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posterior probability
density, we use the AMA to find the optimal proposal
distribution of the model parameters (Haario et al.,
2001). This algorithm is based on the Metropolis
In order to explore the
best
factor for a Gaussian target distribution and a Gaussian
proposal distribution (Gelman et al., 1996). The candi-
date model vector,
m
, is accepted with the acceptance
-
Hastings algorithm with a symmetric Gaussian proposal
distribution centered at the current model, m i , with a
covariance,
probability
=min 1; π m d
π m
i 1 ;
α m
m
5 23
i
1
i , that changes during the sampling process
in such a way that the sampling efficiency increases over
time (Haario et al., 2004).
The AMA is described as follows. First, we assume that
the model vector, m , is sampled by several states, (
d
C
If the candidate model vector is accepted, we consider
m
i 1 . This Bayesian framework
is used to invert the model vector defined at the begin-
ning of this section.
i =
m
; otherwise,
m
i =
m
0 ,
m
,
RMax ), where
0 corresponds to the initial state
m
,
,
m
m
of
and R Max is the maximum number of realizations.
Each candidate point,
m
, is sampled from a pseudoran-
domGaussian proposal distribution,
m
5.2.3 Results with noise-free data
A first-order solution is usually preferred to start the
AMA sampler. There are many ways to find a first-order
localization of the seismic source while ignoring the het-
erogeneity in the seismic velocity or in the electrical
resistivity distribution. In Figure 5.7, we used a cross-
correlation approach using only the electrical field distri-
bution recorded at the 8 stations located at the ground
surface. The cross-correlation is performed using the
approach discussed by Revil et al. (2001).
To begin, using a first-order test solution, we applied
the cross-correlation density algorithm developed by
Revil et al. (2001) and Iuliano et al. (2002). The electrical
field, E ( r ), due to a single dipole with moment d ,is
written as Er = d
N
(0.1), with a mean
i - 1 , and a covariance matrix
point at the present point,
m
given by
0 ,
i = C
if
i
n 0
C
5 22
i + s n ε n I n , f i > n 0
s n K
i =
where
I n denotes the n -dimensional identity matrix,
K
0 ,
i - 1 ) represents the covariance matrix,
Cov(
ε n is a
small positive number that prevents the covariance
matrix from becoming singular,
m
,
m
0 stands for the initial
covariance matrix, and s n = (2.4) 2 / n signifies a parameter
that depends only on the dimension of the vector
C
n
(Haario et al., 2001). This yields an optimal acceptance
m R
G , where G is Green
'
s function of
True position of the seismic source
Maximum of the cross-correlation function
0
0.9
0.8
Figure 5.7 Dipole occurrence probability
(DOP) of the source using a normalized
cross-correlation of a dipole source with the
signals obtained at the time of the source at
the eight stations located at the ground
surface (each pixel is 50 m in x by 1.5 m
in z ). The maximum of the DOP (denoted
by the star symbol, ) can be used to provide
a prior localization of the source assuming
that the subsurface has a constant
(unknown) electrical conductivity. The
filled star, , denotes the true position of the
seismic source. ( See insert for color
representation of the figure .)
500
0.7
0.6
1000
0.4
0.2
1500
0
100
200
300
400
500
600
700
800
900
Position
x
(m)
 
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