Wavelet Variance Analysis for Random Fields

There has been considerable recent interest in using wavelets to analyze time series and images that can be regarded as realizations of certain oneand two-dimensional stochastic processes. Wavelets give rise to the concept of the wavelet variance (or wavelet power spectrum), which decomposes the variance of a stochastic process on a scale-by-scale basis. The wavelet variance has been applied to a variety of time series, and a statistical theory for estimators of this variance has been developed. While there have been applications of the wavelet variance in the two-dimensional context (in particular in works by Unser, 1995, on wavelet-based texture analysis for images and by Lark and Webster, 2004, on analysis of soil properties), a formal statistical theory for such analysis has been lacking. In this paper, we develop the statistical theory by generalizing and extending some of the approaches developed for time series, thus leading to a large sample theory for estimators of two-dimensional wavelet variances. We apply our theory to simulated data from Gaussian random fields with exponential covariances and from fractional Brownian surfaces. We also use our methodology to analyze images of four types of clouds observed over the south-east Pacific ocean.