seismic forward modeling - arizona state...

Introduction

The technique of forward modeling in seismology beginswith the numerical solution of the equation of motion forseismic waves, or more specifically, the numerical compu-tation of theoretical or synthetic seismograms, for a givengeological model of the subsurface. The idea is then tocompare the synthetic seismic traces with real seismic dataacquired in the field. If the two agree to within an accept-able level of accuracy, the given geological model can betaken to be a reasonably accurate model of the subsurface.If not, the geological model is altered, and new synthetictraces are computed and compared with the data. Thisprocess continues iteratively until a satisfactory match isobtained between the synthetics and the real data. Theforward modeling approach is, in a sense, the opposite ofthe inverse modeling approach in which the parameters ofthe geological model are computed from the acquired realdata. Both methods though ultimately have the same goal– the determination of the geological stru c t u re andlithology of the subsurface.

As mentioned above, in forward modeling, one attempts tosolve the equation of motion for seismic waves, which isnothing more than the mathematical expression ofNewton’s second law of motion, i.e., force = mass x accelera-tion, for material particles in a solid body set in motion byelastic waves. It is a second-order partial differential equa-tion for the vector displacement u experienced by a pointor particle in a solid medium due to the passage of a wave.Once u is known (after solving the equation of motion), thedisplacements of geophones can be computed andsynthetic seismograms can be produced. There are anumber of computational methods, and variants thereof,that are commonly used to compute the synthetic seismictraces, such as ray theory, the finite difference method, andthe reflectivity method. All the methods have their indi-vidual advantages and disadvantages, and an in-depthmodeling effort for a given data set can involve the use ofseveral methods. In this article, I will discuss some of thebasic principles and concepts of some of these methods.Many seismologists have contributed to the developmentof these methods, and it is unfortunately not possible torepresent all of them in this article.

Ray theory

Ray theory can be used to compute seismic wave traveltimes and amplitudes along ray paths in a heterogeneousmedium when the frequencies present in the wave are highenough so that the “geometrical optics” approximation canbe used. As a rule of thumb, in order for ray theory to beapplicable, the medium parameters (e.g., the Lamé param-eters λ and µ, or the P and S wave speeds) should not

change very much over distances of the order of the domi-nant wavelength. This rule of thumb also implies that“high frequency” is a relative term – ray theory could alsobe used with frequencies that would be considered “low”in certain situations, as long as the heterogeneity is so weakthat the rule of thumb applies. Under these conditions ofhigh frequency, the travel time T(x) of the wave from thesource to a point x = (x,y,z) in a heterogeneous isotropicmedium obeys the eikonal equation,

a non-linear partial differential equation, where υ = υ(x) isthe seismic wave speed (for either a P or S wave) at thepoint x. The eikonal equation (1) can be obtained by substi-tuting a trial solution for u into the equation of motion andmaking the appropriate high-frequency approximations.

Much recent work has been done on computing traveltimes by solving the eikonal equation, especially using thefinite difference method (see below), which is computa-tionally efficient (see, e.g., Vidale, 1990; van Trier andSymes, 1991; and Kim and Cook, 1999).

To actually trace a ray through a medium, a set of “rayequations” must generally be solved. Consider first a singlei s o t ropic heterogeneous medium in which υ v a r i e ssmoothly with x. For a given wave speed υ(x), the geomet-rical ray path can be determined, i.e., the coordinates of anyand all points x on the ray path can be computed, by solvingthe following system of ordinary diff e rential equations:

Seismic Forward ModelingE. S. Krebes, Department of Geology and Geophysics, University of Calgary, Calgary

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F i g u re 1. A ray path in a vertically heterogeneous medium in which thewave velocity υ varies with depth z as υ = exp(z2). The horizontal andvertical directions are offset and depth, re s p e c t i v e l y. The ray starts at theorigin with a take-off angle of 5.74º and has a travel time of 1.79 s.

where s is the arc length along the ray path. q ≡ ∇ T is called theslowness vector. It points in the direction of travel of the wave,i.e., it is tangential to the ray path at any point on the ray path.Note that the eikonal equation (1) can be written as q•q = q2 =1/υ2. The quantity q = 1/υ is the reciprocal of the wave speed(hence the term “slowness”). For convenience in numericallysolving (2), ds is often replaced by υdT in (2), so that the moreuseful quantity T (the travel time) is used in the equations,rather than the arc length s. Figure 1, which was produced bynumerically solving (2), shows an example of a ray path in amedium in which the velocity υ varies with depth z as υ =exp(z2).

The standard ray tracing equations (2) can be modified anda p p roximated to simplify and facilitate ray tracing in complexsubsurface stru c t u res. For example, consider a geometrical ray Rgoing between two points in a certain region of the subsurface,and suppose that the paths of the nearby adjacent rays in thisregion do not deviate much from the ray R. Such nearby rays arecalled paraxial rays. In this case, first-order Taylor expansions canbe used to approximate the wave speed υ = υ(x) for the paraxialrays. Then, using a coordinate system centred on the ray re s u l t sin a coupled system of linear diff e rential equations for thecomponents of slowness and the coordinates, which are gener-ally easier to solve than (2). Such methods, and modificationst h e reof, are generally called paraxial ray tracing methods ord y n a m i c ray tracing methods (see, e.g., âe r v e n˘, 2001).

Ray amplitudes can also be computed, in the high-frequencyapproximation, by solving the so-called transport equation forthe wave. For example, in the relatively simple case of anacoustic wave, the transport equation for the amplitude A(x) ofthe pressure wave at the point x is

Once T is known (e.g., by solving the eikonal equation 1), A canbe computed, in principle.

Tracing a ray between known source and receiver points in astructurally complex geological model of the subsurface isgenerally done by a ray shooting method, in which the take-offangle of a ray (the angle at which the ray leaves the sourcepoint) is determined iteratively (essentially through trial and

error). In some of the simpler cases however, mathematicalformulas can be developed which allow one to compute, rela-tively easily, the travel times and amplitudes for the rays at anyoffsets without actually “tracing” or “shooting” any rays. Forexample, consider the case of a geometrical ray in a mediumconsisting of a stack of flat homogeneous isotropic horizontalelastic layers, a model commonly used in land exploration seis-mology. Figure 2 shows an example of such a ray. If the sourceis a spherically symmetric point source located on the surface,then the travel time and amplitude of the wave arriving at thereceiver can be computed from relatively simple algebraicformulas as follows.

First, the desired source-receiver offset X is chosen. Then theray parameter p for the ray going from the source to the receiveris computed by numerically solving the equation

for p, where m is the number of ray segments, and hj and υj arethe layer thickness and wave speed, resp., for the jth raysegment (assumed to be known). p is simply the horizontalcomponent of the slowness vector along the ray, i.e., p = qx .Snell’s law states that p is constant along the ray path, i.e., p =sin(θj)/υj , j = 1, …, m, where θj is the angle that the jth raysegment makes with the vertical axis. Consequently, once p isdetermined, the take-off angle θ1 of the ray can be computed ifneeded, as well as all the other angles θj . Equation (4) can besolved easily using any root-finding method, e.g., the Newton-Raphson method. Once p is known, the travel time T of the raycan be computed from

The amplitude A(ω) of the wave at the receiver for a singlefrequency ω can be computed from

where Y is the product of displacement reflection and trans-mission coefficients along the ray path, and L is the geometricalspreading factor which gives the amplitude loss due to thegeometrical spreading of the wavefront, and i is the imaginaryunit. For a ray such as the one in Figure 2, L is given by

A complex exponential is used in (6) for mathematical conven-ience – it is the real part that re p resents the physical sinusoidalwave. Sometimes in practice, L is roughly estimated simply bythe total length of the ray path. However, this estimate can bevery inaccurate in general, differing from the correct L c o m p u t e df rom (7) by a large amount (e.g., a factor of 2 or 3 or more ) .

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F i g u re 2. A typical geometrical ray of m segments in a layered medium.


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If the mathematical form of the source pulse is known, then theamplitude of the waveform (consisting of a superposition offrequencies) at the receiver can also be computed. For example,the displacement u of the receiver is given by an inverse Fouriertransform, i.e.,

where S(ω) is the frequency spectrum of the source pulse, andb, the polarization vector, is a unit vector in the positive direc-tion of displacement (defined by convention). For example, foran incoming P wave traveling in the xz plane, b = [sin(θm), 0,cos(θm)] = [pυm, 0, cm]. If the free surface effect is included, bwould be more complicated – it would have to include the freesurface reflection coefficients (the free surface effect takes intoaccount the fact that the displacement of the receiver is affectednot only by the incoming wave, but also by the waves reflectedoff the surface). Equation (8) is normally computed using a fastFourier transform algorithm.

The above method can then be used to compute synthetic seis-mograms which can be compared with real data to obtain ageological model of the subsurface. For sub-critical offsets, thismethod can also be easily extended to the case of absorbing(dissipative, anelastic) layers – one simply goes through thesame calculations (4)-(8) but with complex-valued andfrequency-dependent wave speeds υj (Hearn and Krebes, 1990).For example, if the quality factors Qj (typically frequency-inde-pendent) for each ray segment are known (Qj = ∞ for no absorp-tion), then υj ≈ Vj (ω)[1 - i/(2Qj )], where Vj is the real wavespeed. One then obtains, by solving (4), a complex-valued p,which one then uses to compute a complex-valued travel time

T from (5). The real part of T, i.e., Re(T), gives the actual traveltime of the wave, and the imaginary part, Im(T), gives theabsorption factor, resulting in an exponential amplitude decaydue to absorption (in (6), eiωT becomes eiωT = e-ωIm(T) eiωRe(T) ). As p iscomplex, Y and L in (6) also become complex, as well as b in (8).These results are then used in (8) to compute the waveform.For super-critical offsets, care must be taken to correctly choosethe signs of the square roots (cj ) in the above formulas, to avoiderrors. For example, the waveform discrepancy at 1.5 kmshown in Figure 12 of Hearn and Krebes, 1990, is due to an erro-neous sign choice, and not due to anelastic effects.

One of the main drawbacks of the ray method is that it is notaccurate near critical offsets. For instance, for the ray in Figure 2,if the incidence angle at the deepest reflection point were nearthe critical angle for the P or S transmitted wave (at which thetransmitted wave becomes evanescent, and propagates along theinterface), then the ray amplitude computed from (6) and (8)above would not be accurate. Corrections can be made to raytheory to improve the accuracy in these cases, but they ofteninvolve elaborate and cumbersome mathematical extensions,and they are often not generally or easily applicable( e . g . ,âe r v e n˘ and Ravindra, 1971; Plumpton and Tindle, 1989;Gallop, 1999; Thomson, 1990; Aki and Richards, 2002). Figure 3shows a set of seismic traces in which basic ray theory ampli-tudes agree well with exact calculations for pre-critical off s e t s ,and reasonably well for post-critical offsets, but not at all forn e a r-critical offsets. Including a mathematical correction to thebasic ray theory result in Figure 3 to include the head wave (therefraction arrival) would have improved the accuracy in thepost-critical zone, and the extent of the zone of inaccuracys u r rounding the critical offset would also have been reduced –but not eliminated. Figure 4 shows a comparison of the ray

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F i g u re 3. Synthetic SH seismograms for two anelastic isotropic half spaces sepa-rated by an interface, with the source and receiver 500 m above the interface. Thedensities, shear wave speeds, and QS values in the upper and lower half spaces are1.5 and 2.0 g/cm3, 1.0 and 2.0 km/s, and 50 and 100, re s p e c t i v e l y. The dominantf requency of the wavelet is 50 Hz. The critical offset is 0.577 km. The solid linetraces give the exact response (computed by the ω-k method of Abramovici et al.,1990) and the dashed line traces are computed by ray theory. The figure is takenf rom Le et al. (2004).

F i g u re 4. Amplitude versus offset curves for the SH case for two elastic isotro p i chalf spaces separated by an interface, with the source and receiver 1000 m abovethe interface. The densities and shear wave speeds in the upper and lower halfspaces are 2.0 and 2.0 g/cm3, and 1.0 and 2.0 km/s, re s p e c t i v e l y. The frequency is20 Hz. The graph shows the exact result computed by numerical evaluation of thegeneralized reflection integral, and the approximate result computed with rayt h e o r y.

theory amplitude and the exact amplitude against offset. A g a i n ,we see that ray theory amplitudes are quite accurate for smallo ffsets corresponding to sub-critical angles, are reasonably accu-rate for super-critcal offsets, but not at all accurate for near- c r i t-ical offsets. For deep reflectors in horizontally layered media, thereflection angles are generally small (well below critical), and raytheory can safely be used to produce synthetic seismograms.H o w e v e r, for shallow reflectors with large velocity contrasts,reflection angles can be large for the larger offsets, in which caseray theory could not be used for near-critical angles and off s e t s .Reflections near critical angles could also occur for deeper re f l e c-tors if they are dipping.

Ray theory is also inaccurate near caustic zones, where raysfrom a single source converge or focus because of changingvelocity gradients, for example. Focusing of wave energy obvi-ously produces high amplitudes (e.g., a magnifying lens canfocus sunlight enough to burn paper), but in the ray theoryapproximation, the computed ray amplitudes are infinite atcaustics, because the geometrical spreading factor L → 0 there.L is a measure of the spreading of the wavefronts, and so whenthey converge rather than spread, L decreases (and vanishes atthe caustic in ray theory).

In addition, as ray theory is a high frequency approximation, orlow wavelength approximation, it cannot be applied accuratelyto structures with thin layers, “thin” meaning thinner than adominant wavelength, roughly speaking.

Ray theory also does not produce the complete or full wavefield: ray-synthetic seismograms do not contain the waveformsfor all the waves arriving at the receiver, but only those for thespecific rays selected by the user. Of course, one may include asmany rays as one wants in a typical computation, but one isoften not sure in advance whether a particular ray will have ahigh enough amplitude to make it worthy of inclusion(although some progress has been made in this regard in deter-mining threshold amplitudes of rays – see, e.g., Hron, 1971,1972).

Ray theory can also be extended and applied to more complexcases involving other types of sources, laterally heterogeneouslayers, interfaces of any dip and strike, curved interfaces, andanisotropic media (e.g., Richards et al, 1991; Chapman, 1985,âe r v e n˘, 2001; Aki and Richards, 2002; P‰enãík et al., 1996).However, the additional mathematical theory and numericalcomputations involved are generally extensive, and the finalresults still generally include the inaccuracies mentioned in thepreceding paragraphs.

Nevertheless, in spite of the drawbacks, basic ray theory is stillwidely used to compute synthetic seismograms, because of therelative simplicity of the simpler versions of the method, thefast computational times, and the fact that the ray paths for allthe events on ray-synthetic seismograms can be identified fromthe event travel times (because the rays comprising thesynthetic are chosen by the user). In particular, ray theory isvery useful for computing travel times, if not always signalamplitudes. However, for more accurate amplitude computa-tions, and for synthetics which include all the possible wavesthan can arrive at a receiver, one must resort to exact or nearly-exact full-wave methods, such as those discussed below.

Generalized Ray Theory

The geometrical ray theory result in (6) and (8) above, anapproximation to the exact wavefield, makes use of only onevalue of the ray parameter p, namely, the value (call it p0) whichsatisfies (4), the equation for X. A more accurate result wouldinvolve an appropriate superposition of waves with differentray parameter values.

A simplistic way to see this is shown in Figure 5, which showsa 4-segment geometrical ray, with ray parameter p0, repre-senting the reflected wave going from a source point at x = 0 toa receiver at a known offset x = X0. The layers are homoge-neous. The plane wavefronts associated with the geometricalray are also shown in Figure 5. In addition, another ray with rayparameter p and with a larger offset X(p) is shown. This raydoes not arrive at the receiver point X0. Nevertheless, note thatthe plane wavefronts associated with this ray also strike thereceiver. Consequently, in this sense, this ray also contributes tothe displacement at the receiver, and should therefore beincluded in the computation of the receiver response if greateraccuracy is desired. In fact, one can then surmise that superim-posing the contributions of all such rays with different p valuesmight give the exact response, i.e., the full wavefield, or at leasta more accurate response, which is in fact the case.

By analyzing this problem mathematically, with the intent ofobtaining the exact response, one obtains a formula expre s s i n gthe displacement of the re c e i v e r, due to the reflection from thesecond interface in Figure 5, as an integral over p, with p g o i n gf rom 0 to ∞ (the superposition of plane waves mentioned in thep revious paragraph). In fact, this integral contains within it notonly the displacement due to the reflected wave but also that dueto the head wave (the refraction arrival) from the second interface– the integral is said to give the generalized re f l e c t i o n response. InF i g u re 5, the geometrical ray path and the ray path associatedwith the head wave, taken together, can be called a generalized ray.The basic geometrical ray formula (6) is actually the lowest-ord e ra p p roximation of the generalized reflection integral.

Mathematical formulas for generalized ray responses can also


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F i g u re 5. A two-layer-over-half-space model in which velocity increases withdepth. The geometrical ray and head wave between the source and receiver areshown, as well as a ray with parameter p arriving at the offset X(p). The figurere p resents a simple way of understanding the concept of a generalized ray.


include multiples and any other wave types than are associatedwith the ray. Typically, they are applied for high frequencies(which in this case means travel times much greater than thewave period, or source-receiver offsets much greater than thewavelength). They are appealing because they give thecomplete (or nearly complete) wavefield at the receiver, andhave been used in the literature to calculate exact synthetic seis-mograms. The fact that they are superpositions of plane wavesmeans that plane wave reflection and transmission coefficientscan be used in the integrands. Note however that in computingsynthetic seismograms for a layered geological model, it is stillup to the user to decide which generalized rays to include inthe computation. If certain generalized rays are left out of thecalculation that should have been included, the synthetics willnot include the corresponding events, which could hamperinterpretation when the synthetics are compared with real data.Also, a disadvantage of generalized ray theory is that the inte-grals are notoriously difficult to evaluate – the integrands oscil-late wildly. However, numerical methods have been developedto evaluate them as efficiently as possible (see, e.g., Aki andRichards, 2002). The exact results in Figures 3 and 4 werecomputed by evaluating generalized ray integrals.

Matrix Methods

When computing synthetic seismograms for the stack of layersin Figure 2 using basic geometrical ray theory or generalized rayt h e o r y, one must select the particular rays to include in thesynthetics. For example, it is common to choose only theprimary reflections (rays which reflect only once beforereturning to the receiver). However, in some cases, one maywish to include multiples (rays which reflect more than once),such as the ray in Figure 2. Clearly, there are an infinite numberof multiples in a stack of layers, but only a finite number of themcan be included in a synthetic based on the ray methodsdiscussed above. The choice of which multiples to includedepends on the problem one is attempting to solve. Ty p i c a l l y,one would wish to include the multiples whose amplitudes areabove a specified threshold, and mathematical formulas havebeen developed to facilitate the choice (Hron, 1971, 1972). A l s o ,it is clear that the more rays (primaries or multiples) that areincluded, the greater the computation time.

However, it is in fact possible to include all the multiples (aninfinite number) by using the so-called propagator matrix method.This is in essence done by replacing an infinite series with afinite expression, in the same sense that the infinite series 1 + x+ x2 + x3 + … is equivalent to the function 1/(1-x).

The basic ideas involved in the propagator matrix method aredescribed in the following paragraphs.

The boundary conditions that seismic waves satisfy at eachinterface are that the x, y and z components of displacment uand traction T (the force per unit area acting across an interfacedue to a wave) must be continuous across the interface, i.e., theirvalues in the layers above and below the interface must matchat the interface. These are called welded-contact boundary condi-tions, and they express the fact that two layers move as a unit atthe interface when disturbed by a wave – they move as if they

were welded together at the interface. T is also sometimes calledthe stress vector because its components are the normal andshear components of the stress tensor. The boundary conditions,when expressed as mathematical equations and solved, yieldthe formulas for the reflection and transmission coefficients. Yin (6) is a product of such coefficients.

For plane wave propagation in the xz plane (where x is offsetand z is depth) of a vertically heterogeneous solid medium(e.g., a stack of horizontal homogeneous layers, or a medium inwhich the medium parameters vary smoothly with depth), thedisplacement u and the traction T can be computed anywherein the medium using the propagator matrix. More specifically, forplane harmonic P-SV waves, u and T at any depth z can becomputed from given values of u and T at some depth z0 byusing the following equation:

where u and T are 2x1 column vectors whose elements are thevector components of displacement (ux and uz) and traction (Tx

and Tz), respectively, and where P(z, z0) is the propagatormatrix, a 4x4 matrix containing the medium parameters, thefrequency, the ray parameter, and the vertical phase factors ofthe plane waves. The column vector f is called the displacement-stress vector.

For a stack of layers, f can be continued through the layersbecause f contains precisely the quantities that are continuousa c ross an interface. In other words, if we know f in the first layer,we can compute f in the last layer (or any layer). Also, for a stackof layers, the propagator matrix P(z, z0) is itself the product of asequence of matrices, known as layer matrices, and their inverses,which contain the parameters for the individual layers.

The propagator matrix method can be used to solve a numberof different types of problems in seismic forward modeling invertically heterogeneous media. For example, it can be used tocompute the reflection and transmission coefficients for a stackof horizontal homogeneous layers (as opposed to the coeffi-cients for a single interface). More precisely, suppose that adowngoing plane wave of a given frequency is incident uponthe top layer of a stack of such layers. The incident wave will betransmitted into the stack and be multiply reflected and trans-mitted inside all the layers. The upgoing wave emerging fromthe top of the stack into the incidence medium will be a super-position of all the upgoing waves (produced by multiple reflec-tion and transmission at all the interfaces of the stack) passingupwards through the top of the stack. The relative amplitude ofthis upgoing wave can be thought of as the reflection coefficientof the stack. It will be a function of the thicknesses, densities andwave speeds of all the layers. Similarly, the downgoing wave inthe medium below the stack will be a superposition of all thedowngoing waves (due to multiple reflection and tranmissionin the interior of the stack) passing through the bottom of thestack. The relative amplitude of this downgoing wave can be thought of as the transmission coefficient of the stack. It is



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important to note that the stack reflection and transmissionresponses contain the contributions of all the multiples (thereare an infinite number of them). The stack coefficients are alsofunctions of frequency, whereas interface coefficients are not(except for dissipative media).

Such stack reflection and transmission coefficients have beenused in the so-called reflectivity method (e.g., Fuchs and Müller,1971) and its variants, which is a popular method for computingsynthetic seismograms for a stack of horizontal layers forf o r w a rd modeling purposes. As an example of an application ofthe reflectivity method, consider a geological model consistingof a single layer overlying a stack of many thin layers, andsuppose one wants to model the reflection response of the thin-layer stack due to a point source in the single upper layer. Thecontinuation equation (9) would be used to obtain the ampli-tude of the wave incident upon the stack, the stack re f l e c t i o nresponse, and to continue it to a receiver on the surface. Then, anintegral over the ray parameter p is performed to obtain thegeneralized reflection response, and also an integral over thef requency ω to obtain the waveform at the re c e i v e r. In practice,the integral over p is often limited to sub-critical values, as thei n t e rest is often just in the small-angle body wave response. Theresulting waveform would contain all the multiples as well asthe primary re f l e c t i o n s .

The reflectivity method can also be combined with othermethods to enhance the output of the modeling process. Forexample, it can be combined with asymptotic ray theory (Hron,1971, 1972) in which multiples with sufficiently large ampli-tudes are ray-traced and included in the synthetics, to give theso-called ray-reflectivity method (Daley and Hron, 1982). Oneadvantage of this combination is that all the events on thesynthetics can be identified (in terms of ray paths).

Matrix methods are also used in a technique sometimes calledthe recursive reflectivity method (e.g., Kennett, 2001). In this tech-nique, mathematical formulas for the stack reflection andtransmission coefficients for a single layer are first developedby systematically adding up the contributions of all the indi-vidual multiple reflections and transmissions in the layer toobtain the total response. The resulting sum is an infinite series,which is replaced by a finite expression using the above-mentioned formula, 1 + x + x2 + x3 + … = 1/(1-x) , where x is aproduct of the internal reflection coefficients of the layer and avertical phase factor. These one-layer stack formulas are thenused to obtain the stack coefficient formulas for a two-layerstack, which are in turn used to obtain the formulas for a three-layer stack, etc. These formulas all have the same mathematicalform, regardless of the number of layers involved. Althoughthe stack coefficients for any size stack can be computed in thisway, it can be shown that a more efficient two-step recursiveprocedure can be developed for computing them. This proce-dure involves applying vertical phase factors to bring the stackcoefficients from the bottom of a given layer to the top of thelayer, then using the stack coefficient formulas to cross theinterface into the bottom of the next layer above the given one.This process is continued recursively.

As mentioned above, an integral over the ray parameter p canalso be performed to obtain the generalized reflection response,and also an integral over the frequency ω to obtain the wave-form at the receiver.

It can be shown that these many-layer stack formulas give thesame result as the propagator matrix method. Although theyare more complicated and involve more computation, theyoffer several advantages over the propagator matrix approach:(a) the reflection and transmission processes of individual wavemodes (P and S) can be followed more easily, (b) P-S conver-sions can be included or left out (which is useful for deter-mining the effect of conversions on the synthetics, (c) usefulapproximate formulas can be developed from the exact stackformulas, e.g., approximate formulas for the case in which onlya single conversion is allowed, (d) if the infinite series is used inthe formulas (instead of the corresponding finite expression), itcan be truncated to include only as many multiples as desired(which is useful for determining the effect of multiples), (e) ifthe infinite series is used, the individual terms have a verysimple dependence on frequency, allowing the formulas to beused with other frequency-based methods, and (f) the terms inthe formulas are directly related to the physical processesoccurring in the stack (the multiple reflections and transmis-sions), allowing more physical insight into the pro c e s s e s ,w h e reas the propagator matrix approach gives only thecombined cumulative effect. If however, only the total responseis required, the propagator matrix method is generally easierand more straightforward to use.

The matrix methods discussed above can also be used forwavefield extrapolation (e.g., Bale and Margrave, 2003), whichis a part of most seismic wave equation migration processes,and also other inversion methods which contain a forwardmodeling stage.

The finite difference method

Another popular seismic forward modeling method is the finitedifference (FD) method, which is a numerical method forsolving partial differential equations. It can be applied to theseismic equation of motion to compute the displacement u atany point in the given geological model, e.g., at the surface (forgenerating synthetic seismograms for comparison withrecorded data), or at some depth z (for doing wavefield extrap-olation or downward continuation, a stage of wave equationmigration). There are a number of variants of the method, andmany small improvements have been made to the basic methodby many researchers.

The main idea behind the FD method is to compute the wave-field u(x,y,z,t) at a discrete set of closely-spaced grid points (xl ,ym , zn , tq ) , with l, m, n, q = 0, 1, 2, 3, 4, 5, …, by approximatingthe derivatives occurring in the equation of motion with finited i ff e rence formulas, and recursively solving the re s u l t i n gdifference equation.

As an example of a finite difference formula for approximatinga derivative, consider the basic definition of a derivative, whichcan be found in any elementary calculus text, i.e.,

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This leads to the forward-difference approximation for dy/dx, givenby

where ∆x is a small finite-sized interval or grid step in the xdirection (adjacent grid points in the x direction are separatedby the amount ∆x). Similarly, the backward-difference approxima-tion is given by

and the central-difference approximation is given by

It can be easily shown, by applying the Taylor series expansionto these formulas, that the error made in using the forward- andbackward-difference approximations in (11) and (12) is of order∆x (i.e., it is linear in the small quantity ∆x), but that the errorfor the central-difference approximation in (13) is of order (∆x)2

(it is quadratic in ∆x). Consequently, the central-differenceformula in (13) is a more accurate approximation to dy/dx.

As an example of the application of the FD method, considerthe 1D partial differential equation

where u = u(x,t). It is known as the one-way wave equationbecause it admits solutions representing waves travelling withconstant speed c in the positive x direction but not the negativex direction. In mathematical terms, the exact solution is u = f(x-ct), where f is an arbitrary function. Since the exact solution of(14) is known, one would not of course need to use the FDmethod to solve (14) in practice – we use it here only as a simpleexample of how to apply the FD method.

There are many ways to obtain a FD approximation to thisequation. For example, one possibility is to use a backward-difference approximation for ∂u/∂x and a forward-differenceapproximation for ∂u/∂t, resulting in the equation

where ∆t is a small finite-sized interval or grid step in the tdirection. Upon re-arrangement, one obtains

which can be used to recursively compute u at the grid point (x,t+ ∆t) if one knows the value of u at the grid points (x,t) and (x-∆x,t). Given some starting values, i.e., initial conditions or i n i t i a lv a l u e s, such as u(x,0) = f(x), where f(x) is an arbitrary function ofx (the waveform at t = 0) and given values of the grid steps ∆xand ∆t, i.e, a value for g, one can then compute the values u(x,∆t) for all x, which are then in turn used to compute the valuesu(x, 2∆t) for all x, etc. In other words, u can be computed at alltimes t by “marching forward in time” in this way.

It can be shown that if one chooses ∆x and ∆t so that g ≤ 1, then(16) is a stable FD scheme, i.e., the errors resulting from makingFD approximations to the derivatives remain finite – they donot grow without limit as the time-marching computationsprogress. In fact, if one chooses g = 1, the resulting simple FDscheme, u(x, t+∆t) = u(x-∆x,t), gives the exact solution to (14).For g < 1 however, the scheme in (16) is first-order accurate inspace (x) and time (t), because the errors in the forward- andbackward-difference approximations used for the derivativesare of first order in ∆x and ∆t.

If one naively uses the forward-difference approximation (11)to approximate both ∂u/∂x and ∂u/∂t in (14), then one obtainsan unstable FD scheme, in which the error grows without limitduring the computations, regardless of the value of g.

In general, for any 3D partial differential equation, such as theelastic equation of motion, it is necessary to choose the correctFD approximations to the derivatives appearing in the equa-tion, and to choose appropriate values for the step sizes ∆x, ∆y,∆z and ∆t, to ensure numerical stability. Often, a preliminarymathematical analysis of the partial differential equation understudy is useful, and sometimes necessary, to determine whichtype of FD approximations to use.

The 3D seismic equation of motion, a second-order partialdifferential equation in the displacement u, can be solved in thesame way, although the resulting difference equation is muchmore complicated than (16). Normally, the difference equationis marched forward in time t, with u being computed at allvalues of x, y and z for each time level. Because the wavefield isthen known everywhere, it is possible to create movies of thew a v e f ronts propagating through the material, which arehelpful in understanding seismic wave propagation.

FD computations are extremely time-consuming. 3D FDcomputer programs for realistic geological models can takemany hours to run. Consequently, it is common practice to doFD calculations in 2D instead, to reduce the computation time.Also, various computational techniques have been developedto improve efficiency. Even in 2D however, computation timesare typically much longer than those of other methods, such asray theory. However, as computer technology advances, 3Dcalculations should become more practical.



(11)

(12)

(13)

(14)

(15)

(16)

(10)




Accuracy of the computed solution is another primary concernin FD computations. One way to generally improve the accu-racy of the FD solution is to use smaller grid step sizes (whilemaking sure that the stability condition is always satisfied, e.g.,g ≤ 1 for (16) ). However, this increases the computation time.In addition, if the grid step sizes are made too small, accuracycan decrease, due to the occurrence of significant roundofferror. Another way to generally improve accuracy, withoutreducing the grid step sizes, is to use higher-order approxima-tions for the derivatives. As indicated above, the forward- andbackward-difference approximations used in (11) and (12) leadto first-order accurate FD schemes, but the central-differenceapproximation leads to second-order accurate FD schemes, andis commonly used in constructing typical FD schemes. Higher-order approximation formulas, involving a series or combina-tion of forward-, backward- or central-diff e re n c eapproximations, can be used as well, but they result in morecomplicated and cumbersome FD schemes and in an increasedcomputation time.

An unwanted artifact of FD calculations is a degradation of theaccuracy of the solution caused by grid dispersion (e.g., Alford etal., 1974). This occurs if the spatial grid steps are chosen toolarge, i.e., if the grid is too coarse. To avoid grid dispersion, atypical rule of thumb is that the spatial grid steps should be nolarger than one-tenth the size of the shortest wavelength in thewavefield, i.e., there should be at least 10 grid points per wave-length. If higher-order FD schemes are used (i.e., if higher-orderderivative approximations are used), this number can bereduced, e.g., 5 grid points per wavelength (e.g., Alford et al.,1974; Levander, 1988). However, this does not necessarily resultin a reduction in computation time, as the higher- o rd e rschemes are more complicated.

The main advantage of using FD methods in seismic modellingis that they produce the full wavefield – all the different typesof waves that exist (reflections, refractions, etc.) generally willappear in the computed solution with correct amplitudes andphases. One difficulty of the method is the question of how toinsert a given type of source of waves. A straightforward wayto do this is to use an exact solution in the vicinity of the source,and then continue the solution with the FD method. Otherways involve modifications of this approach (e.g., Altermanand Karal, 1968; Kelly et al., 1976).

A number of diff e rent variants of the FD method are in use, forexample, the v e l o c i t y - s t ress FD method (e.g., Madariaga, 1976;Virieux, 1986). In this method, the equation of motion, which is as e c o n d - o rder partial diff e rential equation in the particle-displacement u, is re - e x p ressed as a coupled system of first-ord e rpartial diff e rential equations in the components of the particle-velocity (∂u/ ∂t) and the traction T. First-order equations aregenerally easier to solve. In addition, the use of s t a g g e red grids, inwhich the particle-velocity components and the traction compo-nents are computed on diff e rent grids which are shifted or stag-g e red relative to each other by half a spatial grid step in alld i rections, makes for efficient and accurate computation.

Another popular variant of the FD method is the pseudo-spectralmethod (Kosloff and Baysal, 1982). In this method, the time

derivatives are still computed using a FD approximation in thetime domain, but the space derivatives are computed in thewavenumber domain. More specifically, to obtain a spatialderivative, say ∂ux/∂x, at a given time level, a spatial Fouriertransform is performed on the sequence of wavefield values ux(known from a previous stage of the FD computation) along thex direction to obtain the wavenumber (kx ) spectrum of ux(another sequence of numbers). Then this new sequence ismultiplied by ikx which, in the kx domain, is equivalent totaking the derivative ∂ux /∂x (i is the imaginary unit). Finally, aninverse spatial Fourier transform is performed on this alteredsequence to give the differentiated sequence of values ∂ux /∂xin the x domain. The same technique is applied to the othercomponents and in the other 2 spatial directions. The results arethen used to compute all the wavefield values at the next timelevel. The method gives quite accurate results because thespatial derivatives are done in the wavenumber domain,avoiding the inaccuracies associated with FD approximationsto derivatives. Another advantage of this method is that itrequires considerably fewer grid points per wavelength thanthe standard FD schemes in the space-time domains, making itattractive for 3D computations. With today’s computers,running at 2-3 GHz or higher, medium-sized 3D model compu-tations can be done in several hours or less.

Another unwanted artifact of the FD method is the generationof artificial reflections. These are reflections coming from theedges of the numerical grid, rather than from the bona-fidegeological interfaces in the model. For example, to compute avalue for ux at the time t+∆t for a node at the very edge of thegrid, one generally needs to know the value of ux at the earliertime t at a node just off the grid (many FD schemes work thatway). If one naively sets the off-grid value equal to zero, thenstrong artificial reflections from the grid edge are produced,because by setting the off-grid displacement value equal tozero, one is implicitly making the off-grid region into a rigidmedium, resulting in an elastic-rigid interface. The resultingimpedance contrast produces the artificial reflections whichpropagate back into the grid, and appear in the synthetics.

To suppress such artificial reflections, non-reflecting or absorbingboundary conditions are used at the grid edges. In the case ofnon-reflecting boundary conditions, FD schemes which aredifferent from the one used in the interior of the grid (which isbased on the standard equation of motion) are used. Thesealternate schemes are mathematically designed so that the arti -ficial reflections produced have low (ideally zero) amplitudes.Sometimes they involve solving one-way wave equations nearthe grid edges – equations whose solutions are waves whichcan travel in one direction only, i.e., off the grid but not back onto it. In the case of absorbing boundary conditions, one usesschemes which absorb the energy of the artificial reflections.Typically, the grid is enlarged by a small amount – a few layers– and a modified FD scheme which includes the effects of phys-ical absorption or dissipation is solved in the added grid zone.If the FD scheme used in the interior of the grid alreadyaccounts for seismic wave absorption (i.e., if it is based on anequation of motion containing absorption terms – see, e.g.,Emmerich and Korn, 1987; Krebes and Quiroga-Goode, 1994),

Article Cont’d


then it is a simple matter to suppress artificial reflections – onesimply makes Q very small near the grid edges. Q is a measureof the elastic “quality” of the medium – the lower the Q, themore absorbing or dissipative the medium is (Q = ∞ for a perfectly elastic medium in which waves experience noabsorption). As with all computational methods, non-reflectingand absorbing boundary conditions have their limitations, butin general, they work quite well.

Figures 6-16 show examples of synthetic shot records andw a v e f ronts computed with the standard 2D FD method.Figures 6 and 7 were produced by Peter Manning and GaryMargrave, and Figures 8-16 were produced by Louis Chabot, allmembers of the CREWES project at the University of Calgary.

These figures illustrate how various physical effects in seismicwave propagation can be investigated, because the FD methodproduces the full wavefield.

Figure 6 and 7 show 2D synthetic shot records for an explosiveacoustic pressure source buried at depths of 8 m and 18 m,resp., in a horizontal homogeneous elastic layer overlying arigid half-space. The synthetics show the expected events, i.e.,the direct arrival, the dispersive surface wave and the reflec-tion. They also show how the deeper source results in alowering of the resolution and a reduction in the surface waveamplitude.

Figures 8 and 9 again show 2D synthetic shot records for asource consisting of a force directed vertically downward at apoint on the surface (e.g., a hammer blow). The non-symmetricsource generates both a P wave and an SV wave. The geologicalmodel is a horizontal homogeneous elastic layer overlying ahomogeneous elastic half-space. The top of the elastic layer is afree surface. The synthetics show the expected events, i.e., thedirect arrival, the reflected and converted waves, the headwaves (refraction arrivals) and the prominent dispersiveguided waves. They also show the amplitude and phase(polarity) relationships among the wave types. In addition,Figure 9, for the horizontal component of particle motion,shows a polarity reversal in all events as one goes from the leftside of the record to the right side. This happens because thewaves arriving at geophones left and right of centre producegeophone motions whose x components of displacement haveopposite signs. Some numerical artifacts can also be seen inFigure 9 on the lower left and lower right edges of the grid.

Figures 10-16 show a series of “snapshots” of the wavefrontscorresponding to the shot records in Figures 8 and 9, at the



F i g u re 6. A 2D synthetic shot record produced by the finite difference method fora buried pre s s u re source. The horizontal axis is the sourc e - receiver offset in metersand the vertical axis is the 2-way travel time in milliseconds. The geological modelis a flat elastic layer over a rigid half-space. The upper surface of the elastic layeris a free surface (e.g., an air/rock interface). The source depth is 8 m. The figurewas produced by Peter Manning and Gary Margrave, CREWES project, U. ofC a l g a r y.

F i g u re 7. Same as Figure 6, except the source depth is 18 m. The figure wasp roduced by Peter Manning and Gary Margrave, CREWES project, U. ofC a l g a r y.

F i g u re 8. A 2D synthetic shot record for the vertical component of geophonemotion produced by the finite difference method for a surface source. The hori-zontal axis is the sourc e - receiver offset in meters and the vertical axis is the 2-waytravel time in seconds. The model is 3 km x 1 km. The geological model is a flatelastic layer with a free upper surface overlying an elastic half-space. The sourc eis a vertically directed point force located on the free surface. The density, P andS wave velocities in the elastic layer are 2.1 g/cm3, 1000 m/s and 500 m/s, re s p . ,and in the half-space they are 2.3 g/cm3, 1850 m/s and 925 m/s. Figures 8-16 werep roduced by Louis Chabot.

following fixed times: 300 ms, 500 ms, 600 ms, 800 ms, 1000 ms,1200 ms, and 1400 ms. The upper half of each figure shows thehorizontal component of motion and the lower half shows thevertical component. Several interesting wave pro p a g a t i o nfeatures can be seen in the figures – they show (a) that the“hammer blow” source produces both a downgoing P and SVwave; (b) that there is a downgoing head wave connecting thedowngoing P and SV wavefronts emerging from the sourcepoint; (c) the upgoing Pand SV head waves (refraction arrivals)produced by the transmitted P wavefront beyond the criticalangle (when it breaks away from the other wavefronts at theinterface); (d) the expected left-to-right polarity reversal in thehorizontal component; (e) the variation of amplitude alongsome of the reflected and transmitted wavefronts due to thenon-symmetric “hammer blow” source; and (f) the complex

nature of the wavefield produced by a non-symmetric source,even for a simple one-layer model.

Regarding point (e) in the preceding paragraph, the variation ofamplitude observed along the wavefronts is consistent with theradiation pattern for a vertically directed point force (knownfrom theoretical studies of seismic wave propagation). Forexample, this radiation pattern predicts that a downwardshammer blow would result in mainly a large vertical (longitu-dinal) displacement of the medium at points vertically belowthe source point, with little or no horizontal (transverse)displacement at these points (which one might also predictusing physical intuition). This prediction is confirmed, forexample, in the transmitted P wavefront in Figure 12 (thelowermost wavefront in each half, upper and lower, of thefigure) – for the horizontal component, the centre of the trans-mitted P wavefront has zero amplitude, but for the verticalcomponent, it has a high amplitude.

A good way to study Figures 10-16, with the intent of under-standing wave propagation, is to select a certain wave and




F i g u re 9. Same as Figure 8, except it is for the horizontal component of motion.

Figure 10. Fixed-time plot, or "snapshot", of the wavefronts corresponding tothe shot records and geological model of Figures 8 and 9, produced by the finitedifference method, at an elapsed time of 300 ms (the source is activated at timet = 0). The horizontal component of particle motion is shown in the upper halfand the vertical component in the lower half. The locations of the free surface(where the source is) and the interface between the layer and half-space can beinferred from the locations of the incident and reflected wavefronts on this andthe following figures.

F i g u re 11. Same as Figure 10, except the elapsed time is 500 ms.


follow what happens to it from one snapshot to the next (astime progresses). For example, consider the first reflected Pwave – the vertical component. Beginning with Figure 10(lower half), which shows the downgoing high-amplitude inci-dent P wave that has not yet reached the interface, we move toFigure 11 and note the beginnings of the first reflected P wave.Since the incident P wave is reflecting off a medium (the half-space) which has a larger acoustic impedance (density x Pwavevelocity), the vertical component of displacement experiences apolarity reversal (the central peak of the reflected P wavelet iswhite, not black as for the incident P wave). This is just like thepolarity reversal in displacement experienced, upon reflection,by an incident wave on thin string connected to a thick rope.We then follow the reflected P wave upwards in Figures 12 and13, where it then reflects off the free surface. This free surfacereflection can be seen in Figure 14 – note that there is now nopolarity reversal in the vertical component of displacementbecause the wave is reflecting off a medium with a loweracoustic impedance, i.e., the air layer (although there is apolarity reversal in the pressure at the free surface – a compres-sion reflects as a rarefaction, i.e., a dilatation, off a medium witha lower acoustic impedance). We then follow the P wavereflected from the free surface back down to the interface inFigure 15, after which it reflects off the interface with anotherpolarity reversal (due again to the larger impedance in the half-space), which can be seen in Figure 16. Note also the decreasingamplitude of the wave.

FD forward modeling has also been done for solid mediacontaining interfaces that are not in welded contact with eachother (e.g., Slawinski and Krebes, 2002a, 2002b). N o n - w e l d e dc o n t a c t means that the displacement u is no longer continuousa c ross an interface, although the traction T still is. Mediacontaining joints, fractures and faults may in some cases fall intothis category.










Other methods

Forward modeling has also been done with other methods. Forexample, some success has been achieved with the finite elementmethod. This method still involves computations on a grid ormesh, but involves representing the components of the solutionu(x,y,z,t) to the equation of motion as linear superpositions ofappropriately selected basis functions. For example, see the arti-cles in Kelly and Marfurt (1990), as well as Moczo et al. (1997),Kay and Krebes (1999), and the review article by Carcione et al.(2002). The finite element method gives reliable results, and itcan be more flexible and more accurate than FD methods whenthe geological interfaces in the model are irregular, but it is alsomore difficult to implement and apply than the FD method. Avariety of other methods have also been used for models withirregular interfaces (see, e.g., P‰enãík et al., 1996).

Lastly, seismic wave propagation in media containing small-scale heterogeneities such as cracks, fractures and inclusions,which scatter the waves, can be usefully modelled by solvingintegral equations which describe their behaviour (e.g., see thereferences listed by Carcione et al., 2002).

Acknowledgements

Thanks go to Louis Chabot, formerly of the CREWES project inthe Department of Geology and Geophysics at the University ofC a l g a r y, and Peter Manning and Gary Margrave of theCREWES project, for the finite difference synthetics.

ReferencesAbramovici, F., Le, L.H.T., and Kanasewich, E.R., 1990. The solution of Cagniard’sproblem for a SH line source in elastic and anelastic media, calculated using ω-k integrals.Bull. Seism. Soc. Am., v. 80, p. 1297-1310.

Aki, K. and Richards, P.G., 2002. Quantitative Seismology. 2nd edition, UniversityScience Books, Sausalito, California.

Alford, R.M., Kelly, K.R. and Boore, D.M., 1974. Accuracy of finite-differencemodeling of the acoustic wave equation. Geophysics, v. 39, p. 834-842.

Alterman, Z.S. and Karal, F.C. Jr., 1968. Propagation of elastic waves in layered mediaby finite difference methods. Bull. Seism. Soc. Am., v. 58, p. 367-398.

Bale, R.A. and Margrave, G., 2003. Elastic wavefield extrapolation in HTI media.CREWES Research Report, v. 15.

Carcione, J. M., Herman, G. C. and ten Kroode, A. P. E., 2002, Seismic modelling :Geophysics 67, 1304-1325.

âe r v e n˘, V., 2001. Seismic Ray Theory. Cambridge University Press., Cambridge,UK.

âe r v e n˘, V. and Ravindra, R., 1971. Theory of Seismic Head Waves. University ofToronto Press, Toronto, Canada.

Chapman, C.H., 1985. Ray theory and its extensions : WKBJ and Maslov seismograms.J. Geophys., v. 58, p. 27-43.

Daley, P. and F. Hron, 1982. Ray-reflectivity method for SH-waves in stacks of thin andthick layers. Geophys. J. R. astr. Soc., v. 69, p. 527-535.

Emmerich, H. and M. Korn, 1987. Incorporation of attenuation into time-domaincomputations of seismic wave fields. Geophysics, v. 52, p. 1252-1264.

Fuchs, K. and Müller, G., 1971. Computation of synthetic seismograms with the reflectivity method and comparison with observations. Geophys. J. R. astr. Soc., v. 23, p.417-433.

Gallop, J.B., 1999. Asymptotic Descriptions of Seismic Waves. Ph.D. thesis, Universityof Alberta.

Hearn, D. and Krebes, E.S., 1990. On computing ray-synthetic seismograms foranelastic media using complex rays , Geophysics, v. 55, p. 422-432.

Hron, F., 1971. Criteria for selection of phases in synthetic seismograms for layeredmedia. Bull. Seism. Soc. Am., v. 61, p. 765-779.

Hron, F., 1972. Numerical methods of ray generation in multilayered media. Methods inComputational Physics, v. 12, ed. B.A. Bolt, Academic Press, New York.

Kay, I., and Krebes, E.S., 1999. Applying finite element analysis to the memory variableformulation of wave propagation in anelastic media . Geophysics, v. 64, p. 300-307.

Kelly, K.R. and Marfurt, K.J. (eds.), 1990. Numerical Modeling of Seismic WavePropagation. Society of Exploration Geophysicists, Tulsa.

Kelly, K.R., Ward, R.W., Treitel, S., and Alford, R.M., 1976. Synthetic seismograms: afinite-difference approach. Geophysics, v.41, p. 2-27.

Kennett, B.L.N., 2001. The Seismic Wavefield. Volume 1: Introduction and TheoreticalDevelopment. Cambridge University Press, Cambridge, U.K.

Kim, S., and Cook, R., 1999. 3-D traveltime computation using second order ENOscheme. Geophysics, v. 64, p. 1867-1876.

Kosloff, D. and Baysal, E., 1982. Forward modelling by a Fourier method. Geophysics,v. 47, p. 1402-1412.

Krebes, E.S. and Quiroga-Goode, 1994. A standard finite-difference scheme for thetime-domain computation of anelastic wavefields . Geophysics, v. 59, p. 290-296.

Le, L.H.T., Krebes, E.S. and Quiroga-Goode, G.E., 1994. Synthetic seismograms forSH waves in anelastic transversely isotropic media. Geophys. J. Int., v. 116, p. 598-604.

Levander, A.R. 1988. Fourth-order finite-difference P-SV seismograms. Geophysics, v.53, p. 1425-1436.

Madariaga, R., 1976. Dynamics of an expanding circular fault. Bull. Seism. Soc. Am.,v. 66, p. 639-666.

Moczo, P., Bistricky, E., and Bouchon, M., 1997. Hybrid Modeling of P-SV SeismicMotion at Inhomogeneous Viscoelastic Topographic Structures. Bull. Seism. Soc. Am.,v. 87, p. 1305-1323.

Plumpton, N.G. and Tindle, C.T., 1989. Saddle point analysis of the reflected acousticfield, J. Acoust. Soc. Am., v. 85, p. 1115-1123.

P ‰ e n ã í k, I., âe r v e n˘, V., and Klimes, L., 1996. Seismic Waves in LaterallyInhomogeneous Media, Part I. Birkhäuser Verlag, Basel, Switzerland.

Richards, P.G., Witte, D.C. and Ekstrom, G., 1991. Generalized ray theory for seismicwaves in structures with planar nonparallel interfaces. Bull. Seism. Soc. Am., v. 81, p1309-1331.

Slawinski, R., and Krebes, E.S., 2002a. Finite-difference modeling of SH-wave propa-gation in nonwelded contact media. Geophysics, v. 67, p. 1656-1663.

Slawinski, R., and Krebes, E.S., 2002b. The homogeneous finite-difference formulationof the P-SV-wave equation of motion. Stud. Geophys. Geod., v. 46, p. 731-751.

Thomson, C.J., 1990. Corrections for critical rays in 2-D seismic modelling. Geophys.J. Int., v. 103, p. 171-210.

Van Trier, J., and Symes, W.W., 1991. Upwind finite-difference calculations of travel-times. Geophysics, v. 56, p. 812-821.

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Virieux, J., 1986. P-SV wave propagation in heterogeneous media: velocity-stress finitedifference method. Geophysics, v. 51, p. 889-901.

Further ReadingCarcione, J.M., 2001. Wave Fields in Real Media: Wave Propagation in Anisotropic,Anelastic and Porous Media. Pergamon Press, Amsterdam.

Kelly, K.R. and Marfurt, K.J. (eds.), 1990. Numerical Modeling of Seismic WavePropagation. Society of Exploration Geophysicists, Tulsa.

Slawinski, M.A., 2003. Seismic Waves and Rays in Elastic Media.Elsevier/Pergamon, Amsterdam. R

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