A statistical measure of regularity for the study of wind-generated wave field images

The study of water waves has generated a wealth of sophisticated modeling developments in applied mathematics. Empirical observation capabilities have created a need for novel data analysis tools. This article is motivated by consideration of wind-generated wave field image data from a wave tank facility. A quantitative measure of wave field regularity is developed. The methodology is based on decomposition of the wave field into simple plane waves (an adaptation of projection pursuit). The percent variance explained as a function of the number of terms in the plane wave representation is used to define a variogram, and regularity of the wave field is assessed in terms of the weighted difference between the observed variogram and the expected variogram for a completely random field. The proposed regularity measure is illustrated by application to image data from wind-generated waves. The results suggest that the regularity measure is a function of parameters describing the generation of the wave field (wind speed and evolution). An analysis of the statistical behavior of the regularity measure as a function of sample size (image resolution) is carried out. Even though the regularity measure is based on the nonparametric estimation of functions, provided that this estimation is carried out in a consistent fashion, the error in estimation of regularity has a parametric dependence on image resolution.