Summary
Apparent attenuation resulting from interbed multiples is nicely expressed by the nonstationary convolution model proposed by Margrave (1997). Nonstationary convolution is an extension of the conventional convolutional model for nonstationary processes such as time migration, normal moveout corrections, forward and inverse Q filtering, etc. Any nonstationary but linear effect can be included in the nonstationary model by an appropriate modification to the convolutional matrix. By embedding pure propagating wavelets for each earth interface in the convolutional matrix, nonstationary convolution replicates the effects of interbed multiples in the output trace. These include significant time delays of the primary energy (Stewart et al., 1984) and high frequency loss for the transmitted wavelets for highly cyclic sequences such as coal beds (O’Doherty and Anstey, 1971). Because each column vector in the convolution matrix is associated with a primary-only reflection coefficient, the aligned convolution matrix is better defined as a wavelet dictionary. A major goal in data processing is to convert the various time series in the wavelet dictionary into propagating wavelets that are not time-varying. To assist in this task, the wavelet dictionary time series were approximated with mathematically-defined truncated minimum-phase equivalent Gaussian pulses. As a measure of success, nonstationary convolution with the wavelet dictionary provided a much better synthetic match to field data than the conventional synthetic seismogram and it duplicated the results of the exact all internal multiple algorithm defined by Waters (1981).
