Pageoph, Vol. 114 (1976), Birkh/iuser Verlag, Basel

Surface Waves in Isotropic, Laterally Inhomogeneous Media By S. GREGERSENi) Summary - Two different viewpoints of the phase velocities of the elastic surface waves in isotropic, laterally inhomogeneous media have led to inconsistent results. Arguments in terms of surface wave modes give the conclusion that the phase velocity is independent of the propagation direction, while the outcome of calculations based on a constructive interference of body waves in a surface layer is that the phase velocity is dependent on the propagation direction. Both arguments are summarized and an error in the calculations giving dependence is pointed out. The calculations and observations of surface wave amplitude changes in laterally inhomogeneous media are also summarized.

1. I n t r o d u c t i o n

The theoretical analysis of surface wave propagation was developed for laterally homogeneous media, i.e. media with no change in elastic parameters or in interface depths in the horizontal direction along the free surface. In the earth the assumption of lateral homogeneity is good for m a n y purposes and for m a n y regions. But when a surface wave passes regions where the elastic parameters or the interface depths change (e.g. continental margins) the interpretation of phase velocities and amplitudes in terms of lateral inhomogeneity is important. The theoretical problem of elastic waves in a laterally inhomogeneous medium can only be solved exactly in special cases, so the best we can do generally is to make mathematical approximations and try to understand physically the outcome. One question of which there has been debate in the seismological literature is whether the phase velocities back and forth in a two-dimensional inhomogeneous model are equal. The question will be reviewed and an error in the arguments against equality will be pointed out. Afterwards a description will follow of the various attempts at surface wave amplitude calculations and observations that can be found in the literature concerned with laterally inhomogeneous media.

2. L o v e w a v e m o d e s

For isotropic and laterally homogeneous media we use plane wave theory and assume (1) steady state, in which the motion continues throughout all time with the 1) Geodetic Institute, Gamlehave AUe 22, DK 2920 Charlottenlund, Denmark.

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same period of oscillation, (2) infinity of the half-space, in which the motion is the same for all values of x from minus infinity to plus infinity, and (3) no dependence on the y-coordinate perpendicular to the propagation direction. For one of the discrete surface wave modes the phase velocity, c, of a certain period of oscillation is the velocity of a peak, a trough or any other phase. It is related to the horizontal wave number, kx = oJ/c, if the phase factor of the wave is written as e i(~ where t is the time, x is the space coordinate in the direction of propagation, co is the angular frequency. We can either specify c or kx, when we want to describe the propagation of the Love wave. And since the waves satisfy the wave equation we may as well specify the vertical wave number k, at some arbitrary depth. In a laterally homogeneous model of one homogeneous layer over a homogeneous half space the following relations are valid: k~ + k~l = k]l = (o~/t~l)~, k~ - k ~ = k ~ = (~/~)~,

(1)

where the fl's are the S wave velocities, the k's are the absolute values of the wave numbers, and the indices 1 and 2 refer to the layer and to the half space respectively. For any value of x the wave equations in the elastic media and all the boundary conditions have to be satisfied. The resulting period equation gives the relation between the elastic parameters and one of the parameters describing the propagation of the Love waves (c, kx, or k~l). The simplest way of writing the period equation for the plane Love waves in the laterally homogeneous one-layered model is in terms of the vertical wave numbers: tan (kzlH) = ~2k~2 /~1k=1'

(2)

where index 1 refers to the layer, index 2 refers to the half space, the k's are the absolute values of the wave numbers related by equation (1), H is the thickness of the layer, and the/z's are the rigidities. By this period equation we determine the vertical wave numbers, which describe the fall-off with depth (z) of the displacement functions that are possible in the elastic model. Those displacement functions constitute an orthogonal set of eigenfunctions with respect to an inner product which is the elastic energy (ALSoP, 1966; MCGARR and ALSOt,, 1967), when the mean energy is taken over one period of oscillation. An important characteristic of an eigenfunction, which will be used later, is that it propagates without any loss or gain of energy. The eigenfunctions in the one-layered model consist of the following two parts: cos (k~z) in the layer, and cos (k~H) e-kz 2r in the half-space if z is positive downwards. Figure 1 illustrates eigenfunctions for more complicated layered models. The eigenfunctions may also be called the mode shapes with depth of the Love wave modes. From one of the vertical wave numbers we can determine the phase velocity by equation (1) and c = oJ/kx. The same concept of eigenfunctions may be used in a laterally inhomogeneous

Vol. 114, 1976)

SurfaceWaves in Isotropic, Laterally InhomogeneousMedia 60 s period

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Figure 1 Love wave displacementfunctions (upper part) and energy functions (lower part). All curves are normalized to the same maximumabsolute amplitude. model, where the layer thickness and/or the elastic parameters vary only slightly with the x-coordinate within a wave length, e.g. (Kuo and NAFE, 1962). The eigenfunction relevant for a certain point, O, at the surface goes along a wave front through O from the surface downward. That wave front is plane in the laterally homogeneous model, while it may be curved in the laterally inhomogeneous model. For models of homogeneous layers of simple geometry (TAKAHASm, 1964) the wave fronts follow analytic expressions, because we can find a coordinate system in which we can separate the variables of the differential equations. In a one-layered model of hyperbolically varying layer thickness the wave fronts are the ellipsoidal surfaces orthogonal to the surface as well as to the layer interface (TAKAHASHI,1964). For models of complicated geometry or with varying elastic parameters in the propagation direction the separation of the variables of the differential equations is not possible, i.e. it is not possible to define a coordinate system in which the eigenfunctions do not change shape. And when there is a change in shape of the eigenfunctions only part of the incident energy continues in the incident mode, and reflected and transmitted surface wave modes as well as body waves are generated (GREGERSEN and ALSOP, 1974). But we may locally imagine an eigenfunction along an energy front through point O. That energy front is a curve perpendicular to the flow of energy, and

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it is identical to the wave front if the energy is assumed conserved in the local eigenfunction. The flow lines will go through points of equal relative displacement amplitudes for the eigenfunctions through neighbouring points to O. Some curves of displacement and of energy as a function of depth are illustrated in Figure 1. The discontinuities of the energy functions of Figure 1 are caused by the discontinuities in the elastic parameters at the interfaces of the model. If flow lines cross an interface in the model they will be refracted according to Snell's law just as ordinary body wave rays. An approximation to a flow line pattern may be determined between the plane ' eigenfunctions' at two neighbouring points to O, and the energy front is perpendicular to the energy flow lines. In a laterally inhomogeneous model we can at any point, O, determine the approximate eigenfunction as described above, and thereby we determine the vertical wave number (k~l). When kzl at the surface is determined the phase velocity can be calculated from the horizontal wave number of equation (1). That local phase velocity depends solely on the vertical wave number. Consequently it is the same for any horizontal direction in the laterally inhomogeneous model. It is independent of the direction of propagation upslope or downslope or toward increasing or decreasing elastic parameters. The arguments presented are valid for Rayleigh wave eigenfunctions as well as for Love wave eigenfunctions. In a laterally inhomogeneous model there will be an imaginary component of the horizontal wave number, i.e. the surface amplitude is exponentially decreasing or increasing in the propagation direction (KNoPOFF and MAL, 1967). When adjoining energy flow lines in the vertical profile of the model diverge away from each other the amplitude decreases, and when adjoining flow lines converge toward each other the amplitude increases. There may be depth ranges where the flow lines converge and other depth ranges where the flow lines diverge. The observable amplitudes at the surface decrease or increase dependent on whether the flow lines immediately below the surface diverge away from the surface or converge toward the surface. And here the propagation direction makes a difference. For propagation in the one direction there is a decrease, and for propagation in the opposite direction, there is an increase of the surface amplitudes.

3. Constructive interference of S H body waves The plane surface wave modes in a laterally homogeneous model (Figure 2a) may also be treated in terms of a constructive interference of body waves (HASKELL,1953; OFFICER, 1958). For plane Love wave modes there is an equivalence with a constructive interference pattern of plane SH waves and for plane Rayleigh wave modes the equivalence is with plane P and SV waves that interfere constructively. A fundamental reason for the equivalence is that the amplitude-fall-off with depth of a plane Love wave mode in the layer of the model in Figure 2a can be recon-

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0

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Figure 2 Constructive interference of SH body waves in laterally homogeneous (a) and laterally inhomogeneous (b) models of one homogeneous layer over a homogeneous half-space. Solid lines with arrows are SH rays, dots on a line are wave fronts and the dashed line is the vertical reference. 90~ angles are indicated between rays and wave fronts. structed as the interference pattern between up- and down-propagating plane SH waves, which interfere constructively. Ae~(~- k x x - k ~ Ae~(~ ~)

is a plane SH wave that propagates in the positive x and z directions. is a plane SH wave that propagates in the positive x and in the negative z direction.

Both have the wave number kx in the x (horizontal) direction and the wave number k~ in the z (vertical) direction, A is the amplitude, t is the time, and co is the angular frequency. The sum of the two plane SH waves is: Ae~(~t-~,, x) (e~e,~ + e-~k,~) = Ae~C~

x~ 2 c o s k~z,

where the factor cos k~z is the Love-wave-fall-off with depth in the layer of the model in Figure 2a. This constructive interference pattern has no propagation in the z direction. Together with an inhomogeneous wave in the half space (GRmERSEN and ALSOP, 1974; ALSOP,GOODMAN and GREGERSEN,1974), which falls off exponentially at depth, it constitutes the plane Love wave propagating in the x direction. In the laterally inhomogeneous model the equivalence between the mode approach and the body wave approach locally in the model is accepted by the present author. The requirement of constructive interference of non-plane SH waves (Figure 2b) involves the same degree of approximation as that of the local eigenfuncfions. A number of papers by P~:~ (1967), BHATTACHARYA(1970), CHATTERJEE(1972), NEGI and SINGH (1973), SINGH (1974) are based on the concept of constructive interference of SH body waves in models of varying interface depth (Figure 2b) or constant interface depth and varying elastic parameters. The outcome of these papers is that the phase velocity at any point is dependent on the direction of the propagation with respect to the variation in the elastic parameters. And that outcome is erroneous because the calculations are made for a downgoing wave only on one side of the point of interest,

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while the equivalence between the mode approach and the body wave approach was derived for the superposition of up- and downgoing body waves (HASKELL, 1953; OFFICER, 1958). When the down-going wave on one side of the point of interest is considered alone (Figure 2b, ray OQ) elastic parameters and depths on only one side of the point of interest are taken into account. And both sides have to be used. If we calculate a phase velocity for a wave peak for instance from a point just before point O to a point an equal distance after point O, the local phase velocity will be the limit of that phase velocity, when the distance is decreasing toward zero. That local phase velocity will be equally dependent on the elastic parameters and depths on the two sides of our point of interest. So the error in the above mentioned papers leading to the phase velocity dependence on the propagation direction is that elastic parameters and depths on one side only of point O are taken into account in the calculations. If the period equation is to be set up by the body wave approach, some kind of averaging is necessary between calculations for SH waves propagating up to our point of interest from one side (ray PO) and down from our point of interest to the other side (ray OQ), i.e. some kind of averaging between calculations including elastic parameters to the right and to the left of the point of interest. From the point of view of averaging it is apparent that the velocity is independent of the direction of propagation, toward increasing or decreasing elastic model parameters or depths, because the same SH waves are considered one way and the other way in opposite order.

4. Comments on the phase velocity The point of view of eigenfunctions was taken in papers by Kuo and NAFE(1962), KUO and THOMPSON(1963), TAKAHASHI(1964) and by KNOPOFFand MAL (1967). In the paper by Kuo and THOMPSON (1963) the phase velocity independence of the propagation direction is supported by laboratory experiments. Independence was also the result of finite element calculations by DRAKE(1972). In papers on ray tracing of Love waves in a medium of lateral inhomogeneity in two dimensions (x and y) (GJEWK, 1973 ; WOODnOUSE, 1974) the viewpoint is also taken of local eigenfunctions dependent on the local elastic parameters and depths but independent of the direction of propagation. GJEVlK(1974) has introduced the term isotropy of the period equation for the phase velocity independence of the propagation direction. It is argued above that an acceptance of the ideas of local eigenfunctions or local constructive interference of body waves leads to isotropy of the period equation. If observations show phase velocity dependence on the propagation direction, they should be interpreted in terms of interference of reflected surface waves and in terms of mode conversion and scattering of body waves in the laterally inhomogeneous medium, in the same manner as it may happen at a lateral inhornogeneity in the form of a sharp boundary between two models (KNoPoFF and MAL, 1967;

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GREGERSEN and ALSOP, 1974). Such observations should not lead to suggestions of anisotropy of the period equation. The finite difference calculations performed by BOORE (1970) did show dependence on the propagation direction across a zone of lateral inhomogeneity (10~ difference). The present author supports a previous suggestion (BOORE, 1970; DRAKE, 1972) that BOORE'Sresult is caused by interference of higher modes, reflected modes and scattered SH waves, which are generated by the lateral inhomogeneity.

5. Surface wave amplitudes The surface wave propagation in laterally inhomogeneous media can be treated by theoretical methods as well as by laboratory experiments. The theoretical methods include analytical approximations as well as numerical solutions of the wave equations with boundary conditions. The relative simplicity of Love waves makes them best suited for the theoretical treatments. On the other hand Rayleigh waves are well suited for laboratory experiments. Theoretical treatments of Rayleigh waves have the inherent complication that the equations contain two elastic potentials, while for Love waves the elastic displacements are scalar. Theoretical treatments of Rayleigh waves have often been supported by experimental results, while this is not the case fbr Love waves. In a very special model of a lateral inhomogeneity HmUCHI (1932, in Japanese) and SAT(5(1961) have investigated the transmission and reflection. On the two sides of a vertical interface the elastic parameters were chosen such that they were different in such a way that the Love wave eigenfunctions were equal. Here all boundary conditions may be satisfied with Love wave modes alone, and the transmission and reflection coefficients may be calculated analytically as functions of the elastic parameters. Attempts at analytical approximations in more general structures have been based either on a Green's function approach (KNOPOFF and HUDSON, 1964; MAL and KNOPOFF, 1965) or on a variational technique (ALSOP, 1966; MCGARR and ALSOP, 1967). In the Green's function approach the elasticity representation theorem is used, which states that any disturbance in a medium without sources is caused by the disturbances at the boundary of the medium. MAL and KNOPOFF (1965) remind us that they have hereby described the transmission and reflection of the surface waves in terms of Huygens' principle, i.e. outgoing waves are generated by virtual sources at the boundary between two different structures. Only simple models of a step inhomogeneity were treated. GHOSH(1962) used a similar approach on a model as in Figure 2b. He integrated the effect of a gradually changing depth and made approximations to the analytical result. ALSOP (1966) developed a variational approximation method for the Love wave amplitude calculations at a vertical boundary between two different structures. He satisfied the boundary conditions on the vertical interface by Love wave eigenfunctiOns

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as well as possible in a least squares sense. His method was later extended to Rayleigh waves by MCGARR and ALSOP (1967). The other types of theoretical solutions to the propagation problem of surface waves across lateral inhomogeneities are numerical solutions of the elastic differential equations with the relevant boundary conditions. This approach has only recently been taken into use since it requires very fast computers or long computing times. In the finite difference method some pulse is followed through a set of grid points in the model by difference formulae that are approximations to the derivatives in the differential equations. This method has been used in seismological problems by ALTERMAN and KARAL (1968), ALTERMAN and ROTENBERG (1969), BOORE (1970), ALSOI' and GOODMAN(1971), OTTAVIANI(1971), and ALSOP, GOODMANand GREGERSEN (1974). In the paper by BOORE (1970) the surface waves receive the principal interest. The model contains a zone of varying layer thickness between two laterally homogeneous structures. BOORE finds that amplitudes of transmitted Love waves close to the zone of the sloping interface are affected by interfering waves. There is up to 259/0 difference from a composite of the older approximations. BOORE (1970) suggests that he is observing the interference phenomena of waves generated in the zone of lateral inhomogeneity. In the finite element method on the other hand the response is computed to a steady state oscillation of a structure consisting of small elements. LYSMER and DRAKE (1971) and DRAKE (1972) have made computations for models containing complicated intermediate zones between laterally homogeneous structures. The finite element calculation is used for transmission of a surface wave mode through the intermediate zone, and components of the resulting displacement-stress field are taken on the eigenfunctions of the laterally homogeneous structure. A result reported by DRAKE (1972) was that the transmitted energy through a transition zone is the same for propagation one way and the opposite way. Another procedure of transmitting the displacement-stress field through an inhomogeneity and taking components on eigenfunctions has been developed for complicated models with a vertical discontinity between two structures and for propagation non-perpendicular to the margin by GREGERSEN and ALSOP (1974), ALSOP, GOODMANand GREGERSEN(1974), GREGERSENand ALSO~"(1976). The new procedure is an extension of the Green's function approach (KNoPOFF and HUDSON, 1964; MAL and KNOPOFF, 1965) previously mentioned. While the finite element method can treat a complicated intermediate zone, the approximation method developed by ALSOl,and myself is faster, it treats non-perpendicular angles of incidence with the margin, and it treats realistic surface waves with a fall-off with depth proper for the spherical earth. Especially for the higher modes the last point is important. We have made observations and calculations of amplitudes of Love waves transmitted through a realistic continental margin (GREGERSENand ALSOP, 1974 and 1976). The observations are from the ocean bottom seismograph OBS approx. 225 km west of San Francisco and from the BKS station at Berkeley, California. The observations

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and the computations give the same trend of increasing amplitude ratios (continental/ oceanic) with decreasing period. Consequently there must be a difference between the eigenfunctions in the continent and in the ocean which is larger the shorter the period is in the interval 60-20 seconds. So it is essential that the oceanic structure contains a low velocity zone, in which the energy of the short period Love wave fundamental mode can be concentrated (Figure 1). Observations as well as calculations lead to the conclusion that there is almost no dependence on the horizontal angle of incidence with the continental margin for angles of incidence between 0 ~ and 60 ~ For periods in the interval between 20 and 30 seconds there is a significant mode conversion between the fundamental and the first higher modes of the ocean and the continent (GRmERSEN and ALSOP, 1976). The reason is related to the eigenfunctions of the oceanic and continental structures (Figure 1). When the shapes of the two eigenfunctions are similar, the energy coupling coefficient between them is close to one, and when upper and lower parts of the eigenfunctions counteract, the energy coupling coefficient is close to zero. It is not enough to determine the energy coupling coefficient though. Important for the surface wave amplitude at the surface is the distribution of the energy with depth. The variational approximation technique (ALsoP, 1966) mentioned earlier is also extended to complicated structures and to non-normal incidence at the margin (MALISCHEWSKY,1973 and personal communication, 1975). Preliminary results have given good agreement with results obtained by the present author, when the calculated amplitudes are of the same order of magnitude as the incident amplitude, while there is poor agreement when the calculated amplitudes are an order of magnitude smaller than the incident amplitude. The mentioned coupling calculations by the present author and by MALISCHEWSKY(1973) are relevant when the transition zone is sharp measured with the wavelength of the waves in question. Recently ray tracing equations for surface waves and expressions for amplitudes and energy density have been found by WOODHOUSE(1974) and by GJEVIK(1973 and 1974). Essential in these studies have been isotropy of the period equation locally, and an assumption that all the incident energy remains in the mode under consideration in the investigated models of small inhomogeneities within a wavelength. Experimental investigations in nature or in the laboratory of the transmission coefficients of surface waves across inhomogeneities are very sparse. Few observations of amplitude transmission coefficients are obtainable on the earth for structures with really large contrasts between them, such as a continent and an ocean. Also amplitudes of surface waves as well as amplitudes of body waves are dependent on near surface local structure under the receiving station and on focussing effects (MCGARR, 1969b). A set of observations on the propagation of Rayleigh waves across a continental margin was obtained by McGARP, (1969a) for incidence perpendicular to the margin. The period interval of interest was 8-25 seconds. Almost no dependence on period was found for 12-25 seconds, while the data scattered for shorter periods. Observations for Love waves of periods between 60 and 20 seconds by GREGERSEN and

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ALSOP (1974, 1976), similar to MCGARR'S observations, have been mentioned earlier in this paper. For long period Rayleigh waves REITER(1974) has interpreted the scatter in observed amplitude fall-off with time for spheroidal free oscillations of period 300-350 seconds as ' scattering' caused by reflection, refraction and mode conversion. Using the duality between free oscillations and surface wave modes he found a 5 percent 'scattering' on the average for these long period Rayleigh waves for propagation once around the earth's circumference. In the laboratory MCGARR and ALSOP (1967) made Rayleigh wave transmission measurements past a step change at the surface and through a sharp vertical boundary between two homogeneous elastic quarter spaces. The results were in good agreement with the numerical result of the approximation method mentioned earlier in this paper (MCGARR and AESOP, 1967). L. MARTEE (personal communication, 1975) has recently refined the experimental results for the step change, and RYKUNOV and FAM VAN TCHUCH (1972, in Russian) have made experiments in models with a plane surface and a step change in a layer thickness. Another laboratory study of the transmission of Rayleigh waves past a step change was made by DALLY and LEwis (1968). They made a photoelastic analysis. A fringe pattern was generated by the propagating waves and photographically recorded, and by that stresses in the plate model could be evaluated. A counting of fringes gave the relative stress amplitudes between incident and transmitted/reflected waves. The results obtained by DALLY and LEWIS (1968) showed that the surface waves close to the corners (and as far as the models went) were influenced by interference effects. The transmitted Rayleigh waves were made up of three parts: a Rayleigh wave following the surface around both corners of the step, a Rayleigh wave caused by the coupling of the mode shape with depth, and a shear wave converted to a Rayleigh wave at the last corner. The results warn that interference between different waves does take place as expected close to the step.

Acknowledgements

The manuscript was improved because of comments from J. Hjelme and E. Hjortenberg.

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nonhomogeneous Layer with linear velocity gradient, Publications of the Dominion Observatory, Ottawa, 35, No. 1. L. REITER(1974), Reflection, Refraction and Mode Conversion of Long Period Surface Waves and the Measurement of Q- 1for Free Oscillations, Bull. seism. Soc. Am. 63, 1709-1722. L. N. RYKUNOVand FAM VAN TCnUCH(1972), Study of the Influence of Local Inhomogeneities on the Rayleigh Wave Field(in Russian), Izvestiya Akademia.Nauk SSSR, Fizika Zemli, 5, 65-71. R. SAT6(1961), Love Waves Propagated across Transitional Zone, Japanese Journal of Geophysics, 2, 117-134. V. P. SINGH(1974), Love-wave Dispersion in a Transversely Isotropic and Laterally Inhomogeneous Crustal Layer, Bull. seism. Soc. Am. 64, 1967-1978. T. TAKAI-IASHI(1964), Transmission of Love waves in a Halfspace with a Surface Layer whose Thickness varies Hyperbolically, Bull. seism. Soc. Am. 54, 611-625. J. H. WOODHOUSE(1974), Surface Waves in a Laterally Varying Layered Structure, Geophys. J. 37, 461-490. (Received 3rd February 1976)