## 1. Introduction

[2] The spatial modeling of rainfall is a long-standing topic in the environmental sciences and has grown in importance with the realization that a warming world is likely to bring more intense precipitation events, and thus higher risk to infrastructure and populations. The topic is currently a highly active research area, some recent articles being *Yang et al*. [2005], *Cooley et al*. [2007], *Feng et al*. [2007], *Vrac and Naveau* [2007], *Zheng and Katz* [2008], *Van de Vyver* [2012], *Shang et al*. [2011], and *Villarini et al*. [2011]. *Wilks and Wilby* [1999] and *Chandler et al*. [2006] review the earlier literature. The emphasis on rare events means that extreme-value statistics [*Coles*, 2001; *Beirlant et al*., 2004] are widely used to estimate return levels and associated quantities. Classical statistics of extremes [*Katz et al*., 2002] underpins standard approaches to the analysis of annual maximum or partial duration series, using block maxima or peaks over threshold methods, respectively, but tools for spatial analysis that extend the classical extreme-value models have only recently begun to be used. The simplest approach to spatial analysis is to fit extreme-value distributions separately to each of many time series, as for example in *Feng et al*. [2007], and then to ignore any spatial correlation between the individual fits, though *Madsen et al*. [2002] suggest a more sophisticated approach. In some cases, this type of model may be appropriate, but in others involving spatial quantities such as joint return levels or areal rainfall, spatial dependence must be taken into account, and models are then needed that respect appropriate dependence properties of extremal distributions.

[3] Max-stable processes [*de Haan*, 1984; *de Haan and Ferreira*, 2006; *Davison et al*., 2012] extend the generalized extreme-value (GEV) distribution, which is widely used to describe univariate maxima, to the spatial setting, and thus provide consistent multivariate distributions for maxima in arbitrary dimensions. Although proposed some time ago [*Coles*, 1993; *Coles and Tawn*, 1996; R. L. Smith, University of Surrey, unpublished manuscript, 1990, http://www.stat.unc.edu/postscript/rs/spatex.pdf], such processes have been little applied until very recently. *Padoan et al*. [2010] show how composite likelihood methods can be used to fit max-stable processes and illustrate this with U.S. rainfall data. *Shang et al*. [2011] use them to gauge the effect of El Niño–Southern Oscillation on winter rainfall in California, and *Westra and Sisson* [2011] use them to understand how extreme rainfall in Eastern Australia depends on explanatory variables such as the Southern Oscillation index and sea surface temperature. All three papers have the limitations of using block maxima and fitting only a single family of max-stable models, however, whereas in many applications it would be preferable to use threshold exceedances, which make more efficient use of the data, and to be able to compare several model classes. Indeed, *Davison et al*. [2012] found that other max-stable models fit extreme rainfall data appreciably better than the Smith (unpublished manuscript, 1990) model used by *Shang et al*. [2011] and *Westra and Sisson* [2011]. *Huser and Davison* [2013a] show that the Smith model also has theoretical drawbacks. *Renard* [2011] describes another approach to annual maximum rainfall analysis, based on Bayesian hierarchical models using a copula approach [*Sang and Gelfand*, 2010] but mentions that use of max-stable modeling of spatial dependence might constitute an improvement. *Davison et al*. [2012] found that max-stable models indeed provided better estimates of extreme spatial rainfall than did Bayesian and standard copula approaches. The copula approach of *Salvadori and Michele* [2010] is intended for a given network of gauge stations rather than for a truly spatial analysis. *Buishand et al*. [2008] use a rather special max-stable model to simulate daily spatial rainfall in a homogeneous region of North Holland, but their approach would be difficult to generalize to more complex settings.

[4] The contributions of the present paper are to explain how max-stable models for extreme spatial rainfall may be fitted to several simultaneous partial duration series using thresholds and a censored likelihood approach, to fit a variety of models to daily rainfall data in a small spatial domain, and to extend the max-stable models themselves by fitting so-called inverted max-stable models, which allow more flexible forms of tail behavior that are coherent with recent developments in statistics of extremes. We illustrate the ideas using data from 24 rainfall time series over a small upland domain, supplemented by a longer series from a nearby site.

[5] Section 'Study Area and Available Data' describes briefly the study site and details how the data we analyze were collected. Section 'Extreme-Value Theory' presents the main results about extremes used in this paper. Inference tools, which are based on Gaussian models and composite likelihood, are explained in section 'Inference' and applied in section 'Modeling Extreme Rainfall in Val Ferret'. Functions for fitting our models were written in R [*R Development Core Team*, 2012].