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a localized seismic source that is described only by a
moment tensor. Because the electrokinetic coupling
mechanism requires the computation of the relative
water flow vector with respect to the solid phase, we
use the Biot poroelastic theory to describe the elastody-
namic part of the problem (see Chapter 2).
The second point that needs to be discussed is the
choice of the inversion method. Whenever the informa-
tion content of a data set is questioned, it is better to use
Bayesian theory associated with a Markov chain Monte
Carlo (MCMC) sampler techniques rather than a deter-
ministic-based approach (Mosegaard, 2011 and referen-
ces therein and Chapter 4). However, classical Monte
Carlo samplers are not very efficient in retrieving the
posterior probability density of model parameters,
because of the slow convergence of the chain. In the
present work, we will use the adaptive Metropolis algo-
rithm (AMA; see Haario et al., 2001, 2004) for the inverse
modeling, and we will discuss the advantages of this
approach. We will also show how electrical data alone
can be used to localize the seismic source and the hetero-
geneities that are present along the wave fronts between
the source and the receivers.
Joint inversion
smaller uncertainty
All electric-based
source models
All seismic-based
source models
Broad uncertainty
True model
(a)
All electric-based
source models
All seismic-based
source models
5.1.2 Forward modeling
To test our forward and inverse modeling, we created a
two-layer earth model comprised of layers L1 and L2
(L1 corresponds to the upper layer; see Figure 5.2).
The material properties of L1 and L2 are summarized
in Table 5.1. The mechanical property values reported
in Table 5.1 yield a P-wave velocity
c
p
= 4322 m s
−
1
and
an S-wave velocity
c
s
= 2721 m s
−
1
for L1 and a P-wave
velocity
c
p
= 1932 m s
−
1
and an S-wave velocity
c
s
=
1317 m s
−
1
for L2. For the seismic source, we used a
moment tensor source corresponding to a double couple
on a normal fault (90 strike from the north direction and
45 dip angle from the horizontal plane). The seismic
stations and electrodes are colocated close to the Earth
(b)
Figure 5.1
Information content of seismic and electromagnetic
disturbances.
a)
Informative joint inversion case. The joint
inversion of seismic and electrical data may help in reducing
the uncertainty in the joint inversion problem by providing
information that is not contained in the seismic data alone.
b)
Noninformative joint inversion case. All the seismic-based
inverted models are contained in the set of electrical-based
model. This can happen when the signal-to-noise ratio is
much greater for the electrical data than for the seismic data.
relative to the uncertainty in the model parameters
derived by using seismic data alone (Figure 5.1a). There
is, however, the possibility (shown conceptually in
Figure 5.1b) that the electrical information is noninfor-
mative. This is the case where the posterior probability
densities for the model parameters determined from
the seismic data alone are contained inside the posterior
probability densities for the same model parameters
derived from the electrical data alone. Such a case is pos-
sible since the level of noise in the seismic and electro-
magnetic data can be vastly different.
To determine the benefits of incorporating electrical
data, we developed a simple finite-element model to
compute the seismic and electrical fields associated with
s
surface, at a depth of 5 m (as explained in the following
text, we impose a perfectly matched layer (PML) just at
the top surface of the system). The modeling domain
itself corresponds to a two-dimensional slab (1.2 × 1.2)
km
2
. The reference position, O(0,0), is located at the
upper left corner of this domain.
We will use the following assumptions in this section:
(i) the porous material is isotropic, (ii) fully water-
saturated with a connected porosity, (iii) the pore fluid
(water) is a viscous Newtonian fluid and the skeleton is
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