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Horizontal electric ield Ex ( ×10 -7 V m -1 )
Vertical electrical ield Ez ( ×10 -8 V m -1 )
Time source
Time source
0.1
0
-0.1
0.1
0
-0.1
P-S
P
P-S
ST01
ST01
S
S
P
0.2
0
-0.2
0.1
0
-0.1
ST02
ST02
0.1
0
-0.1
0.1
0
-0.1
ST03
ST03
0.1
0
-0.1
0.1
0
-0.1
SC
ST04
ST04
DS
0.1
0
-0.1
0.1
0
-0.1
SC
ST05
ST05
SC
DS
DS
0.1
0
-0.1
0.1
0
-0.1
ST06
ST06
0.1
0
-0.1
0.1
0
-0.1
ST07
ST07
0.1
0
-0.1
0.1
0
-0.1
ST08
ST08
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Time (s)
Time (s)
Figure 5.6 Horizontal and vertical electric field components (the vertical component is taken at a depth of 5 m below the ground
surface). We can see the P
P-waves
produced at the interface between the two materials. The occurrence of the direct field (DS) indicates the source time t s = 0.15 s (see the
horizontal component of the electric field). We can also see the seismoelectric conversion (SC) from the interface L1
-
P and S
-
S coseismic electrical response and the coseismic responses associated with the P
-
S- and S
-
-
L2 at t = 0.22 s.
5.2 Joint inversion of seismic
and seismoelectric data
of the inversion solution nonuniqueness is observed in
the posterior probability distribution of the model para-
meters when dealing with noisy data.
We begin this inversion approach with the general
equation used in the inversion process that relates model
parameter estimations with data from measurements.
This equation takes the form of
5.2.1 Problem statement
Next, the goal is to use seismic and electrical measure-
ments to localize the position of a seismic source associated
with hydraulic fracturing and to characterize its moment
tensor through a full waveform inversion of the seismo-
grams and electrograms. Because we are interested in
the information content of the two types of geophysical
signals, we use a stochastic, Bayesian approach to map
the distribution of errors in the observed data into the
model parameter space (e.g., see Jardani et al., 2010,for
an example related to seismoelectric modeling). This sto-
chastic approach is able to provide information that will
allow the nonuniqueness of the inverse problem to be
statistically characterized. The statistical characterization
d = Km
5 16
where d denotes the N -vector containing N separate
observations (measurements), m is an M -vector contain-
ing M model parameters, and K symbolizes an N × M
matrix that represents Green ' s functions of the system.
The joint Bayesian solution to the inverse problem is
based on combining information from seismograms,
electrograms, and some prior knowledge of the problem.
 
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