'&
$%
Depth and Velocity Estimation in a LateralHeterogeneous Medium
by
Debora CoresUniversidad Simon Bolvar
SIAM Annual meeting- PhiladelphiaJuly 2002
1
'&
$%
OUTLINE
Resumen
The Tomography Problem in Lateral Heterogeneous Medium (LHM)
Historical Overview
Discretized Problem
Numerical Approach
Preliminary Numerical Results
Conclusions
2
'&
$%
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.
3
'&
$%
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.
4
'&
$%
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
5
'&
$%
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.
6
'&
$%
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
)
7
'&
$%
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
'&
$%
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 :