# Prewhitening

From the preceding section, we know that the amplitude spectrum of the spiking deconvolution operator is (approximately) the inverse of the amplitude spectrum of the input wavelet. This is sketched in Figure 2.3-3. What if we had zeroes in the amplitude spectrum of the input wavelet? To study this, apply a minimum-phase band-pass filter (Exercise 2-10) with a wide passband (3-108 Hz) to the minimum-phase wavelet of Figure 2.3-2, as shown in frame (h). Deconvolution of the filtered wavelet does not produce a perfect spike; instead, a spike accompanied by a high-frequency pre-and post-cursor results (Figure 2.3-4). This poor result occurs because the deconvolution operator tries to boost the absent frequencies, as seen from the amplitude spectrum of the output. Can this problem occur in a recorded seismogram? Situations in which the input amplitude spectrum has zeroes rarely occur. There is always noise in the seismogram and it is additive in both the time and frequency domains. Moreover, numerical noise, which also is additive in the frequency domain, is generated during processing. However, to ensure numerical stability, an artificial level of white noise is added to the amplitude spectrum of the input seismogram before deconvolution. This is called prewhitening and is referred to in Figure 2.3-3.

If the percent prewhitening is given by a scalar, 0 ≤ ε < 1, then the normal equations (31) are modified as follows:

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

 ${\begin{pmatrix}\beta r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&\beta r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&\beta r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &\beta r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}},$ (32)

where β = 1 + ε. Adding a constant εr0 to the zero lag of the autocorrelation function is the same as adding white noise to the spectrum, with its total energy equal to that constant. The effect of the prewhitening level on performance of deconvolution is discussed in Predictive deconvolution in practice.