INVERSE PROBLEMS NEWSLETTER

Abstract

The Newsletter will be a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting new forum by sending to our Bristol office a brief account or material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book reviews The Ocean Circulation Inverse Problem C Wunsch 1996 Cambridge: Cambridge University Press 442pp ISBN 0-521-48090-6 £ 35.00 ($54.95) This book presents a detailed and comprehensive description of state-of-the-art techniques for solving the ocean circulation inverse problem. It is suitable for use as a textbook for advanced undergraduate and postgraduate students and as a reference work for practitioners in the field of physical oceanography. It will also be useful to anyone having an interest in making quantitative inferences about the physics of the Earth from limited observations. In chapter 1, the author provides a historical and fundamental theoretical setting for the ocean circulation inverse problem. He illustrates through simple examples the differences between forward and inverse problems and demonstrates that they should be treated on an equal footing. He introduces the distinction between inverse problems and inverse methods and subsequently pursues this theme vigorously throughout the book. In chapter 2, the author presents the basic equations of geophysical fluid dynamics which are used to develop models of the general circulation of the world's oceans. He also describes a number of physical oceanographic observations and illustrates their limitations due to sparse sampling and noise. He clearly shows that his primary goal is to unify these sometimes disjointed theoretical and observational threads through the application of inverse techniques which explicitly deal with the limitations of the measurements. He concludes the chapter by setting up the canonical form of the linearized box balance equations to be solved in subsequent chapters. In chapter 3, the author presents the basic techniques available to solve these equations. At this point, the discussion is largely divorced from oceanographic applications, and this chapter could easily stand alone as a tutorial on the solution of simultaneous equations describing noisy, incomplete observations. Topics covered include matrix and vector algebra, basic statistics and regression, least squares, singular value decomposition and Gauss - Markov estimation. A theme that clearly emerges is that these techniques also provide estimates of the errors in the solutions. The purpose of chapter 4 is to apply the mathematical machinery developed in chapter 3 to the problem of determining the general oceanic circulation in the context of a steady-state model. The strengths of this chapter include an intercomparison among different techniques and their application to real data sets. Interesting discussions are also presented on the estimation of property fluxes and mixing coefficients. Chapter 5 is devoted to a discussion of other useful topics, including inequality constraints, linear programming, empirical orthogonal functions, and a variant of Gauss - Markov estimation called kriging. Some techniques for the solution of nonlinear problems are also described. Finally, chapter 6 addresses the solution of the time-dependent ocean circulation inverse problem. Here the important role of the Kalman filter is emphasized. The Pontryagin principle and adjoint methods, which arise in the context of control theory, are also discussed. Overall, The Ocean Circulation Inverse Problem is a valuable addition to the textbook literature on inverse methods and physical oceanography. The use of simple examples to demonstrate the techniques presented is a key component of a clear and cohesive exposition. The description of a number of observational techniques, in addition to the theoretical development, is also a strength of the book. The excellent graphics and comprehensive list of references will surely impress the reader. From a pedagogical viewpoint, the text could be improved through the addition of homework problems at the end of each chapter. Finally, throughout the book, the author injects his own historical perspective on observational physical oceanography and geophysical fluid dynamics which the reader will find interesting and sometimes amusing. G V Frisk Woods Hole Oceanographic Institution, MA Inverse Problems in Geophysical Applications H W Engl, A K Louis and W Rundell (ed) 1997 Philadelphia, PA: SIAM 303pp ISBN 0-89871-381-1 $81.00 This book contains proceedings of a GAMM - SIAM meeting held in Yosemite in 1995. As the editors state in their introduction, inverse problems constitute an interdisciplinary field of research and `the necessity for collaboration between mathematicians and practitioners in application areas becomes clear'. Indeed, closing the gap between theoreticians and researchers actually inverting data is a major problem in pushing theoretical developments in inverse problems to the stage where they are actually implemented for the analysis of real data. However, it looks like this meeting did not attract the right people to reach this goal. In only two of the 15 chapters are real data shown; the remaining chapters are purely theoretical or show applications of inverse problem algorithms to synthetic examples. Since the application of inverse problem algorithms to real data is often a sobering and humiliating experience, the lack of examples dealing with real data is a distraction from reaching the goal set forth by the editors. This does not, of course, distract from the value of the work presented in this book. Many of the chapters contain very interesting material and are well written. In the chapter `Data Integration via Reflection Tomography' Kurt Marfurt and William May carefully explain how the inverse problem approach chosen in seismic exploration should be related to the goals of the inversion and to the characteristics of the medium under consideration (in this case the Earth). This kind of publication is crucial for promoting an interplay between theory and practice in inverse problems. The chapters in this book are devoted to the inversion of elastic acoustic waves (6 chapters), electrical inverse problems (4 chapters), inverse gravity problems (2 chapters) and a wonderful chapter by Pierre Sabatier on a `Patchwork Approach to Inverse Theory'. The main idea of this chapter is that in many classes of inverse problems the problem is of a local nature, but that long-range interactions exist that couple the local problems in a subtle fashion. As an example Sabatier mentions wave propagation in the ocean. Wave action such as the surf on the beach may appear to be local, but ultimately it is long-wavelength swell (a global phenomenon) that plays a crucial role in generating the surf. This idea of Sabatier is very interesting because the inverse problem community behaves like a patchwork in the same way as the example of waves on the ocean. This patchiness is reflected in the contents of this book. Of course there is a hidden `long-range interaction' between the the different chapters in this book, but the editors have not been able to make this visible to the reader. It is the lack of coherence in this book that would make me hesitant to use it for teaching purposes, despite the fact that it contains excellent chapters. This implies that the book is mostly for the researcher who has a specific interest in one of the topics that are covered. R Snieder Utrecht University An Introduction to Inverse Scattering and Inverse Spectral Problems K Chadan, D Colton, L Päivärinta and W Rundell 1997 Philadelphia, PA: SIAM 198pp ISBN 0-89871-387-0 $39.50 This book of about 200 pages consists of three introductions to inverse electromagnetic scattering, inverse spectral theory and inverse quantum mechanics written by three experts in these fields. Since studying an inverse problem requires a solid knowledge of the theory of the corresponding direct problem, a substantial part of all three chapters is concerned with direct scattering and spectral theory. In his chapter David Colton considers the inverse scattering problem to determine the index of refraction both for a nonabsorbing and an absorbing inhomogeneous medium from the far field pattern for the scattering of time-harmonic electromagnetic waves. The presentation is for the two-dimensional case and the solution method presented is the dual space method of Colton and Monk. The inverse spectal problem considered by William Rundell consists of determining the coefficient of the differential equation in a regular Sturm - Liouville problem from its eigenvalues. Using the Gelfand - Levitan - Marchenko integral operator approach, uniqueness results are proven, and the use of a Newton method for the numerical solution of the inverse problem is discussed. Finally, in the last chapter, Khosrow Chadan considers the inverse problem to recover the potential in the Schrödinger equation from spectral data where he presents the Gelfand - Levitan - Marchenko reconstruction algorithm. He also briefly discusses the application of inverse problems techniques for the investigation of solitons. Lassi Päivärinta starts the book with a collection of mathematical tools, in particular from functional analysis, which are relevant for the subsequent chapters. The book provides a valuable introduction to some of the basic ideas in each of the above areas of inverse problems. Newcomers to the field will find it useful for a first study in inverse problems and experts in one of the above fields may also find it profitable as introductory information on the other fields. Of course, for a more comprehensive study the reader will have to consult the monographs provided in the bibliographies of each of the chapters. The book could have been made more valuable and easier to read by an increased effort to synchronize the style and notation of the three main authors. For example, the potential to be recovered is denoted by n in chapter 2, q in chapter 3 and V in chapter 4. If, for good reasons, the authors could not agree on a uniform notation, at least someone should have explained the commonalities and connections between these quantities to the reader. In this sense, the first chapter should have been extended to illuminate the interrelations between the three areas (or a corresponding concluding chapter should have been added). R Kress Universität Göttingen