Geoscience Reference
In-Depth Information
change associated with the seismic source (Stein &
Wysession, 2003). One example of the source time
function used in the following is a Gaussian function:
The finite-element computations require a mesh grid,
and for this particular problem, we use a 10 × 10 m rec-
tangular mesh grid on a 1000 × 1000 m domain for all
of the computations, where the grid node spacing is small
with respect to the smallest wavelength of the seismic
wave. This dimension corresponds to the coarsest mesh
for which the solution of the partial differential equation
is still mesh geometry independent. The seismic source is
located at
S
(
x
s
= 650 m,
z
s
= 700 m) with a magnitude of
M
w
=3.0 equivalent to
M
0
= 3.98 × 10
13
Nm, and a source
time function is generated (
t
s
=0.15 ms and
f
c
=30 Hz). At
the four external boundaries of the domain, we apply
a 100 m-thick convolutional perfectly matched layer
(C-PML) so the whole domain is actually 1200 × 1200
m so as to avoid edge effects. The sensors, located at
5 m of depth, are just below the C-PML, and therefore,
the sampled fields are not influenced by the PML bound-
ary condition. The receiver arrangement mimics the
acquisition that would be obtained with triaxial geo-
phones and dipole antennas. They are located at the
8 stations as shown in Figure 5.2.
In order to see the advantages of including the electri-
cal field data versus the seismic data inversion alone, we
generated a noisy data set by adding noise with respect to
the maximum amplitude of the seismic and electrical
signals. The construction of the added noise, as discussed
in the previous sections, is performed as follows: given
a wavelet function,
w
(
t
), and reflectivity series,
r
(
t
), the
noise is computed using the following convolution
product: Noise
t
=
wt rt
=FT
−
1
W
2
f
c
2
2
st
=
exp
−
π
t
−
t
S
5 13
+
∞
S
ω
=
stexp i
ω
t dt
5 14
−∞
where
t
S
denotes the initial time of the source.
In 3D, the seismic moment tensor,
M
, is composed of a
set of nine force couples (Aki & Richards, 2002; Stein &
Wysession, 2003). The tensor
M
is equivalent to
M
0
M
ij
given by
M
xx
M
xy
M
xz
M
yx
M
yy
M
yz
M
zx
M
zy
M
zz
M
0
M
ij
=
M
0
5 15
5.1.3 Modeling noise-free and noisy
synthetic data
We use a 2D finite-element-based numerical problem (in
x
and
z
directions) as a demonstration of the concepts. In
this problem, we couple and solve the poroelastodynamic
wave equations and the electrostatic Poisson equation
for the electrical field in the frequency domain. To do this,
we use the multiphysics modeling package COMSOL
Multiphysics 3.5a and the stationary parametric sol-
ver PARDISO (
http://www.computational.unibas.ch/cs/
scicomp/software/pardiso/;
see Schenk & Gärtner,
2004, 2006, Schenk et al., 2007, 2008). The problem is
solved as follows: (i) first, we solve for the displacement
of the solid phase and the pore fluid pressure using the
poroelastodynamic equations, and (ii) then, we compute
the electrical potential by solving the Poisson equation
coupled to the solution of poroelastodynamic part of
the problem. In both cases, the solution in the time
domain is computed by using an inverse Fourier trans-
form of the solution in the frequency (wave number)
domain (see Jardani et al., 2010).
In the frequency domain, we use the frequency range
ω
R
ω
, where
W
(
) denote the Fourier transforms of
w
(
t
)
and
r
(
t
), respectively, and represents the convolution
product. The amplitude of the noise is scaled to 15% of
the maximum amplitude of the seismic and electrical
time series, with the average of the noise equal to zero.
We chose a Gaussian wavelet to simulate the noisy signal
with a characteristic frequency of
f
c
= 35 Hz. The ref-
lectivity series are composed of randomly distributed
numbers in the range [0, 1], as computed from a pseudo-
random number generator. Note that the reflectivity
noise series here are not related to reflections associated
with the layering of the medium. The two distinctly
generated noise time series are added to the noise-free
seismic and electric data.
ω
) and
R
(
ω
1
100 Hz since the seismic wave operates in this range
and the associated electrical field occurs in the same fre-
quency range (Garambois & Dietrich, 2001). Using this
frequency range, we compute the inverse fast Fourier
transform (FFT
−
1
) to get the time series of the seismic
displacements,
u
x
and
u
z
, and the time series of the two
components of the electrical field,
E
x
and
E
z
.
-
5.1.4 Results
Snapshots of the propagation of the seismic wave and
the resulting electrical potential distribution are shown
in Figures 5.3 and 5.4. Note that the electrical potential
