With the aim of generating daily rainfall sequences, Buishand (1977) analysed rainfall series of different climatic regions using alternating renewal processes in discrete time for the occurrence process. In the alternating renewal process, the daily rainfall data is considered as a sequence of alternating wet and dry spells of varying length. The wet and dry spells are assumed to be independent and the distributions may be different for wet and dry spells. Buishand (1977) found the zero truncated negative binomial distributions to be useful for the description of the wet and dry spell sequences. A distribution of rainfall amounts, fitted with shifted gamma distributions, on wet days is then superimposed on the binary process structure. With data series collected from different rainfall stations which are located in different climatic regions (e.g. some in the north Europe, some in the tropical region), Buishand (1977) has made a comprehensive and detailed analysis of the empirical characteristics of the collected daily rainfall data. For example, apart from what we have cited in the overview section, Buishand also found that: (1) there is no evidence for correlation between the rainfall amount on the first day of a wet spell and the length of the preceding dry spell; (2) the first and the last day of a wet spell have smaller means than the other wet days. the smallest mean is found for solitary wet days; (3) there is some evidence for serial correlation of successive rainfall amounts within a wet spell. Because of the diversity of his data sets, Buishand showed that the variability of the model performance in fitting to different type of data series was high. For example, he found that an alternating renewal process performed better than a binary discrete autoregressive moving average (DARMA) process in fitting the data from the Netherlands but the DARMA model looked more promising in tropical and monsoonal areas. Whereas Buishand (1977) has thoroughly developed the alternating renewal process modelling rainfall approach, he pointed out that in the case of long dry spells, the negative binomial distribution sometimes cannot fit the distribution or the length of these spells. He suggested, in such cases, using transition probabilities for the generation of the wet-day series instead of generating lengths of wet and dry spells.
Stern and Coe (1984) employed a Markov chain model for daily occurrences which represents a general approach to the analysis of daily rainfall data from a single site and similar ideas have been presented by other authors, both previously and subsequently. In this approach, rainfall occurrences are characterized by the binary process indicat-
ing whether or not it is raining on a particular day. Like all the empirical statistical rainfall models, the analysis is broken up into separate models for rainfall occurrences and rainfall amounts. In particular, Stern and Coe (1984) used a model with Markov chains fitted to the occurrence of rainy days, and a gamma distribution fitted to the rainfall amounts captured on wet days. The model was fitted to data taken from a 53-year record in Morogoro, Tanzania, and a 37-year record taken from Irbid, Jordan, as examples to assess the model performance. In the use of the models section, the au- thors restricted their attention to a first order, two-state Markov chain to describe the occurrence of rain process, although higher order Markov chains (Appendix B.2) were examined in the model fitting section. One special feature (which has been emphasized by the authors as an important advantage) is the possibility of fitting the model as a Generalized Linear Model (GLM, Nelder and Wedderburn, 1972) framework. The sea- sonal effects of the rainfall occurrence process was taken into account by incorporating a GLM model into the expressions for transition probabilities for the Markov chain. The seasonal effects of rainfall amount was taken into account through the mean of the gamma distribution which was allowed to depend on whether rain had occurred on the previous day, as determined by the Markov chain. Standard statistical tests, e.g. deviances and Pearson’s Q-statistic, were used to assess the model goodness-of-fit. Depending on the result of these tests, for a given site, the model used between 20 to 50 parameters with seasonal effects taken into account. The performance of the model was shown to be good for the intended application, which was agricultural planning.
Foufoula-Georgiou and Lettenmaier (1987) stated that Markov chains have been found, in general, to be inadequate when modelling the clustering dependencies present in daily rainfall occurrences. Foufoula-Georgiou and Lettenmaier (1987), therefore, developed a Markov renewal model for rainfall occurrences in which the time between rainfall occurrences (in number of days) were represented by two different geometric distributions (a two-state process). The transition from one distribution to the other was governed by a Markov chain. The daily rainfall process is modelled by coupling the Markov renewal model with a mixed exponential distribution for the rainfall amount captured on rainy days. The authors defined a rainfall event as a rainy period of consecutive wet days. In the model, rainfall events were classified as either primary or secondary, where a primary event corresponded to the arrival of a front, and a secondary event corresponded to the occurrence of rainfall within the same frontal system. The model thus exhibits clustering as a result of this dependence structure.
This model is different from a Markov chain in that the probability of having a rainy day does not depend on the condition (wet or dry) of the previous day but rather on the number of days since the last rainfall event. The authors pointed out that the model behaves as a Markov chain during a rainy spell. The model uses four parameters for the rainfall occurrence process and then an additional three parameters for the depth (rainfall amount) process, before seasonal effects are taken into account. The model was applied to a 15 years daily rainfall series for Snoqualmie Falls, Washington, USA. The data were divided into five seasons and the parameters were estimated for each season using the maximum likelihood method. The model was found to be successful in preserving the short-term structure of the occurrence process, as well as the distributional properties of the seasonal rainfall amounts.
Acreman (1990) developed a model to generate hourly rainfall series. The observed dry spell sequence and wet spell sequence (Figure 4.2) have been treated as independent of each other and the Pareto and exponential distributions were fitted to the spell lengths of the two sequences, respectively. This is an alternating renewal process model for the rainfall occurrence process. Note that the author has used a continuous distribution to fit the discrete data – the spell lengths are measured in number of hours. Therefore, he had to make continuity corrections in the fitting process. The length of a wet spell, i.e. ‘the duration of an event’ as the author has termed it, was defined as an integer value defining the length of a continuous sequence of wet hours bounded on either side by at least one dry hour. Given an event duration, a gamma distribution, conditional on the event duration, was fitted to the total volume of rainfall observed in the rainfall event. In this way, the strong correlation evident between event depths and durations was taken into account. Furthermore, a beta distribution was used to obtain the rainfall profile (i.e. how rainfall depths were distributed over the duration) of rainfall events. Modelling the rainfall event profile was crucial for retaining the observed annual maximum rainfall property in the fitted model. The model was fitted to data from Farnborough, UK, and the data divided into summer and winter subsamples. There were a total of 22 parameters in the model and some of these parameters were dependent on the season, and others were constant throughout the year. The model was assessed by comparing the monthly rainfall totals and the annual maxima obtained from the simulation samples against the historical values. The results showed that the simulated monthly totals were consistently greater than the historical monthly totals for December to April, and consistently less than the historical values
for July to October. This implied that the data may need to be divided into more seasonal subsamples. The simulated annual maxima at the 1 hour aggregation level fitted the historical annual maxima well up to a 20-year return period1. The simulated
annual maxima at the 24 hours aggregation level showed a slight overestimation when compared to the historical values, although it should be noted that the extreme values had not been used to fit the model.