Seismic characterisation of coal interbed multiples in Cooper Basin, Australia

Apparent attenuation resulting from interbed multiples is conveniently expressed by Margrave’s nonstationary convolution model. A few examples of nonstationary processes are time migration, normal-moveout corrections, and forward and inverse Q filtering. 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 at each earth interface in the convolutional matrix, nonstationary convolution replicates the effects of interbed multiples in the output matrix. These propagating wavelets in highly cyclic sequences, such as coal beds, include significant time delays of the primary energy, high-frequency transmission loss and a decrease of seismic resolution for primary energy contaminated with interbed multiples.
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 short propagating wavelets that are not time varying. To assist in this task, the wavelet dictionary time series were approximated with 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. By studying the computed wavelet dictionary, a time delay of 25.6ms/1000ft (304m) and energy loss of 74dB loss/1000ft (304m) for primary energy were observed beneath the coal beds. The two parameters needed to estimate the Gaussian function from the wavelet dictionary amplitude spectra offer insight for designing future data processing algorithms to correct for the coal bed effects. However, the assumption of minimum-phase spectra for the Gaussian wavelets needs further work or different wavelets need are needed to approximate the wavelet dictionary.