Thin-bed resolution from cepstrum analysis
A method of cepstrum analysis is developed for the purpose of resolving thin-beds. The method relies on the detection of periodic pulses of the cepstra of reflectivity functions, which are isolated by computing a sub-cepstrum and a sum-cepstrum, and highlighted with a discriminator, where the sub-cepstrum of the functions f₁(t) and f₂(t) is the difference between the cepstra of the two functions, the sum-cepstrum of f₁(t) is the sum of the sub-cepstra of f₁(t) and fk(t), k=2,3,4,... , and the discriminator is the product of the sum-cepstrum and the autocovariance of the sum-cepstrum. The technique requires at least two reflected wavelets generated by the same source.
The method was applied to synthetic thin lens models. The method is shown to be sensitive to the ratio of the reflection coefficients at the top and bottom of the thin-bed. Specifically, the resolution depends on the ratio of the reflection coefficients. Optimum resolution is achieved when the reflection coefficients at the top and bottom of the thin-bed are equal in absolute magnitude. In addition, in the noise-free case, the absolute magnitude of the cepstral pulses can be used to determine the absolute magnitude of the ratio of the reflection coefficients. The technique is also sensitive to the sample interval used. The finest sample interval provides the best resolution because it produces the sharpest cepstral pulses and resolves the thinnest beds. The resolution of the method is drastically reduced by random noise, although thin-bed thicknesses are still detectable when the S/N of the synthetic seismic section is 15/1 and the upper frequency of the bandwidth of the noise is 1.1 octaves above the upper frequency of the bandwidth of the source wavelet.