Point process models for rainfall: The case of a modified Bartlett-Lewis model

C. Derzekos, Point process models for rainfall: The case of a modified Bartlett-Lewis model, Diploma thesis, 90 pages, Department of Water Resources, Hydraulic and Maritime Engineering – National Technical University of Athens, Athens, October 2004.



The analysis and simulation of rainfall time series on fine time scales require the use of special types of stochastic models. This necessity is justified from the intermittent character of rainfall on these time scales. Among the successful model types are the point process models. According to these models, the rainfall events are simulated through the generation of clustered point (instantaneous bursts) or rectangular pulses. The position of these pulses on the time axis is achieved by the use of one or more Poisson processes. The implementation of the models mentioned above aims to optimize the fitting of desirable small-scale rainfall statistics for different levels of aggregation. The objective is the determination of a unique optimal parameter set for all considered levels of aggregation. The original rectangular pulses Bartlett-Lewis Model (BLM) was one of the first used in rainfall modeling. Later, Rodriguez-Iturbe et al. (1988) modified it to reproduce the probability of zero rain (or probability dry, PDR) for different levels of aggregation and the new model became known as the Random Bartlett-Lewis Model (RBLM). Koutsoyiannis and Onof (unpublished research) have examined a further modification of RBLM. This modification aimed to introduce a negative correlation between storm intensity and duration. The resulted model is called the Modified Random Bartlett-Lewis Model (MRBLM). The main objective is the improvement of the fitting attained by RBLM. The fitting is meant in the reproduction of mean, variance, lag one auto-covariance and proportion dry, for different levels of aggregation. The purpose of the present study is to examine the behaviour of the MRBLM. Two historical time series are used as case studies. The first one refers to the Denver airport station data (1949-1976), while the second one is referred to the National Technical University of Athens (NTUA) station data (1994-2003). The complexity of the mathematical model, the introduction of non-analytical equations and the presence of many local optimal points create the necessity of applying a direct search ("global") optimization method. A novel optimization algorithm is developed, while a decomposition approach results in the proposal of several simplifications, to the optimization procedure. Additionally, qualitative, semi empirical criteria are developed, to roughly estimate in advance the model efficiency. In the Denver case, the new model attains a 54% improvement in preserving historical rainfall statistics, in comparison to those of RBLM. The simulation results (statistics of synthetic series) confirm this conclusion. However, in the Athens case, the new model, even though results in better approximation of the historical statistics (in comparison to RBLM), in simulations did not give any improvement due to unacceptable ratio of negative parameter values. As a result, RBLM is preferable from the modified model in the Athens case.

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