Discrete spectral analysis is a suite of classic data processing tools aiming to quantify the energy distribution of seismic data over temporal or spatial scales. This chapter starts with the law of decomposition and superposition, which is the foundation of many seismic processing methods. According to Fourier theory, a seismic trace can be expressed as a linear superposition of harmonic functions of different frequencies with appropriate amplitudes and phases, thus enabling the spectral analysis. Because seismic data are in digital form with limited time durations, classic spectral analysis is achieved using discrete Fourier transform (DFT). Readers should pay special attention to the characteristics of the DFT, as these often differ from the continuous Fourier transform. Fast Fourier transform (FFT) is described as an example to improve the computation efficiency in processing.

Discrete spectral analysis is discussed using several examples. Useful processing tricks in one type of processing are often borrowed to solve problems in another type of processing. To decompose seismic traces, for instance, we may use wavelets of fixed shape but varying amplitudes and lengths rather than harmonic functions as the basic building blocks. This enables wavelet decomposition and seismic wavelet analysis, as another application of the law of decomposition and superposition. Yet another usage of the law is in interpolation of digital data, which has many applications in the processing of seismic and non-seismic data.