1. Fundamentals of Signal Processing
Katarzyna Mościńska · Society of Exploration Geophysicists eBooks · 2001
1.0 INTRODUCTION The Fourier transform is fundamental to seismic data analysis. It applies to almost all stages of processing. A seismic trace represents a seismic wavefield recorded at a receiver location. The digital form of a seismic trace is a time series which can be completely described as a discrete sum of a number of sinusoids — each with a unique peak amplitude, frequency, and a phase-lag (relative alignment). The analysis of a seismic trace into its sinusoidal components is achieved by the forward Fourier transform. Conversely, the synthesis of a seismic trace from the individual sinusoidal components is achieved by the inverse Fourier transform. A brief mathematical review of the Fourier transform is given in Appendix A. Seismic data processing algorithms often can be described or implemented more simply in the frequency domain than in the time domain. In Section 1.1, the one-dimensional (1-D) Fourier transform is introduced and some basic properties of time series in both time and frequency domains are described. Many of the processing techniques — single- and multichannel, involve an operand (seismic trace) and an operator (filter). A simple application of Fourier analysis is in the design of zero-phase frequency filters, typically in the form of band-pass filtering. The two-dimensional (2-D) Fourier transform (Section 1.2) is a way to decompose a seismic wavefield, such as a common-shot gather, into its plane-wave components, each with a certain frequency propagating at a certain angle from the vertical. Therefore, the 2-D Fourier transform can describe processes like migration and frequency-wavenumber (f-k) filtering. A common application of f-k filtering is the rejection of coherent linear noise by dip filtering, and attenuation of multiples based on velocity discrimination between primaries and multiples in the f-k domain (Section 6.2).