Spectral Methods in Geodesy and Geophysics

The text develops the principal aspects of applied Fourier analysis and methodology with the main goal to inculcate a different way of perceiving global and regional geodetic and geophysical data, namely from the perspective of the frequency, or spectral, domain rather than the spatial domain. The word "methods" in the title is meant to convey that the transformation of a geophysical signal into the spectral domain can be applied for purposes of analysis as well as rapid computation. The text is written for graduate students; however, Chapters 1 through 4 and parts of 5 can also benefit undergraduates who have a solid and fluent knowledge of integral and differential calculus, have some statistical background, and are not uncomfortable with complex numbers. Concepts are developed by starting from the one-dimensional domain and working up to the spherical domain, which is part of every chapter. Many concepts are illustrated graphically with actual geophysical data primarily from signals of gravity, magnetism, and topography.

Preface Introduction Definitions and Notations Geophysical Motivation Summary Fourier Transforms of Functions on the Continuous Domain Introduction Fourier Series The Fourier Integral Two-Dimensional Transforms in Cartesian Space The Hankel Transform Legendre Transforms From Sphere to Plane Examples of Fourier Transform Pairs Convolutions and Windows on the Continuous Domain Introduction Convolutions of Non-Periodic Functions Convolutions of Periodic Functions Convolutions on the Sphere Filters on the Line, Plane, and Sphere Window Functions Transforms, Convolutions, and Windows on the Discrete Domain Introduction Infinite Sequences Periodic Sequences Cyclic Versus Linear Discrete Convolution Discrete Functions on the Sphere Discrete Filters and Windows Correlation and Power Spectrum Introduction Correlation of Finite-Energy Functions Correlation of Finite-Power Functions Correlation of Periodic Functions Correlation of Functions on the Sphere Stochastic Processes Estimation of the Covariance and PSD Applications in Geodesy and Geophysics Introduction Spectrum of the Potential Function Global Spectral Analysis Local Spectral Analysis Convolutions by FFT Least-Squares Collocation References Exercises