The nature of the non-Gaussianity of primary reflection coefficients and its significance for deconvolution

One of the important properties of a series of primary reflection coefficients is its amplitude distribution. This paper examines the amplitude distribution of primary reflection coefficients generated from a number of block-averaged well logs with block thicknesses corresponding to 1 ms (two-way time). The distribution is always essentially symmetric, but has a sharper central peak and larger tails than a Gaussian distribution. Thus any attempt to estimate phase using the bi-spectrum (third-order spectrum) is unlikely to be successful, since the third-order moment is almost identically zero. Complicated tri-spectrum (fourth-order spectrum) calculations are thus required. Minimum Entropy Deconvolution (MED) schemes should be able to exploit this form of non-Gaussianity. However, both these methods assume a white reflectivity sequence; they would therefore mix up the contributions to the trace's spectral shape that are due to the wavelet and those that are due to non-white reflectivity unless corrections are introduced. A mixture of two Laplace distributions provides a good fit to the empirical amplitude distributions. Such a mixture distribution fits nicely with sedimentological observations, namely that clear distinctions can be made between sedimentary beds and lithological units that comprise one or more such beds with the same basic lithology, and that lithological units can be expected to display larger reflection coefficients at their boundaries than sedimentary beds. The geological processes that engender major lithological changes are not the same as those for truncation of bedding. Analyses of sub-sequences of the reflection series are seen to support this idea. The variation of the mixing proportion parameter allows for scale and shape changes in different segments of the series, and hence provides a more flexible description of the series than the generalized Gaussian distribution which is shown to also provide a good fit to the series. Both the mixture of two Laplace distributions and the generalized Gaussian distribution can be expressed as scale mixtures of the ordinary Gaussian distribution. This result provides a link with the ordinary Gaussian distribution which might have been expected to be the distribution of a natural series such as reflection coefficients. It is also important in the consideration of the solution of MED-type methods. It is shown that real (coloured) primary reflection series do not seem to be obtainable as the deconvolution result from MED-type deconvolution schemes.