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.

Frequency‐wavenumber velocity filtering is often applied to prestack seismic data for the attenuation of coherent noise. Although the process often gives excellent results, it can sometimes result in signal smoothing and distortion and poor attenuation of coherent noise. A slowness adaptive f-k filter reduces signal distortion and improves the attenuation characteristics of the filter. The technique uses a time‐ and space‐variant narrow reject‐band f-k filter. Optionally, coherent noise is compressed before application of the filter. The apparent slowness of coherent noise events is estimated using local t-x slant stacks weighted by coherence. A two‐dimensional (2-D) window is moved across the shot record, and at each point on the record slant stacks are taken through the central sample of the window. The slowness value that produces the maximum stack is assigned to the central sample of the window. In this way, an instantaneous slowness image of the shot record is produced. A one‐dimensional (1-D), high‐pass, finite‐duration impulse‐response (FIR) filter is applied in a spatially and temporally varying way across the record on the basis of the instantaneous slowness values. Before filter application, trace‐to‐trace static and amplitude effects are estimated and removed from the data. This results in compression of coherent noise and improved attenuation after filtering. The filtering process has been applied to low‐fold prestack dynamite data from the Surat Basin, Australia. The results indicate that the technique has good attenuation characteristics and produces minimal distortion of seismic signal. The process, however, is computationally expensive.

Abctract Median filters may be used with seismic data to attenuate coherent wavefields. An example is the attenuation of the downgoing wavefield in VSP data processing. The filter is applied across the traces in the ‘direction’ of the wavefield. The final result is given by subtracting the filtered version of the record from the original record. This method of median filtering may be called ‘median filtering operated in subtraction’. The method may be extended by automatically estimating the slowness of coherent wavefields on a record. The filter is then applied in a time‐ and‐space varying manner across the record on the basis of the slowness values at each point on the record. Median filters are non‐linear and hence their behaviour is more difficult to determine than linear filters. However, there are a number of methods that may be used to analyse median filter behaviour: (1) pseudo‐transfer functions to specific time series; (2) the response of median filters to simple seismic models; and (3) the response of median filters to steps that simulate terminating wavefields, such as faults on stacked data. These simple methods provide an intuitive insight into the behaviour of these filters, as well as providing a semiquantitative measurement of performance. The performance degradation of median filters in the presence of trace‐to‐trace variations in amplitude is shown to be similar to that of linear filters. The performance of median filters (in terms of signal distortion) applied obliquely across a record may be improved by low‐pass filtering (in the t‐dimension). The response of median filters to steps is shown to be affected by background noise levels. The distortion of steps introduced by median filters approaches the distortion of steps introduced by the corresponding linear filter for high levels of noise.

Two‐dimensional median filters can be designed so that they have properties similar to f-k fan filters. This is done by using the coefficients of a truncated impulse response of an f-k filter as the weight coefficients for the weighted median process. The filter is called a median f-k filter and can be used to discriminate between events on the basis of apparent velocity. The filter appears suitable as a poststack coherency filter because it produces less distortion at wavefield terminations than conventional f-k fan filters. One‐dimensional weighted median filters that include negative coefficients are a logical starting point for the analysis of median f-k filters since simple numerical techniques may be used to analyze the behavior of these filters. We show that median filters with negative coefficients do not provide an unbiased estimate of the mean and can misplace the position of steps. Faults on a stacked section may be modeled by steps, and therefore applying a median f-k filter to stacked seismic data could change the position of faults. However, the distortion of steps introduced by median f-k filters is shown to be less than the distortion produced by the corresponding linear f-k filter, and the error in step placement is small. We present simple model examples and a seismic field data example to illustrate differences between linear f-k filters and median f-k filters.

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