Modelling rainfall with a Bartlett–Lewis process: New developments

Abstract

The use of Poisson-cluster processes to model rainfall time series at a range of scales now has a history of more than 30 years. Among them, the Randomised (also called modified) Bartlett-Lewis model (RBL1) is particularly popular, while a refinement of this model was proposed recently (RBL2) (Kaczmarska et al., 2014). Fitting such models essentially relies upon minimising the difference between theoretical statistics of the rainfall signal and their observed estimates. The first are obtained using closed form analytical expressions for statistics of order 1 to 3 of the rainfall depths, as well as useful approximations 5 of the wet-dry structure properties. The second are standard estimates of these statistics for each month of the data. This paper discusses two issues that are important for optimal model fitting of the RBL1 and RBL2. The first is that, when revisiting the derivation of the analytical expressions for the rainfall depth moments, it appears that the space of possible parameters is wider than has been assumed in the past papers. The second is that care must be exerted in the way monthly statistics are estimated from the data. The impact of these two issues upon both models, in particular upon the estimation of extreme rainfall depths 10 at hourly and sub-hourly timescales is examined using 69 years of 5-min and 105 years of 10-min rainfall data from Bochum (Germany) and Uccle (Belgium), respectively. BackgroundThe objective of stochastic rainfall modelling is to provide tools that enable the generation of long realistic series of rainfall.These can then be used as inputs to catchment hydrological models, to erosion models, to sewerage discharge models, or 15 even be used to examine the frequency of outages in telecommunications networks (Connolly et al., 1998;Arnbjerg-Nielsen, 2012;Arnbjerg-Nielsen et al., 2013;Onof and Arnbjerg-Nielsen, 2009;Wang et al., 2010). Depending upon the application, 'realistic' will mean different things. For applications that are related to design, 'realistic' will involve the reproduction of the observed extreme behaviour of the precipitation process at a range of scales.In this paper, we focus upon one particular approach to rainfall modelling, that which is based upon the use of point processes 20 as defining the times at which the building blocks of the model, i.e. rainfall cells, arrive. These cells are conceptual ones, although their typical characteristics are those of Small Mesoscale Areas (SMSA) which are embedded in Large Mesoscale Areas (Burlando and Rosso, 1993). The presence of clustering means that a homogeneous Poisson point process is not an appropriate choice for the underlying process of cell arrivals. Two options are available. The first introduces randomness by having the Poisson rates behave as a continuous-time Markov chain: this defines a Cox (doubly-stochastic) process (see Ramesh 25 (1995); Ramesh et al. (2018)). The second explicitly models the clustering process. This can be done by defining the number cells in a storm as a random variable, wit...

Stochastic investigation of hydrological extremes

“…Rodriguez-Iturbe et al, 1987b). Marani (2003) highlighted the fact that accordingly Poisson-cluster models are expected to underestimate the variance for scales larger than those of calibration, which typically extends from one to few days; which is also confirmed by Onof and Wang (2019) in spite of various amendments to the original model. A remedy proposed by Park et al (2019) is the coupling of the Bartlett-Lewis model with a seasonal autoregressive integrated moving average (SARIMA) model in order to capture the observed long-term rainfall variability.…”

“…Reproduction of the extremes is the most challenging task for this type of models, as they tend to underestimate hourly and sub-hourly extremes , often followed by an overestimation the daily (Onof and Wang, 2019). Inclusion of the skewness in the calibration set along with various reparameterizations of the original model and coupling with disaggregation schemes have been found to contribute to better fitting to the extremes (Cowpertwait, 1998;Kaczmarska et al, 2014;Kossieris et al, 2018;Onof and Wang, 2019).…”

“…Reproduction of the extremes is the most challenging task for this type of models, as they tend to underestimate hourly and sub-hourly extremes , often followed by an overestimation the daily (Onof and Wang, 2019). Inclusion of the skewness in the calibration set along with various reparameterizations of the original model and coupling with disaggregation schemes have been found to contribute to better fitting to the extremes (Cowpertwait, 1998;Kaczmarska et al, 2014;Kossieris et al, 2018;Onof and Wang, 2019). Onof and Wang (2019) argue that it is of high importance to capture the fat-tailedness of the storm intensity distribution as well, by employing a heavier tail distribution in the model.…”