numbers of climate model runs focused on the temporal/spatial scale of interest. Leaving aside the attribution question, these papers provide two specific examples of how observational and climate model data may be combined to estimate the probability of an observed extreme event under current climate conditions and, by extension, how those probabilities might change under scenarios of future climate change.
EXTREMES OF DEPENDENT EVENTS (EVENT CLUSTERS)
We are also concerned with the possibility of extreme events occurring simultaneously in different locations as a result of common meteorological features such as ENSO or Rossby waves. Statistical methods have been developed for this problem, primarily through the methods of multivariate extreme value theory. The bivariate case (where there are just two dependent events) has been particularly highly studied.
In its simplest form, the method used for bivariate analysis is first to perform an analysis of the two variables individually (either a GEV or Generalized Pareto analysis could be appropriate for this) and then to transform the distribution to unit Fréchet form [P(X<x)=e–1/x, x>0] using a probability integral transform. This transformation has the effect of exaggerating the most extreme events so that they stand out sharply on a plot. Traditional measures of dependence, such as correlation, are not readily interpretable in this context, but a number of alternative measures of dependence that are specifically adapted to extreme events have been proposed (Coles et al., 1999).
Going beyond simple characterizations of extremal dependence, there are a number of formal statistical models that have been used to calculate joint probabilities of extreme events. There has been limited practical application of these models to climate data, but we illustrate the possibilities by considering two examples related to earlier discussion.
Example 1. Herweijer and Seager (2008) argued that the persistence of drought patterns in various parts of the world may be explained in terms of sea surface temperature patterns. One of their examples (Figure 3 of their paper) demonstrated that precipitation patterns in the south-west United States are highly correlated with those of a region of South America including parts of Uruguay and Argentina. As an illustration of this, we have computed annual precipitation means corresponding to the same regions that they defined, and we show a scatterplot of the data in the left-hand panel of Figure D-1. The two variables are clearly correlated (r = 0.38; p <.0001). The correlation coefficient is lower than that found by Herweijer and Seager (r = 0.57), but this is explained by their use of a six-year moving average filter, which naturally increases the correlation. However, the feature of interest to us here is not the correlation in the middle of the