The gradual standardization of synthetic seismogram codes for layered structures was reflected in the small number of papers addressing this classical problem e.g. Chen [1993]. However, a similar equilibrium is less evident in forward algorithms for waveform inversion, even for one-dimensional (1-D) structures. Clarke [1993ab] develops the Complete Ordered Ray Expansion (CORE) algorithm, to generate differential seismograms using asymptotic theory applied to a finite collection of rays. The CORE algorithm models waveform perturbations caused by 1-D variations in velocity and discontinuity surfaces. Though the CORE algorithm is readily adaptable to lateral variations in seismic discontinuities, other ray-based synthetic schemes incorporate these explicitly [ Richards et al 1991; Sen and Frazer, 1991]. Gee and Jordan [1992] and Li and Tanimoto [1993] eschew rays in favor of modal kernels for representing differential waveforms at long period, in order to incorporate P-S interaction in the grey area between body waves and free oscillations. At higher frequency, Randall [1994] describes a reflectivity-based algorithm for rapid calculation of differential seismograms for velocity shifts within constant layers.
Because the exact lowpassed seismic response of an elastic earth model can, in principle, be calculated in terms of a finite sum of free oscillations, such studies are often used to test the predictions of proposed models of mantle structure [ Resovsky and Ritzwoller, 1994; Ritzwoller and Lavely, 1994], surface-wave raytracing [ Wang et al 1993], and the accuracy of waveform perturbative schemes [ Um and Dahlen, 1992; Wang and Dahlen, 1994]. Zhao and Dahlen [1993] report an asymptotic method to approximate the free oscillation eigenfrequencies of a spherical earth model, which, when extended to modal eigenfunctions, may lead to dramatic efficiency improvements in coupled-mode algorithms. Tromp and Dahlen [1992ab; 1993b] and Pollitz [1992] extended normal-mode variational formalisms to model the propagation of long-period surface waves in smooth structure.
Some surface wave data argue for significant coupling between modal
dispersion branches, particularly those that sum to form the
fundamental Rayleigh and Love surface waves.
Park and Yu [1992] identify Love-to-Rayleigh scattered energy
that corresponds to seismic records where fundamental spheroidal and
toroidal coupling pairs show anomalous amplitudes.
Comparison of coupled-mode synthetics for different structures argues
strongly for lateral variations in azimuthal anisotropy as the cause of
these observations [ Yu and Park, 1993; 1994].
Anisotropy can generate a ``quasi-Love'' scattered wave, which can
be modeled with differential seismograms based on
Born-approximation perturbation theory [ Su et al 1993].
Tromp [1994] develops an asymptotic JWKB theory for
surface waves in smooth anisotropic structure.
However, observations suggest that strong lateral variations in
upper-mantle anisotropy occur within
1000-km of the ``seaward''
side of some subduction zones, so that surface wave
reflection and transmission at vertical interfaces [ Its and Lee,
1993] may be relevant.
Moreover, Park [1993] derived relations that show
Love-Rayleigh interaction to be sensitive to the lateral scale length of
anisotropy, as well as to the dip of the local symmetry axis, making
model inversion difficult without restrictive assumptions.
It is therefore not surprising that many earthquake seismologists have
deferred the study of anisotropy as long as its influence could be
neglected.
Even in simple layered structures, azimuthal anisotropy complicates
seismogram synthesis by requiring two slowness integrations, rather
than one, to represent faithfully azimuthal variations in group velocity
[ Nolte et al, 1992].
Real seismograms often do not resemble the clean, isolated wave arrivals predicted by simple layered velocity structures, so theoreticians have long recognized the importance of modeling scattered waves in ``rough'' velocity structure. The availability of finite difference codes for two- and three-dimensional geometries, coupled with increases in computer speed, have made direct simulation of specific velocity structures feasible [ Levander and Gibson, 1991; Frankel and Vidale, 1992; Wagner and Langston, 1992; Levander and Hollinger, 1992]. The enormous memory requirements of realistic finite-difference calculations have drawn attention to pseudospectral algorithms [ Witte and Richards, 1990; Chen and McMechan, 1993] which save memory by using a Fourier basis to calculate spatial derivatives. In a more traditional statistical approach to rough velocity structure, Zeng [1993] expressed scattering in the frequency-wavenumber domain, thereby incorporating multiple scattering. The special case of crustal fractures was modeled for data from a geothermal field by [ Gibson et al 1993] using Born theory and perturbation terms derived from the continuum formulae for microscopic aligned cracks. Murakami [1991] and Nihei et al [1994] modeled a similar physical problem, but assumed the cracks to be macroscopic, so that special conditions on the tangential slip along cracks was considered explicitly.
The last few years have seen significant improvements in the ``phase-screen'' method for estimating scattered energy as a wavefront passes through rough lateral structure, an approximation to full finite-difference seismograms that can reduce computation time by more than one order of magnitude. In its original scalar formulation, this method is appropriate for acoustic wave propagation, but does not model wave conversions. Fisk and McCartor [1991] formulated a ``vector'' phase-screen method in which P and S wavefronts are propagated independently in discrete steps through a model. At each step, like-type and converted forward-scattered waves are estimated from distortions of the wavefront phase caused by local velocity fluctuations along the step. The total P and S wavefields are then reassembled for the next step through the model. Although a comparison with finite-difference seismograms in random two-dimensional media showed that this approach has some merit [ Fisk et al 1992], Wu [1994] pointed out physical inconsistencies in the scheme, and proposed an alternate ``thin slab'' approximation for wide-angle forward scattering, in which the physical connection between P and S waves is better represented. Liu and Wu [1994] compared synthetic seismograms calculated by the thin-slab phase-screen, finite difference and eigenfunction expansion methods.