Computational Ocean Acoustics, by Jensen, Kuperman, Porter, & Schmidt. If you are in the field of underwater acoustics you will recognize the names of all four of these authors. Two of them have produced wave propagation codes that are widely used in that community: the KRAKEN mode-based code by Mike Porter, which is not a fully elastic code and so note it may not always be appropriate for shallow water problems depending what you're doing. But it sure runs nice and fast and there are certainly many problems it is appropriate for, especially "deep water" problems. The other is the OASES wavenumber integration code by Henrik Schmidt, which is a fully elastic code appropriate for a wide range of seismic and acoustic problems, but runs slower, plus only the version with 1D bottom, ie 2D propagation, is available for free. In any case the theory behind both codes is explained in this book (note my PhD advisor Bob Odom recommends reading the finite elements chapter before the wavenumber integration chapter to understand OASES), as well as much other material central to the field of underwater acoustics. It's an extremely well-written, very readable (albeit mathematical) textbook that has understandably become a classic in the field. (A B P U)



Introduction to Seismology, by Peter Shearer. By my understanding, the focus of this book was to essentially act as an introduction to Aki & Richards (below), which is an incredibly comprehensive and useful book, but sure as hell isn't an introductory one. Lay & Wallace is the other common seismo textbook, and while also very useful it too is pretty comprehensive for a single introductory class. Shearer's book regularly references pages in A&R (unfortunately the older version but the references are still helpful) and I think does a fantastic job laying out the basics of seismology and its associated mathematics. After learning the fundamentals from this book, then one is ready to delve into A&R. But even now, I still probably go back and look up at least as much stuff in Shearer's intro as I do in A&R. (A B P U)



Quantitative Seismology, 2nd ed., by Keiiti Aki & Paul G. Richards. This is the grandaddy text of theoretical seismology -- amazing to think folks learned this stuff just from a bunch of scattered journal papers before this book first came out in 1981. A&R (as it is commonly called) covers, in great detail, mathematical descriptions of seismic sources and elastic wave propagation. An extremely useful reference text for seismologists and underwater acousticians who deal with shallow water environments, but I must say it was a hell of a couple semesters going through this material for the first time, and that was with the introductory background from Shearer's book (above). If you're a seismo PhD student you will undoubtedly use this text. Note this is the 2nd edition, which removed some material from the first (dual volume) edition that was deemed to be better treated elsewhere nowadays (e.g. inverse theory and so on). Some additional material was added however, mainly extra tidbits here and there relating to nuclear explosion monitoring.
Lastly, a fun trivia note from Paul Richards' website: the picture on the back cover shows Aki and Richards hovering in a bizarre pose in mid conversation. The reason it looks so strange was they were actually hovering over glasses of wine at a party, and the publishers chose to crop out the wine glasses! The before and after versions are shown on Richards' website there. (A B P U)



Computational Statistics Handbook with Matlab, by Wendy Martinez and Angel Martinez. This is a very hands-on book that comes with a free, downloadable Matlab toolkit full of statistical functions which are explained in this book. So the focus is really "how to do it in Matlab" rather than theoretical background (but at least a little bit is given for each subject). I haven't spent a whole lot of time with this book yet, but some of the key subjects it covers include: MCMC sampling, non-parametric regression, simulating random variables given their distribution, and pattern recognition. But again, the focus is really just on using the provided Matlab functions to do the analysis (which I still think is useful); theoretical background must be found in other books. Note there is apparently a 2nd edition coming out in Dec 2007 that you may wish to wait for. Also, I have found that the authors' "infinityassociates" website mentioned in the book (this 1st edition) to obtain the software has been bought out and no longer has anything to do with the book. You must use either the StatLib website or the book's CRC webpage to download the software toolkit (also note it says for Windows but you can use the .m files themselves anywhere). (A B P U)



Applied Optimal Estimation, editted by Arthur Gelb. In spite of being compiled in 1974, this book is still a fantastic introduction and very practical reference for filters and smoothers. As the title implies, it's very applied and focuses on implementation and application rather than rigorous theoretical background. And since it's from 1974, it basically is about Kalman and Extended Kalman filters/smoothers (and variations such as 2nd order EKFs). More modern developments such as unscented KFs and particle filters must be found elsewhere, but of course one must understand the basics first. This book is fantastically readable, and has useful example problems. In the course of my PhD research I have worked up two of those example (tracking) problems into Matlab scripts to create a tutorial webpage (with the Matlab script available) that compares a number of filters and smoothers. (A B P U)



Stochastic Processes and Filtering Theory, by Andrew H. Jazwinski. This one is another old classic text (the publish date was 1970), but was out of print for years and the library kept making me return it. But it was finally just re-released as a nice cheap paperback. This is a standard companion to the Gelb book above, covering more of the theoretical development of linear and linearized filters and smoothers, whereas Gelb by contrast focuses more on implementation. So they're a great combo, besides which so much filtering material in the literature makes references to pages in this book that it's hard not to have it when reading such papers. One of the most useful sections of the book to me compares Kalman filtering to classical least squares based parameter estimation (in the Bayesian context), allowing me to identify the parallels and common terminology between the two methods. (A B P U)



Fundamentals of Geophysics, by William Lowrie. (A B P U)



Dynamic Earth, by Geoffrey Davies. (A B P U)

Parameter Estimation and Inverse Problems, by Richard Aster, Brian Borchers, & Clifford Thurber. For beginners to inversion, I strongly recommend this book above other inverse theory textbooks; there are plenty very useful books on the topic, but this one really gets you up to speed in the subject fast with great hands-on Matlab examples. Then, after you're more familiar with the material, go back and reread the book again - there are tons of handy comparisons between methods with references to deeper treatment of the individual methods elsewhere, and many practicality details that address numerical issues and real world limitations. This book mostly focuses on frequentist inversion but does have a chapter on Bayesian inversion comparing the two approaches. This book comes with a CD with a copy of Per Christian Hansen's (linear) regularization tools for Matlab, used in most of the book's example problems, whose Matlab scripts are also on the CD. Note also the homepage for this book which includes the errata. (A B P U)


Geophysical Inverse Theory, by Robert L. Parker. A classic frequentist text that is very readable - Parker is rigorous and introduces the reader to functional analysis concepts, but injects witty tidbits here and there which keep you interested. This book focuses on the Gram matrix / representers technique, which parameterizes the model with the same number of parameters as there are data points, and requires numerical integration for many real-world problems. But the technique is ideal for gravity inversion and magnetotellurics (MT), and those two topics are explored in detail in the book's examples. So you should be sure to read this book if you are working on gravity or MT inversion. (A B P U)



Inverse Problem Theory, by Albert Tarantola. Very well written book with a probabilistic approach to inversion, in contrast with the frequentist statistical approach focused on in Aster/Borchers/Thurber, Menke, Hansen, and Parker. Tarantola doesn't like to call it "Bayesian" (even though it's the same formulation), for he rederives it differently to address concerns such as Borel's paradox seen in Bayesian statistics. In any case, this probabilistic approach allows for the same type of linearization approach often done for weakly nonlinear problems in the frequentist approach, but additionally provides for a very flexible numerical sampling approach to handle more strongly nonlinear problems. In spite of the focus on the probabilistic approach, this book has many useful comparisons between probabilistic (Bayesian) and frequentist inverse theory. Note he offers a free PDF copy of this book on his webpage (with a request to please buy the book if you find yourself using it and can afford it). For those who are familiar with his 1987 book, this one largely replaces it but leaves out a few bits I thought were really useful in the 1987 version, like expressions for derivatives of elastic displacement with respect to media properties. But many of the other example problems remain. (A B P U)



Markov Chain Monte Carlo in Practice, by W.R. Gilks, S. Richardson, & D.J. Spiegelhalter. Markov Chain Monte Carlo (MCMC) sampling, among other uses, is used with several variations for the numerical sampling of the posterior distribution in Bayesian (probabilistic) inversion. The book is a fantastic supplement to Tarantola's book. The 19 page introductory chapter (16 pages without the refs) is such a fantastic and accessible overview of the subject of MCMC that it would make a very good reading assignment for an inverse theory class that discussed Bayesian inversion. The rest of the chapters are articles by experts in the field, detailing implementation issues like convergence, sampling mechanics, and differences between variations of MCMC such as Metropolis/Hastings vs Gibbs sampling and so on. And important note made in the introduction is that MCMC is not only for Bayesian inversion -- in frequentist inversion one may wish to sample the integral in the data misfit distribution (say if it were some arbitrary distribution other than Gaussian). (A B P U)



Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, by Per Christian Hansen. Very well written book concerning various regularization methods in frequentist inversion, with emphasis on a SVD/spectral approach. The value of formulating regularization in the SVD domain is twofold -- one, it's greatly faster computationally (analogous to speeding up timeseries processing by doing it in the frequency domain), and two, it does a much better job at preserving numerical accuracy. A good chunk of this book's material (but not all) can be found in Hansen's Regularization Tools manual (below under WWW links). And that's free unlike this book! (A B P U)



Geophysical Data Analysis: Discrete Inverse Theory, Revised Edition, by William Menke. Another frequentist classic text; note the revised edition is recommended as the original had numerous typos in the equations. While Parker leaves the Earth model being estimated as a continuous function, Menke (like Aster/Borchers/Thurber and others) discretizes it as his parameterization. The idea is that this discretization is finer than the resolution of the inverse problem solution. This book is the only one in which I've seen much discussion of the data resolution matrix (as opposed to model resolution matrix). (A B P U)



Time Series Analysis and Inverse Theory for Geophysicists, by David Gubbins. As the title implies, this book is sortof two books in one, and so far it is the inverse theory half that I've spent time reading. Unfortunately I found it somewhat scattered and think it would be very difficult to learn the material for the first time from this book. On the other hand however, this book is worth reading after being more familiar with the topic, as there are useful tidbits throughout. Just as one example, this was the only inverse theory book I've found so far that specifically points out that the model resolution matrix is not always symmetric (for example when using 2nd-order Tikhonov regularization). That drove me nuts figuring that out, but wouldn't have if I'd come across this book sooner. (A B P U)



Probability, Random Variables and Stochastic Processes, by Athanasios Papoulis. A classic probability/statistics text; I'm sure many other textbooks cover the same material, I just happen to have and like this one. Actually I have an earlier version than the one shown in this picture. (A B P U)



The C Programming Language, by Brian Kernighan & Dennis Ritchie. (A B P U)