# Signature deconvolution

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

In marine seismic exploration, the far-field signature of the source array can be recorded. The idea is to apply a deterministic deconvolution to remove the source signature, then to apply predictive deconvolution. The convolutional model is given by

 ${\displaystyle x(t)=s(t)\ast w(t)\ast e(t),}$ (41)

where s(t) is the source signature recorded in the far-field just before it travels down into the earth, which has an impulse response e(t). Since s(t) is recorded, an inverse filter can be deterministically designed, as discussed in Inverse filtering, then applied to the recorded seismogram to remove it from equation (41). The unknown wavelet w(t) includes the propagating effects in the earth and the response of the recording system. This remaining wavelet then is removed by the statistical method of spiking deconvolution as discussed in Optimum wiener filters. Compare equation (41) with equation (3a) and note that the old w(t) of equation (3a) is split into two parts — the source signature s(t), which is the known component, and the new w(t), which is the unknown component.

 ${\displaystyle {x(t)=w(t)\ast e(t)}.}$ (3a)

There are two ways to handle s(t). One way is to convert it to its minimum-phase equivalent followed by predictive deconvolution (Figure 2.5-7). Another way is to convert s(t) into a spike followed by predictive deconvolution (Figure 2.5-8). The process involves the following steps:

• Estimate the minimum-phase equivalent of the recorded source signature by computing the spiking deconvolution operator (equation 39) and taking its inverse.

 ${\displaystyle {\begin{pmatrix}r_{0}&r_{1}&r_{2}&r_{3}&r_{4}&r_{5}\\r_{1}&r_{0}&r_{1}&r_{2}&r_{3}&r_{4}\\r_{2}&r_{1}&r_{0}&r_{1}&r_{2}&r_{3}\\r_{3}&r_{2}&r_{1}&r_{0}&r_{1}&r_{2}\\r_{4}&r_{3}&r_{2}&r_{1}&r_{0}&r_{1}\\r_{5}&r_{4}&r_{3}&r_{2}&r_{1}&r_{0}\\\end{pmatrix}}{\begin{pmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\\b_{4}\\b_{5}\\\end{pmatrix}}={\begin{pmatrix}L\\0\\0\\0\\0\\0\\\end{pmatrix}}}$ (39)
• (b) Design a shaping filter to convert the source signature to its minimum-phase equivalent or a zero-delay spike (equation 30).

 ${\displaystyle {\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}g_{0}\\g_{1}\\g_{2}\\\vdots \\g_{n-1}\end{pmatrix}}}$ (30)
• Apply the shaping filter to each trace in each recorded shot record.
• Apply predictive deconvolution to output data from step (c).

The results shown in Figures 2.5-7 and 2.5-8 (panels (c)) should be compared with single-step statistical deconvolution (Figure 2.5-9). Since the source was not minimum-phase in this case, Figure 2.5-9b should be better than Figure 2.5-9d. Is it?

Actually, results of signature processing depend on the accuracy of the recorded signature. One should avoid signature processing of old marine data unless there exists a concurrently recorded source signature. Contemporary marine data almost always include the recorded source signature at each shot location. Figure 2.5-10a shows a recorded water-gun signature with its amplitude and phase spectra. The minimum-phase equivalent is shown in Figure 2.5-10b, and the result of signature deconvolution to convert the recorded waveform to its minimum-phase equivalent is shown in Figure 2.5-10c. While the amplitude spectrum is unaltered, the phase spectrum is minimum phase.

Application of signature processing described in Figure 2.5-10 to a recorded shot gather is shown in Figure 2.5-11. Again, note that signature processing aimed at converting the recorded source signature to its minimum-phase equivalent leaves the amplitude spectrum and autocorrelogram of the shot record unaltered (Figures 2.5-11a,b). We can also observe from the corresponding CMP-stacked sections that the process has not made any impact on the degree of vertical resolution; only a change in phase has taken place (Figure 2.5-12). Following the deterministic step of Figure 2.5-11b, statistical deconvolution is applied to flatten the spectrum (Figure 2.5-11c).