Introduction

This book is about the estimation and construction of geophysical maps. Geophysical maps are used for exploration, most commonly for petroleum and mineral resources, but also for water, archeology, lost treasure, graves, and environmental pollution.

Here we follow physical measurements from a wide variety of geophysical sounding devices to a geophysical map, a 1-, 2-, or 3-dimensional Cartesian mesh that is easily transformed to a graph, map image, or computer movie. A later more human, application-specific stage (not addressed here) interprets and annotates the maps; that stage places the ``$\times$'' where you will drill, dig, dive, or merely dream.

Map estimation is a subset of ``geophysical inverse theory,'' itself a kind of ``theory of how to find everything.'' In contrast to ``everything,'' maps have an organized structure (covariance) that makes their estimation more concrete and visual, and leads to the appealing results we find here.

Geophysical sounding data used in this book comes from acoustics, radar, and seismology. Sounders are operated along tracks on the earth surface (or tracks in the ocean, air, or space). A basic goal of data processing is a map that shows the earth itself, not an image of our data-acquisition tracks. We want to hide our acquisition footprint. Increasingly, geophysicists are being asked to measure changes in the earth by comparing old surveys to new ones. Then we are involved with both the old survey tracks and new ones, as well as technological changes between old sounders and new ones.

To enable this book to move rapidly along from one application to another, we avoid applications where the transform from model to data is mathematically complicated, but we include the central techniques of constructing the adjoint of any such complicated transformation. By setting aside application-specific complications, we soon uncover and deal with universal difficulties such as: (1) irregular geometry of recording, (2) locations where no recording took place and, (3) locations where crossing tracks made inconsistant measurements because of noise. Noise itself comes in three flavors: (1) drift (zero to low frequency), (2) white or steady and stationary broad band, and (3) bursty, i.e., large and erratic.

Missing data and inconsistant data are two humble, though universal problems. Because they are universal problems, science and engineering have produced a cornucopia of ideas ranging from mathematics (Hilbert adjoint) to statistics (inverse covariance) to conceptual (stationary, scale-invariant) to numerical analysis (conjugate direction, preconditioner) to computer science (object oriented) to simple common sense. Our guide through this maze of opportunities and digressions is the test of what works on real data, what will make a better image. My logic for organizing the book is simply this: Easy results first. Harder results later. Undemonstrated ideas last or not at all, and latter parts of chapters can be skimmed.

Examples here are mostly nonseismological although my closest colleagues and I mostly make maps from seismological data. The construction of 3-D subsurface landform maps from seismological data is an aggressive industry, a complex and competitive place where it is not easy to build yourself a niche. I wrote this book because I found that beginning researchers were often caught between high expectations and concrete realities. They invent a new process to build a novel map but they have many frustrations: (1) lack of computer power, (2) data-acquisition limitations (gaps, tracks, noises), or (3) they see chaotic noise and have difficulty discerning whether the noise represents chaos in the earth, chaos in the data acquisition, or chaos in the numerical analysis.

People need more practice with easier problems like the ones found in this book, which are mostly simple 2-D landforms derived from 2-D data. Such concrete estimation problems are solved quickly, and their visual results provide experience in recognizing weaknesses, reformulating, and moving forward again. Many small steps reach a mountain top.