| Abstract: | A crucial step in the use of synthetic seismograms is the estimation of the filtering needed to convert the synthetic reflection spike sequence into a clearly recognizable approximation of a given seismic trace. In the past the filtering has been effected by a single wavelet, usually found by trial and error, and evaluated by eye. Matching can be made more precise than this by using spectral estimation procedures to determine the contribution of primaries and other reflection components to the seismic trace. The wavelet or wavelets that give the least squares best fit to the trace can be found, the errors of fit estimated, and statistics developed for testing whether a valid match can be made. If the composition of the seismogram is assumed to be known (e.g. that it consists solely of primaries and internal multiples) the frequency response of the best fit wavelet is simply the ratio of the cross spectrum between the synthetic spike sequence and the seismic trace to the power spectrum of the synthetic spike sequence, and the statistics of the match are related to the ordinary coherence function. Usually the composition cannot be assumed to be known (e.g. multiples of unknown relative amplitude may be present), and the synthetic sequence has to be split into components that contribute in different ways to the seismic trace. The matching problem is then to determine what filters should be applied to these components, regarded as inputs to a multichannel filter, in order to best fit the seismic trace, regarded as a noisy output. Partial coherence analysis is intended for just this problem. It provides fundamental statistics for the match, and it cannot be properly applied without interpreting these statistics. A useful and concise statistic is the ratio of the power in the total filtered synthetic trace to the power in the errors of fit. This measures the overall goodness-of-fit of the least squares match. It corresponds to a coherent (signal) to incoherent (noise) power ratio. Two limits can be set on it: an upper one equal to the signal-to-noise ratio estimated from the seismic data themselves, and a lower one defined from the distribution of the goodness-of-fit ratios yielded by matching with random noise of the same bandwidth and duration as the seismic trace segment. A match can be considered completely successful if its goodness-of-fit reaches the upper limit; it is rejected if the goodness-of-fit falls below the lower one. |
