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