Process-Driven Direction-Dependent Asymmetry: Identification and Quantification of Directional Dependence in Spatial Fields

Natural processes generate spatial fields which reflect their specific properties. In this paper the effect of the direction of processes on the resulting spatial fields is investigated. This is done by extending the concept of reversibility used for time series to space. A novel copula based measure of asymmetry is defined which is an indicator of directional dependence. Contrary to traditional geostatistics where all points separated by a vector are considered irrespective of its sign, in this study the direction of the vector is also taken into consideration, leading to differences in the dependence corresponding to the vector h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {h}}$$\end{document} and -h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${-}{\mathbf {h}}$$\end{document}. The concept of directional dependence and the corresponding measure of asymmetry are defined using spatial copulas, and are thus independent of the scale of measurement. The result is a bivariate directional third-order moment based measure which can identify the direction in which the processes generating the spatial field acted. A statistical test to find the statistical significance of the asymmetry indicating directional dependence is presented. The methodology is tested on a number of synthetic and observed cases. Precipitation and groundwater quality parameters obtained using numerical models are first investigated. Regular dense grids obtained by numerical simulations show good correspondence between properties of the modeled processes and the new measure introduced. Measured variables observed on sparse irregular networks show similar behavior to the theoretical examples. Mean flow directions in groundwater and advection directions of precipitation fields can be detected from single snapshots. As a further example, dominant wind directions in the Sahara are found by investigating the digital terrain model.