Electrical disturbances associated with seismic sources - The Seismoelectric Method - page 140

measurable conductivities. Manipulation of Maxwell
equations under this formulation yields the electric field,
E=

themselves (e.g., Huang & Liu, 2006). Therefore,
these types of conversions are sensitive to the level
of heterogeneities of the subsurface, and they have
been broadly recognized in the literature as seismo-
electric conversions (e.g., Garambois & Dietrich,
2001). If the subsurface is described with piecewise
constant material properties, these conversions are
created at the interface between homogeneous blocks
and are sometimes called

(in Vm − 1 , where

denotes the electrostatic
potential), and the magnetic field H (in Am − 1 ) is defined
by the following scalar and vectorial Poisson equations:

−∇ ψ ψ

∇ σ ∇ ψ = ∇ J S , 5 9

2 B=

∇ − μ ∇ × J S , 5 10

conversions.
iii Type III corresponds to the coseismic electromagnetic
signals. They are electromagnetic signals propagating
at the same speed as the P- and S-seismic waves and
are due to local fluid flow associated with the passage
of the seismic wave. This flow generates a source cur-
rent density that is locally compensated by the conduc-
tion current density in homogeneous materials. The
type III electrical field is proportional to the acceleration
of the seismic wave with a local transfer function that
depends on the local electrical properties of the soil.
Because electrical source signals (type I in the earlier
nomenclature) are observed at receivers at the moment
when the seismic source occurs (the travel time of the
information is quasi-instantaneous), they provide the time
of occurrence of the seismic event (time zero). We will see
that the electric field information is very useful to charac-
terize a seismic point source location in addition to deter-
mining the seismic source mechanism information.
The bulk body force,

“ interfacial ”

are the electrical conductivity (in S m − 1 )
and the magnetic permeability (in Hm − 1 ) of the medium,
respectively, and where the dynamic streaming current
density, J S , is given by (Jardani et al., 2010; Revil &
Jardani, 2010)

where σ and μ

J S = Q 0 Q 0

V

2

V w =

− i ω ∇ p − ω ρ f u − f 5 11

The insulating boundary condition at the ground
surface is written as n E =0( n denotes the normal unit
vector to the ground surface). Therefore, the normal
component of the electrical field at the ground surface
is strictly equal to zero. This condition respects the non-
radiative nature of the quasistatic limit of Maxwell
equations. Note also that we will use, later on, a PML
boundary condition at the ground surface to simplify
the analysis (see Chapter 4). This assumption is, strictly
speaking, incorrect for the Earth surface, but it is a good
approximation if we consider just the first wave arrivals.
In the following, we will compute the vertical component
of the electrical field at a depth of 5 m.
When Equations (5.9) through (5.11) are coupled to
the poroelastodynamic equations, three types of electro-
magnetic signals are generated by a seismic source. We
use the following nomenclature with Roman numerals
hereinafter:
i Type I corresponds to electromagnetic signals directly
triggered by a seismic source, which moves the fluid
with respect to the solid phase and therefore to the
body force,

, described earlier, is related to
the moment tensor of the seismic source by (Aki &
Richards, 2002)

F

F r = − M ∇ δ r − r S S ω 5 12

where r S ( x S , z S ) denotes the source position, r is the
observation position or receiver position, S ( ω ) represents
the source time function (written here in the frequency
domain), and δ signifies the delta function. The moment
tensor, M , is a quantity that depends on the source
strength and contains all the information about the
source properties. Importantly, the distance between
the source and the receiver needs to be much greater
than the dimension of the fault plane (Aki & Richards,
2002). This forms the “ localized source ” assumption and
is usually used to characterize seismic sources in the far
field. In earthquake seismology, the seismic response is
inverted to retrieve the position of the source; its seismic
moment, M 0 ; the moment tensor components, M ij ;the
time history of the source perturbation, s ( t ); its dominant
frequency, f c ; and to some extent the amount of stress

, as stated in Equation (5.1) (e.g., Gao
& Hu, 2010). Type I signals can be described by a
multipole expansion (the leading term is expected to
be the dipolar or quadrupolar term). Therefore, the
amplitude of type I electromagnetic disturbances
quickly decreases with the distance between the seis-
mic source and the receivers.
ii Type II denotes electromagnetic disturbances gener-
ated at heterogeneities in the electrical, mechanical,
and transport properties (e.g., a geological interface)
during the propagation of

f

the seismic waves