Example of deconvolution with a known wavelet

The top trace of Figure 1 shows a marine reflection seismic trace from northern Scandinavia. The most pronounced feature is a series of multiple reflections from the ocean bottom seen at .6s intervals. These reflections share a similar waveshape that alternates in polarity. (The alternation of polarity will be more apparent after deconvolution.) The alternation of polarity results from a negative reflection coefficient at the ocean surface (where the acoustic pressure vanishes). The spectrum of the top trace has a comb pattern that results from the periodicity of the multiples.

 

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Figure 1 The top half is time functions and the bottom half is the corresponding frequency functions. The top signal is a marine seismogram 4s long. A wavelet windowed between 0.5s and 1s was used to deconvolve the signal with a tiny $\epsilon$ (bottom) and a larger $\epsilon$ (middle). (Calculated by Bill Harlan.)

Letting the raw trace be Y, a filter F was chosen by extracting a portion of Y, that portion from 0.5 to 1 second that is the water bottom reflection. The spectrum of this windowed portion of the trace is like that of the whole trace except the comb pattern is absent (for reasons given in chapter 1). The second trace of Figure 1 shows the result of a damped deconvolution of the data with this wavelet. The reflections now look like bandlimited impulses, and the spectrum appears only slightly broader than before. The third trace of Figure 1 shows the result of deconvolution with a much smaller damping factor. It is noisy at high frequencies.

Experiments with $\epsilon^2$,showed that satisfactory results were found within a large numerical range. The damped deconvolution shown in Figure 1 took $\epsilon$ to be 0.01 times the average over $\omega$ of |F| and for the underdamped example it was 0.00001. Bill Harlan and I each experimented with $\epsilon$varying with frequency but did not get results interesting enough to show.