by haiqiang lan and zhongjie zhang - · pdf fileby haiqiang lan and zhongjie zhang ......

A High-Order Fast-Sweeping Scheme for Calculating First-Arrival

Travel Times with an Irregular Surface

by Haiqiang Lan and Zhongjie Zhang

Abstract Topography-dependent eikonal equation (TDEE) formulated in a curvi-linear coordinate system has been recently established and is effective for calculatingfirst-arrival travel times in an Earth model with an irregular surface. In previous work,the Lax–Friedrichs sweeping scheme used to approximate the TDEE viscosity solu-tions was only first-order accurate. We present a high-order fast-sweeping scheme tosolve the TDEE with the aim of achieving high-order accuracy in the travel-time cal-culation. The scheme takes advantage of high-order weighted essentially nonoscilla-tory (WENO) derivative approximations, monotone numerical Hamiltonians, andGauss Seidel iterations with alternating-direction sweepings. It incorporates high-orderapproximations of the derivatives into the numerical representation of the Hamiltoniansuch that the resulting numerical scheme is formally high-order accurate and inheritsfast convergence from the alternating sweeping strategy. Extensive numerical examplesare presented to verify its efficiency, convergence, and high-order accuracy.

Introduction

In seismic exploration, the problem of irregular topog-raphy makes the fine reconstruction of underground struc-tures rather difficult at the crustal and, especially, the basinscales. In seismic exploration for oil and gas, seismologistsencounter similar problems of undulating topography alongthe line of a survey. In processing seismic data, seismictravel time is a very important constraint in static correction(e.g., Lawton, 1989; Coppens, 2006), Kirchhoff migration(e.g., Gray and May, 1994; Symes et al., 1994), location ofearthquakes (e.g., Kennett and Engdahl, 1991; Aki and Ri-chards, 2002; Bouchon and Vallée, 2003; Doser, 2006), andseismic tomography (e.g., Hole and Zelt, 1992; Piromalloand Morelli, 1997; Doser et al., 1998; Zhang and Thurber,2003; Badal et al., 2004; Zhao et al., 2004; Trampert andvan der Hilst, 2005; Xu et al., 2006, 2010; Wang, 2011).A major challenge for seismologists in these areas is thedifficulty of obtaining a stable and accurate travel time.To date, methods for dealing with this problem are essen-tially based on unstructured grids (e.g., Fomel, 1997; Kim-mel and Sethian, 1998; Sethian, 1999; Sethian andVladimirsky, 2000; Alkhalifah and Fomel, 2001; Rawlinsonand Sambridge, 2004; Qian et al., 2007a,b; Kao et al.,2008; Lelièvre et al., 2010). However, a number of wave-field modeling methods are based on structured grids (e.g.,Fornberg, 1988; Tessmer et al., 1992; Hestholm and Rudd,1994; Nielsen et al., 1994; Tesssmer and Kosloff, 1994;Hestholm and Rudd, 1998; Zhang and Chen, 2006; Appeloand Petersson, 2009; Lan and Zhang, 2011a,b), making itimportant to calculate seismic travel times (for inversion

and migration courses) on these grids. Recently, we devel-oped a method for calculating first-arrival travel times bysolving a topography-dependent eikonal equation (TDEE;Lan and Zhang, 2013), which takes advantage of a trans-formation from Cartesian to curvilinear coordinates. In thismethod, the Lax–Friedrichs sweeping scheme used toapproximate the TDEE viscosity solutions only has first-or-der accuracy (Lan and Zhang, 2013). High-order accuraciesare required to compute other quantities to an acceptableprecision, such as the amplitude and the distribution of ve-locities (e.g., Cerveny et al., 1977; Qian and Symes, 2002b;Luo and Qian, 2011; Luo et al., 2011). We herein advance ahigh-order fast-sweeping scheme to solve the TDEE, inte-grating the advantages of high-order weighted essentiallynonoscillatory (WENO) derivative approximations, mono-tone numerical Hamiltonians, and Gauss Seidel iterationswith alternating-direction sweepings.

First, we briefly summarize the TDEE in a curvilinearcoordinate system. Then, we describe the solutions of theTDEE (in a curvilinear coordinate system) using the high-order Lax–Friedrichs fast-sweeping method and present anumber of numerical examples to demonstrate the accuracyand efficiency of our proposed method. Last, we draw someconclusions.

A Topography-Dependent Eikonal Equation in theCurvilinear Coordinate System

In the Cartesian coordinate system and for an isotropicmedium, the eikonal equation in 2D can be written as

2070

Bulletin of the Seismological Society of America, Vol. 103, No. 3, pp. 2070–2082, June 2013, doi: 10.1785/0120120199

�∂T∂x

�2

��∂T∂z

�2

� S2�x; z�; (1)

where T�x; z� is the time of the first arrival of the seismicenergy that propagates from a point source through amedium that has a slowness distribution of s�x; z�.

To deal with irregular surfaces, we perform a transfor-mation from the Cartesian coordinate system to the curvilin-ear coordinate system (see Appendix A for details). Usingequation (A2) in Appendix A, the eikonal equation (1)can be written in the curvilinear coordinate system as

��qx∂q � rx∂r�T�2 � ��qz∂q � rz∂r�T�2 � s2�q; r�; (2)

where q and r are the curvilinear coordinates, qx denotes∂q�x; z�=∂x (and similarities with other variables). Usingrelationships (A3) to replace the metric derivativesqx; qz; rx; rz, equation (2) can be written as

�1

J2�x2r � z2r�

��∂T∂q

�2

−2

J2�xqxr � zqzr�

∂T∂q

∂T∂r

��1

J2�x2q � z2q�

��∂T∂r

�2

� s2�q; r�: (3)

Equation (3) is known as the TDEE (Lan and Zhang, 2013)and is a member of the Hamilton–Jacobi equations (e.g., Se-thian and Vladimirsky, 2001, 2003; Alkhalifah, 2002; Qianet al., 2007a,b; Alton and Mitchell, 2008). This expressiongives the travel time T�x; z� of a ray that passes through apoint �x; z� in a medium with slowness s�x; z� and can bewritten in a more compact form such that

A�∂T∂q

�2

� B∂T∂q

∂T∂r � C

�∂T∂r

�2

� s2�q; r�; (4)

where A � 1J2 �x2r � z2r�, B � − 2

J2 �xqxr � zqzr�, and C �1J2 �x2q � z2q�. The parameters A, B, and C are topography de-pendent. When the free surface is flat, xr and zq become zero,xq and zr tend to 1, and J � 1, so the eikonal equation (4)reduces to the classical eikonal equation.

A High-Order Lax–Friedrichs Sweeping Scheme forSolving the Topography-Dependent Eikonal Equation

The TDEE is an elliptically anisotropic eikonal equation(e.g., Sethian and Vladimirsky, 2001; Qian et al., 2007a,b),although the medium is isotropic. Errors will occur in calcu-lating travel times if the fast-marching method designed forisotropic eikonal equations are extended to anisotropic eiko-nal equations without taking the differences between theminto account, as demonstrated in a number of studies (e.g.,Sethian and Vladimirsky, 2001, 2003; Qian et al., 2007a,b;Alton and Mitchell, 2008).

A paraxial formulation establishing a relationship be-tween the characteristic direction and the travel-time gradientdirection has been proposed to tackle such anisotropic eiko-nal equations in seismic studies (e.g., Qian and Symes, 2001,2002a,b, 2003). Almost at the same time, Sethian and Vla-dimirsky (2001, 2003) developed a so-called ordered upwindschemes for the static Hamilton–Jacobi equations. As aniterative method for Hamilton–Jacobi equations, the fast-sweeping method has been recently gaining attention. Tsaiet al. (2003) applied the fast-sweeping method to a class ofstatic Hamilton–Jacobi equations and derived explicit up-dated formulas so that the Gauss–Seidel-based sweepingstrategy can be performed easily. Kao et al. (2004) extendedthe fast-sweeping method to approximate the viscosity solu-tions of arbitrary static Hamilton–Jacobi equations by useof the Lax–Friedrichs monotone numerical Hamiltonian.Zhang et al. (2006) extended the first-order Lax–Friedrichsfast-sweeping scheme to a higher order, based on WENOSchemes (Jiang and Shu, 1996; Jiang and Peng, 2000). Here,we introduce the high-order Lax–Friedrichs sweepingscheme (Zhang et al., 2006), as a substitute for the first-ordermethod (Kao et al., 2004), to solve the TDEE for calculatingthe first-arrival travel times (Lan and Zhang, 2013). In thefollowing, we rearrange the TDEE into the Lax–FriedrichsHamiltonian form, and then present the discretization formu-las and the calculation steps for the solution of the TDEEusing the Lax–Friedrichs fast-sweeping scheme.

Lax–Friedrichs Hamiltonian of the Topography-Dependent Eikonal Equation

Consider the rectangular domain �qmin; qmax� × �rmin; rmax�in the curvilinear coordinate system with a uniform dis-cretization �qi; rj�; i � 1; 2;…; m1; and j � 1; 2;…; m2;where qi � �i − 1�hq � qmin, rj � �j − 1�hr � rmin, hq ��qmax − qmin�=�m1 − 1�, and hr � �rmax − rmin�=�m2 − 1�.

The topography-dependent eikonal equation (3) or (4) isa special case of a static Hamilton–Jacobi equation and canbe written as

�H�Tq; Tr� � s�q; r� �q; r�∈Ω;T�q; r� � g�q; r� �q; r�∈Γ∈Ω; (5)

where Ω denotes the computational domain; Γ denotessource points within the computational domain; Tq � ∂T

∂q,

Tr � ∂T∂r, H�Tq; Tr� �

��A�∂T∂q�2 � B ∂T

∂q∂T∂r � C�∂T∂r�2

qis the

Hamiltonian of equation (4); g�q; r� denotes initial valuesfor the source; the meanings of A, B, and C are the sameas defined in equation (4); and s�x; z� is the slowness ofthe medium.

To discretize the Hamiltonian H�Tq; Tr�, we use theLax–Friedrichs numerical Hamiltonian, which is the simplestamong all monotone numerical Hamiltonians, such that

High-Order Fast-Sweeping Scheme for Calculating First-Arrival Travel Times 2071

HLF�u−;u�;v−;v��H�u−�u�

2;v−�v�

2

�−1

2αq�u�−u−�

−1

2αr�v�−v−�; (6)

where u � ∂T=∂x; v � ∂T=∂z; u� and v� are the forwardand backward difference approximations of ∂T=∂x and∂T=∂z, respectively; and αq and αr are artificial viscositiesthat satisfy

αq ≥ maxD≤u≤EF≤v≤K

jH1�u; v�j; αr ≥ maxD≤u≤EF≤v≤K

jH2�u; v�j:

Here, Hi�u; v� is the partial derivative of H with respect tothe ith argument, �D;E� is the value range for u�, and �F;K�is the value range for v�.

Kao et al. (2004) constructed a first-order Lax–Friedrichssweeping scheme for general static Hamilton–Jacobi equa-tions, which we used to solve the topography-dependenteikonal equation (5):

Tn�1i;j �

�1

αq

hq� αr

hr

��si;j −H

�Ti�1;j −Ti−1;j

2hq;Ti;j�1 −Ti;j−1

2hr

�

�αqTi�1;j�Ti−1;j

2hq�αr

Ti;j�1�Ti;j−1

2hr

�: (7)

The superscripts ofT are not included because they depend onthe sweeping directions of the alternating Gauss Seidel typeiterations. We always use the most recent value for T in theinterpolation stencils according to the Gauss Seidel typeiteration.

To construct a high-order scheme, we need to approxi-mate the derivatives Tq and Tr with high-order accuracy. Weuse the popular WENO approximations, developed by Zhanget al. (2006). To illustrate the feasibility of this approach, wetake the third order rather than the fifth order WENO approx-imations; one certainly can replace the third-order WENOwith the fifth or even high-order WENO approximations.The interpolation stencil for the third-order WENO schemeis shown in Figure 1. �Tq�i;j and �Tr�i;j denote approxima-tions for Tq and Tr at the grid point (i, j), respectively. Theapproximation for Tq at the grid point (i, j) when the wind“blows” from left to right is written as

�Tq�−i;j � �1 − w−��Ti�1;j − Ti−1;j

2h

�

� w−

�3Ti;j − 4Ti−1;j � Ti−2;j

2h

�; (8)

where

w− � 1

1� 2r2−;

r− � ε� �Ti;j − 2Ti−1;j � Ti−2;j�2ε� �Ti�1;j − 2Ti;j � Ti−1;j�2

; (9)

in which ε is a small number used to prevent the denomi-nator becoming zero. The approximation for Tq at thegrid point (i, j) when the wind blows from right to left iswritten as

�Tq��i;j � �1 − w��Ti�1;j − Ti−1;j

2h

�

� w�

�−Ti�2;j � 4Ti�1;j − 3Ti;j

2h

�; (10)

where

w� � 1

1� 2r2�;

r� � ε� �Ti�2;j − 2Ti�1;j � Ti;j�2ε� �Ti�1;j − 2Ti;j � Ti−1;j�2

: (11)

�Tr�−i;j and �Tr��i;j are defined in a similar way.Next, we incorporate these high-order derivative ap-

proximations (equations 8–11) into monotone numericalHamiltonians. According to the definitions of �Tq�−i;j and�Tq��i;j, Ti;j − h�Tq�−i;j can be considered as an approxima-tion to Ti−1;j, while Ti;j � h�Tq��i;j can be considered asan approximation to Ti�1;j. Similarly, Ti;j − h�Tr�−i;j andTi;j � h�Tr��i;j can be considered as approximations toTi;j−1 and Ti;j�1, respectively. Replacing Ti−1;j, Ti�1;j,Ti;j−1, and Ti;j�1 with these approximations in equation (7),we derive the following high-order scheme, such that

Tn�1i;j �

�1

αq

hq�αr

hr

��si;j−H

��Tq�−i;j��Tq��i;j2

;�Tr�−i;j��Tr��i;j

2

�

�αq

2Toldi;j �hq��Tq��i;j−�Tq�−i;j�

2hq

�αr

2Toldi;j �hr��Tr��i;j−�Tr�−i;j�

2hr

�(12)

which may be written more compactly as

Tn�1i;j �

�1

αq

hq�αr

hr

��si;j−H

��Tq�−i;j��Tq��i;j2

;�Tr�−i;j��Tr��i;j

2

�

�αq

�Tq��i;j−�Tq�−i;j2

�αr

�Tr��i;j−�Tr�−i;j2

��Told

i;j ;

(13)

where Tn�1i;j denotes the to-be-updated numerical solution for

T at the grid point (i, j) and Toldi;j denotes the current (old)

value for T at the same grid point. When the WENO approx-imations for the derivatives �Tq�−i;j, �Tq��i;j, �Tr�−i;j, and�Tr��i;j, in equation (13) are calculated using equations (8)–(11), we always use the newest value for T in the interpola-tion stencils, according to the Gauss–Seidel type iteration. Of

2072 H. Lan and Z. Zhang

course, because we did not update Ti;j yet, Toldi;j is used in

equations (8)–(11).

Steps for Calculating the First-Arrival Travel TimesUsing the Lax–Friedrichs Fast-Sweeping Scheme

The Lax–Friedrichs Hamiltonian gives a solution depen-dent on all its neighbors in all dimensions, so we need tocarefully specify the values of points outside the computa-tional domain, otherwise huge errors will be introduced forthe points on the computational boundary and then propagateinto the computational domain. Here, we impose the follow-ing conditions on the four sides of the boundary to combineextrapolation, maximization, and minimization and to calcu-late the values for points outside the computational domain,as proposed by Kao et al. (2004):

8>>><>>>:

Tnewii;j � min�max�2Tii�1;j − Tii�2;j; Tii�2;j�; Told

ii;j�;Tnewm1�1−ii;j � min�max�2Tm1−ii;j − Tm1−1−ii;j; Tm1−1−ii;j�; Told

m1�1−ii;j�;Tnewi;jj � min�max�2Ti;jj�1 − Ti;jj�2; Ti;jj�2�; Told

i;jj�;Tnewi;m2�1−jj � min�max�2Ti;m2−jj − Ti;m2−1−jj; Ti;m2−1−jj�; Told

i;m2�1−jj�;(14)

where ii � −1; 0 and jj � −1; 0.The Lax–Friedrichs fast-sweeping method for equa-

tion (5) consists of four steps that include (1) initialization;(2) alternating sweeps; (3) enforcing computational boun-dary conditions; and (4) convergence test. These steps canbe summarized as follows:

• Initialization. Exact or interpolated values T0i;j are assigned

at grid points on or near the source Γ, and these values arefixed throughout the iterations. Large positive values areassigned at all other grid points; these should be larger thanthe maximum value of the true solutions, and they are up-dated in the iterations.

• Alternating sweepings. At iteration n� 1, we calculateTn�1i;j according to equation (13) at all grid points (qi, rj),

1 ≤ i ≤ m1, and 1 ≤ j ≤ m2, except for those that have as-signed values, and update Tn�1

i;j only when its value issmaller than its previous value Tn

i;j. As stated above, thisprocess needs to be performed in alternating sweeping di-rections, which means that it needs four different sweeps inthe 2D case: (a) from lower left to upper right i � 1 : m1,j � 1 : m2; (b) from lower right to upper left i � m1 : 1,j � 1 : m2; (c) from upper left to lower right i � 1 : m1,j � m2 : 1; and (d) from upper right to lower lefti � m1 : 1, j � m2 : 1. In general, in the d-dimensioncase, we need 2d alternating sweeps.

• Enforcing computational boundary conditions. After eachsweep, we enforce computational boundary conditions ac-cording to equation (14), trivially modified depending onthe boundary we are using.

• Convergence test. Given the convergence criterion δ, thealgorithm converges and stops if kTn�1 − TnkL1

≤ δ.

Accuracy and Efficiency Tests

We perform three groups of numerical experiments, withincreasing topographical complexity, to test the accuracy andefficiency of our proposed method for the computation of

first-arrival travel times in the presence of surface topogra-phy. Initially, we consider a group of two models with a flatsurface: the first is a homogeneous medium, and the secondis the SEG/EAGE Marmousi model, which is used to test themethod on media with strong lateral/vertical velocity varia-tions. The second group includes a couple of models withcosine-curved surfaces and a source located at a differentdepth in each model. The third group comprises a model witha much more complex surface topography. We compute theanalytical solutions (see Appendix B for details) and measurethe errors under grid refinement in each case to explore theconvergence of our numerical solutions. In the followingexamples, we consider the iteration convergent if the L1error of the difference between two successive iterationskTn�1 − TnkL1 is less than 10−6. The formula for the L1 er-ror (Qian and Symes, 2002a) is

L1�t� �Pm1

i�1

Pm2

j�1 jtana − tftjm1m2

; (15)

where m1 and m2 are grid point quantities in the horizontaland vertical directions, respectively; tana is the travel timefrom the analytical method; and tft is the travel time fromthe finite-difference scheme. A single iteration includes fouralternating sweepings for 2D problems. Generally, the algo-rithm converges within a few hundred iterations. The veloc-ity is 2000 m=s in all of the homogeneous models presentedin this paper. Initial values are assigned in a circle, whichincludes the source point, and the radius of the circle is2h where h is the mesh size (i.e., the number of grid points

k-2 k k+1 k+2k-1

Figure 1. A stencil of the third-order WENO scheme, φ � q; r.


is fixed in the initialization throughout the mesh-refinementstudy).

Group 1

A flat-surface model with the size of 1:6 km × 1:2 km isconsidered here. We discretize the model using three meshsizes: 160 × 120, 320 × 240, and 640 × 480. The source islocated at coordinate (0.8 km, 1.1 km). We calculate first-arrival travel times using the low- and high-order Lax–Friedrichs fast-sweeping schemes, respectively, and comparethe results with the exact solutions. To save space here, onlythe travel-time contours of the numerical and analyticalsolutions on the 160 × 120 mesh are shown (Fig. 2a), andcomparisons of the L1 errors between the numerical and ex-

act solutions for the different mesh sizes are summarized inTable 1. The L1 errors of the high-order scheme are hundredsof times smaller than those of the low-order method, andonly about twice as many iterations are required.

Next, we use the SEG/EAGE Marmousi model to testthe method on media with strong lateral/vertical velocity var-iations. Because the analytical solution is unavailable, we usethe results calculated by the fast-sweeping method on a finer(2560 × 1920) mesh to approximate the analytical solution.The L1 errors for the full domain are reported in Table 2. Thehigh-order method clearly achieves much better accuracythan the first-order scheme on the same mesh. The numericaland analytical solution travel-time contours on the 640 × 480

mesh (see Fig. 2b) show good agreement between the ana-lytical and high-order scheme solutions, but the results fromthe first-order scheme reveal remarkable errors.

Group 2

The surface of the model used is described by z�x� �1:0� 0:2 cos�1:5πx� km, x∈�−1 km; 1 km�, representingtopography with combinations of one hill and two depres-sions. The computational domain extends to a depth of−1 km. The 100 × 100 mesh boundary-conforming gridsare shown in Figure 3. We focus on the performance of theproposed method for the first-arrival travel-time calculationin a model with more complicated topography and seismicsources at different depths. We place the seismic source atsix different depths (η � 1:1, 0.7, 0.3, −0:1, −0:5, and−0:9 km), and the source is located at (−0:2 km; η).We com-pute the analytical solutions in each case (see Appendix Bfor details) and measure the errors between the calculations

Figure 2. Travel-time contours (in seconds) for the group 1models, (a) homogeneous medium model (b) Marmousi model.The source is located at (0.8 km, 1.1 km) for both models. The com-putations are performed on the 160 × 120 mesh for (a) and the640 × 480 mesh for (b). The black solid line represents the analyti-cal solution, while the blue and red dashed lines represent thenumerical solutions obtained using the first- and high-order Lax–Friedrichs sweeping schemes, respectively.

Table 1L1 Errors and Iteration Numbers for Numerical ResultsCalculated by First- and High-Order Lax–Friedrichs

Sweeping Schemes for the Homogeneous Flat-Surface Model,with a Source Located at (0.8 km, 1.1 km)

MeshL1 Error

(First Order)L1 Error

(High Order)

IterationNumber

(First Order)

IterationNumber

(High Order)

160 × 120 1:0087 × 10−2 8:6768 × 10−5 38 64320 × 240 5:9143 × 10−3 5:4398 × 10−5 64 117640 × 480 3:3920 × 10−3 3:1793 × 10−5 113 222

Table 2L1 Errors and Iteration Numbers for the First- and High-OrderLax–Friedrichs Sweeping Schemes in the Marmousi Model

with a Source Located at (0.8 km, 1.1 km)

MeshL1 Error


(High Order)

IterationNumber

(First Order)

IterationNumber

(High Order)

160 × 120 1:8107 × 10−2 4:3745 × 10−3 40 74320 × 240 1:0973 × 10−2 1:4187 × 10−3 67 130640 × 480 6:2147 × 10−3 5:3735 × 10−4 118 239


using the Lax–Friedrichs fast-sweeping schemes and theexact solutions under grid refinement. The L1 errors of thefirst- and high-order Lax–Friedrichs sweeping schemes andthe respective iteration numbers are shown in Table 3. Thehigh-order scheme gives much smaller errors than does thelow-order method on the same grids.

Figure 4a–f shows the travel-time contours for each test.The numerical solutions of the high-order method on the100 × 100 mesh match very well with the analytical solu-tions for each case and are much better than the results fromthe first-order method on a finer mesh (800 × 800). Theresults of the first-order method on the sparse mesh(100 × 100) reveal notable errors, which should not be useddirectly in light of this consideration. Figure 5a–f displaysthe distributions obtained by comparing travel times calcu-lated by the high-order Lax–Friedrichs sweeping scheme andthe analytical solutions on the 100 × 100 mesh for each test.The errors are larger where the curvature of the topography isgreatest. Figure 6a–f shows a comparison of the numericaland analytical solutions of travel times recorded at the irregu-lar surface for each test, which confirms our conclusions.

Group 3

To test the proposed scheme further, we consider an-other model with a much more complex surface topography.

X1 km−1

Figure 3. The 100 × 100 mesh boundary-conforming grids forthe group 2 models. The inverted black triangles represent thereceivers located at the irregular surface, with horizontal distancesbetween −1 and 1 km.

Table 3L1 Errors and Iteration Numbers for the First- and High-Order Lax–Friedrichs Sweeping

Schemes in the Group 2 Models with Sources Located at Different Depths

Source Location MeshL1 Error


(High Order)Iteration Number(First Order)

Iteration Number(High Order)

(−0:2, 1.1) 100 × 100 2:9857 × 10−2 3:1788 × 10−3 56 51200 × 200 1:9682 × 10−2 1:4779 × 10−3 93 88400 × 400 1:1723 × 10−2 5:3604 × 10−4 161 163800 × 800 6:6531 × 10−3 — 290 —

(−0:2, 0.7) 100 × 100 2:1935 × 10−2 1:6522 × 10−3 44 69200 × 200 1:3471 × 10−2 6:6823 × 10−4 74 122400 × 400 8:1473 × 10−3 1:6854 × 10−4 130 228800 × 800 4:5590 × 10−3 — 238 —

(−0:2, 0.3) 100 × 100 1:9476 × 10−2 9:2037 × 10−4 39 48200 × 200 1:2158 × 10−2 4:1163 × 10−4 64 80400 × 400 7:1047 × 10−3 6:7634 × 10−5 112 147800 × 800 4:0577 × 10−3 — 203 —

(−0:2, −0:1) 100 × 100 2:0109 × 10−2 1:1333 × 10−4 38 40200 × 200 1:2047 × 10−2 1:6507 × 10−5 61 69400 × 400 6:9463 × 10−3 3:9539 × 10−6 104 125800 × 800 4:0412 × 10−3 — 185 —

(−0:2, −0:5) 100 × 100 2:0161 × 10−2 1:2355 × 10−3 43 42200 × 200 1:2622 × 10−2 4:1841 × 10−4 71 73400 × 400 7:2589 × 10−3 4:7859 × 10−5 121 133800 × 800 4:1329 × 10−3 — 218 —

(−0:2, −0:9) 100 × 100 2:7176 × 10−2 1:0876 × 10−3 49 49200 × 200 1:3382 × 10−2 4:3949 × 10−4 81 88400 × 400 7:6445 × 10−3 4:8919 × 10−5 142 164800 × 800 4:4425 × 10−3 — 256 —


Figure 4. Travel-time contours (in seconds) for the group 2 models with different source depths: (a) (−0:2 km, 1.1 km), (b) (−0:2 km,0.7 km), (c) (−0:2 km, 0:3 km), (d) (−0:2 km, −0:1 km, (e) (−0:2 km, −0:5 km), (f) (−0:2 km, −0:9 km). The black solid line representsthe analytical solution, the blue and red dashed lines represent the numerical solutions obtained using the first- and high-order Lax–Friedrichssweeping schemes on the 100 × 100 mesh, and the green dashed line represents the first-order Lax–Friedrichs sweeping scheme numericalsolutions on the 800 × 800 mesh.

Figure 5. Absolute differences, or errors, (in seconds) between travel times calculated using the high-order Lax–Friedrichs sweepingscheme and the analytical solutions on the 100 × 100 mesh for the group 2 models with different source depths, (a) (−0:2 km, 1.1 km),(b) (−0:2 km, 0.7 km), (c) (−0:2 km, 0.3 km), (d) (−0:2 km, −0:1 km), (e) (−0:2 km, −0:5 km), and (f) (−0:2 km, −0:9 km).


The size of the model is 1:6 km × 1:32 km. The surfaceundulation is between 1.1 km and 1.32 km and representstopographies with combinations of two hills and two depres-sions. We discretize the model with meshes of 160 × 120,320 × 240, 640 × 480, and 1280 × 960 (the last is used onlyto solve the first-order scheme). Figure 7 shows the boun-dary-conforming grids of the 160 × 120 mesh. The source islocated at (0.8 km, 1.1 km). We test the accuracy of thenumerical methods by comparing them with the analyticalsolutions (see Appendix B for details) under the mesh refine-ment, and the results are shown in Table 4. The contour plotsof the numerical solutions and the analytical results areshown in Figure 8. Figure 9 shows the error distributions forthe high-order scheme on the 160 × 120 mesh. Comparing

the numerical and analytical solutions of travel times re-corded at the irregular surface (Fig. 10) provide further sup-port to our conclusions.

Concluding Remarks

We have presented a TDEE in a curvilinear coordinatesystem that uses a boundary-conforming grid and maps arectangular grid onto a curved one. We use a high-order fast-sweeping method to solve the TDEE in the curvilinear coor-dinate system. A general procedure is given to incorporatethe high-order approximations into monotonic numericalHamiltonians so that the first-order sweeping scheme canbe extended to a high-order scheme. Extensive numerical

(a) (b)

(c) (d)

(e) (f)

Figure 6. Comparison of analytical and numerical solution travel times at the irregular surface for the group 2 models with differentsource depths, (a) (−0:2 km, 1.1 km), (b) (−0:2 km, 0.7 km), (c) (−0:2 km, 0.3 km), (d) (−0:2 km, −0:1 km), (e) (−0:2 km, −0:5 km), and(f) (−0:2 km, −0:9 km). The black solid line represents the analytical solution, the blue and red dashed lines represent the numerical solu-tions obtained using the first- and high-order Lax–Friedrichs sweeping schemes on the 100 × 100mesh, and the green dashed line representsthe first-order Lax–Friedrichs sweeping scheme numerical solutions on the 800 × 800 mesh.


examples demonstrate that the high-order method yieldshigh-order accuracy in the solution and fast convergenceto viscosity solutions of the topography-dependent equation,which will be helpful in the real-world processing of seismicdata acquired at irregular surfaces, for static correction, pre-stack migration, and tomography.

Data and Resources

All data used in this study are obtained from publishedsources listed in the references.

Acknowledgments

We would like to thank Editor in Chief Diane I. Doser, AssociateEditor Michel Bouchon, and two anonymous referees for their constructivecomments and helpful suggestions, which led to a significant improvementof the early manuscript. Financial support from the Chinese Academy ofSciences (KZCX2-YW-132), the National Natural Science Foundation ofChina (41074033, 40721003, 40830315, and 40874041), the Important Na-tional Science and Technology Specific Projects (2008ZX05008-006), andthe Ministry of Science and Technology of China (SINOPROBE-02-02) aregratefully acknowledged.

X

1.6 km0

Figure 7. The 160 × 120 mesh boundary-conforming grids forthe group 3 model. The inverted black triangles represent the receiv-ers located at the irregular surface, with horizontal distances be-tween 0 and 1.6 km.

Table 4L1 Errors and Iteration Numbers for the First- and High-order

Lax–Friedrichs Sweeping Schemes in the Group 3 Modelwith a Source Located at (0.8 km, 1.1 km)

MeshL1 Error


(High Order)

IterationNumber(FirstOrder)

IterationNumber(HighOrder)

160 × 120 1:1560 × 10−2 4:5952 × 10−4 48 65320 × 240 6:9548 × 10−3 1:0930 × 10−4 82 119640 × 480 3:9093 × 10−3 2:1213 × 10−5 146 2241280 × 960 2:2930 × 10−3 — 288 —

Figure 8. Travel-time contours (in seconds) for the group 3model with the source at (0.8 km, 1.1 km). The solid black linerepresents the analytical solution, the blue and red dashed linesrepresent the numerical solutions obtained using the first- andhigh-order Lax–Friedrichs sweeping schemes on the 160 × 120mesh, and the green dashed line represents the numerical solu-tions of the first-order Lax–Friedrichs sweeping scheme on the1280 × 960 mesh.

Figure 9. Absolute differences, or errors (in seconds) betweentravel times calculated using the high-order Lax–Friedrichs sweep-ing scheme and the analytical solutions on the 160 × 120 mesh forthe group 3 model.


References

Aki, K., and P. G. Richards (2002). Quantitative Seismology, Theory andMethods, W.H. Freeman and Co., San Francisco, California, 685 pp.

Alkhalifah, T. (2002). Traveltime computation with the linearized eikonalequation in anisotropic media, Geophys. Prospect. 50, no. 4, 373–382.

Alkhalifah, T., and S. Fomel (2001). Implementing the fast marching eikonalsolver: Spherical versus Cartesian coordinates, Geophys. Prospect. 49,165–178.

Alton, K., and I. M. Mitchell (2008). Fast marching methods for stationaryHamilton–Jacobi equations with axis-aligned anisotropy, SIAM J.Numer. Anal. 47, 363–385.

Appelo, D., and N. Petersson (2009). A stable finite difference method forthe elastic wave equation on complex geometries with free surfaces,Commun. Comput. Phys. 5, 84–107.

Badal, J., U. Dutta, F. Seron, and N. Biswas (2004). Three-dimensionalimaging of shear wave velocity in the uppermost 30 m of the soilcolumn in Anchorage, Alaska, Geophys. J. Int. 158, 983–997.

Bouchon, M., and M. Vallée (2003). Observation of long supershear ruptureduring the magnitude 8.1 Kunlunshan earthquake, Science 301,no. 5634, 824–826.

Cerveny, V., I. A. Molotkov, and I. Psencik (1977). Ray Method inSeismology, Univ. Karlova Press, Praha, Czech Republic.

Coppens, F. (2006). First arrival picking on common-offset trace collectionsfor automatic estimation of static corrections, Geophys. Prospect. 33,1212–1231.

Doser, D. I. (2006). Relocations of earthquakes (1899–1917) in south-central Alaska, Pure Appl. Geophys. 163, no. 8, 1461–1476.

Doser, D. I., K. D. Crain, R. B. Mark, K. Vladik, and C. G. Matthew (1998).Estimating uncertainties for geophysical tomography, ReliableComput. 4, no. 3, 241–268.

Fomel, S. (1997). A variational formulation of the fast marching eikonalsolver, in SEP-95, Stanford Exploration Project, 127–147.

Fornberg, B. (1988). The pseudo-spectral method: Accurate representationin elastic wave calculations, Geophysics 53, 625–637.

Gray, S. H., and W. P. May (1994). Kirchhoff migration using eikonalequation traveltimes, Geophysics 59, 810–817.

Hestholm, S., and B. Ruud (1994). 2D finite-difference elastic wavemodeling including surface topography, Geophys. Prospect. 42,371–390.

Hestholm, S., and B. Ruud (1998). 3-D finite-difference elastic wavemodeling including surface topography, Geophysics 63, 613–622.

Hole, J., and B. Zelt (1995). 3-D finite-difference reflection travel times,Geophys. J. Int. 121, 427–434.

Hvid, S. (1994). Three dimensional algebraic grid generation, Ph.D. Thesis,Technical University of Denmark.

Jiang, G., and D. Peng (2000). Weighted ENO Schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21, 2126–2143.

Jiang, G., and C. Shu (1996). Efficient implementation of weighted ENOschemes, J. Comput. Phys. 126, 202–228.

Kao, C. Y., S. Osher, and J. Qian (2004). Lax–Friedrichs sweepingscheme for static Hamilton–Jacobi equations, J. Comput. Phys. 196,367–391.

Kao, C. Y., S. Osher, and J. Qian (2008). Legendre-transform-based fastsweeping methods for static Hamilton–Jacobi equations on triangu-lated meshes, J. Comput. Phys. 227, 10,209–10,225.

Kennett, B., and E. Engdahl (1991). Traveltimes for global earthquakelocation and phase identification, Geophys. J. Int. 105, 429–465.

Kimmel, R., and J. A. Sethian (1998). Computing geodesic paths onmanifolds, Proc. Natl. Acad. Sci. Unit. States Am. 95, 8431–8435.

Lan, H., and Z. Zhang (2011a). Three-dimensional wave-field simulation inheterogeneous transversely isotropic medium with irregular freesurface, Bull. Seismol. Soc. Am. 101, no. 3, 1354–1370.

Lan, H., and Z. Zhang (2011b). Comparative study of the free-surface boun-dary condition in two-dimensional finite-difference elastic wave fieldsimulation, J. Geophys. Eng. 8, 275–286.

Lan, H., and Z. Zhang (2013). Topography-dependent eikonal equation andits solver for calculating first-arrival travel times for an irregularsurface, Geophys. J. Int., doi: 10.1093/gji/ggt036.

Lawton, D. C. (1989). Computation of refraction static corrections usingfirst-break traveltime differences, Geophysics 54, 1289–1296.

Lelièvre, P. G., C. G. Farquharson, and C. A. Hurich (2010). Computingfirst-arrival seismic travel-times on unstructured 3-D tetrahedralgrids using the fast marching method, Geophys. J. Int. 184, no. 2,885–896.

Luo, S., and J. Qian (2011). Factored singularities and high-order Lax–Friedrichs sweeping schemes for point-source travel-times andamplitudes, J. Comput. Phys. 230, 4742–4755.

Luo, S., J. Qian, and H. K. Zhao (2012). High-order schemes for 3-Dtravel-times and amplitudes, Geophysics 77, no. 2, T47–T56.

Nielsen, P., F. If, O. Berg, and O. Skovgaard (1994). Using the pseudospec-tral method on curved grid for 2D acoustic forward modeling,Geophys. Prospect. 43, 321–341.

Piromallo, C., and A. Morelli (1997). Imaging the Mediterranean uppermantle by P-wave travel time tomography, Ann. Geofisc. 40, no. 4,963–979.

Qian, J., and W. W. Symes (2001). Paraxial eikonal solvers for anisotropicquasi-P travel-times, J. Comput. Phys. 173, 1–23.

Qian, J., and W. W. Symes (2002a). Finite-difference quasi-P travel-timesfor anisotropic media, Geophysics 67, 147–155.

Qian, J., andW.W. Symes (2002b). An adaptive finite-difference method fortraveltimes and amplitudes, Geophysics 67, 167–176.

Qian, J., and W. W. Symes (2003). A paraxial formulation for the viscositysolution of quasi-P eikonal equations, Comput. Math. Appl. 46, 1691–1701.

Qian, J., Y. T. Zhang, and H. K. Zhao (2007a). A fast sweeping methodfor static convex Hamilton–Jacobi equations, J. Comput. Phys. 31,237–271.

Qian, J., Y. T. Zhang, and H. K. Zhao (2007b). Fast sweeping methodsfor eikonal equations on triangular meshes, SIAM J. Numer. Anal.45, 83–107.

Rawlinson, N., and M. Sambridge (2004). Wave front evolution in stronglyheterogeneous layered media using the fast marching method,Geophys. J. Int. 156, no. 3, 631–647.

Sethian, J. A. (1999). Fast marching methods, SIAM Rev. 41,199–235.

Sethian, J. A., and A. Vladimirsky (2000). Fast methods for the eikonal andrelated Hamilton–Jacobi equations on unstructured meshes, Proc.Natl. Acad. Sci. Unit. States Am. 97, 5699–5703.

Figure 10. Comparison of analytical and numerical solutiontravel times at the irregular surface for the group 3 model withthe source located at (0.8 km, 1.1 km). The black solid line repre-sents the analytical solution, the blue and red dashed lines representthe numerical solutions obtained using the first-and high-order Lax–Friedrichs sweeping schemes on the 160 × 120mesh, and the greendashed line represents the numerical solutions of the first-orderLax–Friedrichs sweeping scheme on the 1280 × 960 mesh.


http://dx.doi.org/10.1093/gji/ggt036

Sethian, J. A., and A. Vladimirsky (2001). Ordered upwind methods forstatic Hamilton–Jacobi equations, Proc. Natl. Acad. Sci. Unit. StatesAm. 98, 11,069–11,074.

Sethian, J. A., and A. Vladimirsky (2003). Ordered upwind methodsfor static Hamilton–Jacobi equations: Theory and algorithms, SIAMJ. Numer. Anal. 41, no. 1, 325–363.

Symes, W., R. Versteeg, A. Sei, and Q.H. Tran (1994). Kirchhoffsimulation migration and inversion using finite-difference travel-times and amplitudes, Annual Report, The Rice Inversion Project. RiceUniversity, http://www.trip.caam.rice.edu/ (last accessed January2013).

Tessmer, E., and D. Kosloff (1994). 3-D elastic modeling with surfacetopography by a Chebychev spectral method, Geophysics 59, 464–473.

Tessmer, E. T., D. Kosloff, and A. Behle (1992). Elastic wave propagationsimulation in the presence of surface topography, Geophys. J. Int. 108,621–632.

Trampert, J., and R. D. van der Hilst (2005). Towards a quantitative inter-pretation of global seismic tomography, in Earth’s Deep Mantle:Structure, Compositon, and Evolution, American Geophysical Mono-graph, Vol. 160, 47–62.

Thompson, J., Z. Warsi, and C. Mastin (1985). Numerical Grid Generation:Foundations and Applications, North-Holland Publishing, Amster-dam.

Tsai, R., L. T. Cheng, S. Osher, and H. K. Zhao (2003). Fast sweeping al-gorithms for a class of Hamilton–Jacobi equations, SIAM J. Numer.Anal. 41, 673–694.

Wang, Y. (2011). Seismic anisotropy estimated from P wave arrival times incrosshole measurements, Geophys. J. Int. 184, 1311–1316.

Xu, T., G. Xu, E. Gao, Y. Li, X. Jiang, and K. Luo (2006). Block modelingand segmentally iterative ray tracing in complex 3Dmedia,Geophysics71, T41–T51.

Xu, T., Z. Zhang, E. Gao, G. Xu, and L. Sun (2010). Segmentally iterativeray tracing in complex 2D and 3D heterogeneous block models, Bull.Seismol. Soc. Am. 100, no. 2, 841–850.

Zhang, W., and X. Chen (2006). Traction image method for irregular freesurface boundaries in finite difference seismic-wave simulation, Geo-phys. J. Int. 167, 337–353.

Zhang, H., and C. H. Thurber (2003). Double-difference tomography: Themethod and its application to the Hayward fault, California, Bull. Seis-mol. Soc. Am. 93, no. 5, 1875–1889.

Zhang, Y. T., H. K. Zhao, and J. Qian (2006). High order fast sweepingmethods for static Hamilton–Jacobi equations, J. Sci. Comp. 29,25–56.

Zhao, A., Z. Zhang, and J. Teng (2004). Minimum travel time tree algorithmfor seismic ray tracing: Improvement in efficiency, J. Geophys. Eng. 1,no. 4, 245–251.

Appendix A

Transformation from Cartesian to CurvilinearCoordinates

For a given topographic surface, the discrete grid shouldconform to the free surface to suppress artificial errors. Sucha grid is termed a boundary-conforming grid (Thompsonet al., 1985; Hvid, 1994), and has been used by a numberof researchers in the simulation of seismic-wave fields (Forn-berg, 1988; Zhang and Chen, 2006; Appelo and Petersson,2009; Lan and Zhang, 2011a,b). A grid of this type may beformed by carrying out a transformation from a (curvilinear)computational space to a (Cartesian) physical space, asshown in Figure A1. Under such a transformation, the curvi-

linear coordinates q and r are mapped to Cartesian coordi-nates within the physical space, with both systems includinga positive upward direction for the vertical coordinate.

After generating the boundary-conforming grid, the Car-tesian coordinates of each grid point may be determined fromthe curvilinear coordinates using the following equations:

x � x�q; r� and z � z�q; r�: (A1)

We then can express the spatial derivatives in the Car-tesian coordinate system (x, z) from the curvilinear coordi-nate system (q, r) following the chain rule, such that

∂x � qx∂q � rx∂r and ∂z � qz∂q � rz∂r; (A2)

where qx denotes ∂q�x; z�=∂x, and similar relationships ofthis type apply in other cases. These derivatives are knownas metric derivatives or simply metrics. We can also find thederivatives of the metrics

qx �zrJ; qz �

−xrJ

; rx �−zqJ

; and rz �xqJ;

(A3)

where J is the Jacobian determinant of the transformation,which can be written as J � xqzr − xrzq.

It is worth noting that even if the mapping equations (A1)are expressed as an analytical function, the derivatives shouldstill be calculated numerically to avoid spurious source termsthat may be caused by the derivative coefficients when theconservative forms of the equations are used (Thompsonet al., 1985). In all of the examples presented here, the metricderivatives are computed numerically using second-order ac-curate finite-difference approximations.

z=z (q, r)x=x (q, r)

x

z

qr

Physical space Computational space

q

r

Figure A1. Mapping between computational and physicalspace in two dimensions.


http://www.trip.caam.rice.edu/





Appendix B

Strategies for Computing the Travel-Time AnalyticalSolutions for the Homogeneous Models withIrregular Surfaces Presented in This Paper

Analytical Solutions for the Group 2 Models

For the models in group 2, we first consider the solutionsfor the source at (−0:2 km, 1.1 km) (Fig. B1a), solutions forother cases are discussed later. Computing analytical solu-tions for a cosinusoid surface is not as easy as for a flat surface.The challenge in this context is the difficulty in seeking theshortest paths from the source to other points within the do-main. According to the paths of the rays emitted from point S,the domain can be divided into three parts: the upper leftpart (surrounded by the arc DM, and the line segments MEand ED), the upper right part (surrounded by the arcGN, andthe line segments NF and FG), and the remaining part. Therays reach the former two parts with bent paths but get to thelast part with straight paths. The first-arrival travel times forthe third part can be calculated easily; that is, the analyticalsolutions are simply the Euclidean distances divided by thevelocity in the medium. However, for the upper left and rightparts, the first-arrival travel times can not be computed soeasily. The left part is taken as an example in the following(it is easy to extend this strategy to the right part).

For points beneath the surface, take point B as an exam-ple, the shortest path for the ray from S to B is SD�DC�CB, where SD and CB are tangents to the cosine curve

−1 −0.5 0 0.5 1−0.2

0.2

0.6

1

1.4

X (km)

Z (

km)

G

S

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

X (km)

Z (

km)

S

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

X (km)

Z (

km)

Z (

km)

Z (

km)

Z (

km)

S

(a) (b) (c)

(d) (e) (f)

Figure B1. The strategy for computing the travel-time analytical solutions for group 2 models. S represents the source for this model. SEand SF are two vectors that are tangential to the cosine curve (irregular surface) at points D and G, respectively. A represents a point on thesurface curve, and B represents a point beneath the surface. CB is tangential to the cosine curve at point C. The cosine curve intersects the leftand right boundaries at points M and N, respectively.

B

F

Figure B2. The strategy for computing the travel-time analyti-cal solution for the group 3 model. S represents the source for themodel. A line that crosses through the source and is parallel to the xaxis intersects the left and right boundaries of the model at points Eand F, respectively. Because the source and the lowest points of thesurface (D and G) are at the same height, the line passes through Dand G as well and is tangential to the surface curve (irregular sur-face) at points D and G, respectively. C is the inflection point ofDM, and B is the inflection point of GN.


(irregular surface) at points D and C, respectively, and arcDC is part of the cosine curve. This path can be demonstratedto be the shortest path by comparing it with others, but that isnot included here. After finding the shortest path, it is easy tocompute its length. The lengths of vectors SD and CB can beobtained using the distance formula between the two points,while the length of DC can be calculated by integrating thearc between the points D and C.

Both arcs MD and NG are strictly concave, so, forpoints on these two arcs, the shortest path consists of an arcand a line segment. Take the point A as an example, the short-est path for the rays emitted from point S arriving at this pointis SD�DA, the length of which can be calculated using themethod mentioned above.

When the source is located at (−0:2 km, 0.7 km) or(−0:2 km, 0.3 km) (Fig. B1b and c, respectively), the solu-tion is similar to the case analyzed above. For the source lo-cated at (−0:2 km, −0:1 km) (Fig. B1d), only a fraction onthe upper right of the domain needs to be treated using themethods described above; for other parts, including when thesource is at (−0:2 km, −0:5 km) or (−0:2 km, −0:9 km)(Fig. B1e and f, respectively), the analytical solutions aresimply the Euclidean distances divided by the velocity.

Analytical Solutions for the Group 3 Model

Generally speaking, and for this model (Fig. B2), we cancalculate the first-arrival travel times in a similar way to the

method described above. To make the solutions easier, weput the source at a special point, the x coordinate of the pointis at the center of the x axis, while the y coordinate is at thesame height as the lowest point of the complex surface (Dand G). Except for the two shaded regions, travel times forother parts can be obtained directly. Travel times in the twoshaded regions can be calculated in a similar way to themethod described above, but with some differences. DMand GN are two complex curves that consist of a convexcurve (CM or BN) and a concave curve (DC or GB), respec-tively. For a point on CM or BN, there should be a linethrough the point tangential to DC or GB. Furthermore,the shortest path from S to the point on CM or BN consistsof three parts (two segments and an arc), while for the pointon DC or GB, the shortest path comprises only two parts(a segment and an arc, as in group 2). For the other shadedregions, the travel times can be computed as described above.

State Key Laboratory of Lithospheric EvolutionInstitute of Geology and GeophysicsChinese Academy of SciencesNo. 19 Beitucheng Western Road 9825#Chaoyang DistrictBeijing 100029, [email protected]

Manuscript received 9 June 2012


by haiqiang lan and zhongjie zhang - · pdf fileby haiqiang lan and zhongjie zhang ......

Documents