# Preface

The difference between theory and practice is smaller in theory than it is in practice. -folklore
We make discoveries about reality by examining the discrepancy between theory and practice. There is a well-developed theory about the difference between theory and practice, and it is called ``geophysical inverse theory''. In this book we investigate the practice of the difference between theory and practice. As the folklore tells us, there is a big difference. There are already many books on the theory, and often as not, they end in only one or a few applications in the author's specialty. In this book on practice, we examine data and results from many diverse applications. I have adopted the discipline of suppressing theoretical curiosities until I find data that requires it (except for a few concepts at chapter ends).

Books on geophysical inverse theory tend to address theoretical topics that are little used in practice. Foremost is probability theory. In practice, probabilities are neither observed nor derived from observations. For more than a handful of variables, it would not be practical to display joint probabilities, even if we had them. If you are data poor, you might turn to probabilities. If you are data rich, you have far too many more rewarding things to do. When you estimate a few values, you ask about their standard deviations. When you have an image making machine, you turn the knobs and make new images (and invent new knobs). Another theory not needed here is singular-value decomposition.

In writing a book on the ``practice of the difference between theory and practice" there is no worry to be bogged down in the details of diverse specializations because the geophysical world has many interesting data sets that are easily analyzed with elementary physics and simple geometry. (My specialization, reflection seismic imaging, has a great many less easily explained applications too.) We find here many applications that have a great deal in common with one another, and that commonality is not a part of common inverse theory. Many applications draw our attention to the importance of two weighting functions (one required for data space and the other for model space). Solutions depend strongly on these weighting functions (eigenvalues do too!). Where do these functions come from, from what rationale or estimation procedure? We'll see many examples here, and find that these functions are not merely weights but filters. Even deeper, they are generally a combination of weights and filters. We do some tricky bookkeeping and bootstrapping when we filter the multidimensional neighborhood of missing and/or suspicious data.

Are you aged 23? If so, this book is designed for you. Life has its discontinuities: when you enter school at age 5, when you marry, when you leave university, when you retire. The discontinuity at age 23, mid graduate school, is when the world loses interest in your potential to learn. Instead the world wants to know what you are accomplishing right now! This book is about how to make images. It is theory and programs that you can use right now.

This book is not devoid of theory and abstraction. Indeed it makes an important new contribution to the theory (and practice) of data analysis: multidimensional autoregression via the helical coordinate system.

The biggest chore in the study of ``the practice of the difference between theory and practice" is that we must look at algorithms. Some of them are short and sweet, but other important algorithms are complicated and ugly in any language. This book can be printed without the computer programs and their surrounding paragraphs, or you can read it without them. I suggest, however, you take a few moments to try to read each program. If you can write in any computer language, you should be able to read these programs well enough to grasp the concept of each, to understand what goes in and what should come out. I have chosen the computer language (more on this later) that I believe is best suited for our journey through the ``elementary'' examples in geophysical image estimation.

Besides the tutorial value of the programs, if you can read them, you will know exactly how the many interesting illustrations in this book were computed so you will be well equipped to move forward in your own direction.