Our goal is to obtain an inverse wavelet filter that has a spectrum that is the inverse of the wavelet's spectrum at low frequencies only, with zero values beyond. Like other methods, we propose to compute the inverse wavelet from autocorrelation of the wavelet. This paper assumes that the reflectivity series is white and the autocorrelation of the seismogram is a scaled version of that of the seismic wavelet (Yilmaz, 1987), and the spectrum of the seismogram is a jagged version of that of the wavelet (Claerbout, 1991). Since the spectrum of the wavelet can be computed from its autocorrelation, and in our seismic case the wavelet spectrum band occupies only the low frequency end of the full spectrum, proper subsampling of the autocorrelation does not incur loss of information. The idea is to compute the inverse wavelet with a lower Nyquist frequency (coarser sampling interval) that contains just the wavelet spectrum. Using a subsampled autocorrelation to compute the inverse wavelet at a coarser sampling interval confines its effective frequency range to the wavelet frequency band.
Suppose we have decided that there is no energy beyond half Nyquist. Then the effective frequency range of the inverse wavelet should be the first half Nyquist interval. Then the inverse wavelet should be computed at a sampling interval that is twice the original sampling interval. This means that we need compute only every other point of the autocorrelation . Next we put these values into a conventional wavelet decon operator design program (Claerbout, 1976, p. 57) to compute the inverse wavelet .The resulting inverse wavelet has a spectrum that is the inverse of the wavelet spectrum, and has a Nyquist frequency that is half of the original Nyquist. We then spread these points out to mimic the original sampling interval, yielding , and use this as the wavelet decon operator by convolving it with the seismogram. As the entries of the filter are periodically zero, the increment step of the convolution calculation can be made larger than one. This has two important effects:
Now we can generalize. Whether there is energy all over the spectrum, or only up to half Nyquist, one-third Nyquist, or one-quarter Nyquist will determine whether we use every point, every other point, every third point, or every fourth point of the autocorrelation, and the corresponding order of zero filling of the computed inverse wavelet. The next important aspect is operator length. We try 2,4,8,16,32,64 non-zero points, compare their results, and select the one that best flattens the wavelet spectrum and does not enhances noise much.
Figure 1 shows (on the left) a common shot gather from a Geco marine dataset, after muting to remove direct wave and head waves, and after tpow=1 spherical divergence correction. The water bottom reflections are so strong that even when we use tpow=2, the deep parts are still not visible; its use only causes the waterbottom multiples to be stronger than its primary. So we stay at tpow=1 correction. On the right of Figure 1 is the output of wavelet decon. On this section, the wavelet is compressed, but the noise is not enhanced much. Figure 2 shows the amplitude spectrum of the 12th trace before and after wavelet decon.
Because there is energy only up to half Nyquist frequency, we compute only every other point of the autocorrelation. We tried and finally made the choice of 16 non-zero points in the autocorrelation and the inverse wavelet filter. The inverse wavelet filter for the 12th trace is shown on the left of Figure 3. For comparison, on the right of Figure 3 is the conventional Burg inverse wavelet. Figure 4 shows the spectra of the two different wavelets. Convolving our inverse wavelet with the seismogram better flattens the wavelet spectrum. While the amplitude spectrum of the noise is smaller than that of the signal, this processing does not significantly enhance random noise.
Here we use time-invariant decon because we concern ourselves only with the hard waterbottom reflections that have a consistent waveform. Other events are too weak to be seen.