Digital filtering is a very commonly used seismic data processing technique, and it has many forms for different applications. This chapter begins by describing three ways to express digital filtering: the rational form, recursive formula, and block diagram. The names of the filters usually come from their effects on the frequency spectrum. In the rational form of a filter, the zeros are the roots of the numerator, and the poles are the roots of the denominator. Using the zeros and poles we can make the pole–zero representation on the complex z-plane as a convenient way to quantify the effect of a digital filter as a function of frequency. The rule of thumb is: poles add, zeros remove, and the magnitude of the effect of the pole or zero depends on their distance from the unit circle. Different types of filtering in seismic data processing are discussed in the chapter using several examples. In particular, f–k filtering is discussed in detail with its typical processing flow. Owing to the widespread application of inverse problem in geophysics, much of the attention is given to inverse filtering, which requires that the corresponding filter be invertible. It can be proven that a minimum-phase filter is always invertible because all of its zeros and poles are outside the unit circle on the complex z-plane. This notion means that the minimum-phase filters occupy an important position in seismic data processing. In general, a minimum-phase wavelet is preferred in seismic data processing because of stability concerns, while a zero-phase wavelet is preferred in seismic interpretation to maximize the seismic resolution. The final section of the chapter prepares the reader with the physical and mathematical background materials for inverse filtering. These materials are fundamental to the understanding of deconvolution, an application of inverse filtering, in the next chapter.

Functionality of a digital filter

A digital filter is represented by a sequence of numbers called weighting coefficients, which can be expressed as a time series or denoted by the z-transform. When the filter acts on an input digital signal which can be expressed as another time series, the filter functions as a convolution with the input signal (Figure 5.1).