a 3d ray tracing approach
DESCRIPTION
An Approach for tracing Rys in 3D mediaTRANSCRIPT
Universidad Simón Bolívar
A 3-D Ray Tracing and Inverse Problem Approach
Debora Cores Carrera
SOVG 2004
Noviembre 14-17, 2004
Universidad Simón Bolívar
OUTLINE
The Ray tracing problem (RT)
The Inverse problem approach (IP)
Brief historical overview
The optimization Solver
Numerical Results for RT and IP
Adavantages of the solver
Conclusions
Full waveform inversion
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The Ray Tracing Problem
Minimize
12
2),,(
n
i i
i
X
X
XX v
l
zyxv
dlT
r
s
r
s
),,( zyxv is the group velocity and is the differential dl
along the ray
The number of layers is given by n
2l
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Tomography Inverse problem
12
2
)(n
i i
ji
j v
lvT
jl2
jl3jl4
jl5
Minimize 22||)(||
2
1vTTobs T
nr vTvTvT ))(),...(()( 1uvl
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Brief Historical Overview
Ray Tracing Approaches
Solving Differential Equations Solving Optimization Problems
•P.L. Jacson (1970)
•H. Jacob (1970)
•R.L. Wesson (1970-1971)
•Julian and Gubbins (1970-1971)
•Pereyra et al. (1980)
•Um and Thurber (1987)
•Prothero et al. (1988)
•Mao and Stuard (1997)
•Cores et al. (2000)
Especially in the 70’s More recently
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Brief Historical Overview
Inverse tomography Approaches
Reconstruction Techniques Damped Gauss Newton
•Bishop et al. (1985)
•Chiu et al. (1986)
•Zhu and Brown (1987)
•Farra and Madariaga (1988)
•Dines and Lytle (1979)
•Ivansson (1983)
•Lines and Treitel (1984)
Conjugate Gradient type methods
Pica et al. (1990)
•Michelena et al. (1993)
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The Optimization Approach used for solving both Problems
The Projected Spectral Gradient (PSG) Method (Raydan et al. (2000))
Considered a low cost and storage technique as any of the extensions of conjugate gradient methods (Polak-Ribiere, Hestenes-Stiefel) for a nonlinear optimization problem.
•Local Storage requirements
•Few floating point operations per iteration
•Fast Local Convergence
•Do not require to solve a linear system of equation per iteration
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Projected Spectral Gradient (PSG) Method
}1,min{0 Mkj
)( kk xfg Where: P is the projection on and }/{ uxlx n
1
1. Given , and
2. If , stop
3. Compute and set :
4. If , then
go to step 5
5.
nx 00 0M
0||)(|| kkk xgxP
kTkjkk dgxfxf )(max)( 1
kkkkkkkkkkk xxsggydxx 111 ,,,
kkkkk xgxPd )(
kTk
kTk
k ys
ss1
)(xfs. t. uxl Min
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Advantages of the Optimization Approach
1. The projection over is simple and has low computational cost
2. The objective function is not monotonicaly decreasing because of step lenght and the non monotone line search (step 4). Implying less function evaluations to converge from any initial point (Global convergence).
3. The step size is not the classical choice for the steepest descent method. It speeds up the convergence of the PSG method.
4. The PSG method is related to the Quasi Newton methods. It can be view as a two point method.
5. The PSG method is competitive and many times out performs the extensions of CG methods (CONMIN and PR+)
6. The method converge to the global minimun if we have an stratified and dipped model with constant velocity between layers
k
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Numerical Results for Ray Tracing
5 layer synthetic model where P-S converted waves velocities are considered
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1. 157 recievers and 3 sources randomly genereted at the surface.
2. The average CPU time for 1 shot is 3 s (from different initial rays).
3. Convergence to the
global minimum is obtained.
5 layer synthetic model where P-S converted wave velocities are considered
Numerical Results for Ray Tracing
Universidad Simón Bolívar
1. 157 recievers and 5 sources randomly generated at the surface.
2. Lateral heterogeneous model :
3. We can not guarantee convergence to the global minumum.
4. The average CPU time for the first shot was 50 s (from different initial rays).
T
T
T
c
b
a
cbyaxyxv
)800,700,500,150,150,500,700,800,0(
,)1,1,1,1,1,1,1,1,0(
,)7.1,5.1,3.1,8.0,8.0,3.1,5.1,7.1,0(
,),(
4 layer synthetic lateral heterogeneous model of complex stratigraphy
Numerical Results for Ray Tracing
Universidad Simón Bolívar
Numerical Results for Ray Tracing
We consider a 5 layer ellipsoidal anisotropic medium,where the velocities are
given by the formula:
Where and denote the polar and azimuthal rotation angles in the
layer i, and j=P,SV,SH, i=1,2,...,2n+1
If the medium is an stratified or dipped model, the approach converges to a
global minimum
),cos()sin()sin()cos()sin(
),cos()sin(
),sin()cos()sin()cos()cos(
,))((
)(
))((
)(
))((
)(11
'
'
'
2],,[
2'
2],,[
2'
2,
2'
iiiiiiiii
iiiii
iiiiiiiii
ijyx
i
ijzx
i
ijz
i
ii
zyxz
yxy
zyxx
v
y
v
x
v
z
lv
i i
Universidad Simón Bolívar
Numerical Results for Ray Tracing
5 layer synthetic ellipsoidal anisotropic medium
157 receivers at the surface and 1
source in the origen.
for i=2,...,n+1
sminv
sminv
sminv
smiv
smiv
smiv
iszy
iszx
isz
ipyx
ipzx
ipz
/)3(*801150)(
,/)3(*501000)(
,/)3(*1001400)(
,/*801350)(
,/*501200)(
,/*1001500)(
],,[
],,[
,
],,[
],,[
,
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Numerical Results for the tomography inversion
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Numerical Results for the tomography inversion
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Numerical Results for the tomography inversion
We fixed CPU time and the
grid size (500x500) to observe
the reduction in the gradient
and the residual during that
period of time
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We used a (20x20)
grid size to measure
the precision of PR+
and PSG
Real velocities Initial velocities
The initial velocities have an error of 50% from the real velocities
Final velocities (PSG) Final velocities (PR+)
The quality of the solution by the 2 methods are almost the same
Numerical Results for the tomography inversion
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Numerical Results for the tomography inversion
1. SIRT has low computational cost per iteration but requires too many iterations and therefore consumes more CPU time.
2. PSG, PR+ and CONMIN reach quickly a good precision (10e-03) when compared to SIRT and Gauss Newton methods.
3. Gauss Newton is fast, in CPU time, for very small size of the grid.
4. The PSG and PR+ methods outperform CONMIN for very large problems.
5. The PSG method is always slightly faster , in CPU time, than PR+.
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Conclusions
1. The PSG method is a simple, global and fast method for large scale problems (Example: inversion and ray tracing).
2. The PSG method reachs quickly to a good precision (For example 10e-02 or 10e-03).
3. The PSG method only requires firts order information.
4. The PSG method does not require exhastive line search which implies less function evaluations per iteration.
5. We also used the method for Full waveform inversion, obtaining very good results.
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Full waveform inversion (for Modified Marmousi model)