Level crossing statistics and Hausdorff‐Besicovitch similarity dimension of generalized rough surfaces

Abstract

Generalized rough surfaces can exhibit a rich variety of relief that cannot be easily described as either periodic or Gaussian. For such surfaces, one can explore the spatial distribution per unit length of occurrence of the mean height (=0 by definition), the rms heights, or the crossing of the relief with a chosen level. An extension of this concept is also possible with joint probability distribution characterization of the distribution of successive level crossing known as Palm distributions. Evaluation of these measures for some test functions and stochastically generated surface relief is presented and the benefits of their use is contrasted with conventional statistical and correlation function techniques. Surfaces that contain a possible infinite hierarchy of scales, e.g., fractal surfaces, are shown to best be characterized by a parameter known as the Hausdorff‐Besicovitch similarity (HBS) dimension. The use of this parameter in describing generalized rough surfaces possessing self‐similar characteristics or multiple scales of roughness is described and the reflection characteristics of such surfaces in optical and acoustical applications are discussed.