4.4 Evolution of point process models
4.4.4 Developments from basic NSRP and BLRP models
Over the last two decades, a number of improvements on the basic NSRP and BLRP models have been made. Other properties such as the probability that a period is dry for the basic NSRP model was found by Cowpertwait (1991), and used to fit the basic NSRP model to a series of 10 years of British data.
Rodriguez-Iturbe et al. (1988) first proposed the random parameter model idea to improve the fit to the proportion dry. In the proposed modified BLRP model, the cell duration distribution parameterη, was allowed to vary randomly between storms. Subsequently, Entekhabi et al. (1989) introduced a modified NSRP model by allowing
the cell duration to vary for each storm according to a gamma distribution (which resulted in having an additional parameter in the model, i.e. six parameters in total). They fitted the model to the Denver summer rainfall data and found that the proportion of dry days derived from the model compared favourably to the historical values. More properties for the modified BLRP model (Rodriguez-Iturbe et al., 1988) were derived by Onof and Wheater (1993, 1994) and they reported a good fit to the proportion of dry periods at all time scales for British rainfall.
Motivated by the meteorological fact of the existence of high intensity convective rain cells within lower intensity large mesoscale areas of precipitation (e.g. Austin and Houze, 1972; Waymire and Gupta, 1981a; Cox and Isham, 1994), Cowpertwait (1994) proposed a generalized point process model for rainfall. In the model, a number
n of cell types are used for the NSRP model with different exponential distributions for both cell durations and cell intensities. The model was fitted using two cell types (eight parameters per season) to hourly data from Farnborough, UK. In comparison with the equivalent basic NSRP model, the most significant improvement achieved with the generalized model was a good fit to the annual maximum aggregated at the 1 hour scale. Another way of improving the representation of rainfall based on the basic NSRP and BLRP specifications is to introduce an explicit dependence between cell duration and intensity within the model (Kakou and Onof, 1996). In addition, Cowpertwait (2004) used superposed independent processes for different storm types to characterize distinct types of precipitation which provides yet another modified NSRP model. An analytical expression for the third-order moment property was derived and used to obtain a better fit to extreme values (Cowpertwait, 1998).
Recently, CIO2007 introduced the Bartlett-Lewis Pulse (BLP) model for fine reso- lution rainfall series. In a BLP model, the pulse arrival process is composed of three Poisson processes, which should account for the rainfall variability over a wide range of time scales and be sufficient for most applications (Koutsoyiannis, 2002). Because of such a complex model structure, on the other hand, a closed form of likelihood function can not be readily obtained for BLP models, so the method of moments approach is adopted for parameter estimation. Potentially, superposition of BLP processes leads to a large number of model properties for fitting among which a subjective choice has to be made.
4.5
Concluding remarks
The empirical statistical models and the point process models are two popular classes of stochastic rainfall models. Both classes have been used for generating continuous rainfall series at a point that preserves the main statistical properties of observed rainfall series. Simplicity (in terms of model fitting) and the intuitive appeal have been counted as the positive points of the empirical approach because the observed discrete rainfall series represent the combined effect of several underlying mechanisms (Stern and Coe, 1984; Foufoula-Georgiou and Lettenmaier, 1987). The major drawback of the empirical models is that they are not based on any known physical process governing rainfall. One important contribution of Waymire and Gupta (1981a,b,c) is that they have shown, through theoretical justification and application examples, that rainfall must be considered in its space-time context and modelled on physical ground in terms of a random field. The single site point process models form the temporal part of this approach and many successful application cases have been reported since the late 1980s, especially for the sub-daily rainfall series.
In general, the specification of empirical statistical models follows an ad hoc pro- cedure. By contrast, the point process models provide a systematic and consistent approach of the mathematical representation of rainfall. The Poisson cluster models have been successfully used to fit rainfall series at various aggregation levels, from 5 minutes to 24 hours.
Parameter estimation is a problem, for which a satisfactory solution has not yet been found, for both the NSRP and BLRP models. Onof et al. (2000) regard this as one of the major areas of possible future improvement.
In both the NSRP and BLRP models, rainfall intensity is a random variable that remains constant throughout the lifetime of a rain cell. Although this may seem unrealistic in continuous time, there are many examples in the literature where NSRP and BLRP models provide a good fit to rainfall data aggregated at the 1-hour level or higher (Cowpertwait, 1994; Verhoest et al., 1997; Onof et al., 2000). fine-scale empirical analysis showed that a typical observed wet spell lasts from a few minutes to a few hours, so that a rectangular pulse approximation is adequate for many practical applications that only require rainfall simulation results at higher aggregation levels. There is a need to develop a more realistic pulse structure for the analysis of fine scale rainfall data (e.g. at 5-minute level), which is required for applications like the design
of stormwater sewerage systems. One approach has been to simulate 1-hour series using a rectangular pulse model and then to disaggregate the series using another stochastic model (Cowpertwait et al., 1996b; Onof et al., 2005). In general, however, it is preferable to have a single model that is capable of representing series over all time scales, since this would provide a more complete description of the physical process as well as reduce model uncertainty. This is a primary motivation for the development of a Bartlett-Lewis Pulse (BLP) model by CIO2007.