Estimation of fractal dimension of random fields on the basis of variance analysis of increments

This paper deals with estimating the fractal dimension of realizations of random fields. The numerical methods in use are based on analysis of the variance of increments. To study the fractal properties, we propose the use of a specific characteristic of random fields called “variational dimension”. For a class of Gaussian fields with homogeneous increments, the variational dimension converges to the Hausdorff dimension. Several examples are presented to illustrate that the concept of variational dimension can be used to construct effective computational methods.