Geoscience Reference
In-Depth Information
m
∗
denotes the computed and scaled (3
M
× 1) vector of
model parameters, and
a
refers to a (3
M
× 1) vector with
only zeroes.
Step 6.
Because the resulting inverted model with
compactness,
m
∗
, is depth weighted, we need to unscale
the model solution to produce the inverted (3
M
×1)
model vector,
m
,by
Thresholding does bias the solution; however, we expect
the solution to be in the vicinity of Hole 9 without
components spread throughout the block volume.
The grid used for the previously described gradient-
based approach, combined with compactness, is actually
pretty coarse (see Figure 5.21). This is to reduce the
computational effort and time needed to find a prelimi-
nary position for the source. As shown in the next
section (see Figures 5.22, 5.23, 5.24, 5.25, and 5.26),
the inversion of the data leads to a source localized in
the vicinity of Hole 9 (as expected). Once this is done,
we switch our inversion to the GA on a refined grid
located in the vicinity of the solution found by the
gradient-based approach. The GA used for this second
inversion phase is described in the next section.
T
m
=
m
x
,
m
y
,
m
z
5 51
m
x
,
y
,
z
=
Ω
−
1
x
,
y
,
z
m
x
,
y
,
z
5 52
This model vector solution is in terms of the current
dipole moment expressed in Am. For compactness,
we initially use a small support parameter of
=10
−
12
.
β
Then for this value, we compute the
“
best
”
value of the
regularization parameter,
, using the L-curve approach.
If the solution is not compact enough, we multiply the
previous value of
λ
5.3.5.3 Inversion phase 2: GA approach
Using the results of the gradient-based approach, we
apply a single dipole GA-based search through a new
and finer kernel matrix with only 360 positions
(Figure 5.27). A single dipole is presumed to represent
the overall effect of water flow during the leak part of
experiment. This assumption is expected to be good
enough to locate the centroid of the leak position within
the block volume where the leak is occurring. The GA is
used as follows: a population of candidate solutions is
used to find the solution of the inverse problem. This
population has to evolve toward solutions that minimize
the data misfit function:
by 10 and repeat the process until
no additional compactness is achieved.
Step 7.
The solution found in step 6 represents a
mathematical solution to Equation (5.46) that represents
the minimum of the objective function. This solution has
the smallest mismatch between the measured data and
the synthetic data forward computed using the inverted
model and as such contains dipoles with finite moments
in every dipole point position in the model vector. How-
ever, we wish to use only the main contributing compo-
nents of this model that best represents the electrical
effects. Based on this, the computed model is thresholded
to keep only the main dipoles that explain most of the
solution. A threshold is applied to the model vector to
zero all of the dipoles with moment values that were
below the final value of
β
P
d
m
=
d
−
Km
2
5 53
where
2
refers to the L2 norm. The evolution starts
from a population of randomly generated individual
solutions in the volume defined by the deterministic
gradient-based algorithm described earlier. For each
generation, the goodness of fit is evaluated through
Equation (5.49), and multiple individuals are stochasti-
cally selected from the current population and modified
to form a new population. This new population is then
used at the next iteration. The process is continued until
a predetermined number of generations have been
reached or a satisfactory data misfit has been reached
for the population.
Themodel vector,
m
, contains 360 × 3 = 1080 elements
(the number 3 represents the three components of the
current dipole moment vector). The 360 positions of
the kernel matrix are positioned on three concentric
. The existence of a large num-
ber of low-magnitude dipoles in the model vector that are
produced by the gradient inversion process represents a
purely mathematical solution to the source inversion.
Therefore, it is considered that these small dipoles do
not represent real sources and are not likely to be phys-
ically present during the event generation process.
Thresholding these widely spread small dipoles removes
the nonphysical and mathematical only contributions to
the solution, allowing the comparison of the principal
elements of the inversion with the real data. This helps
to realistically quantify the inversion process and source
localization error but does not affect the main compo-
nents of the solution. Indeed, we expect the solution
to be rather compact, and not broadly distributed.
β
