Regional estimates of intense rainfall based on the Peak-Over-Threshold (POT) approach
Highlights
► Regional Peak-Over-Threshold (POT) approach is applied to southern Québec rainfall. ► POT series for 5 min to 12 h are defined using a fixed exceedance rate. ► Intra-annual variability of Generalized Pareto Distribution parameters was modeled. ► Models with regional shape and scale parameters are selected for all durations. ► Introduction of spatial covariates (latitude, longitude) improves model performance.
Introduction
Two approaches are usually considered for the statistical estimation of the intensities of rare rainfall events. The most common is based on Annual Maxima (AM) series and consists in extracting from available records the maximum rainfall recorded each year during a specific duration (e.g. 5 min or 1 h; Coles 2001). Based on extreme value theory, it can be shown that these series can be statistically represented by the Generalized Extreme Value (GEV) distribution (Coles, 2001). The Gumbel distribution (a special case of the GEV distribution) is also used to represent AM series. The second approach is called the Peak-Over-Threshold (POT) or Partial-Duration (PD) approach (Coles, 2001, Madsen et al., 1997a, Madsen et al., 1997b, Kingumbi and Mailhot, 2010). The idea is to keep, from recorded series, rainfall values above a predefined threshold. The resulting series are commonly represented by the Generalized Pareto Distribution (GPD) (Madsen et al. 1997a). Other distributions have also been proposed to represent exceedance magnitudes, among others the Weibull (Miquel, 1984, Ekanayake and Cruise, 1993) and lognormal distributions (Rosbjerg et al., 1991). The advantage of the POT approach is that more than one extreme value can be considered each year while only the most extreme value is kept using the AM approach. The POT approach therefore improves the sampling of extreme events. Studies by Madsen et al., 1997a, Madsen et al., 1997b showed that the POT–GPD approach is generally more efficient than the AMS–GEV model. Two reasons explain the fact that the POT approach is less often used than the AM approach. Firstly, the definition of the threshold is, to some extent, arbitrary; secondly, POT series can be autocorrelated (Madsen et al., 1997a).
In Canada, observed rainfall records used to assess the intensity of extreme rainfall are generally short and spatial coverage is sparse. Evaluating the intensity of extreme rainfall events is therefore challenging and efficient approaches are needed to extract the information from available series, especially for events with a return period much longer than the sample size. In this context, the POT approach may be seen as an interesting alternative to AM series. The application of the POT approach implies that a threshold has to be defined. It is also important to account for a possible intra-annual variability of the GPD parameters since meteorological systems generating extreme rainfall events vary in terms of spatial and temporal extents during the year. Taking into account a possible intra-annual variability may improve the homogeneity of the series.
Regional analysis may also be an interesting option. It is based on the hypothesis that local series can be represented by a regional distribution with both regional and local parameters (Hosking and Wallis, 1997). The fact that some parameters have a regional character implies that series from multiple sites can be used to estimate these parameters and therefore reduces uncertainties in these regional parameters and extreme rainfall estimates. Buishand (1991) considered a regional joint likelihood function with an application to AM series of daily precipitation in the Netherlands. He proposed an approach with a common shape parameter of the GEV distribution for all stations within a given region.
Regional analysis of the POT series can be achieved through different means. Madsen et al. (2002) used a generalized least squares regression model that explicitly accounts for intersite correlation to describe the variability of the POT parameters using physiographic and climatic characteristics. These authors showed that, for concurrent events, the correlation was rather small for short duration (10 min.) and more important for longer duration (24 h.). Beguería and Vincente-Serrano (2006) obtained at-site parameters estimates using the method of probability-weighted moments. These sets of parameters were then regressed upon a set of explanatory variables to obtain a spatially explicit probability model. Both approaches consist in the regression of the GPD parameters on explanatory variable after performing at-sites estimations.
The objective of this paper is to develop a general framework for the implementation of a regional version of the POT approach based on the definition of a joint likelihood function as initially proposed by Buishand (1991). This approach is developed through an application to Quebec data. As will be showed, it is essential in this context to take into consideration the intra-annual variability of parameters. Models of increasing complexity in terms of the representation of intra-annual variability, and based on various combinations of regional and local parameters, were compared using the Akaike criterion (Akaike, 1973). Spatial covariates were introduced to describe the spatial dependency of parameters. In our study, explanatory variables are explicitly taken into consideration in the estimations of the GPD parameters through the regional joint likelihood function. The article is organized as follows. Section 2 provides a description of the available data. A general description of the POT approach is presented in Section 3 along with a specific discussion about threshold estimation and how POT series were constructed. Section 4 presents the methodology used to analyze available rainfall series while selected regional models are presented and discussed in Section 5. Section 6 addresses the issue of spatial covariates. Section 7 investigates sensitivity to threshold and model selection and provides interpolated regional maps summarizing main results. Finally, Section 8 presents a summary and conclusions.
Section snippets
Available data
Maximum daily rainfall depths at stations (107 stations for 5-, 10-, 15-, 30-min durations and 109 stations for 1-, 2-, 6- and 12-h durations) located in the southern part of the Province of Québec were considered (Fig. 1; please note that zones Z1, Z2 and Z3 identified in Fig. 1 are defined for demonstration purposes; see Fig. 3). These stations are operated either by the Ministère du Développement Durable, de l’Environnement et des Parcs (MDDEP) or by Environment Canada. Available records
Threshold estimation and construction of POT series
A threshold needs to be defined for the construction of POT series. Various methods can be found in the literature to guide our choice of threshold – the Hill plot (Hill, 1975), the mean excess plot (Davison and Smith, 1990), the bootstrap method based on mean squared error (Danielsson and De Vries, 1997, Danielsson et al., 2001). An alternative is to fix the average number of times the threshold is exceeded over a given period (Dierckx and Teugels, 2010, Kysely et al., 2010, Coelho et al., 2008
Statistical analysis of POT series
Values above the threshold are represented by the GPD for which the probability density function (pdf) is given by (following the notation of Hosking and Wallis, 1997):where ξ (threshold) corresponds to the location parameter , α to the scale parameter , and k to the shape parameter . The function is defined for if k > 0 and if (heavy-tailed).
Regional versus local models with Fourier and monthly parameters
Fig. 5 presents the ranking of the 40 models for 30-min duration events. We can observe that:
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The “local-local” models (local for scale and shape parameters) do not perform very well since the “best” local model ranks 16th (2L0L model).
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The “no mode” models (with no intra-annual variability) are among the “worst” models; they rank successively 35th (0L0R), 36th (0R0R), 38th (0L0L) and 39th (0R0L).
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The monthly parameter models are always outperformed by one of the corresponding Fourier
Regional models with spatial covariates
The selection of regional–regional models means that the only “local information” used in estimating rainfall intensities at a given site is the threshold value. Selected models only have 8 or 10 parameters (plus the local position parameters). Scale and shape parameters include regional Fourier representation of intra-annual variability. This apparent simplicity may hide some spatial structure of the parameters. Spatial covariates were therefore introduced. For the shape parameter, the
Estimation of intense rainfall depths
Rainfall depths for a given duration and return period were estimated at each station by solving the following equation:with , the cumulative distribution function of the GPD, the regional parameters values for calendar day j, the threshold value at station i for calendar day j, T the return period (in MO years) and the mean number of threshold excesses per MO year (six in this analysis). The sum is over the calendar
Summary and conclusions
The POT approach may be seen as an interesting alternative to annual maxima series for the statistical analysis of intense rainfalls. By retaining rainfall values over a given threshold, the POT approach improves the sampling of extremes and therefore possibly provides better estimates of more extreme rainfall events. Regional analysis, where series from various stations are combined under the hypothesis of a common statistical distribution with regional and local parameters, is another
Acknowledgements
This work was funded in part by the Climate Change and Adaptation Program of Natural Resources Canada, by the Consortium Ouranos on Regional Climatology and Adaptation to Climate Change and by the Collaborative Research and Development (CRD) Grants of the Canadian Natural Sciences and Engineering Research Council of Canada (RDCPJ_387169 – 09, owned by A.C. Favre). The authors also thank Ms Catherine Savard of the Ministère du Développement Durable, de l’Environnement et des Parcs (MDDEP) of the
References (37)
- et al.
Tail index and quantile estimation with very high frequency data
J. Empirical Finance
(1997) - et al.
Using a bootstrap method to choose the sample fraction in tail index estimation
J. Multivar. Anal.
(2001) - et al.
Parameter estimation of the generalized Pareto distribution – Part I
J. Stat. Plan. Infer.
(2010) - et al.
A modified Mann–Kendall trend test for autocorrelated data
J. Hydrol.
(1998) - et al.
Statistics of extremes in hydrology
Adv. Water Resources
(2002) - et al.
Frequency analysis of a sequence of dependent and/or non-stationary hydro-meteorological observations: a review
J. Hydrol.
(2006) - et al.
Estimating extremes in climate change simulations using the peaks-over-threshold method with a non-stationary threshold
Global Planet. Change
(2010) - et al.
Towards operational guidelines for over-threshold modeling
J. Hydrol.
(1999) - et al.
Seasonality and duration in extreme value distributions of significant wave height
Ocean Eng.
(2008) - Akaike, H., 1973. Information theory and an extension of the maximum likelihood principle. In: B.N. Petrov and F. Csaki...
A hierarchical approach for the regionalization of precipitation annual maxima in Canada
J. Geophys. Res.
Mapping the hazard of extreme rainfall by peaks over threshold extreme value analysis and spatial regression techniques
J. Appl. Meteor. Climatol.
Assessing the reliability of regional depth-duration-frequency equations for gaged and ungaged sites
Water Resour. Res.
Extreme rainfall estimation by combining data from several sites
Hydrol. Sci. J.
Model Selection and Multimodel Inference. A Practical Information–Theoretic Approach
Multimodel inference – understanding AIC and BIC in model selection
Sociological Methods & Research
Methods for exploring spatial and temporal variability of extreme events in climate data
J. Clim.
An Introduction to Statistical Modeling of Extreme Values
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