Generic probability distribution of rainfall in space: the bivariate model

Abstract

Two variates representing either point or spatially-averaged precipitation amounts are governed by a bivariate probability distribution which has a mixed binary-continuous structure built of six elements: three constants and three functions, two univariate and one bivariate. This bivariate distribution is generic to probabilistic analyzing and forecasting of precipitation in space. This article investigates its structure and properties, constructs models of its continuous elements, and validates the models by testing them statistically on 600 pairs of variates, each representing the 24 h precipitation amount at a rain gauge in the Appalachian Mountains. The bivariate conditional (on precipitation occurrence at both points) distribution is suitably represented by the meta-Gaussian model. Consequently, the bivariate distribution can be constructed from eight elements: four constants and four univariate functions. The three marginal conditional distributions of each variate obey a dominance law uncovered herein through statistical tests. This law is exploited to establish parametric transformations between the marginal conditional distributions. A potential application is demonstrated in real-time forecasting: Given four elements (two probabilities and two marginal conditional distributions) specified by a probabilistic quantitative precipitation forecast (PQPF), and given five climatic parameters, the bivariate distribution of precipitation amounts can be constructed san arbitrary assumptions. Lastly, comparisons are made between the generic bivariate distribution and the bivariate distributions implied by previously published multivariate models of precipitation fields, revealing approximations and flaws imbedded in these models.