Appendix A

Crustal deformation is accompanied by the process of pore water diffusion. The hydrodynamics of pore water was described in the classic papers of Frenkel [1944] and Biot [1956]. We can write the following equation describing the relation between the change in volume strain q=DV/V and the hydrostatic pressure change P in a porous, water-saturated medium [Frenkel, 1944]:

(A1)

where b=(1-K/K₀)/m; b=1+(b-1)K₂/K₀; K₀, K and K₂ are the bulk moduli of rock matrix, dry porous rock, and pore water, respectively; m is the porosity; m_v, k₀ and r_w are the dynamic viscosity, permeability, and density of water, respectively. Note that the magnitude of q depends in general on P. However, for most rock parameter values, the changes in pore pressure have practically no effect on the volume strain. This is why we shall suppose that q depends only on time t and position r, but is independent of P. So q (t, r) acts simply as a source function in equation A1.

Let's compare the first two terms on the left in equation A1, assuming a monochromatic disturbance of frequency w and time dependence e^iwt. Then these terms are -w² P/K₂ and iwm_v P/K₂k₀r_w. If ww _constm_v/k₀r_w then the second term on the left can be ignored. By a similar argument the second term on the right can be ignored relative to the first term. This means that the pore pressure will be described by the inhomogeneous wave equation,

(A2)

From this equation we can define the speed of propagation of compressional waves in pore water:

(A3)

Compare this speed with the speed of compressional waves in the matrix [Landau and Lifshitz, 1986]:

(A4)

where m_s is shear modulus and r _matrix is the density of the matrix. Using typical parameters (Table A1), we find V² _water/V² _matrix1/10.

From equation A2, when a seismic wave of frequency ww _const and wavelength l=2pV_matrix/w propagates through a porous medium, it will drive the pressure fluctuations in the water at the same frequency and wavelength. The second term on the left in equation A2 is therefore essentially w² P/V²_water while the term ² P is of order (2p/l)² P or w² P/V²_matrix. This latter term will be much smaller than the former; ignoring it gives

(A5)

Integration results in

(A6)

(Under conditions of hydrostatic equilibrium, the constants of integration vanish.)

Now consider the situation when ww₀. In this case we can neglect the terms in equation A1 containing the second time derivatives, leading to a diffusion-type equation:

(A7)

It is easy to show that, in this case also, the pressure fluctuations will track the volume fluctuations. To see this we again consider a seismic wave as the driving force. The first term on the left is m_v iwP/k₀r_w or iw _constwP, while the second term is essentially (K₀/br_w)w² P/V² _matrix or (V² _water/V² _matrix)w² P. The latter term can be ignored compared to the former, so

(A8)

Therefore

(A9)

So in order to connect P and q for fast processes when ww _const we should use equation A6 and for slow processes (like water diffusion after sudden changes of volume strain) we should use equation A9.

Appendix B

We present here asymptotic formulas for the components of the above-ground electromagnetic field produced by horizontal and vertical electric and magnetic dipoles embedded in the conducting half-space z>0 (see Figure 2a in Section 4) with conductivity s, electric permeability e and magnetic susceptibility m [Banos, 1966]. The distance r from the coordinate origin is characterized as being in the near zone, intermediate zone or far zone (see Section 4 and Figure 2b). In order to save space, we present results only for the dominant components in the intermediate and far zones. For the near zone, all components are given. The quantities n, k₁ and k₂ are defined in Section 4.

1. Horizontal electric dipole

(a) Near zone

(B1)

(B2)

(B3)

(B4)

(B5)

(B6)

(b) Intermediate zone

(B7)

(B8)

(c) Far zone

(B9)

(B10)

2. Horizontal magnetic dipole

(a) Near zone

(B11)

(B12)

(B13)

(B14)

(B15)

(B16)

(b) Intermediate zone

(B17)

(B18)

(c) Far zone

(B19)

(B20)

3. Vertical electric dipole

(a) Near zone

(B21)

(B22)

(B23)

(b) Intermediate zone

(B24)

(B25)

(c) Far zone

(B26)

(B27)

4. Vertical magnetic dipole

(a) Near zone

(B28)

(B29)

(B30)

(b) Intermediate and far zones

(B31)

(B32)

Appendix C

In Section 4, consideration of the electro-kinetic effect resulted in equation 21b, in which the second and third terms involve surface integrals instead of volume integrals. To see how this comes about, consider the volume integral (see equation 12),

where j⁰ is the electro-kinetic current and the integral is over the crust. If the crust is homogeneous this integral equals zero, since P is a localized function, i.e. vanishes at large distances. Now suppose there is an inhomogeneity across the plane x =0, resulting in a discontinuity in the coefficient C: C = C₁H(x) - C₂H(-x). Then

If P(r, t) P(0)l_cr exp[- rL - ( t - r/VDt)²]H(r - l_c), I can be written in the form

which is the second term in equation 21b. The same reasoning can be applied to the third term.

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