9R54. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. Interdisciplinary Applied Mathematics, Vol 13. - N Bleistein (Center for Wave Phenomena, Colorado Sch of Mines, Golden CO 80401-1887), JK Cohen (Deceased), and JW Stockwell Jr (Center for Wave Phenomena, Colorado Sch of Mines, Golden CO 80401-1887). Springer-Verlag, New York. 2001. 510 pp. ISBN 0-387-95061-3. $79.95.

Reviewed by JG Berryman (Geophys and Global Security Div, LLNL, 7000 East Ave, Mail Stop L-200, Livermore CA 94550).

This group at the Colorado School of Mines has a unique perspective on reflection seismology, as it has been developed for and practiced by the exploration community within the oil industry during the last 25 years. This reviewer thinks that nowhere else in academia are applied mathematicians and geophysicists working so fruitfully in such close proximity, sharing students and joint research seminars, as at CSM.

This book was written by three of the most mathematically- and computationally-inclined members of this group. The intended audience is other applied mathematicians, and/or mathematically-inclined geophysicists. The described work is concerned with sound waves propagating in the earth and how to use proven, practical methods on field data to invert for (or image) geological structures underground or beneath the ocean.

The method of choice is integral equations that can be written as Fourier integral operators or pseudodifferential operators. As is explained very carefully in Chapter 2, the reason for this choice of method is the very practical desire to have a stable inverse for the integral operator used for forward modeling. Fredholm integral equations may or may not have stable inverses in general. But the stability of Fourier transforms, and of their inverse transforms, are very well understood both analytically and numerically. Meanwhile, the previously mentioned extensions, pseudodifferential operators and Fourier integral operators, share many of these stability characteristics. Nevertheless, some regularization methods are often required to deal with issues associated with finite bandwidth field data, but these issues too are largely well understood now.

Having chosen a theoretical framework, the remainder of the book explores a wide range of applications in both 2D and 3D acoustic imaging. High-frequencies, and therefore large wavenumbers, are often assumed, and this leads to eikonal equation solvers and ray theory for finite aperture imagers. In the present applications, the seismic field array determines the pertinent aperture of the imaging system. Asymptotic analysis and stationary phase methods play an important role throughout the book.

These explorations culminate in Chapter 7 where the authors present a general theory of data mapping. For those interested in this topic, this chapter alone justifies the price of the book. The topics covered include perennial issues such as velocity analysis, wave-equation datuming, and data regularization, as well as some discussion of less common ideas such as “uncoverting” of mode-converted waves.

Five Appendices cover topics both more and less mathematical than those treated in the main text. Distribution theory, Fourier transforms of causal functions, and a long (53 pages) appendix introducing ray theory are the main contributions here.

The principal limitations of the book are listed in its Preface. The authors see this book as one at an introductory level and, therefore, do not treat the more advanced topics of elastic wave propagation, anisotropy, mode conversion, elimination of multiple reflections, wavelet deconvolution, and quite a few other topics (including seismic crosswell tomography) that might have required a different formulation from the one preferred here. Nevertheless, the book should not be viewed as introductory in the general sense. The authors assume a fairly high level of understanding by the reader of wave propagation, Fourier transforms, and ray theory (the Appendices help with these topics to some extent). As a textbook for graduate-level studies, this presentation is very appropriate. But this reviewer would not recommend it to undergraduates unless they have very exceptional preparation.

In addition to its potential use as a textbook, the book will be especially useful 1) to researchers in applied mathematics who want to gain a foothold in this very important area of geophysical imaging and 2) to working geophysicists who need to gain a better understanding of the math behind the data processing they routinely do so they can make further improvements. The book has good subject and author indices, as well as professional quality figures throughout.

CSM has also developed the Seismic Unix (SU) package of data processing routines that is freely available for download at http://www.cwp.mines.edu/cwpeodes. The combination of the book and the codes will go a long way to helping both mathematicians and geophysicists catch up to the level of understanding summarized in this very useful volume.

This reviewer recommends that libraries covering wave propagation of all types, and especially those related to geophysical imaging, should have a copy of this book (Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion) available to their patrons. Students and individual researchers may want to have a personal copy because of the very clear exposition and the extensive set of references to recent literature in this continually expanding area of active and topical research.