seismic wave propagation in complex crust-upper mantle media using 2-d finite-difference synthetics

Geophys. J . Int. (1994) 118, 643-670

Seismic wave propagation in complex crust-upper mantle media using 2-D finite-difference synthetics

S. 0. Hestholm,' E. S. Husebye2 and B. 0. Ruud2 ' IBM Bergen Environmental Sciences and Solutions Centre, Thorm@hlensgate 55, 5008 Bergen, Norway 'Institute of Solid Earth Physics, Bergen University, Allkgt. 41, 5007 Bergen, Norway

Accepted 1994 February 1 . Received 1994 January 14; in original form 1993 April 7

SUMMARY A description is given of the numerical FD scheme used to solve the elastic wave equation, including a few remarks on the source functions used. Our FD method has been used for computing synthetic seismograms for 2-D crust/upper mantle models of size 150 x 400 km2, with options for free-surface topography. The strategy was to introduce successively more complex lithosphere models for generating the synthetics; the reference model was laterally homogeneous lithosphere. The interface scattering was visualized through displays of free-surface synthetic waveforms and snapshots for models with a corrugated Moho only and free surface topography only. Near the free surface the latter seems to dominate, in the form of P-to-Rg and S-to-Rg conversions. Lithosphere randomizations were introduced through von KBrmin functions of order Y = 0.3, with rms velocity fluctuations of 3-4 per cent and correlation distances (horizontal and vertical) at 2.5 or 10 km. In case of a medium with only sub-Moho heterogeneities, those with horizontal anisotropy (ax = 10 km; a, = 2.5 km) produced relatively strong Pn and Sn phases. The respective codas were dominated as in most of our experiments by P-to-S and S-to-S scattering wavelets excluding Rg scattering at a free surface with topography. For a medium with crustal heterogeneities, the distortions of the P and S wave trains with distance were clearly demonstrated. For full-scale heterogeneous lithosphere models, characteristic features of the synthetics were quantitatively similar to observational records of local events. Dominant attributes were a pronounced P coda consisting mainly of P- and Rg-scattered wavelets, and a relatively strong S coda consisting mainly of P-to-S and S-to-S scattered wavelets. The P and S waveforms are severely distorted pointing at the futility of reliably picking many secondary arrivals in local event recordings. Most of the scattering wavelets are confined to the crustal waveguide and to surface waves, since coda excitations for sensors at a depth of 100km were weak and, moreover, consisted mainly of S wavelets. This implies that a strong teleseismic P coda does not reflect scattering within the crust in the source region but, rather, a complex source. Observational results from analysis of NORESS and ARCESS local event recordings are also presented. Clearly the lithosphere is not isotropically inhomogeneous. The essence of our 2-D FD synthetic seismogram experiments is that a simple lithosphere model, being moderately heterogeneous, gives rise to complex seismograms which are grossly similar to the observational recordings. In contrast, complex models derived from profiling surveys (but lacking the fine-scale random variations) give simple, 'ray tracing' like synthetics, not necessarily similar to the observed records.

Key words: coda excitation, 2-D finite difference, deformation of body-wave phases, elastic wave equation, free-surface topography, lithosphere models, P-to-Rg and S-to-Rg scattering, synthetics, von KfirmBn randomization.

643

644 S. 0. Hestholm, E. S. Husebye and B. 0. Ruud

1 INTRODUCTION

The understanding of high-frequency (>1 Hz) seismic wave propagation in the lithosphere structural system remains a

1991 12271 3:27: 0.0 NORESS AQ SP Azi: 207.0

l ' ! ' : ' ! ' ! ' ! ' : ' i ' ; ' l

13864 _____lk R

20848 T

12910

t v : r : t ; , ' T i l : ~ : . : , l

0 10 20 30 50 60 70 80 90 Time ( s i

Figure 1. Three-component NORESS recordings of an explosion in the coastal waters of southern Norway. Distance and azimuth are respectively 220 km and 207". Maximum amplitudes (counts) are given to the left of each trace. Note the strong S g / L g arrival in the transverse component, representing P-to-S conversion close to the source. A primary goal of our study is to generate synthetics where the coda waves quantitatively match those in local seismic records.

20 -

15 .-

A

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problem to seismologists and exploration geophysicists alike. The observational manifestations are easy to illustrate: at local distance ranges both P and S waves are often accompanied by prolonged coda waves which are not accounted for by standard structural models (e.g. see Figs 1 and 2). Likewise, array recordings of teleseismic events exhibit P-wave time and amplitude anomalies, even between sensors spaced only 1-2km apart (Haddon & Husebye 1978). Complex layered models in combination with wave propagation schemes tied to ray tracing, Gaussian beams, WKBJ, perturbation methods, etc., cannot realisti- cally model these observational features. The rapid P-wave time and amplitude fluctuations across the large arrays NORSAR (south-east Norway) and LASA (Montana) are reminiscent of scattering. The first successful attempts at seismological modelling of such phenomena, using the first-order (acoustic) scattering theory of Chernov (1960), were Aki (1973) (see also Berteussen et al. 1975, Flatte & Wu 1988, among others). In this context, the scattering medium is described statistically in terms of first- and second-order moments and correlation distance. Naturally, in the solid earth, scatterers must have deterministic locations, as elegantly demonstrated by Aki, Christoffersson & Husebye (1977), using their novel tomographic technique in analysis of array P-time residuals from the NORSAR array.

Efforts to exploit the information potential of the coda waves from small, local earthquakes have met with relatively little success in terms of locating specific scattering

~ ~~~~

50 100 150 200 250 300 350 DISTANCE IN KM

NORSAR 03C 01A 06C 02C 04C

400 450 500 550

Figure 2. Main-line record section of the CANOBE profile. Amplitudes are multiplied by distance and the seismograms are filtered between 0.2 and 1.5 Hz. The records beyond 450 km were recorded by the NORSAR array and have been passed through a 4.75 Hz low-pass filter. Dots indicate first-arrival picks. Reproduced from Cassell et al. (1983).

Sehmic- wave propagation 645

sources. On the other hand, Aki & Chouet (1975) demonstrated that coda duration, decay rates and scattering attenuation can be modelled in terms of a single back-scattering theory for weakly inhomogeneous media in which the Born approximation is valid. Further elaborations on the single scatter theory come from the work of Wu & Aki (1985) and Wu (1985,1988), who split the scattering contributions in two parts, namely, a back-scattering term stemming from impedance perturbations and a forward- scattering term stemming from velocity perturbations. Dainty & Toksoz (1977) considered similar problems using diffusion theory.

The basic problem in most studies of wave propagation in complex media is that scattering theory incorporating the Born approximation is not necessarily valid, while convenient analytical solutions of the elastodynamic wave equation are seldom at hand. It is advantageous to use numerical solutions of the wave equation (e.g. Alterman & Karal 1968). Finite-difference solutions can handle complex media and at the same time properly account for multiple scattering effects, although computer requirements still limit the practical usage of such modelling tools. Most of the works published so far appear to be aimed at testing the validity of the numerical codes used, exploring the practical applicability of the Born approximation in scattering studies and recently investigating the range of ‘scattering’ functions appropriate for crust/lithosphere medium descriptions. Works of relevance here are Macaskill & Ewart (1984), Frankel & Clayton (1986), McLaughlin & Anderson (1987), Dougherty & Stephen (1988), Nikolaev (1989), Toksoz, Dainty & Charette (1991), Sat0 (1991) and Levander & Holliger (1992). An excellent review work on coda waves and scattering is given by Herraiz & Espinosa (1987).

In this study we concentrate on generating FD synthetics with basis in the elastodynamic wave equation, and limit ourselves to 2-D lithosphere models. We present briefly the velocity-stress formulation used in the numerical solution of the wave equation, source functions and inhomogeneous media representation. Scattering is of importance where medium properties are non-uniform within the lithosphere (Flatt6 & Wu 1988), and hence the resulting synthetics are discussed in terms of characteristic features of local event recordings (A < 300 km).

2 THE ELASTODYNAMIC WAVE EQUATION: NUMERICAL SOLUTIONS

The basic equations governing wave propagation in a continuous elastic medium are the momentum conservation and the stress-strain relation. Using the velocity-stress formulation of Achenbach (1984), we have:

(1)

a d a -ujj=A-uI+2p-vj j , l = l , . . . ,J, at 3x1 dXj

where Einstein’s summation convention is used. J is the dimensionality of the problem, p is density, and A and p are

Lam& parameters. f;. are body forces, and uj and ujl are particle velocities and stresses, respectively.

Spatial partial differentiation is achieved through cost- optimized, dispersion-bounded, high-order finite-difference operators on a staggered grid. For time stepping, a leap-frog technique is used. For the numerical dispersion relations, the stability limit and bandwidth introduced by the discretization, the reader is referred to Sguazzero, Kindelan & Kame1 (1989). The essence of their scheme is presented in the Appendix. To our knowledge, this is the first time this spatial differentiation technique has been used in large-scale seismic wave modelling experiments.

2.1 Absorbing boundary conditions

To avoid artificial reflections from the sides and bottom of the grid, absorbing boundary conditions are applied here. Along these edges the velocities and stresses are multiplied by exponentially decreasing terms. This procedure gives satisfying results for our 2-D synthetics. An alternative method is to introduce specific non-reflecting operators for each boundary segment (Higdon 1990, 1991). We experimented with both of these methods, and that of exponential damping proved to be the most efficient one.

2.2 Free-surface boundary conditions and their discretization

On the top surface, we use the vanishing stress condition for a free boundary

n . T = O .

Here n is the outward normal unit vector of the surface and T is the stress tensor, In 2-D, or in the xz-plane, this leads to a closed set of boundary conditions fQr the velocity terms at the free surface, namely:

dv, dv, - dz dx

and

(4)

This gives the velocities at the free surface. In our application a cost-optimized, dispersion-bounded

eigth-order operator of length 8 is chosen as spatial differentiator. It corresponds to a sampling frequency of three grid points per minimum wavelength, and it has a relative error bound of 1.9 per cent for the group velocity. Because of the length of the operator, other schemes have to be used near all boundaries. Second-order staggered finite differences are therefore employed at all four outermost points of the domain.

To discretize the conditions (4) at the free surface, we use second-order, staggered finite-difference operators. Below the surface the operator order gradually increases with depth, i.e. from second to fourth and sixth order, and these central, uniform operators (Fornberg 1988) are employed on a staggered grid (Levander 1988). In this way, a high level of accuracy is maintained for surface waves (Rg) , with phase velocities in the range 2.5-3.5 km s-’ for periods 0.5-4.0 s


(Ruud, Husebye & Hestholm 1993). Free-surface topography has been incorporated by adapting the method of Tessmer, Kosloff & Behle (1992) to our finite-difference scheme. Details on the theory and implementation of this technique is given in a separate paper (Hestholm & Ruud 1994).

2.3 Source representations

In the modelling examples given in this paper, the seismic fields were excited through the use of body force terms in the numerical scheme (see eq. 1). Two types of sources were used. One was an explosion source (pure divergence) which generates only P waves, and the other was a rotation source (pure curl) which generates only S waves. The time history of the force terms was a first-order derivative of a Gaussian function, and the centre frequency of the waves generated was 2.5 Hz.

3 REPRESENTING SCATTERING HETEROGENEITIES IN THE LITHOSPHERE

Traditionally, the earth is often modelled as a simple stratified medium, each of the stratum having constant physical properties. Seismograms from these models tend to match the gross features of observational records but lack the variations in amplitude and traveltime and coda waves accompanying major arrivals. These features are symptoma- tic of scattering and depend on the size, distribution and magnitude of the heterogeneities in the lithosphere. In the literature of wave scattering, heterogeneous media are commonly described in terms of a few physical parameters like thickness of the scattering layer, heterogeneity fluctuation (rms values) and heterogeneity correlation distance a (Herraiz & Espinosa 1987). In case of anisotropic correlation functions we use a, and a, to denote the hprizontal and vertical correlation distances, respectively,

with axla, being the aspect ratio. Like other researchers, we use the term ‘spatial anisotropy’ to signify such a medium, although isotropic stress-strain relations are retained. Heterogeneous media realizations are often represented by random fields where complexities are expressed in terms of a few parameters like a and rms. In this context, the Gaussian, the exponential and von Kirmin correlation functions have been much used by seismologists studying scattering phenomena. The latter appears most suitable for describing lithospheric heterogeneities (Frankel & Clayton 1986; Toksoz et al. 1988; Charrette 1991). By varying the rms, the correlation distances a, and a, and the order of the von Kirmin function v, we may generate several medium realizations (see the model description in Table 1). The parameter v controls the medium power spectrum roll-off with wavenumber (k) for ka > 1. For v = 0.5 the correlation function simplifies to an exponential one, while for v = 0.0 the medium becomes self-similar for ka > 1. ‘Self-similar’ means that the standard deviation of the medium, calculated over equal logarithmic intervals of wavenumber, remains constant over a range of scale lengths (Frankel & Clayton 1986). FlattC & Wu (1988), in their 3-D scattering analysis of NORSAR P waves, used a band-limited power law function similar to that of the exponential function for ka > 1.

In other contexts, alternating high- and low-velocity layering has been introduced in the lower crust in order to conceptionally simulate lamina in seismic reflection records (Sandmeier & Wenzel 1986). More recently, velocity perturbations have been generated using isotropic fractal spatial distributions (e.g. see Levander & Gibson 1991; Charrette 1991). An uncomfortable result in this regard according to Levander & Holliger (1992) is that different lower-crust velocity models can match the same wide-angle data equally well. A recent approach is to mimic the real earth on the basis of geological observations as demonstrated by Holliger & Levander (1992). Their starting

Table 1. Listing of crust and mantle lithosphere parameters, supplementary to those in Fig. 3, for various models used in the analysis. The free surface, Moho and crust/upper mantle columns give 1-D and 2-D von Karman medium parameters, namely the order v , the correlation distance a (for spatial anisotropy a, and a,) and the rms level (km) and velocity (per cent) perturbations. The rightmost column gives the figure numbers where the corresponding synthetics are displayed.

IZI Mod 9

Seismic-wave propagation 647

point was the Ivrea Zone (northern Italy) where the lower crust is presumed to be exposed. From digitization of two geological maps, they obtained a fractal dimension estimate of 2.7 (corresponding to Y = 0.3), an aspect ratio of 4 and a horizontal correlation distance of 500-1000 m for different models (Holliger, Carbonell & Levander 1992).

Due to excessive computing requirements, most finite- difference synthetics are confined to 2-D medium models with the number of grid points seldom exceeding lo6. With signal frequencies up to about 30Hz, as used in reflection profiling, the areal extent of the 2-D model is limited to about 500 km2. The lapse time of interest seldom exceeds 10s. In contrast, we are interested in mimicking seismic-wave propagation in the crustal waveguide with frequencies up to about 5Hz, hence for our synthetics we use models with an areal extent of up to 60000 km2 (150x400) and lapse times of up to 90s. This also motivated the use of the memory and cost-efficient ‘higher-order staggered grid’ scheme introduced in the previous section and with details in the Appendix. In our synthetic experiments we aim at a better understanding of the type of media required to produce the strong coda seen in the real seismograms and also whether secondary phases, usually explained by simple or multiple reflections within the crust, can be adequately described by simple ray paths.

4 SYNTHESIZING W A V E PROPAGATION IN THE CRUST A N D LITHOSPHERE

The problem at hand is perhaps most easily illustrated through display of real seismic records (Figs 1 and 2). Fig. 1 shows three-component NORESS recordings of an explosion in the sea 220 km south of the array. The surprising feature here is that the dominant wavefield amplitude is associated with shear waves (Lg phase) on the transverse component, reflecting P-to-S conversions close to the source. In Fig. 2 a refraction profiling section CANOBE (Cassell et al. 1983) is shown. The coda dominates the wavefield after the PmP-phase arrival (PmP is a P wave reflected from the Moho). We consider it speculative to attempt to identify specific arrivals within the P coda.

4.1 Models considered

The basic model parameters are a crustal thickness of 35 km, and with a constant velocity gradient both above and below the Moho, as illustrated in Fig. 3. P and S velocities are related through the Poisson’s ratio of 0.25, and model densities are calculated from the P velocities via Birch’s (1961) law. In our calculations of synthetics, the above basic model has been perturbed in various ways with details for specific models given in Table 1; two ‘visual’ examples are given in Fig. 4. The free surface is either flat or given an artificial topography based on a 1-D von KBrmzin realization. The topography is restricted to a maximum 200 m rms to ensure stable free-surface boundary conditions in the finite-difference schemes (Hestholm & Ruud 1994). Also Moho can be made corrugated based on a 1-D von KBrm6n realization.

Our model (Fig. 3) has an upper crust low-velocity layer (LVL), since such a feature appears to be common in continental areas (Ruud et al. 1993). The uppermost 400m

are always homogeneous (unperturbed) in order to avoid numerical instabilities close to the free surface. In most models the LVL is homogeneous but it can have velocity perturbations similar to those of the other parts of the lithosphere. The LVL’s lower boundary is flat except in some models used for Rg synthetics. The crust and/or the underlying mantle are either homogenous or inhomogeneous. For transforming a homogeneous medium into an inhomogeneous medium, 2-D velocity perturbations are introduced through a 2-D von KzirmBn realization of order 0.3. As mentioned, the S velocity and density perturbations are tied to those of the P velocity parameter. For models with velocity perturbations, we have usually incorporated horizontal spatial anisotropy. Specific model parameters are listed in Table 1.

5 SEISMIC RESPONSES OF CRUST A N D U P P E R MANTLE HETEROGENEITIES

The crust/lithosphere structural system presented in the previous section can be considered to generate two sorts of scattering: (i) interface/surface scattering tied to corrugations of the two major impedance boundaries, the free surface and the Moho; and (ii) volume scattering tied to random small-scale velocity perturbations throughout the crust and upper mantle. Below we attempt to isolate and illustrate the effects various model perturbations (Fig. 3) have on the crustal wavefield.

5.1 Reference model synthetics

As a guide for judging the extent of scattering dontributions and P- and S-wave train deformations stemming from various kinds of complex media, we have computed synthetics for simple, homogeneous re’ference models for P and S sources, respectively. The Model 3 (Table 1) synthetics are for a source depth of 20 km and are shown in Figs 5 and 6. These synthetics, equivalent to those produced by ray tracing in homogeneous media, are clean and simple in contrast to real records (Figs 1 and 2). In Fig. 7 semblance plots for EO-sensor group synthetics in Fig. 5 are shown; scattering is not apparent either for the P or the S source. This also implies that specific, secondary-phase arrivals are easily identified (appropriate notations are included in the figures).

5.2 The effect of the top crust low-velocity layer (LVL)

The observational evidence for the LVL stems mainly from analysis of fundamental mode Rayleigh (Rg) waves (Ruud et al. 1993). In Fig. 8 we present Rg synthetics for upper crustal models (flat surface) with increasingly complexities of the LVL medium specifications (Fig. 8c). In most cases Rg signal frequencies above 2 Hz are wiped out by structural heterogeneities and/or masked by P-to-S scattering wavelets. Observationally, topography appears to be important in P-to-Rg scattering both in forward and backward modes, as convincingly demonstrated by Bannis- ter, Husebye & Ruud (1990) and Gupta, Lynnes & Wagner (1993). The relative efficiency of P-to-Rg and S-to-Rg conversions naturally depends on the geometry and the relief of the ‘scattering hills’.

648 S. 0. Hestholm, E. S . Husebye and B. 0. Ruud

FREE SURFACE

Is FLAT OR ID von K. (V = .5; o = .I km, a = 5 km)

v = .3; a = 3%; ax = az = 1

ax = 10 km, az = 2.5 krn;

ax = 2.5 km; az = 10 krn [

-- 35km Moho-

D von K (v = .5; o = 1 km. a = 1

ez W n n 2 150km

BLOCK SIZE: 150 x 400 km2

SIP SOURCES AT: x = 75 km, z = 28420 km

GRID SAMPLING =.2 km

CLUSTERS OF 6 SENSORS (UR-.8kn

z = 0; x = 165, 205, 245, 285 & 325

z = 100; x = 85, 125, 165, 205 & 24

(165) (205) (245) (285) (325 km) A 0 60 C,O DO EO

(125) (165) (205) (245 km)

Figure 3. The class of crust and upper-mantle models used in our generation of 2-D finite-difference (FD) synthetic seismic records. Specific model choices, exclusive layer thicknesses and crust and upper-mantle velocity gradients are detailed in Table 1 . Both vertical ( Z ) and horizontal ( R ) component seismograms are extracted at five groups of six sensors (group interval 40 km, sensor spacing 0.8 km) each at the free surface and at a depth of 100 km. The source-receiver geometry is shown in the lowermost box.

5.3 Interface scattering: Moho corrugations and free- surface topography

The synthetics for Model 4 with a corrugated Moho but otherwise identical to the reference model are shown in Fig. 9 and the corresponding EO-sensor group semblance plot in Fig. 10. Regarding Moho corrugations, the corresponding scattering effects are relatively insignificant, although more pronounced for P waves. Comparing Figs 7 and 10 (upper parts, P sources) we see that for the corrugated Moho model the P coda is dominated by S waves generated by asymmetric P-to-S reflection/conversions, in agreement with the results of Paul & Campillo (1988). The amplitude of the converted waves are, however, small compared to the P phases (and, as we will show later, compared to waves scattered by medium heterogeneities) so the effects of a corrugated Moho are unlikely to be easily Observable. Similarly, the synthetics for a homogeneous medium with

topography (Model 5) are shown in Fig. 11 in terms of snapshot displays. The P-to-Rg conversions are observable in the figure (a) as a thin, alternating dark and light band in the upper 1-2 km of the model. The amplitude of the Rg wavelets are relatively weak, amounting to about one-tenth of that of the direct P phase. Observationally, the existence of the Rg wavelets is easily demonstrated using array recordings as shown in Fig. 23 (see also Bannister et a1

The free-surface topography also effectively scatters Rayleigh (Rg) waves through conversion to body waves (Ruud et al. 1993). In this way rough topography seems to be rksponsible for strong apparent attenuation of Rg waves, so such recordings are seldom reported beyond epicentral distances of 100 km for stations/arrays in hilly areas like NORESS (south-east Norway) and GERESS (south-east Germany). Another observable effect is that waveforms of major phases are slightly modified due to interference from

1990).

Seismic- wave propagation 649

P-velocity (km/s)

Figure 4. P-velocity fluctuations in the crust and upper mantle for a 40 x 100 km2 subset of the Fig. 3 basic model. (a) A realization of an isotropic medium with correlation distance 10 km (Model 1 in Table 1). (b) A realization of a medium with so-called spatial anisotropy with correlation distances of 10 km horizontally and 2.5 km vertically (Model 2 in Table 1). Common model features are a top crust low-velocity layer (the black rim in the figures) and a corrugated Moho of enhanced visibility due to the velocity jump across the interface.

free-surface reflected and converted wavelets. For example, free-surface topography, of the order modelled here, leaves across NORESS the range of anomalies in apparent the crustal guided body waves almost undistorted (Fig. 11). velocities of Pn and Pg phases as estimated from incidence angles on three-component instruments, is about f l km s - ’ . We were able to account for about half of this velocity bias 5.4 Crust and upper heterogeneities

using a model with a 2-D NORESS topography scaledhown A major drawback with the above interface scattering by a factor of two. Although the above-mentioned effects models is their inability to excite coda at levels comparable can be quite strong near the surface where Rg waves to real recordings. The best way to judge the effect of propagate (and where most seismometers are located), heterogeneous structures is to compare synthetics for

650 S. 0. Hestholm, E . S. Husebye and B. 0. Ruud

(a) P-source 1 ' : ' : ' : ' : ' : ' : ' ; ' : ' t

724 1 DO

pPmP .I. .. EO R

a32

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4560 co

Sn 7 . . . EO

S R , , . . 2961

Figure 5. (a) The P-source and (b) S-source synthetic seismograms (Z components) for a laterally homogeneous medium, i.e. Model 3 in Table 1. Common features are a source depth of 20 km and that only the first trace from each of the five free-surface sensor clusters $re shown (details in the lower box in Fig. 3). These synthetics, including phase designations, are similar to those generated by ray tracing techniques-sharp and clear P and S phases without coda waves. Adding parts (a) and (b) would give synthetics similar to those stemming from an earthquake source (except for radiation pattern). The synthetics in this figure are used as a reference for comparing synthetics generated for various types of inhomogeneous media, shown in Figs 9, 12, 14 and 16.

progressively more complex models, starting from the simplest. This is demonstrated in Figs 12 (traces) and 13 (semblance) where the synthetics are computed for Model 6 (Table 1) with an inhomogeneous mantle only. Correspond- ing results for Model 7, or an inhomogeneous crust only, are shown in Figs 14 and 15. Finally, synthetics and semblance results for a model with both crust and upper mantle inhomogeneities (Model 8) are shown in Figs 16 and 17.

For the synthetics in Fig. 12 (upper mantle inhomogeneities), the conventional crustal phases are little deformed and remain similar to those in Fig. 5. However, the Pn phase has become relatively strong and most of the energy appears to be delayed 1.0-1.5 s although the first weak arrival begins at about the same time as for the unperturbed model. The relatively strong amplitude of the Pn arrival is taken to reflect that heterogeneities in the

(a) P-source I ' !, ' : ' I ' : ' I ' I ' : ' : ' i

FO

631 JO

1537 HO

k 10 I

1405

1289 JO

I t r ' , ; , ; I ; , ; ! ' I I , : s t o 1'0 20 30 40 50 do 70 a0 90

Time ( 5 )

Figure 6. The (a) P-source and (b) S-source synthetic seismograms (2 components) for a laterally homogeneous medium, i.e. Model 3 in Table 1 . The traces displayed are from the first sensors in each of the five clusters at a depth of 100 km (source-recgiver geometry in Fig. 3). Caption otherwise as for Fig. 5 . These synthetics, including phase designations, serve as a reference for those in Fig. 18.

upper mantle effectively scatter Pn energy, i.e. head waves and diving waves propagating mainly along and just below the Moho (Kennett 1989, 1994). up into the crust. A prominent feature is the coda excitation. Semblance analysis of the P source coda at the EO-sensor cluster (Fig. 13a) implies that the coda as shown almost exclusively consists of P-to-S reflected wavelets from the sub-Moho heterogeneities. The relatively large amplitudes of these wavelets may indicate that our rms velocity fluctuation of 4 per cent is too large. On the other hand, FlattC & Wu (1988), using NORSAR data, obtained a rms velocity estimate of 2 per cent, which may in part reflect the relatively low signal frequencies typical of the teleseismic P waves used in their analysis.

In Fig. 14 we display synthetics for a model where only the crust is inhomogeneous. In comparison to Fig. 5, the Pn phase is essentially intact, while the conventional crustal phases have been strongly distorted. Physically, we take this smearing to reflect the cumulative effect of wave propagation through an inhomogeneous medium. Regarding P-source coda excitation (Fig. 15a), it is dominated by P-to-S converted wavelets in a forward-scattering mode for a P source.


Semblance

n m O 0 U

-20

a 12.5 n v)

Y

‘E 10.0

v

Y % 7.5 .I

0 0 5 5.0 >

2.5

0.0 0.2 0.4 0.6 0.8 1.0

30 40 50 60 70 80 90 Time (s)

Figure 7. Semblance analysis of the P and S synthetics (Model 3) shown in Fig. 5. The amplitudes are rms values in dB relative to maximum and averaged over the six sensors in the E-cluster (distance 250 km from the source) for a time window of I .5 s and updating every 0.5 s. Intersensor spacing is 0.8 km (Fig. 3). The semblance values are also averaged over a 1.5 s window and updated every 0.5 s. For the P-source synthetics (a) there are no P-to-S conversions, while in (b) much of the S-wave energy converts to P by reflection at the free surface and at the Moho discontinuity. Since the model is laterally homogeneous, there is no scattering.


Syntli model

1 2 3

5 6

, 4

(4 ' " " 1 " " : " " ~ " " ~ " " ~ " " ' 14070 - j v ' ' v p

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Baric model pi Ira p2

(km/s) (km) ( W s ) 3.55 - 2.60 1.4 3.55 2.82 1.4 3.55 2.82 1.4 3.55 2.82 1.4 3.55 2 82 1 . 4 3.55

5254 - MOD.6

yon Ki rmin medium Y a rms

(km) . . - - - -

0.5 10 O. lkm 0.0 5 4% 0.7 10 005kiri

0 ' I " 5 ' ''1'0' ' ' '1'5' 2'0' i5' d0 T i m e ' 5 )

(b) I " " ! " " ! " " ! " " : " " : ' " "

5688 h -MOD. 2

5483 MOD. 1

8676 3

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I , . ! . : . . ' . . , . ' . . , . I . . . . ; . . . . I 0 5 ' 1'0 1'5 20 25 30

Time (s) Filtered 0 -1.5 Hz

ICl

Model type - hspace gr.layer LVL LV L LV L LV L

Model dist.

none none none

CI RM

TOPO

-

Figure 8. (a) Synthetic seismograms computed by 2-D finite- difference elastic wave modelling for the six models detailed in (c). In all cases the P source is located at 60 km distance and 2 km depth. The labels to the right of each trace refer to the model numbers. In (b) the same traces are shown lowpass filtered with a cut-off frequency of 1.5 Hz. By comparing models 3 , 4 and 5 we see that the low-frequency Rg waves are almost unaffected by the random variations. The effect of surface topography (Model 6) is more significant. In (c), CI means corrugated interface generated by 1-D randomization of layer thickness, RM means 2-D velocity perturbations in the low-velocity layer (LVL) and TOPO means free-surface topography; 'gr.' means linear increasing velocity in LVL, and tied to the model shear velocities. As before, a Poisson ratio of 0.25 is assumed and the Birch (1%1) velocity-density relation is used. Reproduced from Ruud er al. (1993).

In Fig. 16(a), P synthetics are shown for a model where both the crust and upper mantle are inhomogeneous. To us these synthetics are encouraging in the sense that their essential features are similar to those frequently observed in real recordings. These are a relatively weak Pn followed by a stronger PglPmP, and then a coda of long duration with amplitudes similar to those of the Pn phase. The corresponding S-source synthetics are displayed in Fig. 16(b). The two sets of synthetics demonstrate that both P

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and S waves respond similarly to random medium perturbations. For a heterogeneous crust (Fig. 14b) the mentioned wave-smearing effects appear to be more pronounced for S waves than for P waves. For this model with perturbed velocities in both the crust and upper mantle (Fig. 16b) the Sn phase is difficult to recognize (hidden in the P coda) while the S wave train (consisting of Sg, SmS, sSmS, etc.) is prominent in the seismograms. Snapshots for such media for shallow (2 km) and deep (20 km) P and S sources are shown in Figs 20, 21 and 22. Again, our synthetics are similar to real recordings; a few NORESS and ARCESS records are displayed in Figs 1, 23, 24 and 25. For a most realistic comparison to the real event recordings one could add the P- and S-source synthetics together.

From the corresponding semblance computations (Fig. 17a), we can observe that the first part of the coda is dominated by P wavelets while P-to-S converted waves become dominant at time coincident with Sg arrivals. Since P-to-S scattering is inefficient in the forward direction, these wavelets would mostly arrive after the direct S phases. The reason being that that P-to-S scattering is most effecient in directions perpendicular to the source-receiver ray path, which also explains why their traveltimes would exceed that of Sg phases. This phenomenon is illustrated in Fig. 1; the


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strongest phase is the Lg on the transverse component. The late coda appears to consist of diffusively scattered waves (i.e. energy arriving from a multitude of directions) in the sense that the coda coherence (semblance) is low. The semblance results for an S source are shown in Fig. 17(b). The S-to-P conversion is inefficient while S-to-Rg appear to be efficient in view of the relative strong coda in the 80-90 s time interval.

5.5 Teleseismic coda excitation

From NORESS array record analysis it is clear that most of the teleseismic coda, say the time interval 10-60s after P onset, have apparent velocities similar to the main P phase (Bannister et al. 1990; Dainty 1990). Since local crustal reverberations for these waves are weak (as seen from Fig. 6), the coda must have been generated/scattered mainly in


Figure 11. Snapshot displays of vertical particle velocity at a lapse time of 20 s for (a) P sources and (b) S sources at 20 km depth and embedded in a homogeneous medium with free-surface topography (Model 5 in Table 1). Distances along horizontal axes are relative to the left edge of the model. Particle velocities should be multiplied by lo-" to give realistic values in ms-I. Even for such a simple source and medium, the corresponding synthetics are quite complex. In order to facilitate their interpretations, major upgoing phases are labelled. (Note that the relatively weak sPmS terminates at the Moho at about 170 km.) Downgoing free-surface reflections are not marked due to a lack of a specific nomenclature here for local distances. The efficiency of P-to-Rg conversions at the surface is obvious and appears as a grey/white banding in the top crust of (a) where P-to-S conversions can also be seen.

the source area and it is generally explained in terms of single- and multiple-scattering effects with S-to-P and Rg-to-P conversions being major ingredients. The hypothesis can be tested from our modelling experiments by examining traces from the sensor clusters at depths of 100 km (Figs 3 and 6). Although the models incorporating inhomogeneities in the crust and/or the upper mantle generate a significant amount of coda at the free surface, the coda wavefield at a depth of 100km (Fig. 18) appears relatively weak except. for the S waves, which do not

teleseismic coda. If Rg-to-P scattering is important for teleseismic P-coda generation, one would expect very shallow events (within the upper 4 km of the crust) to have more coda than deeper ones (Ruud et al. 1993). This is in contrast to observation since nuclear explosions (e.g. the SEM event in Fig. 2 in Bannister et al. 1990) have a weak coda compared to other events in the crust. In other words, this would imply that teleseismic coda is not a product of scattering, but mainly of rupture front effects.

5.6 Effects of spatial anisotropy contribute to the teleseismic coda. A comparison with real records (e.g. see Fig. 2 in Bannister et al. 1990) gives further . _ evidence in support of our hypothesis that crust and upper Most of the synthetics displayed above are tied to models mantle inhomogeneities cannot fully account for the with high aspect ratios (a, > a,) which is intended to mimic


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Figure 12. (a) P-source synthetics and (b) for a homogeneous model except that the upper mantle part is inhomogeneous (Model 6 in Table 1). In comparison to Fig. 5 the crustal phases are little deformed, while the Pn and Sn phases on trace EO are strong and appear to be delayed -1 s in time. The most striking feature is the relatively strong coda reflecting first-order back-scattering contributions from sub-Moho inhomogeneities. The P-coda (until the first S arrival) consists mainly of P wavelets, and then S wavelets dominate. These synthetics demonstrate as expected theoretically (Aki 1992) that P-to-S scattering is far more efficient than S-to-P scattering.

a presumed predominance of horizontal layering in the crust and upper mantle. In this section we explore whether differences in spatial anisotropy would provide significant differences in the synthetics, that is, for media with varying aspect ratios. Here we use the terms 'horizontal anisotropy' (HA) for a, > a,, 'vertical anisotropy' (VA) for a, < a,, and isotropy (ISO) for a, = a , . For example, in Fig. 19 we display snapshots of vertical particle velocity at a lapse time of 30s for a P source in this kind of media. Not unexpectedly, the HA model favours vertically propagating scattering wavelets (Fig. 19a), the VA model favours horizontally propagating wavelets (Fig. 19c), while the IS0 model gives somewhat messy results (Fig. 19b). Another feature of interest is that the apparent velocity of the scattering wavelets also reflect the choice of scattering model, i.e. H A models give rise to the relatively high apparent velocities of the scattering wavelets. This hypothesis was confirmed by semblance analysis of sensor

clusters for media models like those used in Fig. 19. S-source synthetics were also generated, and the results here are similar to those discussed above for P-wave synthetics. A notable difference is that due to their relatively shorter wavelengths, the S waves resolve heterogeneities (and surface/interface topography) far better than the P waves, and as a consequence the S-to-Rg scattering both forward and backward (not shown) is efficient (Figs 15 and 17).

5.7 Source depth effects

In the previous sections we have presented and commented upon synthetic seismograms for a large variety of models, but mainly for a source depth of 20 km. In this section we will explore whether the wavefield changes significantly if the source is located close to the free surface. This type of problem is of considerable importance in a seismic surveillance context, namely, discriminating between tectonic-related earthquakes and man-made sources like chemical and nuclear explosions. In Figs 20, 21 and 22 snapshots are shown for P- and S-source depths of 2 and 20 km and at lapse times of 30 s (P) and 20 and 30 s (S). The corresponding distance range shown is 25-240 km (the source is at 75 km).

For a P source (Fig. 20) the focal depth appears to be relatively important; for a shallow source at 2 km, wave energy is 'trapped' in the upper crust which is attributed to the velocity gradient incorporated in the basic model (Fig. 3). There is clear deterioration of the Pg phase (snapshots not shown for 20 s lapse time; as a substitute, see Fig. 11) due to interference with the PmP phase and a contemporary energy loss due to P-to-S conversions. At this distance (Fig. 20) the synthetics exhibit some similarities to the profiling records in Fig. 2; a weak Pn followed by a stronger coda without a clear secondary phase arrival. The S-source synthetics in Figs 21 and 22 are in many respects similar to those for P-type phases due to little energy leakage out of the crustal waveguide and inefficient S-to-P conversions. Obviously, a shallow source is relatively more efficient than a deep source in coda excitation because it can generate more surface waves and body waves which propagate with low velocities in the upper part of the crust. Scattering by surface topography would also contribute to the coda excitation; in Figs 2l(a) and 22(a), S-to-Rg conversions at the free surface appear as alternating light and dark 'rods' with a depth extent of a few kilometres. Likewise, the mentioned smearing of P and S phases are relatively prominent for shallow sources. In view of the above results, and the fact that earthquakes within the upper 3-4 km of the crust are relatively rare, it is not surprising to us that explosion/earthquake source classification is feasible at local distances (Baumgardt & Young 1990; Dysart & Pulli 1990). Even more refined source discrimination should be feasible by introducing relative coda excitation as a discriminant parameter (e.g. see Su, Aki & Biswas 1991). Regarding direct estimation of focal depth, we note that S sources generate strong local depth phases in the P wave train, e.g. the sPg and sPn phases in Fig. 5(b). These phases are disturbed by surface topography, especially at high frequencies, but when properly identified in real earthquake records they would provide valuable information on source depth.


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5.8 Summary remarks on synthetics limited to distances of a few hundred kilometres. The inhomogeneous models we use (Figs 3 and 4) are in some

In this section we have presented a large number of ways complex, but in other contexts simple-essentially a synthetic seismograms and snapshots for a variety of crust one-layer crust over a half-space with velocities increasing and upper mantle models. Such an exercise provides linearly with depth. A major objection against ray tracing improved insights in the many complexities associated with and related techniques for generating synthetic waveforms crustal wave propagation, albeit for computational reasons for stratified media is the failure of coda excitations which,


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Figure 14. (a) P-source synthetics and (b) S-source synthetics for a homogeneous model except that the crustal part is inhomogeneous (Model 7 in Table 1). In comparison to Figs 5 , 9 and 12, the crustal phases like Pg and S g ( L g ) are strongly deformed. In (a) the relatively high coda levels for the P source imply again that P-to-S scattering is relatively efficient. Also, (b) demonstrates why Sn phases are reported relatively seldom: they are simply lost in the P coda.

as demonstrated here, are an integral part of our synthetics. The real challenge to us is whether these synthetics quantitatively match those of real recordings as dealt with in the next section.

6 REAL EVENT RECORDINGS: ARRAY DATA ANALYSIS

In the preceding section we dealt exclusively with synthetic seismograms; here these results are discussed in the context of NORESS and ARCESS array recordings of a few local explosions and earthquakes. A problem is that our synthetics have dominant signal power in the 1-4Hz band, compared to the 4-16Hz for the real records. With this handicap in mind, we will address some of the scattering phenomena highlighted in the previous section.

6.1 Topography: Rayleigh (Rg) wave excitation

In the synthetics, P-to-Rg scattering was prominent for models with topography. In cases of shallow sources (2 km),

Rg excitation was also efficient (Ruud et al. 1993). Observationally, the Rg is rather elusive, but nevertheless the synthetics strongly suggest that there should be Rg wavelets in P coda for local events recorded in hilly areas.

In our search for Rg wavelets in NORESS and ARCESS local event recordings, the primary P wave was removed by subtracting the appropriate P beam from the individual sensors prior to the semblance analysis (Bannister et al. 1990). The outcome of such an experiment is shown in Fig. 23. Obviously, the P coda contains a significant amount of Rg- scattering wavelets (see also Gupta et al. 1993) thus supporting our synthetic results. Similar results were obtained in the analysis of ARCESS local event recordings. The reason for a general Rg elusiveness appears to be that both explosions and earthquake signals contain most energy at high frequencies (4-8Hz) while Rg phases are seldom observed for frequencies above 2 Hz.

6.2 Moho corrugations and Pn/Sn propagation efficiencies

From our synthetics we have that Moho corrugations are of minor importance in a scattering context. Kvaerna & Doornbos (1991) have proposed scattering from topographic relief along the Moho as an explanation for a delayed and anomalous high amplitude Pn phase seen in some NORESS records. Our synthetics suggest that scattering by velocity heterogeneities below the Moho (Fig. 12) would be a more efficient way to generate such wave features. Both Pn and Sn are relatively weak phases, and the latter is often lost (not observed nor reported) in the P coda. Our synthetics based on inhomogeneous media easily reproduce these observational features (Fig. 16).

Observationally, the Pn amplitudes may vary considerably and explanations here range from geological compositions, via tectonic reworkings of an area, to complex multilayered I-D velocity models (Gajewski et al. 1990; Kennett 1989, 1994). In our synthetic modelling it was sufficient to introduce sub-hloho velocity perturbations in order to ensure sufficiently strong Pn- and Sn-phase amplitudes. In other words, Pn- and Sn-phase velocities and amplitudes are not necessarily a good diagnostic for tectonic or compositional features.

6.3 Spatially anisotropic inhomogeneous media

In the synthetics, coda characteristics reflecting spatially anisotropic inhomogeneous media with different aspect ratios are rather evident (Fig. 19). In essence, back- scattered waves tend to propagate in the direction of fastest velocity variation. The semblance analysis of synthetic S coda for models with a, = 10 km and a, = 2.5 km gave velocities in the range 4.0-4.8 km s-', which increased slightly with increasing lapse times. This velocity range is equivalent to an incident angle range of 45"-60". As P - t o 4 scattering is more efficient than S-to-P, we would expect that with increasing lapse times S wavelets would tend to dominate the coda independently of source type. Aki (1992) lists comprehensive supporting evidence for this view. In the analysis of real data, we used local event recordings and the results here are displayed in Figs 24 and 25. The S-coda velocities are in the range 4.0-5.0 km s-' and often increase


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with lapse time. Coda velocities are roughly similar for the and 25. This is clearly seen in the NORESS event (Fig. 25) two arrays, but in exceptional cases ARCESS exhibits while for the ARCESS event (Fig. 24) scattering in the velocities exceeding 5.0 km s-’ (see also Dainty & Toksoz event azimuth direction appears to dominate. In the 1990). A common feature is the lack of P wavelets in the S synthetics, back-scattered waves travelling towards the coda. For real recordings, back-scattered waves can arrive source were weak to moderate at most receiver groups, from any direction in the late S coda, as seen from Figs 24 although most pronounced for models with dominating


character of these arrivals, as has been noted by others (Kennett 1989, 1994). Likewise, the often presumed identifications of phases like pPn, PnPn, etc., from 'within' the Pn-Pg wave train is seldom justified unless based on objective wave-propagation criteria; even the simple models which we have used are complex in this respect.

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vertical correlation distance. The discrepancy here may be explainable in terms of signal-frequency differences; the local event recordings contain more energy at high frequencies compared to our synthetics. As is well known, back-scattering is most pronounced for high-frequency signals, that is ka < 1 (Charrette 1991).

6.4 Smearing of local/regional phases like Pg and Sg/Lg

Such smearing is obvious from the synthetics presented here (Figs 14 and 16) and also from the observational records displayed above. The notable feature is the shift of energy from forward to the backward parts of these signals. The physical explanation is that these phases represent a sum of phases having slightly different travel paths and hence different traveltimes. For inhomogeneous media and long propagation paths these implied interference effects are further amplified. Thus, the generic descriptions Pg and Sg used for crustal phases and likewise Pn and Sn for mantle phases are not necessarily an accurate description of the

7 DISCUSSION

In the foregoing sections we have presented 2-D FD synthetic seismograms and snapshots for a suite of progressively more complex lithosphere models. In this section we will discuss the relevance of this exercise, that is, to what extent do we mimic seismic-wave propagation in the crustal waveguide. A proper starting point may be the numerical scheme used for solving the elastic wave equation. First, no intrinsic attenuation is included nor is any geometrical spreading correction introduced (modelling is in 2-D while seismic recording is in 3-D). The intrinsic Q-values are likely to be very high for most parts of the crystalline crust so for the distance ranges considered here this effect is expected to be of minor importance. Another drawback is that our 2-D synthetics are unable to model scattering from heterogeneities outside the vertical plane through the source and receivers, an effect which is important for signal coherence. Otherwise, the above shortcomings in our numerical scheme are considered to be of little significance for the propagation/scattering effects we have studied. The snapshots are instructive in this respect.

Our basic lithosphere model (Fig. 3, no randomization) is typical of continental areas and in our opinion the same applies to the way this model has been perturbed. First, both from scattering and tomographic studies, velocity fluctuations in the lithosphere amount to 2-4 per cent. From analysis of NORSAR and NORESS array recordings, Charrette (1991) favours a differentiation of the lithosphere velocity fluctuations with 3 per cent in the crust (with a self-similar correlation function) and 2 per cent sub-Moho lithosphere (with a Gaussian correlation function). Also FlattC & Wu (1988) prefer to divide the lithosphere into layers with different rms and correlation functions. There is no such differentiation in our lithosphere and for two reasons: (i) forward modelling is not well suited for uniquely determining physical parameters of complex media, and (ii) our sub-Moho sampling is only through the late coda and by the Pn phase which only appears on the EO-sensor clusters.

In scattering studies, heterogeneous media are often represented by random fields (e.g. Capon 1974; Wu & Aki 1985). and such an approach was adopted here. With scale lengths much smaller than the model area (Fig. 3), the complex 2-D velocity function (von KirmAn) was expressed using a few low-order statistical moments, namely, the mean, variance and correlation distance. In analytical studies the Gaussian autocorrelation function is popular (the existence of higher-order derivatives), but it is rather smooth in the spatial domain with the consequence of little back-scattering (ka > 1). However, von Karmin realizations of order 0.5 (exponential), 0.3 (Kolmogorov turbulence) and 0.0 (self-similar) give spatial spectra of a roughness considered representative of the real earth even for a 'mesoscale' correlation distance of 10km (for a more extensive discussion, see Charrette 1991).

660 S . 0. Hestholm, E . S . Husebye and B. 0. Ruud

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Most of the scattering studies depend on analytical insight in seismic-wave scattering processes, but their methods (see Sat0 1991) and in order to obtain meaningful relevance to real earth scattering remains problematic (Wu results, many strong assumptions have to be made. The & Aki 1985; Levander & Holliger 1992). Considerations most common ones are the validity of the Born of the above type motivated us to undertake a visualization approximation (first-order scattering) and isotopic random of wave propagation and scattering in inhomogeneous media media (no lithospheric layering nor a free, topographic as an alternative to trying to estimate specific medium surface). In other words, these studies provide an in-depth parameters. In this context, free-surface topography


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apparently produced the strongest effect in the synthetics, particularly in the P and S coda. As mentioned in the foregoing section, P-to-Rg and S-to-Rg scattering both in forward and backward directions are very efficient for such models. The other model factor of striking importance for the synthetics was found to be the extent of the stratification in the lithosphere or the dominance of horizontal anisotropy. As demonstrated in Fig. 12, strong Pn-to-Sn phases require a sort of sub-Moho layering as also noted by Kennett (1989, 1994) and Bertil et al. (1989) among others. Likewise, upper crustal waveguide effects are easily explained as a combination of a shallow source and a velocity gradient in the crust. Regarding spatial anisotropy, the apparent velocities of coda wavelets for a medium with dominant horizontal correlation distance should be significantly higher than for other aspect ratios (Fig. 19).

There is a significant difference between synthetic seismogram analysis and that of real-event recordings; in the

latter case we have no control over the medium parameter specifications. An exception here is the effect of surface topography. Using data from the NORESS array, Bannister et al. (1990) and Gupta et al. (1993) have demonstrated that P-to-Rg and S-to-Rg scattering are very efficient as confirmed from the synthetics (Fig. 23). Regarding the strength of Pn and Sn phases compared to Pg and S g l L g , there is in the case of NORESS recordings a considerable variation, which moreover is clearly azimuth-dependent. The two options here are to consider this variability as specific medium realizations or that the local lithosphere medium properties are non-isotropic. In view of the complex tectonic history of southern Norway, we prefer the latter explanation, and do not need to invoke sub-Moho compositional differences to explain Pn and Sn amplitude variabilities as suggested by Gajewski et nf. (1990).

Regarding apparent coda velocities, the observational picture is rather blurred. For some events the early S coda had apparent velocities in the range 3.8-4.0 km s-', which increased up to 4.8 km sC1 for later parts of the coda. For other events, the apparent coda velocities were relatively stationary in the interval 4.0-4.5 s-'. Like Dainty & Toksoz (1990), we also found some abnormal ARCESS events with apparent coda velocities above 5.0 km s-I. These coda results (see also Figs 24 and 25) testify towards the variability in the seismic-event recordings for signal frcquencies above 1 Hz. They are really not unexpected in view of our synthetic results and, most importantly, there is no need to invoke large-scale structural or compositional changes in the lithosphere for their explanation. After all, our basic model is simple: essentially a crustal layer over a half-space. By introducing small velocity and interface perturbations, we can generate synthetic seismograms which encompass dominant observational event recording features, namely, 'smearing' effects in the main P- and S-phase arrivals and a relatively strong coda aominated by P-to-S scattering wavelets.

Finally, most seismic coda studies and synthetic seismogram analyses for heterogeneous media aim at solving forward problems, i.e. computing the seismic responses of predefined and simplified lithosphere models. The corresponding inverse problems have proved far more difficult to solve, namely, that of extracting characteristics medium parameters from observational data (see Sat0 1991; Wennerberg 1993). For example, in cases of coda analysis, most observational parameters extracted are tied to coda envelope decay rates for lapse times well into the S coda. The specific coupling between decay rates and medium parameters remains unclear. Another manifestation of heterogeneous medium-wave propagation is in terms of teleseismic P-signal deformations in terms of amplitude fluctuations and phase shifts. Recently, Wu (1989) introduced the joint transverse angular coherence function and the concept of stochastic tomography for estimating power spectra of lithosphere heterogeneities which has been tested as mentioned on NORSAR array data (Flatte & Wu 1988; see also Wu & Xie 1991). Besides the examples given above, most scattering studies give much weight to 'visual correlations' between observational and theoretical results. Our study as reported here is of the latter kind. Simply, our full-scale lithosphere model (no. 9 in Table 1) was to us too complex to attempt retrieving the many medium parameters


Figure 19. Snapshot displays of vertical particle velocity for a lapse time of 30 s for the model range of 50-150 km for P sources at 20 km depths (see also Fig. 11 caption). The purpose is to illustrate the effect of different types of spatial anisotropy on the late P coda for media with (a) ‘horizontal anisotropy’, (b) isotropic and (c). ‘vertical anisotropy’ (Models 1, 2 and 10 in Table 1). For such idealized models, the P coda would give information on the medium anisotropy as defined in this study. Similar results are expected for an S source.

by some sort of inversion scheme. Furthermore, this kind of problem is difficult in cases of relatively high-frequency seismic waves travelling in the crustal waveguide. Despite such shortcomings, the approach of visualizing lithosphere wave propagation in terms of finite-difference synthetic waveform, snapshots and semblance velocity decomposi- tions of the wavefield are in our opinion instructive, and probably would remain so for years to come.

8 CONCLUSIONS

In this study we have presented a comprehensive set of 2-D finite-difference synthetics seismograms and snapshots aimed at visualizing seismic-wave propagation in the lithosphere. In comparison to many other similar studies, our medium model of 150 X 400 km2 is very large, and can also incorporate free-surface topography. Our research strategy was aimed at isolating specific propagation effects like a corrugated Moho and a non-flat surface (interface scattering) and crust and/or upper mantle heterogeneities

(volume scattering). The inhomogeneous media realizations were tied to von KBrman distributions of order 0.3 and with rms velocities at 3 per cent and correlation distances in the range 2.5-10 km. Our theoretical results were supplemented with a few observational ones stemming from stacking and semblance analysis of NORESS and ARCESS array recordings. Major findings were as follows.

(1) Finite-difference synthetics for a homogeneous lithosphere with no corrugated interfaces are very similar to those produced by ray tracing-this was our reference model.

(2) A corrugated Moho has relatively little effect on lithospheric wave propagation, and besides is not likely to be observable using real recordings.

(3) The free-surface topography has a rather dramatic effect on the wavefield near the surface particularly in terms of P-to-Rg and S-to-Rg scattering, both in forward and backward directions. Also the P wave train is deformed; for NORESS we could account


Figure 20. Snapshot displays of vertical particle velocity at a lapse time of 30 s (Model 9 in Table 1 ) for P sources at depths of (a) 2 km an (b) 20 km (see also Fig. 11 caption). Since the medium models are roughly similar (Models 8 and 9 in Table 1). the snapshots in (b) are comparable to the traces in Fig. 16(a). Note that a shallow source apparently produces more wave energy in the upper part of the crust, while in both (a) and (b) the Pn phase is weak and rather blurred.

for about half of the observed slowness anomalies using a model with real topography but scaled down by a factor of two.

(4) Sub-Moho heterogeneities contribute to the P and S coda mainly through P-to-S and S-to-S back- scattering. The strength of the Pn and Sn phases depends, in addition to a sub-Moho velocity gradient, strongly on sub-Moho heterogeneities and/or layering. Such features can be modelled by introducing spatial anisotropy in the upper mantle with a dominant horizontal correlation distance.

(5) Crustal heterogeneities generate a significant amount of coda, and strongly distort the P and S wave trains at least for travel distances exceeding 100 km. In this

regard, reading of P and S arrival times to the nearest 1/10 of a second for local events does not seem meaningful.

(6) Teleseismic coda excitations (sensor recordings at a depth of 100 km) proved inefficient as most of the coda wavelets were of the S type. In other words, S-to-P and Rg-to-P scattering are relatively weak and the teleseismic P coda seemingly stem mostly from source rupture front effects.

(7) Coda excitation and apparent velocities of the coda waves depend on spatial anisotropy (i.e. on the aspect ratios of the vertical and horizontal correlation distance). For example, relatively high apparent S coda velocities would be expected for


Figure 21. Snapshot displays of vertical particle velocity at a lapse time of 20s (Model 9 in Table 1) for S sources at depths of (a) 2 km and (b) 20 km (see also Fig. 11 caption). In comparison to Fig. l l (b ) (same configuration), we see that only the strongest phases are retained intact. The SmS phase is well preserved in the middle crust, but strongly weakened at the surface-where the seismometers are located. For the shallow source in (a) relatively much wave energy is confined to the top crust. Fig. 16(b) traces compare to the (b) snapshot.

models with dominant horizontal correlation distance.

(8) The synthetic wavefields are different for models with source depths at 2 and 20 km respectively; the former one is clearly more complex and the coda appears stronger. This effect is ascribed to the ability of a shallow source to excite surface waves more efficiently and to the fact that the velocity gradient tends to concentrate wave energy in the upper part of the crust.

(9) A comparison between characteristic 2-D FD synthetic features as described here and the corresponding ones derived from analysis of real data was also undertaken. First, as expected, it was easy to identify Rg wavelets in the P coda for local event

recordings both at NORESS and ARCESS. Besides R g , the P coda is void of S waves. S coda velocities are mainly in the range 4.0-4.5 km s-' as expected from the synthetics. Moreover, with increasing lapse times, the S wavelets arrive from all directions typical of diffuse scattering (Figs 24 and 25).

(10) Modelling realism: our lithospheric models produce synthetics which in terms of coda excitation, wavelet composition and duration, plus the deformation of P and S signals, qualitatively match corresponding observational features. However, the local lithosphere is obviously not isotropically inhomogeneous as structural fabrics reflecting past tectonic deformation seemingly have to be invoked to account for the large variability in the observational results.


Figure 22. Snapshot displays of vertical particle velocity at a lapse time of 30 s (Model 9 in Table 1) for S sources at depths of (a) 2 km and (b) 20 km (see also Fig. 11 caption). This figure is directly comparable to Fig. 21, except for a lapse time difference of 10 s, roughly equivalent to a distance difference of 35 km for S g / L g . The increase in seismogram complexities with even a small increase in epicentrc distance is obvious.


NORESS DATA 1986 / 36 / 17 : 54 : 30.0 AZI : 304"

'b) 15 - 30 sec .a) BEAMS -

V : 7.0 - - -0.2

-

w 0.0

Az : 90 V :3.0 --.

I I

5 i5 is Time 1.1

FILTER : 2 - 4 HZ U x f s I km t U x (s / km )

Figure 23. Beamforming and P-coda semblance analysis for a local earthquake recorded by the NORESS array at an epicentre distance of -400 km. (a) The upper trace is the P beam, while the two lower traces are residual beam traces aimed at enhancing Rg-scattered wavelets. Residual beam means that the P- beam trace was subtracted from the individual sensor records prior to the stacking. The P coda is dominated by P wavelets having a source-region spatial location as shown in (b) for the 15-30 s interval of the coda. The outcome of the residual beam search for P-to-Rg scattering is shown in (c) and (d) for the time intervals 15-30 s and 25-40 s, respectively. The circles in these two figures are for an apparent velocity of 3.33 km S C ' . which is typical for Rg phases. Results in (c) and (d) agree reasonably well with those reported by Bannister er al. (1990), i.e. the Bronkeberget (AZ - 80") and Skreikampen (AZ - 235") scattering sources. The latter results were derived from relatively low-frequency teleseismic signals.


- 0 . d E Y

01 . - 5 0.:

O.(

-0.:

-0.4

ARCESS DATA : 199 1 / 103 / 21 : 18 : 36.0 AZI : 2590

T

z

k , 0 20 , a ' 60

1

-0.4 -0.2 C

FILTER : 2 - 4 HZ

Time (sl * d' 55 - 70sec C

- -0.2 0 0.2.

Ux ( s I km )

0.4

0.2

0.0

-0.2

0.4

.0.2

-0.0

c - 0 . 2

. -0.4 4

Figure 24. S-coda semblance analysis of a local earthquake at an epicentre distance of -85 km recorded by the ARCESS array. (a) The rotated, 2-4 Hz bandpass-filtered traces for the centre three-component station A 0 are shown. The analysis results are in terms of contoured semblance values in horizontal slowness space for the respective time intervals of (b) 35-50 s, (c) 45-60 s and (d) 55-70 s. The circles indicate apparent velocities of 5.0 and 3.33 km sC'. Clearly, most of the late S coda consists of S wavelets with propagation velocities around 4.0 km SC '. Similar results are obtained in our synthetic record semblance analysis. Note that the preferential wave-scattering direction coincides with the evident azimuth direction.


NORESS DATA 1989 / 100 / 21 : 14 : 13.0 AZI : 2250

0.4 E Y

cn . - 5 0.;

O.(

-0.

-0.4

FILTER :

Figure 25. S-coda semblance analysis of a local earthquake recorded by the NORESS array at an epicentre distance of -25 km. The time intervals analysed are (b) 15-30 s, (c) 25-40 s and (d) 35-50 s. Otherwise caption as for Fig. 24.


ACKNOWLEDGMENTS

Stimulating discussions with Drs A. Dainty (Phillips Laboratory, Hanscom AFB, MA) and A. Kame1 (IBM Bergen Environmental Sciences and Solutions Centre, Bergen, Norway) are hereby acknowledged. The many valuable comments by an anonymous referee were much appreciated. The work reported here was supported by the Air Force Office of Scientific Research, USAF under Grant F49620-92-5-0510 (B.O.R.) and Contract F49620-89-C-0038 (E.S.H.). The IBM Bergen Environmental Sciences and Solutions Centre is acknowledged for generous use of their IBM 3090/600S VF computer.

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162-178.

APPENDIX

Spatial partial differentiation is achieved by cost-optimized, dispersion-bounded, high-order finite-difference operators on a staggered grid. For time stepping, a leap-frog technique is used. The discretization of the elastodynamic eqs (1)-(3) with two staggered numerical space differentiators, 6 *, applied as in Levander (1988) to stresses and particle velocities leads to

pj+{V,+(t + At/2) - V:(t - At/2)}

= A t ( F ; ( t ) + 6:Sii(t) + a;sf'(t)) I = 1 l#j

j , 1 = 1, . . . , J

q,(t + At) - Sjj(t) J

= IAt 6;V:(t + At/2)

+ 2pAtb,:Vf(t + At/2), ,=l

j , 1 = 1, . . . , J

S$'(t + At) - Sz' ( t )

= p$'At{6fV:(t) + 6;Vt(t)}, j , l = 1 , . . . , J , j # l

with

Vf(t) = vi(x + hi/2, f ) , F T ( t ) =$(x + hj/2, t ) ,

Sij(t) = uji(x, t ) ,

pi' = p ( x + hj/2), P = P ( X )

S;+(r) = uil(x + hi/2 + h,/2, f),

I = k(x ) , and pLi;+ = p(x + hj/2 + h1/2).

aq axi - (X + hi/2),

q(X + ( I - l)hi) - q(x - lh,)

h i

L-12

6 i q ( x ) = c 4 - 1 I= 1

aq dXi

=--- (X - hj/2).

Here x is position with xi its coordinate in the jth direction; hi is the unit vector in the jth direction; k, p and Sjj are defined at the nodes of the Cartesian mesh; p; , Vf and FT are defined at the links connecting the nodes; and S;' and p;+ are defined at the centres of the 'plaquettes'. 6* are numerical differentiators of coefficients { d g - l}. q is velocity or stress, and L* is the length of the operatgr.

For the numerical dispersion relations, the stability limit and bandwidth introduced by the discretization, the reader is referred to Sguazzero et al. (1989). The computational cost (in floating point operations) of an elastic numerical simulation is given by

C,, = (J2(nip + nLp + 2) + 75 - l}(ZV~~m)J'la~l.

Here n , a 2 is the number of grid points per shortest wavelength needed to model the wavefield: n $ = {3(Lf/2) - l} is the number of floating point operations needed to compute the spatial derivative; and Nn is the number of shortest wavelengths in each direction of the computational domain. J = 1, 2 or 3 is the dimension of the problem, a,,, = c AtlAx is the Courant number of this explicit scheme, and c is the relevant propagation speed. n, = n/B, , where 0, is the bandwidth over which the relative error in the group velocity is bounded by a threshold. The entity n , is, among others, a function of the coefficients {d& l}.

The method of spatial partial differentiation is achieved by optimizing the cost function C,, with respect to {dgPl}, subject to a chosen upper bound in the relative error of the group velocity (Sguazzero et al. 1989). This process leads to the optimized operators utilized in our code.

seismic wave propagation in complex crust-upper mantle media using 2-d finite-difference synthetics

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