A Procedure For the Systematic Interpretation of Body Wave

Geophys. J . Int. (1991) 104, 41-72

A procedure for the systematic interpretation of body wave seismograms-I. Application to Moho depth and crustal properties

Timothy J. Clarke’ and Paul G. Silver2 ‘ Center for Earthquake Research and Information, Memphis State University, Memphis, TN 38152, USA ’Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road, NW Washington, DC 20015, USA

Accepted 1990 July 18. Received 1990 July 18; in original form 1990 February 12

S U M M A R Y Teleseismic body waves represent a natural tool for the investigation of Earth structure, but most studies have used only a very small part of the available information. We present a general procedure for the interpretation of body wave seismograms, which allows us to incorporate information from the entire set of body wave arrivals, and which is based on a new method for the calculation of synthetic seismograms. These ‘Complete Ordered Ray Expansion’ (CORE) seismograms are both complete, in the sense of incorporating all significant energy contributions, and interpretable, in the sense that we can identify all such contributions explicitly.

We show how our procedure can form the basis for a waveform inversion scheme, exploiting the natural division of body wave arrivals into main phases associated with smooth structure, and families of interaction phases associated with discontinuities. We present the results of applying this technique to the Moho beneath various RSTN and DWWSSN stations, showing that it is possible to obtain very consistent values of crustal thickness from single deep focus events, using phases associated with both the P and S arrivals. By incorporating such arrivals in a linearized waveform inversion, we also obtain estimates of a mean Poisson’s ratio for the crust.

Key words: body waves, Moho, Poisson’s ratio, synthetic seismograms, waveform inversion.

1 INTRODUCTION

Teleseismic body waves have provided us with most of our information about the Earth’s interior, illuminating both its average radial structure and its more subtle laterally heterogeneous features. The various techniques utilized for analysing these waves are now very familiar; they include the use of absolute and differential traveltimes, absolute and differential amplitudes, forward waveform modelling, and most recently the direct inversion of waveforms. The number of problems that can be attacked using body waves is surprisingly great, from the large-scale 3-D structure of the lower mantle (e.g. Dziewonski 1984) to the detailed properties of mantle discontinuities. Nevertheless, such studies typically use only a small fraction of the information available in a body wave seismogram. Indeed, the Dziewonski study used only the primary arriving P-wave.

The various techniques applied to the study of discontinuities within the Earth have been similarly limited in their use of the information contained in the body wave seismogram. Until recently, such studies have focused on

particular boundary interaction phases such as specific reflected or converted arrivals. Although these arrivals are generally small, there are certain conditions where a particular phase is strongly excited in a given distance range, or occurs in an otherwise quiet portion of the seismic record, and where it may thus be observed. Best studied are the precursors to P’P’ resulting from bottom side reflections from the 400 and 670km discontinuities (Whitcomb & Anderson 1970; Sobel 1978; Husebye, Haddon & King 1977; Nakanishi 1986). Other examples include S to P conversions (Faber & Muller 1980, 1984; Sacks & Snoke 1977; Bock & Ha 1984; Baumgart & Alexander 1984) and P to S conversions (Vinnik 1977; Vinnik, Avetisjan & Mikhailova 1983; Paulssen 1988), as well as precursors to PP (Bolt, O’Neill & Qamar 1968; Bolt 1970; Wajeman 1990).

More recently, some progress has been made in extracting more of the information from the body wave portion of the seismogram. In particular, Revenaugh & Jordan (1989) have used long-period transverse-component multiply reflected ScS arrivals and their associated interaction phases to image

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42 T. J . Clarke and P . G . Silver

mantle discontinuities. By exploiting the reinforcement of a set of kinematically equivalent arrivals they have been able to extract significant information about the location and properties of the 400 and 670 km discontinuities, as well as identifying energy possibly associated with other mantle discontinuities. Although this approach has already yielded much important information about the elastic structure of the transition zone, it has some fundamental limitations that restrict the types of seismological problems that may be attacked. The use of long-period data restricts the vertical resolution achieved, particularly in terms of constraining the sharpness of the transition. In addition, the restriction to almost normal incidence shear wave energy on the transverse component means that only the shear wave impedance contrast at the boundary can be imaged, and the individual behaviour of the seismic velocities and density is impossible to separate.

The major disadvantage of such approaches, however, lies in their restriction to a specific family of rays. Ideally one would like to allow the geometry of a particular source-receiver path to dictate which of the vast number of interaction phases associated with the discontinuity being studied have significant amplitudes, and are observable. In addition, one should make use of the full three-component properties of the wavefield. In particular, the inclusion of all of the P-SV interactions increases by one to two orders of magnitude the number of phases that may be exploited. By incorporating these phases and exploiting the large range of ray parameters available, it is possible, at least in principle, to separate density contrast across a discontinuity from the jumps in seismic velocities (Clarke 1984). This would enable us to make a direct estimate of those parameters that are most often determined by laboratory studies of mantle minerals. Such an approach, taking advantage of the excellent resolution afforded by broad-band data, is most appropriate for studying the various zones of rapid transition within the Earth; the Moho, the transition zone of the mantle, and the D” region of the core-mantle transition. These regions remain the Earth’s least well-determined features, although, paradoxically, they are probably the most significant in providing information about the chemical and dynamical processes occurring deep within the Earth.

In this paper we present such a procedure for the analysis and inversion of body wave seismograms, in which we are able, in principle, to incorporate all of the information pertaining to both the smooth structure of the Earth as well as the elastic properties of specific boundaries. Our approach is directed by an analysis of the way in which various features, such as discontinuities, influence the observed seismic wavefield. The body wave portion of a teleseismic seismogram consists of a very large number of discrete ‘main phases’, each corresponding to a specific path of propagation within the Earth, and t o a particular wave type or combination of wave types. Each such arrival then interacts in a variety of ways with a given boundary, to produce an entire family of ‘interaction phases’, including reflected and converted arrivals, the majority of which will have amplitudes so small that they are essentially unobservable. For any particular source-receiver geometry and focal mechanism, however, there will be a subset of these arrivals that d o have non-negligible amplitude, and

which could thus potentially furnish significant information about the boundary properties.

Underlying both our scheme for identifying energy associated with boundary interaction phases, as well as our inversion procedure for boundary properties, is a new method for calculating synthetic seismograms (Clarke & Silver 1986, 1988a. 1988b). By making use of symbolic manipulation of the full wavefield expressions of Reflectivity (Kennett 1983) to generate ray expansions for a layered Earth model, we are able to generate seismograms that are both complete, in the sense of including all non-negligible energy contributions, and interpretable, in the sense that we are able t o keep track of all such contributions explicitly. Such a method is ideal for flexible wavefield manipulation and ultimately for the inversion for elastic structure.

To demonstrate the effectiveness of our procedure we have chosen to limit the present application to a study of perhaps the best documented seismic discontinuity, the Moho. As the subject of a preliminary study this has several advantages; not only does the Moho present a relatively large contrast in elastic properties, but the fact that the crust is thin relative to mantle length scales means that the family of interaction phases for each main phase is close in time to its associated progenitor. At the same time, the closeness in traveltime between arrivals requires maximal resolution to obtain boundary properties, requiring us to exploit fully the information content of broad-band seismograms.

In Section 2 below we describe our method for calculating synthetic seismograms, referred to as the Complete Ordered Ray Expansion method, or ‘CORE’. We describe its role in solving the forward problem of generating seismograms where all energy associated with a particular discontinuity is identified and ‘tagged’. We discuss the domain of applicability of the C O R E method, together with its limitations with respect to more ‘exact’ formulations such as Reflectivity .

In Section 3 we describe the inverse problem of determining the properties of a boundary from the observed seismogram, by means of a waveform inversion procedure. The problem is constructed in such a way that the entire body wave seismogram can be modelled, first by performing an inversion for the ‘main’ phases and then for the smaller boundary interaction phases. The first step may also be viewed as an automatic procedure for extracting multiple traveltime estimates from a single seismogram. While our scheme allows us to model both traveltimes and amplitudes, we restrict our treatment in this study to the traveltime problem, regarding the amplitude perturbations as statistical fluctuations that may be interpreted as a source of noise. Clearly, the next step is t o model these amplitude perturbations deterministically, but we leave this to a later report.

In Section 4 below we show the results of applying our method to the problem of elucidating Moho properties. We have confined ourselves to determining those parameters affecting the traveltimes for reflected and converted phases from the Moho; that is, t o boundary location and the average P and S velocities of the crust. We show that it is possible to obtain very consistent estimates of crustal thickness at a numbcr of RSTN and DWWSSN stations,

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Systematic interpretation of body wave seismograms 43

using individual deep focus events with a range of focal mechanisms and source time histories. In addition, we show how reliable estimates of Poisson's ratio for the crust may be made on single seismograms simply by consideration of P-coda phases.

2 THE F O R W A R D PROBLEM

2.1 The Complete Ordered Ray Expansion (CORE) seismogram

Our scheme for generating synthetic seismograms is a hybrid method that incorporates features of the Reflectivity and WKBJ methods of seismic synthesis. As with the WKBJ method we simplify our analysis to asymptotic solutions within a few, smoothly varying regions, coupled at a small number of first-order discontinuities. At the same time our method exploits the algebraic structure of Reflectivity to generate an exhaustive set of rays completely automatically. This allows us not only to generate body wave synthetic seismograms that in most cases are extremely good approximations to the full Reflectivity response, but also to manipulate the wavefield in ways that are ideal for studying discontinuities. For example we can generate 'boundary synthetics', consisting of rays identified with a given discontinuity, or focus attention on certain types of interactions (either conversions or transmissions), or even specify rays that interact with a boundary at a particular point along the way, such as the source region, mid-point, or receiver region.

In calculating Reflectivity seismograms, a stratified earth model consisting of the triplet of elastic functions density, P-wave velocity and S-wave velocity, is approximated, via an earth-flattening transformation, by a stack of uniform layers. An expression for the complete plane wave response W, is evaluated at a large number of slowness and frequency values, and a double transform performed (Fourier for the frequency transform and Hankel for the wavenumber) to obtain the three-component time domain displacements at a number of stations. This expression

W,=W(I - R F R ) - ' F ( l - RgLRZ)-'(Cu + RLLCD), (1) (Kennett 1983, p. 168) involves reflection matrices R F , REL, RZ, the free surface reflection matrix R, the transmission matrix T F , a free surface operator W and source vectors C,, and C,. The subscripts U and D on the reflection and transmission matrices denote up and down respectively, while the superscripts refer to a portion of the stratification. Thus R F is the downward reflection matrix for the region between the surface and the source depth, and RE the reflection matrix for the deeper structure. The two matrix inverse terms in (1) represent reverberation operators for those portions of the model above and below the source depth.

Equation (1) expresses the plane wave response W, in terms of the reflection and transmission matrices for portions of a stratified model. These matrices may be calculated from the reflection and transmission coefficients for individual interfaces, together with phase terms for the uniform layers, by a recursive procedure (Kennett & Kerry 1979) whereby a single uniform layer is added to the top of a

pre-existing stack. The reflection and transmission matrices for the augmented model may be expressed in terms of those for the original model, together with the interfacial coefficients for the additional layer. The downward reflection matrix Rf, for the model with the extra layer is thus given in terms of that, R,, for the original model, with the reflection and transmission matrices ru, r,, t,, t, for the superposed interface as

RL = rD + t,R,(I - rURD)-'tD.

By repeating this process, starting at the bottom of the stack, the reflection and transmission matrices for portions of the model represented in (1) may thus be constructed from properties of the individual layers (Kennett 1983).

Although equation (1) involves two matrix inverses, its numerical evaluation is relatively simple, since the matrices involved are only 2 x 2 for the P-SV system. It is, however, possible to write (1) as an infinite sum of terms, by expanding each reverberation operator using the identity

(I - M)-'= I + M + M ~ + . . . (3) The reverberation operator in (2), and in the equivalent

expressions for the other reflection and transmission matrices, may be expanded in the same way. By truncating each expression after a fixed number of terms, an approximation to the total wavefield is obtained, as a finite sum of terms. This 'ray series' involves reflection and transmission matrices for individual interfaces, as well as phase matrices representing propagation through the individual layers. Each term of the series consists of a product of such matrices, and represents a potential ray path (or set of ray paths, when the different wave types are taken into consideration) within the medium. Thus the first two rays in the expansion (l), obtained by taking only the first term, or unit matrix, in each reverberation series, are given by

W, = w C u + \iiTTSsR&LCD. (4)

The physical interpretation is obtained by reading each term from right to left: the first is the wave transmitted directly upwards from the source, generally referred to as p or s, while the second corresponds to the turning ray P or S.

In our scheme, we make use of symbolic manipulation of such truncated expansions of the wavefield, to generate sets of geometric rays. Several computer packages are now available, such as MACSYMA, which are designed in part to provide assistance with tedious mathematical tasks such as simplifying or expanding unwieldy algebraic expressions. It would be possible to use such a package in the present situation, but it is computationally more efficient to write a simple algebraic manipulation program specifically for the problems encountered in generating such wavefield expansions. We use such a scheme, described in Appendix A, to expand the expression in (1) into a sum of ordered matrix products, truncating the matrix inverse for each reverberation operator at some finite order. In addition, we expand the reflection and transmission matrices themselves, each of which refers to some major portion of our stratified model, into the more fundamental reflection and transmission matrices for individual layers.

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While Reflectivity represents an ‘exact’ method of generating synthetics, a drawback of representing the medium as a large number of uniform layers is that an enormous number of rays is generated, most of whose amplitudes are negligible. Two approximations, those used in constructing the WKBJ seismogram (Chapman 1978), lead to a more compact formulation. First, we model the medium as a few smoothly varying regions, separated by first-order discontinuities. In this case the basic algebraic unit is a reflection or transmission matrix from one of these smooth regions or from an interface separating such regions. Depending on ray parameter, which dictates whether the individual wave types propagate through, turn within or are evanescent for a given region, either the reflection matrix R,, or the transmission matrices T,,, T,, or both, o r neither will exist. Second, we make the approximation, corresponding to asymptotic ray theory, that the two wave types propagate independently within a smoothly varying layer, and are coupled only at the interfaces. Thus we neglect partial reflections and conversions within a single layer, and concentrate on the highest order asymptotic behaviour.

Having generated an appropriate ray expansion, the process of calculating the seismogram is straightforward. Although our scheme facilitates the computation of a WKBJ seismogram for each ray, in practice this is rarely necessary, and we use the highest order geometric ray theory instead. In fact, ray theory has the distinct advantage of being free of the spurious truncation phases that often accompany WKBJ seismograms, a feature that is especially important in studying boundary interaction phases, which are typically low in amplitude and in the vicinity of much larger arrivals. For the ith component of ground motion, each ray may be written as a frequency-independent complex amplitude AiJ and frequency-dependent wavelet w,( w ) arriving at time T,. Thus the Fourier-transformed displacement u,( w ) may be expressed as the ray sum

N

u , (w) = A,w,(w)e-i”7;. / = 1

We assume that an earthquake at r, can be modelled as a synchronous source (Silver & Jordan 1983); that is, that each element of the moment tensor M-possesses the same source time history, M(r,, o) = fi M,Mf(r,, w ) . Here MT is the total scalar moment, while the tensor M, with unit Euclidean norm, is referred to as the source mechanism and f(ro, w ) is the normalized moment-rate function. The form of w j ( w ) is then taken to be the product

where s,(w) is the source time function [the volume integral of f(ro, w ) with the appropriate kernel], and a , (o ) is a causal attenuation operator. For a receiver at r, A,, can be written as the product

1.

A,/ = V % q m / ( r , ro)R/(r> ro) n B,,,, (7) 1 - 1

where 4 is the (frequency independent) source excitation due to M, RcJ(r, ro) is a receiver operator transforming a unit wave into displacement components and a free surface operator, and Gl(r, ro) is the geometrical spreading factor.

For a layered medium this is given by

where p/ is the (flattened Earth) ray parameter for the jth ray, and X , = X , ( p , , r, ro) is the horizontal distance function for the flattened Earth model, while X ; stands for the derivative of Xi with respect to ray parameter. Finally B,,, is the appropriate combination of reflection and transmission coefficients (assumed frequency independent for the present application) for interaction with the fth discontinuity.

2.2 Computational procedure

Our ray generation procedure therefore consists of several steps. The first stage is the calculation, using symbolic manipulation of the wavefield expressions, of a series of matrix expressions representing ray paths, up to some specified order of reverberation within the structure. At this point the wave type for each segment of the ray is not specified. Next a set of critical ray parameters is calculated, consisting of values just above and below the P- and S-wave slownesses a t the model interfaces. These, together with zero, represent the largest and smallest slowness values for which particular sets of rays exist. For each ray parameter, all possible combinations of wave types for each ray path are considered, within the restriction of a prescribed maximum number of conversions, and the ray descriptions stored. Next the range limits for which each ray exists are calculated. If a station lies within these limits, approximate values for its traveltime and ray theoretic amplitude are calculated, using a restricted grid of slowness values. The rays for each station are then sorted and ordered by amplitude, traveltime or a combination of the two, and a subset selected which includes all non-negligible arrivals. Within this subset, the actual ray parameter for the arrival is calculated iteratively, and an accurate traveltime and amplitude computed, together with a value of t* based on an assumed Q model. Finally, by adding together all such contributions, and including the appropriate attenuation operator and source wavelet (see Section 3 below) for each ray as well as the instrument response, we obtain a synthetic seismogram.

We are thus able to generate ‘complete’ asymptotic body wave seismograms up to a specified order of interaction within the structure, by a method that is both relatively simple and very efficient, while a t the same time keeping track of the physical properties and ray trajectory of each individual arrival. There are several advantages to this approach. Rather than decide which phase or family of phases to use, we can allow the particular source-receiver geometry and focal mechanism to dictate which interaction phases are efficiently excited, and make use of those phases in our inversion. In addition, once the original ray generation and sorting has been performed, the most time-consuming part of the process, it is very simple to handle perturbations to the rays and hence to the seismogram as a function of model perturbations; this is particularly important in the inversion for elastic properties. Perhaps even more important for the inverse problem is that this representation allows maximum flexibility in manipulat- ing the wavefield, enabling one t o focus on a particular

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discontinuity, wave type, kind of interaction, or geographic region.

2.3 Extension to 3-D

For simplicity, we have formulated the forward problem for a layered model. Since our method is based on ray theory, however, it is straightforward to extend the procedure to a fully 3-D model. Such an extension would be of little practical use, however, since the number of rays generated initially is vast, and the computing time involved in ray tracing would be excessive. Instead we make use of two alternative approximations, which enable us to generate reasonably accurate synthetic seismograms in a reasonable time. The first, and less restrictive, approximation is to generate the initial suite of rays using a spherically symmetric model, and make the assumption that the same subset that contributes to the seismogram in this case will still dominate the response for the laterally heterogeneous model. This ‘ray conservation principle’ will be reasonable if the lateral heterogeneity is not so extreme as to severely disrupt the ray paths, or to generate large numbers of extra caustics. Having retained only the significant phases, we then restrict the full 3-D ray tracing to this smaller set of arrivals. In the current application, however, the fact that we limit

our attention to traveltime perturbations, and ignore amplitude variation, allows us to impose a more severe restriction. We model the effect of laterally heterogeneous structure by calculating traveltime perturbations to the arrival times of the wavelets according to Fermat’s principle, so that the assumed path remains unchanged. This is roughly equivalent to the approach of Woodhouse & Dziewonski (1984) who performed a waveform inversion of surface wave data for Earth structure using an asymptotic approximation to the phase perturbation, but leaving the amplitudes unchanged.

2.4 Accuracy of synthetic seismograms

In assessing the reliability and accuracy of our scheme, we need to address two distinct questions; which phenomena are predicted by a ray expansion and how accurate is such a representation. The answer to the first question is simpler; so long as we stay away from surface waves and more generally guided or channelled phases (diffracted waves, head waves) we would expect ray theory t o describe the wavefield relatively well. The second question requires more careful evaluation of the approximations involved. We will not attempt such an analysis here: for now, we simply note that comparisons of our C O R E seismograms with Reflectivity synthetics show very good agreement. One would expect such good agreement except in those situations such as caustics where geometric ray theory breaks down. In the problem under study, involving seismograms at teleseismic distances, the vast majority of rays will be sufficiently far from critical regions to be adequately modelled by geometrical options. In the few cases where ray theory is clearly inadequate, such as for phases with turning points close to mantle discontinuities, our procedure still provides us with a means to automatically identify such arrivals, allowing us to compute

their properties with a higher order theory. The most compelling evidence for the adequacy of our method, however, is provided by its success in reproducing not only the main phases of the various observed seismograms described below, but also many extremely small and subtle features of the data.

3 T H E INVERSE PROBLEM

In developing an inversion procedure for body wave seismograms we seek to exploit the structure of such a seismogram. In particular, we note that there is a natural division of the observed energy into two classes; large main phases, associated with smooth structure, and, linked with such phases, families of boundary interaction phases, generally with small amplitudes and associated with the rough structure (or discontinuities).

In our inversion scheme we attempt to exploit this structure, by treating a hierarchy of problems, starting with the gross features of the seismogram, and proceeding to more subtle properties of the waveform. Thus we make use of a three step procedure, which is presently applied separately to each source-receiver pair. In the first stage we obtain an estimate of the wavelet function w,(w). The second stage is to obtain by linearized waveform inversion a smooth 2-D velocity ‘path’ model that correctly predicts the traveltimes of the main phases, and thus, at least partially, accounts for lateral heterogeneity. Finally, the boundary interaction phases are used to invert for boundary properties. This last stage can also be broken up into two parts; in the first step we perturb traveltimes only, to invert for the simplest boundary properties, and in particular location, while in the second step we incorporate amplitude information to constrain physical properties at the discontinuity. In this paper we restrict our attention to the traveltime problem: the use of amplitudes will be the subject of a later report.

The general formulation of our inversion procedure is given in this section, while the procedure actually used for the specific application to the Moho is examined in Section 4 below. Although we focus our attention here on waveform inversion, we should note that our procedure is general, and may be applied to any problem involving the identification and characterization of body wave phases, and in particular of low-amplitude arrivals.

3.1 Estimation of the wavelet

In order to maximize the information content and resolution of broad-band seismograms, it is necessary to match the complexity in the waveform as closely as possible over the entire frequency band. For this, we require an accurate representation of the wavelet, which we estimate from the data. For each source-receiver pair we extract the time-domain wavelet wo(t) from a reference phase, such as P, that is either isolated or at least dominant in a localized interval. We then make the simplifying assumption that all phases have the same source time function, s,(w) = s(w) , and differ only in their degree of attenuation. We expect this approximation, which effectively ignores directivity as well as neglecting the corner frequency effects on the source spectra of P- and S-waves, to be appropriate for the

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deep-focus events used in the present report. The wavelets for P-related phases are thus related via a differential attenuation operator u by

where Sf: = r: - f;. The values of f * are calculated using a standard earth model such as PREM (Dziewonski & Anderson 1981).

The method of wavelet estimation, which is described in Appendix B is, in effect, a deconvolution procedure for removing the instrument response, that is stabilized by assuming the source pulse is time-localized. Fig. 1 shows the wavelets we have obtained from the direct P-waves for the six events used in this study, as recorded at the station RSON. Note that they are all one-sided, as expected on physical grounds, and reproduce the complexities of the broad-band seismograms in spite of their very simple shapes.

3.2 Fitting the main phases

From the ray expansion equation (5) we can write the first-order perturbation in the seismogram 6 u ( w ) (we suppress the component index for simplicity) due to perturbations in A, and T, as

where b i / = 6Al/Al. We will regard the term in 6Al as a source of noise q(w) and use the expression

N

6 u ( w ) = -iw c bT,A,w/(w)e-"'.c+ ~ ( w ) (1 1) / = I

to invert for the differential traveltimes 6T,. The justification for this simplification requires some explanation. For the main phases the expression (7) for A,(w) depends of five factors, each of which may produce variations in amplitude. Probably the largest sources of error are the source terms; MT, the value of seismic moment assumed, and the source excitation S,. The receiver operator R,, may also be a large soutce of uncertainty. The geometrical spreading factor C, is expected to vary slowly, while. for the main phases, the factors Bi,, are primarily transmission coefficients. which are close to unity in most cases.

We wish to both minimize the variance of the effective noise q(w) , and to require that it has mean close to zero so that our least-squares estimates are unbiased. The mean is given by

which will be zero if (bA, ) = 0. The variance is N

( l t ) ( ~ ) ~ I ) = c ( 6A,bA; )w, (w)w; (w)e - ' " (~ -T* ) (13) / , k = l

where the asterisk denotes complex conjugate. Assuming ( b i l 62;) = 6,,d then the variance can be written

so that one seeks to reduce the size of a2. We assume that C? is the combination of an error in scaling the entire seismogram, due to incorrect source characteristics, and the fluctuations in amplitude due to an incorrect structural model. Assuming these fluctuations are small on average, the scaling error can be minimized by constraining the data and synthetic to have the same L , norm.

In relating (11) to velocity perturbations we make a second approximation, invoking Fermat's principle that to first order, the traveltime may be calculated along a reference ray path r". Thus the differential traveltime bT, for the mth phase, calculated with respect to a reference velocity model I+,, may be written as

where 60 = u,'(u - u,J is the relative velocity perturbation. This may be expanded in a set of orthonormal basis functions Qi and 0, of radius r and distance A , as

where the basis functions will generally be sines and cosines. Substituting the representation (16) into (15) and defining KVrn by

LI = -1 G1(+-wr)@,(A) (17) rk

yields

bTm = C cq ~ m t / 1

1.1

which for simplicity we will subsequently write as

bT, = 2 c,K,,. (18)

6u(w) = -iw 2 c,KmlAm~m(w)e-'WTm + q(w).

I

Then expression (1 1) becomes

( 19) 1.m

This expression forms the basis of a linearized least-squares problem for the coefficients c,. Of course 60 actually represents two functions, 60, and 60.$, and any mixed-type phase will depend on both.

For the linearized inversion to be successful it must be carried out within a frequency band where the approximation (11) is valid, that is, where wbTm << 1 for each phase. In general this condition will not be satisfied initially for the full broad-band response, and it will be necessary to iterate. Similar procedures have been described by Chapman & Orcutt (1985) in connection with the waveform inversion of refraction data, as well as by Nolet, van Trier & Huisman (1986), who refer to the technique as adaptive pre-conditioning. Fortunately it is not necessary to recalculate the kernels for each iteration if the underlying velocity model remains linearly close t o the starting model, in the sense of Fennat's principle. In fact, the only non-linearity arises from our insistence on using waveform inversion, rather than extracting traveltimes directly. Thus by iteration, starting with severely low-passed data and synthetics, a large initial bT, may be successively reduced

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~~~

83 153

85121

841 14

841 1 1

8400 1

84066

Source Time Functions

*,

0 1 2 3 4 :set)

40000 Y k

MOO0

40000

8 [sec)

0 4

roo00

-40000 -mono O k

8 [secl

0 4

2.0 3.0k 1.0 :::;k :::;b -10000 -100w

-300w 0.0

4 6 0 4 8 12 0 4 8 12 (secl Isecl (secl A0 0 & 2 .04E:I;I

O.ME.00

.s.mc.m

30

M -5.mE.04 - irnE.05 . l . a y - f f i -

.1.50E.05 . I . y Y . f f i - 10

0

0 10 20 0 10 20 lsecl [secl 4 8

(secl 0

M ::mi:!&#;k - I ~ E . o ~ 3 mE.M

I D

10 [secl 10 20 Isecl 20 0 4 8 12 0 lsecl

0

Fwre 1. Source time functions at station RSON (Table 1) for the six events (Table 2), derived from the P-wave. First column shows source time function, second shows function convolved with instrument response. Third column shows observed P-wave for comparison.

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48 T. J . Clarke and P . G. Silver

making it possible to work up to the full broad-band response.

At the end of this process, we obtain a 2-D path model 6fi,(r, A) for the nth source-receiver path. We have thus performed a crude tomographic inversion using that set of ray paths present within a single seismogram. The primary purpose of the model, however, is to match the main phase traveltimes. The velocity model itself is of secondary importance and need only be smooth enough to satisfy the linearity requirements. The waveform inversion can also be seen, however, as an automated procedure for obtaining multiple traveltime estimates from a single seismogram; estimates which may be subsequently used for a 3-D traveltime inversion.

We illustrate this procedure in Fig. 2 for that portion of the seismogram including the direct P and S arrivals, for the seismogram recorded at the station RSNT (Table 1) for event 84066 (Table 2). The top panel shows a comparison of data (below) and synthetic (above) for vertical (bottom) and radial (top) components, for the reference model (PREM), with the primary arrivals identified. The bottom panel gives the final perturbed seismogram for this source-receiver pair. Not only are the main phases P, p P , sP, PP, S and ScS lined up, as well as some smaller arrivals such as PcP, SKS

(the small phase between S and ScS) and ScPPcP, but even the details of many of the phases are extremely well

Table 1. Stations used in this study. Statioiis used i n this s tudy

Station

RSCP RSNT RSSD RSNY RSON COL TOL

Network Latitude

RSTN 35.6 RSTN 62.5 RSTN 44.1 RSTN 44.6 RSTN 50.9

DWWSSN 64.9 DWWSSN 39.9

Longitude

-85.6 -114.6 -104.0 -74.5 -93.7 -147.8 -4.1

Locatton

Cumberland Plateau, TN, USA Northwest Territory, Canada Black Hills, SD, USA Adirondack, NY, USA Red Lake, Ont, Canada College Outpost, Alaska Toledo, Spain

Table 2. Events used in this study. Events used in this s tudy

Event Latitude Longitude Depth (km) mb Locaiion

83153 -9.5 -71.2 599 5.9 South America 85121 -3.2 -71.3 539 6.0 South America 84001 33.4 137.3 374 6.5 Izu Bonin 84066 29.4 138.9 457 6.3 Izu Bonin 84111 50.0 148.8 581 6.0 Sea of Okhotsk 84114 47.5 146.7 415 6.0 Sea of Okhotsk

- 0 8 , , , , , . , , , . . , . , , . , l . . . . . l . I I I . I . . . , I I . . I I I l . I . I . ~ ~ ~ ~ . ~ ~ , I

700 800 900 (fz$O 1100 1200 1300 1984 66 2 27 33 6 STN RSNT DIST 71 5 BAZ 298 1

R

Z

-0 0

700 800 900 (;:$O 1100 1200 1300 1984 66 2 27 33 6 STN. RSNT DlST 71 5 BAZ 298 1

(b)

Figure 2. Results of waveform inversion for 2-D smooth perturbation, for event 84066 recorded at station RSNT. Top panel shows data (lowe trace for each component) and synthetic (upper trace) for reference (starting) model PREM. The largest main phases are indicated, as well a Moho interaction phases P P 6 S and s k P . Lower panel shows same comparison for final model, after 40 iterations.

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predicted. In addition the Moho interaction phases s f iP (vertical component) and P P 6 S (radial) are also clearly seen (see Appendix D for naming convention). The part of the seismogram that is less well matched consists of multiple P phases such as PPP (triplicated) and PPPP, as well as sPPP (triplicated), which have shallow turning points and are strongly affected by details of upper mantle structure. Despite this, many of the phases are still clearly visible, and the relative time shifts will provide us with extremely useful information about the transition zone. We note that this one seismogram yields at least 15 traveltimes that may subsequently be used in conjunction with a 3-D model.

3.3 Fitting the boundary interaction phases

In this step, we once again use Fermat's principle. The total change in traveltime 6T: for the mth phase due t o the perturbations ahj of the N boundaries (positive upward) is

N

In this and subsequent equations, all summations are given explicitly. There are four classes of interaction with a boundary (upward and downward, reflection or transmission), and a typical phase has multiple interactions. Using I,m, to denote the number of interactions of the Ith type, we can express the kernel Q , as

These 16 cases correspond to the four types of interaction with the boundary, for all combinations of the two wave types, and can be written concisely as

91 - 40 (upward transmission), -(ql - qo) (downward transmission), -(qI + qo) (topside reflection), 41 + 40 (bottomside reflection)

(22) Qlj =

(Dziewonski & Gilbert 1976). The quantities q1 and qo are the vertical slownesses for the incident wave and outgoing (reflected or transmitted) wave respectively, evaluated above or below the j th boundary for the appropriate wave types.

In general the locations of the various discontinuities within the Earth will vary laterally, and 6h will be a function of A for each source-receiver pair. It may thus be parametrized like (16) as

N &,(A) = 2 dj,oi(A).

i = l

Such a parametrization is necessary if we are to simultaneously consider phases that interact with a boundary near the source, bounce point and receiver.

We may form the first-order seismogram by substituting (20) and (23) into (11) and following a development essentially the same as (15-19). We illustrate the process by showing the special case where each phase interacts with a boundary once only; the generalization to multiple interactions is straightforward. Defining KZ, = QmjQi(Am,),

where Amj is the location of the interaction point, we can express the traveltime perturbation as

i.1

which for simplicity we write as

6T: = d,K:,. I

We thus obtain the first-order perturbation to the seismogram as

6u(w) = - i u 2 d , K ~ , A , , , ~ ~ ( o ) e - ~ " ~ ~ 1. m

which forms the basis for a least-squares problem for d,. As in the case of the main phases, we treat the amplitude perturbations as an effective noise term. The main difference is that for the boundary interaction phases the coefficients Bijl may now represent significant sources of error. To minimize this error, we again normalize the seismograms to the synthetics, as in Section 3.2, but this time with the main phases removed from both data and synthetics. This subtraction, which is trivial for the synthetics, is also possible for the data since the wavelet function for the synthetics is determined from the data, and is thus matched exactly, while the first step in our inversion ensures that the main phases are exactly lined up.

There is one further complication, arising from the fact that it is not possible to decouple entirely the effect of smooth perturbations from boundary perturbations. In fact, any perturbation to a boundary changes the arrival times not only of the interaction phases, but also to a lesser extent the main phases. Our intention, however, is that the smooth perturbation results in agreement between observed and predicted traveltimes for the main phases. To be consistent, we should require this to be true even after the perturbation to boundary properties. This physical requirement imposes an important additional constraint on the inversion procedure; namely, that any perturbation 6h, must leave the traveltimes of the main phases constant. That is,

where the index m is restricted to main phases. This is equivalent to applying an additional smooth perturbation to the model that serves to exactly offset the change in traveltime due to 6h for the main phases. The explicit form of this constraint is given in Appendix C.

The solution of (26) requires two additional considera- tions that were not present when solving (19). First, it is possible that in the reference model Poisson's ratio is incorrect in a localized layer like the continental crust. In this case, the traveltimes for P and S interaction phases will be inconsistent. As shown in Section 4 below, we can exploit this discrepancy and invert for a perturbation in average 6v, and 6u, and thus Poisson's ratio.

Second, the use of an iterative linearized solution to (26) has the major drawback that the initial frequency band must be relatively low. Since boundary interaction phases are typically very small and near unrelated main phases, some of which may not be accurately modelled, the inversion procedure is strongly affected by these phases a t low frequencies. In the present application we have solved this

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problem by considering perturbations to only one boundery , and by assuming that 6h is a constant function of A. While this would normally be a severe limitation, it is not if we specialize to phases that are unambiguously associated with either the source, receiver or bounce point. In this case, (26) is a function of only one parameter, and we can thus perform a 1-D non-linear search for the minimum misfit. Having nearly lined up the boundary interaction phases, we can then apply the linearized inversion at much higher frequency to invert simultaneously for the three parameters ah, Sv, and Sv,, thus retaining the resolution in the broad-band seismograms. With only a limited ray parameter range, it is possible to constrain only two of the three parameters. One of these is Poisson's ratio, while the other may be regarded as an apparent 6h given a fixed velocity (6v, or Sv,) or vice versa. In Section 4 below, we utilize this procedure to invert for apparent depth, and for Poisson's ratio for two of the stations.

4 APPLICATIONS TO THE MOHO

The applications addressed in this section are designed to illustrate some of the properties of our method. as well as suggesting future applications. At this stage, we are primarily concerned with extracting single record estimates of Moho properties as a way of demonstrating the information content available in only one seismogram. Clearly it is advantageous eventually to combine records from several paths by stacking, but we have left this analysis to a subsequent report. We have examined the Moho under selected stations of the RSTN and DWWSSN network (Table 1) using three pairs of deep focus events, with magnitudes ranging from Mh = 5.9 to 6.5 (Table 2). These are two South American events, two Sea of Okhotsk events and two Izu Bonin events that are nearly in the same location and nearly the same size, enabling us to distinguish between deterministic signal and noise, and to identify azimuthal variation.

Synthetic seismograms were generated using the CORE method and the model PREM, modified to have a 40 km thick crust more appropriate for the continents. Moment tensors were taken from the Harvard CMT solutions while the hypocentres were taken from the NEIS PDE listings; in the inversion for smooth properties, however, hypocentral parameters can be allowed to vary simultaneously with the velocity structure.

We have performed the non-linear inversion for Moho depth 6h for all of the stations listed in Table 1. In addition, we have performed the linearized inversion for Sh, 6ucpruS' and S v ~ ' for the stations RSON and RSNT, which have the simplest receiver structures, and which have the smallest depth perturbation in the non-linear step. This second condition enables us to use the linear approximation (26) with minimal error. We have restricted ourselves to the use of those phases that interact only on the receiver side; P , PcP, S, ScS, and SKS. While the phases associated with PcP are generally weak, there are a set of five interaction phases associated with P that may be utilized; PmS, PPliiP, PSrES, and the kinematic equivalents PPliiS and PSrEP (see Appendix D). All three of the S related phases are usually well excited and there are a variety of phases, both before and after that may be used to constrain boundary

properties. For S, the possible arrivals are the precursoi SmP, the coda phases SPliiP, SSliiS and the kinematic equivalents SSriiP and SPliiS. Analogous arrivals will be present for ScS and SKS, although those for S are usually better isolated. Thus there are potentially five P-related and 15 S-related phases on one seismogram, not counting phases with multiple interactions.

The same analysis may be applied to the source-side and mid-point Moho location. For example, one can use phases that unambiguously interact with the source-side Moho, such as the precursors PAP, SAP, sAS, sASCS and sASKS as well as the kinematic equivalents such as sSliiSS and sSS6iS that interact with the source- and receiver-side respectively. Similar precursors (e.g. PAP) and kinematic equivalents (such as PPliiPP and P P P f i P ) will also exist for the mid-point. Our procedure allows us to automatically identify the various categories of phases, and hence to prefrom inversions in which only those members of a specific class are perturbed.

In all the results we show below, we have first inverted for a path model to fit the main phases using the procedure discussed in Section 3.2. For the P coda, this reduces to matching the traveltimes of P and PcP, while for the S coda we have direct S either alone or combined with the other two phases. Although we will examine both the P and S coda, we will emphasize the results for the P coda, as the analysis is simpler for several reasons. The higher frequency content provides better time resolution, while the steeper angles of incidence of the interaction phases reduce the errors in using Fermat's principle, and allow us to perturb only average crustal velocities in the linearized inversion. Ultimately, the combination of the two codas will be highly valuable in a more detailed examination of the crust, as they provide distinctly different ranges of ray parameter. Indeed, it is possible using such a combination of phases to resolve the crustal thickness and velocities individually.

4.1 Boundary location estimates from direct P RSNT

Before performing the inversion, we deconvolve the data to displacement to enhance temporal resolution, and remove the direct P arrival, using the estimate of the wavelet function. In addition we rotate from vertical and radial components to directions parallel and perpendicular to the calculated direction of motion for the P wave, a procedure suggested by Vinnik (1977). We will refer to these as the P and S components respectively, as the P ( S ) (at the receiver) interaction phases will primarily end up on the P ( S ) component. Fig. 3 shows the steps in this procedure, for event 84001 (Izu Bonin) recorded at the station RSNT. Fig. 3(a) shows the deconvolved data (lower traces) and synthetics (upper traces), (b) the result of rotating into P and S components, and (c) the same components after the P-wave has been removed.

The results of the inversion are shown in Fig. 4. We plot a series of synthetic traces for different Moho depth perturbations Sh, relative to a reference depth of 40 km, with a positive perturbation corresponding to a shallower depth. The data trace is placed at the value producing the smallest L, norm residual between data and synthetic. All

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IC i

-.- % h 10 H50 fi6O

F i r e 3. Stages in the processing of seismograms before inversion for Moho properties. (a) shows initial deconvolved data (lower trace for each component) and synthetic (upper trace). (b) shows data and synthetics rotated into P and S components. (c) shows result of subtracting the P-wave from both components.

phases of non-negligible amplitude are identified at the bottom of the plot. The time shown is with respect to the origin time of the event. As we will see, for these six deep focus events, in addition to Moho phases, there is a surprising variety of phases from the 400 and 650km discontinuities that are predicted to arrive in the P -wave coda. This figure shows the presence of all of the expected boundary interaction phases. P P f i P is particularly prominent on the P component (PcP is also shown), while PmS, PPrFiS and PSfiS are seen on the S component. The calculated synthetics for the minimum residual perturbation of 4.0 km show a remarkably good fit to the observed traces, suggesting that the crustal structure under RSNT is extremely simple. 84001 is one of the larger events and possesses a complicated wavelet function, as indicated by Fig. 1.

In Fig. 5, we show the equivalent plot for the South

American event 85121, which has a much shorter duration. Again, all expected phases are visible with very clear PmS and PPfiS (concurrent with, but of larger amplitude than, PSf iP) and P P f i P arrivals. The minimum residual is at 6 km. Note that several phases from deeper discontinuities are predicted to be present: S6P, pas. In Fig. 6 we plot the misfit as a function of perturbation 6h at station RSNT for each event. Each panel displays the misfit for the two components separately (triangles for P , dashed line for S), together with the total residual (solid line). All three curves are calculated with respect to an initial misfit, taken to be the residual with the boundary interaction phases set to zero. Not only is there a well-defined minimum in all cases, but there is very little scatter. Based on the six events, we obtain 6h = 3.7 f 0.6 km, suggesting that the data are capable of providing excellent precision and that there is little azimuthal variation in Moho properties.

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52 T. J. Clarke and P. G. Silver

Fwe 4. Inversion of p-coda at RSNT for Moho depth, for event 84001. Bottom panel shows component in P-direction, top panel shows perpendicular 'S' component. Reference depth is 40 km, and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below. Note prominent PPfiP ( P component); PmS, PPriis, PSkS (S component).

minimum residual a t 6 0 km

06 0 0

-0 8

1.0

0.0

Figure 5. Inversion of P-coda at RSNT for Moho depth, for event 85121 (cf. Fig. 4).

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Moho at RSNT - 84001 (P) Moho a t RSNT - 841 1 1 (P) Moho at RSNT - 83153 (P)

- 0

._ Y)

Q,

TJ

2 2000 .-

u -5 u -4 L o ? B -11) $ -6 g -2000 cr -8 Ill

-20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20

Depth perturbotion (km) Depth perturbotion (km) Depth perturbation (krn)

Moho at RSNT - 84066 (P) Moho nt RSNT - 84114 (P) Moho at RSNT - 85121 (p)

-15000

-20 -10 0 I0 20

0

rJ p -150

$ -200 in

-250 -20 -10 0 10 20 -20 -10 0 10 20

Depth perturbotion (km) Depth perturbotion (km) Depth perturbotmn (km)

F p 6. P-coda relative residuals for Moho perturbation at RSNT for all six events. Each panel shows three traces; dotted line is residual for S component, line with triangles indicates residual for P component, solid line is total residual. Each residual has a reference value subtracted.

PPriiP on the P component. Fig. 8 for event 83153 shows the phase P P 6 P on the P component as well as a clear

RSON

Figure 7 shows a comparison of synthetic and observed PmS on the S component. The phase S6P is predicted to be seismograms for event 84066, which has a particularly as large as PPriiP; there does appear to be a phase present complex source time function (Fig. 1). For this event the at that time although slightly delayed. There is also an phases PmP and PSriiS are seen on the s component and unmodelled phase beginning at t = 596 s on the P

Figure 7. Inversion of P-coda at RSON for Moho depth, for event 84066. Reference depth is N k m , and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below.

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54 T. J . Clarke and P. G. Silver ~~ -~

83153 RSON I 3 2" minimum residual a t 0 O k m 20 0

S-data

0 6

0 0 -20 0

Figure 8. Inversion of P-coda at RSON for Moho depth, for event 83153 (cf. Fig. 7). In addition to Moho phases, note S6P from 650 km discontinuity.

component. The residual plots (Fig. 9) show very consistent minima. The mean perturbation at RSON from these six events is 6h = 1.0 f 0.4 km.

co L

The seismograms at COL are clearly more complicated than those at the previous two stations. Given the tectonically

complex environment, this is not unexpected. Event 84111 (Fig. 10) shows a clear PmS (both components) and PPriiS. There are six predicted phases from the 400 and 650km discontinuities, one of which, p&' appears to be present on the P component. There are also several unmodelled phases, such as the one at t = 337 s on the S component (between PmS and PPriiS) and before PPriiP on the P

~ ~~

Moho at RSON - 84001 (P) Moho at R50N - 8 4 1 1 I (PI Moho at RSON - 83153 (P)

Depth perturbotlon (km) Depth perturbation (km)

Moho a t RSON - 84066 (P) Moho at RSON - 84114 (P) Moho a t RSON - 85121 (P)

. . . - n 2 0

; -1000 .-

L g -2000

z -3000 3

- 40 U w Ln -20 - 4000

-20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 Depth perturbotion (krn) Depth perturbation (km) Depth perturbotion (km)

Figure 9. P-coda relative residuals for Moho perturbation at RSON for all six events (cf. Fig. 6).

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20 0

S-data

O d

0 0 -20 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , . , . I . . . . . I v - - 360 370 390 400 S-syn

u380 (see1 d * L / - : 20 0

Fgare 10. Inversion of P-coda at COL for Moho depth, for event 84001. Reference depth is 40 km, and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below. Note Moho phases P P k P (P component); PmS, P P t S (S component).

20 0

S-data

1 0 0

0 -20 0

S-syn

20 0

P-data

1%

0

1%

-20 0

P -syn 530 540 550 ~ 560 570

Figure 11. Inversion of P-coda at COL for Moho depth, for event 84066. In addition to Moho phases, note apparent P6P (P component).

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56 T. J . Clarke and P. G. Silver

component, suggestive of other structure. Event 84066 shows a very clear PhS, as well as PPliiP (Fig. 11). Again, there are unmodelled phases such as the large arrival at the end of the P seismogram. The only phase predicted to arrive with significant energy is the precursor to P P . P6P, an underside reflection from the 650 km discontinuity, which appears to be delayed by 2 s with respect to the synthetic. As seen on Fig. 12, the residual plots are more complicated, in the case of 84001 and 84066 producing secondary minima. In spite of this complexity, however, C O L yields very consistent absolute minima. The mean perturbation at C O L from five events is 6h = 10.2 f 0.9 km.

RSNY

The results for RSNY are less consistent, reflecting both greater structural complexity and decreased signal-to-noise ratio for the events recorded. The South American events, as illustrated by 85121 (Fig. 13) have simple S components and well-defined residual minima of 6 h = -6 km (83153) and 6h = -7 km (85121) as seen in Fig. 14. However, both events from the Kurils (841 11, 841 14) show well-developed double minima, primarily as a result of the P component. The S components give minima consistent with those from the South American events, and using these values we obtain 6h = -7.5 f 1.0 km. This shallower secondary minimum may actually represent a mid-crustal reflector, possibly related to the mid-crustal high-velocity layer found by Owens, Taylor & Zandt (1987).

RSSD

The most complex results are found for station RSSD. The only event that gives well-behaved results is event 85121

shown in Fig. 15 (83153 was not available), with a southeast back azimuth. In fact it is one of the simplest seismograms in the entire data set. All expected boundary interaction phases appear to be present. with an especially prominent PmS. The value of bh = -10 km for this event is consistent with previous results for this area (Steinhart & Meyer 1961). The only clear, unmodelled phase, is at t = 569 s on the S component. All of the other events, which come in from the northwest, produce completely different seismograms. The most distinctive feature of all of them is a very early arrival on the S component, which for 84114 has been modelled as PmS from a Moho at 6h = 12 km (Fig. 16) and is also seen on 84066 (Fig. 17). This provides a consistent feature in the S component residual plots (Fig. 18) and suggests the presence of a mid-crustal reflector that has an impedance contrast at least as large as the Moho. The fact that the crustal structure is complicated at RSSD has been pointed our previously (Owens et al. 1987). This is not entirely surprising, as RSSD sits on the boundary between the Trans Hudson Orogen to the southeast and the Wyoming Craton to the northwest. Since those portions of the crust being traversed by incoming waves from the two azimuths will differ in distance by more than 100 km, we may actually be sampling two distinctly different geologic terrains.

TOL

While we have only one record for TOL (for event 83153), we present it because it provides a well-behaved result and illustrates that in some cases, one seismogram is sufficient to constrain Moho depth. This record (Fig. 19) exhibits a large P h S on the S component and a P P G P arrival on the P component. The value of 6h = 8 km (Fig. 20) is consistent with previous estimates of crustal depth near this location (Meissner 1986).

Moho dt C O L - 94001 (P ) Moho a t COL - 84111 (P) Moho ot COL - 83153 ( P I

Depth perturbation ( h m ) Depth perturbation (km) Depth perturbation (km)

Moho at COL - 84066 (P ) Moho at COL - 84114 (p)

Depth perturbotion (km) Oepth perturbation (km)

Figure U. P-coda relative residuals for Moho perturbation at COL for five available events (cf. Fig. 6).

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85121 RSNY minimum residual at - 7 0 km

0 6

0 0

20 0

0 8

D I

D O

500 510 520 + 530

Figure 13. Inversion of P-coda at RSNY for Moho depth, for event 85121. Reference depth is 40 km, and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all signficant phases are identified below.

The perpendicular direction is referred to as P. Due to the 4.2 Moho depth from S-wave phases lower frequency content of S-wave signals, the data have

Next we examine the S-related boundary interaction phases. not been deconvolved. In fact, the undeconvolved velocity As in the case of the P-wave, the data and synthetics are seismograms have approximately the same frequency rotated in the vertical plane, this time into components content as the deconvolved P seismograms. We use a parallel to and perpendicular to the calculated S V direction. separate source wavelet s ( w ) for S since the PREM Q

Moho at RSNY - 841 1 1 :P) Moho o t RSNY - 83153 (l')

Depth perturbation (km) Depth perturbotion ( k m j

Moho a t RSNY - 85121 (P) Moho at RSNY - 8 4 1 1 4 (P)

Depth perturbotion (krn) Depth perturbotion (kmi

Figure 14. P-coda relative residuals for Moho perturbation at RSNY for four available events (cf. Fig. 6). Total residuals give different minima, but S residuals are more consistent.

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Figure 15. Inversion of P-coda at RSSD for Moho depth, for event 85121. Reference depth is 40 km, and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below.

model gives values of t * too low for broad-band noise levels and check assumptions about the procedure. seismograms with predominantly continental paths. One of these is the assumed value for Poisson's ratio. As we

The results from S are useful as a consistency check on will see, the S-coda inversions are usually dominated by the the depth estimates from P , since we can evaluate relative P component and thus 6 ~ ~ ' ' ~ ~ while the P-coda inversions

minimum residual at 1 1 0 k m ,-- 20 0

S-data

-20.0 15

-

:: -20.0

Figure 16. Inversion of P-coda at RSSD for Moho depth, for event 84114 (cf. Fig. 15).

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84066 RSSD I A 20 0

- - \ S-data

P-data

-20.0

Figure 17. Inversion of P-coda at RSSD for Moho depth, for event 84066 (cf. Fig. 15)

discussed earlier are usually determined by the S component and thus &IT'. The two inversions may consequently give different apparent depths.

Second, the phases associated with S differ in ray parameter by a factor of fi. This is especially important for SPriiP, which has a bottoming depth in the vicinity of the Moho, leading to the possibility that the Fermat's principle may be invalid. As we will see the phases SmP and SPliiP often yield incompatible results, suggesting such a

problem. As a result, in several cases we have used a shorter time window, and estimated 6h from the single phase SmP, which should be the most compatible with the P-wave results. In many cases this phase is strongly excited, and our CORE modelling allows us to identify it unambiguously.

RSON We have applied the procedure to four of the six available earthquakes. Events 84001 and 84066 have S, ScS, and SKS

Moho o t RSSD - 84001 (P) Moho at RSSD - 84111 (P) Moho o t RSSD - 85121 (P)

- 2 4

e o ?

v g - 4 3 cr in

-a

g -500

g -1000 8 - 1500

-20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20

Depth perturbotion (km) Depth perturbabon (krn) Depth perturbation [km)

Moho a t RSSD - 84066 (P) Moho 01 RSSD - 841 14 (P)

$ -100 & -1500 g -120

; -500 2 -1000

-2000 v,

-l4:2o -10 o 1 0 20 -20 -10 0 10 20 Depth perturbobon (km) Depth perturbation (km)

Figure 18. P-coda relative residuals for Moho perturbation at RSSD for five available events (cf. Fig. 6).

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83153 TOL I A = 7 3 l o , minimum residual a t 8 0 km

-20 0

S-syn

20 0

0 4

0 0

-0 4

P-data

-20 0

Figure 19. Inversion of P-coda at TOL for Moho depth, for event 83153. Reference depth is 40 km, and synthetics are plotted for perturbation depths between -20 and +20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below.

too close together and were not used. Of the four, the two from the Kurils (84111 and 84114) exhibit simple seismograms (Fig. 21 for 84114), while those from South America are more complicated (Figs 22 and 23). The Kuril events give similar results to P , with 6h = -1.0, 0.0 km with simple residual minima (Fig. 24). Inversions using SmP alone give 6h = -2.0, 3.0 km. In Fig. 21 a prominent SP+S phase is seen on the S component, as well as SmP on the P component.

Figure 24 illustrates that events 83153 and 85121 generate complicated misfit curves (although similar to each other). There is clearly an incompatibility between phases. While inversions using SmP alone yield 6h = 3.0 for both events, using all phases gives 6h = 2.0, 9.0. The seismograms for these events illustrate the inconsistency. In particular, the minimum of 9.0 km for 85121 does not fit the SmP phase, and the misfit is dominated by the large SP+S on the S

1 Moho o t TOL - 83153 (P)

20

80

40

0

.40 -20 -10 0 10 20

I Depth perturbation (km)

Figure 20. P-coda relative residuals for Moho perturbation at RSSD for event 83153 (cf. Fig. 6).

component. These results suggest either an error in using Fermat’s principle or that these phases are suggesting a problem with the assumed lower crustal velocities. The contrast between the South American and Kuril events does not appear to be a distance effect, since all four events are in the distance range 62.9”-69.2”. If we restrict ourselves to the SmP inversion, the average 6h for all four events is 1.75 km, which compares favourably with the P -coda results of 6h = 1.0 f 0.4.

There are two unmodelled features of these seismograms that should be mentioned. First, at the end of all four P component seismograms, there is a large phase that is not in the synthetic. For 84114 the arrival S P is predicted by PREM to be triplicated by the discontinuity at 200 km depth, with large contributions from the second two arrivals. The unmodelled phase has the same general form but it is shifted by 4-5s. The other seismograms also predict S P arrivals near the interval, but with nowhere near the observed amplitude and with arrivals that again are several seconds early. Since this arrival is clearly sensitive to subtle details of the velocity structure just beneath the Moho, its precise traveltime and amplitude are not accurately predicted. The large observed amplitude, however, may well be indicative of a high velocity gradient or second discontinuity in this part of the mantle. Such a structure has been suggested by several previous studies (Hales 1969; Zandt & Randall 1985). Second, all four seismograms show an unmodelled arrival on the P component about 6 s after SmP. The only arrival consistently expected at that time is SmP Pf iP , although this phase is predicted to be very small. The predicted amplitude, however, is strongly dependent on crustal P velocity and thickness, with small changes in either

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84114 RSON : I : A = 69.2" : minimum residual at 0.0 k%

om lbm

0

1120 1130 1140 1150 1160 (set)

"om

0

20.0

P-data

-20.0

P -syn

S

20.0

-data

-20.0

plllue 21. Moho inversion at RSON using S-wave interaction phases for event 84114. Bottom panel shows component in S direction, top panel shows perpendicular 'P' component. Reference depth is 40 km, and synthetic traces are shown for perturbation depths between -20 and t20 km. Data traces are inserted at perturbation giving smallest total waveform residual, and all significant phases are identified below. Note prominent SPrFiS (S component) and SmP (P component).

20 0

P-data

emu

dOL0 0 -20 0

P -syn

20 0

S-data

UID

0 -20 0

Figore 22. Inversion of S 'coda' at RSON for Moho depth, for event 83153 (cf. Fig. 21). Note also SPrFiP, SSrFiS (S component).

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s O.OOE+OO m

minimum residua: a t 9 0 k m

- -200

m \ . -

1

Figure 23. Inversion of S 'coda' at RSON for Moho depth, for event 85121 (cf. Fig. 21).

giving a much larger phase. Alternatively, it could be S P i P , where x is a mid-crustal reflector. More detailed work would be required to distinguish between these possibilities.

RSNT

The inversions for RSNT (Figs 25 and 26) give on average a shallower Moho than the P-coda data. Three events are

consistently shallow with values of 6h of 10, 9 and 8 km for 84114, 85121 and 83153. The fourth event, 84111, yields 6h = 3 km, with an average of 7.5 km. One of the problems with 84111, which may explain the somewhat different value, is that the S-wave has a very long and complicated wavelet in excess of 18s in duration. Silver & Chan (1986) have previously noted the unusual features of this arrival and have attributed it to slab multipathing. For the inversion

Moho at RSCIN - 84111 ( S ) Moho at RSON - 83153 (s)

Depth perturbotion (km) Oepth perturbotion (km)

Moho a t RSON - 81114 (s) Moho at RSON - 85121 (5)

5.00€+09 0.00E+00

2 - 1 00E+09

2 -?.OOE+09 L

0

v) m $ -5.00€+09

ul -20 -10 0 10 20 Depth perturbotion (km) Depth perturbotion (km)

Figure 24. S-coda relative residuals for Moho perturbation at RSON for four available events. Each panel shows three traces; dotted line is residual for P component, line with triangles indicates residual for S component, solid line is total residual. Each residual has a reference value subtracted.

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FIgnre 25. Moho inversion at RSNT using S-wave interaction phases for event 85121 (cf. Fig. 21). Note the three main phases S, SKS, ScS and associated Moho phases SmP, SKSmP, ScSmP, ScSPrW.

restricted to SmP, the average is reduced to 2.75 km, which the equivalents of SmP and SP6iP for S, ScS and SKS. is more compatible with the P-wave results of 3.7 km. Indeed, the predicted P component is fairly complex, and is

An example of a record with a multitude of boundary in general agreement with the data. The phases observed interaction phases present is 85121 (Fig. 25). The P are SmP, SKSmP, ScShP and ScSPfiP. The apparently component is predicted to have six S-related phases, namely shallower depth using all phases suggests either a

Moho at RSNT - 84111 (S) Moho ot RSNT - 83153 ( S )

D ; O.OE+OO

B -1.OEt09 TI

0 6 O.OE+OO

,$ -2.OE+09

20 -10 0 10 20 -20 -10 0 10 20 - S.OE+ 0%

Depth perturbation (km) Depth perturbation (km)

Moho at RSNT - 841 14 ( S ) Moho at RSNT - 85121 (5 )

I .

0 a o o E + o o ~ 1.OE+09

g -1.OEt09

.- .___ o -2.OEt09 $ -3.OE+09 $ -4.0€+09

-5.OE+09 Wl

-20 -10 0 10 20 -20 -10 0 10 20 Depth perturbation (km) Depth perturbation (km)

Figure 26. S-coda relative residuals for Moho perturbation at RSNT for four available events (cf. Fig. 24).

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Figure 27. Moho inversion at RSNY using S-wave interaction phases for event 83153 (cf. Fig. 21).

breakdown of Fermat's principle or a change in Poisson's ratio. Assuming that the P-coda results are dominated by us (through the S component phases) and the S-coda results by u p (through the P component phases), then the apparent change in depth of 3.8km would suggest an increase in Poisson's ratio of nearly 30 per cent from the reference value of 0.264.

Moho 0 1 RSNY - 83153 (5)

GOOE+OO

f -5 .GOE+08

rn -20 -10 0 I0 20

Depth perturbotion (km)

Moho nt RSNY - 8 5 1 2 1 (S)

- ; 2 0 E + 0 9 D YI O.OE+OO ' - 2 . O E + 0 9 f' -4.OE+09

$ -6 OE+09

-8.OE+09

D

-20 - I 0 0 I G 20

Depth perturbation (km)

ffigure 28. S-coda relative residuals for Moho perturbation at RSNY for South American events (cf. Fig. 24).

RSNY

For station RSNY, we have records from the two South American events, one of which is shown in Fig. 27. They both yield very simple minima (Fig. 28) and are relatively self-consistent, giving 6h = 0 km (83153) and 4 km (85121). However, they are significantly shallower than the P results, especially for these same events (Fig. 14). The S k P inversion yields 6h = -6 and -11 km, which are more consistent with the P-wave results (-7.5 km). The nature of this inconsistency is apparent. On the P component (Fig. 27) while the larger amplitude SPtEP is well fit, SmP is clearly misfit, so that the two phases are inconsistent for any assumed depth. If we tried to account for this depth discrepancy solely by adjusting Poisson's ratio, it would require an unreasonable increase of about 75 per cent. While some of this could represent a real change, much, if not most of this discrepancy must be due to the incorrect modelling of S P 6 P .

Other stations

The results for COL and RSSD appear to have similar problems with the phase SPtEP. For COL, we have results for the Kuril events and they yield high-signal seismograms (Fig. 29) and simple minima (Fig. 30). However, they give values of 6h that are consistently deeper than the P-wave results; -14 and -10 km for 84111 and 84114 respectively. As in previous cases, S m P and S P 6 P are not consistent, and the inversion weights the higher amplitude SPtEP more heavily. Nevertheless the fits to the waveforms are striking. This inconsistency could be resolved by an unreasonable decrease in Poisson's ratio and, as with RSNY, again argues

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120

M 0

-80

M 1)

0 -m

680 ,\690 S-syn a m n o t :

a w p * 'E " 4 "

,E iL t ;BE %

' " a 660 C5O a >: \: $ $ a n a :<$ xrn~ % )E a x cnEn a

" m u ) u)

\E v) 0

I

Figure 29. Moho inversion at COL using S-wave interaction phases for event 84111 (cf. Fig. 21).

for the inadequate modelling of SPriiP. This is most probable for this station, since the real Moho depth is probably 10 km away from the reference value, which would have a profound effect on the properties of this phase. The results using only SmP are +2 km and +11 km for 84111 and 84114 respectively.

Moho a t COC - 84111 (S)

- a

y1

2 I O E t 0 5

E 0 0 E t 0 0

0 -1.OE+05 I----',,,;, , , , & -2.OEc05 , v) ..

-20 -10 0 10 20

Depth perturbation (km)

Moho a t COL - 84114 (s)

- h % - T % - Y o Depth perturbation (krn)

Figure 30. S-coda relative residuals for Moho perturbation at COL for Sea of Okhotsk events (cf. Fig. 24).

Station RSSD for 85121 (Figs 31 and 32) yields 6h = 2 km, which is significantly shallower than the P-coda results. This is clearly the result of the dominant phase present in both seismograms. The interpretation of this seismogram is complicated by the presence of the phase SmPP which will bottom near the Moho. The P component is not being fit at all and the S component has interpreted this phase as SPrES. Again, we have inverted using SmP alone, giving a value of -12 km which is consistent with the P-wave results of -10 km for this station.

We have one very clean record for RSCP from 83153 which gives excellent results, (Figs 33 and 34), with a well-defined minimum at 6h = -14 km. The solution is dominated by the phase SPfiP, but a smaller SmP is also seen. Using SmP alone also yields 6h = -14 km suggesting that in this case these phases are not incompatible. The depth obtained is consistent with previous estimates in the area, which vary substantially, but suggest a value between 50 and 55 km (Owens et al. 1987; Prodehl, Schlittenhardt & Stewart 1984).

4.3 Linearized inversion for CNS~PI velocities

The results in the previous section suggest that there may be substantial variations in Poisson's ratio between the various stations. This has motivated an attempt to perform the linearized waveform inversion described above. We have restricted ourselves to consideration of the P-coda, since we have seen that the S-coda, particularly SPriiP, is not being modelled adequately. The smaller ray parameters of the P phases and the greater distance of their turning points from the Moho simplifies the analysis. The non-linear inversion

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Figure 31. Moho inversion at RSSD using S-wave interaction phases for event 85121 (cf. Fig. 21).

for 6h constitutes a starting point from which three parameters are inverted for further variations in 6h, and perturbations in mean crustal velocities 6u,, 6u,. For a constant ray parameter, there are effectively two pieces of information in the P-coda; namely the P and S traveltimes through the crust, primarily by the phases P P 6 P and P S 6 S . Thus, only two parameters can actually be constrained. The first is Poisson’s ratio u and the second is further variations in 6h, given a fixed value of 6u, or 6u,. This is enforced by taking only two eigenvalues in the generalized inverse.

We are primarily concerned here with the inversion for u as this has the potential for providing new and important information about variations in crustal properties at various sites. We first note that d In u l d In R = 3, where R is the ratio up/us . Thus, reasonable variations in Poisson’s ratio of 10-20 per cent correspond to changes in velocity of 3-7

O.OE+OO

$ - 4 0 E + I O v) -6.OE+10

-20 -10 0 10 20 Depth perturbaOon ( k r n )

Figure 32. S-coda relative residuals for Moho perturbation at RSSD for event 85121 (cf. Fig. 24).

per cent and are represented as variations in traveltime of about 1 s or less. This is equivalent to variations of 1-3 km in Moho depth. Because of the excellent consistency in Moho depth we have obtained for RSON and RSNT, less than 1 km for the standard error in the mean, we have focused on these stations for the linearized inversion. In addition. we have used those events with the highest signal-to-noise ratios and simplest wavelet functions: 84111, 83153, and 85121.

The results for two of the source receiver pairs 83153-RSNT, 83153-RSON are shown in Figs 35 and 36. The seismograms have been low-passed with a filter that is flat to 0.25Hz and then decreases from full response at 0.25 Hz to zero response at 1 Hz using a cosine taper. The starting values of 6h were the average values obtained from the non-linear inversion. The seismograms have been rotated into P and S components, and the P-wave has been removed. These figures show the P-wave coda for the initial model (top panel) and the final model after 20 iterations. Note in Fig. 35 that for RSNT, PmS and PPliiS and P S 6 S are all late with respect to the synthetic, whereas P P 6 P is nearly lined up or perhaps slightly early. The perturbed model has lined up most of the phases, although P S 6 S is still slightly misfit. The change in u is +27 per cent for RSNT. The phase PcP is predicted to have significant amplitude on the P component, but is not present in the data. This is an artifact of including both P and the much smaller PcP in the source wavelet. For RSON (Fig. 36) the initial fit is much better, although the S phases are slightly early (see in particular P A S ) and the resulting perturbation in u is much smaller, -4 per cent. The phase S6P is predicted to be the dominant arrival on the P component. It

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20 0

- .~ l P-syn - 3 m

810 820 84 0 850

20 0

S-data

-20 0 Loom

0

-1- S-syn 8

Figure 33. Moho inversion at RSCP using S-wave interaction phases for event 83153 (cf. Fig. 21).

does appear to be present in the data although late by about 1 s and reduced in amplitude.

The average u for the three events is + 19 and +5 per cent for RSNT and RSON respectively, from a reference value of 0.264 from PREM. This corresponds to u's of 0.277 and 0.314. The uncertainty in the mean estimates is about f5-10 per cent suggesting a real difference in u between these two stations, and a value from RSNT that is significantly greater than PREM.

5 CONCLUSIONS

In this study, we have presented a general strategy for the interpretation of body wave seismograms. We have shown that it may be applied to a wide variety of seismological problems, such as the large-scale 3-D structure of the mantle, the nature of seismic discontinuities, and the

I Moho a t RSCP - 83153 ( S )

-20 -10 0 10 20 Depth perturbation (km)

I

Figure 34. S-coda relative residuals for Moho perturbation at RSCP for event 83153 (cf. Fig. 24).

properties of localized layers. The technique provides us with maximal flexibility to manipulate the entire body wave portion of the seismogram, extracting information relating to any specified part of the Earth, or to a particular type of interaction. Our procedure enables us to utilize all the available information in a three-component broad-band seismogram, and is thus ideally suited to the detection of small signals, such as boundary interaction phases. It is also ideal for the analysis of seismograms from portable experiments, where deployment time is necessarily short and it is important to extract the maximum information from the recorded traces.

We have illustrated the method with one particular application; Moho depth and average crustal velocities. Our results for the Moho show conclusively that it is possible to determine receiver Moho depth using only a single three-component seismogram. The consistency between measurements is remarkable, implying a precision of single record estimates of 1-2 km. A comparison of results from the two stations RSNT and RSON demonstrates that a difference of 4 km in Moho depth is easily resolvable. As shown in Fig. 37, the values we obtain for the seven stations were we have at least one good record agree well with previous estimates of Moho depth. The Moho study most closely related to our own is that of Owens et nl. (1987), who used a receiver function analysis (Langston 1979) to invert the P-coda for crustal structure under the RSTN stations using a stack of some 50 events. By exploiting both the radial and vertical components, rather than using the vertical component to deconvolve the radial, we gain additional information, particularly the phase PPf iP which provides a strong constraint on the two-way P traveltime

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Figure 35. Linear inversion of P-coda (P-wave removed) for crustal parameters at station RSNT, for event 83153. Each panel shows data (lower trace for each component) and synthetic (upper trace), with all significant phases identified below. Low-pass filter applied, with linear decrease between 0.25 and 1.OHz. Upper panel shows comparison for initial model: lower panel shows comparison for final model after 20 iterations. Change in Poisson's ratio is +27 per cent.

through the crust. It is this constraint, when combined with information from the other phases, in particular PSriiS, that allows us to derive a value for Poisson's ratio. Only a few precise estimates of Poisson's ratio presently exist worldwide (Holbrook, Mooney & Christensen 1990), and the fact that this parameter represents perhaps the most important seismological constraint on crustal composition provides a strong motivation for obtaining such measurements.

We have shown results for one particular study, but more important than this specific application is the potential of our technique as a tool for future innovation. Through the

Figure 36. Linear inversion of P-coda for crustal parameters at station RSON, for event 83153 (cf. Fig. 35). Change in Poisson's ratio is -4 per cent.

insights it provides into the structure of the body wave seismogram, enabling us to unravel the complexities of propagation paths within the Earth, it provides a pathway for solving a wide range of seismological problems.

ACKNOWLEDGMENTS

Both authors acknowledge support from the National Science Foundation, through grant EAR-8708195. The first author acknowledges the support of a post-doctoral fellowship from the Carnegie Institution of Washington. The authors also wish to thank an anonymous reviewer for a careful review, and Guust Nolet for his comments on the manuscript, as well as Tanya George for her assistance in drafting the figures. The first author also wishes to thank Dr Salman Kazmi, without whose professional assistance this work would not have been possible.

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[I .- E 40

A

A

A

A 3

I I I I I I I I I I I I I I I 30 4 0 50 6 0

Est imated crustal depth (km)

Figure 37. Comparison of Moho depth values at seven stations from present study with previous estimates. Values for crustal thickness (km) are (with previous estimate and source in parentheses); RSSD: 50 (45; Steinhart & Meyer 1961), RSON: 40 (42; Owens et al. 1987), RSCP: 54 (50; Meissner 1986), RSNY: 48 (47; Owens et al. 1987), RSNT: 36 (38; Barr 1971), TOL: 32 (35; Meissner 1986), COL: 30 (30; Mooney, Beaudoin & Cotton 1989).

REFERENCES

Barr, K. G., 1971. Crustal refraction experiment: Yellowknife 1966, 1. geophys. Res., 76, 1929-1947.

Baumgardt, D. R. & Alexander, S. S., 1984. Structure of the mantle beneath Montana Lasa from analysis of long-period, mode-converted phases, Bull seism. SOC. Am. , 74, 1683-1702.

Bock, G. & Ha, J., 1984. Short-period S-P conversion in the mantle at a depth near 700 km, Geophys. J. R. astr. SOC., 77,593-615.

Bolt, B. A., 1970. PdP and PKiKP waves and diffracted PcP waves, Geophys. J . R. ask. SOC., 20,367-382.

Bolt, B. A., O'Neill, M. & Qamar, A, , 1968. Seismic waves near 110": is structure in the core or upper mantle responsible?, Geophys. J . R. astr. SOC., 16, 475-487.

Chapman, C. H., 1978. A new method for computing synthetic seismograms, Geophys. 1. R. astr. SOC., 54, 431-518.

Chapman, C. H. & Orcutt, J. A., 1985. Least-squares fitting of marine seismic refraction data, Geophys. 1. R. astr. SOC., 82, 339-374.

Clarke, T. J., 1984. Full reconstruction of a layered elastic medium from P-SV slant stack data, Geophys. J. R. astr. SOC., 78, 775-793.

Clarke, T. J. & Silver, P. G., 1986. The detection and characterization of discontinuities using estimates of body-wave vector displacement, EOS, Trans. Am. geophys. Un., 67, 1095.

Clarke, T. J. & Silver, P. G., 1988a. Body wave reflections and conversions from mantle discontinuities, EOS, Trans. Am. geophys. Un. , 69, 494a.

Clarke, T. J. & Silver, P. G., 1988b. Inversion of body wave phases and the characterization of mantle discontinuities, EOS, Trans. Am. geophys. Un., 69, 1333.

Dziewonski, A. M., 1984. Mapping the lower mantle: Determina- tion of lateral heterogeneity in P velocity up to degree and order 6, J. geophys. Res., 89, 5929-5952.

Dziewonski, A. M. & Gilbert, F., 1976. The effect of small aspherical perturbations on traveltimes and a re-examination of the correction for ellipticity, Geophys. 1. R. astr. SOC., 44,

Dziewonski, A. M. & Anderson, D. L., 1981. Preliminary Reference Earth Model (PREM), Phys. Earth planet. Inter.,

Faber, S. & Miiller, G., 1980. Sp phases from the transition zone between the upper and lower mantle, Bull. seism. SOC. Am. ,

Faber, S. & Miiller, G., 1984. Converted phases from the mantle transition zone observed at European stations, 1. Geophys., 54,

Hales, A. L., 1969. A seismic discontinuity in the lithosphere, Earth planet. Sci. Lett. ~ 7, 44-46.

Holbrook, W. S., Mooney, W. D. & Christensen, N. I. , 1990. The seismic velocity structure of the lower continental crust, in Physical Properties of the Lower Crust, ed. Fountain, D.M., Elsevier, Amsterdam, in press.

Husebye, E. S., Haddon, R. A. W. & King, D. W., 1977. Precursors to P'P' and upper mantle discontinuities, J. Geophys., 43, 535-543.

Kennett, B. L. N., 1983. Seismic Wave Propagation in Stratified Media, Cambridge University Press, Cambridge, UK.

Kennett, B.L.N. & Kerry, N.J., 1979. Seismic waves in a stratified half space, Geophys. J. R. astr. SOC., 57, 557-583.

Langston, C. A, , 1979. Structure under Mount Rainier, Washington, inferred from teleseismic body waves, J . geophys. Res., 84, 4749-4762.

Meissner, R., 1986. The Continental Crust, Academic Press, San Diego, CA.

Mooney, W. D., Beaudoin, B. C. & Cotton, J. A, , 1989. Crustal structure across the northwestern boundary of the Yukon- Tanana terrane, central Alaska: Results from TACT' 1987, EOS, Trans. Am. geophys. Un., 70, 1339.

Nakanishi, I . , 1986. Seismic reflections from the upper mantle discontinuities beneath the mid-Atlantic Ridge observed by a seismic array in Hokkaido Region, Japan, Geophys. Res. Lett.,

Nolet, G., van Trier, J. & Huisman, R., 1986. A formalism for nonlinear inversion of seismic surface waves, Geophys. Res. Lett., 13, 26-29.

Owens, T. J., Taylor, S. R. & Zandt, G., 1987. Crustal structure at Regional Seismic Test Network stations determined from inversion of broadband teleseismic P waveforms, Bull seism. SOC. Am. , 77,631-662.

Paulssen, H., 1988. Evidence for a sharp 670-km discontinuity as inferred from P to S converted waves, 1. geophys. Res., 93,

Prodehl, C., Schlittenhardt, J. & Stewart, S. W., 1984. Crustal structure of the Appalachian highlands in Tennessee, Tectonophysics, 109, 61-76.

Revenaugh, J. & Jordan, T. H., 1989. A study of mantle layering beneath the western Pacific, J. geophys. Res., 94, 5787-5813.

Sacks, I. S. & Snoke, J. A., 1977. The use of converted phases to infer the depth of the lithosphere-asthenosphere boundary beneath South America, J. geophys. Res., 82, 2011-2017.

Silver, P. G. & Jordan, T. H., 1983. Total moment spectra of fourteen large earthquakes, J. geophys. Res., 88, 3273-3293.

Silver, P. G. & Chan, W. W., 1986. Observations of body-wave multipathing: evidence for lower mantle slab penetration beneath the Sea of Okhotsk, 1. geophys. Res., 91,

Sobel, P., 1978. The phase P'dP' as a means for determining upper mantle structure, Thesis, Faculty of the Graduate School of the University of Minnesota.

Steinhart, J. S. & Meyer, R. P., 1961. Explosion studies of continental structure: Washington, D.C., Carnegie Institution of Washington Publication, vol. 622, p. 409, Washington, DC.

7-17.

25, 297-356.

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13 787-13 802.

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Vinnik, L. P., 1977. Detection of waves converted from P to SV in the mantle, Phys. Earth planet. Inter., 15, 39-45.

Vinnik, L. P., Avetisjan, R. A. & Mikhailova. N. G., 1983. Heterogeneities in the mantle transition zone from observations of P-to-SV converted waves. Phys. Earth planet. Inter..

Wajeman, N., 1990. Detection of underside P reflections at mantle discontinuities by stacking broad band data, J. geophys. Res., in press.

Whitcomb. J . H. & Anderson, D. L.. 1970. Reflection of P’P’ seismic waves from discontinuities in the mantle, J. geophys. Res., 75, 5713-5728.

Woodhouse, J . H. & Dziewonski, A. M.. 1984. Mapping the upper mantle: Three-dimensional modeling of Earth structure by inversion of seismic waveforms. J. geophys. Res.. 89, 5953-5986.

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33, 149-163.

APPENDIX A: THE F O R W A R D PROBLEM

Writing the wavefield expression (1) as

wo = W W u z u + WDxD), ( A l l

we seek to generate an expansion of each of the matrices W,, W, as a ray series, with each term consisting of a product of reflection and transmission matrices for portions of our model. Dividing the model into M smooth regions, separated by discontinuities, we order the matrices associated with reflection and transmission at the interfaces, together with propagation as upward-travelling, downward- travelling and turning rays through the smoothly varying layers, as the set of K matrices

By truncating the expressions for W,, W, at some order r, corresponding physically to limiting the amount of reverberation permitted within the structure, we obtain a finite series of N , rays, which we write as Wu, W,. The ith such ray will involve some subset of 5, say 5, = { S j k , k = 1, K , } , where the K, matrices will usually include some repetitions of elements of S. The required expansion for Wu may now be written as

and similarly for W,. The problem reduces to finding an appropriate repre-

sentation for such expressions, and the rules corresponding to their addition and multiplication. We represent W, by a 2-D integer array A, where each column of A represents one term in the sum, and consists of an ordered set of integers identifying the individual matrices, while the entire matrix represents a sum of such terms. Then addition corresponds to combining all columns from two matrices within a single matrix, and performing any necessary simplification, while multiplication may be taken term by term, with each term corresponding to a product of two matrix products, and hence represented by an operation in which the integers from the second matrix column are appended to the end of the column from the first matrix. As an example, consider multiplying together the expressions I + RU and I + R, to

obtain the result I + R, + R, + RuR,. Assuming the matrices I, Ru, R, are represented by the integers 1, i , j respectively, and denoting the operation corresponding to multiplication by @ , we obtain

1 i j i . . .

=(; 8 8 ; ;::). . . . . . . . . .

By making repeated use of these simple rules, we are able to generate representations for the truncated reverberation operators to some prescribed order, and ultimately for the wavefield (1). The fact that we are dealing with matrices that d o not commute actually simplifies the procedure, since we d o not need to gather together terms. The major limitation of such a scheme is computer storage, since the number of rays grows very rapidly with the number of model layers. Our method is relatively efficient in this respect, since each matrix product actually contains an entire family of physical rays, corresponding to different assignments of wave type for the various layers.

APPENDIX B: ESTIMATION OF T H E B R O A D - B A N D WAVELET

One of the essential steps in broad-band waveform modelling is the estimation of the wavelet function, w,(w), defined in (6) for the j th phase as

w j ( w ) = s,(w)aj(w), (B1)

where s,(w) is the source time function and a,(w) is the attenuation operator for a particular source-receiver path.

We describe here a stable time-domain procedure for estimating a single isolated wavelet w(t), although the generalizations to multiple phases on a single seismogram or to the simultaneous consideration of multiple seismograms through a source/attenuation model is straightforward. Estimating w(f) is really no more than seeking a stable form of deconvolution, since the observed wavelet can be written

where Z(t) is the instrument response. We expand w(t) in a set of basis functions &(t) defined over the time interval [O, TI as

In general we choose these basis functions t o be sines and cosines (Fourier series)

cos ( n n f / T ) , sin ( n n t / T ) , n = 0, 1, . . . , N .

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Substituting (B3) into (B2), we obtain

2N

= c %@At), n - 0

where @ , ( t ) = @,(I) * [ ( t ) . Since the unknown coefficients ui are linearly related to the seismogram, they may be obtained by solving a standard least-squares problem.

The two parameters that need to be specified are N and T. N can be chosen by calculating the squared misfit c2 for a

range of N and then choosing an optimum value after which there is no significant reduction 6'. The specification of an interval [0, T ] outside which the signal is identically zero serves to stabilize the low-frequency component of w(t ) , by imposing the constraint that the pulse is time-localized. This effectively provides an extrapolation of w ( t ) to frequencies below the pass band of the instrument. In determining an appropriate value for T we strike a balance between the requirement that it be large enough that the entire wavelet is adequately modelled, and the fact

5

4

20000 1 'I

6 12 18 (secl

0

40000

20000

-230W

40000 O k 0 6 12 18

20000 40000 :~~~~k 6 12 18

Isecl 0

20coo

" 5 12 18 n ( s e c )

2ilooc

40000

6 12 18 ( s e c l

0

Figure B1. Result of using successively larger numbers of basis functions [N increasing from 4 (bottom) to 8 (top)] in inverting for P-wave source function (event 85121 recorded at station RSON). First column shows source time function, second shows function convolved with instrument response. Third column shows observed P-wave for comparison.

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12 T. J . Clarke and P. G . Silver

that too large a value leads to a less compact representation of the pulse involving more basis functions.

As seen in Fig. 1 the procedure produces pulses that are one-sided and smoothly go to zero at the ends of the interval, as would be expected on physical grounds. This is not imposed as a constraint, but follows from the parametrization, since the presence of a step in the displacement pulse at the ends of the interval would lead to a large excursion in the predicted seismogram for most broad-band velocity sensors, and serve to increase the misfit. A useful feature of this procedure is that as N is increased, it is possible to easily associate features of the wavelet function with features in the observed seismogram. Fig. B1 shows a series of source functions (left), incorporating increasing numbers of basis functions from bottom to top, demonstrating that apparently subtle changes in the displacement wavelet can produce large changes in broad-band seismograms.

APPENDIX C: B O U N D A R Y PERTURBATION CONSTRAINTS

Consider the (distance independent) boundary perturbation, given by (25) as

where KZ, is the kernel for the perturbation in traveltime for the lth boundary and mth phase, and dl the perturbation coefficients. We seek a smooth model perturbation that, for the main phases, produces a traveltime perturbation 6Tm equal and opposite in sign,

6Tm = -6TZ,

so that these remain fixed. We consider a model of the form (16) whose traveltime perturbations are given by (18) as

6Tm = Kmlcl. I

Applying the constraint (C2) for the main phases, and using (C3) and (Cl), we obtain

C KmlCl = - C K.Z d, 1 (C4)

or in matrix form

KC= -KBd,

where K and KB are restricted to the subspace of main phases. Solving (C5) by generalized least squares, we obtain c as

c = -(K'K)+K'KBd, (C6) where t denotes the generalized inverse, and ' the transpose. Thus, the traveltime perturbation t o each phase, including the boundary interaction phases, due to the smooth model is

(C7) where K("I') spans the space of all phases. Then the change in traveltime for any phase due to both smooth and

6T = K("ll)c = -K("II)(K'K)tK'KBd

boundary perturbations is

6T, + 6T = [K""'" - K("")(K'K)'K'KH]d. (C8) Thus for each phase (C8) gives the change in traveltime for the boundary perturbation d, subject to the constraint that the main phase traveltimes remain fixed.

A P P E N D I X D: N A M I N G CONVENTIONS

With the large number of phases comprising a CORE seismogram, it is essential to have a compact naming convention that is unique and consistent with present conventions for main phases. First, as is standard, P- and S-waves leaving downward from the source are written P , S, while those leaving upward are designated p and s. Lower case is not used for any other purpose. There are four possible interactions with a boundary; topside reflection ("), bottomside reflection ( A ) , downward transmission ( '), and an upward transmission ( '). The boundary in question is specified by name: (m) for Moho, (c) for core-mantle boundary, (4) for 400km discontinuity, (6) for 650km discontinuity, (i) for the inner-core-outer-core boundary, and the interaction by means of the accent. Examples are:

E: upward reflection from CMB, m: upward transmission through Moho.

For each interaction, there are four possible wave type combinations. In the case of transmissions, only changes in wave type are explicitly declared. In accordance with present convention, reflections at the free surface are implied (boundary not named and downward reflection assumed). Examples are:

PmS: P to S conversion at the Moho, P P M : direct P followed by a P reflection at the surface,

S upward reflection and conversion at the Moho, p i s : P upward from the source, bottomside reflection and

conversion from the 400 km discontinuity, P4S: P downward from the source, downward trans-

mission/conversion across the 400 km discontinuity, sPriiPP: S upward from the source, conversion t o P at the

surface, upward P reflection at the Moho, P reflection at the surface,

StPCS: SKS. The correspondence with the standard convention is complete if (c) alone is understood to mean (E), ( K ) means (tPc'), (KI'K) means (?PrP'), (KZK) means (tPiPiPck), ( K J K ) means ( t P i S h ) and so on.

While such a naming scheme is very general and especially compact for the consideration of first-order interactions, there is still an ambiguity because transmissions of the same phase type across boundaries are not noted explicitly. This becomes a problem in the case of triplicated arrivals where the same name can be assigned to a wave that bottoms above and below a given boundary. This same ambiguity exists in the standard naming convention. One simple solution is to include transmission interactions only in order to remove the ambiguity. For example, the P-waves triplicated by the 400 km discontinuity would be written as P (bottoming above the boundary) PdP (reflecting from the boundary) and P i P 4 P (bottoming below the boundary).

Dow



A Procedure For the Systematic Interpretation of Body Wave

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