aki and larner 1970

.•OURNAL OF GEOPHYSICAL RESEA•tCH VoL. 75, No. 5, FEBRUARY 10, 1970

Surface Motion of a Layered Medium Having an Irregular Interface Due to Incident Plane $H Waves

KEIITI AKI AND KENNETH L. LARNER

Department o] Earth and Planetary Sciences Massachusetts Institute o! Technology, Cambridge, Massachusetts 02139

A practical method is devised to calculate the elastic wave field in a layer-over-half-space medium with an irregular interface, when plane waves are incident from below. This method may be used for studying the interface shape of the M discontinuity, for example, using the observed spectral amplitude and phase-delay anomalies due to teleseismic body waves. The method is also useful for the engineering-seismological study of earthquake motions of soft superficial layers of various cross sections. The scattered field is described as a superposition of plane waves, and application of the continuity conditions at the interface yields coupled integral equations in the spectral coefficients. The equations are satisfied in the wave-number domain when the interface shape is made periodic and the equations are Fourier transformed and truncated. Frequency smoothing by using complex frequencies reduces lateral interferences associated with the periodic interface shape and permits comparison of computed results with those obtained from finite bandwidth observations. Analyses of the residuals in the interface stress and displacement, performed for each computed solution, provided esti- mates of the errors. The relative root-mean-square residual errors were generally less than 5% and often less than 1% for problems in which the amplitude of the interface irregularity and the shortest wavelength were comparable. The method is applied to several models of 'soft basins,' 'dented M discontinuity' and 'stepped M discontinuity.' The results are compared with those derived from the flat-layer theory and from the ray theory. In addition to vertical interference effects familiar in the flat-layer theory, we observe the effects of lateral interference as well as those of ray geometry on the motion at the surface.

INTRODUCTION

This paper describes a practical method of interpreting seismic observations on the surface of a layer-over-half-space with an irregular interface, as shown in Figure 1. We simplify the problem by assuming that the medium is uniform in the y direction, and the depth • to the interface is a function of x alone. We are in-

terested in the motion at the surface, when plane waves are incident parallel to the x-z plane. The case of SH waves is discussed in this paper, and those of P and SV waves will be reported in a subsequent paper by the present authors (see abstract, Larner and Aki [1969]).

The principle of our method is not new, but goes back to Rayleigh [1907, 1945], who studied scattering of plane waves by gratings. In our method, the scattered wave field is represented as a linear combination of plane waves with discrete horizontal wave numbers (includ- ing inhomogeneous waves), where the coeffi-

Copyright (•) 1970 by the American Geophysical Union.

cients are determined in such a way that the boundary condition is approximately satisfied. There are several approximate ways to satisfy the boundary condition. Rayleigh used an iterative approximation, expanding the boundary condition in a power series of the amplitude of corrugation. The same approximate method has been applied to seismic problems by Sato [1955]; Abubakar [1962a, b, c]; Dunkin and Eringen [1962]; Asano [1960, 1961, 1966]; and Levy and Deresiewicz [1967]. Meecham [1956] used a variational method in which the bound-

ary condition is satisfied in the least-squares sense. The method presented here is closer to Meecham's but takes advantage of the fast Fourier transform algorithm [Cooley and Tukey, 1965] and satisfies the boundary condition in the wave-number domain.

Another unique feature of our method is the use of complex frequency. The time function of our incident waves has the form e -•*. By making the frequency (o a complex number, we are able to compute the spatial distribution of

933

934 AKI AND LARNER

_ r"' Free Surface

Half-space:

Incident

Haskell method [Haskell, 1953]. It is important to know how the resonance conditions are

affected by the lateral variation of layer thickness. We shall show in the present paper some results of the successful application of our method to the above problems.

FORIV•ULATION OF •ROBLEIV•

Fig. 1. Schematic cross section displaying the 'layered-medium configuration and coordinate axes.

amplitude and phase-delay anomalies smoothed over a certain frequency band, whose width may be chosen appropriate for a given problem.

There are two major problems that have motivated us to develop the present method. One is the use of amplitude distribution and phase-delay anomalies of teleseismic body waves in the study of crustal structure. If the Mohoro- vicic discontinuity is not a horizontal plane, but has an irregular shape, the amplitude of teleseismic body waves at the surface will show a spatial variation owing to focusing and defocusing. So far, only the ray theoretical approach has been undertaken [cf., Mechler and Rocard, 1967; Mack, 1969; Mereu, 1969]. The ray theory is inadequate in problems that involve wavelengths comparable to the linear dimensions of the interface irregularity. We are applying our method to the Montana LASA data in order to determine a more unique pic- ture of the Moho shape under the array. The result will be reported in a separate paper [Alii and Larner, 1969].

The other problem is the so-called 'ground motion' problem. The spectrum of seismic motion at the earth's surface shows peaks and troughs owing to constructive and destructive interferences within the surface layers. Earth- quake seismologists are interested in this problem because they can utilize the shape of spectrum for studying the crustal layering. The engineering seismologists are concerned about this problem because the ground motion can be amplified significantly a•t the resonance frequencies.

The case of $H waves in horizontally uniform layers has been extensively studied by Kanai and his colleagues [Kanai, 1952; Kanai ei al., 1959]. Cases involving P and $V waves have been studied by Haskell [1960, 1962], Phinney [1964], and others, using the Thomson-

As shown in Figure 1, our medium consists of a homogeneous, isotropic layer with shear velocity fi• and density p• overlying a homogeneous isotropic half-space with shear velocity fi2 and density p2. The interface depth • is a function of x alone. Our problem is to find the motion at the free surface z -- 0, when plane SH waves polarized in the y direction with freqency • are incident parallel to the x-z plane at the angle 0; from the z direction. This is a two-dimensional problem, and the solution is independent of y.

Let the displacement in the upper layer be u• (x, z) and that in the lower medium, u,x, z). For e -'•' time dependence, they may be expressed as

u(x, z) = + where ko and Vo are the x and z components of the wave number of the incident waves, respectively, and

The signs of radicals ro and • are chosen so that, when the frequency • is real, the first term in the right hand side of (2) expresses the incident waves coming from z -- -F •, and the second term expresses the regular waves scattered back toward z -- -F• and inhomo-

geneous waves attentuating toward z -- -F•. (The extension to the case of complex frequency is discussed below.)

Questions have been raised [Uretsky, 1965] concerning the validity of expressions such as (1) and (.2) as descriptions of the wave fields

SURFACE MOTION OF LAYERED MEDIUM 935

near the interface. The essential difficulty is that for •(x) < z < •a where • is the maximum of •(x), the scattered field in the lower medium may include waves locally propagating upward, as indicated in Figure 2. Therefore (2) does not adequately represent the wave field near the interface, and the boundary conditions at the interface cannot be satisfied

rigorously. The discrepancy is greatest in problems involving large interface slopes, particularly when the source wavelength is smaller than the amplitude (• -- sr) of the interface irregularity, where •m is the minimum of •(x): In many problems, however, the upward-propagating scattered waves near the interface may be negligibly small, so that (2) may be satisfactory for practical purposes. If so, small residuals or discontinuities left at the interface

would generate only small observable motions at the free surface. Therefore our approach is to assume that equation 2 is a good approximation to the wave field near the interface, and

to test the adequacy of this assumption by evaluating the residuals of the boundary conditions in each practical problem.

The boundary conditions are the vanishing stress at the surface z = 0, and the continuity of stress and displacement at the interface. The stress-free requirement at the surface is satisfied if we put B,(k) = As(k) so that equation 1 becomes

u:(x, z) = 2 f;• A•(k)e The interface conditions are

cos •,•z dk (5)

= r(x)] (o)

iz(Ou/On) = tz.(Ou.lOn) (7)

Rayleigh Ansatz Error:

Interfoce

Inc/dent ' / /' • Wove / / "

Fig. 2. ,Schematic illustration of the region near the interface displaying causes of the Ray- leigh ansatz error.

where • is the rigidity of_..the upper layer, • is that of the lower medium, and O/On repre- sents the space derivative in the direction normal to the interface.

0 - n.grad = nx(a/ax) q-n,(a/az) (8)

On

nx = --(dg'/dx)E1 q- (df'/dx)•] -•/• (9)

n, = [1 -Jr-(d•'/dx)2] -1/2 Using (2), (5), and (8)in (6) and (7), we get

[Al(k)gll(k , x) + A2(k)g12(k , x)]e '• dk

f:oo ml(k)

where

-- hl(X)e '•ø.* (10)

g:•(k, x) -Jr- A•.(k)g22(k, x)]e '• dk

= h•.(x)e '•øx

•11(k, x) = 2 COS 121•'(X )

gx•.(k, x) - --e

g•.•(k, x) -- 2tzl[ikn.. cos

-- •,xn, sin •q•'(x)] (11) g•.•(k, x) - --tz•.[ikn•, q- i•,n,]ei"•r (•)

= e

= in.o] By solving the above two integral equations for A•(k) and A•(k), we can determine the wave field in any part of the medium.

APPROXIMATE SOLUTION

In order to solve the integral equations 10, we convert them into infinite-sum equations by assuming a periodicity in the interface depth • (x); that is,

•'(x q-- mL) = •'(x) rn = 4-1, 4-2, --- (12)

In the examples that we consider in this paper, the interface is plane except for a localized interval in the x coordinate. If L is taken lon[ as compared to the length of interface irregularity, the effect of repeated irregularities at distances of mL can be neglected. As will be shown later, this effect can be easily diminished

936 AKI AND LARNER

by making the frequency complex in such a way that the imaginary part of frequency is large enough for waves to attenuate over the travel distance L.

When •'(x) is periodic, h• (x) and g• (k, x) are also periodic because they depend on x only through •' and d•'/dx. Let us multiply equation 10 by e -•'•. Then we have

[ •(•) •(•, x) + •(•) •(•. x)]• '• a•

= hi(x) j = 1, 2 (13)

where • = k -- ko. Since the right hand side of (13) is periodic in x, the left hand side likewise must be. Since g•i(k, x) are also periodic, • can only take such values that •tisfy

or

•L = 2wn n = 0, •1, •2, --.

Thus, the integral equation 13 must be re- placed by the infinite-sum equations

where

An (i) (ii)

g.,

= A,(k•)Ak,, i= 1, 2

= g,i(k,•, x) i, j = 1, 2

k., = ko-]- 2z-niL (rs)

Ak,, = 2•r/L

where

Gnm(il ) i fo • - L gn (il)(x)½ 2•'i(n-m)x/L dx

G (•) 1 •• - g.(•:'(x)e :"("-m'•/• dx (18) nm L

H•m - L hi(x)e -:•i•/r dx The matrix G•? ø should be well behaved as long as the variation of •(x) is small. In the case of constant •, for example, aH G,• øø vanish except when n -- m, and all H• vanish except when m -- 0. Thus, all A• (ø vanish except when n -- 0, which corresponds to re- fiected and refracted w•ves with the same wave

number, ko, as that of incident waves. In other words, there are no diffusely scattered waves in this case.

Using the coefficients A, (ø determined by solution of equations 17, we write the approximate solution for the displacement field •

, iknx u•(x, z): • • •(• cos •(l•z e (in l•yer)

(1•> N

(in ha, lf-sp•ce)

where

27rn

We now approximate the infinite-sum equations by the finite-sum equations,

• [ A,, (•' g,,(•"(x) + A. (:' g,,(i:, (x)]e:.... /'•

= hi(x) j = 1, 2 (16)

Instead of solving the above equation directly in x, we first take the Fourier transform of both sides by multiplying (1/L)e -2'•'m'/• and in- tegrating over 0 < x < L. Then we have 4N + 2 simultaneous linear equations,

•'. [A. (I•G.•(•' + A. ('• G.• (•'•] = H•

j = 1,2, m = --N, ... ,0, ... , +N (•7)

(i) j = 1, 2

SMOOTHING BY THE USE OF COMPLEX

FREQUENCY

The solution obtained above is the response to incident waves having the time function e -•'. When m is real, the solution corresponds to the steady-state case; that is, the response to sinusoidal oscillations lasting from the infinite past. In this case, the frequency spectrum is a discrete line. In practical problems, we are always dealing with a signal of finite length, which has a continuous spectrum over a certain frequency band. If we wanted to cover the frequency range with the solutions for line spectra, we would need an infinite number of such solutions. A simple and effective way of

SURFACE MOTION OF LAYERED MEDIUM

avoiding this problem is to make the frequency complex. The usefulness of such a procedure in the seismogram synthesis was pointed out by Phinney [1965].

By making the frequency complex, we are now looking at a response to incident waves having the nonstationary shape of an ex- ponentially increasing oscillation, in which amplitudes are increased by a factor e at every time interval r -- 1/•,. Here, •, is the imaginary part of • and must be taken positive for the reasons explained later.

Consider the contributions of disturbances at

various parts of the medium to the motion at an observation point at t -- 0. The contribution from a travel distance x must have originated at that source point at t -- --x/j3, where/• is the propagation velocity of the waves. If x//• >> r, the amplitudes at the source that caused the disturbance were negligibly small, and therefore the contribution from that dis-

turbance may be neglected. For example, if we take 1/•, smaller than L//•, the undesirable effect of the repeated interface shape can be removed.

The frequency spectrum for the complex • case is no longer a discrete line, but is a continuous spectrum with the bandwidth propor- tional to •o,. Therefore, the solution should correspond to a smoothed one over that frequency band. By this procedure, we lose in frequency resolution but gain in stability and economize computation time for application to transient waves.

The spatial distributions of amplitude and phase calculated for complex frequency may be compared with those that we would obtain by spectral analysis of transient records pre- multiplied by the time window of the shape i(t) -- e -q', which should be properly delayed at each station according to the arrival time of source waves.

When the imaginary part of • is introduced, the length of the wave-number vector •/• becomes complex, where /• is the medium velocity. However, with this introduction the components of the wave-number vector are not uniquely determined. For their unique deter- mination, an additional condition is required. The condition that is appropriate to our problem is that the incident waves have constant

amplitude along the wave front. In other words,

937

we request. that not only the real part but also the imaginary part of phase k•x --VoZ be constant along the wave front. That is,

(o)x - ½o)Z

+ i{Im (ko)x -- lm (•o)Z} = constant

for z = xtan Oo •- constant

where Oo is the incidence angle. It, then, follows that

R e (ko) I m (ko) = = tan Oo (20)

Re (•o) Im (•o)

Im (ko) Im (uo) w, = - -

where e is real positive and may be a function of •o•, the real part of •o.

According to the periodicity argument sur- rounding equation 13, the integral variable k can take only the discrete values given by

t•,,= k,,--ko

= 2•rn/L (n = O,-4-1,-4-2...) Thus the wave numbers k,• in the infinite-sum

equation 14 must all have imaginary parts equal to the imaginary part of ko. In the complex k plane, this means that, when a plane source wave is given in terms of complex o• and ko, periodicity imposes the requirement that the summation in (14) be over discrete complex values of k• that are equispaeed along a line through ko parallel to the real k axis.

Let us consider the complex k plane in more detail, first discussing the branch point locations and remarking on the choice of signs of the vertical components of wave number (equations 3 and 4). Since all the integrands in (10) are even functions of v•, we need not worry about the sign of v•. The sign of v, is so chosen that the backscattered waves attenuate toward z --

+o•. In other words, we use the top sheet [Lapwood, 1040] of the complex k plane where Im(v,) is positive. The branch cut is defined by Im(w_) -- 0, which is a part of a hyperbola, as is well known. It is also well known that

Re(v,) is positive below the hyperbola and is negative above in the first quadrant of k plane for positive •,.

The locations of branch lines and summation

points k. are shown in Figure 3. The e line,

938

x x

AKI AND LARNER

Ira(k)

•f branch line I rn (•): 0

• Im (•,) > 0 • Re (•) < 0

Im (z•) >0 • Re(•z)•O • •/B• •

k-n • k-2 k_ I k 0 k I k• ..... k n I ,

• • Re(k)

Fig. 3. Locations of the summation wave numbers kn on the top sheet of the complex k plane for the case •o -- •o•(1 q- ie). The solid square is the branch point, the open circle is the source wave number, and the crosses are the locations ks.

which satisfies equation 21, is also indicated. The plane waves corresponding to the points lying on this line have constant amplitude along a wave front and have a real unambiguous incidence angle 0o as defined in equation 20. The incidence waves satisfy this condition, but the scattered waves do not unless e -- 0. Thus

whenever • is complex, the computed wave amplitudes A,'" and A, ('• pertain to waves whose propagation directions are uncertain. In fact, for ks to the right of the dashed hyperbola in Figure 3, Re(v,) < 0 so that those waves attenuate downward toward z = q-oo while

appearing to propagate upward. Thus it is difficult to attach simple physical meaning to the waves that contribute to our frequency smoothed solutions.

ANALYSIS OF ERRORS AND RESOLUTION

Sources o] errors. In a previous section we cited the existence, in our method, of an intrinsic error attributable to the incomplete description of the wave field near the interface (Ray- leigh ansatz error, Uretsky [1965]; Meecham [1956]). Other errors that can arise include those attributable to truncating the infinite- sum equations, to introducing the periodic interface shape, and to smoothing with the use of complex frequency. The list of errors can also include those occurring in the numerical calculations. The sizes of these errors depend upon critical parameters such as the shape and

characteristic lengths of the interface irregularity relative to the wavelength, and propagation direction of the incident plane wave.

Since no exact solutions exist for this problem, we have used relative methods to estimate the

accuracy of our solutions. The various error criteria considered are based upon evaluation of the displacement and stress discontinuities determined at the interface. A relation that follows from the conservation of energy provides another check of accuracy. However, this error measure is a comparatively insensitive one that is equivalent to a weighted integral of the boundary condition residuals. We obtain a more meaningful estimate of the accuracy of the computed surface motion by using a representation theorem in which the residuals at the interface are taken as sources.

Residuals ai the inter/ace. To estimate the errors in the wave field associated with the interface discontinuities, let us first consider the Kirchhoff integral solution to the scalar Helm- holtz equation. If u is a solution to the equation that is valid at all points of a region R enclosed by the boundary S, then it may be expressed as the integral,

' ' On

- z') z') 1 ' On


The variables x' and z' are coordinates on the

boundary S, and O/On is the spatial derivative in the outward direction normal to S. The func-

tion G satisfies the inhomogeneous wave equation

•v•6 + pJ'6 = -•(•- z) ,(z- z') (23)

throughout R. Here, p is the mass density and B(x) is the Dirac delta function.

Now, let S consist of the free surface, the irregular interface, and vertical surfaces con- necting the free surface and interface at x -- --+m. The enclosed region is the layer (medium 1). The exact solution u• to our problem satisfies the scalar Helmholtz equation within the layer, as does our approximate solution Therefore, the difference Am_• -- Ul• -- u• must satisfy (22). By assuming that a small amount of attenuation exists within the layer, we can neglect the contributions to motion from sources along the vertical surface at in- finity. We now take the Green's function G to be that solution to (23) whose normal derivative vanishes along the free surface. The stress-free requirement at the free surface was imposed upon both the exact and the approximate solutions to our problem, and, hence, upon the difference Au•. With our choice of the Green's

function, the free-surface contribution to the Kirchhoff integral is zero; the error in surface motion can therefore be expressed in terms of the errors in displacement and stress ,along the interface. We have

= f [•(•, o; x', •-)a•-,•(x', •-) nterfaee .

- • •u•(•', •) o•(•, o; x', D] d• (24) On

where

(.Ou•N(x', •') Ou•(x', •').) /Xr•(x', D = •, \ On - On is the error in the y component of stress along the interface.

The Green's function that satisfies (23), the free surface condition, and the radiation condition consistent with the e -'•' time dependence is given by

a(x, z; x', z')

i IHo(• - 4• where

• = [(•- •,)'- + (z- z,)'-] •'•

•, = [(• - •,), + (• + z,)'-] •,• Ho (•) is the zero order Hankel function of the

first kind. Along the free surface, z -- 0; this becomes

G(x, 0; x', •') = •Uo r where

r = [(X- X') •' -]- By using (24) we could compute the exact

free-surface motion, provided that we knew the errors in stress and displacement at the interface. However, we do not know these errors; instead, we know only the differences (residuals) between the approximate solutions in the layer and in the half-space. These residuals are just the differences between the real errors (at the interface) in the layer and in the half-space. That is, at the interface

• = u,y- u•.N = Au•N- AU•.N (26) • = r•N- r,.N = Ar•- Ar•

where • and • are the displacement and stress residuals, respectively; and Au• and Ar• are •he real errors • •he half-space at •he •terface. In order •o make a low order estimate of

the error in free-surface motion, we shall assume tha• •he residuals are comparable to the actual errors at the interface. That is, at the interface,

Using a and • as so•ees in (24) yields estimated error in displacement at the free surface,

a(•, o) = •

+ a cos(y-- a) ' dS (27)

where

a = dip angle of the interface.

94O AKI AND LARNER

TABLE 1. Root-mean-square (rms) Errors and Conservation of Energy Errors (•) for the Examples Shown in the Figures

Figure 2N -{- 1 deg • c/•, s• L/W rms Error

4 53 0 0.1 0.5 0.3 5 0.112 79 0 O. 1 O. 5 O. 3 5 O. 0053

5 79 40 O. 1 O. 67 2.0 32 O. 038 41 40 O. 1 O. 67 2.0 32 O. 054 25 40 O. 1 O. 67 2.0 32 O. 074

6 65 0 0.0 0.67 0.3 5 0.0019 65 0 0.01 0.67 0.3 5 0. 00039 65 0 0.1 0.67 0.3 5 0. 00034

7 65 0 0.1 0.2 0.05 5 0.012 8 65 0 0.1 0.5 0.05 2.5 0. 062 9 79 0 0.01 0.25 0.15 13 0.017

10 65 0 0.0 0.25 0.3 5 0. 00055 65 51 0.0 0.25 0.3 5 0.0013

11 53 42 0.1 0.67 0.3 5 0. 0040 53 48 0.1 0.67 0.3 5 0. 0031 53 55 0.1 0.67 0.3 5 0. 0041 53 64 0.1 0.67 0.3 5 0. 0075 53 78 0.1 0.67 0.3 5 0. 0225 41 89.9 0.1 0.67 0.3 5 0. 0205

12 53 42 0.1 0.67 0.3 5 0. 0185 53 48 0.1 0.67 0.3 5 0. 0091 53 55 0.1 0.67 0.3 5 0. 0053 53 64 0.1 0.67 0.3 5 0. 0078 53 78 0.1 0.67 0.3 5 0. 0092 41 89.9 0.1 0.67 0.3 5 0.0061

13 79 50 0.1 0.67 2.0 32 0. 048 79 40 0.1 0.67 2.0 32 0. 038 79 32 0.1 0.67 2.0 32 0. 027 79 18 0.1 0.67 2.0 32 0.012 79 9 0.1 0.67 2.0 32 0.019 79 0 0.1 0.67 2.0 32 0. 027 79 --9 0.1 0.67 2.0 32 0.032 79 --18 0.1 0.67 2.0 32 0.035 79 --32 0.1 0.67 2.0 32 0. 052 79 --40 0.1 0.67 2.0 32 0.049 79 --50 0.1 0.67 2.0 32 0.075

5 X 10 -• ß ß .

ß , .

ß . ,

ß , ,

ooo

7 X 10 -6 3 X 10 -•

ß , ,

ß , ,

ß . ,

ß . .

ß , o

ß . ,

ß , o

ß , o

* Positive values imply source wave incoming from the lower left.

_ tan_• (x - .x'). (RM SE)" H,(') = first order Hankel function of the

first kind.

The relative importance of the dimensionless residuals •/p, and •w//•,, in influencing surface motion, is displayed in (27). For example, in the case of a shallow depth of interface (w•'/l•, • 1), the surface motion is determined primarily by the displacements at the interface because H,(')(x) dominates H0(•)(x) for small values of x.

As a measure of residuals at the interface, we define relative root-mean-square (rms) error as the following:

2

(28) UlNi(.0

where •i and •i are the displacement and stress residuals, respectively, at position j along the interface; u•Ni and u,.N• are the computed values of displacement- at position j along the interface in the layer and half-space; and •z/i and •,.• are the computed stresses at those positions. The positions are equispaced along the x direction, and M is several times larger than N,


the truncation index in (19). The rms error values for the cases presented in this paper are listed in Table 1.

Conservation o] energy. In discussing the reflection of plane acoustic waves from corrugated boundaries Meecham [1956] and Heaps [1957] used a relationship derived from the conservation of energy as a check on the accuracy of their computations. The conservation of energy statement is that, for problems involving real •, the time averaged net flux of energy across a plane at large depth in the ha]f- space must be zero. Below the deepest point on the interface, the exact solution can be completely represented by a superposition of plane waves,

U2(X• z) --eikøx+iYøg-•- E Cneiknx+i•'•½g)g n-- --c•

The mathematical statement of the conserva-

tion of energy requirement is that

cos 0n/cos 0o - I (30)

where 0• is the acute angle between vertical and the direction of propagation of the nth order scattered plane wave. The summation is over all regular plane wave orders (the inhomogeneous waves, i.e., those that decay exponenti- ally away from the interface, are insensible at large depth).

When the coefficients A• •, determined using our approximate method (see equation 19), replace the exact solution coefficients C• in the left side of (30), the right side becomes I -t- 8. In the appendix, we demonstrate that the error 8 can be expressed as a weighted integral of the interface residuals. This conservation of

energy error measure is more easily and accurately evaluated than is the rms error.

The error 8 is included in Table I for those

examples involving real •. The small values of 8 confirm the accuracy of the numerical computations as well as the fact that the wave equations are indeed satisfied.

Example of the truncation error. Our method for solving these wave-scattering problems is made feasible by the speed and large core memory of modern digital computers. Even so, computer time and storage constraints require

that we truncate the infinite-sum equations to include an upper limit of 79 scatter orders (corresponding to the 2N -]- I wave numbers). This truncation imposes the principal limitation on the accuracy of our solutions. For a given medium configuration (characterized by the layer thickness, the medium parameters, and the shape of the interface irregularity), the truncation error is dependent upon the following quantities:

1. c/X, the ratios of the amplitude of the interface irregularity to the wavelengths in- volved (c

2. s• -------- Id•/dxl•, the maximum slope of the interface.

3. 0o, the angle of incidence. 4. L/W, the ratio of the periodicity length

for the interface shape to the width of the anomalous zone.

5. •, the ratio of the imaginary to real parts of frequency.

For a fixed number of scatter orders, the truncation error increases with increasing values of c/X, s•,, 0o, and L/W. Since the intrinsic (Rayleigh ansatz) error also increases with and s•, we cannot always distinguish between the two errors. The quantities c/X and s•, in- fluence the rates of decrease for the amplitudes of the higher order wave-number terms. To our surprise, the intrinsic error does not depend on the angle of incidence 0o, and the truncation error depends only slightly on 0o, as will be shown later. We find that, with 79 scatter orders, the rms error is generally less than 1% when c/X and s• are less than unity. Thus we are able to study problems involving reasonably irregular interface shapes and wavelengths comparable to and larger than the size of the irregularity.

One means for studying the truncation error, for a given scattering problem, is to compare solutions obtained using different truncation numbers. Figure 4 displays one such example. The problem configuration is shown at the bottom of the figure. A wave of wavelength 50 km is incident vertically upon a basin 5 km deep by 50 km wide. The interface periodicity length is 256 km in all the examples described in this paper (the use of an explicit unit of distance is simply for convenience; of course the solutions are unchanged when all lengths are scaled

942 AKI AND LARNER

3.0 ---- 53 coefficents

E • • • •:• -- 79coefficents 2.0 ee ß

• io

00- I I I I I I, I

Surface Position (kin)

--- layer- (53) / [half-space (õ3)

-- 3'0 I- •.1oyer and half-space

'•5 1.0 ' 'i'•1 • O0 I • GO 80 I00 120 140 IGO 180


=0.7 km/sec • = 2.0 grn/cm s

I vertical = 3.5 kmYsec incidence 2.8 gm/cm $

X, I = I0 krn Irn•/Re• =0.1

Fig. 4. Spatial distribution of the normalized amplitudes of free-surface displacement and the amplitudes of interface displacement (arbitrary units), displaying the effect of truncation of the infinite-sum equations œor a soft-basin problem. The normalization is made with respect to the displacement that would be obtained for the fiat- layer problem in the absence of the interface irregularity.

equally). The interface shape has the form of a single cycle cosine. Away from the basin, the layer thickness is 0.01 km. Solutions were obtained using 53 and 79 scatter orders, respectively. The curves in the center of the figure are spatial distributions of interface displacement amplitude. When 53 coefficients are used, the solution in the layer (dashed curve) oscillates about the one in the half-space (solid curve). When 79 coefficients are used, the interface displacement distributions are indistinguishable, in the figure, from the solid curve. The rms error for these two cases (Table 1) are 11.2% and 0.5%, respectively. Obviously, since the rms error is so small for the 79 coefficient example, the Rayleigh ansatz error is insignificant in this problem.

Note that dividing k• -- ko -F 27rn/L by k• yields sin 0• -- sin 0o -k nX/L, n ---- O, ñ1, ñ2-.. where 9• is the acute angle from vertical associated with the nth order scattered wave

(the angle 9• may be complex), and X -- 27r/k• is wavelength. Because the wavelength in the half-space is five times larger than that in the layer for the example shown in Figure 4, the wave amplitudes in the half-space solution

pertain to scattered waves covering a broader span of directions than do the wave amplitudes in the layer. However, the wave amplitudes are more basically functions of 10• -- 0o] than of In[; hence the half-space wave amplitudes decay more rapidly with increasing in I than do those in the layer, and thus may explain the interesting finding that when an insufficient number (53) of scatter orders is used, the error in the half-space remains small, i.e., the solution in the half-space suffers little error due to truncation of the wave-number spectrum.

The normalized amplitudes of displacement at the free surface are shown at the top of Figure 4. The normalization is made with respect to the amplitude that would be observed in the same problem if the layer had uniform thickness given by the thickness away from the interface irregularity (0.01 km in this case). Note that the surface motion computed using 53 scatter orders departs very little from the more accurate solution. Thus, in this case, the rms error is a pessimistic measure of the accuracy of the computed motion at the free surface. The errors in the layer decay rapidly away from the interface because these errors occur predominantly in the higher order wave-number terms, which are attenuated due to our smoothing with the use of complex frequency.

The dots in Figure 4 are values of surface displacement computed using the Thomson- Haskell approximation assuming that the layer has uniform thickness equal to the local thickness. Further comparisons with the fiat-layer theory are described in a later section.

Example o/ the inherent error. In some problems, we encounter relatively large interface residuals that cannot be reduced by increasing the number of scattered wave orders. In fact, these residual errors increase as more scatter orders are included. This type of error is probably the intrinsic Rayleigh ansatz error. Uretzky suggests that the intrinsic error is manifested by asymptotic behavior of the series representation in equations like (19). That is, as N increases from small values, the series approximation (19) first approaches to, then diverges from, the true solution. This is the behavior exhibited in the example shown in Figure 5.

In this problem, a 10-km wavelength wave is incident at 0o -- 40 ø from vertical upon a


layer 25 km thick, with a severe irregularity (step) in which the thickness varies 5 km as a half-cycle cosine wave over 4 km, and thus has a ma.ximum gradient of almost 2. The stresses at the interface (solid curve for the layer and dashed curve for the half-space) are displayed in the center of the figure for three solutions obtained using 2N •- I -- 25, 41, and 79 coefficients. The rms stress residuals for

these solutions are 0.061, 0.060, and 0.064, respectively. The sizes of these rms errors do not reflect the large residual localized at the step although they are determined predominantly by it. In the 25-coefficients case, the truncation error is manifested as an oscillation of stresses

in the layer as in the example shown in Figure

4. As N increases, the oscillations in the layer diminish as does the stress rms error slightly. As N increases further, however, the stress rms error again increases while the spatial oscillations in the half-space become more rapid and the oscillation amplitudes become larger and more concentrated near the step. The number of coefficients beyond which the displacement residuals become divergent is generally different from that beyond which the stress residuals diverge. In these cases involving 25, 41, and 79 coefficients the rms displacement residuals declined from 0.0816 to 0.0480 and 0.0145, respectively.

It is difficult to assess the effect of the irre-

movable localized residuals in these step prob-

Lo 0.5

Free Surface D•splacement 2N + I: 4 1,79 /-',, --- 2N +1:25

8 2N+I --79

layer

--- half -space

-I00 -80 -60 -40 -20 0 20 40 60 80 I00 km

25 km 2;).5 km layer ß •1 = 3.0 km/sec • PI =2.8gm/cm3 • 5km•

half space

•2: 4.0 km/sec •: 3.3 gm/cm 3

•2: I0 km •: 3 98 sec

Fig. 5. Spatial distributions of the normalized amplitudes of free-surface displacement and interface stress displaying the stress residuals f, or three solutions computed using 2N •- 1 -- 25, 41, and 79 coefficients. The normalization is made with respect to the displacement that would be obtained for the fiat-layer problem, and stresses are expressed in equivalent units of displacement by multiplying by (1/p•o.,o).

944 AKI AND LARNER

I Free_ Surface Displacement

0.5

o.o ! I I I I I I ! I I I I 0 ZO 40 60 80 I00 120 140 160 180 200 220 240km

/,• Free Surface

I t 3' I

Layer ,,81 = 3.0 km/sec 51 •-- 46-•- km =6•- X I

PI = Z.8 gm/cm :5 • • , Half -space, ,82 =4.0 km/sec

P2 =3'3gm/cm3 I Vertical Incidence

Xz = I0 km

1.0

1.0

1,0

•n •o

(db) -I0-

o &--O.I

-30 o

_o40, o

o -50- o

• -60- I I I

_ • Inhomogeneous woves in

half-space

ß

ß

ß ß _

_ 000 *•000 ß ß

_

I I I -30 -20 -I0 0 I0 20 $0

Scotter Ord.er Number (n)

Fig. 6. (Upper portion) Spatial distributions of the normalized amplitude of free-surface displacement, for a dented M discontinuity problem, displaying the smoothing effects of the use of complex frequency. The normalization is the same as that described in Figure 4. (Lower portion) Wave-number spectrum of spectral amplitude ratio versus scatter-order number; ß is the ratio of the imaginary to real parts of frequency.

lems upon the computed displacements at the free surface. The stepped M discontinuity examples presented below must be considered in that light. The fact that the stresses in the layer (41 coefficient case) roughly reflect the step shape of the interface suggests that the errors in the layer may not be so severe as the residuals suggest.

The irremovable errors are larger only in problems involving steeply sloping interfaces, and in these problems particularly when the wavelength is small as compared with the amplitudes of the interface anomalies. For less severe interface shapes such as those in the dented M discontinuity problems presented below, the rms residuals are small and well dis- tributed over the length of the interface. In those cases, the errors continually decrease as 2N •- 1 increases to the maximum value of 79.

Surprisingly, we find that the existence of shadow zones in our problems does not imply

large inherent errors. We find that if the truncation errors are small when Oo -- 0, they remain small when Oo increases even to grazing incidence (0o -- •/2). The inherent error ap- pears to be independent of Oo.

Example o• the smoothing e•ect o• complex •requency. Figure 6 demonstrates the smoothing of the spatial distribution of computed free- surface displacements effected by the use of complex frequency. The problem configuration consists of a half-space overlain by a layer 6•/• wavelengths thick with a 5-km depression in the interface. The depression has a cosine shape for one cycle (50 km). A 10-km wavelength wave is incident vertically. The top curves are the computed free-surface displacement amplitude (each normalized to the respective fiat-layer solution as described previ- ously) for three values of e, the ratio of imaginary to real part of frequency. The amplitude scale applies to the e -- 0.1 curve; the other


curves are displaced upward. The use of complex frequency severely damps the oscillations along the limbs of the e ----- 0 distribution. Those oscillations have an 8-km wavelength, characteristic of the cutoff wavenumber (N -- 32 in these examples), and are thus caused by the sharp cutoff of the wavenumber spectrum. The inner two- or three-side lobes in the top curve are actual lateral wave interference effects. The

use of complex frequency smooths out these side lobes.

For real frequency, the layer vibrates in a resonance condition where the thickness is 6• wavelengths (see the discussion of the soft basin cases below). Over the depression, the amplitude distribution displays these characteristics of fiat-layer interference. When the frequency becomes complex, these vertical-interference effects deteriorate with the results that the

main side lobes are less deep and the amplitude at the center of the anomaly is increased. The introduction of complex frequency alters the surface displacement distribution from one that is dominated by fiat-layer interference effects to one dominated by wave focusing and defocusing effects (the later-arriving multiple reverberations are de-emphasized).

The bottom portion of Figure 6 contains wave-number spectra for the e ---- 0 and e = 0.1 eases. These are plots (on a deeibel scale) of the spectral amplitude ratios IA,J"!/IAo'"I versus the scatter order number n. A• (" is the com-

plex amplitude of waves in the layer (equation 19), with horizontal wave number given by 2•-n/L. For normal incidence, the spectra are symmetric about n -- 0; thus, the e -- 0 amplitudes are plotted only to the right and the e = 0.1 amplitudes only to the left of the n ---- 0 line. For e = 0, the amplitudes do not decay sufficiently rapidly out to the cutoff order number, thus eausing the 8-km oscillations already mentioned.

Although these amplitude ratios are small, a significant degree of wave-number coupling (diffraction) is indicated. On the basis of simple ray theory refraction, neglecting reverberations in the layer, we expect contributions only from waves whose In[ < 3. However, we require the larger wave number (larger n) waves to adequately.. satisfy the boundary conditions. We interpre• the wave-number coupling as diffraction primarily on the basis of similar computa-

tions (not shown here) for problems involving scattering at the irregular interface between two half-spaces. In those problems, layering effects are absent. We find that the spectral amplitude ratios decay more slowly with increasing k/k• for problems involving longer wavelengths relative to the dimensions of the interface irregularity.

SOFT B,•srN PROBLEM

Let us consider ground motion at the surface of a soft medium basin when plane $H waves are incident from below. This problem has been studied by other investigators primarily under the assumption that the basin structure consists of horizontally fiat layers having the same stratification as the one directly beneath the observation point. In other words, a problem of three dimensions has been treated as if it were one-dimensional. We wish to test the

validity of this assumption. In a horizontally fiat-layered medium, the surface motion, due to an incident plane wave, is determined by the interference between upgoing and down- going waves that have the same phase velocity in the horizontal direction. This phenomenon of vertical interference can be completely described by the Thomson-Haskell method. If the interface is not plane, scattered waves with horizontal phase velocities different from that of the incident wave are generated, and the lateral interference can become as important as the vertical one.

It will be shown that, so long as the interface slope is small, the fiat-layer theory using the stratification directly beneath each observation point gives a satisfactory result. As the interface becomes more irregular, the effect of lateral interference becomes important. We shall show an extreme example, where the lateral interference of pseudo-Love waves gives rise to a large amplitude variation across a basin that has a uniform thickness over a finite area. We shall also observe that the lateral interference

becomes more significant when the direction of incident waves becomes closer to the horizontal.

The first example is illustrated in Figure 7. The velocity and density of half-space are 15 and 1.4 times those of the soft basin, respectively, as shown in the figure; therefore the impedance ratio is 7. The width of the basin is 50 km, and the depth is 1 km at the deepest

946 AKI AND LARNER

8 i.o.

/• approximate solution ? ß * ß ß flat layer response

/

.,' \. : .e.e.e.e e'e/ e".-e..e•e• e

I I I I I I I I I I I00 120 140 160 180

Surface Position (km)

/9• = 0.7 km/sec p• = 2.0 gm/cm 3

= 3.5 km/sec =2.8 gm/cm 3

I vertical incidence

vertical scale

horizontal scale =5.0

- •o,,.-• (db)

i, -30 -20 -I0 0 I0 20 30

Scoffer Order Number (n)

Fig. ?. (Upper portion) Normalized displacement amplitude at the free surface oœ a soft basin (maximum depth is % wavelength); the solid line is the solution computed by the present method; the dots are amplitudes computed by the local Thomson-I-laskell approximation using local layer parameters. (Lower portion) Wave-number spectrum of spectral-amplitude ratio versus scatter-order number.

point. The shape of the interface irregularity is a cosine form for one cycle.

The smooth curve in the top of Figure 7 shows the amplitude of surface motion, calculated by our method, when plane waves whose wavelength is 5 km in the layer (25 km in the half-space) are incident vertically. The imaginary part of frequency is set at 10% of the real part. The dots in the same figure are amplitudes calculated by the fiat-layer theory assuming, at each point, that the basin structure is a horizontal layer having a constant thickness equal to that directly beneath the point. Agreement between the dots and the smooth curve is excellent, demonstrating that the fiat- layer assumption is a good one in calculating the surface motion for this case.

The waves propagating vertically (n -- 0) dominate the scattered waves in this case, as indicated in the wave-number spectrum shown at the bottom of Figure 7. However, the

spectral amplitudes decay slowly with increasing I nl indicating significant wave-number coupling attributable both to diffraction and multiple reverberations in the layer.

In the second example, the incident wavelength in the layer is chosen as exactly twice the greatest depth of the basin, as shown in Figure 8. The fiat-layer theory predicts a resonance condition at depths that are (2n -- 1)/4 times the wavelength and an antiresonance condition at depths that are n/2 times the wavelength, where n is an integer. In this example, antiresonance occurs where the basin depth is maximum, and resonance occurs where the depth is half the maximum. Figure 8 shows that the solution obtained by our method agrees well with the prediction by the fiat-layer theory, although the wave-number spectrum for this case indicates larger amplitude scattered waves than in the previous example.

In the third example, the basin structure consists of a long section of uniform thickness bounded by short sections of rapidly changing thickness. As shown in Figure 9, the uniform

• zoL ,/ I ,,/

oF_..-" '"-..._ . 80 I00 120 140 160 180

Surface Position (km)

verticol scale 0.7 km/sec horizontal scale = 2.5 2.0 gm/cm 3 I vertical incidence

3.5 km/sec :>.8 gm/cm 3

Im• O. I /Re• =

o ...-•... (db) Love wove ,_] -iO'Jc't I-,- Love wove region •-]"•" / '%.... I-- region -.... :•..'..: -2o T v:.-.... :.

I I"" I I-•ø1 - I I ""1 -40 -30 -20 -I0 0 I0 20 $0 40

Scatter Order Number (n)

Fig. 8. (Upper portion) Normalized displace- merit amplitude at the free surface of a soft basin (maximum depth ]s % wavelength in the layer). The dots are amplitudes computed by the local Thomson-Haskell aproximation using local layer parameters. (Lower portion) Wave-number spectrum of spectral-amplitude ratio versus scatter- order number.


section is 1 km deep and 80 km wide. This section is flanked on each side by a sloping interface, having the shape of cosine for a half cycle over a horizontal distance of 10 km. The wavelength of waves in the layer is exactly 4 times the thickness of the uniform section, satisfying the resonance condition mentioned before. Note that in this example the imaginary part of frequency is 1% of the real part.

The fiat-layer theory predicts the amplitude distribution as shown by the dashed line at the top of Figure 9. The solution by our method gives amplitudes that fluctuate considerably above and below the curve predicted by the fiat-layer theory. The wave-number spectrum at the bottom shows secondary peaks at the horizontal wave number corresponding to Love waves in the uniform portion of the basin. The Love wave region indicated in Figure 9 includes those horizontal wave numbers that are

between o•//•2 and o•//•, where o• is the angular frequency and/• and/•2 are the shear velocities in the layer and in the half-space, respectively. We interpret the large amplitude variation across the basin as due to lateral interferences

between the primary waves (n -- O) and scattered pseudo-Love waves. This example demonstrates that if we try to explain the

observation on a two-dimensional structure by a theory appropriate for a one-dimensional structure, the observed and theoretical amplitudes can differ by a factor of 2.

The last example in this section (Figure 10) compares the motions of a soft basin for waves of long wavelength (20 km in the layer) incident from two different directions. For the

zero degree incidence angle case, we see again that the fiat-layer theory predicts the result calculated by our method. For the 51 ø incidence angle case, however, we see a significant discrepancy between the theoretical and calculated solutions. The fiat-layer theory predicts a con- siderable decrease of amplitude with increase of incidence angle. On the other hand, our solution shows a nearly equal or even slightly greater amplitude for the larger incidence angle. This result is intuitively acceptable because, for wavelengths comparable to the width of the basin, the motion of a soft basin is perhaps determined primarily by the vibration of the basement, relatively independent of the incidence direction for the wave that forces that

vibration. We may interpret this result as implying that relatively more energy is trapped in the basin as the propagation direction of incident waves becomes closer to horizontal.

8.0 6.0

4D

2.0

o.o I I I I I I I I I I I I I 60 80 I00 120 140 160


approximate solution riot layer response

I i 180 200

=0.7 km/sec 2.0 gm/cm 3

=3.5 km/sec = 2.8 gm/cm 3

Im•/Re• =0.01

c/),1 = I/4

• vertical incidence

o_Jr_ (db) Love wave---- I '1' •-• Love wave

region •-I _10' T. I--- region ß ,• -2o T o• ..o •

.: "•;-3o '..':" :..

I" I I I-4ø• I I I %1 -40 -30 -20 -,0 0 •0 20 30 40

Scotter Order Number (n)

vertical scale

horizontal scale 5.0

Fig. 9. (Upper portion) Normalized displacement amplitude at the free surface of a fiat soft basin (maximum depth is % wavelength in the layer). The dots are amplitudes computed by the local Thomson-Haskell approximation using local layer parameters. (Lower portion) Wave-number spectrum of spectral-amplitude ratio versus scatter-order number.

948 AKI AND LARNER

7.0

•.o /• eo.O- •.o --- •o-'•" 4.0 / , e•, © © © riot ioye, response(•o=O*) 3.0 •, x• x x , riot layer

_

I00 120 140 160 080 Surface Position

-• • = 0.7 / t ,•:) = •,. 5

P2=2.8

Real •

X I = 20 km

Fig. 10. Normalized displacement amplitudes at the free surface of a soft basin for two directions of incidence of long wavelength (100 km in the half-space) waves. The dots are amplitudes computed by the local Thomson-Haskell approximation using local layer rameters.

DENTED AND STEPPED M DISCONTINUITY

If the M discontinuity and the interfaces in the earth's crust are horizontal planes, the amplitude observed at the surface due to teleseismic body waves will not show spatial variation within a small area in which the incident

waves may be regarded as plane waves. We are interested, in this section, in possible amplitude anomalies that may be caused by irregular shapes of these interfaces. This problem is somewhat different from the previous problem of a soft basin because the distance from the

observation point to the irregular interface is greater. In this problem, the ray-geometrical effects, such as focusing and defocusing, start playing an apparently significant role.

In the example presented below, the imaginary part of frequency •o• is sufficiently large so that the exponential window e-"•' is down to 1/e in 3.98 seconds. This time is short compared with the travel time through the layer (10 seconds for one-way vertical path); therefore, the effects of multiples are nearly absent in these examples.

Figure 11 shows results for a case in which the interface, located at a nominal depth of 25 km, has a depression with depth 5 km and width 50 km. The shape of dent is a cosine for one cycle. The velocities and densities of the layer and half-space roughly correspond to

with wavelength 10 km in the half-space (7.5 km in the layer) are incident from below at various incidence angles. The top traces are free-surface displacement amplitudes, for each incidence angle, normalized to the respective amplitudes obtained for the plane layer case without the dent. The phase delays are likewise relative to the phases calculated in the plane layer case. The amplitude and phase-delay scales are shown for the uppermost curve in each case.

The solid circles are projections of the trough of the interface depression along geometric ray paths. The maxima of flux density calculated by ray theory occur at nearly these same positions in this case (these locations are indicated by the arrows). The double arrows for the 8o -- 64 ø curves denote the intersections of caustics with

the free surface. No ray theory arrows are shown for the 8o = 78 ø and 89.9 ø cases be-

cause the ray-theory solutions have shadow zone gaps. The rms errors (Table 1) are small even for the case 8o ---- 89.9 ø. Note that the amplitude variations and the breadths of the phase-delay anomalies increase with increasing incidence angle and that the qualitative shapes of the anomalies are consistent with one another

while 8o changes. The dashed curves for the 8o ---- 55 ø are

amplitudes and phase delays computed using those of the crust and upper mantle, respec-• ,•.•the ray theory. The amplitude is very large tively, as shown in the figure. The plane waves near 149 km because the focal region is• near


the free surface. The discontinuities at 120 and

170 are artificial manifestations of the sensitivity of the ray-theory solution to the second derivative of interface shape. Our solutions are rea- sonable, smoother versions of the ray-theory solutions. When the incident wavelength is reduced to 5 km in our method, the amplitude peak at 149 km increases, and the depression at 164 km moves to the left toward the ray- theory lobe. Also, the phase-delay curve then nearly coincides with the ray-theory solution. These consistent suites of curves and the good accuracy suggest that the solutions are valid even to grazing incidence. Note that the 0o ---- 89.9 ø amplitude curve does not normalize to unity away from the anomaly. The reason

is that waves at nearly grazing incidence have 'seen' the repeated depressions rather than a 25-kin-thick fiat layer.

The next example is the case of a rise as shown in Figure 12. The media parameters for this case are identical to those for the preceding one. The arrows indicate the positions where the minima of flux intensity and maxima of phase advance calculated by ray theory occur. The solid circles indicate the projected positions of the crest of the rise along the geometric ray paths. The residual errors are again small. The amplitude variations are less simple than those in the previous case. However, the 8o -- 55 ø curve compares well with the ray-theory solution except at the artificial discontinuities

2.0

80 = 89.9

80=78 ø OoO

80 =64 ø

80 = 55 ø

80 = 48 ø ,

0.25-

0.0

ø-0.25

0

1.0

1.0

1.0

8o= 42 ø

8 o = 89.9' 80 = 78' 80 = 64'

80 = 55'

8o = 48 ø

8o= 42 ø

i i I i 20 40 60 80

• 0.0 0.0

• 0.0 • 0.0

i00 120 140 160 180

DISTANCE (kin)

0.0

0.0

I 200 220 240 260

CRUST

MANTLE 89'9ø'-'-•" 78

42 ø

30 km t Cl = 3.0 km/sec

25km pi =2.8 gm/cm 3

C2= 4.0 km/sec p2 = 3.3 gm/cm 3

X 2 = I0 km • =0,1

Fig. 11. Spatial distributions of normalized amplitudes and phase delays for the free- surface displacements in the downward-dented M discontinuity problem as functions of the angle of incidence 0o. The dots show the projected positions of the trough of the depression along the geometric ray paths. The arrows show the positions of peak flux intensities and phase delays predicted from ray theory (excluding multiple reflections). The dashed curves are ray theoretical solutions.

950 AKI AND LARNER

1.5

1.0-

0.õ

0.0

- •o = 8 9.9 e

•o ='78ø e o = 64 ø 8o:55 ø eo=4e ø 80=42 ø

._. 0.25 "' 0.0

•: 0.25

80 = 89.9 ø eo = 78 ø

80 =64ø 8o =55ø 8o = 48 ø 8o =42 ø

i i i . 20 40 60

0.0

0.0

• 0.0 • 0.0

I I I I I i I J I iI 80 i00 120 140 160 180 200 220 240 260

DISTANCE (kin)

$ • C m = 3.0 km/sec CRUST 20 km 25 km

• • Pm =2'8 gm/cm' MANTLE 89.9 ø I_ 50 krn C2=4.0 krn/sec

78o- -.-'""• r" ,o 2:3.3 gm/cm •

64 ø '• •,2 = I0 km ,:;.// .:o.,

42 •

Fig. 12. Spatial distributions of normalized amplitudes and phase delays for the free-surface displacements in the upward-dented M discontinuity problem as functions of the angle of incidence Co. The dots show the projected positions .of the minimal flux intensities and phase delays, respectively, predicted from the ray theory (excluding multiple reflections). The dashed curves are ray theoretical solutions.

in the ray-theory solution. The phase delays compare very well with ray-theory predictions. Again the anomalies increase with incidence angle. Such amplitude and phase-delay anomalies for near-grazing incidence may be useful as interpretation tools in refraction seismology.

The last example is the case of 'stepped' M discontinuity as shown in Figure 13. The height of step is 5 km, the depth to the center of step is 25 km. The shape of step is a cosine for a half cycle (with wavelength 8 km) connected to half wavelength cosines on both sides. The solid circles again indicate the projected position of the center of step along geometrical ray paths. Incident waves have the same wavelength and the same imaginary to real part ratio as those in the preceding two examples. When the waves are incident from the side of thinner crust

to that of thicker crust, we find a general agree-

ment between the projected points of center of step and the maximum of amplitude calculated by our method.

The above three examples illustrate that the amplitude-distribution and phase-delay anomaly observed at the surface are sensitive functions

of the interface shape. The change of amplitude distribution with the change of incidence angle is especially diagnostic of the depth of the irregular interface. We believe that our method can serve as a new useful tool in the study of regional geophysics.

Conclusion. Our method provides a practical means for the study of aspects of the wave fields peculiar to the simple two-dimensional models discussed. The amplitudes and slopes of the interface irregularities that we study are larger than those allowed in the iterative approximation method of Rayleigh or in various


perturbation methods [Gilbert and Knopo[l, 1960; Herrera, 1964; Mclvor, 1968]. The computational scheme is straightforward and more easily applied than is the method of Banaugh [1962]. (Also, see Sharma [1967]). Banaugh's method is based upon an integral representation for the solution and, in principle, can be applied to scattering of waves from obstacles of arbitrary shape without suffering from the Rayleigh ansatz error. However, for comparable computation effort, when his method is applied to problem geometries similar to those treated in this paper, errors arise that are more severe than our truncation errors.

The method was programmed for the IBM 360-65 at the M.I.T. computation center. Com- putation time is controlled by the number of scattering orders representing our solution. For a given problem configuration and frequency, we economize by handling cases involving various source-wave directions concurrently. Typical computer time for problems involving the maximum number of scattering orders is 4 minutes for the first incident-wave direction and 20 seconds for each additional direction.

Although we showed no examples in this paper, the computer program is applicable to problems involving multilayered (homogeneous,

isotropic) media where all but one of the interfaces are plane and parallel. This includes the problem of variable topography. The method is readily extended to include problems involving additional irregular interfaces; however, we must sacrifice either accuracy or roughness of the irregular interface in the problems that can be solved under the present computational constraints.

APPENDIX

The error in conservation o/ energy. When frequency is real, if u is a solution to the homogeneous scalar Itelmholtz equation throughout a region R, then the real and imaginary parts are individually solutions, throughout R, to the same equation. A consequence of Green's theorem is that

Im fsu(u* Ou) as = o (A1) where Im denotes the imaginary part of, the asterisk denotes complex conjugate, and S is the surface enclosing R.

Let us apply (A1) to the approximate solution u,N in the region of the half-space bounded by the interface, the plane at large constant z, and vertical surfaces at x = 0 and x = L (L is

1.5

1.0

0.5 O.C

-40 -20 0 20

9_2.5 km

Incident wave directions

40 60 km i !

layer

half-space

00-' 50 ø I.O

'o" ,•• 1.0 3_.=•.•j•_ •.0 18• 1.0

9•••'••.•,._._- 1.0 Oø • 1.0

-60 -40 -20 0 20 40 60 km I I I I I I I

•1=3.0 km/sec

•2 = 4.0 km/sec P2 = 3'3 gm/cm$ )'1 = 7.5 km Tma•/Reu• =0.1

Incident wove d•rections

Fig. 13. Spatial distributions of the normalized amplitudes of free-surface displacement for the stepped M discontinuity problem as functions ,of the angle of incidence Co. The dots show the projected poition of the center of the step along the geometric ray paths.

952 AKI AND LARNER

the period of the interface shape). For convenience, we make the horizontal component of wave number ko (equation 2) of the source wave equal to 2•'t/L where t, an integer, is the number of wavelengths across L. With this choice, both the exact and approximate solutions have the period L in the horizontal direction; consequently, the integrals over the vertical portions of $ at ß = 0 and ß = L cancel. Equation A1 becomes

Imfo [arge z

U2N :• T2N dx

f• x=L =im u•.N* r2x dS---- tz2• (A2) =0

interface

Equation A2 defines the quantity 7. Similarly, let us apply (A1) to the approxi-

mate solution Ul• in the region of the layer bounded by the free surface, the interface, and the vertical surfaces at x -- 0 and x ---- L.

Again, the integrals along the vertical surfaces cancel. Also, the integral along the free surface vanishes because of the stress-free condition.

We have, then

fx x----L Im UlN* rlN dS = 0 (A3) =0

interface

Using (26) and (A3), we rewrite the integral over the interface in (A2) as

fx x----L /z2r/ = Im [a* q -- u,•* • =0

interface

- a, rl] aS (A4)

That is, V is an integral of stress and displacement residuals weighted by the approximate solution in the layer. Since v vanishes with the residuals, it is a measure of the accuracy of the approximate solution.

The integral along the plane at large z is readily evaluated when we replace u2•,r,r by an expression obtained from the computed solution, equation 19. Because the solution is periodic for our choice of ko, we can use the orthogonalfry condition,

fo L 2•' • (n,-n)x/L dx = L when m = n

fo L 2 •' i (m-n) e dx = 0 when m • n

Recalling that the inhomogeneous waves do not contribute when z is large, we find

n• COS 0n (2) 2 cos 0o IA,• I = 1

r//•2 ---- i +.• (A5) •-o•L cos 0o The quantity 8 is the departure from the conservation of energy requirement ascribab]e to the residuals at the interface. The error 8 is

more easily and accurately evaluated, by means of summation over the scattered wave orders on

the left side of (A5), than is the rms error. The error 8 is not a sensitive one for two reasons.

First, the amplitudes and phases of the residuals generally oscillate along the interface, tending to produce small values of V; second, this error measure does not have preferential weighting of the larger residuals (localized anomalies) as does the rms error criterion.

We note that the conservation of energy criterion provides a measure of the error in the solution for the half-space only, and states nothing about the solution in the layer. To see this, we replace • in (A3) by the exact so]u- tion •, and note that u•: Au• + • at the interface. We find that

fx •----L =0

interface

[u•* /xr2• q- /Xu2•r•

q- Au2•* Ar2•] dS

This integral for V is independent of the approximate solution in the layer; it vanishes with the errors Au•r and A• in half-space solution.

When frequency is complex, the real and imaginary parts of u no longer individually satisfy the homogeneous scalar wave equation. Application of Green's theorem yields a volume integral on the right side of (A1). In that case, we could attach no physical meaning to an error measure based upon the integral 7.

Acknowledgments. An outline of the method described in the present paper was conceived by the senior author (Keiiti Aki), when he was working at the National Center for Earthquake Research, U.S. Geological Survey, Menlo Park, California, under a Taper appointment. lie owes


a pleasant and profitable summer to L. C. Pakiser and his colleagues at Menlo Park. The fast Fourier transform program used in the present paper was written by Ralph Wiggins of the Massachusetts Institute of Technology. The authors thank Dr. Wiggins and Dr. David Boore for suggesting valuable improvements over the original manuscript. A part of the computation was done at the computation center, M.I.T.

This research was supported partly by the National Science Foundation under grant GA- 4039, and partly by the Advanced Research Proj- ect Agency; it was monitored by the Air Force Office of Scientific Research under contract AF 49(638)-1632.

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(Received June 23, 1969.)

aki and larner 1970

Documents

excluding

higher order

corrugated

plane sh waves

scatter order

local layer

layer theory

rayleigh ansatz