Effect of random noise on deconvolution
We assume that the noise component in the recorded seismogram is zero (assumption 4). The autocorrelation of ideal random noise is zero at all lags except the zero lag (Figure 2.1-5). Therefore, the effect of random noise on deconvolution operators should be somewhat similar to the effect of prewhitening. Both effects modify the diagonal of the autocorrelation matrix, making it more dominant [equation (32)].
However, the noise component also slightly modifies the nonzero lags of the autocorrelation. Compare the autocorrelograms of traces (b) in Figures 2.4-24 and 2.4-31. In Figure 2.4-24, an isolated minimum-phase wavelet was considered, while in Figure 2.4-31, random noise was added to the same wavelet. The output wavelet shape from spiking deconvolution of the noisy wavelet using a 128-ms operator is similar to the output from spiking deconvolution of the wavelet without noise, using the same operator length but with, say, 20 percent prewhitening. This result has practical importance — prewhitening is equivalent to adding perfect random noise to the system. Since a recorded seismogram always contains some amount of random noise, only a minute amount, say 0.1 percent, of the white noise needs to be added to the seismogram for numerical stability.
The effect of random noise on the performance of deconvolution is examined further in Figures 2.4-32 and 2.4-33. These results should be compared with their noiseless counterparts in Figures 2.4-9 and 2.4-13, respectively. Observe that the noise component has a harmful effect on deconvolution. For example, when comparing Figures 2.4-9 and 2.4-32, note that the deconvolution result from the noisy seismogram has spurious spikes (for instance, between 0.5 and 0.6 s), which could be interpreted as genuine reflections. Noisy field data, which yield better stack when not treated by deconvolution, have been noted. Only by testing can we determine whether deconvolution performs satisfactorily on data with a severe noise problem.
Figure 2.1-5 A random signal with infinite length has a flat amplitude spectrum and an autocorrelogram that is zero at all lags except the zero lag. The discrete random series with finite length shown here seems to satisfy these requirements. What distinguishes a random signal from a spike (1, 0, 0, …)?