# Spiking deconvolution

The process with type 1 desired output (zero-lag spike) is called spiking deconvolution. Crosscorrelation of the desired spike (1, 0, 0, …, 0) with input wavelet (x0, x1, x2, …, xn−1) yields the series (x0, 0, 0, …, 0).

The generalized form of the normal equations (30) takes the special form:

 ${\displaystyle {\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}g_{0}\\g_{1}\\g_{2}\\\vdots \\g_{n-1}\end{pmatrix}}}$ (30)

 ${\displaystyle {\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}}}$ (31)

Equation (31) was scaled by (1/x0). The least-squares inverse filter, which was discussed in Inverse filtering, has the same form as the matrix equation (31). Therefore, spiking deconvolution is mathematically identical to least-squares inverse filtering. A distinction, however, is made in practice between the two types of filtering. The autocorrelation matrix on the left side of equation (31) is computed from the input seismogram (assumption 6) in the case of spiking deconvolution (statistical deconvolution), whereas it is computed directly from the known source wavelet in the case of least-squares inverse filtering (deterministic deconvolution).

Figure 2.3-2  Starting with wavelet (a), autocorrelogram (d) is computed to derive spiking deconvolution operator (e). This operator and its inverse (h) are minimum-phase. The inverse of the deconvolution operator has the same amplitude spectrum as that of the original wavelet. [Compare (i) to (b).] Its phase spectrum is simply the sign-reversed phase spectrum of the spiking deconvolution operator. [Compare (j) to (g).] If operator e were convolved with original wavelet a, then output k results. Although the output spectrum nearly is flat (l), it is far from being a spike at t = 0 (n). This desired output (n) is obtained if the input is a minimum-phase wavelet (h), rather than a mixed-phase wavelet (a).

Figure 2.3-2 is a summary of spiking deconvolution based on the Wiener-Levinson algorithm. Frame (a) is the input mixed-phase wavelet. Its amplitude spectrum shown in frame (b) indicates that the wavelet has most of its energy confined to a 10- to 50-Hz range. The autocorrelation function shown in frame (d) is used in equation (31) to compute the spiking deconvolution operator shown in frame (e). The amplitude spectrum of the operator shown in frame (f) is approximately the inverse of the amplitude spectrum of the input wavelet shown in frame (b). (The approximation improves as operator length increases.) This should be expected, since the goal of spiking deconvolution is to flatten the output spectrum. Application of this operator to the input wavelet gives the result shown in frame (k).

Ideally, we would like to get a zero-lag spike, as shown in frame (n). What went wrong? Assumption 7 was violated by the mixed-phase input wavelet shown in frame (a). Frame (h) shows the inverse of the deconvolution operator. This is the minimum-phase equivalent of the input mixed-phase wavelet in frame (a). Both wavelets have the same amplitude spectrum shown in frames (b) and (i), but their phase spectra are significantly different as shown in frames (c) and (j). Since spiking deconvolution is equivalent to least-squares inverse filtering, the minimum-phase equivalent is merely the inverse of the deconvolution operator. Therefore, the amplitude spectrum of the operator is the inverse of the amplitude spectrum of the minimum-phase equivalent as shown in frames (f) and (i), and the phase spectrum of the operator is the negative of the phase spectrum of the minimum-phase wavelet as shown in frames (g) and (j). One way to extract the seismic wavelet, provided it is minimum phase, is to compute the spiking deconvolution operator and find its inverse.

In conclusion, if the input wavelet is not minimum phase, then spiking deconvolution cannot convert it to a perfect zero-lag spike as in frame (k). Although the amplitude spectrum is virtually flat as shown in frame (l), the phase spectrum of the output is not minimum phase as shown in frame (m). Finally, note that the spiking deconvolution operator is the inverse of the minimum-phase equivalent of the input wavelet. This wavelet may or may not be minimum phase.