carcione et. al. 2002 seismic modeling

7/27/2019 Carcione Et. Al. 2002 Seismic Modeling

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GEOPHYSICS, VOL. 67, NO. 4 (JULY-AUGUST 2002); P. 1304 1325, 10 FIGS., 1 TABLE.10.1190/1.1500393

In honor of the new millennium (Y2K), the SEG Research Committee is inviting a series of review articles and tutorials thatsummarize the state-of-the-art in various areas of exploration geophysics. Further invited contributions will appear duringthe next year or twoSven Treitel, Chairman, SEG Research Subcommittee on Y2K Tutorials and Review Articles.

Y2K Review Article

Seismic modeling

Jose M. Carcione, Gerard C. Herman, and A. P. E. ten Kroode

ABSTRACT

Seismic modeling is one of the cornerstones of geo-

physical data processing. We give an overview of themost common modeling methods in use today: directmethods, integral-equation methods, and asymptoticmethods. We also discuss numerical implementation as-pects and present a few representative modeling exam-ples for the different methods.

INTRODUCTION

Seismic numerical modeling is a technique for simulatingwave propagation in the earth. The objective is to predict theseismogram that a set of sensors would record, given an as-sumed structure of the subsurface. This technique is a valuabletool for seismic interpretation and an essential part of seismicinversionalgorithms.Anotherimportant application of seismicmodeling is the evaluation and design of seismic surveys. Thereare many approaches to seismic modeling. We classify theminto three main categories: direct methods, integral-equationmethods, and ray-tracing methods.

To solve the wave equation by direct methods, thegeologicalmodel is approximated by a numerical mesh, that is, the modelis discretized in a finite numbers of points. These techniquesare also called grid methods and full-wave equation methods,the latter since the solution implicitly gives the full wavefield.Direct methods do not have restrictions on the material vari-ablity and can be very accurate when a sufficiently fine grid isused. Furthermore, the technique can handle the implementa-tion of differentrheologiesand is well suitedfor thegenerationof snapshots which can be an important aid in the interpreta-

tion of the results. A disadvantage of these general methods,however, is that they can be more expensive than analyticaland ray methods in terms of computer time.

Manuscript received by the Editor January 16, 2001; revised manuscript received January 22, 2002.OGS, Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy. E-mail: [email protected] University of Technology, Department of Applied Analysis, Mekelweg 4, 2628 CD Delft, The Netherlands. E-mail: [email protected] Research SIEP, Post Office Box 60, 2280 AB Rijswijk, The Netherlands.c 2002 Society of Exploration Geophysicists. All rights reserved.

Integral-equation methods are based on integral represen-tations of the wavefield in terms of waves, originating frompoint sources. These methods are based on Huygens princi-

ple, formulated by Huygens in 1690 in a rather heuristic way(see Figure 1). When examining Huygens work closer, we cansee that he states that the wavefield can, in some cases, be con-sidered as a superposition of wavefields due to volume pointsources and, in other cases, as a superposition of waves due topoint sources located on a boundary. Both forms of Huygensprinciple are still in use today and we have both volume inte-gralequations and boundary integral equations, eachwith theirown applications. We briefly review bothmethods. Thesemeth-odsare somewhat more restrictive in their application than theabove direct methods. However, for specific geometries, suchas bounded objects in a homogeneous embedding, boreholes,or geometries containingmany small-scale cracks or inclusions,integral-equation methods have shown to be very efficient andto give accurate solutions. Due to their somewhat more ana-lytic character, they have also been useful in the derivation ofimaging methods based on the Born approximation. [For ex-ample, see Cohen et al. (1986) and Bleistein et al. (2001) for adescription of Born inversion methods.]

Asymptotic methods or ray-tracing methods are very fre-quently used in seismic modeling and imaging. These methodsare approximative, since they do not take the complete wave-field into account. On the other hand, they are perhaps themost efficient of the methods discussed in this review. Espe-cially for large, three-dimensional models the speedup in com-puter time can be significant. In these methods, the wavefieldis considered as an ensemble of certain events, each arriving ata certain time (traveltime) and having a certain amplitude. Wediscuss some of these methods, as well as some of their proper-ties. Asymptotic methods, due to their efficiency, have playeda very important role in seismic imaging based on the Born ap-

proximation for heterogeneous reference velocity models. An-other important application of these methods is the modelingand identification of specific events on seismic records.

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Seismic Modeling 1305

In this overview, we discuss the above methods in some de-tail andgive theappropriatereferences. In order not to getlostin tedious notations, we discuss only the acoustic formulationso we can concentrate on the differences between methods.All methods discussed here have been generalized to the elas-tic case and, where appropriate, references are given. We also

present a few examples illustrating the applicability of the dif-ferent methods. Since the applicability regions of the methodsare rather different and, largely, nonoverlapping, the modelsare different but typical for each technique. Hopefully, theywill provide thereader with some idea of thetypesof problemsor geometries to which a particular modeling method is bestsuited. The modeling methods discussed here have in commonthat they are applicable to various two- and three-dimensionalgeometries. This implies that we do not discuss methods espe-cially suited for plane-layeredmedia, despite the fact that thesemethodsarecertainlyatleastasimportantandoftenusedastheones we discuss here. For an excellent overview of these plane-wave summation (or slowness) methods, see Ursin (1983).

DIRECT METHODS

We considerfinite-difference (FD),pseudospectral(PS), andfinite-element (FE) methods. These methods require the dis-cretization of the space and time variables. Let us denotethem by (x, t) = (j d x, n dt), where d x and dt are the grid spac-ing and time step, respectively. We first introduce the appro-priate mathematical formulations of the equation of motion,and then consider the main aspects of the modeling as fol-lows: (1) time integration, (2) calculation of spatial derivatives,(3) source implementation, (4) boundary conditions, and (5)absorbing boundaries. All these aspects are discussed by usingthe acoustic- and S H-wave equations.

Mathematical formulations

For simplicity, we consider the acoustic- and S H-waveequations which describe propagation of compressional andpure shear waves, respectively.

FIG. 1. Illustration by Huygens of Huygens principle, stating,in French, that each element of a light source, like the Sun, acandle or a glowing charcoal, causes waves, with the elementas centre [taken from the 1690 Trait e de la lumi `ere (Huygens,1990)].

Pressure formulation

The pressure formulation for heterogeneous media can bewritten as (Aki and Richards, 1980, 775)

L2p + f = 2p

t2, L2 = c2

1

, (1)

where is the gradient operator, p is the pressure, c is thecompressional-wave velocity, is thedensity, and f isthe bodyforce. This is a second-orderpartial differentialequationin thetime variable.

Velocity-stress formulation

Instead of using the wave equation, wave propagation can alsobe formulated in terms of a system of first-order differentialequations in the time and space variables. Consider, for in-stance, propagation of S H-waves. This is a two-dimensionalphenomenon, with the particle velocity, say v2, perpendicularto theplaneof propagation. Newtonssecond lawand Hookeslaw yield the velocity-stress formulation of the S H-waveequation (Aki and Richards, 1980, 780):

v

t= Hv + F, (2)

where

v = [v2, 32, 12], F = [ f, 0, 0], (3)

Hv = A vx1

+ B vx3

, (4)

A =

0 0 1

0 0 0

0 0

, B = 0

1 0

0 0

0 0 0

, (5)and denotes stress and is the shear modulus.

The solution to equation (2) subject to the initial conditionv(0) = v0 is formally given by

v(t) = exp(tH)v0 + t0

exp(H) F(t ) d, (6)where exp(tH) is called the evolution operator, because ap-plication of this operator to the initial condition vector (or tothe source vector) yields the solution at time t. We refer to Has the propagation matrix. The S H and acoustic differentialequations are hyperbolic (Jain, 1984, 251; Smith, 1985, 4) be-cause the field has a finite velocity.

Variational formulation

The most standard finite-element method uses the waveequation (1) as a starting point. We consider a volume Vbounded by a surface S. The surface Sis divided into Sp, wherepressure boundary conditions are defined, and Sdp , the part onwhich normal accelerations (or pressure fluxes) are given. As-sume a small pressure variation p which is consistent with theboundary conditions. If we multiply equation (1) by p, inte-grate over the volume V and by parts (using the divergencetheorem), we obtain

V

1

p p d V =

V

p

c22p

t2d V +

V

fp

c2dV

+

Sdp

p

n p d S, (7)


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1306 Carcione et al.

where n is the normal to the surface S. This variational for-mulation derived from the wave equation is equivalent to aGalerkin procedure (Zienkiewicz, 1977, 70; Hughes, 1987, 7).

Time integration

Thenumericalsolutionof thewave equationrequiresthe dis-cretization of the time variable by using finite differences (anexceptionto this is formedby thespectral methods; seebelow).The basic idea underlying FD methods is to replace the partialderivatives by approximations based on Taylor series expan-sions of functionsnear thepoint of interest. Forwardand back-warddifference approximationsof the timederivatives (Smith,1985, 7) lead to explicit and implicit FD schemes, respectively.Explicit means that the wavefield at present time is computedfrom the wavefield at past times. On the other hand, in implicitmethods, the present values depend on past and future values.Unlike explicit methods, implicit methods are unconditionallystable but lead to a great amount of computation arising fromthe need to carry out large matrix inversions. In general, thedifferentialformulationof thewave equation is solvedwith ex-

plicit algorithms, since the time step is determined by accuracycriteria rather than by stability criteria (Emerman et al., 1982).

Eigenvalues and stability

Wave equations used in seismic exploration and seismologycan be expressed as v = Hv, where H is the propagation matrixcontaining the material properties and spatial derivatives (thedot denotes time differentiation) [e.g., equation (2)]. We nowaddress the stability aspects of finite-difference schemes andtherefore consider the eigenvalue problem of wave propaga-tion. Assume constant material properties and a plane-wavekernel of the form exp(i k x i t), where k is the wavenum-ber vector, x is the position vector, and is the angularfrequency (which can be complex valued in the case of attenu-ation). Substitution of the plane-wave kernel into the wave

equation yields an eigenvalue equation for the eigenvalues =i .Fortheacoustic-andS H-waveequations, theseeigen-values lie on the imaginary axis of the -plane. For instance, in1-D space, the eigenvalues corresponding to equations (1) and(2) are = i kc, where c is either the compressional- or theshear-wave velocity. There are other equations of interest inseismicmodelingin whicheigenvaluesmightlie in theleft-hand-plane. Some of these are discussed below.

Consider an anelastic medium described by a viscoelasticstress-strain relation. Wave attenuation is governed by mate-rial relaxation times, which quantify the response time of themedium to a perturbation (lossless solid materials respond in-stantaneously, i.e., the relaxation time is zero). For a viscoelas-tic medium with moderate attenuation, the eigenvalues have asmall negative real part causing the waves to be attenuated. In

addition, when solving the equations in the time domain, thereare eigenvalues with a large negative part and close to thereal axis that are approximately given by the reciprocal ofthe relaxation times corresponding to each attenuation mech-anism. Then, the domain of the eigenvalues has a T shape(see Tal-Ezer et al., 1990). If the central frequency of theserelaxation peaks is close to the source frequency band,or equivalently, if the related eigenvalues are close to theimaginaryaxisof the-plane, an explicit scheme performs veryefficiently.

The situation is critical for porous media, where the eigen-valuecorrespondingto the slow compressional waveat seismicfrequencies (a quasi-static mode)has a verylargenegativepart,which is related to the location of Biot relaxation peaks, usu-ally beyond the sonic band for pore fluids like water and oil(Carcione and Quiroga-Goode, 1996). When the modulus of

the eigenvalues is very large compared to the inverse of themaximum propagation time, the differential equation is said tobe stiff (Jain, 1984, 72; Smith, 1985, 198). Due to the presenceof the quasi-static slow wave, the low-frequency Biot differ-ential equations are hyperbolic/parabolic. Although the bestalgorithm would be an implicit method, the problem can stillbe solved with explicit methods (see below).

Time and space discretization of the wave equation withan explicit scheme (forward time difference only) leads toan equation of the form vn+1 = Gvn , where G is called theamplification matrix. The von Neumann condition for stabil-ity requires max |gj | 1, where gj are the eigenvalues of G(Jain, 1984, 418). It can be shown that this condition does nothold for all dt when explicit schemes are used, and that im-plicit schemes do not have any restriction. For instance, ex-

plicit fourth-order Taylor and Runge-Kutta methods requiredt| max| < 2.78(Jain,1984, 71), implying very small time stepsfor verylarge eigenvalues. Implicit methods are A-stable (Jain,1984, 118), meaning that the domain of convergence is the leftopen-half-plane.

Classical finite differences

Evaluating the second time derivative in equation (1) at(n + 1) dt and (n 1) dt by a Taylor expansion, and summingboth expressions, yields

2pn

t2= 1

dt2

pn+1 + pn1 2pn 2

L=2

dt2

(2)!

2pn

t2

.

(8)The wave equation (1) provides the higher order time deriva-tives, using the following recursion relation:

2pn

t2= L2

22pn

t22+

22 fn

t22. (9)

This algorithm, where high-order time derivatives are re-placed by spatial derivatives, is often referred to as the Lax-Wendroff scheme (Jain, 1984, 415; Smith, 1985, 181; Dablain,1986; Blanch and Robertsson, 1997). A Taylor expansionof theevolution operator exp(dtH) is equivalent to a Lax-Wendroff-scheme.

The dispersion relation connects the frequency with thewavenumber and allows the calculation of the phase velocitycorresponding to each Fourier component. Time discretization

implies an approximation of the dispersion relation, which inthecontinuous case is = ck, with the angular frequency. As-suming constant material properties and a 1-D wave solutionof the form exp(i kx i n dt), where k is the wavenumber and is the FD angular frequency, yields the following dispersionrelation:

2

dtsin

dt

2

= ck

1 2 L=2

(1) (ck dt)22

(2)!. (10)


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The FD approximation to the phase velocity is c = / k. Us-ing equation (10) with second-order accuracy [neglect O(dt2)terms], the FD phase velocity is

c = c|sinc()| , =f dt, (11)

where = 2 f and sinc() = sin( )/( ). Equation (11) in-dicates that the FD velocity is greater than the true phase ve-locity. Since should be a real quantity, thus avoiding expo-nentially growing solutions, the value of the sine function inequation (10) must be between 1 and 1. This constitutes thestability criterion. For instance, for second-order time integra-tion this means ck dt/2 1. The maximum phase velocity, cmax,and the maximum wavenumber (i.e. the Nyquist wavenumber,/d xmin) must be considered. Then, the condition is

dt s

d xmin

cmax

, s = 2

. (12)

A rigorous demonstration, based on the amplification factor,is given by Smith (1985, 70; see also Celiaand Gray, 1992, 232).In n-D space, s = 2/(n) and, for a fourth-order approxima-

tion (L = 2) in 1-D space, s = 23/ . Equation (12) indicatesthat stability is governed by the minimum grid spacing and thehigher velocities.

Let us now consider the presence of attenuation. Time-domain modeling in lossy media can be described by vis-coelastic stress-strain relations. This requires the use of the so-called memory variables, one for each relaxation mechanism(Carcione et al., 1988). The introduction of additional differ-ential equations for these field variables avoids the numericalcomputation of the viscoelastic convolution integrals. The dif-ferential equation for a memory variable e has the form

e

t= a be, b > 0, (13)

where is a field variable (e.g., the dilatation) and a and b arematerial properties (b is approximately the central frequency

of the relaxation peak). Equation (13) can be discretized byusing the central differences operator for the time deriva-tive[dt(e/t)n = en+1 en1] andmean value operator forthememoryvariable(2en = en+1 + en1).Theseapproximationsareused in the Crank-Nicolson scheme (Smith, 1985, 19). This ap-proach leads to an explicit algorithm

en+1/2 = 2 dt a2 + b dt

n +

2 b dt2 + b dt

en1/2 (14)

(Emmerich and Korn, 1987). This method is robust in termsof stability, because the coefficient ofen1/2, related to the vis-coelastic eigenvalue of the amplification matrix, is less than 1for any value of the time step dt. The same method performsequally well for wave propagation in porous media (Carcioneand Quiroga-Goode, 1996).

Splitting methods

Time integration can also be performed with the method ofdimensional splitting, also called Strangs scheme (Jain, 1984,444; Mufti, 1985; Bayliss et al., 1986; Vafidis et al., 1992). Con-siderequation (2)and denote a derivative withrespectto a vari-able by the subscript ,. The 1-D equations v,t = Av,x1 andv,t = Bv,x3 are solved by means of one-dimensional differenceoperators Lx1 and Lx3 , respectively. For instance, Bayliss et al.

(1986) use a fourth-order accurate predictor-corrector schemeand the splitting algorithm vn+2 = Lx1 Lx3 Lx3 Lx1 vn , where eachoperator advances the solution by a half time step. The max-imum allowed time step is larger than for unsplit schemes,because the stability properties are determined by the 1-Dschemes. Splitting is also useful when the system of dif-

ferential equations is stiff. For instance, Biots poroelasticequations can be partitioned into a stiff part and a nons-tiff part, so that the evolution operator can be expressed asexp(Hr + Hs )t, where r indicates the regular matrix and sthe stiff matrix. The product formulas exp(Hrt) exp(Hs t) andexp( 1

2Hs t)exp(Hrt)exp(

12

Hs t) are first- and second-order ac-curate,respectively. Thestiff partcan be solved analytically andthe nonstiff part with an standard explicit method (Carcioneand Quiroga-Goode, 1996). Strangs scheme can be shown tobe equivalent to the splitting of the evolution operator forsolving the poroelastic equations.

Predictor-corrector schemes

Predictor-corrector schemes of different order find wide ap-

plication in seismic modeling (Mufti, 1985; Bayliss et al., 1986;Vafidis et al., 1992; Dai et al., 1995). Consider equation (2) andthe first-order approximation

vn+1 = vn + dt Hvn, (15)known as the forward Euler scheme. This solution is givenby the intersection point between the tangent ofv at t= n dtand the line t= (n + 1) dt. A second-order approximation canbe obtained by averaging this tangent with the predicted one.Then, the corrector is

vn+1 = vn + dt2

(Hvn + Hvn+1). (16)

This algorithm is the most simple predictor-corrector scheme(Celia and Gray, 1992, 64).A predictor-correctorMacCormack

scheme, second-order in time and fourth-order in space, is usedby Vafidis et al. (1992) to solve the elastodynamic equations.

Spectral methods

As mentioned before, a Taylor expansion of the evolutionoperator exp(dtH) is equivalent to a Lax-Wendroff-scheme.When thenumber of terms in equation (8)is increased, a largertime step can be used while retaining high accuracy. Taylor ex-pansions and Runge-Kutta methods, however, are not the bestin terms of accuracy. The evolution operator in equation (6)can be expanded in terms of Chebyshev polynomials as

v(t) =M

k=0CkJk(t R)Qk

H

Rv0, (17)

where C0 = 1 and Ck = 2 for k= 0, Jk is the Bessel function oforder k,and Qk are modified Chebyshevpolynomials. R shouldbe chosen larger than the absolute value of the eigenvalues ofH (Tal-Ezer et al., 1987). This technique allows the calcula-tion of the wavefield with large time steps. Chebyshev expan-sions are optimal since they require the minimum number ofterms. The most time-consuming part of a modeling algorithmis the evaluation of the terms L2p in equation (1) or Hv inequation (2) due to the computation of the spatial derivatives.


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A Taylor-expansion algorithm needs N= tmax/dt of such evalu-ations to compute the solution at time tmax. On the other hand,the number of evaluations using equation (17) is equal to thenumber of terms in the Chebyshev expansion. Numerical testsindicate that M is comparable to N for second-order finite dif-ferencing, but the error of the Chebyshev operator is prac-

tically negligible for single-precision programming (Tal-Ezeret al., 1987). This means that there is no numerical dispersiondue to the time integration.

When the wave equation is used, which is second order intime [see equation (1)], the rapid-expansion method (REMmethod) is twice as efficient since the expansion then containsonly even-order Chebyshev functions (Kosloff et al., 1989). Asimilar algorithm for the viscoelastic wave equation is devel-oped by Tal-Ezer et al. (1990). These methods are said to havespectral accuracy, in the sense that the error of the approx-imation tends exponentially to zero when the degree of theapproximating polynomial increases.

Algorithms for finite-element methods

In theFE method, which canbe derived from thevariationalformulation (7), the field variables are evaluated by interpola-tion fromnodal (grid) values. For a second-order isoparametricmethod (Zienkiewicz, 1977, 178; Hughes, 1987, 118), the inter-polation can be written as

p(xi ) = P, (18)where P is a column vector of the values p(xi ) at the nodes,and is a row vector of spatial interpolation functions, alsoreferred to as shape and basis functions. The approximation toequation (7) is obtained by considering variations p accordingto the interpolation (18). Since p =P, and P is arbitrary,theresult is a setof ordinary differentialequations at thenodalpressures P (Zienkiewicz, 1977, 531; Hughes, 1987, 506):

KP + M 2

Pt2

+ S = 0, (19)

where K is the stiffnessmatrix,M isthe mass matrix, andS isthegeneralized source matrix. These matrices contain volume in-tegrals that areevaluated numerically (see also theThefinite-element method section). The matrix M is often replaced bya diagonal lumped mass matrix M such that each entry equalsthe sum of all entries in the same row ofM (Zienkiewicz, 1977,535). In this way, the solution can be obtained with an explicittime-integration method, suchas the central differencemethod(Seron et al., 1990). This technique can be used with low-orderinterpolation functions, for whichthe error introducedbythe algorithm is relatively low. When high-order polynomials(including Chebyshev polynomials) are used as interpolationfunctions, the system of equations (19) is generally solved withimplicit algorithms. In this case, the most popular algorithmis the Newmark method (Hughes, 1987, 490; Padovani et al.,1994; Ser on et al., 1996).

Finally, numerical modeling can be performed in the fre-quency domain. The method is very accurate but generallyexpensive when using differential formulations, because it in-volves the solution of many Helmholtz equations (Jo et al.,1996). It is used more in FE algorithms (Marfurt, 1984; Santoset al., 1988; Kelly and Marfurt, 1990).

Calculation of spatial derivatives

The name of a particular modeling methodis usually derivedfrom the algorithm for computing the spatial derivatives. Thefollowing sections briefly review these algorithms.

Finite differences

FD methods use either homogeneous or heterogeneous for-mulations to solve the equation of motion. In the first case,the motion in each homogeneous region is described by theequation of motion with constant acoustic parameters. For thismethod, boundary conditions across all interfaces must be sat-isfied explicitly. The heterogeneous formulation incorporatesthe boundary conditions implicitly by constructing FD repre-sentationsusing the equation of motion for heterogeneousme-dia. The homogeneous formulation is of limited use becauseit can only be used efficiently for simple, piecewise homo-geneous geometries. The heterogeneous formulation, on theother hand, makes it possible to assign different acoustic prop-erties to every grid point, providing the flexibility to simulatea variety of complex subsurface models (e.g., random mediaor velocity gradients). Heterogeneous formulations generallymake use of staggered grids to obtain stable schemes for largevariations of Poissons ratio (Virieux, 1986). In staggered grids,groupsof field variables andmaterialproperties aredefined ondifferent meshes separated by half the grid spacing (Fornberg,1996, 91). Thenewly computed variables are centered betweenthe old variables. Staggeringeffectively halves the grid spacing,increasing the accuracy of the approximation.

Seismic modeling in heterogeneous media requires the cal-culation of first derivatives. Consider the following approxi-mation with an even number of points, suitable for staggeredgrids:

p0

x= w0(p 1

2 p 1

2) + + w(p+ 1

2 p 1

2), (20)

with weighting coefficients w. The antisymmetric form guar-antees that the derivative is zero for even powers ofx . Let ustest the spatial derivative approximation for p =x and p =x3.Requiring that equation (20) be accurate for all polynomialsup to order 2 yields the approximation (p 1

2 p 1

2)/d x , while

for fourth accuracy [the leading error term is O(d x4)]the weights are obtained from w0 + 3w1 = 1/d x and w0 +27w1 = 0,giving w0 = 9/(8 dx ),and w1 = 1/(24 dx ) (Fornberg,1996, 91). To obtain the value of the derivative at x = j dx , sub-stitute subscript 0 with j, + 1

2with j + + 1

2and 1

2with

j 12

. Fornberg (1996, 15) provides an algorithm for com-puting the weights of first and second spatial derivatives forthe general case, i.e., approximations which need not be evalu-ated at a gridpoint such as centered and one-sided derivatives.He also shows that the FD coefficients w in equation (20)are equivalent to those of the Fourier PS method when ap-proaches the number of grid points (Fornberg, 1996, 34).

Let us now study the accuracy of the approximationby considering the dispersion relation. Assuming constantmaterial properties and a 1-D wave solution of the formexp(i k j dx i t), the second-order approximation gives thefollowing FD dispersion relation and phase velocity:

2 = c2k2sinc2(), c = c|sinc()|, (21)


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where = K d x , with k= 2 K [equation (21) can be derivedin the same manner as equation (11)]. The spatial dispersionacts in the opposite sense of temporal dispersion [see equa-tion (11)]. Thus, the FD velocity is smaller than the true phasevelocity.

Staggered grids improve accuracy and stability, and elim-

inate noncausal artifacts (Virieux, 1986; Levander, 1988;Ozdenvar and McMechan, 1997; Carcione and Helle, 1999).Staggered grid operators are more accurate than centered dif-ferences operators in the vicinity of the Nyquist wavenumber(e.g., Kneib and Kerner, 1993). Thevelocity-stress formulationin staggered grids constitutes a flexible modeling techniquebecause it allows one to freely impose boundary conditions(see the Boundary conditions section) and is able to directlyyield all the field variables (Karrenbach, 1998).

However, there is a disadvantage in using staggered gridsfor anisotropic media of symmetry lower than orthorhombic.Staggering implies that the off-diagonal stress and strain com-ponents are not defined at the same location. When evaluat-ing the stress-strain relation, it is necessary to sum over a lin-ear combination of the elasticity constants (cI J, I, J= 1, . . . 6)multiplied by the strain components. Hence, some terms ofthe stress components have to be interpolated to the locationswhere the diagonal components are defined (Mora, 1989).

A physical criterion to improve accuracy is to compute theweights w in equation (20) by minimizing the relative error inthe components of the group velocity cg = /k (the velocityof a wave packet). This procedure, combined with grid stagger-ing and a convolutional scheme, yields an optimal differentialoperator for wave equations (Holberg, 1987). The method isproblem dependent because it depends on the type of waveequation. Igel et al. (1995) obtained high accuracy with opera-tors of small length (eight points) in the anisotropic case. Thetreatment of the P-SV case and more details about the finitedifference approximation are in Levander (1989).

The modeling can be made more efficient by using hybrid

techniques, for instance, combining finite differences with fa-ster algorithms such as ray-tracing methods (Robertsson et al.,1996), integral-equation methods (Stead and Helmberger,1988), and reflectivity methods (Emmerich, 1989). In this way,modeling of the full wavefield can be restricted to the target(e.g., the reservoir), and propagation in the rest of the model(e.g., the overburden) can be simulated with faster methods.

Pseudospectral methods

The pseudospectral methods used in forward modeling ofseismic waves are mainly based on the Fourier and Chebyshevdifferential operators. Gazdag (1981) first and Kosloff andcoworkers later applied the technique to seismic explorationproblems (e.g., Kosloff and Baysal, 1982; Reshef et al.,1988). Mikhailenko (1985) combinedtransforms methods (e.g.,Bessel transforms) with FD and analytical techniques.

The sampling points of the Fourier method are xj = jd x = j xmax/(Nx 1) (j = 0, . . . , Nx 1), where xmax is themaximum distance and Nx is the number of grid points. Fora given function f(x), with Fourier transform f(k), first andsecond derivatives are computed asf

x= ik f,

2 fx 2

= k2 f, (22)

where k is the discrete wavenumber. The transform f to thewavenumber domain and the transform back to the space do-main are calculated by the fast Fourier transform (FFT). Thederivatives of two real functionstwo adjacent grid lines ofthe computational meshcan be computed by two complex(direct and inverse) FFTs. The two functions are put into the

real and imaginary parts, the FFT is performed, the result ismultiplied by ik, and the inverse FFT gives the derivatives inthe realand imaginaryparts. Staggeredoperators that evaluatefirst derivatives between grid points are given by

Dx =k(Nx )

k=0i kexp(ik dx/2) (k)exp(ikx), (23)

where k(Nx ) = /dx is the Nyquist wavenumber. The standarddifferential operator is given by the same expression, withoutthe phase shift term exp(ik dx/2). The standard operator re-quires the use of odd-based FFTs (i.e., Nx should be an oddnumber). This is because even transforms have a Nyquist com-ponent which does not possess the Hermitian property of thederivative (Kosloff and Kessler, 1989). When (x) is real, (k)

is Hermitian (i.e., its real part is even and its imaginary partis odd). If Nx is odd, the discrete form ofk is an odd function,therefore i k (k) is also Hermitianandthe derivative is real. Onthe otherhand, thefirst derivative computedwith the staggereddifferential operator is evaluated between grid points and useseven-based Fourier transforms. The approximation (23) is ac-curate up to the Nyquist wavenumber. If the source spectrumis negligible beyond the Nyquist wavenumber, we can considerthat there is no significant numerical dispersion due to the spa-tial discretization. Hence, the dispersion relation is given byequation (10), which for a second-order time integration canbe written as

= 2dt

sin1

ck dt

2

. (24)

Because k should be real to avoid exponentially growing solu-tions, the argument of the inverse sine must be less than one.This implies the stability condition kmaxc dt/2 1, which leadsto c dt/dx 2/ , since kmax = /dx ( is calledthe Courantnumber). Generally, a criterion < 0.2 is used to choose thetime step (Kosloff and Baysal, 1982). The Fourier method hasperiodic properties. In terms of wave propagation this meansthat a wave impinging on the left boundary of the grid willreturn from the right boundary (the numerical artifact calledwraparound).

The Chebyshev method is mainly used in the velocity-stressformulation to model free surface, rigid, and nonreflectingboundary conditionsat the boundaries of the mesh. Chebyshevtransforms are generally computed with the FFT, with a lengthtwice of that used by theFourier method(Gottlieb andOrszag,1977, 117). Since the sampling points are very dense at theedges of the mesh, the Chebyshev method requires a 1-Dstretching transformation to avoid very small time steps [seeequation (12)]. Because the grid cells are rectangular, mappingtransformationsare alsoused for modelingcurved interfacestoobtain an optimal distribution of grid points (Fornberg, 1988;Carcione, 1994a) and model surface topography (Tessmer andKosloff, 1994).

The Fourier and Chebyshev methods are accurate up tothe maximum wavenumber of the mesh that corresponds to


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a spatial wavelength of two grid points (at maximum grid spac-ing for the Chebyshev operator). This fact makes these meth-ods very efficient in terms of computer storage (mainly in 3-Dspace) and makes the Chebyshev technique highly accurate forsimulating Neumann and Dirichlet boundary conditions, suchas stress-free and rigid conditions (Carcione, 1994a, b) (see the

Boundary conditions section).

The finite-element method

The FE method has two advantages over FD and PS meth-ods, namely, its flexibility in handling boundary conditions andirregular interfaces. On the basis of equation (18), considerthe 1-D case, with uniform grid spacing d x , and an elementwhose coordinates are X1 and X2 (X2 X1 = d x) and whosenodal pressures are P1 and P2. This element is mapped intothe interval [1, 1] in a simplified coordinate system (the ref-erence Z-system). Denote the physical variable by x and thenew variable by z. The linear interpolation functions are

1=

1

2

(1

z), 2=

1

2

(1+

z). (25)

If the field variable and the independent (physical) variableare computed by using the same interpolation functions, onehas the so-called isoparametric approach (Hughes, 1987, 20).That is,

p = 1 P1 + 2 P1, x = 1X1 + 2X1. (26)Assembling the contributions of all the elements of thestiffness matrix results in a centered second-order differencingoperator if the density is constant. When the density isvariable, the stiffness matrix is equivalent to a staggered FDoperator (Kosloff and Kessler, 1989).

FE methods have been used to solve problems in seismol-ogy, in particular, propagation of Love and Rayleigh waves in

the presence of surface topography (Lysmer and Drake, 1972;Schlue, 1979). FE applications for seismic exploration require,in principle, more memoryandcomputertimethanthe studyofsurface waves (as used in modeling soil-structure interaction).In fact, the problem of propagation of seismic waves from thesurface to the target (the reservoir) involves the storage oflarge matrices and much computer time. During the 1970s and1980s, the effort was in rendering efficient existing low-orderFE techniques rather than proposing new algorithms. More-over, it turns out that besides the physical propagation modes,there are parasitic modes when high-order FE methods areused(Kelly and Marfurt, 1990). These parasiticmodes are non-physical solutions of the discrete dispersion relation obtainedfrom the von Neumann stability analysis. For instance, for a2-D cubic element grid, there are ten modes of propagation:two corresponding to the P- and SV-waves, and eight parasiticmodes of propagation.

This was the state of the art at the end of the 1980s. In the1990s. Seron et al. (1990, 1996) further developed the compu-tational aspects of low-order FE to make them more efficientfor seismic exploration problems.

High-order FE methods became more efficient with the ad-vent of the spectral element method (SPEM) (Seriani et al.,1992; Padovani et al., 1994; Priolo et al., 1994; Komatitsch andVilotte, 1998). In this method, the approximation functional

space is based on high-order orthogonal polynomials havingspectral accuracy (i.e., the rate of convergence is exponentialwith respect to the polynomial order). Consider, for instance,the 2-D case and the acoustic wave equation. The physical do-main is decomposed into nonoverlappring quadrilateral ele-ments. In each element, the pressure field p(z1,z2), defined on

the square interval [1, 1] [1, 1] in the reference system Z,is approximated by the product

p(z1,z2) =N

i=0

Nj=0

Pi j i (z1)j (z2). (27)

In this expansion, Pi j are the nodal pressures and i areLagrangian interpolants that satisfy the relation i (k) = ikwithin the interval [1, 1] and vanish outside (ik denotes theKronecker delta and k stands for z1 and z2). The Lagrangianinterpolants are given by

j () =2

N

Nn=0

1

cj cnTn(j )Tn(), (28)

where Tn are Chebyshev polynomials, j arethe Gauss-Lobattoquadrature points, and c0 = cN = 0, cn = 1 for 1 n N. TheChebyshev functions are also used for the mapping transfor-mation between the physical world X and the local system Z.Seriani et al. (1992) used Chebyshev polynomials from eighthorder to fifteenth order. This allows up to three points perminimum wavelength without generating parasitic or spuriousmodes, and computational efficiency is improved by about oneorder of magnitude compared to low-order FE. If the mesh-ingof a geological structureis as regular as possible (i.e., with areasonable aspectratio forthe elements), thematricesare wellconditioned andan iterativemethod such as theconjugate gra-dient uses less than eight iterations to solve the implicit systemof equations.

Source implementation

The basic seismic sources are a directional force, a pressuresource, and a shear source, simulating, for instance, a verticalvibrator, an explosion, or a shear vibrator. Complex sources,such as earthquakes sources, can be represented by a set ofdirectional forces [e.g., a double couple (Aki and Richards,1980, 82)].

Consider an elastic formulation of the wave equation, thatis, P- and S-wave propagation (Kosloff et al., 1984). A direc-tional forcevector has components fi = a(xi )h(t)im , where a isa spatial function (usually a Gaussian), h(t) is the time history, denotes the Kronecker delta function, and m is the sourcedirection. A pressure source can be obtained from a potentialof the form = a(xi )h(t) as fi = /xi . A shear source is oftheformf

= A, where A is a vectorpotential. In 2-Dspace,

A = (0, A) with A = a(xi )h(t). In velocity-stress formulations,the source can be introduced as described above or in the con-stitutive equations (stress-strain relations) such that a pressuresource implies 11 = 22 = 33 initially at the source location,and shear sources result from a stress tensor with zero trace(e.g., Bayliss et al., 1986).

Introducing thesource in a homogeneous region by imposingthe values of the analyticalsolution should handle the singular-ity at the source point. Many FD techniques (Kelly et al., 1976;


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Virieux, 1986) are based on the approach of Alterman andKaral (1968). The numerical difficulties present in the vicin-ity of the source point are solved by subtracting the field dueto the source from the total field due to reflection, refraction,and diffractions in a region surrounding the source point. Thisprocedure inserts the source on the boundary of a rectangular

region. The direct source field is computed analytically. Thismethod has been recently used by Robertsson and Chapman(2000) to inject a numerical wavefield into a FD mesh. Theirapproach allows for efficient computation of many common-shot experiments after alterations of the seismic model in asubarea containing the target (e.g., the reservoir).

When solving the velocity-stress formulation with pseu-dospectral (PS) algorithms and high-order FD methods(Bayliss etal., 1986), thesourcecan beimplementedin onegridpoint in view of the accuracy of the differential operators. Nu-merically(in 1-Dspaceand uniform grid spacing),the strengthof a discrete delta function in thespatial domainis 1/d x , whered x is the grid size. Since each spatial sample is represented bya sinc function with argument x /d x (the spatial integration ofthis function is precisely d x), the introduction of the discrete

delta will alias thewavenumbers beyondthe Nyquist (/dx ) tothe lower wavenumbers. However, if the source time-functionh(t) is band limited with cut-off frequency fmax, the wavenum-bers greater than kmax = 2fmax/c will be filtered. Moreover,since the wave equation is linear, seismograms with differenttime historiescan be implemented by convolvingh(t)withonlyone simulation obtained with (t) as a source (a discrete deltawith strength 1/dt).

The computation of synthetic seismograms for simulatingzero-offset (stacked) seismic sections requires the use of theexploding-reflector concept (Loewenthal et al., 1976) and theso-called nonreflecting wave equation (Baysal et al., 1984). Asource proportional to the reflection coefficients is placed onthe interfaces and is initiated at time zero. All the velocitiesmust be halved in order to get the correct arrival times. The

nonreflecting condition implies a constant impedancemodel toavoid multiple reflections, which are, in principle, absent fromstacked sections and constitute unwanted artifacts in migrationprocesses.

Boundary conditions

Free-surfaceand interface boundary conditionsare the mostimportant in seismic exploration and seismology. Althoughin FE methods the implementation of traction-free bound-ary conditions is natural (simply do not impose any constraintat the surface nodes), FD and PS methods require a specialboundary treatment. However, some restrictions arise in FEand FD modeling when large values of the Poissons ratio (orVP /VS ratio) occur at a free surface.

Consider first the free-surface boundary condition. The clas-sical algorithm used in FD methods (e.g., Kelly et al., 1976)is to include a fictitious line of grid points above the sur-face, and use one-sided differences to approximate normalderivatives and centered differences to approximate tangen-tial derivatives. This simple low-order scheme has an upperlimit of VP /VS 0.35, where VP and VS are the P-wave andS-wave velocities. Moreover, the method is inaccurate due tothe use of one-sided differences. The use of a staggered differ-ential operator and radiation conditions of the paraxial type

(see below) is effective for large variations of Poissons ratio(Virieux, 1986).

The traction-free condition at the surface of the earth canbe obtained by including a wide zone on the upper part of themesh containing zero values of the stiffnesses [the so-calledzero-padding technique (Kosloff et al., 1984)]. Whereas for

small angles of incidence t his approximation yields acceptableresults, for larger angles of incidence it introduces numericalerrors. Free surface and solid-solid boundary conditions canbe implemented in numerical modeling with nonperiodic PSoperators by using a boundary treatment based on character-istics variables (Kosloff et al., 1990; Kessler and Kosloff, 1991;Carcione, 1991; Tessmer et al., 1992; Igel, 1999). This methodis proposed by Bayliss et al. (1986) to model free-surface andnonreflecting boundaryconditions.The method canbe summa-rized as follows (Tessmer et al., 1992; Carcione, 1994b). Con-siderthe algorithmfor theS H-waveequation (2).Most explicittime integration schemes compute the operation Hv (v)old,where H is defined in equation (2). The vector (v)old is thenupdated to give a new vector (v)new that takes the boundaryconditionsinto account. Consider the boundaryx3 = 0 (e.g., thesurface)and that theincidentwaveis incident on this boundaryfrom the half-space x3 > 0. Compute the eigenvalues of matrixB: / = c and 0. Compute the right eigenvectors of ma-trix B, such that they are the columns of a matrix R, whereB = RR1, with the diagonal matrix of the eigenvalues. Ifwe define the characteristics vector as c = R1v, and considerequation (2) corresponding to the modes traveling along thex3-direction,

c

t= c

x3, (29)

the incoming and outgoing waves are decoupled. Two of thecharacteristic variables, components of vector c are v2 + 32/Zand v2 32/Z, with Z= c. The first variable is the incomingwave and the second variable is the outgoing wave. Equating

the new and old outgoing characteristic and assuming stress-free boundary conditions (32 = 0), the update of the free- sur-face grid points is v12

32

new

=

1 0 Z1

0 1 0

0 0 0

v12

32

old

. (30)

It can be shown that this application of the method of char-acteristics is equivalent to a paraxial approximation (Claytonand Engquist, 1977) in one spatial dimension.

Robertsson (1996) presents a FD method which does notrely on mapping transformations and therefore can handle ar-bitrary topography, although with a staircase shape. The free-surface condition is based on the method of images introducedby Levander(1988). This method is accurate andstable for highvalues of the Poisson ratio. An efficient solution to the stair-case problem is given by Moczo et al. (1997), who propose ahybrid scheme based on the discrete-wavenumber, FD, and FEmethods. These modelingalgorithmsincludeattenuation basedon memory-variable equations (Emmerich and Korn, 1987;Carcione et al., 1988; Carcione, 1994b; Robertsson et al., 1994).

Interface boundary conditions are satisfied explicitly in ho-mogeneous modeling (Kelly et al., 1976). At the interface


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between a solid anda fluid, thenormal particle velocity (ordis-placement, depending on the formulation) and normal stresscomponents are continuous when crossing the interface; in asolid/solid boundary, both horizontal and vertical particle ve-locity components and normal stresses are continuous. How-ever, heterogeneous modeling is preferred when the interfaces

have arbitrary shape. In this case, spurious diffractions arisefrom an inappropriate modeling of curved and dipping inter-faces (the so-called staircase effect). Irregular interfaces andvariable grid spacing are easily handled by FE methods, since,in principle, grid cells can have any arbitrary shape. Whenusing FD and PS algorithms, an averaging method can beused to reduce spurious diffractions arising from the stair-case effect. Muir et al. (1992) used effective media theorybased on Backus averaging to find the elastic constants at thefour grid points of the cell. The modeling then requires ananisotropic stress-strain relation. Zeng and West (1996) obtainsatisfactory results with a spatially weighted averaging of themodelproperties (slownessaveraging, mainly), and ZhangandLeVeque (1997) present a method based on the modificationof the FD scheme in the vicinity of an interface to satisfy the

boundary conditions. Similarly, algorithms based on rectangu-lar cells of varying size allow the reduction of staircase diffrac-tions and the number of grid points (Moczo, 1989; Oprsal andZahradnik, 1999). When the grid points are not chosen in ageometrically regular way, combinations of 1-D Taylor seriescannot be used, and 2-D Taylor series must be applied (Celiaand Gray, 1992, 93).

Absorbing boundaries

The boundaries of the numerical mesh may produce non-physical artifacts which disturb the physical events. These arti-facts are reflections from the boundaries or wraparound as inthe case of the Fourier method. The two main techniques usedin seismic exploration and seismology to avoid these artifactsare the sponge method and the method based on the paraxial

approximation.The classical sponge method uses a viscous boundary or a

strip along the boundaries of the numerical mesh, where thefield is attenuated (Cerjan et al., 1985; Kosloff and Kosloff,1986). When we consider the pressure formulation, equa-tion (1) can be written as a system of coupled equations andmodified as

t

p

q

= 1

L2

p

q

+

0

f

, (31)

where is an absorbing parameter. The solution to this equa-tion is a wave traveling without dispersion but whose ampli-tude decreases with distance at a frequency-independent rate.A traveling pulse will thus diminish in amplitude without achange of shape. In the frequency domain, this implies thatthe stiffness is divided by the factor i + , whereas the den-sity is multiplied with the same factor. Therefore, the acousticimpedance remains real valued. This can be adjusted in sucha way that the reflection coefficient is zero at the onset of theabsorbing strip. An improved version of the sponge method isthe perfectly matched-layer method or PML method used inelectromagnetism (Berenger, 1994) and interpreted by Chewand Liu (1996) as a coordinate stretching. It is based on a(nonphysical) modification of the wave equation inside the

absorbing strips, such that the reflection coefficient at thestrip/model boundary is zero, and there is a free parameter toattenuate the wavefield. The improvement implies a reductionof nearly 75% in the strip thickness compared to the classicalmethod.

The sponge method can be implemented in FE modeling by

including a damping matrix C in equation (19):

KP + C P t

+ M 2P

t2+ S = 0, (32)

with C = M + K, where and are the damping parameters(e.g., Sarma et al., 1998).

For approximations based on the one-way wave equation(paraxial) concept, consider the acoustic wave equation on thedomain x 0. At the boundary x = 0, the absorbing boundarycondition has the general form

Jj=1

(cos j )

t c

x

p = 0, (33)

where |j | < /2 for all j (Higdon, 1991). Equation (33) pro-vides a general representation of absorbing boundary condi-

tions (Keys, 1985; Randall, 1988). The reason for the successof equation (33) is the following. Suppose that a plane waveis hitting the boundary at an angle and a velocity c. In 2-Dspace, such a wave can be written as p(x1 cos +x3 sin + ct).When an operator of the form (cos ),t c,x1 is applied tothis plane wave, the result is zero. The angles j are chosen totake advantage of a priori information about directions fromwhich waves are expected to reach the boundary.

Consider now the approachbased on characteristicvariables(discussed in the Boundary conditions section) and apply itto the S H-wave equation (2) in the plane x3 = 0. The outgo-ing characteristic variable is v2 32/Z. This mode is left un-changed (new = old), while the incoming variable v2 + 32/Z isset to zero (new= 0). Then, the update of the boundary gridpoints is v12

32

new

= 12

1 0 Z1

0 2 0

Z 0 1

v1232

old

. (34)

These equations are exact in one dimension (i.e., for wavesincident at right angles). Approximations for the 2-D case areprovided by Clayton and Engquist (1977).

Example

Modeling synthetic seismograms may have different pur-poses, for example, to design a seismic experiment ( Ozdenvaret al., 1996), to provide for structural interpretation (Fagin,1992), or to perform a sensitivity analysis related to the de-tectability of a petrophysical variable, such as porosity, fluid

type,fluid saturation, etc. Modeling algorithms canalsobe partof inversion and migration algorithms.

Model and modeling design

Designing a model requires the joint collaboration of geolo-gists, geophysicists, and log analysts when there is well infor-mation of the study area. The geological modeling proceduregenerally involves the generation of a seismic-coherence vol-umeto define themain reservoirunitsand theincorporation of


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fault data of the study area. Seismic data require the standardprocessing sequence and prestack depth migration supportedby proper inversion algorithms when possible. A further im-provement is achieved by including well-logging (sonic- anddensity-log) information. Since the logs have a high degreeof detail, averaging methods are used to obtain the velocity

and density field at the levels of seismic resolution.In planning the modeling with direct methods, the following

steps should be followed.

1) From the maximum source frequency and minimum ve-locity, find the constraint on the grid spacing:

d x cmin2 fmax

. (35)

This implies that the spacing should not exceed half thesmallest wavelength in order to avoid aliasing. Theactualgrid spacing depends on the scheme chosen. For instance,an FD scheme which is second-order in time and fourth-orderin space wouldrequire 58grid points perminimumwavelength.

2) Find thenumber of grid pointsfrom thesizeof themodel.3) Allocate additional grid points for each absorbing strip

at the sides, top, and bottom of the model. For instance,the standard sponge method requires four wavelengths,where the wavelength is given by = 2cmax/fd and fd isthe dominant frequency of the seismic signal.

4) Choose the time step according to the stability condi-tion (12) and accuracy criteria. Moreover, when possi-ble, the modeling algorithm used requires testing againstknown analytical solutions to verify its correctness.

5) Define the source-receiver configuration.

Simulation

The 2-D model we consider is shown in Figure 2 with theproperties indicated in Table 1. The low velocities and low

FIG. 2. Geological model. See Table 1 for material properties.

quality factors of medium 7 simulate a sandstone subjectedto an excess pore pressure. All the media have a Poisson ratioequal to 0.2, except medium 7 which has a Poisson ratio of 0.3,corresponding to an overpressure condition.The 2-D modelingalgorithm (Carcione, 1992) is based on a fourth-order Runge-Kutta time-integration scheme and the Fourierand Chebyshev

methods to compute the spatial derivatives along the horizon-tal and vertical directions, respectively. This allows the mod-eling of free-surface boundary conditions. Since the mesh iscoarse (two points per minimum wavelength), Zengand Wests(1996) averaging method is applied to the slownesses to avoiddiffractions due to the staircase effect (the density and therelaxation times are arithmetically averaged). The mesh has135 129 points, with a horizontal grid spacing of 20 m anda vertical dimension of 2181 m with a maximum vertical gridspacing of 20 m. Stress-free and nonreflecting boundary condi-tions of the type of equations (30) and (34) are applied at thetop and bottom boundaries, respectively. In addition, absorb-ing boundaries of the type of equation (31) with a length of 18grid points are implemented at the sides and bottom bound-ary. The source is a vertical force (a Ricker wavelet) applied

at 30-m depth with a maximum frequency of 40 Hz. The wave-field is computed by using a time step of 1 ms with a maximumtime of 1 s (the total wall-clock time is 120 s for an Origin 2000computer with four CPUs).

The seismogram recorded at the surface is shown inFigure 3, where the main event is the Rayleigh wave (groundroll) traveling with wave velocities between the shear veloc-ities of media 1 and 2, approximately. The reflection eventcorresponding to the anticlinal structure can be clearly seenbetween 0.6 and 0.8 s.

INTEGRAL-EQUATION METHODS: HUYGENS PRINCIPLE

As a first step in discussing integral-equation methods, let usconsider Huygensformulation illustrated in Figure 1. Huygens

statesthat thewavefieldoriginatingfrom theflame canbe con-sidered as a superposition of waves dueto pointsourceslocatedin the flame. We can also formulate this in a somewhat moremathematicalway. Consider thescalar wave equation fora ho-mogeneous medium (i.e., with constant density and constantspeed of sound c = c0),

1c20

2

t2

p = q, (36)

Table 1. Material properties for the geological model inFigure 2.

Medium VP (km/s) VS (km/s) Q P QS g/cm3

1 2.6 1.6 80 60 2.12 3.2 1.96 100 78 2.33 3.7 2.26 110 85 2.34 4 2.45 115 90 2.45 4.3 2.63 120 92 2.56 4.5 2.75 125 95 2.67 3.2 1.7 30 25 2.38 4.6 2.82 150 115 2.69 4.8 2.94 160 120 2.710 5.4 3.3 220 170 2.8


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which follows from equation (1) with q = c20 f. The inte-gral representation for this scalar wave equation is given by(Courant and Hilbert, 1937, 403)

p(x, t) =

G(x, xs , t t) q(xs , t) dxs dt, (37)

where x is the position vector. Since the velocity is constant,the Greens function G is given by

G(x, xs , t) =(t |x xs |/c0)

4 |x xs |. (38)

The Greens function is the mediums response to a pointsource, and satisfies the wave equation (36) for a point sourcegiven by q = (x xs )(t). If we substitute equation (38) intoequation (37), we obtain the integral representation

p(x, t) =

D

q(xs , t |x xs |/c0)4 |x xs |

dxs , (39)

where D is the region in space where the source term q ispresent. Equation (39) is the mathematical formulation of

Huygens principle illustrated in Figure 1. The integration is asummation over volumetric wave field densities, q/4 |x xs |.Each element of that distributionresides at a source pointxsat time t and arrives at the point x at the retarded time t|x xs |/c0. The volumetric wave field density is given by theinitial source distribution, q, weighted by the so-called 3-Dspreading factor 4 |x xs |. In order to derive the wave equa-tion and its corresponding integral representation above, oneneeds integral and differential calculus which was not yet de-veloped at the time of Huygens.

Domain integral-equation method

Apart from radiation from sources, as illustrated by equa-tion (39), domain-integral representations can also be usedto formulate and solve scattering problems. Consider, for in-stance, the case of a scattering object V with a sound speed

FIG. 3. Seismogram of the vertical particle velocity.

c(x) embedded in a surrounding medium with constant soundspeed c0. Both scattering object and embedding medium havethe same (constant) density 0. The wave equation then reads,

1

c22

t2p = q, (40)

with c = c0 outside V. This can be rewritten in the followingcontrast formulation:

1c20

2

t2

p = q

1

c20 1

c2

2p

t2. (41)

The wave equation we obtain in this way is very similar toequation (36), the only difference being an additional sourceterm dueto the presence of theobject. Therefore,we canwritethe integral representation, analogous to equation (39), in theform

p = pinc + psc , (42)

where the incident field pinc

, which would have been presentdue to the source q in the absence of the object, is given by theright-hand side of equation (39) and the scattered field is givenby

psc (x, t) =

V

1

c20 1

c(x)2

2

t2p(x, t R/c0)

4Rdx,

(43)

with R =|x x| the distance between observation point x andintegration point x. With the aid of equations (42)(43), thefield outside the object V can be represented in terms of thesources q and the field values inside the object. In order todeterminethefieldvalues insidetheobject, wecan takex insideV in equation (42), substitute psc of equation (43), and obtain

an integro-differential equation for the unknown field p insidethe object V. So far, the acoustic case has been presented herefor the sake of simplicity. The elastic case has been discussedby Pao and Varatharajulu (1976). For an overview of differentmethods for acoustic, electromagnetic, and elastic fields, seeDe Hoop (1995).

In the above, a homogeneous embedding medium was con-sidered. This method is also applicable to heterogeneous em-bedding media, provided the Greens function (the wavefielddue to a point source) can be determined. This is the case forlayered mediausing plane-wavesummation techniques(Ursin,1983).

For special geometries, the volume integral equation ob-tainedherecan besolved with theaid of separation techniques.In general, however, these methods are not applicable and t heequation has to be solved numerically. This can be done withthe method of moments (Harrington, 1968). The first step inapplying thismethod is discretization of the unknown functionwith the aid of expansion functions, followed by a weighting ofthe integral equation usingappropriateweightingfunctions. Toillustrate this procedure, we apply it to the integral-equationformulation given by equations (42)(43). First, we subdividethe volume V of the scatterer into smaller volumes Vm withcenters xm (m = 1, . . . M) and assume that the pressure is con-stant in each subvolume. If we then enforce the equality sign at


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the center points at times tj , we obtain the following relation:

p

xn, tj = pincxn, tj+ M

m=1

Vm

1

c20 1

c(x)2

2t p

x, tj Rnc04Rn dx, (44)

with n = 1, 2, . . . M and Rn the distance between xn and x. Ifwe express the time derivative under the integral in backwardtime differences, we obtain an explicit time-stepping algorithmwhere new values in each point can be computed from previ-ously computed values. In this way, no matrix inversion is re-quired. A disadvantage of this method, however, is that onehas to be aware of potential instabilities and that previouslycomputed values in all points have to be stored. If the objectis large compared to the pulse width of the field, this approachbecomes impractical. An alternative is to formulate the prob-lem in the frequency domain and solve the resulting system ofequations separately for each frequency. This circumvents thestorage and stability problems but can still be inefficient if theobject is large compared to the wavelength due to the fact that

the system matrix is full (in contrast to FD methods). There-fore, this type of volume integral-equations is especially suitedto the case of objects of modest size.

Boundary integral-equation methods

In Figure 4, Huygens principle is illustrated in a differentform. Each point on a wavefront can be considered as thesource of waves propagating away from that point. In order toobtain a mathematical formulation corresponding to the situa-tion of Figure 4, we again consider the wave equation, but thistime in a domain D containing a bounded object with bound-ary S(D is the regionoutside S).On S, we prescribean explicitboundary condition, for example a pressure-release boundarycondition. We then have the following relations:

1c20

2

t2p(x, t) = q(x, t), (x D), (45)

FIG. 4. Illustration of Huygens principle showing that eachpoint on a wavefront acts as secondary source for sphericalwaves [taken from the 1690 Trait e de la lumi`ere (Huygens,1990)].

and boundary condition

p(x, t) = 0 (x S). (46)

As in the previous section, the pressure in D can now be de-composed into two contributions,

p = pinc + psc , (47)

where the incident field, pinc, is again given by the right-handside of equation (39), andthe scatteredfield, accounting forthepresence of the object, is given by the integral representation

psc (x, t) =

S

n p(x, t R/c0)4R

dx, (48)

where n is the outward pointing normal vector on S. This canbe derived by applying theboundarycondition (46) to theinte-gral representation for the scalar wave equation in a boundeddomain (see, for instance, Bennett and Mieras, 1981). Theinte-gration over Sis in fact a summation of point-source wavefields

located on Sand is therefore a mathematical representation ofthe same idea illustrated in Figure 4. With the aid of equa-tions (47)(48), the field ouside the object can be representedin terms of sources q and certain field values at the boundaryof the object. In order to determine these field values, we canlet the point of observation approach the object and enforcetheboundarycondition (46). In this way, we obtainthe integralequation

pinc(x, t) =

S

n p(x, t R/c0)4R

dx (x S).(49)

From this equation, the field quantity n p on S can be de-termined. In general, this integral equation has to be solvednumerically using methods similar to those discussed for the

case of the volume-integral equation. In the discretization pro-cedure, proper care has to be taken of the singularity of theGreensfunction atx = x. Here,we have only discusseda scalarexamplefor the pressure-releaseboundarycondition. The elas-tic formulation for the above problem is given by Tan (1976).Scattering of elastic waves by cracks or cavities has also beendiscussed by Bouchon (1987) and, more recently, elastic scat-tering by hydrofractures was discussed by Pointer et al. (1998).The application of integral-equation techniques for studyingwave propagation in media with a number of cracks was dis-cussedby Liuet al.(1997),whereas wave propagation in mediacontaining large numbers (several thousands) of small crackswas studied by Muijres et al. (1998). In particular, the case ofmany small cracks can be solved very efficiently with the aidof boundary-integral equations sinceeach crackis representedby only one unknown coefficient which represents the jump innormal velocity.

Boundary-integral equations are also very well suited to ac-curately model geometries with either explicit boundary con-ditions or discontinuities in properties. For instance, scatteringof elastic waves by a rough interface between two solids wasdiscussed by Fokkema (1980). The geometry of irregular bore-holes was considered by Bouchon and Schmidt (1989), andthe radiation of seismic sources in boreholes in layered andanisotropic media was discussed by Dong et al. (1995).


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Integral-equation modeling example

In order to investigate theeffect of small-scale scattering ob-jects like cracks or inclusions, integral equations can be useful.If the cracks are small with respect to the dominant seismicwave length, each crack only adds one unknown to the prob-

lem, which implies that largenumber of crackscan bemodeled.In Figure 5, 4000 cracks are randomly positioned in a homoge-neous embedding medium with sound velocity c0 = 3000 m/s.After choosing the wave form of the incident pressure field (inthe frequency domain) and solving the relevant integral equa-tion numerically for each frequency, the resulting total trans-mitted field at the receiver is shown in Figure 6 in comparisonwith the incident field. This method is described in more detailin Muijres et al. (1998).

ASYMPTOTIC (RAY-TRACING) METHODS

The integral representations discussed in the previous sec-tion areespeciallyuseful if themedium is homogeneous. In thecase of heterogeneous media, the computation of the Greensfunction is tedious. In that case, asymptotic methods can be an

FIG. 5. Model consisting of 4000 cracks having a width of 1 meach. The speedof soundin theembeddingmedium is3000m/s.The receiver location is indicated with o; the incident wave isa plane pressure wave propagating downwards.

FIG. 6. Normalized incident pressure field recorded by the re-ceiver in the absence of cracks and transmitted total field ac-counting for the presence of the cracks.

attractive alternative. Before we discuss the asymptotic meth-ods, we first return to equation (38), the Greens functionfor a homogeneous medium. Clearly, this function describesa spherical wavefront which propagates at speed c0. An initialdisturbance of the medium at t= 0 and x = xs is spreading inspace as time proceeds; at time t, it has arrived at the sphere

|x xs |= c0t. The function(x, xs ) = |x xs|/c0 (50)

measures the time needed for the disturbance to travel fromthe source location xs to the point x. It is called the travel-time function. The amplitude of the disturbance is given by thefunction

A(x, xs ) =1

4 |x xs|. (51)

With these twodefinitions, we canrewritethe Greens functionfor a constant velocity medium as

G(x, xs , t) = A(x, xs )(t (x, xs)). (52)

In what follows, we work with the temporal Fourier transformof the Greens function, which is denoted by G(x, xs , ). Forthe constant velocity case, it is of the form

G(x, xs , ) = A(x, xs )ei(x,xs ). (53)In general, the velocity of course varies with position x. Ingeophysical applications, this is predominantly caused by tran-sitions from one geological formation to another, but there arealso other causes, such as varying geopressure or fluid con-tent of the rocks. From physical intuition, one would expectthat the solution of the wave equation (36) still behaves likea propagating wavefront in this case. This can be made moreprecise mathematically by studying the high frequency behav-ior of G(x, xs , ). It can be shown that in the limit (hence the name asymptotic) G(x, xs , ) is still of the form

of equation (53), where (x, xs ) and A(x, xs ) are general trav-eltime and amplitude functions. The wavefront is no longerspherical in the heterogeneous case, and the traveltime andamplitude functions are no longer given by the simple explicitrelations (50) and (51) above.

Eikonal and transport equations

We illustrate the use of asymptotic methods by constructingthehigh-frequencybehavioroftheGreensfunctionG(x, xs , )away from the location of the source (i.e., for x = xs ). We startby substituting the expression (53) into the Fourier transformof the wave equation. This equation is given by

(i )2

c(x)2 G(x, xs , ) = 0 (54)

and is known as the Helmholtz equation. The result is

((i)2[()2 c2(x)]A+ i[2A + A] + A)ei = 0. (55)

Dividing out the factor ei , we obtain a polynomial in i,which is equated to zero. Since this equation has to hold for all, all the coefficients of the polynomial have to be zero.


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From the coefficient of (i)2, we find that the traveltimefunction has to satisfy the so-called eikonal equation

()2 = c2(x). (56)Similarly, from the coefficient ofi we find

2A + A = 0. (57)This equation is called the transportequationfor the amplitudefunction. These two equations, when supplemented with suit-able initial conditions, determine the traveltime and amplitudefunction uniquely. For a constant velocity, this leads to an am-plitude given by equation (51) for which A = 0. For spatiallyvarying velocity, A = 0, which implies that the coefficient of(i )0 in equation (55) is not equal to zero. Therefore, one isforced to replace the simple solution (53) by a power series in(i )1, i.e.,

G(x, xs , ) =k0

(i)kAk(x, xs)ei (x,xs ). (58)

To determine the equations for the higher order amplitudesAk+1 (k 0), one substitutes this expression in the Helmholtzequation andequates thecoefficientsof allpowers of to zero.Besides the eikonal equation (56)for (x, xs ) andthe transportequation (57) for A0(x, xs ), one finds for k 0 the followinghigher order transport equations:

2Ak+1 + Ak+1 = Ak. (59)This showsthat a solution of thewaveequation of theform(58)can be constructed by first solving the eikonal equation for(x, xs ), subsequently the transport equation for A0(x, xs ), andthen solving the higher order transport equations recursivelyfor Ak+1(x, xs ) (k 0).

So we arenaturally ledto an infinite powerseries (58) in 1/,which converges for || 0 0. If0 = 0, it converges for all

frequencies; if 0 = , the series is meaningless. If the seriesconverges uniformly in x for a given frequency, it is a solutionof the Helmholtz equation for that frequency. The fact thatthe series (58) in general diverges for small frequencies meansthat we have only constructed the solution of the Helmholtzequation for large frequencies. Therefore, ones speaks of anasymptotic solution of the Helmholtz equation.

Let us try to assess the consequence of knowing only a high-frequency solution for the time-domain solution. To this end,we take theinverse Fouriertransform ofG(x, xs , ), which canbe written as

G(x, xs , t) =1

2

00

G(x, xs , )ei t d

+

1

2 ||>0 G(x, xs , )eit d. (60)

The first term is not known (at least not by the procedure out-lined above) unless 0 happens to be zero. The only thing thatcan be said about it is that it must be a smooth function oft, x and xs , since differentiation can be done under the integralfor an integral over a finite interval. Equivalently, the secondterm of equation (60) is representative for the singular (i.e.,nonsmooth) behavior of the solution. The smooth part of thesolution has a frequency dependence that must decay faster

than any algebraic power of and can therefore not be repre-sented by a power series.

If w() is a smooth taper function satisfying, e.g.,

w() =

0 if|| 01 if

|

| 20,

(61)

we can write

G(x, xs , t) 1

2

w()G(x, xs , )e

it d, (62)

where the sign means that we have neglected a smooth func-tion. The taper function effectively acts as a high-pass filter.We can now substitute the series (58) in the right-hand side ofthis relation. The term with k= 0 transforms into A0w(t )[which equals A0 (w )(t), a high-pass version of a propa-gating wavefront]. Similarly, the term with k= 1 transformsinto A1 (w H )(t), where H is the Heaviside step function,given by

Ha(t)

= 0 ift < a

1 ift a.(63)

This is again a propagating wavefront. The Heaviside functionis discontinuous at the wavefront (x, xs ) = t, but the discon-tinuity of the Heaviside function is of course less severe thanthe singularity of the delta function. Similarly, the higher orderterms, which can be obtained by repeatedly integrating withrespect to t, become less and less singular.

Usually, onebreaks offthe solution after thefirst term. Fromtheabove, it is clear that this means that oneonly considersthemost singular part of the solution. Moreover, for the constantvelocity case, the first term coincides with the full solution.

Constructing the asymptotic solution

Before discussing numerical methods, we first explain howto construct a solution from a theoretical point of view by themethod of characteristics. Numerical methods for solving theeikonal and transport equations relyheavily on this theoreticalconcept.

Method of characteristics: seismic rays

The method of characteristics is in fact a general solutionmethod for first-order partial differential equations, of whichthe eikonal equation is an example [see, e.g., Courant andHilbert (1966) for a classical reference]. The general idea isto construct curves x() along which the partial differentialequation reduces to an ordinary differential equation.

To explain this concept in the case of the eikonal equation,we again first consider the simplest case of a constant velocity.

In that case thewavefronts (i.e., thesurfacesof constant travel-time t) are spheres of radius ct centered at the source locationxs [see equation (50)]. Moreover, choosing an arbitrary point xon a wavefront = t, the line from xs to x is orthogonal to thatwavefront and hence in the direction of the gradient(x, xs ).Therefore, this line can be parameterized as

x() = xs + (x, xs ) = xs +

c

x xs|x xs |

, (64)


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where we have scaled the parameter in such a way that

dx

d= = 1

c

x xs|x xs|

. (65)

Differentiating the traveltime function (x, xs ) along this linex(), we obtain

d

d= dx

d = ()2 = 1

c2, (66)

which is indeed an ordinary differential equation for the trav-eltime. Given the line and this equation, the traveltime func-tion is easily retrieved. Integrating with respect to , onefinds = /c2. Moreover, from equation (64) one finds that = c|x xs |, and hence (x, xs ) =|x xs |/c.

In conclusion, we can say t hat, for the constant velocity case,it is sufficient to have the family of lines originating from thesourcelocation(these arethe characteristic curvesin this case)and the ordinary differential equation (66) along these lines tosolve for the traveltime function.

In the general case of a spatially varying velocity, we wouldalso like to construct curves x() satisfying dx/d

= . Un-

fortunately, to construct these curves, one would have t o knowthe solution of the eikonal equation up front, whereas thewhole purpose of constructing characteristics is to find this so-lution. To circumvent this problem, we differentiate once morewith respect to :

d2x

d2= d

d

= dxd

()= ()= [()2/2]= [c2/2]. (67)

Hence, we find a second-order equation for the curve x()which does not depend on the solution . This second-orderequation is equivalent to the system of first-order equations

dx

d= p,

(68)dp

d= [c2/2].

The solution curves (x(), p()) of this system are calledbicharacteristic curves. They belong to the phase space{(x, p) | x, p R3}. The curves x(), which live in the configu-ration space R3, are called the characteristic curves or, briefly,the characteristics of the eikonal equation. In seismology, theyare also referred to as (seismic) rays.

The system (68) can be solved if we specify initial conditions

x(0) and p(0). We choose

x(0) = xs , p(0) =1

c(xs )

sin 1 cos 2sin 1 sin 2cos 1

. (69)Thus, all rays start at xs for = 0 and leave the source locationin the direction specified by the angles 1 and 2. The solutioncurves depend smoothly on the initial conditions xs , 1, and 2.

If the source location is fixed, we write x = x(1, 2; ) andp = p(1, 2; ) todenote thedependenceon thetake-off anglesof the ray at xs .

In fact, the solution curve x(1, 2; ) defines a map(1, 2, ) x(1, 2; ) which can be seen as a coordinatetransformation provided, of course, that the Jacobian

J(1, 2, ) =(x, y,z)

(1, 2, )(70)

does not vanish. Notice that for the constant velocity case, thisis just the t ransformation from spherical to rectangular coordi-nates, and the Jacobian indeed does not vanish away from thesource location.

In order to construct a solution (x, xs ) of the eikonal equa-tion, we supplement (just as in the constant velocity case) theequations (68) with the equation

d

d= 1

c2, (0) = 0. (71)

Integrating this equation along a characteristic curvex(1, 2

;), one finds

(1, 2; ) =

0

d

c2(x(1, 2; )). (72)

If the Jacobian Jintroducedin equation (70)is indeed nonzero,this function can also be seen as a function ofx:

(x(1, 2; ), xs ) = (1, 2; ). (73)Notice that equation (68) is a system of Hamilton equationsfor the Hamiltonian H, given by

H(x, p) = (p2 c2(x))/2. (74)This system can be written as

dx

d =

H

p

,

(75)dp

d= H

x.

Let H(1, 2, ) = H(x(1, 2; )).Then, because of theseequa-tions, H is independent of:

H

= dx

d H

x+ dp

d H

p= 0. (76)

Moreover, because of the initial conditions (69), H(1, 2, 0) =0. Therefore, the Hamiltonian vanishes identically along thebicharacteristics:

H(1, 2, ) = 0. (77)From the definition of H, we see that this means noth-ing else than that the length of the tangent vectorp(1, 2; ) = dx(1, 2; )/d is always 1/c(x). For this reason,p is often called a slowness vector in geophysics.

The only thing left to show is that the function (x, xs ) thatwe have constructed in this waydoes indeed satisfy theeikonalequation. Since we have just shown that p2 = c2, it is sufficientto show that p =, just as in the constant velocity case. Aproof of this relation can be found in Courant and Hilbert(1966).


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Transport along the rays

Now let us turn our attention to the transport equation (57).This equation can also be rewritten as an ordinary differentialequation along the rays. The gradient of the traveltime func-tion in a point x(1, 2; ) is given by the momentum vectorp(1, 2; ) which, according to Hamiltons equations (68), isequalto thetangentvectordx/d. Therefore,the termA represents the derivative d A/d of the amplitude along a ray.Hence, we can cast the transport equation into the form

1

A

d A

d= 1

2. (78)

In fact, there is a simpler form of the transport equation whichuses the identity

= 1J

d J

d. (79)

This follows from the definition (70) of the determinant J andHamiltons equations ( Cerveny et al., 1977; Cerveny, 1985,1987). Using this identity, we find

d

d(

J A) = 0. (80)

In other words, the quantity

J A is equal to a constant Calong a ray. The constant C can be found by requiring thatthe amplitude A approaches the constant velocity amplitudeA given by equation (51) for 0 [see, e.g., Bleistein (1984)].The result is

C = 14

sin 1

c(xs). (81)

So we get

A(x, xs)=

1

4 sin 1c(xs )J(1, 2; ) . (82)This result shows that the amplitude of the Greens functioncan be calculated from the determinant of the matrix

Q =

x

1,

x

2,

x

. (83)

If we also introduce the matrix

P =

p

1,

p

2,

p

, (84)

we can derive from Hamiltons equations that the matrices Qand P satisfy the system of ordinary differential equations

d Q

d =P,

d P

d =Qt

(c2/2). (85)

Solving the Hamilton equations (68) together with the sys-tem (85) is called dynamic ray tracing [see, e.g., Cerveny et al.(1977) and Cerveny (1985, 1987) for a standard reference]. Itenables us to calculate the traveltimes and amplitudes neces-sary for the computation of the asymptotic Greens function.The elements of thematrices Q and P areusedin many asymp-totic calculations, e.g., in true amplitude Kirchhoff migration[see, e.g., Bleistein et al. (2001) for a recent reference].

Caustics

In the previous section, we saw that the determinantJ(1, 2, ) defined in equation (70) plays an important role.If it is nonzero, the transformation (1, 2, ) (x1,x2,x3) isa coordinate transformation, which enables us to consider

the traveltime function in equation (72) and the amplitudefunction A in equation (82) as functions of (x1,x2,x3) ratherthan of (1, 2, ). Moreover, J occurs explicitly in the expres-

sion (82) forthe amplitude. A pointx0((0)1 ,

(0)2 ,

(0)), forwhich

J((0)1 ,

(0)2 ,

(0)) vanishes, is called a caustic point. From equa-tion (82), it is clear that we are in serious trouble here, sincethe amplitude A(x0,xs ) would be infinite. Because of this, it isimpossible to find a solution of the simple form Aei for theGreens function at a caustic point.

Let us analyze from a geometrical point of view the vanish-ing of the determinant. To this end, we consider the collectionof all solutions of the ray equations {x(1, 2; ), p(1, 2; )}in phase space, the 6-D space spanned by all (x, p). Clearly,this collection is a smooth 3-D subset of phase space, whichis parameterized by the take-off angles 1, 2 and the flow pa-

rameter

along the bicharacteristics. The tangent vectors inthe direction of the 1, 2, and -coordinate axes are givenby (x/1, p/1), (x/2, p/2), and (x/,p/), re-spectively. Projecting these vectors to the 3-D physical space,we get x/1, x/2, and x/. The condition J= 0 meansnothing else than that these vectors are dependent, i.e., thatthey span a space of dimension lower than three. This impliesthat the space, spanned by these tangent vectors, has at leastone direction which is purely vertical, i.e., in the p-direction.

As a consequence, the set of bicharacteristics may turnin phase space. This situation is illustrated in Figure 7. Thesolid curve in this figure is a cartoon of the 3-D set of bichar-acteristics in phase space. The point x0 is a caustic point; it isright underneath a point (x0, p0) where the set of bicharacter-istics has a vertical tangent direction. The point x exemplifiesa point in configurationspacewhereseveralrays intersect. No-

tice that at the point of intersection the two rays have differ-ent directions given by the slowness vectors p(

(1)1 ,

(1)2 ,

(1))and p(

(2)1 ,

(2)2 ,

(2)). Also, at such a point x there are twodistinct traveltimes (

(1)1 ,

(1)2 ,

(1)) and ((2)1 ,

(2)2

(2)) de-pending on whether the journey from xs to x

is undertakenalongthefirstorthesecondray.Sooncetherearecaustics,there

FIG. 7. Turningof theset of bicharacteristics in phase space. Atthe caustic point x0, the set of bicharacteristics has a verticaltangent direction.


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are several rays in general, all with different slowness vectorsp(i), passing through thesamepoint in physical space. The trav-eltime function (1, 2, ) becomes a multivalued function inx-coordinates: (i)(x) = ((i)1 ,

(i)2 ,

(i)). Because of this, andbecause of the fact that the gradient vectors (i) jump dis-continuouslyfrom onebranch of thetraveltime function to the

other, we cannot speak of a classical solution of the eikonalequation anymore. Still, the different branches are importantfor imaging in complex geologies [see, e.g., ten Kroode et al.(1998) and Operto et al. (2000)].

Figure 8 illustrates the concept of caustics and multivaluedtraveltime functions in the case of two spatial dimensions. Thefigure shows a fan of rays originating from a common sur-face location (xs , 0). The velocity model is a smoothed ver-sion of the well known Marmousi model (Versteeg and Grau,1991). Also indicated are some wave fronts (i.e. surfaces ofconstant traveltime) perpendicular to the rays. Clearly, thereare region

carcione et. al. 2002 seismic modeling

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