Effect of random noise on deconvolution

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

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)].

{\begin{pmatrix}\beta r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&\beta r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&\beta r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &\beta 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}},

(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, …)?
Figure 2.4-9 Test of operator length where n = operator length, α = prediction lag, and ε = percent prewhitening. (a) Reflectivity, (b) impulse response, (c) seismogram with unknown, minimum-phase source wavelet.
Figure 2.4-13 Test of operator length where n = operator length, α = prediction lag, and ε = percent prewhitening. (a) Reflectivity, (b) impulse response, (c) seismogram with unknown, mixed-phase source wavelet.
Figure 2.4-24 Test of percent prewhitening for a single, isolated input wavelet where n = operator length, α = prediction lag, and ε = percent prewhitening. (a) Impulse response, (b) seismogram with minimum-phase source wavelet.
Figure 2.4-31 Effect of random noise on deconvolution performance. Input seismogram (b) associated with reflectivity (a) contains a single, isolated wavelet (at around 0.2 s) buried in random noise. Here, n = operator length, α = prediction lag, and ε = percent prewhitening.
Figure 2.4-32 Effect of random noise on deconvolution performance. (a) Reflectivity, (b) impulse response. Input seismogram (c) with unknown, minimum-phase source wavelet is noise contaminated. Here, n = operator length, α = prediction lag, and ε = percent prewhitening.
Figure 2.4-33 Effect of random noise on deconvolution performance. (a) Reflectivity, (b) impulse response. Input seismogram (c) with unknown, mixed-phase source wavelet is noise contaminated. Here, n = operator length, α = prediction lag, and ε = percent prewhitening.