Vs	Yes, from resolved coseismic moveout	Potentially with array geometry	kinematic observable
G, Ku, H	Yes	No without density	modulus scales with ρV²
ν	Yes	Only if both Vp and Vs are reliable and isotropy applies	depends on velocity ratio
Young’s modulus	Yes, dynamic	No without density and elastic assumptions	derived from G and ν
Drained Kd	Conditionally	No	requires Biot, porosity, grain and fluid moduli
Static settlement modulus	Indirectly	No	dynamic-to-static relationship is site/strain dependent
Undrained shear strength su	Possible empirical correlation	No	strength is not fixed by elastic wave speed
Effective friction angle φ′ / cohesion c′	Only contextual	No	strength parameters require stress-path testing/calibration
CBR / bearing capacity	Only via local correlations	No	not a constitutive ES observable
Lithology	Supports classification	No	many materials share overlapping elastic/electrical responses

Best geotechnical use. Use seismoelectrics to create a high-resolution, mechanically informed stratigraphic model: identify elastic interfaces, estimate P/S velocities where moveout is resolved, calculate small-strain dynamic stiffness with density, and use Biot theory to interpret fluid/frame effects. Then calibrate those dynamic properties to the static or strength parameters required by the project.

Integrated source module

Formation Thermal Information

Application section. It tracks temperature sensitivity through viscosity, conductivity, permittivity, density, fluid modulus and electrokinetic coupling, emphasizing time-lapse and calibrated rather than standalone thermometry.

Source interpretation policy. SOURCE labels equations or relationships directly present in the supplied material. DERIVED labels algebraic sensitivity or inversion expressions developed here from those source equations. EXTERNAL CALIBRATION marks information that the seismoelectric trace does not uniquely provide.

1. What “formation thermal information” means

A seismoelectric trace does not sense temperature in isolation. It senses a coupled fluid-solid-electrical state in which several controlling parameters are temperature dependent.

Thermal sensitivity

A measurable seismoelectric quantity changes when the temperature-dependent fluid state changes. This is the weakest but most defensible claim.

Relative thermal change

Repeated measurements at the same station show a reproducible change that can be interpreted as $\Delta T$ after a baseline calibration removes static geology and instrumentation.

Absolute temperature

A numerical $T$ is estimated by fitting the seismoelectric observables to laboratory or site-specific property-versus-temperature functions while constraining salinity, pH, saturation, permeability and pore geometry.

Most important interpretive rule. The supplied theory supports thermally sensitive electrokinetic and poroelastic response, but it also emphasizes non-uniqueness: one amplitude can reflect hydraulic, electrical, chemical or elastic changes. A thermal interpretation becomes defensible only when those competing variables are controlled or constrained (Revil & Mahardika, 2013; Revil et al., 2014; Jouniaux & Zyserman, 2016).

Figure 48. Thermal information enters the seismoelectric observation through several temperature-sensitive fluid and electrochemical properties. The source archive explicitly connects

C_s

\epsilon_f,\zeta,\eta_f,\sigma_w

, the dynamic transition to

\eta_f,\rho_f,k,\phi,\alpha_\infty

, and poroelastic storage to fluid properties. The inversion is therefore multivariate rather than a direct thermometer.

2. Where temperature enters the coupled physics

The Pride-type transport formulation links electrical current and relative pore-fluid motion. In the supplied theory, the electrokinetic coefficient appears in both the mechanically driven electrical current and the reciprocal electro-osmotic fluid flux. Temperature is not an extra independent wave mode; it changes the coefficients of this coupled system by changing the fluid and interfacial state (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012).

Equation 181 - Coupled transport pair SOURCE

J = \sigma(\omega)E + L(\omega)X_p

q = L(\omega)E + k(\omega)X_p/\eta_f

Meaning. The observed electric field depends on Ohmic conduction and mechanically driven streaming current. Relative fluid flow depends on permeability and viscosity. If temperature changes $\eta_f$ , $\sigma$ or the interfacial coupling $L$ , the seismoelectric transfer changes even if the pore geometry is unchanged.

Source provenance: reciprocal transport structure in the supplied advanced-theory material, based on Pride & Haartsen (1996) and the electrokinetic tutorials/reviews.

Fluid-state pathway

Temperature changes the state of the pore fluid. The supplied material specifically requires temperature-corrected viscosity and permittivity when interpreting $C_s$ , and it requires fluid temperature when converting intrinsic permeability to hydraulic conductivity.

Electrochemical pathway

The zeta potential is not a temperature-only variable. It also depends on mineral surface chemistry, pH and electrolyte composition. This is a major reason thermal inversion from amplitude alone is non-unique.

Electrical pathway

Pore-water conductivity enters the streaming-potential coefficient and the bulk-conductivity model. A thermal change that changes fluid conductivity can therefore mimic or oppose changes caused by viscosity or surface chemistry.

Poroelastic pathway

The Biot storage modulus contains the pore-fluid bulk modulus. The supplied material notes that $K_f$ may be calculated from fluid composition, pressure and temperature; therefore a thermal state can also perturb seismic velocity/timing through the fluid contribution to poroelastic stiffness.

3. From grounded-dipole voltage to calibrated observables

A grounded dipole measures a potential difference, not temperature. The first step is therefore always electrical calibration and geometry correction. The source material defines the voltage as the line integral of the electric field and gives the short-dipole approximation (Pride & Haartsen, 1996; Jouniaux & Ishido, 2012).

Equation 182 - Grounded-dipole voltage SOURCE

V_{AB}(t)=\phi(B,t)-\phi(A,t)=-\int_A^B \mathbf{E}(\mathbf r,t)\cdot d\mathbf l

The recorded channel integrates the component of electric field along the electrode baseline.

Equation 183 - Short-dipole field estimate SOURCE

E_{\parallel}(t)\approx-\frac{\Delta V(t)}{\ell}

Use the surveyed electrode separation $\ell$ , correct instrument gain and polarity, and preserve the raw trace. A thermal inversion performed on uncalibrated counts is not physically meaningful.

Figure 49. Thermal information is extracted only after the grounded-dipole voltage has been converted to a calibrated electric field and linked to a known or independently modeled mechanical source. The thermal-sensitive quantities are derived transfer observables, not raw voltage counts.

For a P-dominated low-frequency coseismic window, the supplied theory gives an approximate local relation between electric field and solid-frame acceleration:

Equation 184 - Low-frequency coseismic transfer SOURCE

E_{\parallel}\approx-\rho_f C_s a_{s,\parallel}

Field use. If the acceleration is measured or modeled from a calibrated source, estimate $C_s$ from many samples/frequencies rather than one peak ratio.

Supported in the supplied material through the low-frequency transfer treatments summarized from Jouniaux & Zyserman (2016) and Bordes et al. (2015).

Equation 185 - Effective field estimate of streaming coupling SOURCE

\widehat C_s=-\frac{E_{\parallel}}{\rho_f a_{s,\parallel}}

Use a complex spectral regression or weighted estimate over a coherent low-frequency band. This estimated coupling coefficient is one of the principal thermal-sensitive observables.

4. Thermal sensitivity of the streaming-potential coefficient

The most direct temperature-sensitive equation in the supplied electrokinetic theory is the Helmholtz-Smoluchowski low-frequency limit. It is not a temperature equation, but it exposes exactly which temperature-dependent quantities control the measured coupling.

Equation 186 - Helmholtz-Smoluchowski low-frequency limit SOURCE

C_s\approx\frac{\epsilon_f\zeta}{\eta_f\sigma_w}

εf fluid dielectric permittivity

ζ zeta potential

ηf dynamic viscosity

σw pore-water conductivity

Source instruction. The supplied advanced-theory material explicitly says to measure fluid conductivity and temperature and use viscosity and permittivity at that temperature. It also warns that pH/surface chemistry strongly affect $\zeta$ .

Beamish (1999); Jouniaux & Ishido (2012); Jouniaux & Zyserman (2016).

Equation 187 - Logarithmic thermal sensitivity of coupling DERIVED

\frac{d\ln|C_s|}{dT}=\frac{d\ln\epsilon_f}{dT}+\frac{d\ln|\zeta|}{dT}-\frac{d\ln\eta_f}{dT}-\frac{d\ln\sigma_w}{dT}

This is obtained by taking the logarithm of Equation 6 and differentiating with respect to temperature. It provides a compact way to build a site-specific thermal calibration from laboratory slopes.

Critical limitation. If salinity, pH or mineral surface chemistry also changes, the measured $C_s$ change cannot be assigned to temperature alone.

Equation 188 - Small-change thermal approximation DERIVED

\Delta\ln|C_s|\approx S_C\,\Delta T,\qquad S_C\equiv\frac{d\ln|C_s|}{dT}

If a laboratory or field calibration provides $S_C$ over the temperature range of interest, a repeat survey can estimate $\Delta T\approx\Delta\ln|C_s|/S_C$ .

Figure 50. The coupling coefficient is thermally sensitive because several of its constituent variables are temperature dependent. The same diagram also explains the non-uniqueness: a change in

C_s

is not uniquely a temperature change unless conductivity and interfacial chemistry are constrained.

4.1 A robust time-lapse coupling ratio

Equation 189 - Baseline/repeat coupling ratio DERIVED

R_C=\frac{C_s(T_2)}{C_s(T_1)}=\frac{\epsilon_2\zeta_2\eta_1\sigma_{w1}}{\epsilon_1\zeta_1\eta_2\sigma_{w2}}

Because the ratio compares the same station and acquisition geometry, static instrument gain and much of the fixed geological geometry can cancel after careful calibration. The remaining ratio is still sensitive to chemistry and saturation, so those must be stable or monitored.

5. Dynamic transition frequency as a thermal-sensitive observable

The supplied dynamic seismoelectric theory introduces a characteristic frequency separating quasi-static viscous flow from stronger inertial/dynamic behavior. Bordes et al. use this transition in laboratory transfer-function analysis, and the supplied permeability workflow explicitly rearranges it to estimate intrinsic permeability.

Equation 190 - Characteristic angular frequency SOURCE

\omega_c=\frac{\phi\eta_f}{\alpha_\infty k_0\rho_f}

Below this scale, permeability is approximately static; around and above it, amplitude and phase become more frequency dependent.

Bordes et al. (2015), as summarized in the supplied advanced-theory and permeability-calculation documents.

Equation 191 - Characteristic frequency SOURCE

f_c=\frac{\phi\eta_f}{2\pi\alpha_\infty k_0\rho_f}

Temperature enters through $\eta_f(T)$ and $\rho_f(T)$ . If $k_0,\phi,\alpha_\infty$ are fixed and known, a measured change in $f_c$ is a thermal-sensitive measure of $\eta_f/\rho_f$ .

Equation 192 - Thermal sensitivity of corner frequency DERIVED

\frac{d\ln f_c}{dT}=\frac{d\ln\eta_f}{dT}-\frac{d\ln\rho_f}{dT}\quad\text{if }\phi,\alpha_\infty,k_0\text{ are constant}

This derived expression shows why a time-lapse corner-frequency shift can be a powerful thermal observable in a formation whose pore geometry and intrinsic permeability do not change during the monitoring interval.

Equation 193 - Time-lapse corner-frequency ratio DERIVED

R_f=\frac{f_{c2}}{f_{c1}}=\frac{\eta_2\rho_1}{\eta_1\rho_2}\quad\text{for fixed }\phi,\alpha_\infty,k_0

With a laboratory calibration of $\eta(T)$ and $\rho(T)$ for the actual formation fluid, this ratio can be mapped to a temperature change without re-estimating the static rock parameters at every repeat survey.

Figure 51. Conceptual time-lapse use of the dynamic transition. The figure intentionally does not assign a universal direction or magnitude to the thermal shift; the actual mapping must use the formation-fluid property calibration. A change in permeability would also move the corner and must be ruled out or jointly estimated.

5.1 The permeability-temperature trade-off

Equation 194 - Permeability inversion from the same corner SOURCE

\widehat k_0=\frac{\phi\eta_f}{2\pi\alpha_\infty\rho_f\widehat f_c}

This equation is why temperature and permeability cannot generally be estimated independently from $f_c$ alone. If viscosity is unknown because temperature is unknown, then an error in assumed temperature maps directly into the permeability estimate.

Identifiability stop rule. Do not claim an absolute temperature from

f_c

if permeability is also unknown and unconstrained. The same measured corner can be explained by multiple combinations of

k_0

and

\eta_f(T)

6. Optional poroelastic thermal sensitivity

The supplied theory also exposes a second, weaker thermal route through the pore-fluid bulk modulus and density. This is not unique to seismoelectrics; it is a poroelastic/seismic sensitivity that can be incorporated into a joint interpretation if the acquisition provides reliable timing or velocity information.

Equation 195 - Biot storage modulus SOURCE

M=\left[\frac{\alpha-\phi}{K_s}+\frac{\phi}{K_f}\right]^{-1}

The source material states that for a clean single-fluid saturated medium, $K_f$ may be calculated from fluid composition, pressure and temperature. A thermal change can therefore modify storage and the effective compressional response.

Equation 196 - Effective fast-P velocity limit SOURCE

V_P\approx\sqrt{\frac{H}{\rho_b}},\qquad H=K_u+\frac{4G}{3}

If a calibrated poroelastic model maps $T\to K_f(T),\rho_f(T)\to K_u(T),\rho_b(T)$ , a measured velocity or travel-time change can provide an additional thermal constraint. In practice this term is commonly confounded by saturation, stress and frame changes, so it should be used as a joint constraint rather than a standalone thermometer.

7. Time-lapse thermal extraction: the strongest practical strategy

Time-lapse acquisition is especially attractive because many fixed quantities cancel or can be held constant: dipole length, electrode positions, source geometry, porosity, tortuosity and intrinsic permeability can remain approximately unchanged over a monitoring interval. The goal is then not to infer every property from one trace but to measure repeatable changes in calibrated observables.

baseline survey at T₁→calibrate Cs₁ and fc₁→repeat survey→calibrate Cs₂ and fc₂→form ratios→map through site thermal calibration

Equation 197 - Source-normalized transfer function DERIVED FROM SOURCE WORKFLOW

H_{SE}(\omega,t_s)=\frac{E(\omega,t_s)}{A_s(\omega,t_s)}

$t_s$ is survey epoch. Normalizing to the source/mechanical spectrum is essential because a changing impact amplitude must not be mistaken for a thermal change.

The supplied theory explicitly defines

H_{SE}=E/A_s

and recommends calibrated source/mechanical transfer estimation.

Equation 198 - Time-lapse normalized spectral change DERIVED

R_H(\omega)=\frac{H_{SE}(\omega,t_2)}{H_{SE}(\omega,t_1)}

Amplitude and phase of $R_H$ identify frequency bands that changed between surveys. Thermal inversion should be restricted to frequencies with stable source calibration and good repeatability.

Figure 52. Time-lapse design turns a highly non-unique absolute inverse problem into a change-detection problem. The baseline does not eliminate chemistry/saturation effects automatically, but it greatly reduces static geometry and pore-structure uncertainty.

7.1 Linearized dual-observable thermal estimate

Equation 199 - Coupling-derived temperature change DERIVED

\widehat{\Delta T}_C\approx\frac{\ln(C_{s2}/C_{s1})}{S_C}

$S_C$ must come from a laboratory/site calibration that preserves the relevant fluid chemistry and mineral surface condition.

Equation 200 - Corner-derived temperature change DERIVED

\widehat{\Delta T}_f\approx\frac{\ln(f_{c2}/f_{c1})}{S_f},\qquad S_f=\frac{d\ln\eta_f}{dT}-\frac{d\ln\rho_f}{dT}

Valid only when intrinsic permeability, porosity and tortuosity remain unchanged or are independently corrected.

Equation 201 - Weighted dual-observable estimate DERIVED

\widehat{\Delta T}=\frac{\widehat{\Delta T}_C/u_C^2+\widehat{\Delta T}_f/u_f^2}{1/u_C^2+1/u_f^2}

This is a simple inverse-variance combination when the two thermal estimates can reasonably be treated as independent. A full covariance model is preferable when they share common calibration uncertainties.

8. Absolute-temperature inversion

Absolute formation temperature requires a calibrated forward model. The seismoelectric data contribute measured $C_s$ , spectral shape, $f_c$ and possibly travel-time/velocity observables. The calibration contributes the temperature dependence of the pore fluid and interfacial chemistry.

Route A: coupling calibration

Use a laboratory relation $C_s^{pred}(T;\mathrm{chemistry},pH,\mathrm{mineral})$ . This route is only credible if salinity/surface chemistry are constrained.

Route B: corner-frequency calibration

If $k_0,\phi,\alpha_\infty$ are independently known, use $f_c$ to recover the fluid mobility ratio $\eta_f(T)/\rho_f(T)$ .

Route C: joint thermal fit

Fit $C_s$ , $f_c$ , phase and any reliable poroelastic timing together. This is the preferred route when several observables are available.

Equation 202 - Viscosity implied by a measured corner when permeability is known DERIVED

\widehat\eta_f=\frac{2\pi\alpha_\infty k_0\rho_f\widehat f_c}{\phi}

Equation 22 does not itself produce temperature. It produces a fluid-viscosity constraint. Convert that viscosity to temperature only using a calibrated $\eta_f(T)$ relation for the actual pore fluid at relevant pressure/composition.

Equation 203 - Coupling-implied thermal property combination DERIVED

\frac{\epsilon_f(T)\zeta(T)}{\eta_f(T)\sigma_w(T)}\approx\widehat C_s

Absolute temperature is obtained by finding values of $T$ for which the calibrated property combination matches the measured coupling, subject to chemistry and surface-conduction constraints.

9. Joint inversion and identifiability

A defensible thermal inversion should expose nuisance parameters rather than bury them. The supplied archive repeatedly recommends joint interpretation and warns that unconstrained inversion is non-unique.

Equation 204 - Thermal forward data vector DERIVED

\mathbf d_{pred}(T,\mathbf m_n)=\left[\ln|C_s|,\ \ln f_c,\ \angle H_{SE},\ t_{IR}\ \text{or}\ V_P,\ldots\right]^T

$\mathbf m_n$ contains nuisance or jointly estimated parameters such as permeability, salinity, pH/zeta model, saturation, porosity, tortuosity and conductivity corrections.

Equation 205 - Regularized thermal inverse objective DERIVED

\Phi(T,\mathbf m_n)=\|W_d[\mathbf d_{obs}-\mathbf d_{pred}(T,\mathbf m_n)]\|^2+\lambda^2\|W_m[\mathbf m_n-\mathbf m_{prior}]\|^2

This follows the regularized joint-inversion form already used in the supplied advanced workflow. The thermal variable is introduced as one component of the model rather than assumed to be the sole explanation of the data.

Figure 53. A change in the seismoelectric response has several plausible causes. Independent constraints convert an underdetermined attribution problem into a thermal inversion. The supplied source material strongly supports this joint-interpretation philosophy.

9.1 A practical identifiability matrix

Observable	Thermal sensitivity	Main confounders	What must be constrained
Low-frequency Cs	Through εf, ηf, σw and possibly ζ	salinity, pH, mineral surface, saturation, surface conduction	fluid chemistry and electrochemical model
Corner frequency fc	Through ηf/ρf	intrinsic permeability, porosity, tortuosity	k0 or a strong prior; φ and α∞
Phase / full HSE	Through dynamic fluid/electrical coefficients	source spectrum, instrument response, mixed wave modes	source and receiver response, model selection
Vp / interface timing	Through fluid modulus/density within poroelastic model	stress, saturation, frame moduli, geometry	rock-physics model and stable geometry
Raw amplitude only	yes, but non-specific	almost every acquisition and formation parameter	insufficient alone

10. Mapping thermal information with depth

Thermal information becomes a formation model only after the thermal-sensitive observables are assigned to depth. The source material treats interface-generated electrical responses as being triggered near the one-way seismic travel time to a contrast.

Equation 206 - Interface-response timing SOURCE

t_{IR}=\int_{source\rightarrow interface}\frac{ds}{V_P(s)}+t_{EM}\approx t_{P,\,to\,interface}

For shallow scales the electromagnetic delay may be small relative to seismic travel time. The source material emphasizes that this is a one-way seismic timing relation for interface conversion, not a conventional two-way reflection formula.

Haartsen & Pride (1997); Haines et al. (2007); Dupuis et al. (2009); Schakel et al. (2011).

Equation 207 - Simple vertical depth assignment SOURCE

z\approx V_P t_{IR}

Use a calibrated velocity model. If temperature itself changes $V_P$ appreciably, depth and thermal inversion should be iterated because the thermal state then affects the travel-time-to-depth conversion.

Figure 54. Thermal-sensitive observables become a vertical formation model only after events or windows are assigned to depth using a calibrated seismic velocity model. If repeated stations are occupied laterally, the same procedure can build a 2-D/3-D thermal-change model, subject to the spatial sensitivity of the acquisition.

11. Signal processing and thermal bias

Thermal interpretation is unusually sensitive to processing consistency because it depends on subtle amplitude, phase and spectral changes. The supplied acquisition/processing material recommends instrument-response correction, source-trigger control, individual-shot rejection, stacking, power-line treatment, coherence analysis and careful window selection.

Processing step	Why it matters thermally	Recommended implementation
Instrument gain/phase correction	An electronics change can look like a change in Cs or spectral roll-off.	Use the same calibrated acquisition chain or remove each response explicitly.
Source normalization	Changing source amplitude/bandwidth can mimic a thermal transfer change.	Use a known source signature or a calibrated mechanical model for each epoch.
Shot rejection then stacking	Random electrical noise inflates uncertainty; contaminated hits bias spectra.	Save individual shots, reject source/electrical contamination, then stack accepted records.
Power-line suppression	Notches remove frequency information and can perturb phase near fc.	Preserve raw data; estimate/subtract narrow harmonics only where required; exclude affected bins from thermal fitting.
Time gating	Different windows change spectral shape and can move fitted fc.	Use identical physical event windows and tapered edges at baseline and repeat surveys.
Coherence / repeatability mask	Thermal fitting in noise-dominated bins creates false changes.	Mask low-coherence or non-repeatable bands before fitting Cs/fc.
Reflection / mode contamination	Mixed events change amplitude and phase independently of temperature.	Use timing and forward modeling; do not interpret a mode-mixed spectral shift as thermal.

12. Practical field implementation

Establish a baseline thermal calibration

At a representative location or core, measure the actual fluid chemistry and temperature while acquiring seismoelectric data. Build laboratory/site curves for the quantities used in the thermal model: at minimum the effective $C_s(T)$ or its component properties, and - if using the spectral route - the fluid properties entering $f_c(T)$ .

Freeze geometry and acquisition

Use the same dipole spacing/orientation, grounding approach, source location, trigger definition, acquisition gain and processing sequence for repeat surveys.

Record enough repeated impulses

The supplied field workflow recommends individual source records so contaminated hits can be removed before stacking. Thermal monitoring should quantify shot-to-shot variability because the desired thermal change may be smaller than a single-shot amplitude fluctuation.

Recover Cs and spectral shape

Convert voltage to electric field, source-normalize, estimate the low-frequency plateau/coupling and fit the dynamic corner only if the usable bandwidth spans the transition.

Check non-thermal state variables

Record water chemistry, pH, conductivity and saturation indicators wherever practical. A chemistry change can invalidate a thermal calibration even when the acquisition is perfect.

Invert for thermal state with uncertainty

Prefer a time-lapse or joint fit. Report temperature only when the calibration and nuisance parameters make it identifiable; otherwise report a thermal-sensitive change index rather than a false absolute temperature.

Figure 55. Practical implementation. Thermal extraction is a calibrated inverse workflow, not a direct voltage conversion. External controls are part of the measurement model because the supplied theory identifies them as parameters that also control seismoelectric amplitude and frequency response.

13. Uncertainty and acceptance tests

Equation 208 - First-order uncertainty propagation SOURCE FRAMEWORK

u_y^2\approx\sum_i\left(\frac{\partial y}{\partial x_i}\right)^2u_{x_i}^2

The supplied advanced workflow uses first-order uncertainty propagation. For thermal inversion, include source calibration, electric-field calibration, spectral-fit uncertainty, fluid-property calibration, chemistry and any rock-property priors.

Equation 209 - Linearized uncertainty of coupling-derived ΔT DERIVED

u_{\Delta T_C}^2\approx\left(\frac{u_{\ln R_C}}{S_C}\right)^2+\left(\frac{\ln R_C}{S_C^2}u_{S_C}\right)^2

This makes the calibration requirement explicit: even a very precise field ratio cannot yield a precise temperature if the laboratory thermal slope $S_C$ is poorly known.

Equation 210 - Linearized uncertainty of corner-derived ΔT DERIVED

u_{\Delta T_f}^2\approx\left(\frac{u_{\ln R_f}}{S_f}\right)^2+\left(\frac{\ln R_f}{S_f^2}u_{S_f}\right)^2

Add permeability-change uncertainty if $k_0$ cannot be assumed constant.

Acceptance test	Pass condition	If it fails
Source repeatability	Source-normalized transfer is stable in a control interval.	Do not interpret amplitude/spectral change thermally.
Coupling plateau	A coherent low-frequency band supports a stable Cs estimate.	Report only broader spectral change, not Cs-derived temperature.
Dynamic corner	Usable bandwidth spans both sides of fc and the fit is stable to window/band changes.	Do not use the fc thermal route.
Chemistry stability	σw, pH and relevant fluid chemistry are stable or explicitly modeled.	Thermal attribution from Cs is non-unique.
Permeability stability	k is independently known or demonstrably unchanged for a time-lapse fc interpretation.	Jointly invert k and T or do not report T from fc.
Baseline calibration	Property-vs-temperature model represents the actual fluid/mineral system.	Report a thermal-sensitive index, not absolute temperature.
Processing stability	Result persists under reasonable windows, taper and noise masks.	Classify the inferred thermal change as processing-sensitive.

14. What the source material does not support

No universal direct thermometer. The supplied archive does not establish a universal conversion from seismoelectric voltage amplitude, phase or frequency directly to degrees Celsius. Any such formula would require additional empirical assumptions not present in the source material.

Not supported: T from one raw amplitude

The same amplitude can change because of source strength, electrode coupling, salinity, zeta potential, saturation, porosity, permeability, geometry or wave-mode mixing.

Not supported: T and k from fc alone

They trade through viscosity in the characteristic-frequency equation unless one is independently constrained.

Not supported: assuming ζ is fixed everywhere

The supplied theory explicitly warns that surface chemistry and pH alter zeta potential.

Not supported: ignoring temperature when reporting Kh

The supplied permeability workflow specifically says to report the fluid temperature and chemistry used because viscosity depends strongly on temperature.

What is supported. Use seismoelectric data to estimate temperature-sensitive coupling and dynamic-response parameters; combine them with calibrated fluid/electrochemical property functions and independent geological constraints; then report either a thermal-change estimate or an absolute temperature with an uncertainty that reflects the non-thermal degrees of freedom.

15. Calibration-slope thermal calculator

This calculator intentionally contains no built-in universal water-property coefficients. Enter slopes measured or adopted for the actual formation fluid/mineral system. It implements the derived linearized Equations 19-21 and therefore illustrates the inversion without inventing a temperature law that is absent from the supplied source material.

Coupling ratio RC = Cs₂/Cs₁

Calibrated SC = d ln|Cs|/dT [1/°C]

Corner ratio Rf = fc₂/fc₁

Calibrated Sf = d ln fc/dT [1/°C]

Uncertainty u(ΔTC) [°C]

Uncertainty u(ΔTf) [°C]

ΔT from coupling -

ΔT from corner -

Weighted ΔT -

The example numbers are only placeholders to demonstrate the calculator. They are not source-derived thermal coefficients and must be replaced by the user's calibrated values before scientific use.

Integrated source module

Groundwater Quality, Salinity and NAPL/DNAPL Contamination

Application section. It treats salinity as an electrochemical/conductivity problem and NAPL/DNAPL as a multiphase saturation problem, with contaminant identity remaining dependent on chemistry and site calibration.

Source policy used in this document. Equations marked source relation are present in the supplied theory material. Equations marked derived diagnostic are algebraic rearrangements or time-lapse ratios built from those relations. They are not presented as published contaminant-specific laws. Statements about LNAPL/DNAPL density behavior are general hydrogeologic definitions, not a claim that seismoelectric data alone distinguish the two.

Contents

Why water quality affects seismoelectric data
What the grounded dipole measures
Salinity and dissolved ionic pollution
pH, surface chemistry and non-unique water-quality effects
NAPL/DNAPL as a two-fluid problem
Contaminant interfaces and depth
Spectral and effective-charge information
Practical environmental workflow
Time-lapse plume monitoring
Confounders and failure modes
What can be reported defensibly
References from supplied material

1. Why water quality affects seismoelectric data

A seismoelectric signal is generated because seismic forcing causes pore fluid to move relative to the mineral frame, advecting electrical double-layer charge. Pollution matters when it changes the fluid, the mineral-fluid interface, the connected water pathways, or the mechanical/hydraulic state of the pores.

Figure 56. Environmental contaminants are not sensed by name; they are sensed through the physical properties they modify. This is why the method is sensitive but inherently non-unique.

Equation 211 - Coupled electrokinetic transport pair source relation

J = σ(ω)E + L(ω)X_p
q = L(ω)E + k(ω)X_p/η

The electrical current density contains ordinary conduction plus mechanically driven streaming current; the fluid flux contains electro-osmotic and pressure/inertial terms. A contaminant can therefore alter the signal by changing σ, L, k, η or the mechanical forcing.

Source provenance. Pride & Haartsen (1996); Jouniaux & Ishido (2012); Schoemaker et al. (2012), as summarized in the supplied advanced-theory document.

2. What a grounded dipole measures in an environmental survey

A grounded dipole measures a potential difference, not contaminant concentration. Its spacing and orientation determine the component and spatial averaging of the electric field. Shorter dipoles preserve locality; longer dipoles can increase signal voltage but average over more heterogeneous ground.

Equation 212 - Grounded-dipole voltage source relation

V_AB(t)=φ(B,t)-φ(A,t)=-∫_A^B E(r,t)·dl

For a short straight dipole, E_∥≈-V_AB/ℓ. Environmental interpretation should therefore preserve electrode spacing, polarity, contact condition and receiver gain.

Source provenance. Supplied dipole-geometry note; Pride & Haartsen (1996); Jouniaux & Ishido (2012).

Figure 57. Conceptual single-dipole environmental geometry. The dipole records the integrated electrical response produced by mechanically driven pore-fluid motion and interface conversions; interpreting the signal as salinity or NAPL/DNAPL requires a petrophysical model.

3. Salinity and dissolved ionic pollution

The supplied material explicitly lists water conductivity / salinity as a major control on seismoelectric amplitude because it changes current paths and how streaming voltage is expressed. In the simplest low-frequency, thin-double-layer limit, the streaming-potential coefficient is inversely proportional to pore-water conductivity.

Equation 213 - Helmholtz-Smoluchowski low-frequency limit source relation

C_s ≈ ε_fζ /(η_fσ_w)

σw pore-water conductivity

ζ zeta potential

ηf fluid viscosity

εf fluid permittivity

Cs streaming-potential coefficient

If all other factors were fixed, increasing salinity/conductivity decreases the voltage-per-pressure coupling. In real groundwater, however, salinity can also alter zeta potential and double-layer structure, so the simple inverse scaling is a calibration law, not a universal concentration meter.

Source provenance. Beamish (1999); Jouniaux & Ishido (2012); Jouniaux & Zyserman (2016).

Equation 214 - Pore-water conductivity inferred from calibrated coupling derived diagnostic

σ̂_w ≈ ε_fζ /(η_fĈ_s)

This is a direct algebraic rearrangement of Equation 3. It is useful only if ζ, η_f and ε_f are calibrated or independently constrained. It should not be used where surface conduction, clay effects or changing pH dominate.

Equation 215 - Time-lapse conductivity ratio derived diagnostic

σ_w,2/σ_w,1 = (ε₂ζ₂η₁C_s,1)/(ε₁ζ₁η₂C_s,2)

If temperature and surface chemistry are stable enough that ε,η,ζ are effectively constant, then approximately σ_w,2/σ_w,1≈C_s,1/C_s,2. This makes repeat surveys more defensible than attempting absolute salinity from one acquisition.

Figure 58. Idealized inverse dependence of low-frequency coupling on pore-water conductivity. The physics is source-supported; a site-specific salinity calibration is still required.

3.1 Bulk conductivity is not the same as pore-water conductivity

Equation 216 - Bulk conductivity with surface contribution source relation

σ₀ = σ_w/F + σ_s

A clay-rich or low-salinity formation may carry appreciable current along mineral surfaces. Therefore an electrically conductive zone is not automatically saline groundwater, and a low-conductivity zone is not automatically NAPL.

Source provenance. Jouniaux & Zyserman (2016); Revil et al. (2014).

Equation 217 - Archie formation factor source relation

F = φ^-m

For a clean porous medium, the formation factor links pore geometry to electrical conduction. It should be calibrated locally, especially in clay-rich, fractured or mineralized formations.

4. pH, surface chemistry and broader water-quality effects

Water quality is broader than salinity. The source material identifies pH and mineral surface chemistry as controls on surface charge and zeta potential. Because ζ appears directly in the electrokinetic coupling, a pollution plume that changes pH or mineral-fluid chemistry can change the seismoelectric response even if its conductivity change is modest.

Equation 218 - Coupling coefficient from streaming potential source relation

L = -σ₀C_s, |L| ≈ ε_f|ζ|/(η_fF)

This highlights two distinct observables: C_s is strongly affected by pore-water conductivity, while L depends on electrochemical surface properties, viscosity and pore geometry. Comparing both can help distinguish a conductivity-driven change from a coupling/surface-chemistry change.

Source provenance. Jouniaux & Zyserman (2016).

Figure 59. Two different contaminant mechanisms can produce similar electrical-amplitude changes. Joint interpretation is therefore essential.

Pollutant-specific limit. The supplied source archive does not provide validated direct calibrations for nitrate, arsenic, PFAS, dissolved solvents, metals, landfill leachate, acid mine drainage or other named contaminants. Such pollution may be seismoelectrically visible if it changes conductivity, pH, saturation, fluid properties or interfaces, but chemical identification still requires sampling or a site-specific laboratory calibration.

5. NAPL and DNAPL: treat the problem as two immiscible fluid phases

NAPL means non-aqueous phase liquid. In general hydrogeologic usage, LNAPL is less dense than water and commonly accumulates near the water table; DNAPL is denser than water and can migrate downward through saturated media. The seismoelectric relevance is not the label itself but that a second immiscible fluid changes water saturation, capillary state, effective compressibility/density, electrical connectivity, relative permeability and the distribution of mobile excess charge.

The supplied archive contains Jardani & Revil (2015), specifically on two immiscible fluid phases, plus two Zhao, Sun & Nie (2023) papers devoted to oil-wet porous media containing oil and water. These sources support treating hydrocarbon contamination with a multiphase seismoelectric model rather than a single-fluid saline-water model.

Equation 219 - Unsaturated / partially water-filled Archie relation source relation

σ_b-σ_s ≈ σ_wφ^mS_wⁿ

Loss of connected water-filled pore pathways lowers the water-conduction term. In a water-NAPL system, decreasing S_w may therefore reduce bulk conduction even when the remaining water has unchanged salinity. This is one reason a resistive anomaly cannot by itself prove NAPL: dry/unsaturated material can produce a similar electrical effect.

Source provenance. Revil & Mahardika (2013); Revil et al. (2014); Bordes et al. (2015).

Equation 220 - Water saturation estimate from calibrated Archie parameters source-derived inversion

Ŝ_w = [(σ_b-σ_s)/(σ_wφ^m)]^1/n

The supplied theory stresses that this is an electrical-petrophysical saturation estimate used to constrain the seismoelectric model, not a rule that derives NAPL concentration directly from ES amplitude.

Equation 221 - Saturation-dependent mobility source relation

k(S_w) = k₀k_r(S_w), K_h=k(S_w)ρ_fg/η_f

A second fluid phase changes the connected hydraulic mobility of water. The seismoelectric signal depends on the water phase that actually moves and transports charge, so relative permeability and capillary connectivity matter.

Source provenance. Revil & Mahardika (2013); Revil et al. (2014); Jardani & Revil (2015).

Figure 60. General hydrogeologic LNAPL/DNAPL geometry combined with source-supported two-phase seismoelectric parameters. The ES record responds to the property contrasts; density class is not determined from waveform shape alone.

5.1 Why fluid compressibility can matter

Equation 222 - Biot storage modulus source relation

M=[(α-φ)/K_s + φ/K_f]^-1

In a single-fluid saturated formation, K_f is the pore-fluid bulk modulus. The supplied theory explicitly warns that gas or multiphase saturation requires an effective-fluid model and makes storage strongly saturation-dependent. NAPL/DNAPL can therefore alter both the electrical and mechanical sides of the coupled response.

Source provenance. Pride & Haartsen (1996); Revil et al. (2014).

Equation 223 - Effective P-wave speed in the low-loss limit source relation

K_u=K_d+α²M, H=K_u+4G/3, V_P≈√(H/ρ_b)

Changes in fluid modulus and density can shift seismic travel times and reflection coefficients. These mechanical changes can reinforce contaminant-boundary detection but also introduce trade-offs with lithology.

6. Contaminant boundaries can generate interface seismoelectric responses

Layered seismoelectric theory predicts electromagnetic conversion when a seismic wave reaches a contrast in elastic, hydraulic, electrical or electrokinetic properties. A saline front, a fresh/saline boundary, a saturation front, or a water-NAPL transition can therefore act as a candidate converter if the property contrast is strong enough.

Equation 224 - First-order interface-response timing source relation

t_IR=∫_{source→interface} ds/V_P(s)+t_EM ≈ t_{P,to interface}

In a simple vertical constant-velocity case, z≈V_Pt_IR. This can be used to map the top or base of a contaminant-related property contrast if the event is correctly classified and the velocity model is known.

Source provenance. Haartsen & Pride (1997); Haines et al. (2007); Dupuis et al. (2009); Schakel et al. (2011).

Figure 61. A contaminant-related property contrast can act as an interface converter. Event timing provides depth information; event amplitude is controlled by multiple contrasts and should not be treated as contaminant concentration by itself.

7. Spectral information, effective charge and dynamic response

Environmental contamination may change not only low-frequency amplitude but also the frequency dependence of coupling. This matters when viscosity, permeability, saturation and electrical properties vary enough to move the dynamic transition or change the transported excess charge.

Equation 225 - Effective excess-charge coupling source relation

C_EK⁰ = - Q̂_v⁰k₀ /(η_wσ₀)

This form shows why contaminant effects are intertwined: a change in conductivity, viscosity, permeability or the charge carried by flowing water can all alter the measured coupling.

Source provenance. Revil & Mahardika (2013); Revil et al. (2014); Jougnot & Solazzi (2021).

Equation 226 - Frequency-dependent effective-charge coupling source relation

C_EK*(ω) = - Q̂_v*(ω)k*(ω)/[η_wσ*(ω)]

Amplitude and phase may contain information beyond one low-frequency number. But the same spectrum can trade off among effective charge, dynamic permeability and electrical conductivity, so independent constraints remain valuable.

Source provenance. Jougnot & Solazzi (2021).

Equation 227 - Characteristic transition frequency source relation

ω_c=φη_f/(α_∞k₀ρ_f), f_c=ω_c/(2π)

NAPL/DNAPL can alter effective viscosity/density and phase mobility; salinity can alter conductivity but does not enter this particular characteristic frequency directly. Comparing coupling amplitude with transition-frequency behavior can therefore help determine whether a change is mainly electrochemical or hydraulic/mechanical.

Source provenance. Bordes et al. (2015).

Figure 62. A pollution interpretation is stronger when several observables change consistently with one physical model.

8. Practical environmental characterization workflow

Baseline the clean site

Record repeatable source/receiver data before contamination if possible, or at an uncontaminated reference station with comparable geology.

Calibrate the electrical channel

Convert counts to volts and then to electric field using actual dipole spacing. Record contact impedance and preserve polarity.

Characterize the source

Use a known source wavelet or mechanical reference. A single electric amplitude without source normalization is not a water-quality metric.

Separate event classes

Distinguish source transients, coseismic energy, interface conversion and reflected energy by timing, repeatability, polarity and geometry.

Estimate coupling

Use a P-dominated, low-frequency, high-SNR window to estimate C_s or the empirical transfer function.

Build water-quality hypotheses

Test whether the observed change is better explained by σ_w, ζ, S_w, k, fluid properties or a new interface.

Use site chemistry

Measure conductivity, temperature and pH; where contamination is suspected, use laboratory chemistry to constrain pollutant identity.

Use petrophysics

Constrain porosity, formation factor, surface conduction, saturation and permeability rather than allowing them all to float in one inversion.

Validate against samples

Use wells/cores or other direct observations to calibrate the site-specific mapping from ES parameters to contaminant state.

Equation 228 - Low-frequency coseismic field / acceleration relation source relation

E_∥ ≈ -C_sρ_fa_s,∥

This is the primary route from calibrated ES data to an effective coupling coefficient. For a single electric receiver without a geophone, the mechanical acceleration must be supplied by a calibrated source/forward model, as described elsewhere in the supplied material.

Source provenance. Jouniaux & Zyserman (2016); Bordes et al. (2015).

Equation 229 - Empirical seismoelectric transfer function source relation

H_SE(ω)=E(ω)/A_s(ω)

Use cross-spectral estimates and coherence rather than unstable pointwise division where the mechanical spectrum is small. Time-lapse changes in H_SE are generally more useful for contamination monitoring than raw voltage changes.

Source provenance. Bordes et al. (2015); Schoemaker et al. (2012).

Figure 63. Recommended workflow: infer physical state first; attach pollutant identity only after calibration against direct environmental information.

9. Time-lapse monitoring is usually stronger than one-time contaminant identification

A repeat survey over unchanged geology cancels many nuisance parameters. The most valuable environmental use may therefore be monitoring plume movement, remediation, saline intrusion, freshwater flushing or phase redistribution.

Equation 230 - Normalized time-lapse coupling change derived diagnostic

R_C(ω,t)=H_SE(ω,t)/H_SE(ω,t₀)

Values departing systematically from one indicate a change in the coupled formation state after source and receiver normalization. Interpretation still requires checking whether the change comes from conductivity, saturation, temperature, pH or hydromechanical evolution.

Equation 231 - Interface migration from repeated timing derived diagnostic

Δz ≈ V_PΔt_IR + t_IRΔV_P

This first-order differential of z≈V_Pt_IR shows how a moving salinity/saturation front can be tracked, provided the velocity model is stable or updated.

Figure 64. Time-lapse surveys can map migration or remediation more robustly because lithology and many geometric factors remain fixed.

10. Confounders and failure modes

Observation	Possible environmental cause	Non-contaminant alternative	Required check
\|Cs\| decreases	higher pore-water conductivity / salinity	zeta-potential change, temperature, surface conduction	EC, pH, temperature, mineralogy
bulk conductivity decreases	NAPL replacing connected water	drying, lower salinity, lithologic change	water saturation + pore-water EC
new interface ES event	salinity or saturation front	geological layer or water table change	baseline survey + velocity model
spectral corner shifts	fluid viscosity/mobility or phase change	permeability heterogeneity, source spectrum	source normalization + hydraulic constraints
polarity changes	coupling/sign change across chemistry or saturation contrast	electrode reversal, geometry, wave-mode change	polarity metadata + repeat geometry
large flat electrical transient	possible interface conversion	source electronics / cultural electrical noise	dummy-source and isolation tests

Hydrocarbon non-uniqueness. The supplied basic-theory material explicitly warns that hydrocarbons do not have a unique seismoelectric signature. A NAPL/DNAPL interpretation must therefore be the result of a constrained two-phase model, not pattern matching to one amplitude or waveform.

11. What can be reported defensibly

Strongest products

Relative change maps in calibrated coupling; interface depth from one-way timing; conductivity/salinity trend when ζ/temperature/surface conduction are controlled; water-saturation or two-phase probability when Archie/petrophysical parameters are calibrated; and plume migration from repeat surveys.

Weak or unsupported products

A unique pollutant name from a single trace, exact contaminant concentration without site calibration, DNAPL versus LNAPL solely from waveform shape, or direct water-potability certification. Those claims are not supported by the supplied seismoelectric theory.

Recommended environmental use. Treat seismoelectrics as a high-sensitivity bridge between mechanics and pore-fluid electrochemistry. Use it to locate and monitor anomalous fluid-property or saturation zones, then use chemistry and site petrophysics to determine whether the anomaly represents salinity, an ionic contaminant plume, NAPL/DNAPL, remediation effects or a natural geological change.

Part V

Integrated interpretation and reporting

A unified decision framework for combining observables, priors, uncertainty, identifiability and defensible reporting.

Integrated synthesis

From a voltage trace to a defensible formation model

The preceding sections are deliberately detailed and application-specific. This final synthesis reconnects them into a single interpretation hierarchy: classify the measured event first, calibrate geometry and timing second, constrain the mechanical/electrical state third, and only then invert for formation properties that the data are actually sensitive to.

1. The interpretation hierarchy

Triggered seismic source→mechanical wavefield→relative pore-fluid motion→charge transport→grounded-dipole voltage→event classification→joint formation model

The same raw seismoelectric record can contain source-zone coseismic energy, primary interface conversion, reflected coseismic energy, secondary conversions and electrical noise. These components have different kinematics and different parameter sensitivities. The central rule across the supplied material is therefore to avoid mapping one amplitude directly to one geological property without a forward model and the independent constraints required by that model.

2. What each observable is best suited to constrain

Target product	Primary seismoelectric observable	Important supporting constraints	Main ambiguity to control
Interface depth	Primary interface-conversion time; reflection timing can provide a consistency check	P-wave velocity model, trigger/latency, geometry	One-way timing is not an independent velocity measurement in a single fixed-dipole geometry
Permeability	Calibrated coseismic transfer and, when resolved, dynamic transition frequency	Source signature or mechanical channel, porosity, tortuosity, fluid density/viscosity, conductivity	Coupling, conductivity, effective charge and permeability trade off spectrally
Fractures	Localized interface conversions and travel-time/moveout localization	Geometry, velocity model, borehole or crosshole control where available	Lithologic/electrical boundaries can mimic a fracture response
Elastic stiffness	Coseismic P/S timing and interface contrast	Bulk density and wave-mode identification	Dynamic/undrained moduli are not automatically static design moduli or strength
Thermal change	Time-lapse coupling and transition-frequency changes	Fluid-property calibration, conductivity, chemistry, stable pore structure	Temperature is entangled with salinity, zeta potential, saturation and permeability
Salinity / water quality	Coupling and conductivity-sensitive changes, interface response, time-lapse ratios	Water chemistry, temperature, pH, electrical petrophysics	No universal pollutant-specific waveform or concentration law
NAPL / DNAPL effects	Saturation-dependent coupling, electrical connectivity and multiphase interfaces	Resistivity, saturation model, geology, chemistry/fluid sampling	A resistive or low-coupling anomaly alone does not identify hydrocarbon type

3. Minimum quality-control sequence

Preserve the raw, individually triggered records and reject source/electrical artefacts before stacking.
Calibrate dipole spacing, polarity, channel gain, source time zero and any recorder latency.
Classify events by timing, moveout where available, polarity, geometry, repeatability and source-control tests.
Keep primary interface, local coseismic and reflected/secondary event windows separate until a forward model explicitly calls for them to be fitted together.
Estimate properties only in bandwidths with adequate signal-to-noise ratio and model sensitivity; test results against alternative windows, filters and plausible nuisance parameters.
Report measured quantities, direct calculations and model-dependent inversions separately, with uncertainty and stop rules.

Unified conclusion. Seismoelectrics are most powerful as a coupled constraint on mechanics, fluids and electrochemistry. The method becomes less defensible as interpretation moves from measured timing and transfer functions toward uniquely named formation properties without independent constraints.

References

Consolidated bibliography from the supplied source documents. Duplicate bibliographic entries were merged by their source reference identifiers; DOI/publisher links were retained where present.

S. R. Pride, M. W. Haartsen (1996). Electroseismic wave properties. Journal of the Acoustical Society of America 100 . DOI / legal source
M. W. Haartsen, S. R. Pride (1997). Electroseismic waves from point sources in layered media. Journal of Geophysical Research: Solid Earth . DOI / legal source
D. Beamish (1999). Characteristics of near-surface electrokinetic coupling. Geophysical Journal International 137, 231-242 . DOI / legal source
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Abstract

Fundamental seismoelectric theory

Core Theory: From Pore-Scale Physics to Field Inversion

1.Scope, terminology, and the complete physical chain

Generation

Observation

Inference

2.Pore-scale origin: the electrical double layer

Surface charge

Mobile counter-charge

Relative flow

3.What is observed: coseismic fields, interface conversion, and reciprocity

Coseismic field

Interface response

4.Coupled continuum framework and sign conventions

5.Governing electromagnetic and poroelastic laws

2.1 Maxwell equations in the frequency domain

2.2 Biot variables: the fluid moves relative to the frame

6.Electrokinetic transport, streaming potential, and Onsager reciprocity

3.1 Streaming potential and open-circuit condition

7.Formation factor, bulk conductivity, and surface conduction

8.Frequency dependence, dynamic permeability, and transition frequency

9.Effective excess charge formulation

10.From coupled equations to observable wave signatures

7.1 Low-frequency coseismic P-wave transfer

7.2 Interface conversion

11.Unsaturated and two-phase extensions

12.Field acquisition: source, electrodes, mechanical channels, and geometry

9.1 Electrical channels

9.2 Mechanical channels

9.3 Source reference and electrical hygiene

9.4 Geometry as a filter

Practical measurement view

6.1 Why two electrodes?

6.2 Why stack many shots?

13.Signal processing and event-separation logic

Shot-gather diagnostics

Ask about moveout

Ask about polarity

Ask about simultaneity

Ask what the source itself did

14.General parameter derivation from field data

15.Single-dipole near-source inversion: from ΔV(t) to coupling and permeability

Required independent inputs

Inversion chain

Additional equations unique to the single-dipole implementation

Reproducible inversion sequence

Single-dipole failure modes and validation

16.Forward modeling, inversion, and non-uniqueness

17.Parameter-to-measurement crosswalk

18.Worked end-to-end workflow: shallow saturated sand over a contrasting layer

19.Assumption failures, identifiability limits, and defensible reporting

Surface conduction is not negligible

The data are not below fc

Multiple wave modes occupy the window

Saturation is patchy

The interface is not planar

The electrical event is source-generated

Interpretation boundaries

20.Conclusions

Acquisition geometry and resolution

Grounded-Dipole Geometry and Lateral Sensitivity

1. Scope and practical claim

2. What the grounded dipole actually measures

3. Why one electrode belongs next to the source

It maximizes the potential contrast

It anchors the measurement in time and space

It biases the measurement toward the target interval

4. Why about 2 m is often the best spacing

5. Why dipole-field geometry is the controlling factor

Geometric concentration

Geometric selectivity

6. Why the data mainly represent the geology directly below the source

Rapid field decay

Early-time gating

Lack of lateral aperture

7. Why lateral geology contributes very little

7.1 No lateral receiver aperture

7.2 Lateral contributions are geometrically disadvantaged

7.3 Early-time gating favors vertical rather than lateral sensitivity

The data are not below f_c