A conceptual stochastic model for rainfall, based on a Poisson-cluster process with rectangular pulses representing rain cells, is further developed. A method for deriving high-order moments is applied to obtain the third-moment function for the model. This is used with second-order properties to fit the model to January and July timeseries taken from a site in Wellington, New Zealand. It is found that the parameter estimates may follow two solution paths converging on an optimum value over a bounded interval. The parameter estimates are used with the model to simulate 200 years of hourly data, and parametric tests used to compare simulated and observed extreme rainfalls. These show good agreement over a range of sampling intervals. The paper concludes with a discussion of the standard errors of the model parameter estimates which are obtained using a non-parametric bootstrap.
The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the
x
-
y
plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to
n
with different parameters for the different cell types, so that the random variables of an arbitrary cell, e. g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.
A conceptual stochastic model of rainfall is proposed in which storm origins occur in a Poisson process, where each storm has a random lifetime during which rain cell origins occur in a secondary Poisson process. In addition, each cell has a random lifetime during which instantaneous random depths (or ‘pulses’) of rain occur in a further Poisson process. A key motivation behind the model formulation is to account for the variability in rainfall data over small (e.g. 5 min) and larger time intervals. Time-series properties are derived to enable the model to be fitted to aggregated rain gauge data. These properties include moments up to third order, the probability that an interval is dry, and the autocovariance function. To allow for distinct storm types (e.g. convective and stratiform), several processes may be superposed. Using the derived properties, a model consisting of two storm types is fitted to 60 years of 5 min rainfall data taken from a site near Wellington, New Zealand, using sample estimates taken at 5 min, 1 hour, 6 hours and daily levels of aggregation. The model is found to fit moments of the depth distribution up to third order very well at these time scales. Using the fitted model, 5 min series are simulated, and annual maxima are extracted and compared with equivalent values taken from the historical record. A good fit in the extremes is found at both 1 and 24 hour levels of aggregation, although at the 5 min level there is some underestimation of the historical values. Proportions of time intervals with depths below various low thresholds are extracted from the simulated and historical series and compared. A tendency for underestimation of the historical values is evident at some time scales, with a close fit being obtained as the threshold is increased.
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