Fundamentals of Seismic Wave Propagation , ChrisChapman Cambridge U. Press, New York, 2004. $75.00 (608 pp.). ISBN 0-521-81538-X

The universe of seismic wave propagation is divided into overlapping worlds. Scientists who approach the problem from the low-frequency (30 seconds to an hour) global-Earth perspective naturally adopt a formalism based on normal-mode theory in a spherical Earth. Their formalism is well suited for studying part of the wave field containing surface waves and, more generally, long-period waveforms. For those interested in studying the propagation of high-frequency (10 seconds to several tens of hertz) elastic-body waves, a more suitable framework is ray theory, which allows for natural geometrical definitions of seismic phases and expedient computations of their travel times, in realistic media, in Cartesian (flat-layer) or spherical geometry.

Although nearly all introductory seismology textbooks and even most advanced ones aim to describe the entire observed wave field, and therefore cover the basics of both worlds, they do so at the expense of in-depth, consistent, and comprehensive presentation of the state of the art in either approach. In recent years, researchers have made significant advances in the development of numerical methods that are becoming a powerful tool for accurately reproducing observed seismograms in a complex, three-dimensional Earth. However, the methods remain computationally intensive, and seismologists will need to continue to use ap-proximate methods that provide not only faster results but also insights into wave-propagation physics.

Fundamentals of Seismic Wave Propagation by Chris Chapman covers the mathematical development of asymptotic ray theory for seismic waves. It focuses on the specific case of Cartesian geometry as related to local and regional wave propagation on Earth and particularly for situations relevant to the petroleum industry. The author invokes only briefly Earth’s flattening approximation as a means to convert from one reference system to the other. But to show the broader applications of the developments presented, the only realistic example that Chapman illustrates—no doubt intentionally—concerns core phases, a global problem that requires spherical geometry.

Chapman’s book represents a synthesis of his life’s work in theoretical ray seismology and attempts to present, in a consistent mathematical fashion, the status of a field to which he has made many original contributions. In this elegant framework, the author successively builds the fundamentals of asymptotic, kinematic, and dynamic ray theory; boundary conditions and their consequences at medium reflection and transmission interfaces; and the response of stratified media in the frequency and wavenumber domain. He also presents several methods—ranging from the classical Cagniard–de Hoop and the WKBJ (named after physicists Gregor Wentzel, Henrik Kramers, Marcel Louis Brillouin, and geo-physicist Harold Jeffreys) techniques to more practical spectral approaches—to re-cover the wave field in the time domain.

The mathematical developments in the text are first presented in the simplest case, usually that of acoustic media, and are then repeated for the general case of anisotropic elastic media and the particular case of isotropic media. This progression allows readers to first grasp the principles before gradually building more sophistication.

The first four chapters can be studied independently from the book’s core material as they provide a good reference to the seismologist’s basic theoretical tool kit. Chapters 2, 3, and 4 include a simple and clear layout of the basic ray-related concepts, a discussion of the mathematical tools used—principally transforms—and a review of continuum mechanics and elastic waves. Although the material presented in the bulk of the book concerns the simple situation of stratified media, chapter 10 is devoted to techniques that allow the extension of ray theory beyond its theoretical limits: the Maslov approach, Born scattering theory, and the Kirchoff surface-integral method—each of which helps solve a different shortcoming of ray theory. An appendix provides additional useful mathematical background.

Chapman’s book is most suitable for a graduate theoretical course in high-frequency seismology and could be compared with Quantitative Seismology: Theory and Methods (W. H. Freeman, 1980) by Keiiti Aki and Paul Richards. Chapman’s text may serve a similar role for high-frequency seismology as Theoretical Global Seismology (Princeton U. Press, 1998) by Francis A. Dahlen and Jeroen Tromp does for normal-mode theory. Both texts expand beyond the already sophisticated basics of the field as presented in Aki and Richards’s book.

Fundamentals of Seismic Wave Propagation covers more or less the same ray-theory material found in Brian L. N. Kennett’s two-volume The Seismic Wavefield (Cambridge U. Press, 2001 and 2002). Yet Chapman stresses more the mathematical elegance of seismic wave propagation and places little emphasis on actual observations. Nonetheless, mathematicians and mechanicians will certainly appreciate the author’s elegant presentation of the field.