A growing list of recommended textbooks and helpful papers, Q&A list, related web links, and lecture notes, all on aspects of geophysical inverse theory.

Introductory material:

A Conceptual Introduction to Geophysical Inversion (NEW!), by Andrew Ganse, 12Mar2012. PDF file, presented in the UW Earth & Space Sciences brown bag series. No math in this one, just an overview level talk, basically the graphical version of the primer below.
A Geophysical Inverse Theory Primer, by Andrew Ganse, draft 31Mar2008. This document (PDF file) is ten pages long, contains no equations, and aims to provide an overview of the main concepts in inverse theory. By giving a summary at a high-level, the goal is to introduce the subject to the new user, and place the different concepts and solution methods in perspective with each other before delving into mathematical details. Feedback greatly appreciated via email!
An introduction to geophysical inversion, with comparisons to analytical inversion, by Andrew Ganse, 17Oct2007. Powerpoint file, presented on invitation to the UW Math Dept Inverse Problems seminar series, posted here on request of several colleagues. Many more equations than above; you would definitely want to start with the above primer first if you are new to inverse theory.

Textbooks:

These are a subset of the books listed on my Bookshelf page. Note the (A B P U) links at the end of each synopsis are links to the book in each of the following online bookstores when available: Amazon, Barnes & Noble, Powell's Books, and the University of Washington Bookstore. (Please click book images in the carousel to select brief paragraph overviews of each book.)
• A Reading List in Inverse Problems, Brian Borchers, 1998.
A very useful bibliography of inverse theory references, both classic and modern, with brief 1-2 sentence summaries.
• Resolution of seismic waveform inversion: Bayes versus Occam, Gouveia & Scales, 1997.
A fantastically helpful paper comparing and contrasting Bayesian and Occam/classical inversion techniques. Explains what happened to the regularization bias and tradeoff parameter in the Bayesian approach. (By the way, you'll find the Bayesian interpretation of the resolution matrix in Tarantola's book above.)
• Inverse Problems in Geophysics, Snieder & Trampert, 2000.
At 73 pages long, it's somewhere between an article and a book. The whole thing is well-written and useful, but if nothing else I strongly recommend the first five pages as a particularly good conceptual overview of geophysical inverse theory.
• Estimation of information content and efficiency for different data sets and inversion schemes using the generalized singular value decomposition, Th. Günther, S. Friedel & K. Spitzer, 2003.
A handy addendum to the discussion of the GSVD based inversion and resolution material in section 5.5 of the Aster et al textbook, which describes resolution for higher-order (eg 2nd) Tikhonov regularization. This paper simply clarifies the GSVD material for the case when the model norm is ||L(m-m0)||2, instead of just ||Lm||2. This comes up for example in a locally linearized (nonlinear) problem where we want to smooth the actual model rather than the model perturbation that is solved for at each iteration. This nonlinear inversion topic via the regularized, iterative Gauss-Newton method is discussed further in section 10.1 of the Aster et al book. (And that's the nonlinear method we learned in ESS523 class.) You can do the problem without the GSVD stuff, it just goes faster in GSVD (analogous to how sometimes timeseries processing is faster in frequency space - GSVD space is in fact a type of spectral space), and the GSVD filtering approach may give some more intuitive feel for stabilizing the problem.
• Regularization Tools User's Manual, Hansen, 2001.
The first half of this user's manual for this very useful software is a very well written treatise on regularization of linear inverse problems. This is a .zip file which contains a postscript file.
• Monte Carlo sampling of solutions to inverse problems, Mosegaard & Tarantola, 1995.
This and the next paper address a useful Monte Carlo technique for handling nonlinear inversions, including the non-Gaussian statistics of their solution. This is an ideal nonlinear technique when your problem is small/fast enough that you can run your forward problem many many times in a reasonable amount of computation time.
• Resolution Analysis of General Inverse Problems through Inverse Monte Carlo Sampling, Mosegaard, 1998.
• The Role of Nonlinearity in Inverse Problems, Snieder, 1998.

• Papers by Roel Snieder (his own pubs webpage)
• Per Christian Hansen's website including links to his pubs and his Regularization Tools for Matlab.
• Papers by Klaus Mosegaard (his own pubs page)
• NMT Inverse Theory class website cotaught by Rick Aster and Brian Borchers at New Mexico Tech. These guys are coauthors of the textbook listed at the top of this page.
• Homepage of Albert Tarantola (including a PDF copy of his textbook above!)
• Autodiff.org, a web resource for automatic differentiation. Derivatives are a requirement for a number of inversion related computations, and yet you're often stuck with computing finite difference approximations excruciatingly slowly, or taking lots of time deriving analytic expressions manually for the few problems you can do that for. By contrast, automatic differentiation (AD) is a software approach which runs over your forward problem code module and gives you a corresponding Nth derivative code module, to use just as you would use an analytic derivative expression that you coded up. The only catch is your forward problem must be able to be called as a function with arguments (be that in C/C++, Fortran, or Matlab).
• Samizdat Press, repository of free books, lecture notes, & software, with a particular emphasis on math and physics material (including seismology & inverse theory).

Andy's Lab Lecture Notes and Tutorial Matlab scripts from class:
These grew out of TAing for UW-ESS523, Geophysical Inverse Theory, for professor Ken Creager in Fall quarter 2005.
• LAB #1: `reviewtut.m` : A review tutorial script covering Matlab scripting fundamentals and some linear algebra review. This is a long script in which everything is initially commented out. Read and uncomment it bit by bit, running local sections of it to learn key concepts. It ends with an assignment to do a parabolic curve fitting to noisy data.
• LAB #1.5: `covstattut.m` : A tutorial exploring the meaning of covariance and related statistical concepts. As with reviewtut.m above, the idea is to read and uncomment it bit by bit.
• LAB #2: `lab2.txt` : An extension to the curve fitting in lab #1; now we compute statistical measures of the quality of the curve fit we'd found in lab #1.
• LAB #3: `lab3.pdf` : Linear inversion of several density profiles of the Earth using the Gram matrix / inner product approach we learned in class and in Parker's book.
• LAB #4: `lab4.pdf` : Parameter estimation of earthquake source location given arrival times at known receiver station locations - a weakly nonlinear problem.
• LAB #5: `lab5.pdf` : Objective surfaces for linear and nonlinear problems and relation to statistics of inverse solution. Plotting objective surface and confidence ellipses for last week's weakly nonlinear earthquake location problem.
• LAB #6: `lab6.pdf` : Use of a smoothing matrix - ie higher-order Tikhonov regularization - in Occam's inversion and related methods for inversion of weakly nonlinear problems. In this lab the problem is to estimate a cubic spline curve fit (the smoothest possible curve) to a set of random data points, as a simple example that you can adapt to your own inverse problems.
• LAB #7: `lab7.pdf` : (Getting rewritten, available soon) Supplemental lecture not given in class: Global methods for estimating the inverse of a not-so-weakly nonlinear problem (also important for comparing against results of the iterated linearized method we learned for weakly nonlinear problems). Includes very brief overviews of uniform Monte Carlo, simulated annealing, genetic algorithms, and Bayesian Metropolis sampling (a type of Markov-Chain Monte Carlo or MCMC). This last is the most rigorous in terms of statistics for the solution of a nonlinear inverse problem, and is great if you can computationally afford it (eg it's not meant for global tomography). The Tarantola book and papers above expand fully on this topic.

Geophysical Inversion Q&A session from CSM 2004 Summer School
The professors/students questions-and-answers list from the "2004 Mathematical Geophysics and Uncertainty in Earth Models" summer school at the Colorado School of Mines in Golden, CO.