The societal and commercial benefits of wave propagation research can be measured most directly by its contribution to seismology's principal extra-scientific applications: 1) earthquake hazard mitigation, 2) the discrimination of underground nuclear explosions from earthquakes and quarry blasts, and 3) shallow crustal imaging for resource exploitation and environmental monitoring. In each of these areas there are useful spinoffs of recent US research in wave propagation theory, but in some cases one can also identify missed opportunities. fields in knowledge in the Earth
Theoretical seismology can be used to predict the likely ground motion at a particular location for a seismic event on a prescribed fault plane with a prescribed size, so that public officials can anticipate loci of enhanced structural damage and adjust building codes to reflect predicted shaking levels. The previous development of finite-difference computer codes for media that varies in two- and three-dimensions has benefitted investigators greatly, as demonstrated by the wide range of earthquake scenarios investigated in the literature [ Frankel, 1993; Graves, 1993; Saikia, 1993]. Asymptotic methods were also applied to hazard forecasts. Gaussian beam methods were developed for teleseismic seismology to broaden the sensitivity of ray theory to lateral structure gradients, and were adapted by Qu et al [1994] to estimate where focussing of short-period surface waves could cause enhanced shaking in the Los Angeles Basin. As most damage to structures occurs atop sedimentary basins, and is especially severe where local resonances amplify shaking, the modeling of basin resonances is relevant. Although there are recent efforts to transfer concepts and tools from variational mechanics to estimate the resonances of irregular basins [ Rial and Ling, 1992; Zhao and Dravinski, 1994], there is much less attention devoted to this problem than is devoted to whole-earth free oscillations. The predictions of resonant frequencies and of brute-force wavefield calculations depend critically on knowledge of the underlying seismic velocity structure, especially near the surface, which is often poorly known. Though many cities might balk at the cost of seismic reflection crews crawling up and down its major thoroughfares, Field and Jacob [1993] note that little attention has been devoted to modeling the resonances observed in ambient seismic noise, a much cheaper and less disruptive measurement to make. The relationship between ambient-noise and strong-motion amplifications has thusfar not drawn much attention from theoretical seismologists.
Much US research in theoretical seismology is supported by
government agencies that seek to improve our ability to detect and
locate seismic events worldwide, and to distinguish underground
nuclear explosions from earthquakes and quarry blasts.
Because the scattered wave field can obscure the aspects of the seismic
source mechanism that are useful for discrimination, methods for
incorporating three-dimensional structure into synthetic seismogram
algorithms, such as those described in the previous section, contribute
to the goal of verifying treaties that limit underground nuclear tests.
Several studies have modeled the effects of near-source tectonic
structure on the conversion of explosion-generated P waves to the
S waves and surface waves more characteristic of faulting sources
[ Frankel and Leith, 1992; McLaughlin et al 1992;
Stevens et al 1993].
More distant from the source, effort has focussed on the propagation
of the 
phase, a short-period (
sec) overtone
surface wave that is largely trapped within the continental crust.
The 
wave was originally mistaken for a Love wave in the
granitic crust by early researchers, and its detailed waveform is
difficult to model.
However, its amplitude can be used empirically for detection,
discrimination, and as a stable measure of explosion yield [
Hansen et al 1990].
This continues to motivate numerical studies of 
propagation
[ Xie and Lay, 1994; Saikia, 1994; Gibson and
Campillo, 1994].
In the post-Cold-War regime of worldwide nonproliferation
monitoring, less emphasis is likely to be placed on specific
source-receiver paths, such as between the former Soviet test site at
Novaya Zemlya to the Norwegian NORSAR seismic array, and more
emphasis placed on more general wave propagation behavior in e.g.,
the Middle East, the South American craton, and the tectonically
active lithosphere of southern Eurasia.
The relationship between basic earthquake research and applied research in exploration seismology often does not follow standard models for academia-industry interaction, as their theoretical perspectives have, to some extent, diverged over time. In fact, data processing methods for stacking and migrating large data volumes, developed largely by industry, have been borrowed by academic investigators of the Earth's mantle. Nevertheless, where basic and applied seismology have similar problems to solve, there is significant feedback in the literature. Improved confidence in tomographic inversions of seismic traveltimes has been a goal of many groups. In the shallow crust, large three-dimensional variations in seismic velocity can lead to large deflections of the shortest-time raypath. In the academic community, competing interpretations of P travel-time data sets from regional seismic arrays spurred the development of iterative nonlinear inversion schemes [ Ellsworth et al 1991; VanDecar, 1991; Lees and Shalev, 1992; Lees, 1992; Scott, 1992; Scott et al 1994]. A computational bottleneck for these algorithms is raytracing through interim three-dimensional models of velocity structure. Although fast raybending algorithms [e.g. Um and Thurber, 1987] were typically used in these applications, such techniques cannot guarantee to find the shortest path in all structures. To overcome this shortcoming, researchers have developed direct, rather than perturbative, methods for calculating traveltimes. One approach is finite-difference numerical calculation of the eikonal equation, first proposed by Vidale [1988; 1990], which is particularly useful for large numbers of receivers. This approach avoids interim calculations required by formal raytracing, though its ability to calculate only the first arrival makes problematic the resolution of triplications [ Nowack, 1992], where more than one ray connects the source and receiver. A more abstract approach, based on graph theory, was proposed by Moser [1991] and extended by Fischer and Lees [1993]. Exploration seismologists have been quick to exploit the eikonal-equation approach, proposing improvements and special cases [ van Trier and Symes, 1991; Schneider et al 1992; Qin and Schuster, 1993; Coultrip, 1993].
Another connection, more tenuous, between basic and applied seismic wave propagation theory in the early nineties involved borehole waves in inhomogeneous media. The ``tube wave'' associated with the borehole must be accounted for when interpreting seismic data, and has been modelled using both a three-dimensional finite-difference approach [e.g. Yoon and McMechan, 1992] and formal perturbation theory [e.g. Sinha et al, 1994]. Ellefsen et al [1991ab] considered difficulties in modeling the tube wave when the borehole cross-section is noncircular. Perturbations to fluid-solid boundaries had caused difficulties for global seismologists in the 1970's, leading to a series of papers in the free oscillation literature. As a result Ellefsen et al [1991a] was able to utilize Woodhouse and Dahlen [1978], which specified the proper variational terms for non-spherical internal fluid-solid boundaries in the Earth.
Despite these examples of cross-fertilization, the published literature suggests less interplay between earthquake and exploration wave propagation studies than is often evident overseas. Although significant exceptions exist, US researchers tend to be more specialized, and the recent contraction of domestic exploration research has dampened the incentives for cross-disciplinary projects. It has been more common for non-US research groups to develop theoretical tools common to both mantle and crustal imaging problems, for instance, the ray-based waveform synthesis in anisotropic structures derived by Thomson et al [1992] and applied by Kendall and Thomson [1993] and Guest et al [1993]. By contrast, interaction is less evident between US seismologists concerned with mantle anisotropy caused by oriented olivine and US seismologists concerned with anisotropy caused by thin-layered shales.
Specialization may inhibit the US wave propagation community in the future, as scientific and technological advances often thrive in a more interactive environment.