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Depth and Velocity Estimation in a LateralHeterogeneous
Medium
by
Debora CoresUniversidad Simon Bolvar
SIAM Annual meeting- PhiladelphiaJuly 2002
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OUTLINE
Resumen
The Tomography Problem in Lateral Heterogeneous Medium (LHM)
Historical Overview
Discretized Problem
Numerical Approach
Preliminary Numerical Results
Conclusions
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Resumen
In this work we present the depth and velocity estimation
problem as a set ofnon-linear constrained optimization problems,
where few depth and velocityparameters are needed to get the
approximated estructure of the model. Onthe other hand, to solve
this optimization problem we use the recentlydeveloped Spectral
Projected Gradient Method (SPG), that allow us to handlebox
constraints, which correspond to velocity bounds. This
optimizationtechnique is a low optimization technique that only
requieres first orderinformation and have some computational
advantages. We present somenumerical results that show the
computational advantage and performace ofthe SPG for solving this
nonlinear inverse problem.
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PROBLEM: The Tomography Problem in LHM
Minimizef(S) =
12
kT
r
T (S)k
22
T : IR
6n+6m
! IR
nsnrn travel time function,
T = (T
1
(S); T
2
(S); : : : ; T
nsnrnn
(S)) where,
T
i
(S) =
Z
Ray
i
1
V (x; y; z)
dl
i
T
r
2 IR
nsnrn real traveltime vector.
p 2 IR
6n3n parameters inLHM.
n is the number of layers
ns is the number of sources
nr is the number of receivers.
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Historical Overview
In a 2D MediumGauss Newton Approach:Levenberg and Marquardt
methodusing Gauss Seidel with SuccessiveOverrelaxation: T. Bishop
et al, 1985solved the problem in a 2D laterally var-ing
mediaLevenberg and Marquardt Methodwith SVD descomposition : T.Zhu
and L. Brown, 1987; Farra andMadariaga, 1988.Low Storage Opt.
Techniques:Spectral Gradient Method: Castillo,Cores and Raydan ,
2000.
In a 3D MediumGauss Newton Approach:Levenberg and Marquardt
methodusing SVD descomposition: Bishopet al., 1985
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DISCRETIZED PROBLEM: Travel time function
The travel time function ray j reflecting in the layer k, can be
represented asthe sum on i of the straight segments of the ray.
T
i;j;k
(S) =
P
k+1
i=2
l
i;j;k
v
i;j;k
+
P
2n+1
i=2n+2k
l
i;j;k
v
i;j;k
where
l
i;j;k
=
p
(x
i;j;k
x
i1;j;k
)
2
+ (y
i;j;k
y
i1;j;k
)
2
+ (f
i;j;k
f
i1;j;k
)
2
v
i;j;k
= a
i
(x
i;j;k
+ x
i1;j;k
)
2
+ b
i
(y
i;j;k
+ y
i1;j;k
)
2
+ c
i
for k = 1; : : : ; n, i = 2; : : : ; k + 1; 2n+ 2 k; : : : ; 2n+
1 (the straightsegments of the ray) and j = 1; : : : ; ns nr n.
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DISCRETIZED PROBLEM: depth parameters
We only consider dip interfaces in the problem. Assume that for
any interface
i = 1; : : : ; n+ 1 we have three points, let us denote them
by
P
1
i
= (p
1i
; q
1
i
; k
1
i
)
T
, P
2
i
= (p
2i
; q
2
i
; k
2
i
)
T and P 3i
= (p
3i
; q
3
i
; k
3
i
)
T
.
Then the plane generated by these points can be written as:
f
i;j;k
=
1i
(x
i;j;k
p
1i
)
2i
(y
i;j;k
q
1
i
)
3i
+ k
1
i
where,
1
i
= (q
2
i
q
1
i
)(k
3
i
k
1
i
) (k
2
i
k
1
i
)(q
3
i
q
1
i
)
2
i
= (k
2
i
k
1
i
)(p
3i
p
1i
) (p
2i
p
1i
)(k
3
i
k
1
i
)
3
i
= (p
2i
p
1i
)(q
3
i
q
1
i
) (q
2
i
q
1
i
)(p
3i
p
1i
)
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DISCRETIZED PROBLEM: Model parameters
Result: For different values of k1i
; k
2
i
; k
3
i
, i = 1; : : : ; n+ 1, we can
generate any dip plane in
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GENERAL DISCRETIZED PROBLEMMin
12
kT
r
T (S)k
22
=
P
ns
i=1
P
nr
j=1
P
nk=1
(T
r
i;j;k
T
i;j;k
(S))
2
s:t: L S U
where,
T :
