Modelling and simulation of seasonal rainfall

Abstract

We propose and justify a model for seasonal rainfall using a copula of maximum entropy to model the joint distribution and using gamma distributions to model the marginal monthly rainfalls. The model allows correlation between individual months and thereby enables a much improved model for seasonal variation. A central theme is the principle of maximum entropy which we use to find the most parsimonious representation for the underlying distributions—using the minimum possible number of parameters to model the relevant physical characteristics. A particular emphasis is the use of the gamma distribution to model the marginal monthly rainfalls. We wish to simulate monthly and seasonal rainfall at Kempsey, NSW during February-March-April. Our first task is to explain why we choose to model monthly rainfall totals using the gamma distribution. The principle of maximum entropy (Jaynes, 1957a, 1957b) states that, subject to precisely stated prior data, the probability distribution which best represents the current state of knowledge is the one with largest entropy. We will use this principle to argue that the gamma distribution is the best distribution to represent monthly rainfall totals provided the means of the observed monthly totals and the natural logarithm of the observed monthly totals are both well-defined and finite. This is true if the observed totals are always strictly positive. Our second task is to devise a graphical representation that displays the gamma distribution in the simplest possible way—as a straight line. We will use this procedure to compare simulated data from the chosen gamma distribution to the observed data. Our third task is to use the simple graphical representation above to compare the observed monthly rainfall to simulated monthly rainfall generated by the chosen gamma distribution. Our conclusion will be that there is no significant statistical difference between the simulated data and the observed data. Our fourth task is to to demonstrate the goodness-of-fit for the observed monthly rainfall data to the selected gamma distributions for each month. To do this we used two kinds of Q-Q plot. Firstly we plot simulated quantiles from the gamma distribution against theoretical quantiles to determine 95% confidence intervals and then plot observed quantiles against theoretical quantiles. Once it has been decided that the monthly rainfallXi can be modelled by a gamma distributionXi ( i; i) with Fi(x) = F i; i (x) then the observed data setfxi;jgj=1;2;:::;N can be transformed into a corresponding data setfui;j = Fi(xi;j)gj=1;2;:::;N for each i = 1; 2;:::;m. This has the effect of removing seasonal factors from the observed data and also preparing for the use of a copula of maximum entropy to model the joint distribution of the monthly rainfall totals. The next step in the modelling process is to construct a joint probability distribution for the entire three-month time period. Past studies of rainfall accumulations over several months (Katz and Parlange, 1998; Rosenberg et al., 2004; Withers and Nadarajah, 2011) have observed that for models with independent marginal distributions the seasonal variance is often too low. It has been suggested that this happens because there is an overall positive correlation between the individual monthly totals. Since the observed data shows positive correlation for February-March-April at Kempsey our aim will be to construct a joint distribution so that the desired marginal distributions are preserved and so that the grade correlation coefficients match the observed rank correlation coefficients. We construct the joint distribution using a checkerboard copula of maximum entropy (Piantadosi et al., 2012a, 2012b). Finally we compared the observed rainfall to rainfall generated by three different models (a) a maximum likelihood gamma distribution that models seasonal rainfall but does not generate individual monthly rainfalls. (b) a checkerboard copula of maximum entropy with marginal gamma distributions that preserves the observed rank correlation coefficients and (c) a joint distribution with independent marginal gamma distributions. We conclude that the copula of maximum entropy provides an excellent model for rainfall simulation.