AFGL-TR-88-0141

00 Lg-Wave Propagation in Heterogeneous Media

T0

S Brian L. N. Kennett

Australian National UniversityResearch School of Earth SciencesGPO Box 4, CANBERRA ACT 2601Australia

6 April 1988

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Lg-wave Propagation in Heterogeneous Media (Unclassified)

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1I.(SJPPLEMENTARY NOTATIONThis research was supported by the Defense Advanced Research Projects Agency

JA

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)FIELD GROUP SUB-GROUP Wave Propagation

Lg waves Heterogeneity Coupled Modes

19. ABSTRACT (Continue on reverse if necessary and identify by block number)The Lg wave phase which is of considerable interest for nuclear

discrimination proble.as is normally observed after propagation throughconsiderable distances. This phase is dominantly guided in the crustalwaveguide, which is known to be a region with very considerable horizontalvariability in properties.

The effect of heterogeneous crustal structures on Lg waves has beendetermined by using a "coupled-mode" technique in which the local seismicwavefield in the real medium is expressed as a combination of the modal Peigenfunctions of a stratified reference structure. Departures of theseismic properties in the medium from those of the reference medium leadto coupling between the amplitude coefficients in the modal expansion.The evolution of these modal weighting factors with horizontal positionare described by a coupled set of ordinary differential equations. K

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19. ABSTRACT (continued)

This approach provides a calculation scheme for studying guided wavepropagation over extended distances, at frequencies of 1 Hz and above.The heterogeneity models which have been used are two-dimensional andcalculations are carried out for one frequency at a time.

A sequence of models with varying levels of heterogeneity have beenconsidered in order to determine the merits and limitations of thecomputation scheme. The coupled mode technique works well withheterogeneous models in which the local seismic velocities differ from thestratified reference model by up to 2 per cent and there are nosignificant distortions of the main discorcinuities (e.g. the crust-mantleboundary). The approach can be used for higher levels of heterogeneity andwith distortzz 4nterfaces but a large number of modes needs to beconsidered with consequent high computation cnsts. If the level ofheterogeneity is not too large then the interaction between modes can be

restricted, rather than extending over the whole mode set, with consequent Sreduction in computation cost.

One of the major effects of crustal heterogeneity is to introduce thepossibility of smearing out the main amplitude peak in the Lg wave trainover a band of group velocities. As a result, an effective measure of theenergy content of the Lg waves will be to consider the integrated v,

amplitude along the traces beween group velocities of 3.6 and 3.3 km/s. SThe effects of heterogeneity vary between different parts of the Lg wavetrain and the representation of the wavefield in terms of modalcontributions allows a detailed analyis in terms of the group velocitycomponents.

The computational scheme is currently being extended to try to allowfor mode-coupling in three-dimensions for boi Love ana RayLeighcomponents of Lg waves. When heterogeneity transverse to the prupagationpath is relatively weak a good approximation to the propagationcharacteristics can be produced.

S N

SUMMARY

The Lg wave phase which is of considerable interest for nuclear discrimination problemsis normally observed after propagation through considerable distances. This phase isdominantly guided in the crustal waveguide, which is known to be a region with veryconsiderable horizontal variability in properties.

The effect of heterogeneous crustal structures on Lg waves has been determined by usinga 'coupled-mode' technique in which the local seismic wavefield in the real medium isexpressed as a combination of the modal eigenfunctions of a stratified reference structure.Departures of the seismic properties in the medium from those of the reference medium lead tocoupling between the amplitude coefficients in the modal expansion. The evolution of thesemodal weighting factors with horizontal position are described by a coupled set of ordinarydifferential equations.

This approach provides a calculation scheme for studying guided wave propagation overextended distances, at frequencies of I Hz and above. The heterogeneity models which havebeen used are two-dimensional and calculations are carried out for one frequency at a time.

A sequence of models with varying levels of heterogeneity have been considered in orderto determine the merits and limitations of the computation scheme. The coupled modetechnique works well with heterogeneous models in which the local seismic velocities differfrom the stratified reference model by up to 2 per cent and there are no significant distortionsof the main discontinuities (e.g. the crust-mantle boundary). The approach can be used forhigher levels of heterogeneity and with distorted interfaces but a large number of modes needs "to be considered with consequent high computation costs. If the level of heterogeneity is nottoo large then the interaction between modes can be restricted, rather than extending over thewhole mode set, with consequent reduction in computation cost.

One of the major effects of crustal heterogeneity is to introduce the possibility of smearingout the main amplitude peak in the Lg wave train over a band of group velocities. As a result,an effective measure of the energy content of the Lg waves will be to consider the integratedamplitude along the traces beween group velocities of 3.6 and 3.3 km/s. The effects ofheterogeneity vary between different parts of the Lg wave train and the representation of thewavefield in terms of modal contributions allows a detailed analyis in terms of the groupvelocity components.

The computational scheme is currently being extended to try to allow for mode-couplingin three-dimensions for both Love and Rayleigh components of Lg waves. Whenheterogeneity transverse to the propagation path is relatively weak a good approximation to thepropagation characteristics can be produced.

ACCeossion For

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CONTENTS

Summary iii

Research Objectives 1

Research Status and Achievements 1

Studies of Two-Dimensional Heterogeneity in the

Crust and Mantle I

Studies of Three-Dimensional Heterogeneity 10

Discussion 10

References 11

Appendix: Model Evolution Equations 13

Distribution of Research Results 19

a) Publications 19

b) Presentations 19

-i.f

RESEARCH OBJECTIVES

The aim of this work is to develop techniques for handling guided wave propagation inthree-dimensionally heterogeneous media, especially for Lg waves, in order to improve theunderstanding of the nature of regional seismic phases and the way in which thecharacteristics of the seismic source can be modified by propagation to the receiver. Suchinformation will be valuable in assessing the effect of geological structure on the behaviourof potential discriminants between earthquakes and underground nuclear explosions.

Most of the regional phases of interest for nuclear discrimination problems areobserved after propagation through considerable distances. These phases travel throughthe crust and the uppermost mantle, which from a variety of studies are known t- beregions with very considerable horizontal variability in properties.

What is therefore needed is a computational procedure which will allow the tracking ofguided seismic wave propagation through a horizontally heterogeneous crust for distancesup to 1000 km or more. Such a method should be able to be used with three-dimensionalvariations in seismic properties superimposed on the normal increase in seismic wavevelocities with depth. A suitable candidate is the coupled mode scheme described byKennett (1984) which has already been used with some success in understanding thecharacteristics of Lg wave behaviour (Kennett & Mykkeltveit 1984).

RESEARCH STATUS AND ACHIEVEMENTS

During the past year, the work on Lg phases has mostly been carried out using theA coupled modes method introduced by Kennett (1984). Firstly, the limitations of the

calculation scheme have been explored for two-dimensional heterogeneity models and anumber of approximations have been explored with the aim of improving computationalperformance. Secondly, an attempt has been made to extend the methods to more generalthree-dimensional models of heterogeneity.

For regional S wave trains, a representation in terms of a limited number of discretemodes gives a economical computational description for horizontally stratified media. Ateach frequency only a limited number of such modes need be considered. In a laterally

"" heterogeneous medium, at a single frequency, it is possible to represent the seismicdisplacement and traction fields within the varying medium as a sum of contributions fromthe modal eigenfunctions of a reference structure, with coefficients which vary withposition. The evolution of these modal expansion coefficient terms with horizontal positioncan be described by a set of coupled partial differential equations in the horizontalcoordinates (an outline of the theoretical development is given for completeness in theAppendix). The cross-coupling terms between different modes depend on the departures ofthe heterogeneous structure from the stratified reference model. These differences are notrequired to be very small but must not be such as to completely change the nature of thecrustal wave guide. Thus, it is possible to accommodate substantial localised change inseismic velocities and density, but more difficult to allow for shifts of more than 2-3 km inthe position of major interfaces such as the crust-mantle boundary. Maupin & Kennett(1987) give an extended discussion of the circumstances in which the coupled modeapproach can be applied to two-dimensional heterogeneous models.

Studies of Two-dimensional Heterogeneity in the Crust and MantleWhen the heterogeneity within the medium is two-dimensional, the calculations can be

recast as the solution of non-linear differential equations of Ricatti equation for thereflection and transmission matrices connecting the modal expansion coefficients atdifferent positions. The advantage of this rearrangement is that, for each frequency, theboundary conditions on the differential equations are simplified for a generallyheterogenm..o;L medium.

* W-J.. -I *.Y~~' Y - .A;' V %l~

If we adopt a reference medium which does not vary with horizontal position, theindividual mode contributions propagate independently in that structure. However, oncethe properties of the true medium differ from the reference, the independence of modalpropagation is lost. The effect of the heterogeneity enters into the differential equations forthe modal expansion coefficients via a coupling matrix whose dimensions are dictated bythe number of modes included in the calculation.

The choice of the number of modes is of considerable importance. All the significantwavenumber components for the seismic phase of interest should be included, as well as anallowance for steeper angles of propagation than would be present in the reference medium,in order to allow for scattering effects. The demands of computation are that the number ofmodes should be kept as small as possible, since the computation time depends on thesquare of the number of modes. On the other hand if the .runcation is too tight, a poorrepresentation of the displacement field can ensue and the results are of limited use.

In figure 1 we illustrate the reference model (ARANDA) used for most of the Lg mode ecalculations, and the modal eigenfunctions for the 22 Love modes included in calculationsat 1.5 Hz. The cut-off criterion was based on the effective depth of penetration of themodes. The fundamental mode and first 15 higher modes are confined to the crust, andare the main contributors to Lg. Mode 16 begins to have significant displacement in the %mantle, and the remaining modes carry their main energy in the mantle and represent the Snphase. The truncation is made at a phase velocity of 4.54 km/s, corresponding in thismodel to a penetration depth of about 100 km. Features in the wave propagation processcorresponding to higher phase velocities than 4.54 km/s cannot be represented by thesecoupled mode calculations. 0

In order to give a good account of the guided wave propagation, we must make surethat all the significant wavenumber components for the seismic phase of interest areincluded, as well as making an additional allowance for steeper angles of propagation inorder to allow for scattering effects. This means that we cannot just use those modeswhich have the bulk of their energy within the crust but must also include waves withhigher phase velocity whose energy in the stratified reference model ties dominantly in theupper mantle. At 1.5 Hz as shown in figure 1, the fundamental mode and first 15 highermodes are confined to the crust, and are the main contributors to Lg. However, we need tocarry at least the 22 modes illustrated if we are to allow for wave interaction processesgenerating waves travelling at angles up to 40" from the horizontal in the midcrust and eventhen we will not give a full account of waves travelling at steeper angles to the horizontal.

An indication of the way in which the coupled mode scheme is likely to behave whenconfronted with heterogeneity is provided by looking at the expansion of the modal eigen-functions of a particular stratified structure in terms of the eigenfunctions of a differentreference structure. The situation is illustrated in figure 2 for two cases: firstly, where the -crust-mantle interface is maintained fixed but the crustal velocity is varied, and secondlywhen the crust-mantle interface is moved by 1 km. The pattern of modal behaviour israther different in the two situations. For a change in crustal velocity (fig 2a), the modaleiigenfunction shapes remain similar in the varied model and so the expansion coefficentsare largest for the original mode and its immediate neighbours and decay quite rapidly withincreasing mode separation. A I km shift in the crust-mantle bounary has a morepronounced effect (fig 2b), now the modal expansion coefficients alternate in sign anddecay less rapidly with increasing mode separation. This arises because the structure of thedispersion equations and therefore, the nature of the modal eigenfunctions, is much more 0sensitive to the position of a major discontinuities than to the details of the velocitydistribution between them. A consequence is that for a perturbed interface many modes areneeded in a modal expansion to give an accurate representation of the displacement andtraction fields.

2J

Figure 1: ARANDA reference model and Love mode eigenfunctiorns with phase velocity lessthan 4-54 kn/s at frequency 1 Hz.

p km/s Love Modes 1.5 Hz

0 2 3 4 5

0

40

60 6 7 8 3"10n 4 14r 0 V 8 1320 2

Figure 2: Illustration of the expansion of the cigenfunctions of a varied model in terms of those for areference structure. The expansion coeffiecients are shown for a span of 8 modes. The reference modeland the varied model, indicated by the chain dotted line, are shown in the insert.a) change in velocity, b) change in depth to crust-mantle boundary

1.0 - 3 4 5

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40

VA 60 D.pt, km0

-.2

-. 4 -3 -2 -1 0 1 2 3 4 5Difference in Lode Number

| 1.0s

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44

a 40V.4 k=7 $ 40U

-.4 L -3 -2 -1 0 1 2 3 4 5Difference in Mode Number

3

Significant perturbations of the major interfaces in the model thus lead to ex:ensivecoupling between modes and can induce relatively steep angles of propagation. If suchfeatures are required in the models to be considered, then a broad sweep of modes must betaken for the reference model (for the Lg case at 1.5 Hz, at least 30). With the benefit ofhindsight, i: would appear that the calculations of K,.aett & Mykkeltveit (1984) on theinfluence of graben structures lie at the limit of acceptable model variations for the modalset adopted.

The coupled mode representation is well suited to heterogeneous models where thebehaviour consists of relatively random variations about the properties of the referencemodel (see e.g. fig 3). In this case, a single incident mode will interact with a limitednumber of neighbouring modes so that the coupling matrix is diagonal dominated with alimited effective bandwidth.

With the aid of the coupled mode technique, a study has been made of the way inwhich the modal field for Lg waves is affected by various levels of distributedheterogeneity for frequencies up to 2Hz over paths of 1000 km long. A sequence ofdifferent heterogeneity models were constructed by specifying the seismic velocities in avertical section at 40 km intervals and then using bicubic spline interpolation within eachlayer. For the distributed heterogeneity models the velocity values were generated with a

*random perturbation to the reference value within a prescribed range of variation. Thevelocity values were then smoothed horizontally by applying a moving average over 3points. This imposes a typical horizontal scale length of around 100 km, and the verticalscale varies within the model becoming larger in the upper mantle. The velocity distributionis arranged to return to the reference structure at the ends of the model (0 and 1000 km) sothat a direct physical interpretation of the node coupling results in terms of reflection andtransmission can he made. In figure 3 we illustrate a set of models (D, E, F) withincreasing crustal heterogeneity. Model D is constricted to have about ±1 per centheterogeneity in the crust with a horizontal scale length of about 100 krm, and ±1 per centheterogeneity has been introduced in the mantle as well. Models E and F retain the samemantle heterogeneity structure as model D but have higher crustal heterogeneity amplitudes,±2 per cent for model E and ±5 per cent for model F. We shall also consider model C withe same crustal structure as D, but no heterogeneity below the crust-mantle boundary.

*b After propagating a large horizontal distance in a heterogeneous model the energy-" originally in a single incident mode will no longer be confined to that mode, and indeed

some energy may be reflected back by the heterogeneity, with again the possibility ofconversion between modes. When the level of heterogeneity is relatively low (of the orderof 1 per cent deviations from the stratified reference model - as in model D) transmissioneffects dominate and the level of reflected energy is negligible. Even with rather largerlevels of heterogeneity the reflection from the whole model is small but, as pointed out byMaupin & Kennett (1987), it is necessary to retain apparently reflected waves in the courseof the computation in order to get an accurate calculation of transmission effects.. TheRicatti equations are much simpler when reflection can be neglected and so computationtime can be reduced.

For any particular structure, it is therefore necessary to undertake a preliminarycalculation including both reflection and transmission effects before it can be assumed thatreflection can be ignored. The control of numerical accuracy in the course of thecalculations is rather difficult, although for perfectly elastic models there is the possibilityof monitoring the constancy of the total energy in reflection and transmission associatedwith an individual incident mode. To preserve full numerical precision quite small stepshave to be taken in the horizontal direction: at 1.5 Hz the step should be no larger than 0.5km if both reflected and transmitted waves are considered. For a number of models asomewhat larger step length gave reasonable results for the case of transmission alone.

4P

.. . .. .... .~9

Figure 3: Sequence of heterogeneous velocity models used in studies of Lg wave propagation. Heaviershading indicates negative perturbadions away from the ARANDA reference model.

20:-

200. 400. 600. $00. 1000.

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200 400 600 -. .. 1000.~

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Figure 4: Representation of the transmission matrix at I Hz after passage through 1000 km of model D.Amplitude values less than 1 are left blank.

TransmissionMode No

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150I

2

3

4

5

6

7

10

1

12

1314 "

0 10 20 30 40 50 60 70 80 90 100

For an incident mode at 0 lan the effect of 1000 km of heterogeneity is that the energyoriginally in the mode is no longer confined to that particular mode. In figure 4 we showthe transmission matrix for the 15 mode set appropriate to the ARANDA reference model at1 Hz, after passage through the heterogeneity model D. Each column of the matrixcorresponds to the incidence of a single mode of amplitude 100 at 0 km. We see that thebehaviour is diagonal dominated, but that significant mode conversions can occur. For theLg type modes (0-9) the bandwidth for significant interaction is typically ± 2 modes. Thestrong interaction of modes 1 and 2 arises because of their similar shape in the top 10 km.For the Sn modes (11-15), the eigenfunctions are very similar and there is stronginteraction due to mantle perturbation extending to 200 km (the lower portion is not shownin fig 3). However the separation from the Lg modes is striking; there is very little I

interaction except for mode 10 which shares some of the characteristics of each group.As the frequency increases the number of modes need to cover the same phase velocity

range increases, which causes some problems in combining the results from differentfrequencies. The increased size of the differential equation system also means an increasein computation time as the square of the number of modes if all mode interactions have tobe considered. Fortunately, present tests indicate that for the same model the bandwidth ofinteraction increases only slowly with frequency. For the Lg waves propagating throughmodels similar to D with about 1 per cent heterogeneity, computations can be restricted to aband of 5 modes either side of the target mode without appreciable error. At 1.5 Hz with22 modes included, this bandwidth restriction has the effect of reducing computation timeby nearly a factor of three.

The required bandwidth increases with the level of heterogeneity in the model (seeTable 1). For model D in figure 3, a bandwith of ±5 modes around the target mode is

6

sufficient for both Lg and Sn modes. However to account for the- interaction betweenmodes for model E it is desirable to have at least a bandwidth of ±7 modes included in thecalculation. For the highest level of heterogeneity illustrated in fig 3 (model F) a bandwidthof ±9 modes is definitely insufficient for full accuracy. For very heterogeneous modelsthere is also the possibility of high angle propagation effects and so the size of the mode sethas to be increased. Thus although the coupled mode approach is not confined to low levelheterogeneity, it is at is most effective when the structure does not deviate too far from the I

reference. A convenient working limit would seem to be about 2 per cent deviation as inmodel E.

In order to understand the effect of propagation through a heterogeneous model on thenature of regional phases we need to understand the characteristics of the surface wavemodes in the reference model. In figure 5 we display the group slowness behaviour as afunction of frequency up to 2 Hz for the first 40 Love modes on the ARANDA referencestructure. The advantage of displaying the group slowness (the reciprocal of the groupvelocity) is that the various arrivals can be recognised in the time relationship that theywould have on a seismic record.

The Sn phase can be recognised in figure 5 as the superposition of many slownessmaxima and minima at a slowness around 0.22 s/km (4.5 km/s). All 40 modes contributeto this tightly defined band of slownesses which clearly separates from the other classes ofarrival. The onset of the Lg phase at a slowness of 0.286 s/km (3.5 km/s) is associatedwith the superposition of a number of relatively broad slowness minima arising fronrelatively low order modes. Following this onset is a more tangled skein of maxima andminima with slownesses less than 0.304 s/km (i.e. group velocities greater than 3.3 kmn/s),associated with about four modes at each frequency, which will provide the main Lg Sarrival. Somewhat later, we have a sequence of sharply defined slowness maximaassociated with each mode in turn as the frequency increases. These maxima represent theAiry phase for each mode and will be significant contributors to the Lg coda: they arisephysically from multiple reflections at angles to the vertical just greater than the criticalangle for the crust-mantle boundary.

Figure 5: Group slowness beviour as a function of frequency for the first 40 Love modes of the ARANDAreference model illustrated in figure 1.

2.0

1.5

Ur 1.0

0):L

,20 .30 ,40

Group slowness s/km •

"-" ... ...

From the modal eigenfunction patterns displayed in figure 1, we can recognise thatthose modes (3-6) which contribute most to the onset of Lg at 1.5 Hz have the bulk of theirenergy confined to the middle and upper crust; The main Lg arrivals come from modes7-10 which sample the whole crust. The Airy phases for modes 14, 15 arise just beforethe transition to energy transport in the mantle.

The effect of varying the heterogeneity level in the structural models can be wellillustrated by comparing the energy distribution across the modal sequence for a singleincident mode. In Table 1 we show this energy distribution for a set of incident mode at 0km, at a frequency of 1.5 Hz, chosen to represent different parts of the Lg wave phase.We consider horizontal transmission through 1000 km of structure for four differentmodels C, D, E and F with increasing levels of heterogeneity. Model C has 1 per cent ofhetrogeneity confined to the crust. Models D, E and F are illustrated in figure 3, and havethe same heterogeneity model in the mantle with about 1 per cent variation from thereference model (ARANDA, fig 1), but increasing levels of crustal heterogeneity (±1 percent for D, ±2 per cent for E, and ±5 per cent for F).

At 1.5 Hz, mode 4 is a principal contributor to the onset of the Lg wave train. FromTable I we see that for the incidence of this single mode at 0 Ian, the majority of the energyis carried in the original mode over the full 1000 km propagation path for the structureswith up to 2 per cent variation away from the reference model (C,D,E). However, theproportion of energy in the original mode decreases as the crustal heterogeneity increases.As expected the mantle heterogeneity has very little influence on this group of interactingmodes which is principally confined to the crust. Once we reach the highest level ofheterogeneity (model F) the pattern of the energy distribution is markedly changed, theenergy maximum is shifted over two modes to mode 6 which has a slightly different groupvelocity and energy is spread widely across the whole suite of crustal modes. There is verylittle interaction with the fundamental or first higher modes because, as can be seen fromfigure 1, their energy is confined to the near surface and so is only sensitive to a small partof the total crustal heterogeneity.

For a mode in the main Lg wave arrivals (mode 9), we see that for the two models withrelatively modest levels of heterogeneity (C,D) the energy is concentrated over threeneighbouring modes centered on the incident mode. These three modes have groupvelocities varying by about 0.15 km/s and so the effect of the this level of heterogeneitywill be to give a more diffuse maximum in the Lg wave amplitude. The presence of mantleheterogeneity now has a slight effect since it is possible to couple into modes with someenergy penetration into the uppermost mantle. For the higher levels of crustal heterogeneitythe incident energy gets spread out over a broader range of modes (6 are significant formodel E and at least 10 for model F) and the effect on the Lg wavetrain will be moreprofound. For modest levels of heterogeneity (around 1 per cent) the best measure of Lgwave strength will be the integrated energy in a time window spanning group velocities ofabout 3.6-3.3 km/s for a typical continental situation.

Similar results arise for the Lg wave coda (incident mode 14) though this case, A:ierethe incident mode is most sensitive to lower crustal structure, is not as strongly affected bythe heterogeneity as for mode 9. Even so, with increasing heterogeneity levels therepresentation of the propagation via the modes of the reference structure requires extensivemode coupling. For the higher heterogeneity levels there is significant energy shift into Sntype modes, which we can explain physically as occurring from local changes in the criticalangle at the crust-mantle boundary. Modes like 14 travel close to the critical angle in thereference model and so a slight change of conditions can lead easily to energy transmissioninto the mantle.

81.

Table 1: Energy distribution across the modes of the ARANDA model at 1.5 Hz, aftertransmission through 1000 kIn of heterogeneous structure

a) Incident Mud, 4 (onset of Lg) -

Relative Mode Number. -3 -2 -1 0 1 2 3 4Model: C .000 .032 .048 .689 .109 .078 .003 .023

D .000 .032 .044 .689 .109 .078 .006 .023

E .000 .040 .212 .563 .032 .036 .084 .002

F .001 .098 .082 .201 .057 .408 .071 .024

b) Incident Mode 9 (main Lg arrival)

Relative Mode Number: -4 -3 -2 -1 0 1 2 3 4Model: C .006 .014 .006 .336 .360 .240 .010 .000 .000

D .006 .017 .006 .336 .348 .250 .012 .000 .000

E .008 .023 .144 .090 .423 .160 .078 .004 .000

F .078 .102 .003 .303 .260 .053 .068 .048 .026

c) Incident Mode 14 (coda of Lg)Relative Mode Number. -4 -3 -2 -1 0 1 2 3 4

Model: C .000 .000 .001 .032 .846 .090 .001 .000 .000

D .000 .000 .001 .029 .828 .109 .001 .000 .000E .000 .000 .020 .014 .397 .476 .073 .001 .005

F .001 .008 .006 .084 .672 .109 .006 .020 .014

0 ~ NJN.

f..

aU - -

For incident modes 4 and 9, the nature of the energy spread induced across the modesof the reference structure has been to principally couple Lg type modes together and so thepattern of energy spread shows a skew towards higher mode numbers for mode 4 andlower numbers for mode 9. Whereas, for incident mode 14, the coupling is largely into athe immediate neighbouring modes with similar character and also into the Sn modes.

We should note that even for the moderately heterogeneous model D (fig 3) there hasbeen a significant modification of the distribution of energy between modes after passagethrough 1000 km of structure. Such modifications over a band of frequencies will lead totheoretical seismograms which will differ significantly from the predictions for thehorizontally stratified reference model.

Studies of Three-dimensional HeterogeneityIn addition to the work on two-dimensional structures, efforts have been made over the

past year to extend the mode coupling techniques to three-dimensional models. Forrelatively weak heterogeneity it has proved possible to cnnstruct a approximatedevelopment which should be able to give a good represenation of the Lg wave field whenmost significant energy travels close to a direct path between source and receiver, and somultipathing is not too important. This approach will require the solution of partialdifferential equations for the coupled modes in a spatial swath about the direct path andboth Love and Rayleigh wave modes will need to be considered at the same time (cfSnieder 1986). Thus even at 1 Hz at least 30 coupled partial differential equations will berequired. The number of modes needed at each frequency will be at least double that forthe two-dimensional case and also now calculations will need to be carried for a number ofoffset points in the transverse direction to the path. This requirements will place heavydemands on computational resources in future work and as a result considerable effort isbeing expended on the design of suitable algorithms for the calculations.

DISCUSSION

The current research effort has been directed towards determining the numericalcharacteristics of the Kennett (1984) mode-coupling scheme for guided wave propagationover extended distances at frequencies of 1 Hz and above. The existing models aretwo-dimensional and calculations are carried out for one frequency at a time.

The results of the studies of heterogeneous crustal models show that reflection fromdistributed heterogeneity is not very important. However, both reflection and transmissionprocesses should be included, if at all possible, in the numerical calculations in order toensure the accuracy of estimates of modal transmission. The coupled mode techniqueworks well with heterogeneous models which differ from a stratified reference model-by upto 2 per cent with a limited bandwidth of interaction between modes. The approach can beused for higher levels of heterogeneity but a large number of modes needs to be consideredwith consequent high computation costs.

One of the major effects of crustal heterogeneity is to introduce the possibility ofsmearing out the main amplitude peak in the Lg wave train over a band of group velocities.As a result, an effective measure of the energy content of the Lg waves will be to consider

the integrated amplitude along the traces beween group velocities of 3.6 and 3.3 km/s. Theinfluence of mantle heterogeneity on Lg wave propagation was found to be quite small.However, any conversion into Sn type modes will give arrivals which will have apparentgroup velocities of between 4.2 and 3.7 km/s, depending on the position where conversionoccurs. Such arrivals will appear in a portion of the seismic record which would bepredicted to be very quiet in stratified medium calculations, and as a result may appear to bemore prominent than expected.

10-,/'

The costs of performing the coupled mode calculations, at fixed step length in thehorizontal direction, would rise as the square of the number of modes employed (andtherefore approximately as the frequency) if full modal coupling is allowed. Theintroduction of a limited bandwidth of interaction reduces the cost but still it will be difficultto push the calculations to very high frequency. At 2 Hz, about 35 modes are needed for atypical continental model to give an adequate representation of the effects of heterogeneityon either Love or Rayleigh waves, and this would increase to about 70 modes for 4 Hzpropagation. As the wavelength reduces the horizontal step length in the solution of the-Vdifferential equations needs to be reduced to maintain accuracy. This gives a furtherincrease in computational cost approximately linear in frequency. The numerical algorithmsused for the solution of the matrix Ricatti equations for reflection and transmission workwell with large numbers of modes, but the differential equations are non-linear and there isthe possibility of loss of precision if very large numbers of coupled equations areconsidered.

Nevertheless the coupled mode approach to the calculation of guided wave propagationhas the potential to enable the character of Lg wave propagation to be determined forclasses of model which cannot easily be treated by any other means. Both deterministic andstatistical models of heterogeneity can readily be incorporated into the computation scheme. 0The coupled mode scheme does therefore allow an assessment of structural effects onregional phase propagation which is complementary to existing techniques usinghorizontally stratified models.

The next phase of work will be directed towards using the experience from thetwo-dimensional models to design a computational scheme for mode-coupling inthree-dimensions for both Love and Rayleigh waves. This scheme will take advantage of 0approximations such as limited bandwidth of interaction to reduce total computation time.We will also pursue the problem of generating effective and consistent theoreticalseismograms by combining mode-coupling results from a number of frequencies.

REFERENCES •

Kennett B.L.N. (1972) Seismic waves in laterally inhomogeneous media,Geophys JR astr Soc, 27, 301-325

Kennett B.L.N. (1984) Guided wave propagation in laterally varying media -I. Theoretical development, Geophys J R astr Soc , 79, 235-255 OP

Kennett B.LN. (1986) Wavenumber and wavetype coupling in laterally heterogeneousmedia, GeophysJR astr Soc, 87, 313-331

Kennett B.L.N. & Mykkeltveit S. (1984) Guided wave propagation in laterally varyingmedia - I Lg wave in northwestern Europe, Geophys J R astr Soc, 79, 257-267

Maupin V. & Kennett B.L.N. (1987) On the use of truncated modal expansions in laterallyvarying media, Geophys J R astr Soc, 91, 837-851 0

Snieder R. (1986) 3-D linearized scattering of surface waves and a formalism for surface J%wave holography, Geophys J R astr Soc, 84, 581-605 i%

I1t

APPENDIX: Modal evolution equations

1. ro-diuensional probJees

We present here a oriet derivation of the coupled modetechnique introduced by Kennett (1984).

We will firstly restrict attention to a two-dimensionaliyheterogeneous structure and work with a fully anisotropic mediumfor which a very compact notation is available. The heterogeneityis assumed to be invariant in the transverse direction to the wavevector h , and so the direction of this vector will remainconstant even in the anisotropic case where the ray directionassociated with a particular mode does not lie along 9

We consider the heterogeneity as a deviation from theproperties of a reference model. This perturbation is notnecessarily small but must be bounded in space. We seek arepresentation of the displacement field in terms of asuperposition of the modal eigen+unctions of the reference modelwith modal coefficients which vary with position.

We work in a cartesian coordinate system, with propagationin the .Z-Z plane and introduce the displacement vector

=(ax, , . We also need to specify the traction field.In the case of a modal wavetrain propagating in the x -directionwe are concerned with continuity of traction on vertical planesand so we take -(r,, z, ) In addition the meterialcontinuity requirements mean that we also need to be able to referto the traction on a horizontal planE ( jr,', , ) . Forthe 2-D situation we need a number of combinations of elasticmoduli which may bu represented compactly in terms of matrices

such that

where the Cei are the anisotropic moduli.The equations of motion and the stress-strain conditions can

be cast into a form where the derivatives with respect to thepropagation direction r appear only on the left hand side ofthe equations

in the presence of a volume force contribution IThe differential operators 4.. do not depend on the horizontalderivatives of the material properties and for the 2-D case navethe explicit form

(2)

where we have written -1 for P/P and set

The unclosed brackets in (2) indicate that the operator acts toits right. ALross an interface Z = eo , the z; component o.

13

if"I

I

will be continuous, but Z , are not required to becontinuous. On the other hand 4 will be continuous and canbe represented in terms of f, and 4 as

= 4, 4C, ( (4)

The equations (I)-(3) apply in an arbitrarily heterogeneous mediumbut by themselves provide no direct link to the surface wave casewe wish to consider.

When the heterogeneous medium does not deviate too stronglyfrom the reference medium, we can envisage that it may be possibleto find a representation in terms of the surface wave modes of thereference structure. We here consider the case of a fixed,stratified, reference medium as used in the examples of Kennett(1984) and Kennett & Mykkeltveit (1984).

At each position X we consider cutting the actual structurealong i vertical plane and then weld on the reference structure toensure continuity of displacement * and traction 4 . In thisreference structure we can now represent the displacement andtraction at x as a superposition of modal contributions andwrite

( .&) Z) C(5)/i .) ~ (4 .where the A are the displacement eigenfunctions for the

reference medium and the horizontal tractions Ilf are derivedfrom A The sum is to be taken over all relevant modes atfrequency W In order to give a full representation of thedisplacement this sum will normally involve both forward andbackward travelling waves (i.e. both positive and negative 4 ).As discussed by Kennett (1984) we can arrange the deepest partof the model of the reference medium so that (5) is a sum overan infinite set of orthogonal modes, and achieve a completerepresentation of the seismic wavefield. For example, theconstructive interference of sufficient numbers of surface wavemodes will synthesise body wave phases (see e.g. the examples inchapter 11 of Kennett (1983)). However, especially at higherfrequencies, this requires a very large number of modes. In thevarying medium we wish to concentrate on the surface wave fieldand so would like to be able to work in terms of a restricted (andnot too large) set of modes. Such a truncation of the expansion(5) imposes limitations on the size and character of theheterogeneity which can be tackled.

We recall that r, , T, are, in general, not continuous'acrosshorizontal interfaces and so if the material discontinuities inthe heterogeneous model do not coincide with those in thereference model, we have the problem of representing a jump in

across an interface by a superposition of continuoustraction vectors t A satisfactory fit -an be achieved withmany modes, but the accuracy of the representation may be limitedwith a restricted mode set (cf. Fourier series). As a result itis desirable that the major discontinuities in both theheterogeneous and reference models should be coincident.

14

In the reference structure the modal contributions (5)satisfy the coupled equations

r(6)0

in the absence of any applied force, with coefficients, - C. '.) - We have written 4*

to indicate the form of the differential operators for thereference medium.

In a laterally varying medium we have to require the modal

coefficients to vary with position. On expanding thedifferential operators 4 as 4' o, JA , to separate off thereference medium contribution, we find

These equations may be rewritten as

in hich we can recognise the left hand side as having the same.form as in (6), but we have in addition, on the right side, a termwhich is equivalent to a generalised volume force applied to thereference structure.

As well as the equation of motion we require the wavefield tosatisfy certain continuity conditions at internal amd externalboundaries where the elastic moduli are discontinuous. The mostgeneral case of an interface in the heterogeneous medium will betitled and displaced from those of the reference medium. Continuityof displacement is assured at any interface from the continuityproperties of the displacement eigenfunctions. We also requirethat the tractions normal to any interface should be continuous.We will assume that all such surfaces do not vary rapidly in thehorizontal direction. Then for an interface in the

laterally varying medium described by the function A4) , we may

work to first order in the slope ( A. ), and require

where I denotes the identity matrix and the square bracketsindicate the jump in the enclosed quantities across the interface,evaluated from bottom to top.

If we now separate out the contribution from the propertiesof the reference medium from the local variations, we may write

" -" C" Z

The difference terms such as i' in (10) indicate thediscrepancy between the combinations of elastic moduli for the

15

~A4

heterogeneous and reference models. In the absence of any lateralheterogeneity the expression on the left hand side of (10) wouldvanish. The presence of the heterogeneity is thus equivalent toinducing a traction discontinuity into the reference medium, alongthe line of the interface. Such a traction discontinuity isequivalent to a localised volume force along the interface

/ r,,) - - z -/,&)) I

(see e.g. section 3.1 of Aki & Richards (1980)).We can therefore represent the combination of the equations of

motion and the boundary conditions in the laterally varyingmedium by means of the following system of equations

where the force term is a summation of interface contributions

-. / .9Co114[f eowz Y-. '-(4.

With the abstraction of this force contribution the interfaceconditions on the wavefield are reduced to

[~ ~ ~~~~~- Z.4 #1=~ ~4)( ~7'.

where the subscript ft indicates the jump at any interface.These constraints will be automatically satisfied by therepresentation (5). Equation (11) can be further simplifiedbecause the modal eigenfunctions are just the free solutions forthe reference structure and so the left hand side of the equationvanishes. This leaves us with

r ~~*r

We now exploit the orthogonality relation between differentmodal eigenfunctions for the reference medium

~~~~Z. V I(z )(,)1=§ (3

to get a set of coupled first order differential equations +or themodal coefficients / , appropriate for both elastic andanelastic heterogeneity. These equations can be written in theform

d /. f4-' 0 -4 (4 )&.4'I : ,4::r

where we have written

16

r A5 '

Equation (14) can be simplified by integration by parts, to yield 0

r .7 a. (15)

In which we see that there is no explicit interface term in thecase of. a flat boundary ( . - 0 ). However, the modificationof an interface will appear through the integral term in (15),where the properties of the laterally varying medium differ fromthe reference and there will be a significant contribL.'-on to theintegral which will vary with horizontal position. In afirst-order perturbation theory the whole of this effect would be A

projected onto a specific interface term, but here we are able tomake a more detailed allowance for the behaviour.

The combination of the modal representation (5) with theimposition of continuity of traction at each interface thusleads to the set of coupled equations

Z- 1. Z (16)

for the modal coefficients, where the coupling coefficientsbetween the modes can be written as

-44

2'. 4 f . < (17)

The coupled first-order equations (16) are not very easy tosolve because we have a two-point boundary value problem with bothreflected and transmitted modes to be determined. Kennett (1984,section 3) has shown that an effective procedure is to work with ithe reflection and transmission matrices for a sequence a+ modelsencompassing increasing portions of the heterogeneous medium.This leads to an initial value problem for two coupled matrixRicatti equations. These nonlinear differential equations for thereflection and transmission matrices can be readily solvedntmer i ca I y.

17

A fs-

2 Three-dimensional Problems

In a medium whose properties vary in three dimensions,it is possible to generate evolution equations for the

displacement and horizontal tractions with respect to thehorizontal coordinates. The analysis parallels the two-dimensionalcase, but now one has to take account of variation, in both thematerial properties and the wavefield, transverse to the sirectionof current interest.

if we concentrate on the evolution in the X -direction we find

(, . - C -,, /-,/3Z

(18)

where -

In equation (181, the terms have been written in an order which

emphasises those contributions which appear in the two-dimensional-formulation. Previously we were able to eliminate the dependence ondepth Z and convert from evolution equations in displacement andtraction to equations in the modal expansion coefficients bymaking use of the modal orthogonality properties for modestravelling, in the same (or apposite) directions. With 3-D

heterogeneity we now have the possibility of modes travelling indifferent directions and also coupling between Love and Rayleighmodes. Unfortunately, we cannot write down any simple form for

the representation of the wavefield as a sum of modalcontributions.

If we consider a situation where the heterogeneity is not toostrong and concentrate attention on propagation dominantly alongthe r direction, we can generate a useful approximation bynegl-ecting the angular dependence of modal interaction. By thismeans, we obtain a generalisation of the 2-D equations with a weakdependence on the transverse ( _y ) direction. The partialdifferential equations in x for the modal coefficient are then tobe integrated over a swath surrounding the direct propagation pathfrom source to receiver.

18

DISTRIBUTION OF RESEARCH RESULTS

a) PUBLICATIONS:

Lg-wave propagation in a heterogeneous crust,(B.L.N. Kennett) Inpreparationfor submission to the Bulletin of the Seismological Society of America

b) PRESENTATIONS:

On representing surface wave propagation in laterally varying media by truncated modalexpansions

DARPA/AFGL Seismic Research SymposiumNantucket, Massachusetts, June 1987

Guided waves in laterally heterogeneous mediaSeminar, Hawaii Institute of Geophysics, August 1987

. .

.,p

.,.

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