On spatial extremes: with application to a rainfall problem

arXiv:0807.4092v1 [stat.AP] 25 Jul 2008 The Annals of Applied Statistics 2008, Vol. 2, No. 2, 624–642 DOI: 10.1214/08-AOAS159 c ⃝Institute of Mathematical Statistics, 2008 ON SPATIAL EXTREMES: WITH APPLICATION TO A RAINFALL PROBLEM By T. A. Buishand, L. de Haan1 and C. Zhou Royal Netherlands Meteorological Institute (KNMI ), Erasmus University Rotterdam and University of Lisbon and Erasmus University Rotterdam and Tinbergen Institute We consider daily rainfall observations at 32 stations in the province of North Holland (the Netherlands) during 30 years. Let T be the to- tal rainfall in this area on one day. An important question is: what is the amount of rainfall T that is exceeded once in 100 years? This is clearly a problem belonging to extreme value theory. Also, it is a genuinely spatial problem. Recently, a theory of extremes of continuous stochastic processes has been developed. Using the ideas of that theory and much com- puter power (simulations), we have been able to come up with a reasonable answer to the question above. 1. Introduction. When a damaging flood has occurred extreme rainfall statistics are frequently used to answer questions about the rarity of the event. Engineers often need extreme rainfall statistics for the design of struc- tures for flood protection. A typical question is, for example, what is the amount of rain in a given area on one day that is exceeded once in 100 years? Or, more mathematically, what is the 100-year quantile of the total rainfall in the area on one day? In this paper this question is investigated for a low-lying flat area in the northwest of the Netherlands. The area is shown in Figure 1. Because it roughly covers the province of North Holland, it will shortly be indicated as North Holland. There are 32 rainfall stations in the area for which daily data were avail- able for the 30-year period 1971–2000. Only the fall season, that is, the months September, October and November, is considered. In this season the likelihood of flooding and its impact are relatively large. Because of the restriction to the fall season, it is reasonable to assume stationarity in time. Stationarity in space, except for location and scale, is also assumed. Received January 2007; revised January 2008. 1Supported in part by the FCT project PTDC/MAT/64924/2006. Key words and phrases. Spatial extremes, max-stable process, areal reduction factor. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Statistics, 2008, Vol. 2, No. 2, 624–642. This reprint differs from the original in pagination and typographic detail. 1 2 T. A. BUISHAND, L. DE HAAN AND C. ZHOU Fig. 1. The study area: North Holland. Since we have to extrapolate from a 30-year to a 100-year period, our problem is an extreme value problem. There is also a clear spatial aspect. Engineers often make use of areal reduction factors (ARFs) to convert quantiles for point rainfall to the corresponding quantiles of areal rainfall. ARFs have been derived empirically by estimating the areal rainfall as a function of point rainfall measurements [e.g., Natural Environment Research Council (NERC) (1975), Bell (1976)] or by statistical modeling [e.g., Bacchi and Ranzi (1996), Sivapalan and Bl¨oschl (1998), Veneziano and Langousis (2005)]. The latter requires assumptions on distributions, spatial correlation and/or scaling behavior. The resulting ARF for the 100-year quantile is generally very uncertain. Some attempts have been made to estimate ARFs from weather radar data [Allen and DeGaetano (2005), Stewart (1989)]. One difficulty is that the raw rainfall intensities from the radar reflectivities need to be adjusted for systematic deviations from the values observed at the rainfall stations. Another difficulty is that archived radar data cover a relatively short time period (in the Netherlands only 10 years). Regional Climate Model (RCM) simulations driven by weather reanalysis data are a potential source for areal aggregated rainfall. A reanalysis is an es- timate of the state of the atmosphere based on observations and a numerical weather forecast. The RCM is necessary to increase the spatial resolution. SPATIAL EXTREMES: RAINFALL APPLICATION 3 Various 40-year simulations have been performed recently with spatial reso- lutions of 50 km × 50 km and 25 km × 25 km, in particular, within the frame- work of the EU-funded project ENSEMBLES (www.ensembles-eu.org). In addition to the limited length and rather coarse resolution for our applica- tion, there are systematic differences between simulated and observed rain- fall. For the KNMI RCM driven by ERA40 reanalysis data, Leander and Buishand (2007) report differences up to 20% in seasonal average rainfall for the river Meuse basin, situated south of the Netherlands. For North Hol- land, the differences can even be larger because it is much smaller than the Meuse basin and it is surrounded by water. Statisticians have used ma

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