A 3D ray tracing Approach

Universidad Simón Bolívar

A 3-D Ray Tracing and Inverse Problem Approach

Debora Cores Carrera

SOVG 2004

Noviembre 14-17, 2004


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


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


Tomography Inverse problem

12

2

)(n

i i

ji

j v

lvT

jl2

jl3jl4

jl5

Minimize 22||)(||

2

1vTTobs T

nr vTvTvT ))(),...(()( 1uvl


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


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)


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


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


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


Numerical Results for Ray Tracing

5 layer synthetic model where P-S converted waves velocities are considered


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



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




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



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)(

],,[

],,[

,

],,[

],,[

,


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


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




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+.


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.


Full waveform inversion (for Modified Marmousi model)

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A 3D ray tracing Approach