All theory, my friend, is gray,
But green is the lustrous tree of life.
Mephistopheles, as quoted in Goethe's Faust

In many areas of the physical sciences, spectral analysis finds frequent and extensive use. Examples abound in oceanography, electrical engineering, geophysics, astronomy and hydrology. Spectral analysis is a well-established standard -- its use in so many areas in fact facilitates the exchange of ideas across a broad array of scientific endeavors. As old and as well-established as it is, however, spectral analysis is still an area of active on-going research (indeed, one of the problems for both practitioners and theorists alike is that important methodological advances are spread out over many different disciplines and literally dozens of scientific journals). Since the 1960s, developments in spectral analysis have had to take into account three new factors:

• Digital data explosion. The amount of data routinely collected in the form of time series is staggering by 1960s standards. Examples include exploration seismic data, continuous recordings of the earth's magnetic field, large seismology networks, real-time processing of sonar signals and remote sensing from satellites. The impact on spectral analysis is that fast  and digital  processing has become very important.
• Enormous increase in computational power. In retrospect, it is clear that much of the spectral analysis methodology in vogue in the 1960s was influenced heavily by what could be calculated with commonly available computers. With the computational power now widely available on today's computers, we are finally in a position to concentrate on what should be calculated.
• New awareness of spectral complexity. In the 1960s spectral analysis was designed to be applied primarily to processes with relatively simple spectra. It has become clear since then -- particularly from geophysical applications -- that real data can have spectra with intricate detail and an enormous dynamic range. Hence even small improvements in methodology can be important for interpretations.

Our intent in writing this book is to provide a graduate-level introduction to spectral analysis that covers both well-accepted methodology and some important recent advances addressing these new factors. We place special emphasis on the multitaper technique due to Thomson (1982) because of its potential for routinely handling spectra with intricate structure. At the risk of appearing to agree with Mephistopheles, our treatment is slanted away from theory and toward practical applications. However, it is our experience that successful application of spectral analysis requires a solid understanding of its theoretical underpinnings. We have thus endeavored to present a satisfying balance between the underlying theory and its applications. Examples are given of the application of spectral analysis techniques to real data sets, as well as to synthetic data when pedagogically appropriate. We have tried to weave together the best of the algorithmic innovations from the electrical engineering literature with the more useful statistical inference results from the statistics literature.

There are several distinctive features to this book.

• A thorough review of the necessary Fourier theory is given. A major source of confusion to those learning or using spectral methods is meshing discrete time computing with theory developed for continuous time or frequency. Hence, in Chapter 3 we look at results for continuous time and frequency, discrete time and continuous frequency, and discrete time and discrete frequency. This review also includes spectral concentration measures, providing the background to the nonparametric multitaper spectral analysis method of Chapter 7.
• Computational considerations are extremely important to the practitioner. We have given formulae in terms that can be immediately and easily coded (for example, the sampling interval is explicitly included). Choosing bandwidths for smoothing depends on finding computable bandwidth measures for both smoothing windows and the time series of interest. We give such measures and demonstrate their utility. Tapering is often given little coverage in books on time series and spectral analysis. We have found it to be extremely useful in practice, and Chapter 6 develops the structure and behavior of tapered and smoothed spectral estimators, with several examples.
• The multitaper spectral analysis method -- currently proving to be so powerful in many areas of science where spectra with high dynamic range are commonplace -- is studied in Chapter 7, with some necessary computational details supplied in Chapter 8. Many scientists are initially intimidated by this method due to its apparent  complexity. In reality, it is conceptually simple, as we show with a simple heuristic interpretation. In addition, we provide formal justification via both matrix algebra and projection methods.
• In general most significant developments in harmonic analysis of data containing sinusoids (spectral lines) in noise have taken place in the electrical engineering literature. Chapter 10 reflects this, but builds in newer statistical tests for sinusoids. It considers in detail problems of spectral line splitting and frequency shifting, and gives a method for reshaping the spectral continuum around the location of a spectral line.
• A large number of exercises, of varying levels of difficulty, are provided at the end of the chapters. Many of these exercises support results in the main text (instructors wishing to obtain a solution guide for the exercises should contact the authors for details).

This book grew out of a graduate course given by each of us to students of statistics, electrical engineering and the physical sciences at the University of Washington. The book is similarly best suited to graduate students and researchers. The emphasis is, naturally enough, very strongly in the frequency domain, but time domain methods are introduced as a natural complement where necessary and/or useful. While the more traditional forms of spectral analysis have been necessarily included in detail, the emphasis of the book is on methodology developed in the 1970s on up to the current time. Because of their statistical tractability, we also stress nonparametric spectral analysis techniques. While parametric autoregressive moving average (ARMA) models are often convenient for mathematical purposes, they are often very inconvenient or unsatisfactory when used by themselves for analysis of spectra with high dynamic range and a complicated shape, as often occur in studies in the physical sciences. As we discuss in Chapter 9, simple parametric models (such as pure autoregressive) can, however, often be used to good effect for prewhitening, prior to nonparametric analysis.

Most of the algorithms discussed in this book have been coded in Common Lisp and can be run under the various implementations of this language available on a number of different computers (e.g., Macintosh Common Lisp). This code and most of the data sets used in this book can be obtained through e-mail. Details are given on page xxii.

A complementary volume to this one is currently under preparation and is scheduled to include topics such as spectral analysis of multivariate time series, higher order spectra, spectral estimation for irregularly sampled time series, robust spectral estimation, estimation of power-law spectra, spectral ratios and the simulation of stationary processes.

The book was written using Donald Knuth's superb typesetting system TeX as implemented by Blue Sky Research in their product TeXtures for Apple Macintosh computers. With one exception, all of the figures in this book were created using the plotting system GPL written by W. Hess, who provided us with excellent technical support. The majority of the computations necessary for the various examples and figures were carried out using PiTSSa, a Lisp-based object-oriented program for interactive time series and signal analysis that was developed in part by one of us (Percival) with support from the Office of Naval Research, the Naval Observatory and the Naval Research Laboratory. Without these software tools, this book would certainly never have existed.

We thank S. Murphy, J. Harlett, and R. Spindel of the Applied Physics Laboratory, University of Washington, for providing discretionary funding at critical times to help us start and finish this book. We are much indebted to those who have commented on the manuscript or supplied data to us, namely, B. Bell, W. Dunlap, J. Filliben, W. Fox, P. Guttorp, A. Jessup, R. D. Martin, D. McCarthy, E. McCoy, H. Nhu, C. Siedenburg and B. Walter. Particular thanks are due to C. Greenhall, whose careful and thorough critiques substantially improved this manuscript in a number of areas. We thank C. Andersen and P. Magassy for carefully proofreading earlier versions of the manuscript. We are also very grateful to the many graduate students who suffered through incomplete early versions of various chapters, giving us valuable critiques of the manuscript and exercises and spotting numerous errors. Any remaining errors are, of course, our responsibility, and we would be pleased to hear from any reader who finds a mistake (our `paper' and electronic mailing addresses are listed below). Finally we thank our families for their patience over the seven year period this book was being written.

Don Percival, Applied Physics Laboratory, Box 355640, University of Washington Seattle, WA 98195-5640;