## 1. Introduction

Analysis of the characteristics of extreme precipitation over large areas has received considerable attention, mainly due to its implications for hazard assessment and risk management (Easterling *et al.*, 2000). Hazardous situations related to extreme precipitation events can be due to very intense rainfall, or to the persistence of rainfall over a long period of time. Reliable estimates of the probability of extreme events are required for land planning and management, the design of hydraulic structures, the development of civil protection plans, and in other applications.

The extreme value (EV) theory provides a complete set of tools for analysing the statistical distribution of extreme precipitation, allowing for the construction of magnitude–frequency curves (Hershfield, 1973; Reiss and Thomas, 1997; Coles, 2001; Katz *et al.*, 2002). Derived statistics such as quantile estimates (the average expected event for a given return period) have been widely used to express the degree of hazard related to extreme precipitation at a given location. In most cases, the analysis of extreme events has been reduced to the daily intensity, although other important aspects, including event duration and magnitude, have recently been incorporated into the analysis of extreme precipitation (Beguería *et al.*, 2009).

An important assumption of the classical EV theory refers to the stationarity of the model, which implies that the model parameters do not change over time. However, climatic series are known to be non-stationary, and hence many studies have been devoted to analysing the occurrence of temporal trends and cycles, including the frequency and severity of extreme events (Smith, 1989; Karl *et al.*, 1995; Karl and Knight, 1998; Groisman *et al.*, 1999). In addition, several models have highlighted a likely increase in the frequency of extreme events under modified greenhouse gas emission scenarios (Meehl *et al.*, 2000; Groisman *et al.*, 2005; Kyselý and Beranová, 2009) and so advances in methodologies for assessing such changes are needed.

The importance of these issues has stimulated the development of techniques to identify trends in extreme events. A large number of studies are based on analysing the time variation of statistics defined by fixed magnitudes or quantiles (Karl *et al.*, 1995; Groisman *et al.*, 1999; Brunetti *et al.*, 2004; López-Moreno and Beniston, 2009). These approaches can provide information about changes in high precipitation values, but not necessarily about the most extreme precipitation events, which, by definition, occur intermittently. For this reason, other approaches have been developed to analyse changes and trends in hydrological extremes based on parametric approaches, which are based on an extension of the EV theory, namely, the non-stationary extreme value (NSEV) theory (i.e. Coles, 2001). NSEV methods allow analysis of the time dependence of extreme precipitation within the EV theory context, enabling estimation of the time evolution of the most extreme precipitation events. The NSEV theory is the most reliable framework for analysing the time variation of extreme events, providing the means to address important issues including the occurrence of trends and cycles in data series, as well as co-variation with other climatologic and meteorological factors.

Most applications of the NSEV theory, however, have been based on block maxima data, in which only the *n* highest observations are retained at regular time intervals. The annual maxima method is a common example of this approach. The resulting data series are fitted to EV distributions such as the Gumbel distribution or the generalized extreme values (GEV) distribution (Hershfield, 1973), and there are examples of non-stationary analysis of extreme rainfall using the GEV distribution (Katz, 1999; Nadarajah, 2005; Pujol *et al.*, 2007). There has been criticism of the waste of useful information when the block maxima approach is applied to datasets containing data in addition to the maxima (Coles, 2001; Beguería, 2005). As an alternative, the peaks-over-threshold (POT) approach is based on sampling all observations exceeding a given threshold value, and fitting the resulting data series to the exponential or the GP distributions (Cunnane, 1973; Madsen and Rosbjerg, 1997).

There are some studies demonstrating non-stationarity on POT data. Li *et al.* (2005) used a split-sample approach to demonstrate temporal changes on extreme precipitation in Western Australia. Hall and Tajvidi (2000) used a moving kernel sampling approach to analyse non-stationarity on the GP parameters of extreme temperature and wind speed data. Nonparametric approaches (split sampling and moving kernel sampling) are interesting as a preliminary tool for exploring the presence of non-stationarity on POT data, but do not provide a functional relationship between the GP distribution parameters and time. Parametric methods to account for this relationship have been developed on the basis of the maximum likelihood estimation (MLE) method (Smith, 1999; Coles, 2001), but the examples of their application to hydroclimatic data are relatively scarce. One of the first applications of the MLE method to model time dependence on temperature and precipitation POT data is due to Smith (1999). Chavez-Demoulin and Davison (2005) used a penalized likelihood approach and generalized additive modelling (GAM) to fit the co-variation between GP distribution parameters and other co-variates. Nogaj *et al.* (2006) used MLE for fitting linear and quadratic trends to the parameters of temperature extremes over the North Atlantic region. A similar approach was used by Laurent and Parey (2007) and Parey *et al.* (2007) for temperature extremes in France. Méndez *et al.* (2006) analysed long time trends and seasonality of POT wave height data. Yiou *et al.* (2006) analysed trends of POT discharge data in the Czech Republic. Abaurrea *et al.* (2007) used MLE to model co-variation between the scale parameter of a GP distribution and mean temperature data to check for trends of POT temperature series. Applications of the NSEV theory to POT precipitation data have been very scarce, mostly due to difficulties inherent to the irregular and clustered character of rainfall time series. The study by Smith (1999) is one of the very few references, and it does not incorporate recent advances in POT analysis of precipitation data such as declustering (i.e. using series of rainfall events instead of the original daily series), nor compares the MLE method with other alternative methods such as the nonparametric ones.

Whether there are trends in extreme precipitation records is currently of major interest in climate change studies, and the evaluation of observational datasets at regional scales is still a key research focus for better understanding of the implications of climate change for precipitation extremes. In this study, we explain the principles of application of NSPOT analysis to time series of extreme precipitation events, considering both their intensity and magnitude. One nonparametric (moving kernel sampling) and one parametric (MLE) methods are presented and practical aspects on their use are discussed. We illustrate the use of these techniques with a case study based in the northeast sector of the Iberian Peninsula, which has a strong Atlantic-Mediterranean climate gradient. The purpose of the study is twofold: (1) to expand the current knowledge of the temporal evolution of extreme precipitation in the Iberian Peninsula and (2) to demonstrate the application of NSPOT analysis to the detection of changes in the most extreme precipitation events.