A renewal cluster model for the inter-arrival times of rainfall events

A statistical model, based on a renewal cluster point process, is proposed and used to infer the distributional properties of dry periods in a continuous-time record. The model incorporates a mixed probability distribution in which inter-arrival times are classified into two distinct types, representing cyclonic and anticyclonic weather. This results in rainfall events being clustered in time, and enables objective probabilistic statements to be made about storm properties, e.g. the expected number of events in a storm cluster. The model is fitted to data taken from a gauge near Wellington, New Zealand, by maximising the likelihood function with respect to the parameters. The Akaike Information Criteria is used to select the best fitting distributions from a range of candidates. The log-Normal distribution is found to provide the best fit to the times between successive storm clusters, whilst the Weibull distribution is found to provide the best fit to the times between successive events in the same storm cluster. Harmonic curves are used to provide a parsimonious parameterisation, allowing for the seasonal variation in precipitation. Under the fitted model, the interval series is transformed into a residual series, which is assessed to determine overall goodness-of-fit

A continuous stochastic disaggregation model of rainfall for peak flow simulation in urban hydrologic systems

In the paper by Durrans et al. (1999), an algorithm proposed by Ormsbee (1989) is recommended for the stochastic disaggregation of hourly rainfall in continuous flow simulation studies of urban hydrologic systems. However, Durrans et al. found that the method produced a “severe negative bias” in the maximum rainfall intensity of the disaggregated series, so that peak flows in urban systems are likely to be under-estimated by the model. Here we develop a method for disaggregating hourly data to 5min series, which addresses the problem of negative bias. A regression equation is derived for the ratio of the maximum 5min depth to the total depth in the hour. Thus, for any given hourly depth this ratio can be simulated and multiplied by the hourly depth to obtain a 5min maximum. The temporal location of the maximum within the hour can be randomly placed using an appropriate distribution function, e.g. based on a geometrical construction as developed by Ormsbee (1989). The model is developed and tested using 5min rainfall data taken from Lund (1923-39) and Torsgatan (1984-93), Sweden. The results support the use of the model in urban drainage applications

A stochastic spatial-temporal disaggreation model for rainfall

A stochastic model for disaggregating spatial-temporal rainfall data is presented. In the model, the starting times of rain cells occur in a Poisson process, where each cell has a random duration and a random intensity. In space, rain cells have centres that are distributed according to a two dimensional Poisson process and have radii that follow an exponential distribution. The model is fitted to seven years of five-minute data taken from six sites across Auckland City. The historical five-minute series are then aggregated to hourly depths and stochastically disaggregated to five-minute depths using the fitted model. The disaggregated series and the original five-minute historical series are then used as input to a network flow simulation model of Auckland City’s combined and wastewater system. Simulated overflow volumes predicted by the network model from the historical and disaggregated series are found to have equivalent statistical distributions, within sampling error. The results thus support the use of the stochastic disaggregation model in urban catchment studies

A model of rainfall based on finite-state cellular automata

The purpose of this paper is to demonstrate that a finite state cellular automata model is suitable for modeling rainfall in the space-time plane. The time-series properties of the simulated series are matched with historical rainfall data gathered from Whenuapai, NZ. The spatial scale of the model cells in related to land-area by optimizing the cross-correlation between sites at lag 0 relative to rainfall data collected from Auckland, NZ. The model is shown to be adequate for simulation in time, but inadequate in spatial dimension for short distances

The stochastic generation of rainfall time series

PhD ThesisThe purpose of this project was to propose and validate a stochastic rainfall time series model for the UK, where the model is to be applied to the design of sewer systems. After reviewing the literature, the Neyman-Scott Rectangular Pulses model was selected as being potentially suitable for the project. Some mathematical properties for the model were derived, and used to fit the model to 10 years of hourly rainfall time series. The model performed well, and so could be used with reasonable confidence for the remaining part of the project. A full investigation was carried out to find an optimum combination of historical rainfall statistics to be used to fit the model to hourly rainfall time series. A method of fitting the model to daily rainfall time series was also required. It was found that the hourly rainfall statistics used to fit the model to the hourly rainfall time series could successfully be predicted from daily rainfall statistics. Regression equations were developed so that the mean and variance of the maximum daily rainfalls could be predicted using the parameters of the model. These regression equations were included in the fitting procedure when the model showed a poor fit to the historical daily maxima, so that the model was then able to closely match the historical maxima. The model was fitted to rainfall data taken from 112 sites scattered throughout the UK. The parameters of the model were regressed on site characteristics (e. g. altitude, distance from coast, etc), so that the model could be used to generate hourly rainfall time series at sites lacking in data. Finally, a method of disaggregating the generated hourly rainfall time series to 5 minutely time series was developed and tested.Water Research Centre (WRc Engineering, Swindon)

Regionalised spatiotemporal rainfall and temperature models for flood studies in the Basque Country, Spain

A spatiotemporal point process model of rainfall is fitted to data taken from three homogeneous regions in the Basque Country, Spain. The model is the superposition of two spatiotemporal Neyman–Scott processes, in which rain cells are modelled as discs with radii that follow exponential distributions. In addition, the model includes a parameter for the radius of storm discs, so that rain only occurs when both a cell and a storm disc overlap a point. The model is fitted to data for each month, taken from each of the three homogeneous regions, using a modified method of moments procedure that ensures a smooth seasonal variation in the parameter estimates. Daily temperature data from 23 sites are used to fit a stochastic temperature model. A principal component analysis of the maximum daily temperatures across the sites indicates that 92% of the variance is explained by the first component, implying that this component can be used to account for spatial variation. A harmonic equation with autoregressive error terms is fitted to the first principal component. The temperature model is obtained by regressing the maximum daily temperature on the first principal component, an indicator variable for the region, and altitude. This, together with scaling and a regression model of temperature range, enables hourly temperatures to be predicted. Rainfall is included as an explanatory variable but has only a marginal influence when predicting temperatures. A distributed model (TETIS; Francés et al., 2007) is calibrated for a selected catchment. Five hundred years of data are simulated using the rainfall and temperature models and used as input to the calibrated TETIS model to obtain simulated discharges to compare with observed discharges. Kolmogorov–Smirnov tests indicate that there is no significant difference in the distributions of observed and simulated maximum flows at the same sites, thus supporting the use of the spatiotemporal models for the intended application

A doubly stochastic rainfall model with exponentially decaying pulses

We develop a doubly stochastic point process model with exponentially decaying pulses to describe the statistical properties of the rainfall intensity process. Mathematical formulation of the point process model is described along with second-order moment characteristics of the rainfall depth and aggregated processes. The derived second-order properties of the accumulated rainfall at different aggregation levels are used in model assessment. A data analysis using 15 years of sub-hourly rainfall data from England is presented. Models with fixed and variable pulse lifetime are explored. The performance of the model is compared with that of a doubly stochastic rectangular pulse model. The proposed model fits most of the empirical rainfall properties well at sub-hourly, hourly and daily aggregation levels

A Regionalised Neyman-Scott Model of Rainfall with Convective and Stratiform Cells

International audienceA single-site Neyman-Scott Poisson cluster model of rainfall, with convective and stratiform cells, is fitted to data for 112 sites scattered throughout the UK using harmonic variables to account for seasonality. The model is regionalised by regressing the estimates of the harmonic variables on site dependent variables (e.g. altitude) to enable rainfall to be simulated at any ungauged site in the UK. An assessment of the residual errors indicates that the regression models can be used with reasonable confidence for urban sites. Furthermore, the regional variations of the model parameter estimates are found to be in agreement with meteorological knowledge and observation. Simulated I h extreme rainfalls are found to compare favourably with observed historical values, although some lack-of-fit is evident for higher aggregation levels

Local generalised method of moments: an application to point process-based rainfall models

Long series of simulated rainfall are required at point locations for a range of applications, including hydrological studies. Clustered point process-based rainfall models have been used for generating such simulations for many decades. These models suffer from a major limitation, however, their stationarity. Although seasonality can be allowed by fitting separate models for each calendar month or season, the models are unsuitable in their basic form for climate impact studies. In this paper, we develop new methodology to address this limitation. We extend the current fitting approach by allowing the discrete covariate, calendar month, to be replaced or supplemented with continuous covariates that are more directly related to the incidence and nature of rainfall. The covariate-dependent model parameters are estimated for each time interval using a kernel-based nonparametric approach within a generalised method-of-moments framework. An empirical study demonstrates the new methodology using a time series of 5-min rainfall data. The study considers both local mean and local linear approaches. While asymptotic results are included, the focus is on developing useable methodology for a complex model that can only be solved numerically. Issues including the choice of weighting matrix, estimation of parameter uncertainty and bandwidth and model selection are considered from this perspective

Lifting the veil: richness measurements fail to detect systematic biodiversity change over three decades

While there is widespread recognition of human involvement in biodiversity loss globally, at smaller spatial extents, the effects are less clear. One reason is that local effects are obscured by the use of summary biodiversity variables, such as species richness, that provide only limited insight into complex biodiversity change. Here, we use 30 yr of invertebrate data from a metacommunity of 10 streams in Wales, UK, combined with regional surveys, to examine temporal changes in multiple biodiversity measures at local, metacommunity, and regional scales. There was no change in taxonomic or functional a-diversity and spatial b-diversity metrics at any scale over the 30-yr time series, suggesting a relative stasis in the system and no evidence for on-going homogenization. However, temporal changes in mean species composition were evident. Two independent approaches to estimate species niche breadth showed that compositional changes were associated with a systematic decline in mean community specialization. Estimates of species-specific local extinction and immigration probabilities suggested that this decline was linked to lower recolonization rates of specialists, rather than greater local extinction rates. Our results reveal the need for caution in implying stasis from patterns in a-diversity and spatial b-diversity measures that might mask non-random biodiversity changes over time. We also show how different but complementary approaches to estimate niche breadth and functional distinctness of species can reveal long-term trends in community homogenization likely to be important to conservation and ecosystem function