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.

Stochastic Interpolation of Aquifer Properties Using Fractional Lévy Motion

“…This paper describes a new and more realistic model-based technique for conditional simulation of aquifier properties in unobserved regions. The unique aspect of the new approach is the use of a recently introduced model for heterogeneity [Painter and Paterson, 1994;Painter et al, 1995] based on L6vy-stable probability distributions. Specifically, permeability and porosity variations are modeled using fractional L6vy motion (fLm) [Taqqu, 1987], a generalization of fractional Brownian motion [Mandelbrot and Van Ness, 1968].…”

Statistical Significance of Periodicity and Log-Periodicity With Heavy-Tailed Correlated Noise

“…Expression (14) expresses the fact that the product of the probability to be of the order of or larger than x max by the total number of points is of the order 1, which defines x max . Figure 6 shows the maximal Lomb peak P N (ω) averaged over 50,000 realizations of 100 points as a function ofσ 2 given by expression (13). Hereσ 2 is varied by changing the power law exponent from α = 0.1 to 1.9.…”

“…There is no closed analytical expression for the probability function of the stable distributions with a few exceptions, e.g., Cauchy distribution (α = 1) and Gaussian distribution (α = 2) [13]. Stable distributions are given in terms of their characteristic function [14].…”

Estimating the distribution of primary reflection coefficients

“…Alternatively, nonparametric methods do not assume any pdf form. The histogram, perhaps the simplest method of this group, has been used as an auxiliary tool in the analysis of the distribution and spectral properties of seismic log data (Todoeschuck et al, 1990;Painter et al, 1995;Jones and Holliger, 1997). As is well known, the histogram presents Manuscript some drawbacks: the results depend on the bin size and the origin of the bins.…”

“…Some authors in the geophysical community focus on estimating the pdf by fitting the sample data to a predefined model (Painter et al, 1995;Walden and Hosken, 1986). These strategies belong to the class of parametric methods, because a finite set of control parameters are fitted to the data.…”