Higher-order statistics: improved event resolution?

David C. Henley and Arnim B. Haase


In any area where near-surface earth structure causes one or more small time delays in seismic energy transmission, a seismic trace can be represented as a bandlimited reflectivity signal convolved with a short sequence of scaled copy spikes, each spike representing the delay due to a particular travel path variation. If the pattern (distribution) of spikes or static time shifts can be determined, its effects can be removed from the seismic trace by an explicit deconvolution operation. Accurate estimation of the spike pattern (static distribution function), however, is a key difficulty.

The cross-correlation between a multi-shifted seismic trace and a "pilot" reflectivity trace can be used to estimate the embedded static shift distribution function. For a trace consisting of the simple convolution of a reflectivity series and a short sequence of scaled static delay spikes, with no additive noise, the cross-correlation can perfectly extract the spike sequence. The cross-correlation is bandlimited by the mutual spectra of the input trace and pilot trace, however, and any noise present in the input time series degrades the resolution of the underlying spike sequence, the amount of degradation being inversely proportional to the bandwidth and to the S/N ratio. Hence, we seek an improved alternative to the cross-correlation as an event detector.

The cross-bicorrelation function is a so-called "higher-order statistic" which decomposes the correlation between events on two time series into a two-dimensional correlation function, of which selected individual profiles may show better resolution between closely spaced or weak events than the conventional cross-correlation. Furthermore, the cross-bicorrelation function can be spectrally "normalized", which increases its resolution further.

Preliminary work shows that individual slices of the cross-bicorrelation function can, indeed have greater resolution than a cross-correlation. Selecting the appropriate slice of the function requires extra knowledge or interpretation, however, and the presence of any substantial amount of noise degrades the cross-bicorrelation function, reducing its advantage over the cross-correlation. The frequency-normalized cross-bicorrelation, however, appears to provide improved resolution; and there are tricks which may help to preserve this resolution in the presence of noise.

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