Seismogram synthesis using normal mode superposition: the locked mode approximation

Transcription

1 Geophys. J. R. astr. SOC. (1981) 66, Seismogram synthesis using normal mode superposition: the locked mode approximation Danny J. Harvey Cooperative Institute for Research in the Environmental Sciences, University of ColoradolNOAA, Boulder, Colorado 80309, USA Received 1980 September 4; in original form 1979 March 29 Summary. A normal mode superposition approach is used to synthesize complete seismic codas for flat layered earth models and the P-SV phases. Only modes which have real eigenwavenumbers are used so that the search for eigenvalues in the complex wavenumber plane is confined to the real axis. In order to synthesize early P-wave arrivals by summing a number of trapped modes, an anomalously high velocity cap layer is added to the bottom of the structure so that most of the seismic energy is contained in the upper layers as high-order surface waves. Causality arguments are used to define time windows for which the resulting synthetic seismograms are close approximations to the exact solutions without the cap layer. The traditional Thomson- Haskell matrix approach to computing the normal modes is reformulated so that numerical problems encountered at high frequencies are avoided and numerical results of the locked mode approximation are given. Introduction Historically, normal mode theory has been restricted in its applications to low-frequency bandwidths; namely to low-frequency spherical earth normal modes and Rayleigh and Love surface waves for flat layered structures. In this study we will consider the modal synthesis of high-frequency, undispersed body waves, which constitute the first seismic arrivals and contain the most detailed information about the source and the structure. A variety of methods have been used to synthesize the P and S body waves for laterally homogeneous structures, and these methods differ in the way in which the two inverse transform integrals are evaluated. The asymptotic methods evaluate either one or both of these integrals analytically by approximating the integrand function with suitable asymptotic expansions, and the resulting solution is a decomposition of the complete solution in terms of rays. These ray theories work well in synthesizing particular phases, but they can be cumbersome to use when trying to compute the complete solution, especially in certain distance ranges. In particular, for ranges of 0 to 1000 km with a detailed crust and upper mantle model, ray theories require the a prion specification of a large number of ray

2 38 D. J. Harvey paths to synthesize the complete solution for the P and S body waves (a typical example of this is the Pg coda which is seen in the western United States). Another category of body wave synthesis techniques which have been used extensively are the numerical methods. These are collectively referred to as the reflectivity method and they share in common the numerical evaluation of both inverse transform integrals. Owing to recent developments (Kennett & Kerry 1979; Kennett 1980; Cormier 1980), the reflectivity method can now be used to compute complete seismic codas for arbitrary frequency bandwidths and source depths and for vertically inhomogeneous structural models. The major disadvantage of the reflectivity method is the computational expense which is proportional to the product of frequency bandwidth and maximum range. The modal representation of body waves combines features of both the asymptotic methods and the reflectivity method. By locating the poles of the integrand function in the complex frequency-wavenumber space one can analytically evaluate one of the inverse transform integrals in terms of a residue series without making asymptotic approximations. If a sufficient number of residue terms are summed, this will produce a complete solution within some time window as with the reflectivity method. The advantages of the modal method over the reflectivity method are that it should be more accurate since one of the numerical integrals has been eliminated and it should also be more computationally efficient since the number of residues within some wavenumber bandwidth will be less than the necessary number of numerical integration points within the same bandwidth. Three problems have plagued previous efforts to synthesize the P-wave seismic coda using normal mode superposition: (1) The dispersion function that is computed from the usual Thomson-Haskell matrix formulation is numerically unstable at frequencies above (about) 1 Hz. (2) The eigenfunction computations using the traditional propagator matrix approach become numerically unstable for high frequencies as the source depth is increased. (3) For a typical earth structure, the eigenvalues located on the real wavenumber axis account for the trailing part of the seismic coda and a branch cut integral contribution must be evaluated in order to synthesize the early part of the coda. Knopoff (1964) and Dunkin (1965) reformulated the dispersion function computation to eliminate the numerical instabilities, and more recently Abo-Zena (1 979) rediscovered this work and applied it to compute successfully Rayleigh wave dispersion curves at frequencies up to 20 Hz. Recently we have been able to overcome the remaining difficulties and have successfully synthesized the entire seismic coda for near-regional distances. In this paper we will summarize the theory for normal mode superposition and detail the steps we took to overcome the second and third problems mentioned above. In the present development we consider only the P-SV phases for a flat layered earth model, for which the layers are isotropic and homogeneous. We first derive the dispersion function and the eigenfunctions by considering the free vibration problen. The excitation of the various modes due to a buried point source is then considered and an outline of the overall computational procedure is given. Finally we present numerical results and discuss the merits and problems of the normal mode method in comparison to other synthesis methods. Normal mode superposition - the theoretical basis In the following sections we present a summary of the steps one must take to compute the normal mode eigenvalues and eigenfunctions. Since this subject has been treated in the past by a number of researchers, we have omitted much of the detailed analyses and give only the

3 Seismogram synthesis 39 major steps. Basically, the analysis given here closely follows that of Ben-Menahem & Singh (1972) and Harkrider (1964) who show how to synthesize Rayleigh waves for a general point source. We do present, however, a number of new results and methods that constitute the basis for a successful mode representation of a complete synthetic seismogram. Coordinate systems and layer conventions In order easily to match the boundary conditions at the horizontal layer interfaces, we use a cylindrical coordinate system which is shown in Fig. 1 along with the layer numbering conventions. Since we will be writing solutions of the wave equation in each individual layer and then matching boundary conditions throughout the stack, we employ both a global coordinate system and a set of local coordinates, each relative to an individual layer. The origin of the global coordinate system will be at the free surface with the positive z-axis pointing down. The origin of a local coordinate system will be at the top of the layer, with the radial and azimuthal coordinates being the same for all of the coordinate systems. We distinguish between the global and local vertical coordinates by using an unsubscripted or unsupercripted z for the global coordinate and a superscripted c@) for local coordinates where p is the layer index. The depth of the bottom of the pth layer in global coordinates is z = h ( p) whereas the thickness of the p th layer (!$" = h@) - dp - I)). We use this dual representation for all of the functions of z as well. Whenever a function of z appears without a layer superscript it is understood that the argument will be in global coordinates and whenever a layer superscript does appear, then the argument of the function will be in local coordinates. Thus for some function, f(z) GLOBAL COORDINATE FREE SYSTEM ORIGIN 7 SURFACE - z = h(') = 0 UYER 1 HALF SPACE N Figure 1. Coordinate systems and layer numbering conventions.

4 40 D. J. Harvey The basic solution We follow Ben-Menahem & Sin& (1 972) and write the Fourier transformed solutions of the elastic wave equations in the pth layer in terms of the vector cylindrical harmonics, P, B and C as: ulp)(r, 6, {(P); w) = m=+m 00 C Jo tiag (r, 6, {(p); w, k)k dk m=-m = radial, transverse and vertical components of the displacement in the pth layer; TAp)(r, 6, w) = f'ac(r, 6, w, k)k dk = radial, transverse and vertical components of the traction across a horizontal plane in the pth layer. and We define the vector cylindrical harmonics as follows: 1% exp (im6) The Greek subscripts are used here as tensor indices which range from one to three. We wish to impose the boundary conditions of continuity of displacements and tractions at layer interfaces, zero tractions across the free surface, and no sources at z = 00. Since we have expressed the solution in a separable form and since the vector cylindrical harmonics

5 Seismogram synthesis 41 are orthogonal, in order for these boundary conditions to be met we must impose them on the z-dependent integrand factors in equations (2). Thus we can express the boundary conditions by the following: (1) Continuity of displacements at the layer interfaces: x, (P-1) (5 (P - 1) ) p - k p - 1 ) =x&~(p))is(p)=o (2) Continuity of tractions at the layer interfaces: XA%- 1) (5(P-1))I$P-O= p-1) =X,(P,,(p)p= (3) Zero tractions at the free surface: for 1 < p 5N. (4a) for 1 pi^. (4b) [X$(O)] = [O]. (4) The Sommerfeld radiation condition (no sources at infinity) applies to: (4c) xam (N)(S(N)) and X&$?({(N)). (4d) We specify a point source in layer p = s and at r = 0 and z = zo, (5'") = We shall solve the problem of forced vibrations by adding a particular solution to the homogeneous or unforced solution in the source layer. The form of the particular solution will be the same as equations (1) with the only difference being that the z-dependent integrand factors will be different from those of the homogeneous solution. Now, the particular solution exists only in the source layer and in all the other layers only the homogeneous solution given by equations (1) exist. We can thus express the total solution in the source layer as follows: Where the particular solution integrand factors are denoted by a P superscript and are given by : '&A (r, 9, {('); w, k) = wf2({('); w, k)p,, (r, 9; k) t wp2 (c('); w, k)b,, (r, 9; k) and + wp2 (p; w, k)c,, (r, 9; k) (6a) The boundary conditions given by equations (4) must be modified at the upper and lower source layer interfaces as follows: (1) Continuity of displacements at the upper and lower source layer interfaces:

6 42 D. J. Harvey (2) Continuity of tractions at the upper and lower source layer interfaces: X&-')(g(s-l)) = xo(s2(0) t wp2 (0) X,(S,+')(O) = xj$ (p) -t w2,' (p). We now define two sets of vertical eigenfunctions from the homogeneous solution z-dependent integrand factors corresponding to the P-SV and SH waves as: [RE(z)l = Similarly we separate the P-SV and SH components of the particular solution z-dependent integrand factors as follows: The R superscripts correspond to the Rayleigh-wave solutions and the L superscripts correspond to the Love-wave solutions. In this paper we wil only consider the P-SV case and will develop the SH case in a separate study. With this in mind we drop the R superscript in order to minimize notational clutter. We also note that the azimuthal subscript, m, has been dropped from the eigenfunctions, [El, since they are rn-independent. The eigenfunction solutions and the propagator matrices The eigenfunction solutions for a single layer are given by Ben-Menahem & Singh and are: [E(p)({(P))] = [B(p)(q(P), ~'(p))] [C(P)(v(P), v'(p); {@))I [A@)] (1 0) where [A(p)] is a four-component vector of constant coefficients that are adjusted to satisfy the boundary conditions, [B(P)] is a (4 x4) matrix which is a function only of the horizontal phase velocity and the layer elastic parameters, and [C(p)] is another (4x4) matrix which is z-dependent. The values of q(p) and v'(~) are the P- and S-wave vertical phase velocities and are: c (J 1 - (c/a'p')t

7 Seismogram synthesis and v (~) and are the P- and S-wave vertical wavenumbers and are given by: The C matrix can be expressed as follows: 0 0 lo and We can rewrite equation (10) in terms of initial values of the eigenfunctions at the tops of the layers and in so doing we define the propagator matrix, [A(P)], [E(P)(5'P))] = [A(P)(+P))] [E(P'(O)] where from equations (10,13 and 14) it is easy to show that: [A(P)(p')] = [B(P)(@), 9'(P))] [c(p)(v(p), vw; {(P))][B(P)(p, 9'(P))]-1. The propagator matrix allows us to compute eigenfunctions at the bottom of a layer in terms of eigenfunctions at the top of the layer so that in global coordinates: [E(h(P')] = [A(P)(C;(P))] [E(h'P-'))]. (17) We can start at any layer interface, p, and propagate the eigenfunctions to any other interface, 4, (where p < 4) by applying the boundary conditions expressed by equations (4) and by repeating equation (17) and in so doing we define the interlayer or total propagator matrix, [At9."'1, [E(h(q))] = [A(9*p)] [E(h(p))]. (18) (1 5) (16)

8 44 D. J. Harvey The interlayer propagator matrix is defined as follows: [A(qpP)] = fi [A(q+P+I-I) I=p +I (t (q+p+1-1) The [B] matrix and the layer propagator matrix, [A], are given in Appendix A. 11. The dispersion function, eigenvalue and ellipticity computations In order to compute the eigenvalues we need to apply the boundary conditions given by equations (4c and d) of zero tractions across the free surface and the Sommerfeld radiation condition in the infinite half-space at the bottom of the stack. The Sommerfeld radiation condition in the half-space can be ensured by setting two of the constant coefficients in [A(N)] to zero, so that the upward propagating terms, pcn) and Q"), are eliminated from the solution of the ordinary differential equation in the half-space. More precisely, the Sommerfeld radiation condition requires that: A(N) = AS" = 0 and thus, By using the interlayer propagator matrix in equation (20) and extracting the first and third rows we can define the (2 x 4)D matrix, [:] = [D(p)] [E(h(p))] where and I 4 D(P) = 1 ([B(N)]-l),kA$'-l.P) k=l whenn-1 >p?o and j=1,2,3,4. (22b) 4 D$') = 1 ([B(N)]-1)3kA$-1~P) k=l We evaluate equation (21) at the free surface and apply the remaining boundary condition of vanishing tractions at the surface which requires that:

9 Seismogram synthesis 45 The Rayleigh wave dispersion function, RF(o, k), is equal to the determinant of the (2 x 2) submatrix of [D] shown in equation (23) and clearly equation (23) can only hold if: We find the eigenvalues by varying w and k until the dispersion function is zero. For a flat layered medium the eigenvalues will form discrete curves in the (a, k) space, which are the Rayleigh-wave dispersion curves, for values of c less than the S-wave velocity in the halfspace. We consider the frequency to be the independent variable and the corresponding wavenumber eigenvalues, for a particular frequency, w, and for the nth mode, to be Rkn(w). Thus, RF(U, Rkn (a)) = 0. In order to compute the ellipticity, R ~, ( ~ we ), evaluate equation (23), D(0) =- El(o) I W,Rkn(W) -- Dg)(w, Rkn(U)) (a, Rkn(U)) D$)(o, Rkfl(W)) Dg)(w, Rkfl(W))' The eigenvalue numerical instability So far we have followed the traditional Thomson-Haskell matrix formulation which has been used for many years to compute Rayleigh dispersion curves. At this point we diverge from the traditional approach which suffers from numerical instabilities at frequencies consistent with body wave synthesis. Recently, Abo-Zena (1979) has described a method by which the Rayleigh dispersion function can be computed without loss of precision at arbitrarily high frequencies and, briefly, the method can be described as follows. If one were to expand algebraically the elements of the D matrix in terms of the transcendental functions, P and Q, given by equations (14) and then use this expansion in computing the dispersion function from equation (24), one would find that a number of these expansion terms would exactly cancel. Unless this cancellation is done explicitly before the dispersion function computation is coded in a digital computer program, then the cancellation will be done numerically in the chain of arithmetic executed by the computer. Unfortunately, these terms which cancel grow exponentially with frequency and at high frequencies they are so large that the terms which do not cancel are lost in the computer word roundoff. This is the source of the numerical instability in the Thomson-Haskell formulation and Abo-Zena has explicitly cancelled these terms so that they never appear in the computer code. In order to do this in a straightforward manner, Abo-Zena defines the Y matrix which we give here in terms of the previously defined D matrix, Dg)D$) for j, k = 1,2,3,4. The elements of the (4 x 4) Y matrix constitute all of the possible subdeterminants of the D matrix of order two and clearly the Y matrix is anti-symmetric. We also note that from equation (24) it is obvious that: RF(w, k) = Y~\')(U, k). As we shall see the Y matrix is also handy for computing the eigenfunctions. The Y matrix can be computed recursively at each layer interface using the following relation, (25) (26) (28)

10 46 D. J. Harvey The key to Abo-Zena s method is the way in which he computes this recursion relation and we refer the reader to his paper for the details. Although Abo-Zena s method as given in his paper works very well to cure the dispersion function numerical instability, it does so at considerable expense over the Thomson- Haskell method in terms of computer time. We have carried out the algebra given by Abo- Zena further and have come up with a much more efficient algorithm which also retains the numerical stability. We omit the details of this rather tedious derivation and give the algorithm in Appendix B. We might also point out that although Abo-Zena claims that his method is successful when using single precision arithmetic, our experience in using his method on a broad variety of structures and frequency bands indicate that in general one would be advised to use double precision arithmetic. This becomes more important as the structure depth or the number of layers increases and it is especially important when computing the eigenfunctions. The eigenfunction numerical instability: the problem and the cure In order to synthesize Rayleigh waves for buried sources one must compute the eigenfunctions and these computations also exhibit numerical stability problems. Normally, one would find an eigenvalue by zeroing the dispersion function as computed by Abo-Zena s method. This value is then used to compute the ellipticity so that the surface values of the eigenfunctions are defined. The total propagator matrix to the desired depth is then computed and this matrix is used in equation (18) to compute the eigenfunctions at the desired depth. We tried this straightforward approach and found that in many cases the eigenfunctions became numerically unstable and the problem is more pronounced as the frequency is increased. Basically, what is happening is that small roundoff errors in the four eigenfunctions are effectively amplified by subsequent propagator matrix multiplications until the errors get bigger than the correct values of the eigenfunctions. These errors can be thought of as unstable, drifting errors with respect to depth since there is no inherent mechanism in the computations to stabilize them. Thus the computed eigenfunctions at half-space depths generally no longer meet the Sommerfeld radiation condition. In order to illustrate this problem we use as an example the only structure for which we could readily obtain an exact analytic expression for the eigenfunctions, namely an infinite, homogeneous half-space. Using the propagator matrix from Appendix A (equations A7) and replacing the hyperbolic trigonometric functions with the exponential functions P and Q, we can write the first displacement eigenfunction as follows: Where we have dropped the layer index and, P(z)=exp - 6 : ( )

11 Seismogram synthesis 47 and is the ellipticity for the fundamental mode and RC1 is the eigenphase velocity which, of course, is frequency-independent. Since the value of RC1 is less than the S-wave velocity of the half-space, the functions 9 and 9' are real and positive and thus the exponential functions P and Q have real and positive arguments. If we replace the ellipticity with an analytic expression in 9, 9' and y we find that we can reduce equation (30) to: The cause of the eigenfunction numerical instability can be seen when one compares equation (30) with equation (31). The Sommerfeld radiation condition requires that the growing exponential solutions vanish which of course is the case with equation (31). This comes about because the terms which multiply the exponential functions P and Q in equation (30) are identically zero. In order to show this one must substitute an explicit analytic expression for the ellipticity which in general is impossible to derive but for this simple case is easy to derive. If, however, one were to code equation (30) in a computer program to compute the eigenfunction, then the terms which multiply the growing exponentials would be computed numerically and instead of being identically zero, they would be of the order of the computer word roundoff error. These small but finite terms would then be multiplied by the growing exponential functions and at some depth the eigenfunction error terms would overcome the correct decaying solutions. Since the arguments of the functions P and Q are directly proportional to frequency, this numerical problem becomes more pronounced for a given depth as the frequency increases. One can see from equation (30) that the type of algebraic cancellation used by Abo- Zena to eliminate the dispersion function instability can only be realized with the eigenfunction computations if an explicit analytic solution for the ellipticity can be derived. In general such an expression cannot be derived and so in general we are stuck with using equation (30) or a more complicated version for a buried layer. We were able to solve this problem by adding an additional constraint on the eigenfunction computations. From equations (21) we can see that they constitute a linear system of two equations in the four eigenfunctions and thus we can solve for two of the eigenfunctions in terms of the other two. We can express the two traction eigenfunctions in terms of the two displacement eigenfunctions as follows: In our search through the literature it appears that Dunkin (1965) came up with a similar relation to equation (32), but he did not use it in the manner we have here to control eigenfunction stability. This relation can be used to compute the ellipticity in terms of the elements of the Y matrix which gives, y$'(a, Rkn(a)) -- YG'(u, Rkn(u)) REn(u) = - y$)(w, Rkn(W)) y$)(a, Rkn(a))' In order to apply equation (32) to the half-space example problem we must divide the half-space into a number of pseudo-layers all of which have the same elastic parameters. The procedure we follow to compute numerically stable eigenfunctions is as follows: (1) We evaluate the Y matrix at each pseudo-layer interface. (2) The ellipticity is computed from equation (33) which defines the surface values of the eigenfunctions. (33)

12 48 D. J. Harvey x L I b- a W 0 -L 5- z I Y c a 0 w (1 10 OISPLACEMENT (a) TRACTION 10 I I 1 L 0 E S n a W 10 I#\/: LINEAR -Is r15 LINEAR -15 Figure 2. Radial displacement and shear traction eigenfunctions for an infinite homogeneous half-space. For each eigenfunction there is a plot of the computed eigenfunction on a linear scale and the error between the computed solution and the exact solution on a log scale. (a) No pseudo-layer interfaces. (b) One pseudo-layer interface indicated by the dotted line. (c) Three pseudo-layer interfaces indicated by the dotted lines. (C) (3) We compute the two displacement eigenfunctions at the next pseudo-layer interface using the first two rows of the pseudo-layer propagator matrix. (4) Equation (32) is used to compute the two traction eigenfunctions at the pseudolayer interface instead of using the last two rows of the propagator matrix. (5) Steps (3) and (4) are repeated to the desired depth. The reason this method works is that implicit in equation (32) are the interface boundary conditions and the Sommerfeld radiation condition at the bottom of the structure. Assuming a small error exists in the displacement eigenfunctions which could come about due to the amplification of roundoff errors by the growing exponential propagator terms, then equations (32) effectively introduce small compensating terms in the traction eigenfunctions so that the factors which multiply the growing exponential terms stay small.

13 C Seismogram synthesis 49 5 I Y I c a w D 10 DISPLACEMENT I (a) TRACTION I DISPLACEMENT (b) TRACTION 0 Y I I- a w 0 10 DISPLACEMENT (C) TRACT I ON Figure 3. Radial displacement and shea traction eigenfunctions for a structure with layer interfaces at 5 and 10 km depth. (a) Computed eigenfunctions using the traditional propagator matrix approach. (b) Equation (32) is used to evaluate the traction eigenfunctions instead of the propagator matrix at the layer interfaces indicated by the dotted lines. (c) Two pseudo-layer interfaces are added at 2.5 and 7.5 km indicated by the dotted lines. Without such a correction the error terms grow exponentially with depth until they overpower the correct solutions. This method introduces a feedback mechanism into the computations which controls the error and ensures that it stays small with depth. We graphically show how well this method works in Figs 2 and 3. The half-space problem we have discussed is shown in Fig. 2 which shows plots of the radial displacement and shear traction eigenfunctions as a function of depth at a frequency of 5 Hz. The S-wave velocity

14 50 D. J. Harvey for this half-space was four kms-' and Poisson's ratio was For each of the eigenfunctions there are two traces, the left trace is the computed eigenfunction plotted on a linear scale and the right trace is the difference between the computed trace and the exact solution from equation (31) plotted on a logarithmic scale. Fig. 2(a) shows the numerical instability due to the growing propagator term which causes the eigenfunctions to 'blow up' at about 5 km depth. As can be seen from the error plots this instability is due to amplification of the initial error of about lo-'' which is the double precision roundoff error. In Fig. 2(b) we have placed a pseudo-layer interface at 5 km depth and then applied the eigenfunction stabilization method described previously. Although the eigenfunctions do not 'blow up' they are clearly in error around 5 km depth and they are showing signs of instability once again at about 10 km depth. In this case we have allowed the error to become large before we applied the correction at 5 km, but even in the case where the eigenfunction computations are obviously in error the use of equation (32) forces stability. From this figure we can see that the pseudo-layer thickness must be less than 5 km and in Fig. 2(c) we have used 2.5 km thick pseudo-layers. In this case the error never gets large and the use of equation (32) at 2.5 km intervals keeps the error small. These figures show that this stabilization method is only accurate if the error is not allowed to become large, i.e. the pseudo-layer thickness is below some critical value which will be inversely proportional to frequency. From a practical standpoint we will not be very interested in computing eigenfunctions for a half-space and so we show computed eigenfunctions for a more typical structure in Fig. 3. This particular structure is two 5 km thick layers over a half-space and the eigenfunctions correspond to a high-order mode at 5 Hz frequency. In these figures we have omited the error plots since there is no exact solution to compare against. Once again we show the computed eigenfunctions using the normal approach in Fig. 3(a) and as with the half-space case they 'blow up' at about 5 km. In Fig. 3(b) we have applied equation (32) at the structural layer interfaces at 5 and 10 km, but as before we might suspect that the errors have become too large for accurate computations of the eigenfunctions at and below 5 km depth. We add two pseudo-layer interfaces at 2.5 and 7.5 km and the results can be seen in Fig. 3(c). If we continue to add pseudo-layers to the structure and make the pseudolayer thicknesses smaller the resulting eigenfunctions do not change from those shown in Fig. 3(c) and thus we conclude that these computed eigenfunctions are good approximations to the exact solutions. The forced vibration problem - source excitation The source z-dependent integrand functions, given by equation 9(a), have the same form as the eigenfunctions, given by equations (10, 11, 12, 13 and 14), everywhere except at the source depth where a discontinuity in at least one of the functions is required. We follow Harkrider (1964) and define the source jump vector, [Z$)], as follows: where 6 +(") = 6 (S) + E 6-w = 6'") - 6(") is the vertical source location measured from the top of the source layer and e is a positive infinitesimal. Using the source jump vector we can relate the source integrand components at the top of the source layer to those at the bottom by:

15 Seismogram synthesis 51 Applying the bcundary conditions as with the eigenfunctions we can solve for the total solution z-dependent integrand functions, [F,$')], at the surface, and is the total propagator matrix from the surface to the source depth. The total solution functions, F, become eigenfunctions when the source jump vector is zero. We can solve for the surface values of the F vejtor-in terms of the Y matrix, where We can now write the integral equation for the displacements at the surface due to a buried point source. From equations (1, 3 and 38) and only considering the Rayleigh contribution, this gives: where RF = y(0) 12 (0, k). The method one uses to evaluate the integrals given by equations (40) is one of the major topics in theoretical seismology and in this paper we will approximate the integrals with the residue contributions due to the Rayleigh poles along the real wavenumber axis. Normally this approximation would give good results only for the surface waves, but we will show how this approximation can be made to give good results for body waves as well as with a slight modification of the structure. Thus, considering only the residue contributions, we can write the solutions to equations (40) as follows:

16 52 D. J, Harvey The cylindrical Bessel functions become cylindrical Hankel functions of the second kind due to the extension of the integration path to -00 along with the symmetry properties of the integrands (Hudson 1969). We can further reduce equations (41) to a compact form which does not involve the propagator matrix to the source, [Z], but instead depends on the eigenfunctions evaluated at the source depth. This was done by Harkrider (1964) and we omit the details here, and P and B are modified vector cylindrical harmonics evaluated at the eigenvalues and are given by : Given the eigenvalues and eigenfunctions the evaluation of equation (42) is rather simple since it only involves the evaluation of Hankel functions and trigonometric functions. This assumes that one knows the form of the source jump vectors and these vectors can be very tedious to derive but fortunately these derivations can be found in the literature for a variety of sources. Harkrider (1964) gives the jump vectors for an explosion source, a vertical vector source and a horizontal vector source. Ben-Menahem & Singh (1972) give the jump vectors for several point dislocation double couple sources and Kennett & Kerry (1979) give the vectors for a general moment tensor representation of the source. There is also a connection between the cylindrical jump vectors and the spherical harmonic coefficients for a source field which has been expanded in terms of spherical harmonics and this is given by Bache & Harkrider (1976). We made use of this to compute the jump vectors due to the relaxation source which is described by Stevens (1980). The locked mode approximation So far we have described how numerically stable spectra of flat layered earth normal modes can be computed for a point source at arbitrary depth and for arbitrarily high frequencies.

17 Seismogram synthesis 53 At this point we address the problem of how one can use the residue contributions to approximate the complete integral solutions given by equations (40). We start by first briefly describing the properties of the integrand surface in the complex wavenumber plane. Many researchers have investigated the wavenumber integrand functions of equations (40) (e.g. Ewing, Jardetzky & Press 1957; Aki & Richards 1980). These functions are four valued due to the dual valued square root functions of phase velocity, t#&n) and in the bottom half-space (see Appendix A). Thus the Riemann surface will be four leaved and branch points occur where or I $'(~) are zero. We define branch cuts as the loci along which the real parts of 4 and 9' are zero. Fig. 4 shows the branch points, branch cuts and pole locations for real frequency and assuming no attenuation. We have omitted the mirror image in the second quadrant since the integrand topography in this region is not important for evaluating equations (40). Our approach to solve the wavenumber integrals is to deform the contour of integration so that it encircles the poles and then use the residue contributions to approximate the solutions, and we show the deformed contour as r in Fig. 4. We can replace the integrals from 0 to t 00 with integrals from - to t m by replacing the Bessel functions with Hankel functions of the second kind (Hudson 1969). The contribution of the integration path at Ik I = vanishes so that the wavenumber integral equations (40) are equal to the residue contributions given by equation (42) plus a branch cut integral contribution which can be seen in Fig. 4. If we ignore the branch cut contribution, the resulting synthetic seismograms will be missing the early part of the coda, namely the P-wave arrivals and the early S-wave arrivals and this is because the cut-off phase velocity for modes with real eigenwavenumbers is the S-wave velocity of the bottom half-space. We could move the bottom of the structure down for some fmed range and for a realistic earth model this would result in increasing the S-wave velocity of the bottom half-space. As the cut-off phase velocity increases the branch point would recede towards the origin in the complex wavenumber plane, new poles would appear on the real wavenumber axis, and the resulting synthetic seismograms would contain earlier arrivals. The new poles that appear by increasing the S-wave velocity of the bottom half-space were originally 'leaking' modes (Gilbert 1964; Aki & Richards 1980) and were located off the real wavenumber axis on the other Riemann sheets. Increasing the S-wave velocity of the bottom half-space thus has the effect of converting leaking modes to trapped, or locked, modes. 'ik r Contour of Integration 0 Raylelgh Poles 0 Branch Points AIUWW Branch Cuts Figure 4. Displacement spectra integration contour for real frequency and no structural attenuation.

18 54 D. J. Harvey The essence of the locked mode approximation is to place an unrealistically high velocity half-space at the bottom of the structure and we refer to this high velocity half-space as the cap layer. We then require that this cap layer be placed at a depth such that the earliest P-wave reflection from the top of the cap layer arrives after the seismic coda of interest. As the S-wave velocity of the cap layer increases to infinity the entire seismic coda, including the earliest P-wave arrivals, can be synthesized using only the locked mode residue contributions. In the limiting case we can impose a boundary condition of total reflectivity at the top of the cap layer which is equivalent to specifying zero displacements along this interface. This changes the layered half-space problem into a layered plate problem and it causes the wavenumber integrand function to be single valued. Thus for the layered plate problem no branch cuts exist, there is only one sheet to the Riemann surface, a finite number of poles are located on the real wavenumber axis, and an infinite number of poles are located on the negative imaginary axis. There are a number of problems with the locked mode approximation which we will discuss in detail in the following sections where we will show with numerical results how and under what circumstances these problems can be overcome. The problems fall into three categories which we list below: (1) Spurious reflections off of the cap layer. (2) Time wrap-around of the synthetic seismogram in the time domain due to sampling in the frequency domain. (3) The truncation effects due to cutting off the mode sum to some finite number of modes (phase velocity filtering). Computational procedures We have implemented the locked mode method as described in this paper on a DEC PDPl 1 / 70 minicomputer using FORTRAN computer programs. This computer has an available core size of 64K bytes (1 byte = 8 bits) for program instructions and 64K bytes for data. A high speed, large volume disc is also necessary in order to handle the rather large intermediate data files (these files are sometimes in excess of several megabytes in size). Single precision floating point variables for this computer are 32 bits long and have a numerical precision of about eight decimal digits, and double precision variables are 64 bits long with a precision of 16 decimal digits. Because of the limited core space for both data and program instructions, we divided the computations into five separate computer programs which communicate via four intermediate data files. A brief description of each program and the data necessary to run the program is given below. (1) The first program searches for poles over some specified phase velocity range and for some fvred set of frequencies. Normally the frequencies are equally spaced from zero to some upper cut-off frequency so that the resulting spectra can be transformed to the time domain using a fast Fourier transform program without using any sort of spectral interpolation. The other input parameters are the structural parameters which consist of the P-wave velocity, S-wave velocity, density and thickness of each layer. We assume no attenuation at this point. The output from this program is a data fie which consists of the eigenphase velocities of all of the modes within the prescribed phase velocity window and at the prescribed frequencies. In order to avoid numerical problems we found it necessary to do all of the arithmetic in double precision. The eigenvalue search constitutes the major expense in computer time and it also required the greatest amount of time to develop.

19 Seismogram synthesis 55 (2) The second program computes the amplitude term given by equation (44), the group velocity, and the complex shift of the eigenvalue due to structural attenuation for each mode-frequency. The input eigenvalues are read from the data file generated by the first program and the only other inputs are the values of Q for each layer and at each frequency (it is quite easy to account for frequency-dependent Q). We assume that the shift in eigenvalues due to Q can be approximated using first-order perturbation theory. We compute &F/& numerically (we found this to be the cheapest method) at the original real eigenvalue and for real P- and S-wave velocities; we then compute the dispersion function at the real eigenvalue but using the complex P- and S-wave velocities for each layer, and finally we apply a first-order Taylor s expansion of the dispersion function to compute the shift of the eigenphase velocity. Once again all of the arithmetic for this program is done in double precision. (3) The third program computes a set of eigenfunctions for each mode-frequency. The input parameters are the depths for which the eigenfunctions are to be evaluated. This is the first point in the computational procedure where the source-receiver geometry is constrained since the source and receiver locations must be at depths corresponding to the computed eigenfunctions. All of the arithmetic is done in double precision, but the variables on the output file are stored in single precision. We found that subsequent computations could be done in single precision without undue adverse effects. (4) The fourth program computes the receiver displacement spectra for a given set of receiver locations and source parameters using equation (42). Input parameters are the range, azimuth and depth for each receiver, source depth, and the source jump vector for each mode-frequency and for each value of m, the azimuth index. All three components of the receiver displacement vector are computed for each receiver and the spectra are written on an output file. Phase velocity filtering, group velocity filtering and mode number filtering are done at this point. All of the computations for this program are done in single precision. (5) The fifth program allows for seismic instrument convolution, spectral amplitude filtering, and computes the resulting time domain signal for each component of each receiver. A fast Fourier transform subroutine is used to transform the filtered spectrum to the time domain. Graphics software is used in this program so that the resulting synthetic seismograms can be displayed. The first three programs are basically source-independent (with the exception of the source depth) and, for a given structure, only the last two programs need be executed to account for changes in source parameters or source-receiver geometry. Numerical examples In order to check the validity of the locked mode approximation we decided to try to match a wave propagation problem which had been solved exactly. We chose the problem of a vertical vector point source with a step function time history applied at the free surface of an infinite homogeneous halfspace, Lamb s problem. A solution of Lamb s problem for the case of a material with a Poisson s ratio of one-quarter was derived by Pekeris (1955) and is given in terms of elliptic integrals. We placed the cap layer at a depth of 100 km and gave it a shear wave velocity of 12 km s-. The 100 km thick layer above had a shear wave velocity of 4 km s- and a Poisson s ratio of 0.25 (a P-wave velocity of about 7 km s- ). Our useful range was restricted to less than about 100 km since for ranges greater than that the cap layer reflections would interfere with the desired solution. Fig. 5 shows the synthetic surface displacements from

20 Lamb s Problem I 56 D. J. Harvey Figure 5. A comparison of the modal solution of Lamb s problem with the exact solution (from Pekeris 1955). the locked mode method at a range of 75 km and with a bandwidth of 8 Hz along with the exact solution from Pekeris. The match between the two solutions for the radial displacement is quite good up until the arrival of the fundamental Rayleigh wave. After the arrival of the Rayleigh wave, which for Lamb s problem is a logarithmic singularity, the modal solution drifts away from the correct solution, which asymptotically approaches the static offset due to the step function source. At first this may seem to be unusual, since one might expect a modal solution of Lamb s problem to work quite well in synthesizing the surface wave and not so well in synthesizing the P and S waves, but the opposite conclusion must be drawn here. The reason for this is the sampling in the frequency domain which causes wrap-around in the time domain. This is a basic problem which is shared by all frequency domain synthesis methods and so even if we were to synthesize only the Rayleigh wave using an infinite half-space as the structure, we would see a similar drifting error as we see it in Fig. 5. The wrap-around error for Lamb s problem is pronounced at low frequencies due to the large static offset that results from the step function source. The vertical displacement match between the two solutions is not so good although the major features of the P-, S- and Rayleigh waves do agree between the two solutions. Once again we can see the low-frequency drifting error in the modal solution when we compare the modal Rayleigh wave with the exact solution, which is a step function. We also see more low-frequency error in the P- and S-waves than we did for the radial displacement. The most obvious error in the modal solution is the large acausal reverse step that immediately precedes the P-wave arrival. This is due to an upper phase velocity cut-off of 8 km s-

21 Seismogram synthesis 57 which we imposed when summing the modes. This phase velocity filtering causes a spurious arrival to precede the P-wave and we refer to this arrival as the truncation phase, since it is due to truncating the mode sum. We will discuss the effects of phase velocity filtering in more detail, but for now we note that the P-wave arrival and the rest of the seismic coda does not seem to be affected by the truncation phase even when the phase velocity cut-off is set close to the P-wave velocity. The last feature that we would like to point out is the small bump near the end of the trace. This is the first cap layer reflection and as can be seen it does not interfere with the earlier seismic coda. There is an apparent flaw in the locked mode method which is a consequence of both the wrap-around problem and the presence of the cap layer. We can ensure that the first cap layer reflection arrives after some given time in the seismic coda at a given range by placing the cap layer at sufficient depth. After the first cap layer reflection arrival, the coda will be contaminated by a host of multiple reverberations associated with the cap layer, and so one might think that this part of the coda would be rendered useless. For the case of a rigid cap layer these multiple reflections would continue for infinite time and slowly decrease in amplitude to account for geometric spreading. Since we sample in the frequency domain one would expect multiple reflections due to the cap layer to wrap around into the earlier part of the coda and thus contaminate the entire seismogram. This does happen with the locked mode method when we use a rigid cap layer and when we include a large number of modes with very high phase velocities in the mode sum. If, however, we apply a narrow phase velocity filter in the form of a mode truncation based on phase velocity we do not notice a wrap-around cap layer reflection problem. This is the reason for the phase velocity filter which was used for Lamb s problem; originally a prominent cap layer reflection appeared half way between the P- and S-wave arrivals due to wraparound, but the phase velocity filter removed the reflection at the expense of producing a rather large truncation phase. This effect can be easily explained when we consider the horizontal phase velocity of a cap reflection body wave. Normally cap layer reflections will arrive at the surface with steep angles of incidence and thus they will have high horizontal phase velocities, and the later cap layer reflections which correspond to multiple reflections will have even higher phase velocities. Thus a low pass phase velocity filter with a cut-off set slightly above the highest P-wave velocity will tend to pass most of the seismic coda and remove the cap layer reflections. We will demonstrate this in a more quantitative manner in the next example. After checking the locked mode method with the exact solution of Lamb s problem we proceeded to compute synthetic seismograms for a more realistic Earth structure. We chose a six-layer Southern California crust and upper mantle structure which we obtained from Kanamori & Hadley (1975) and the structural parameters are shown in Fig. 6. We computed the eigenvalues and eigenfunctions for two versions of the structural model. The first version had a cap layer with a shear wave velocity of 15 km s-l at a depth of 110 km and was sampled over a frequency range of 8 Hz with a sampling interval corresponding to a 64 s time window. The second version had a cap layer with the same shear wave velocity but at a depth of 190 km and was sampled over a frequency Iange of 4 Hz with a sampling interval corresponding to a 128 s time window. We refer to the first version as the thin structure and the second version as the thick structure, and we computed the mode parameters for these two similar structural models so that we could see the effects of cap layer depth and frequency sampling interval. Fig. 7 shows the dispersion curves over a bandwidth of 1 Hz for the thin structure. The dispersion curves tend to flatten near the various P- and S-wave layer velocities and these flattened regions when joined together account for the undispersed body wave arrivals. It

22 58 D. J. Harvey Southern California Structure East of the Son Andreas Fault 9 I I I 1 I I 1 I I - 1 P-wave Velocity (krn/sec) S-wave Velocity (krn/sec) - Density (grn/cc) I I - 1 I I I I 1 21 I I I I Depth (krn) Figure 6. A Southern California crust and upper mantle structure (from Kanamori & Hadley 1975). This structure is used in the subsequent examples. should be noted though that each high-order mode samples all of the body wave arrivals and the frequency content of each arrival increases with increasing time. Thus a single high-order mode in the time domain will appear to be dispersed but when all of the modes are summed the undispersed body waves will appear. Another consequence of this characteristic of the high order modes is that the modes try to cross each other, and in the process the dispersion curves develop very sharp bends and kinks to avoid this crossing. This results in irregular pole spacing at a given frequency and this accounted for the difficulty in the development of the pole searching algorithm. 1 Dispersion Curves for a Southern California Structure h 0 n \ E x f r - 0 > a m S 0 n../ t l I I 1 I I I I 1 1 I.O Frequency (Hertz) Figure 7. Dispersion curves for the structure shown in Fig. 6 with a cap layer at 110 km depth.

23 Seismogram synthesis 59 In order to test the effects of phase velocity filtering and to verify that we could use this filtering to decontaminate the seismograms of cap layer reflections, we computed a number of seismograms at a fmed range and for a fmed source, for the thin structure, varying only the phase velocity filter cut-off, and compared these against a seismogram at the same range for the thick structure. The results of these computations can be seen in Fig. 8 where we plot vertical displacements at the surface for the P-waves and the first part of the S-wave coda. These seismograms were computed for a range of 200 km and the source was placed at a depth of 7 km. The source model we used was a relaxation source (Stevens 1980) with a cavity radius of 1 km and a prestress corresponding to a thrust event. The individual traces are labelled with a number which is the phase velocity filter cut-off and either thin or thick which specifies the structural model used. We would expect the seismograms for the thin structure to be heavily contaminated with cap layer reflections since the range is almost twice the cap layer depth and this is the case for the top trace of Fig. 8. The vertical dashed lines show the ray arrival times of two prominent cap layer reflections, the earlier arrival being the first P-wave reflection, and the numbers beside each line are the horizontal phase velocities of the two reflection body waves. In addition to these two reflections there are a number of other spurious arrivals including one prior to the first P-wave arrival which is a wrapped-around multiple cap layer reflection. As we decrease the phase velocity filter cut-off these spurious arrivals appear to vanish, as can be seen in the successive seismograms below the top trace. The individual cap layer reflections are filtered out when the phase velocity cut-off drops below the horizontal phase velocity associated with that reflection. At the lowest phase velocity cut-off of 8.5 km s-l the resulting seismogram appears to be devoid of cap layer reflections, with the exception of a small bump at the second indicated reflection arrival. In order to check this Phase Velocity Filtering Effects I ' thi 9.0 lhin 8.5 thin I 15.0 thick1 n I I i I I I 1 '16.29 y Time (sec) Figure 8. A comparison of modal synthetic surface displacements for various phase velocity cut-offs and as the cap layer depth is changed. Dashed lines indicate cap layer reflection arrivals.

24 60 D. J. Harvey we computed a seismogram using the thick structure for which the cap layer is at a depth of 190 km, and this is shown as the bottom trace of Fig. 8. The first P-wave cap layer reflection is indicated by the vertical dashed line but because of the high phase velocity of this arrival it does not appear in the seismogram. We overlaid the bottom two traces on a light table and found that with the exception of the small bump at about 53 s for the thin structure, the two seismograms matched perfectly. If cap layer reflections were present in either of the two bottom traces of Fig. 8 these traces could not possibly match. Not only is the cap layer depth different for these two seismograms but the frequency sampling interval is different as well so that wrapped-around reflections would appear at different times. Also the phase velocity filter cut-off is different between the two seismograms and so we cannot attribute the matching to some filtering effect. Phase velocity filtering appears to be a very powerful tool for the locked mode method and it allows us to extend the usable range limit as we can see with this example. The final two figures show typical results of the locked mode method for two different distance ranges and source types. Fig. 9 shows a set of vertical displacements at distances of SO to 150 km for a strike-slip relaxation source with a radius of 0.5 km and for a bandwidth of 8Hz. The source is 7km deep in the thin structure and the phase velocity fdter cut-off is 9 km s-'. The first several traces show the truncation phase prior to the first P- wave arrival but this spurious phase moves away from the P-wave arrival and decreases in amplitude as the distance increases. Many ray arrivals can be seen in this figure which correspond to rather complex ray-paths and one can see the way in which certain phases wax and wane over narrow distance ranges, such as the phase between 6 and 8 s. Modal synthetic Pg codas are shown in Fig. 10 for ranges of 300 to 450 km and for an explosion source with an elastic radius of 0.6 km and at a depth of 1 km. These synthetics 25 Vertical Displacements for a Strike Slip Relaxation Source I I i I I 50 h E v 75 D 0 LL C Reduced Tlms (sac) Reduction Velocity = 6.20 km/sec Figure 9. Modal synthetic P-wave codas for a strike-slip source at a depth of 7 km.

25 Vertical Seismograms for an Explosion Point Source I I I I 1 I 1 Seismogram synthesis h E 1 v g 375 C E I I I I I I Reduced Time (sec) Reduction Velocity = 8.02 krn/sec Figure 10. Modal synthetic Po codas for an explosion source at I km depth. were generated using the thick structure and they have a bandwidth of 4 Hz. We filtered the vertical displacements with the response of a Benioff 1 s short-period seismograph to produce the resulting seismograms. The most obvious feature is the strong P, phase and trailing coda which is observed throughout the western United States. Unlike the previous example there are no clearly discernible ray arrivals contained in the coda and it is quite likely that a very large number of rays would be necessary to reproduce this coda. We offer this as an example of seismic coda synthesis without using lateral scattering (although we certainly would not try to argue that lateral scattering is not important). A comparison of the locked mode method with other methods of seismogram synthesis In order for any new seismogram synthesis method to be generally useful to the seismological community, the advantages and disadvantages of the method must be identified, and the parameter space within which the method will produce accurate results must be determined. In this section we will qualitatively discuss this by comparing the locked mode method against two broad categories of synthesis techniques which are used for laterally homogeneous structures; the ray theoretical methods and the reflectivity methods. We feel that the only way a definitive, quantitative comparison could be made would be to execute representative computer programs for the three techniques on the same computer and for the same wave propagation problem. One would then evaluate the relative merits of the techniques based on the accuracy of the results, the amount of computer time and space needed to produce the results, and the amount of human interaction needed to specify the input parameters and to run the programs. We have not done this here but we hope to do

26 62 D. J. Harvey this in a future study. With this in mind we limit our discussion to the general advantages and disadvantages which we perceived when we started this work and which we still believe to be true after having implemented the method. We use the term ray theoretical methods to encompass all of the ray solutions ranging from geometrical or optical ray solutions to the variety of solutions which fall under the category of generalized ray theory. All ray theoretical methods are fundamentally different from the locked mode method in that the solution of the elastic wave equation is decomposed in terms of ray solutions in the former case and in terms of eigensolutions in the later case. The ray decomposition of the complete solution is quite appropriate under certain circumstances, namely for high frequency synthesis at long distance ranges, but for ranges less than 1000 km it is not so clear that ray theoretical methods offer the best way to compute synthetic seismograms. The fundamental difficulty with ray theories for this distance range is that one must specify the individual ray-paths which will make up the complete solution and typically a large number of rays will arrive within the desired time window, as we demonstrated in the examples. It can be a painstaking process to specify all of the necessary parameters for a large number of rays, whereas it is quite simple to specify which modes to sum. Thus in terms of human time we feel that the locked mode method is preferable to ray theoretical methods for distances less than 1000 km. In terms of computer time the basic trade-off is the number of rays necessary for the solution versus the equivalent number of modes. As the distance increases the number of modes at a given frequency increases in a roughly linear manner since the cap layer must be moved deeper. At the same time the number of rays in a given time window will decrease and so there must be some crossover point in terms of distance at a given frequency. Unlike the ray theoretical methods, the reflectivity methods are quite similar to the locked mode method. The difference between the two is that the wavenumber integral is evaluated numerically, with the reflectivity method, instead of using the residue contributions as with the locked mode method. The reflectivity method in its original form (Fuchs & Muller 1971) suffered from two basic problems; the wavenumber integrand function was numerically unstable at high frequencies as the source depth was increased, and there was an upper wavenumber cut-off due to poles appearing on the real axis. Recently the numerical problem has been solved (Kennett & Kerry 1979) and the wavenumber cut-off has been eliminated by deforming the contour of integration off the real axis (Cormier 1980) or by moving the poles off the real axis by introducing attenuation to the structure (Kennett 1980). It is now possible to use the reflectivity method to compute complete seismic codas at arbitrarily high frequencies and for arbitrary source depth and so one might ask what the advantages of the locked mode method over the reflectivity method would be. Originally we thought that this advantage would be in terms of computational efficiency and we still believe this to be the case. The wavenumber integrand function is oscillatory near the poles, and the wavelength of these oscillations wiu be equal to twice the average pole spacing in a particular region of the complex wavenumber plane. In order for a numerical integration of this function to produce accurate results the function must be sampled a sufficient number of times over the wavelength of the oscillations. Also one must evaluate the entire integrand function including the numerator components and the Bessel functions at each desired range for each sampled wavenumber. If one uses residue contributions, however, to compute the integral then the numerator components and the Hankel functions need only be evaluated at the eigenwavenumbers. It would be very unlikely that a numerical integration could achieve sufficient accuracy if the integrand function were to be evaluated twice over an oscillation of that function and a more reasonable sampling rate would be 10 or more samples per oscillation.

27 Seismogram synthesis 63 This is the reason why we think that the locked mode method will prove to be efficient more computationally both in the size of the data space needed for storing the eigenvalues and eigenfunctions (or equivalently the sampled integrand function components) and in the amount of computer time needed to compute the final synthetic seismograms. This argument assumes that the eigenvalues have already been located, and it is true that a large amount of computer time is consumed in the search for the poles, but even including this computer time we feel that at worst the locked mode method will break even with the reflectivity method. We have determined from a large number of mode-frequency computations that the dispersion function is evaluated about 15 times for each pole located, and so over one oscillation of the integrand function the dispersion function would be evaluated about 30 times in the process of locating the two poles associated with that oscillation. The expense of this searching procedure is certainly a disadvantage of the locked mode method but we must weigh against this the following facts. The computation of the dispersion function is much more efficient than the computation of the entire wavenumber integrand function. Even though the dispersion function must be sampled 30 times over one oscillation only the two eigenwavenumbers need to be stored, and this whole procedure needs to be done only once for a given structure. Conclusions We have demonstrated the feasibility of synthesizing early P- and S-wave arrivals using normal mode superposition. We think that for distances less than 1000 km the locked mode method will prove to be more practical than the reflectivity method or the various ray theories for synthesizing complete seismic codas. There are several potential improvements to the method which we hope to pursue. Vertically inhomogeneous layers could be accounted for by changing the form of the dispersion function and eigenfunction solutions. The rest of the computational procedures would remain the same and this would greatly increase the flexibility of the structural model. Another potential improvement would be to reverse the order of integration and use the residue sum to evaluate the frequency integral. This offers a computational advantage in that many factors in the dispersion function are frequency-independent, and for a fuced phase velocity the computation of the dispersion function as frequency is varied would be cheaper than vice versa. The analytic evaluation of the frequency integral should also eliminate time wrap-around problems. The problem with this approach is the effect on the solution of sampling in the phase velocity domain (we chose our present course because the effects of frequency sampling are well understood). Acknowledgments 'Ihis research was supported by the Earthquake Hazards Reduction Program, USGS Contract No and the Advanced Research Projects Agency, Department of Defense, Grant No. AFOSR I especially wish to thank Dr C. B. Archambeau who originally interested me in the project and who provided help, inspiration and criticisms at the appropriate times. References Abo-Zena, A., Dispersion function computations for unlimited frequency values, Geophys. J. R. astr. SOC., 58, Aki, K. & Richards, P. G., In Quantitative Seismology Theory and Methods, W. H. Freeman, San Francisco.