ray tracing and velocity estimation

Universidad Simón Bolívar

Objective : Find Oil

Mathematical methods used for Knowing the earth model and then finding oil.


Velocity estimation as an Inverse Problem and the Ray Tracing

ProblemDebora Cores Carrera

[email protected]

CIMPA 2012

Caracas-Venezuela

Universidad Simón BolívarOUTLINE

The ray tracing problem (RT)

Expressions for the velocities

Brief historical overviewEllipsoidal velocity (homogeneous anisotropic medium)

Fullwave inversion (FI)

The inverse Problem to solve

Models for solving the inverse problem (IP)Seismic Reflection tomography inversion (SRTI)

Constant velocity (homogeneous medium)

The optimization SolverNumerical Results for RT, SRTI and FIConclusions

velocity (heterogeneous medium)


The Inverse Problem

Estimate some parameters that define the subsoil in order to describe the layers of the earth model.


The Inverse Problem

Unknown: the velocities ijv Known: seismic line (travel times for each source

and receiver)


The Travel Time Function or Ray Tracing Problem (RT)

Minimize12

2),,(),,(

n

i i

iX

X

XX v

lzyxv

dlzyxTr

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


Seismic Reflection Tomography Inverse problem (SRTI)

12

2)(

n

i i

ji

j vlvT

jl2

jl3jl4

jl5 Tnr vTvTvT ))(),...(()( 1

Minimize 22||)(||

21)( vTTvf obs

uvl ££

))(()()( obsT

T TvTvJvf


Fullwave Inverse problem (FI)

rxsx

,);,();,()( obssrsr xtxpxtxpvp

)(),('21 vpvp

),()(' 1 vpVvp p

MinimizeMinimize

where,

p

pV

ijijij uvl

is the pressure wavefield in the receiver position at time t, generated by a source , is the velocity

wavefield matrix , and the matrix is a covariance operator.

v


Fullwave Inverse Problem (FI)

);,();,()(

1.)();,()(

122 ss

s xtxsxtxpx

xt

xtxpxv

0);0,( sxxp

0);,();0,(t

xtxpxxp ss

is a function described by the source, is the density of the medium.The full wave equation is solved with the staggered-grid finite difference scheme (Luo and Schuster 1991, Savic 1995)

)(x

The pressure wavefield is a function must satisfy the wave equation:

);,( sxtxp


Expresions for the velocityHomogeneous isotropic

medium: velocity does not change with position or direction.

.ctev

1v2v

Homogeneous anisotropic medium: velocity changes with direction.

v

Heterogeneous isotropic medium: the velocity changes with psotion.

zxzyxv 23),,(For example:

For example: 2D ellipsoidal


),cos()sin()sin()cos()sin(

),cos()sin(

),sin()cos()sin()cos()cos(

,))((

)())((

)())((

)(11

'

'

'

2,

2'

2,

2'

2,

2'

iiiiiiiii

iiiii

iiiiiiiii

ijy

i

ijx

i

ijz

i

ii

zyxzyxy

zyxx

vy

vx

vz

lv

General Ellipsoidal Velocity


A general travel time equation12

22

,

2

2,

2

2,

2

))(()'(

))(()'(

))(()'(),,(

n

i ijy

i

ijx

i

ijZ

iXX v

yv

xv

zZYXT r

s

iiiii yxy cossin'

izyxz iiiiiiii cossinsinsincos'

where,

izyxx iiiiiiii sincossincoscos'


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



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)



Fullwave inversion approaches

Conbining travel time inversion and wave equation

techniques

Using multiscale descomposition techniques for findind long wavelength components first and then

recursively refine them to get shorter scales.

•Pratt and Goultry (1991)

•Zhou et al. (1995)

•Charara (1996)

•Korenaga et al. (1997)

•Primiero (2002)

• Dessa and Pascal (2003)

•Kolb, Collino and Lailly (1986)

•Pica, Diet and Tarantola (1990)

•Bunks et al. (1995)


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

•Do not require to solve a linear system of equation per iteration


Projected Spectral Gradient (PSG) Method

)( kk xfgWhere: P is the projection on and }/{ uxlx n

1. Given , and

2. If , stop

3. Compute and set :

4. If , then

go to step 5

5.

nx0 0 0M

0||)(|| kkk xgxP

kTkjkk dgxfxf )(max)( 1

kkkkkkkkkkk xxsggydxx 111 ,,,

kkkkk xgxPd )(

kTk

kTk

k ysss

1

)(xfs. t. uxlMin


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

Numerical Results for Ray Tracing in an Isotropic Homogeneous Medium

Cores, Fung and Michelena, “A fast and global two point low storage optimization technique for tracing rays in 2D and 3D isotropic media”, Journal of Applied geophysics 45, 273-278, 2000.


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

cba

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 in an Isotropic Heterogeneous Medium

Cores, Fung and Michelena, “A fast and global two point low storage optimization technique for tracing rays in 2D and 3D isotropic media”, Journal of Applied geophysics 45, 273-278, 2000.

Universidad Simón BolívarNumerical Results for Ray Tracing in an Ellipsoidal Anisotropic Medium

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

Theorem: If the medium is an stratified or dipped model, this optimization model 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

ijy

i

ijx

i

ijz

i

ii

zyxzyxy

zyxx

vy

vx

vz

lv

i i

Universidad Simón BolívarNumerical Results for Ray Tracing in an Ellipsoisal Anisotropic Medium

5 layer synthetic ellipsoidal anisotropic medium

157 receivers at the surface and 1

source in the origen.

for i=2,...,n+1

sminvsminvsminv

smivsmivsmiv

isy

isx

isz

ipy

ipx

ipz

/)3(*801150)(,/)3(*501000)(,/)3(*1001400)(

,/*801350)(

,/*501200)(

,/*1001500)(

,

,

,

,

,

,

Cores and Loreto, “A generalized two point ellipsoidal anisotropic ray tracing for converted waves”, Optimization and Engineering 8, 373-396, 2007.


We used a 20x20

grid size to measure

the precision of PR+

and GSG

Real velocities Initial velocities

The initial velocities have an error of 50% from the real velocities

Final velocities (GSG) Final velocities (PR+)

The quality of the solution by the 2 methods are almost the same

Numerical Results for the Homogeneous Tomography Inversion

Castillo, Cores and Raydan, “Low cost optimization techniques for solving the nonlinear seismic reflection tomography”, Optimization and Engineering 1, 155-169, 2000.

Universidad Simón BolívarNumerical Results for Ellipsoidal Anisotropic Tomography Inversion

kmvvkm izijZ 5)()(2.0 ,

kmvvkm ixijx 5)()(2.0 ,

Bounds on the unknown parameters: For i=2,…,2n+1

kmvvkm iyijy 5)()(2.0 ,

3010 i

92 i

Stopping criteriun:06

210))(( kkk XXfXP

M=8 (SPG)


Square mesh Radial mesh


i2 1.5 1.5 1.7 1.69 1.9 1.86 1.49 1.69 1.93 2 1.91 2.3 2.22 2.5 2.37 1.81 2.06 2.214 3 2.86 2.8 2.85 3.3 3.21 3.01 2.81 3.555 2.7 2.83 2.9 2.85 3.1 3.19 2.73 2.88 2.916 1.8 1.89 2 2.07 2.3 2.42 2.01 2.27 2.597 1.3 1.29 1.6 1.61 1.8 1.84 1.29 1.6 1.81

izv )(iyv )(ixv )(

Square mesh ns=2 nr=5 Radial mesh ns=5 nr=16

apixv )( ap

iyv )( apizv )( ap

ixv )( apiyv )( ap

izv )(


inversion

i

2 5 5.84 20 21.49 8.46 20.13 7 7.36 15 15.46 4.93 13.464 3 6 25 20.31 2 25.135 3 5.78 25 19.14 6 20.436 7 7.47 15 14.29 4.47 127 5 5.01 20 18.94 7.64 19.86

apii i

api

api

api

Square mesh ns=2 nr=15 Radial mesh ns=5 nr=16

.


Numerical Results on the Anisotropic Tomography Inversion

• This is a highly nonlinear problem that has many solutions, so regularization of the problem and priori information is required.

• The SPG optimization method gets a good precision for estimating the velocities using small number of rays.

• The problem for obtaining a better estimate of the polar angle vector is not the optimization scheme used, it depends on the seismic data acquisition.

• Increasing the number of rays, the error in the velocity vector and in the azimuthal angle vector can be reduced, but the CPU time increase.

• None of the mesh distribution used here give enough information for obtaining a good estimate of the polar angle vector ( ). May be the travel time information is not appropiate for estimating fracture orientation.

Meza and Cores, “Seismic velocity estimation and fracture orientation in orthorombic media”, in preparation

Universidad Simón BolívarNumerical Results for Full Waveform Inversion (for Modified Marmousi model)


Numerical results for Fullwave Inversion


Numerical Results for Fullwave Inversion

The solution obtained is sufficiently close to the global minimum. Even though this model does not represent real data, we hope our methodology will be accurate enough in real cases with a broader frequency spectrum.

Zeev, Savasta and Cores, “Non monotone spectral projeted gradient method applied to full waveform inversion”, Geophysical Prospecting 54, 1-10, 2006.


Conclusions• To estimate velocities from seismic data can be done by

solving a non linear least squares problem (inverse problem) via tomography formulation or full wave formulation.

• Poor approximations of the wave propagation velocities in the earth models could introduce distorsions on the final images of the subsoil that can have enormous economic impact.

• Any optimization technique that solves the non linear least squares problem could be used to estimate velocities from seismic data.

• Since the inverse problems presented here are considered large scale optimization problems, any low cost and storage optimization techniques are desirable in these cases.


Conclusions• The PSG method is a simple, global and fast method for

large scale problems (Example: seismic inversion and ray tracing).

• The PSG method reachs quickly to a good precision (For example 10e-02 or 10e-03).

• The PSG method only requires firts order information.• The PSG method does not require exhastive line search

which implies less function evaluations per iteration.• We also used the SPG method for Full waveform inversion,

obtaining very good results.


Ellipsoidal Velocity0)),(( 21 UWIG

Where and are the polar and azimuthal phase angles. Solving the eigenvalue problem:

)sin()cos( 121

)sin()sin( 122

)sin( 13

1 2

ljlj

ijklik CG ggå=

=3

1,


Ellipsoidal Velocity

)(sin)(sin))((

1

)(sin)(cos))((

1)(cos))((

1)(

1

12

22

2,

12

22

2,

12

2,

2

];[

];[

iiijNMO

iiijNMO

iijZi

ZY

ZX

v

vvv

Approximating the eigenvalues of the Christoffel equation and using the Byun Transformation, Contreras et al. in 1997 obtained an ellipsoidal group velocity:

is the group velocity in the layer delimited by interfaces i-1 and i. is the i-th component of the normal move out velocity in the symmetry plane [X,Z] with wave propagation mode j=P,SV or SH .

ivijNMO ZX

v )( ,];[


Ellipsoidal velocity

i

iiiiiii l

yxfyxf )()()cos( 1,11,

1

i

iiiii l

yyxx 21

21

1)()(

)sin(

21

21

12

)()()sin(

iiii

iii

yyxxyy

21

21

12

)()()cos(

iiii

iii

yyxxxx


A More general ellipsoidal velocity

Tiiiap

Tiii zyxRRzyx ),,()',','(

ii

ii

pRcos0sin

010sin0cos

The distance segment between two consecutive points at interfaces i-1 and i,

)',','( iii zyx

1000cossin0sincos

ii

ii

aR


Ellipsoidal Velocity

2,

2

2,

2

2,

2

))(())(())((11

][][ ijNMO

i

ijNMO

i

ijZ

i

ii YZXZv

yv

xv

zlv

1iii zzz1iii yyy

For j=P,SV,SH and i=2,…,2n+1

where,

1iii xxx


Numerical results for the tomography inversion


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


Advantages of the Optimization Approach1. The projection over is simple and has low computational

cost

2. The objective function does not decrease monotonicaly 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 Secant 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 of the ray tracing problem, if we have an stratified and dipped model with constant velocity between layers

k


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+.

Conclusions

ray tracing and velocity estimation

Documents

velocity changes

group velocity

velocity wavefield matrix

time t

xthe pressure wavefield

wave equation

earth model

velocities ijv