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Introduction

The technique of forward modeling in seismology beginswith the numerical solution of the equation of motion for

seismic wa ves, or more specifically, the nu merical compu -

tation of theoretical or synthetic seismograms, for a given

geological model of the subsurface. The idea is then to

compare the synth etic seismic traces with r eal seismic data

acquired in the field. If the two agree to within an accept-

able level of accuracy, the given geological model can be

taken to be a r easonably accurate m odel of the subsur face.

If not, the geological model is altered, and new synthetic

traces are computed and compared with the data. This

process continues iteratively until a satisfactory match is

obtained between the synthetics and the real data. The

forward modeling approach is, in a sense, the opposite of

the inverse modeling approach in which the parameters of

the geological model are computed from the acquired real

data. Both methods though ultimately have the same goal

– the determ ination of the geological stru c t u re and

lithology of the subsu rface.

As mentioned above, in forwar d mod eling, one attempts to

solve the equation of motion for seismic waves, which is

nothing more than the mathematical expression of

New ton’s second law of motion, i.e., force = mass x accelera-

tion, for m aterial particles in a solid bod y set in m otion by

elastic waves. It is a second -order p artial differential equa-

tion for the vector displacement u experienced by a point

or particle in a solid med ium d ue to the passage of a wave.

Once u is known (after solving the equ ation of motion), the

displacements of geophones can be computed and

synthetic seismograms can be produced. There are a

number of computational methods, and variants thereof,

that are commonly used to compute the synthetic seismic

traces, such as ra y theory, the finite difference method , and

the reflectivity method. All the methods have their indi-

vidual advantages and disadvantages, and an in-depth

mod eling effort for a given d ata set can involve the u se of

several method s. In this article, I will discuss some of the

basic principles and concepts of some of these methods.

Many seismologists have contributed to the development

of these methods, and it is unfortunately not possible to

represent all of them in this article.

Ray theory

Ray theory can be used to compute seismic wave travel

times and am plitudes along ray paths in a heterogeneous

medium when the frequencies present in the w ave are high

enoug h so that the “geometrical optics” app roximation can

be used. As a rule of thumb, in order for ray theory to be

app licable, the med ium p aram eters (e.g., the Lamé param -

eters λ and µ, or the P and S wave speeds) should not

change very mu ch over distances of the order of the d om

nant wavelength. This rule of thumb also implies t

“high frequency” is a relative term – ray theory could abe used with frequencies that would be considered “lo

in certain situations, as long as th e heterogeneity is so we

that the rule of thumb applies. Under these conditions

high frequency, the travel time T(x) of the wave from

source to a point x = (x,y,z) in a heterogeneous isotro

medium obeys the eikonal equation,

a non-linear partial d ifferential equation, wh ere υ = υ(x

the seismic wave speed (for either a P or S wav e) at

point x. The eikonal equation (1) can be obtained by subtuting a trial solution for u into the equation of motion a

making the appropriate high-frequency approximations

Much recent work has been done on computing tra

times by solving the eikonal equ ation, especially using

finite difference method (see below), which is compu

tionally efficient (see, e.g., Vidale, 1990; van Trier a

Symes, 1991; and Kim an d Cook, 1999).

To actually trace a ray throug h a med ium, a set of “r

equations” mu st generally be solved. Consider first a sin

i s o t ropic heterogeneous medium in which υ v a r

smoothly with x. For a given wa ve speed υ(x), the geom

rical ray path can be determined, i.e., the coordinates of aand all points x on the ray p ath can be computed , by solv

the following system of ord inary d iff erential equations:

Seism ic Forw ar d M odelin gE. S. Krebes, Department of Geology and Geophysics, University of Calgary, Calgary

28 CSEG RECORDER April 2004

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Figure 1. A ray path in a vert ically heterogeneous medium in whichwave velocity υ varies with depth z as υ = exp(z2). The horizontal vertical directions are offset and depth, respectively. The ray starts atorigin 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 p ath. q ≡ ∇ T is called the

slowness vector. It p oints in the d irection of travel of the wa ve,

i.e., it is tangential to the ray pa th at any p oint on the ray p ath.

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 numerically

solving (2), ds is often rep laced by υdT in (2), so that th e more

useful quantity T (the travel time) is used in the equations,

rather than the arc length s. Figure 1, which was produced by

numerically solving (2), shows an example of a ray path in a

medium in which the velocity υ varies with depth z as υ =

exp(z2).

The stand ard ray tracing equations (2) can be mod ified and

approximated to simp lify and facilitate ray tracing in complex

subsurface structures. For examp le, consider a geometr ical ray R

going between tw o points in a certain region of the subsurface,

and su pp ose that the paths of the nearby ad jacent rays in this

region do not deviate much from the ray R. Such nearby rays are

called paraxial rays. In this case, first-order Taylor expansions can

be used to app roximate the wav e speed υ = υ(x) for the paraxial

rays. Then, using a coordinate system centred on the ray results

in a coupled system of linear diff erential equations for the

components of slowness and the coordinates, which are gener-

ally easier to solve tha n (2). Such m ethods, and mod ifications

thereof, are generally called paraxial ray tracing methods or

dynamic ray tracing method s (see, e.g., âerven˘, 2001).

Ray amplitudes can also be computed, in the high-frequency

approximation, by solving the so-called transport equation for

the wave. For example, in the relatively simple case of an

acoustic wav e, the transport equation for the amplitud e A(x) of

the pressure wave at the point x is

Once T is known (e.g., by solving th e eikonal equation 1), A can

be comp uted , in principle.

Tracing a ray between known source and receiver points in a

structurally complex geological model of the subsurface is

generally done by a ray shooting method, in which the take-off

angle of a ray (the angle at which the ray leaves the source

point) is determined iteratively (essentially through trial and

error). In some of the simpler cases however, mathemati

formulas can be developed which allow one to compute, re

tively easily, the travel times and am plitud es for the rays at a

offsets without actually “tracing” or “shooting” any rays. F

example, consider the case of a geometrical ray in a mediu

consisting of a stack of flat homogeneous isotropic horizon

elastic layers, a model comm only used in land exploration s

mology. Figure 2 show s an exam ple of such a ray. If the sou

is a spherically symm etric point source located on th e surfa

then the travel time and amplitude of the wave arriving at

receiver can be computed from relatively simple algebr

formulas as follows.

First, the desired source-receiver offset X is chosen. Then

ray parameter p for the ray going from th e source to the recei

is comp uted by nu merically solving the equ ation

for p, where m is the number of ray segments, and h j and υ j

the layer thickness and wave speed, resp., for the jth rsegment (assumed to be known). p is simply the horizon

component of the slowness vector along the ray, i.e., p =

Snell’s law states that p is constant along the ray path, i.e.,

sin(θ j)/ υ j , j = 1, …, m, where θ j is the angle that the jth r

segment makes with the vertical axis. Consequently, once p

determ ined, the take-off angle θ1 of the ray can be computed

needed, as well as all the other angles θ j . Equation (4) can

solved easily using an y root-finding m ethod, e.g., the New t

Raphson method. Once p is known, the travel time T of the r

can be computed from

The amplitude A(ω) of the wave at the receiver for a sin

frequency ω can be computed from

where Y is the product of displacement reflection and tra

mission coefficients along the ra y p ath, an d L is the geom etri

spreading factor which gives the amplitude loss due to

geometrical spread ing of the wavefront, and i is the imagina

un it. For a ray such as the one in Figure 2, L is given by

A complex exponential is used in (6) for mathem atical conv

ience – it is the real part th at represents the physical sinu soi

wav e. Sometimes in practice, L is roughly estimated simply

the total length of the ray p ath. How ever, this estimate can

very inaccurate in gen eral, differing from the correct L compu

f rom (7) by a large amou nt (e.g., a factor of 2 or 3 or more).

April 2004 CSEG RECORDER

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Seismic Forward ModelingContinued from 8

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Figure 2. A typical geometrical ray of m segments in a layered medium.




Article Cont’d

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If the ma thematical form of the source pu lse is known , then the

amplitude of the waveform (consisting of a superposition of

frequencies) at the receiver can also be compu ted. For example,

the displacement u of the receiver is given by an inverse Fourier

transform, i.e.,

where S(ω) is the frequency spectrum of the source pulse, and

b , the polarization vector, is a unit vector in the p ositive d irec-

tion of displacement (defined by convention). For example, for

an incoming P wave traveling in the xz plane, b = [sin(θm), 0,

cos(θm)] = [p υm, 0, cm]. If the free surface effect is included, b

wou ld be more complicated – it would h ave to includ e the free

surface reflection coefficients (the free surface effect takes into

accoun t the fact tha t the displacement of the receiver is affected

not only by th e incoming wav e, but also by the w aves reflected

off the surface). Equation (8) is norm ally compu ted using a fast

Fourier transform algorithm.

The above method can then be u sed to compu te synthetic seis-

mograms which can be compared with real data to obtain a

geological model of the subsurface. For sub-critical offsets, this

method can also be easily extended to the case of absorbing

(dissipative, anelastic) layers – one simply goes through the

same calculations (4)-(8) but with complex-valued and

frequency-depend ent w ave speeds υ j (Hearn and Krebes, 1990).

For example, if the quality factors Q j (typically frequency-ind e-

pend ent) for each ray segment are known (Q j = ∞ for no absorp-

tion), then υ j ≈ V j (ω)[1 - i / (2Q j )], where V j is the real wave

speed. 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 actua l tra

time of the wave, and the imaginary part, Im(T), gives

absorption factor, resulting in an exponential amplitude dec

du e to absorp tion (in (6), eiωT becomes eiωT = e-ωIm(T) eiωRe(T) ). As p

complex, Y an d L in (6) also become com plex, as well as b in

These results are then used in (8) to compute the wavefor

For super-critical offsets, care must be taken to correctly cho

the signs of the square roots (c j ) in the above formu las, to av

errors. For example, the waveform discrepancy at 1.5 k

shown in Figure 12 of Hearn and Krebes, 1990, is due to an er

neous sign choice, and not d ue to anelastic effects.

One of the main d raw backs of the ray m ethod is that it is

accurate near critical offsets. For instan ce, for th e ray in Figure

if the incidence angle at the d eepest reflection point w ere n

the critical angle for the P or S transmitted w ave (at which

transmitted wave becomes evanescent, and propagates along

interface), then the ray amp litude comp uted from (6) and

above wou ld not be accurate. Corrections can be mad e to r

theory to imp rove the accuracy in these cases, but th ey of

involve elaborate and cumbersome mathematical extensio

and they are often not generally or easily applica

(e.g.,âerven˘ and Ravind ra, 1971; Plum pton and Tindle, 19

Gallop, 1999; Thomson , 1990; Aki and Richards, 2002). Figur

shows a set of seismic traces in w hich basic ray theory am p

tud es agree w ell with exact calculations for pre-critical off s

and reasonably w ell for post-critical offsets, but not at all

near-critical offsets. Includ ing a math ematical correction to

basic ray theory result in Figure 3 to include the h ead w ave (

refraction arrival) wou ld have improv ed the accuracy in

post-critical zone, and the extent of the zone of inaccura

surround ing the critical offset wou ld also have been red uce

but not eliminated. Figure 4 show s a comparison of the

Seismic Forward ModelingContinued from Page 29

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Figure 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, respectively. 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).

Figure 4. Amplitude versus offset curves for the SH case for two elastic isotrhalf spaces separated by an interface, with the source and receiver 1000 m abthe interface. The densities and shear wave speeds in the upper and lower spaces are 2.0 and 2.0 g/cm3, and 1.0 and 2.0 km/s, respectively. The frequenc20 Hz. The graph shows the exact result computed by numerical evaluation ogeneralized reflection integral, and the approximate result computed with theory.



theory amp litud e and the exact amplitude against offset. Again,

we see that ray theory am plitudes are qu ite accurate for small

offsets correspond ing to sub-critical angles, are reasonably accu-

rate for super-critcal offsets, but not at all accurate for near-crit-

ical offsets. For deep reflectors in horizon tally layered m edia, the

reflection angles are generally small (well below critical), and ray

theory can safely be used to prod uce synthetic seismogram s.

However, for shallow reflectors with large velocity contrasts,

reflection angles can be large for the larger offsets, in which case

ray theory could not be u sed for near-critical angles and off sets.

Reflections near critical angles could also occur for deeper ref lec-

tors if they are d ipping.

Ray theory is also inaccurate near caustic zones, where rays

from a single source converge or focus because of changing

velocity grad ients, for examp le. Focusing of wave energy obvi -

ously produces high amplitudes (e.g., a magnifying lens can

focus sunlight enough to burn paper), but in the ray theory

approximation, the computed ray amplitudes are infinite at

caustics, because the g eometrical spreading factor L¡ 0 there.

L is a measu re of the spread ing of the wav efronts, and so w hen

they converge rather than spread, L decreases (and vanishes at

the caustic in ray th eory).

In add ition, as ray theory is a high frequency appr oximation, or

low wav elength app roximation, it cannot be applied accurately

to structures with thin layers, “thin” meaning thinner than a

dominant wavelength, roughly speaking.

Ray theory also does not produce the complete or full wave

field: ray-synthetic seismogram s do n ot contain the w aveforms

for all the w aves arriving at the receiver, but on ly those for the

specific rays selected by th e user. Of course, one m ay includ e as

many rays as one wants in a typical computation, but one is

often not sure in advance whether a particular ray will have a

high enough amplitude to make it worthy of inclusion(althou gh some progr ess has been mad e in this regard in d eter-

mining threshold amplitudes of rays – see, e.g., Hron, 1971,

1972).

Ray theory can also be extended and applied to more complex

cases involving oth er types of sour ces, laterally heterogeneou s

layers, interfaces of any dip and strike, curved interfaces, and

anisotropic media (e.g., Richards et al, 1991; Chapman, 1985,

âerven˘, 2001; Aki and Richards, 2002; P‰enãík et al., 1996).

However, the additional mathematical theory and numerical

computations involved are generally extensive, and the final

results still generally includ e the inaccuracies mentioned in the

preceding p aragraphs.

Neverth eless, in spite of the d raw backs, basic ray theory is still

widely u sed to comp ute synth etic seismogram s, because of the

relative simplicity of the simpler versions of the method, the

fast comp utational times, and the fact that the ray p aths for all

the events on ray-synthetic seismograms can be identified from

the event travel times (because the rays comprising the

synthetic are chosen by the user). In particular, ray theory is

very useful for computing travel times, if not always signal

amplitudes. However, for more accurate amplitude computa-

tions, and for synthetics which include all the possible waves

than can arrive at a receiver, one must resort to exact or nearly-

exact full-wave method s, such as those discussed below.

Gen eralized Ray Theory

The geometrical ray theory result in (6) and (8) above,

approximation to the exact wavefield, makes use of only o

value of the ray parameter p, namely, the value (call it p0) wh

satisfies (4), the equation for X. A more accurate result wou

involve an appropriate superposition of waves with differ

ray param eter values.

A simplistic way to see this is shown in Figure 5, which sho

a 4-segment geometrical ray, with ray parameter p0, rep

senting the reflected wav e going from a source point at x = 0

a receiver at a known offset x = X0. The layers are homo

neous. The plane wavefronts associated with the geometri

ray are also shown in Figure 5. In add ition, another ra y with r

parameter p and with a larger offset X(p) is shown. This

does not arrive at th e receiver point X0. Nevertheless, note t

the plane wavefronts associated with this ray also strike

receiver. Consequently, in this sense, this ray also contributes

the displacement at the receiver, and should therefore

includ ed in th e compu tation of the receiver respon se if grea

accuracy is desired. In fact, one can then su rmise that sup eriposing the contribu tions of all such rays with different p valu

might give the exact respon se, i.e., the full wavefield, or at le

a more accurate r esponse, which is in fact the case.

By analyzing this problem mathem atically, with the intent

obtaining the exact response, one obtains a formu la express

the d isplacement of the receiver, du e to the reflection from t

second interface in Figure 5, as an integral over p, with p go

f rom 0 to ∞ (the superposition of plane w aves mentioned in

previous parag raph ). In fact, this integral contains w ithin it n

only the displacement du e to the reflected wave but also that d

to the head wave (the refraction arrival) from the second interf

– the integral is said to give th e generalized ref lection response

Figure 5, the geometrical ray pa th and the ray p ath associatwith the head w ave, taken together, can be called a generalized r

The basic geometrical ray formu la (6) is actually the lowest-or

approximation of the generalized reflection integral.

Mathematical formulas for generalized ray responses can a


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Article Cont


Figure 5. A two-layer-over-half-space model in which velocity increases depth. The geometrical ray and head wave between the source and receivershown, as well as a ray with parameter p arriving at the offset X(p). The firepresents a simple way of understanding the concept of a generalized ray.




includ e multiples and any oth er wave typ es than are associated

with the ray. Typically, they are applied for high frequencies

(which in this case means travel times much greater than the

wave period, or source-receiver offsets much greater than the

wavelength). They are appealing because they give the

complete (or nearly complete) wavefield at the receiver, and

have been used in the literature to calculate exact synthetic seis-

mogram s. The fact that they are sup erpositions of plane w aves

means th at p lane wa ve reflection and transmission coefficients

can be used in the integrands. Note however that in compu ting

synthetic seismogram s for a layered geological mod el, it is still

up to the user to decide which generalized rays to include in

the computation. If certain generalized rays are left out of the

calculation that should have been included, the synthetics will

not include the corresponding events, which could hamper

interpretation wh en the synthetics are compared w ith real data.

Also, a disadvantage of generalized ray theory is that the inte-

grals are n otoriously difficult to evalu ate – the integrand s oscil-

late wildly. How ever, numerical methods have been dev eloped

to evaluate them as efficiently as possible (see, e.g., Aki and

Richards, 2002). The exact results in Figures 3 and 4 werecomputed by evaluating generalized ray integrals.

Matrix Methods

When comp uting syn thetic seismogram s for the stack of layers

in Figure 2 using basic geometrical ray theory or gen eralized ray

theory, one mu st select the p articular rays to includ e in the

synthetics. For example, it is common to choose only the

primary reflections (rays which reflect only once before

returning to the receiver). How ever, in some cases, one m ay

wish to includ e m ultiples (rays w hich reflect more than once),

such as the ray in Figure 2. Clearly, there are an infinite num ber

of multiples in a stack of layers, but only a finite num ber of them

can be included in a synthetic based on the ray methodsdiscussed above. The choice of which multiples to include

dep ends on the pr oblem one is attempting to solve. Typically,

one would wish to include the mu ltiples whose amplitudes are

above a specified threshold, and math ematical formu las have

been d evelop ed to facilitate th e choice (Hron , 1971, 1972). Also,

it is clear that th e more rays (prim aries or multiples) that are

included, the greater the computation time.

However, it is in fact possible to include all the multiples (an

infinite nu mber) by u sing the so-called propagator matrix method.

This is in essence done by replacing an infinite series with a

finite expression, in the sam e sense that the infinite series 1 + x

+ x2 + x3 + … is equivalent to the fun ction 1/ (1-x).

The basic ideas involved in the propagator matrix method are

described in the following paragrap hs.

The boundary conditions that seismic waves satisfy at each

interface are that the x, y an d z components of d isplacment u

and traction T (the force per un it area acting across an interface

du e to a w ave) mu st be continuou s across the interface, i.e., their

values in the layers above and below the interface must match

at the interface. These are called welded-contact boundary condi-

tions, and they exp ress the fact that tw o layers move as a u nit at

the interface when disturbed by a wave – they move as if they

were w elded togeth er at the interface.T is also sometimes cal

the stress vector because its components are the normal a

shear compon ents of the stress tensor. The boun dar y conditio

when expressed as mathematical equations and solved, yi

the formulas for the reflection and transmission coefficients

in (6) is a product of such coefficients.

For plane w ave propagation in the xz plane (wher e x is offan d z is depth) of a vertically heterogeneous solid mediu

(e.g., a stack of horizontal homogen eous layers, or a med ium

which the medium parameters vary smoothly with d epth),

displacement u and the traction T can be compu ted anywh

in the medium u sing the propagator matrix. More specifically,

plane harmonic P-SV waves, u an d T at any depth z can

compu ted from given values of u an d T at some depth z0

using the following equ ation:

where u an d T are 2x1 colum n vectors wh ose elements are

vector comp onents of displacement (ux and uz) and traction

an d Tz), respectively, and where P(z, z0) is the propaga

matrix, a 4x4 matrix containing the medium parameters, t

frequency, the ray parameter, and the vertical phase factors

the plane w aves. The column v ector f is called the displaceme

stress vector.

For a stack of layers, f can be continued through the lay

because f contains precisely the quantities that are continuo

across an interface. In other w ords, if we know f in the first lay

we can comp ute f in the last layer (or any layer ). Also, for a st

of layers, the propa gator matrix P(z, z0) is itself the product o

sequence of matrices, known as layer matrices, and their inverswhich contain the p arameters for the individual layers.

The propagator matrix method can be used to solve a nu m

of different typ es of problems in seismic forward mod eling

vertically heterogeneous m edia. For example, it can be used

compu te the reflection an d transm ission coefficients for a sta

of horizontal homogeneous layers (as opposed to the coef

cients for a single interface). More precisely, suppose tha

downgoing plane wave of a given frequency is incident up

the top layer of a stack of such layers. The incident wav e will

transmitted into the stack and be multiply reflected and tra

mitted inside all the layers. The upgoing wave emerging fr

the top of the stack into the inciden ce medium will be a sup

position of all the u pgoing w aves (produ ced by m ultiple reftion and transmission at all the interfaces of the stack) pass

up ward s through th e top of the stack. The relative amplitud e

this upgoing wave can be thought of as the reflection coeffici

of the stack . It will be a function of the thicknesses, densities a

wav e speeds of all the layers. Similarly, the d own going w ave

the medium below the stack will be a superposition of all

downgoing waves (due to multiple reflection and tranmissi

in the interior of the stack) passing through the bottom of t

stack. The relative amplitude of this downgoing wave c

be thought of as the transmission coefficient of the stack . It

Article Cont’d


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important to note that the stack reflection and transmission

responses contain the contributions of all the multiples (there

are an infinite number of them). The stack coefficients are also

functions of frequency, whereas interface coefficients are not

(except for d issipative med ia).

Such stack reflection an d transm ission coefficients h ave b een

used in the so-called reflectivity method (e.g., Fuchs a nd Müller,1971) and its variants, which is a p opu lar method for compu ting

synthetic seismograms for a stack of horizontal layers for

f orward m odeling pu rposes. As an example of an application of

the reflectivity meth od, consider a geological mod el consisting

of a single layer overlying a stack of many th in layers, and

supp ose one wants to m odel the reflection response of the thin-

layer stack d ue to a point source in the single upp er layer. The

continuation equ ation (9) would be u sed to obtain the amp li-

tud e of the wave incident u pon th e stack, the stack ref lection

response, and to continue it to a receiver on the su rface. Then, an

integral over the ray parameter p is performed to obtain the

generalized reflection response, and also an integral over the

f requency ω to obtain the w aveform at the r eceiver. In p ractice,

the integral over p is often limited to su b-critical values, as the

interest is often just in the small-angle body wave response. The

resulting w aveform wou ld contain all the mu ltiples as well as

the primary ref lections.

The reflectivity method can also be combined with other

methods to enhance the output of the modeling process. For

example, 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 the

so-called ray-reflectivity method (Daley and Hron, 1982). One

advantage of this combination is that all the events on the

synthetics can be identified (in term s of ray path s).

Matrix methods are also used in a technique sometimes calledthe recursive reflectivity method (e.g., Kennet t, 2001). In this tech -

nique, mathematical formulas for the stack reflection and

transmission coefficients for a single layer are first developed

by systematically adding up the contributions of all the indi-

vidual multiple reflections and transmissions in the layer to

obtain the total respon se. The resulting su m is an infinite series,

which is replaced by a finite expression using the above-

mentioned formula, 1 + x + x2 + x3 + … = 1/ (1-x) , wher e x is a

prod uct of the internal reflection coefficients of the layer and a

vertical phase factor. These one-layer stack formulas are then

used to obtain the stack coefficient formulas for a two-layer

stack, wh ich are in tu rn u sed to obtain the formu las for a three-

layer stack, etc. These formulas all have the sam e math ematical

form, regardless of the number of layers involved. Although

the stack coefficients for any size stack can be comp uted in this

way, it can be shown that a more efficient two-step recursive

procedure can be developed for computing them. This proce-

du re involves app lying vertical pha se factors to bring the stack

coefficients from the bottom of a given layer to the top of the

layer, then using the stack coefficient formulas to cross the

interface into the bottom of the next layer above the given on e.

This process is continued recursively.

As mentioned above, an integral over the ray parameter p c

also be performed to obtain the generalized reflection respon

and also an integral over the frequency ω to obtain the wa

form at th e receiver.

It can be shown that these many-layer stack formulas give

same result as the propagator matrix method. Although th

are more complicated and involve more computation, thoffer several advantages over the propagator matrix approa

(a) the reflection and transmission processes of ind ividual w a

mod es (P and S) can be followed m ore easily, (b) P-S conv

sions can be included or left out (which is useful for det

mining the effect of conversions on the synthetics, (c) use

approximate formulas can be developed from the exact sta

formulas, e.g., app roximate formulas for the case in w hich o

a single conversion is allowed , (d) if the infinite series is used

the formu las (instead of the correspond ing finite expression)

can be truncated to include only as many multiples as desir

(which is useful for determining the effect of multiples), (e

the infinite series is used, the individual terms have a v

simple dependence on frequency, allowing the formulas to

used with other frequency-based methods, and (f) the terms

the formulas are directly related to the physical proces

occurring in the stack (the multiple reflections and transm

sions), allowing more physical insight into the proc ess

w h e reas the prop agator matrix approach gives only

combined cu mu lative effect. If how ever, only the total r espon

is required, the propagator matrix method is generally eas

and more straightforward to use.

The matrix methods discussed above can also be used

wavefield extrapolation (e.g., Bale and Margrave, 2003), wh

is a part of most seismic wave equation migration process

and also other inversion methods which contain a forwa

modeling stage.

The finite difference method

Another p opu lar seismic forward mod eling meth od is the fin

difference (FD) method, which is a numerical method

solving partial differential equations. It can be applied to

seismic equation of motion to compute the displacement u

any point in the given geological model, e.g., at the surface (

generating synthetic seismograms for comparison w

recorded data), or at some depth z (for doing w avefield extr

olation or downward continuation, a stage of wave equat

migration). There are a number of variants of the method, a

many small improvements have been mad e to the basic meth

by man y researchers.

The main idea behind the FD method is to compu te the wa

field u (x,y,z,t) at a discrete set of closely-spaced grid p oints (

ym , zn , tq ) , with l, m, n, q = 0, 1, 2, 3, 4, 5, …, by ap proximat

the derivatives occurring in the equation of motion with fin

diff e rence formulas, and recursively solving the re s u l ti

difference equ ation.

As an exam ple of a finite difference formula for ap proximati

a d erivative, consider th e basic definition of a derivative, wh

can be found in any elementary calculus text, i.e.,

Article Cont




This leads to th e forward-difference approximation for dy/dx, given

by

where ∆x is a small finite-sized interval or grid step in the x

direction (adjacent grid points in the x direction are separated

by the amount ∆x). Similarly, th e backward-difference approxima-

tion is given by

and the central-difference approximation is given by

It can be easily show n, by ap plying the Taylor series expansion

to these formu las, that the error m ade in u sing the forward- and

backward -difference app roximations in (11) and (12) is of ord er

∆x (i.e., it is linear in the small quantity ∆x), but that the error

for the central-difference approximation in (13) is of order ( ∆x)2

(it is quadratic in ∆x). Consequently, the central-difference

formula in (13) is a more accurate app roximation to dy/dx.

As an example of the application of the FD method, consider

the 1D p artial differential equation

where u = u(x,t). It is known as the one-way wave equation

because it admits solutions representing w aves travelling with

constant speed c in the p ositive x direction but not the negative

x direction. In m athema tical terms, the exact solution is u = f (x-

ct), where f is an ar bitrary function. Since the exact solution of

(14) is known, one would not of course need to use the FD

method to solve (14) in pr actice – we u se it here only as a simple

example 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-difference

approximation for ∂u / ∂t, resulting in th e equation

where ∆t is a small finite-sized interval or grid step in the t

direction. Upon re-arrangement, one obtains

wh ich can be used to recursively comp ute u at the grid point

t+ ∆t) if one know s the value of u at the grid points (x,t) and

∆x,t). Given som e starting values, i.e., initial conditions or inivalues, such as u(x,0) = f (x), wh ere f (x) is an arbitrary fun ction

x (the waveform at t = 0) and given values of the grid steps

an d ∆t, i.e, a va lue for g, one can then comp ute the values u

∆t) for all x, which are then in turn u sed to compute the valu

u(x, 2∆t) for all x, etc. In other word s, u can be compu ted at

times t by “marching forward in time” in this way.

It can be show n that if one chooses ∆x an d ∆t so that g ≤ 1, th

(16) is a stable FD scheme, i.e., the errors resulting from m aki

FD approximations to the derivatives remain finite – they

not grow without limit as the time-marching computatio

progress. In fact, if one chooses g = 1, the resulting simple

scheme, u(x, t+∆t) = u(x-∆x,t), gives the exact solution to (1

For g < 1 however, the scheme in (16) is first-order accuratespace (x) and time (t), because the errors in the forward- a

backward-difference approximations used for the derivativ

are of first order in ∆x an d ∆t.

If one naively uses the forward-difference approximation (

to app roximate both ∂u / ∂x an d ∂u / ∂t in (14), then one obta

an unstable FD scheme, in which the error grows without lim

du ring the compu tations, regardless of the value of g.

In general, for any 3D par tial differential equation, such as t

elastic equation of motion, it is necessary to choose the corr

FD approximations to the derivatives appearing in the eq

tion, and to choose appr opriate values for the step sizes ∆x,

∆z an d

∆t, to ensure numerical stability. Often, a preliminamath ematical analysis of the partial differential equation u n

study is useful, and sometimes necessary, to determine wh

type of FD approximations to u se.

The 3D seismic equation of motion, a second-order part

differential equation in the d isplacement u , can be solved in

same way, although the resulting difference equation is mu

more comp licated than (16). Norm ally, the difference equati

is marched forward in time t, with u being computed at

values of x, y an d z for each time level. Because the wavefield

then known everywhere, it is possible to create movies of t

w a v e f ronts prop agating through the material, which

helpful in understanding seismic wave propagation.

FD computations are extremely time-consuming. 3D F

computer programs for realistic geological models can ta

many hours to run. Consequently, it is common practice to

FD calculations in 2D instead, to red uce the comp utation tim

Also, various computational techniques have been develop

to imp rove efficiency. Even in 2D h owever, comp utation tim

are typically mu ch longer than those of other methods, such

ray theory. However, as computer technology advances,

calculations shou ld become m ore pr actical.

Article Cont’d


Continued on Page

(11)

(12)

(13)

(14)

(15)

(1

(10)





Continued on Page

Accuracy of the compu ted solution is another p rimary concern

in FD computations. One way to generally improve the accu-

racy of the FD solution is to u se smaller grid step sizes (while

making sure th at the stability condition is always satisfied, e.g.,

g ≤ 1 for (16) ). How ever, this increases the compu tation time.

In addition, if the grid step sizes are made too small, accuracy

can decrease, due to the occurrence of significant roundoff

error. Another way to generally improve accuracy, without

reducing the grid step sizes, is to use higher-order approxima-

tions for the d erivatives. As indicated above, the forward - and

backward -difference app roximations used in (11) and (12) lead

to first-order accurate FD schemes, but the central-difference

app roximation leads to second-order accurate FD schemes, and

is comm only used in constructing typical FD schemes. Higher-

order approximation formulas, involving a series or combina-

tion of forwa rd -, backw ard - or centra l-diff e re n c e

approximations, can be used as well, but they result in more

complicated and cum bersome FD schemes and in an increased

computation time.

An u nw anted artifact of FD calculations is a degrad ation of the

accuracy of the solution caused by grid dispersion (e.g., Alford e t

al., 1974). This occurs if the spatial grid steps are chosen too

large, i.e., if the grid is too coarse. To avoid grid dispersion, a

typical rule of thu mb is that the sp atial grid steps should be n o

larger than one-tenth the size of the shortest wavelength in the

wavefield, i.e., there shou ld be at least 10 grid p oints per w ave-

length. If higher-order FD schemes are used (i.e., if higher-order

derivative approximations are used), this number can be

redu ced, e.g., 5 grid p oints per w avelength (e.g., Alford et al.,

1974; Levand er, 1988). How ever, this does not necessarily resu lt

in a reduction in compu tation time, as the higher- o rd e r

schemes are more complicated.

The main ad vantage of using FD method s in seismic modelling

is that they produce the full wavefield – all the different types

of waves that exist (reflections, refractions, etc.) generally will

appear in the computed solution with correct amp litudes and

phases. One difficulty of the method is the question of how to

insert a given type of source of waves. A straightforward w ay

to do this is to use an exact solution in the vicinity of the source,

and then continue the solution with the FD method. Other

ways involve modifications of this approach (e.g., Alterman

and Karal, 1968; Kelly et al., 1976).

A number of diff erent variants of the FD method are in use, for

example, the velocity-stress FD method (e.g., Madariaga, 1976;

Virieux, 1986). In th is method , the equ ation of mot ion, which is a

s e co n d-o rder partial diff e rential equation in the particle-

displacement u , is re-expressed as a coup led system of first-order

partial d iff erential equations in the comp onents of the part icle-

velocity (∂u / ∂t) and the traction T. First-order equations are

generally easier to solve. In ad dition, the u se of staggered grids, in

which the particle-velocity components an d the traction compo-

nents are comp uted on diff erent grids w hich are shifted or stag-

gered relative to each other by h alf a spatial grid step in all

directions, makes for efficient and accurate computation.

Another p opu lar variant of the FD method is the pseudo-spectral

method (Kosloff and Baysal, 1982). In this method, the time

derivatives are still comp uted using a FD app roximation in t

time domain, but the space derivatives are computed in

wavenumber domain. More specifically, to obtain a spat

derivative, say ∂ux / ∂x, at a given time level, a spatial Four

transform is performed on the sequence of wavefield va lues

(known from a p revious stage of the FD compu tation) along

x direction to obtain the wavenumber (k x ) spectrum of

(another sequence of numbers). Then this new sequence

multiplied by ik x which, in the k x domain, is equivalent

taking the derivative ∂ux / ∂x (i is the imaginary un it). Finally,

inverse spatial Fourier transform is performed on this alter

sequence to give the differentiated sequ ence of values ∂ux /

in the x domain. The same technique is applied to the oth

componen ts and in the other 2 spatial directions. The results

then used to compute all the wavefield values at the next ti

level. The method gives quite accurate results because

spatial derivatives are done in the wavenumber doma

avoiding the inaccuracies associated with FD approximatio

to derivatives. Another advantage of this method is that

requires considerably fewer grid points per wavelength th

the stand ard FD schemes in the space-time domains, makingattractive for 3D computations. With today’s compute

run ning at 2-3 GHz or higher, medium -sized 3D mod el comp

tations can be done in several hou rs or less.

Another unwanted artifact of the FD method is the generati

of artificial reflections. These are reflections coming from

edges of the numerical grid, rather than from the bona-f

geological interfaces in the model. For example, to comput

value for ux at the time t+∆t for a node at the very edge of

grid, one generally needs to know the value of ux at the earl

time t at a node just off the grid (many FD schemes work t

way). If one naively sets the off-grid value equ al to zero, th

strong artificial reflections from the grid edge are produc

because by setting the off-grid displacement value equalzero, one is implicitly making the off-grid region into a rig

medium, resulting in an elastic-rigid interface. The result

impedance contrast produces the artificial reflections wh

propagate back into the grid, and appear in the synthetics.

To suppress such artificial reflections, non-reflecting or absorb

boundary conditions are used at the grid edges. In the case

non-reflecting boundary conditions, FD schemes which

different from the on e used in the interior of the grid (wh ich

based on the standard equation of motion) are used. The

alternate schemes are math ematically designed so that the ar

ficial reflections p rodu ced hav e low (ideally zero) amp litud

Sometimes they involve solving one-way wave equations n

the grid edges – equations whose solutions are waves whcan travel in on e direction only, i.e., off the grid but not back

to it. In the case of absorbing boundary conditions, one u

schemes which absorb the energy of the artificial reflectio

Typically, the grid is enlarged by a sma ll amount – a few lay

– and a mod ified FD scheme wh ich includ es the effects of ph

ical absorption or dissipation is solved in the added grid zo

If the FD scheme used in the interior of the grid alrea

accounts for seismic wave absorption (i.e., if it is based on

equation of motion containing absorption terms – see, e

Emmerich and Korn, 1987; Krebes and Quiroga-Goode, 199

Article Cont




then it is a simple m atter to su pp ress artificial reflections – one

simply makes Q very small near the grid edges. Q is a measur e

of the elastic “quality” of the medium – the lower the Q, the

more absorbing or dissipative the medium is (Q = ∞ for

a perfectly elastic medium in which waves experience no

absorption). As with all compu tational method s, non-reflecting

and absorbing boundary conditions have their limitations, but

in general, they work quite w ell.

Figures 6-16 show examples of synthetic shot records and

wa vef ronts computed with the standard 2D FD method.

Figures 6 and 7 were produced by Peter Manning and Gary

Margrave, and Figures 8-16 were p rodu ced by Louis Chabot, all

members of the CREWES project at the University of Calgary.

These figures illustrate how various physical effects in seism

wave propagation can be investigated, because the FD meth

prod uces the full wav efield.

Figure 6 and 7 show 2D synthetic shot records for an explos

acoustic pressure source buried at depths of 8 m and 18

resp., in a horizontal homogeneous elastic layer overlying

rigid h alf-space. The synthetics show the exp ected events, ithe direct arrival, the dispersive surface wave and the refl

tion. They also show how the deeper source results in

lowering of the resolution and a reduction in the surface wa

amplitude.

Figures 8 and 9 again show 2D synthetic shot records fo

source consisting of a force directed vertically downward a

point on the surface (e.g., a hamm er blow). The non -symmet

source generates both a P wav e and an SV wave. The geologi

model is a horizontal homogeneous elastic layer overlying

homogeneous elastic half-space. The top of the elastic layer i

free surface. The synthetics show the expected events, i.e.,

direct arrival, the reflected and converted waves, the he

waves (refraction arrivals) and the prominent dispersguided waves. They also show the amplitude and ph

(polarity) relationships among the wave types. In additi

Figure 9, for the horizontal component of particle moti

shows a polarity reversal in all events as one goes from the l

side of the record to the right side. This happens because

waves arriving at geophones left and right of centre produ

geophone motions whose x components of displacement ha

opposite signs. Some numerical artifacts can also be seen

Figure 9 on th e lower left and lower right ed ges of the grid.

Figures 10-16 show a series of “snapshots” of the wavefro

corresponding to the shot records in Figures 8 and 9, at t

Article Cont’d


Continued on Page

Figure 6. A 2D synthetic shot record produced by the finite difference method fora buried pressure source. The horizontal axis is the source-receiver offset in meters

and 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. of Calgary.

Figure 7. Same as Figure 6, except the source depth is 18 m. The figure wasproduced by Peter Manning and Gary Margrave, CREWES project, U. of Calgary.

Figure 8. A 2D synthetic shot record for the vertical component of geophmotion produced by the finite difference method for a surface source. The hzontal axis is the source-receiver offset in meters and the vertical axis is the 2-travel time in seconds. The model is 3 km x 1 km. The geological model is a elastic layer with a free upper surface overlying an elastic half-space. The souis a vertically directed point force located on the free surface. The density, P S wave velocities in the elastic layer are 2.1 g/cm 3, 1000 m/s and 500 m/s, reand in the half-space they are 2.3 g/cm3, 1850 m/s and 925 m/s. Figures 8-16 wproduced by Louis Chabot.



following fixed times: 300 ms, 500 ms, 600 ms, 800 ms, 1000 ms,

1200 ms, and 1400 ms. The upp er half of each figure sh ows th e

horizontal component of motion and the lower half shows the

vertical component. Several interesting wave propa g at io n

features can be seen in the figures – they show (a) that the“hamm er blow” source produ ces both a downgoing P and SV

wave; (b) that there is a downgoing head wave connecting the

down going P and SV wavefronts emerging from the source

point; (c) the up going Pand SV head wav es (refraction arrivals)

produ ced by the transmitted P wavefront beyond the critical

angle (when it breaks away from the other wavefronts at the

interface); (d) the expected left-to-right polarity reversal in the

horizontal component; (e) the variation of amplitude along

some of the reflected and transmitted wavefronts due to the

non-symmetric “hammer blow” source; and (f) the complex

nature of the wavefield produced by a non-symmetric sour

even for a simple one-layer model.

Regarding point (e) in the preceding pa ragrap h, the variation

amp litude observed along the w avefronts is consistent with

radiation pattern for a vertically directed point force (kno

from theoretical studies of seismic wave propagation). F

example, this radiation pattern predicts that a downwa

hammer blow would result in mainly a large vertical (longi

dinal) displacement of the medium at points vertically bel

the source point, with little or no horizontal (transver

displacement at these points (which one might also pred

using physical intuition). This prediction is confirmed,

example, in the transmitted P wavefront in Figure 12 (

lowermost wavefront in each half, upper and lower, of

figure) – for the horizontal comp onent, the centre of the tra

mitted P wavefront has zero amplitude, but for the vert

component, it has a high amplitude.

A good w ay to study Figures 10-16, with the intent of und

standing wave propagation, is to select a certain wave a


Article Cont


Continued on Page

Figure 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 half and 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 t his andthe following figures.

Figure 11. Same as Figure 10, except the elapsed time is 500 ms.




follow what happens to it from one snapshot to the next

time progresses). For example, consider the first reflected

wave – the vertical component. Beginning with Figure

(lower half), wh ich show s the down going high-amp litude in

dent P wave that has n ot yet reached the interface, we move

Figure 11 and note the beginnings of the first reflected P wa

Since the inciden t P wav e is reflecting off a med ium (the ha

space) which has a larger acoustic imped ance (density x Pwa

velocity), the v ertical compon ent of d isplacement experience

polarity reversal (the central peak of the reflected P wav ele

wh ite, not black as for the incident P wav e). This is just like

polarity reversal in displacement experienced, u pon reflecti

by an incident wave on thin string connected to a thick ro

We then follow th e reflected P wave u pw ards in Figures 12 a

13, where it then reflects off the free surface. This free surfa

reflection can be seen in Figure 14 – note that th ere is now

polarity reversal in the vertical component of displacem

because the wave is reflecting off a medium with a low

acoustic impedance, i.e., the air layer (although there i

polarity reversal in the p ressure at the free surface – a compr

sion reflects as a rarefaction, i.e., a d ilatation, off a mediu m wa lower acoustic impedance). We then follow the P wa

reflected from the free surface back down to the interface

Figure 15, after which it reflects off the interface with anoth

polarity reversal (du e again to the larger imped ance in the ha

space), wh ich can be seen in Figure 16. Note also the decreas

amplitude of the wave.

FD forward modeling has also been done for solid me

containing interfaces that are not in w elded contact with ea

other (e.g., Slawinski and Krebes, 2002a, 2002b). Non-wel

contact means that the displacement u is no longer continuo

across an interface, although the traction T still is. Me

containing joints, fractures an d fau lts may in som e cases fall i

this category.



Article Cont’d


Figure 13. Same as Figur e 10, except the elapsed time is 800 ms.



Continued on Page




Article Cont


Other methods

Forward modeling has also been done with other methods. For

example, some success has been achieved w ith the finite element

method. This method still involves computations on a grid or

mesh, but involves representing th e componen ts of the solution

u (x,y,z,t) to the equation of motion as linear superpositions of

app ropriately selected basis functions. For example, see the arti-cles in Kelly and Marfurt (1990), as w ell 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 it

can be more flexible and more accurate than FD methods w hen

the geological interfaces in th e mod el are irregular, bu t it is also

more difficult to implement and apply than the FD method. A

variety of other methods have also been used for models with

irregular 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 solving

integral equations which describe their behaviour (e.g., see the

references listed by Carcione et a l., 2002).

Acknowledgements

Thanks go to Louis Chabot, form erly of the CREWES project in

the Departm ent of Geology and Geophysics at the University of

C a l g a r y, and Peter Manning and Gary Margrave of the

CREWES project, for the finite d ifference synth etics.

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Anelastic and Porous Media . Pergamon Press, Amsterdam.

Kelly, K.R. and Marfu rt, K.J. (eds.), 1990. Numerical Modeling of Seismic WPropagation. Society of Exploration Geoph ysicists, Tulsa.

Slawinski, M.A., 2003. Seismic Waves and Rays in Elastic Me

Elsevier/ Pergamon, Amsterdam. R

apr04 seismic forward modeling

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