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.

• 

  Figure 2.3-3  Prewhitening amounts to adding a bias to the amplitude spectrum of the seismogram to be deconvolved. This prevents dividing by zero since the amplitude spectrum of the inverse filter (middle) is the inverse of that of the seismogram (left). Convolution of the filter with the seismogram is equivalent to multiplying their respective amplitude spectra — this yields nearly a white spectrum (right).

• 

  Figure 2.3-4  (a) Minimum-phase wavelet, (b) after band-pass filtering, (c) followed by deconvolution. The amplitude spectrum of the band-pass filtered wavelet is zero above 108 Hz (middle row); therefore, the inverse filter derived from it yields unstable results (bottom row). The time delays on the wavelets in the left frames of the middle and bottom rows are for display purposes only.

See also

• Spiking deconvolution
• Wavelet processing by shaping filters
• Predictive deconvolution

External links

find literature about Prewhitening