On the distribution of seismic reflection coefficients and seismic amplitudes

Abstract

Reflection coefficient sequences from 14 wells in Australia have a statistical character consistent with a non‐Gaussian scaling noise model based on the Lévy‐stable family of probability distributions. Experimental histograms of reflection coefficients are accurately approximated by symmetric Lévy‐stable probability density functions with Lévy index between 0.99 and 1.43. These distributions have the same canonical role in mathematical statistics as the Gaussian distribution, but they have slowly decaying tails and infinite moments. The distribution of reflection coefficients is independent of the spatial scale (statistically self‐similar), and the reflection coefficient sequences have long‐range dependence. These results suggest that the logarithm of seismic impedance can be modeled accurately using fractional Lévy motion, which is a generalization of fractional Brownian motion. Synthetic seismograms produced from our model for the reflection coefficients also have Lévy‐stable distributions. These simulations include transmission losses, the effects of reverberations, and the loss of resolution caused by band‐limited wavelets, and suggest that actual seismic amplitudes with sufficient signal‐to‐noise ratio should also have a Lévy‐stable distribution. This prediction is verified using poststack seismic data acquired in the Timor Sea and in the continental USA. However, prestack seismic amplitudes from the Timor Sea are nearly Gaussian. We attribute the difference between prestack and poststack data to the high level of measurement noise in the prestack data. Many of the basic statistical techniques upon which seismic deconvolution and wavelet estimation are based presume implicitly that the underlying probability distribution has finite variance. Sample variance, autocorrelation, and even sample means are not reliable for infinite‐variance Lévy‐distributed variables. Although the seismic amplitudes and reflection coefficients are bounded and cannot truly have a Lévy‐stable distribution, some of their statistical properties are similar to those of Lévy‐stable variables. In other words, they behave as if they were following a Lévy‐stable distribution, which suggests that methods developed for Gaussian or near‐Gaussian variables may not be the most appropriate. The highly non‐Gaussian nature of the seismic amplitudes and the reflection coefficients should be considered in the design of seismic deconvolution and inversion techniques.