Maximum entropy methods for generating simulated rainfall

We desire to generate monthly rainfall totals for a particular location in such a way that the statistics for the simulated data match the statistics for the observed data. We are especially interested in the accumulated rainfall totals over several months. We propose two different ways to construct a joint rainfall probability distribution that matches the observed grade correlation coefficients and preserves the known marginal distributions. Both methods use multi–dimensional checkerboard copulas. In the first case we use the theory of Fenchel duality to construct a copula of maximum entropy and in the second case we use a copula derived from a multi–variate normal distribution. Finally we simulate monthly rainfall totals at a particular location using each method and analyse the statistical behaviour of the corresponding quarterly accumulations. 1 Modelling accumulated rainfall It has been usual to model both short–term and long–term rainfall accumulations at a specific location by a gamma distribution [16, 11, 3, 4]. Some authors [14, 5] have, however, observed that simulations in which monthly rainfall totals are modelled as mutually independent gamma random variables generate accumulated bi–monthly, quarterly and yearly totals with much lower variance than the observed accumulations. It is reasonable to surmise that the variance of the generated totals will be increased if the model includes an appropriate level of positive correlation between individual monthly totals. We use a typical case study to show that this is indeed the case. More generally, the problem we address is how to construct a joint probability distribution which preserves the known marginal distributions and matches the observed grade correlation coefficients. We propose two alternative ways in which this could be done. Both methods use multi–dimensional copulas. 1.1 Multi–dimensional copulas An m–dimensional copula where m ≥ 2, is a continuous, m–increasing cumulative probability distribution C : [0, 1]m 7→ [0, 1] on the unit m–dimensional hyper–cube with uniform marginal ∗Research Fellow, Centre for Industrial and Applied Mathematics, Scheduling and Control Group, University of South Australia, Mawson Lakes, SA 5095. Email: [email protected]. †Emeritus Professor, Industrial and Applied Mathematics, Scheduling and Control Group, University of South Australia, Pooraka, SA 5095. Email: [email protected]. ‡Laureate Professor and Director Centre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Distinguished Professor, King Abdulaziz University, Jeddah 80200, Saudi Arabia. Email: [email protected]. §Managing Director and Principal Consultant Statistician, Data Analysis Australia Pty Ltd, Adjunct Professor of Statistics, University of Western Australia, 97 Broadway, Nedlands WA, 6009. Email: [email protected].