Comparison of travltime computation and ray tracing methodsBernard Law1, Daniel Trad1

1. Consortium for Research in Elastic Wave Exploration Seismology (CREWES), University of Calgary

Motivation

Travel times and ray paths of the propagation ofseismic body wave in heterogenous media are used inseismic tomography, imaging and inversion processes.In this study, we review the seismic ray theory, basicprinciples of the fast marching, wavefront constructionand paraxial method. We analyze their differences andsimilarities to investigate the effectiveness of thesemethods in refraction tomography and seismic imaging.

Seismic ray theory

High frequency approximation of the solution ofelastodynamic equation leads to solutions in differentforms. For kinematic ray tracing, the solution leads tothe eikonal equation and the ray equations.I Elastodynamic equation:

σij ,j + fi = ρul (1)

I Eikonal equation:(∂T∂x

)2+(∂T∂y

)2+(∂T∂z

)2=

1c2 (2)

I Ray equations:

d~xds

= c~q (3)

d~qds

= ~5[1c] (4)

Grid based methods

Finite difference solution to eikonal equation(Vidale 1988):Eikonal equation:(∂T

∂x

)2+(∂T∂z

)2= s(x , z)2 (5)

Average finite difference approximation of∂T∂x

and∂T∂z

:

∂T∂x

=12(t0 + t2 − t1 − t3) (6)

∂T∂z

=12(t0 + t1 − t2 − t3) (7)

Substitude (6) and (7) into (5):

t3 = t0 +√

2(hs)2 − (t2 − t1)2

(8)

Fast marching method and upwind scheme(Sethian and Popovici 1999):

Grid based method

I Fast marching method ( Sethian and Popovici1999 )

Sethian and Popovici 1999

Marmousi model Travel time

Refraction ray tracing

Ray based method

I Ray shooting method ( Cerveny and Hron 1980 )

I Loop on take off angles θi, φi

Initial values at first depth stepd~xds

= (sinθicosφi, sinisinφi, cosθi)

~q =1

c(Xs,Ys,Zs

d~xds

I Loop on steps until end is reached1: Solve ODE (2) for next ~x2: Solve ODE (3) for next ~q

I Next stepI Next take off angle

I Wavefront construction method (Vinje 1993)I Shoot out rays of equal time step. End points of these rays

form the new wavefrontI Uses new wavefront to propagate new rays for another time

stepI Geometric spreading can be computed from ratio of

cress-sectional areaaI Multiple values can be computed when ray paths cross

Ray based method

I Wavefront construction method

I Paraxial method (Beydoun and Keho 1987)

I Dynamic ray tracing equations

Qi =∂xi

∂γ;Pi =

∂pi

∂γ(9)

dQi

ds= C(, k)QkPi + cPi (10)

dPi

ds=

∂2

∂xi∂xk(1c)Qk (11)

I Paraxial ray tracing equationsdδxi

ds= c,kδxkpi + cδpi (12)

dδpi

ds=

∂2

∂xi∂xk(1c)xk (13)

I Geometrical spreading

dσ = Q1Q2dγ1dγ2 (14)

I Paraxial ray traveltimes

T (~x+~h) = T (~x)+T,j(~x)hj+12

T,jk(~x)hjhk

(15)T,j = Pj (16)

T,jk = PjnQ−nk1 (17)

Comparison of ray tracing methods

I Source is placed at the depth of 2500mI Receivers are placed at the surface

Rays from WFC and paraxial methods can cross-over in area withcomplex velocity structure and result in multi-arrivals at the samegrid point. Multi-arrivals can be used for different branches oftraveltime including most energetic arrival.

Comparision of ray tracing methods

Caustics are removed in WFC to output minimum travel times

Traveltimes overlaid on finite difference synthetic shot record

I Travel times from WFC and fast marching are almost identicalI Travel times from paraxial method agree with the other methods

at most locations except at locations where rays diverge

Summary

I Fast marching methodI Advantages: Unconditional stable. Can handle turning rays.

Does not have shadow zones problem.I Disadvantages: Does not compute ray paths and amplitude.

Cannot compute multi-arrivals.I Application: refraction tomography

I Wavefront construction methodI Advantages: Stable if appropriate velocity smoothing is

applied; however accuracy can decrease with increasingsmoothing. Can handle turning rays. Does not have shadowzones problem. Can compute multi-arrivals and amplitude.Faster than fast marching method, if large time step size isused.

I Application: refraction tomography and depth imaging

I Paraxial methodI Advantages: More accurate traveltime interpolation in the

vicinity of central ray than classical ray shooting method.Compute multi-arrivals and amplitude.

I Disadvantages: Cannot handle turning ray.I Application: depth imaging,gaussian beam migration

Acknowledgements

We thank the sponsors of CREWES and NSERC(Natural Science and Engineering Research Council ofCanada) through the grant CRDPJ 461179-13 forsupporting this project.

www.crewes.org