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)

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 essentially the same formulation), for he rederives it differently to address concerns such as Borel's paradox seen in Bayesian philosophy. 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)

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)

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)

Markov Chain Monte Carlo in Practice, by W.R. Gilks, S. Richardson, & D.J. Spiegelhalter. Markov Chain Monte Carlo (MCMC) sampling, among other uses, can be used for numerically sampling the posterior distribution in Bayesian (probabilistic) inversion. The book is a fantastic and rigorous supplement to Tarantola's book. The 19 page introductory chapter (16 pages without the refs) is such an 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. An 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)

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)