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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