Our route

Centrally, this book teaches how to recognize adjoint operators in physical processes (chapter ), and how to use those adjoints in model fitting (inversion) using least-squares optimization and the technique of conjugate gradients (chapter ).

First, however, we review convolution and spectra (chapter ) discrete Fourier transforms (chapter ), and causality and the complex $Z=e^{i\omega}$ plane (chapter ), where poles are the mathematically forbidden points of zero division. In chapter  we travel widely, from the heaven of theoretically perfect results through a life of practical results including poor results, sinking to the purgatory of instability, and finally arriving at the ``big bang'' of zero division. Chapter  is a collection of solved problems with a single unknown that illustrates the pitfalls and opportunities that arise from weighting functions, zero division, and nonstationarity. Thus we are prepared for the keystone chapter, chapter , where we learn to recognize the relation of the linear operators we studied in chapters 1-3 to their adjoints, and to see how computation of these adjoints is a straightforward adjunct to direct computation. Also included in chapter  are interpolation, smoothing, and most of the many operators that populate the world of exploration seismology. Thus further prepared, we pass easily through the central theoretical concepts of least-squares optimization, basic NMO stack, and deconvolution applications in chapter .

In chapter  we see the formulation and solution of many problems in time-series analysis, prediction, and interpolation and learn more about mathematical formulations that control stability. Chapter  shows how missing data can be estimated. Of particular interest is a nonstationary world model where, locally in time and space, the wave field fits the model of a small number of plane waves. Here we find ``magical'' results: data that is apparently undersampled (spatially aliased) is recovered.

Hyperbolas are the reflection seismologist's delight. My book could almost have been named Hyperbolas and the Earth. That book includes many techniques for representing and deforming hyperbolas, especially using various representations of the wave equation. Here I repeat a minimal part of that lore in chapter . My goal is now to marry hyperbolas to the conjugate-gradient model-fitting theme of this book.

Having covered a wide range of practical problems, we turn at last to more theoretical ones: spectra and phase (chapter ), and sample spectra of random numbers (chapter ). I have begun revising three theoretical chapters from my first book, (hereinafter referred to as FGDP), which is still in print. Since these revisions are not yet very extensive, I am excluding the revised chapters from the current copy of this book. (My 1985 book, (hereinafter referred to as IEI), deserves revision in the light of the conjugacy methods developed here, but that too lies in the future.)

Finally, every academic is entitled to some idiosyncrasies, and I find Jensen inequalities fascinating. These have an unproved relationship to practical echo analysis, but I include them anyway in a brief concluding chapter.