Estimating the Correlation Function of a Self-affine Random Medium

The medium covariance function is of principal importance in refraction travel-time tomographic inversion, especially when estimating the accuracy of the seismic model, its relation to the geological structure, or the covariance matrix describing the statistics of synthetic travel times. The medium correlation function for the travel-time tomography should be obtained from travel times. Since a geological structure contains heterogeneities of all sizes, very similar on various scales, a selfaffine random medium is a mathematical model very suitable for approximating the statistics of a geological structure. A particular class of self-affine random media, composed of a heterogeneous mean value and a stationary self-affine random function, is considered. The self-affine random function is assumed to be realized in terms of a white noise filtered by the power-law spectral filter of amplitude proportional to a reasonable power of the wavenumber. The corresponding power-law medium correlation function depends on two parameters: the Hurst exponent and the reference standard deviation. The corresponding geometrical travel-time covariances are derived. The geometrical travel-time variances are proportional to the power of ray lengths. A method designed to estimate the parameters of the medium correlation function using field travel times is proposed, and applied to data from the Western Bohemia region. The determination of the Hurst exponent from field travel times is very difficult and sensitive to numerical parameters selected for the inversion. The medium correlation functions with the values of the Hurst exponent like N ¼ 0:1 or N ¼ 0:2 are equally acceptable to statistically describe the travel times measured in the Western Bohemia region. On the other hand, for the fixed Hurst exponent, the determination of the reference standard deviation of slowness is easy and reliable.