Let k = (k0, k1,…, kN−1) denote the least-squares prediction filter for prediction distance α. The prediction-error filter results directly from the prediction filter. As we saw in equation 12 of Chapter 10, the prediction-error filter is f = (1, 0, 0,…, 0, − k0, −k1,…, −kN−1). The prediction-error operator has α−1 zeros that lie between the leading coefficient, which is 1, and the negative prediction-filter coefficients. These α−1 zeros constitute the gap.
Let us begin by examining a prediction-error filter, which is in fact a deconvolution filter. The associated prediction-error series is the deconvolved signal. (See Appendix K, exercise 34, at the end of this chapter for a further description of the prediction filter and the prediction-error filter.) A prediction-error filter must be causal. A successfully deconvolved signal shows improved seismic resolution and provides an estimate of the reflectivity series. Depending on a specified prediction distance α, we distinguish between two types of predictive deconvolution: (1) spiking deconvolution, for which the prediction distance equals one time unit, and (2) gap deconvolution, for which the prediction distance is greater than one time unit.
Let B(Z) be the Z-transform of a minimum-phase wavelet b. Then A(Z) = 1/B(Z) is the Z-transform of the inverse wavelet a = b−1. For prediction distance α, the head of b is h = (b0, b1, …, bα−1) and the tail is t = (bα, bα+1, …). For both the head and tail, the first coefficient is at time 0. Thus, the
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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing (SEG Geophysical References Series No. 15), covers the basic ideas and methods used in seismic processing, concentrating on the fundamentals of seismic imaging and deconvolution. Most chapters are followed by problem sets. Some exercises supplement textual material; others are meant to stimulate classroom discussions. Text and exercises deal mostly with simple examples that can be solved with nothing more than pencil and paper. The book covers wave motion; digital imaging; digital filtering; various visualization aspects of the seismic reflection method; sampling theory; the frequency spectrum; synthetic seismograms; wavelets and wavelet processing; deconvolution; the need for continuing interaction between the seismic interpreter and the computer; seismic attributes; phase rotation; and seismic attenuation. The last of the 15 chapters gives a detailed mathematical overview. Digital Imaging and Deconvolution, nominated for the Association of Earth Science Editors award for best geoscience publication of 2008–2009, will interest professional geophysicists, graduate students, and upper-level undergraduates in geophysics. The book also will be helpful to scientists and engineers in other disciplines who use digital signal processing to analyze and image wave-motion data in remote-detection applications. The methods described are important in optical imaging, video imaging, medical and biological imaging, acoustical analysis, radar, and sonar.