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where
V / V represents a relative variation of volume
during the passage of the seismic wave. The compressibil-
ity of the fluid is defined, in isothermal conditions, as
Δ
Seismoelectric conversion
Seismic source
Ground surface
β f = 1
1
V
V
∂p T
K f =
1 99
Seismic wave
Medium I
Propagation
and K f denotes the bulk modulus (in Pa). Therefore, the
pressure perturbation is related to the displacement by
Interface
Medium II
Oscillating dipole moment
p =
K f θ
=
K f
u
1 100
Figure 1.20 The seismoelectric conversion (called sometimes the
interface response) results from the generation of an unbalanced
source current density at an interface during the passage of a
seismic wave. The divergence of the source current density at
the interface is mathematically similar to an oscillating dipole
moment generated at the interface in the first Fresnel zone.
The star represents the seismic source.
Equation (1.100) corresponds to Hooke
'
s law for a
fluid and is valid for
1 .
Equation (1.100) corresponds to a constitutive equation.
To get the field equation for the pressure perturbation p ,
we need to combine Equation (1.100) with a conserva-
tion equation. Newton
small deformation
θ
s law provides the required con-
servation equation for the momentum
'
interested in the information content associated with
these conversions. However, the magnitude of these con-
versions decreases very quickly with the distance from
the interface. In Chapters 2 and 3, we will provide a gen-
eral theory of the coseismic field and seismoelectric
conversion in saturated and partially saturated cases,
respectively, and we provide in Section 1.4.2 a simple
modeling approach based on the acoustic approximation.
A third effect corresponds to the EM fields generated
directly by a seismic source. We will see that these EM
fields (especially the electric component) can be combined
with the radiated seismic fields and used to localize the seis-
mic source and characterized the moment tensor of the
source. These effects will be fully explored in Chapter 5.
−∇
p =
ρ f u
1 101
where ρ f denotes the density of the fluid (assumed to be
constant) and ü corresponds to the acceleration of the
material. As we need to express the divergence of the
fluid displacement in terms of fluid pressure, we want
to take the divergence of Equation (1.101). This yields
2
2 p = ρ f
−∇
t 2
u
1 102
2 p
2 p = ρ f
K f
1 103
t 2
2 p
c f
1
2 p
t 2 =0
1 104
1.4.2 Simple modeling with the acoustic
approximation
1.4.2.1 The acoustic approximation in a fluid
We consider a fluid with its viscous effects considered to
be negligible in the momentum conservation equation.
The propagation of a seismic wave in this fluid can be
described in terms of a pressure perturbation p ( r , t ) (true
pressure minus the equilibrium pressure) or in terms of a
fluid displacement u ( r , t ). The volume strain θ is related
to the displacement by
where
1 2
K f
ρ f
c f =
1 105
Equation (1.104) corresponds to the wave equation
for the fluid pressure with the velocity given by
Equation (1.105). If we wish to determine the displace-
ment of the fluid, we can, for instance, consider a wave
with a sinusoidal time dependence given by
θ = Δ
V
V
=
u
1 98
p r , t = p ω r exp i ω t
1 106
 
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