The purpose of this paper is to consolidate recent advances in the analysis of independent sub-annual maximum wind speeds in simple and in mixed climates. The paper consolidates the joint model for extremes in mixed climates of Gomes and Vickery (1978) with the Method of Independent Storms (MIS) for continuous data (Cook, 1982), and its developments, IMIS and XIMIS (Harris, 1999, 2009), and with the method of n-day maxima for daily maxima and peak-over-threshold (POT) data (Simiu and Heckert, 1996) and its development, LM&S (Lombardo et al., 2009). The consolidated methodology is demonstrated by re-analysing wind speed data from previously published studies.
1.1. Methods of obtaining independent sub-annual maxima
The Method of Independent Storms (MIS) has been available since 1982 (Cook, 1982). The original extraction methodology has not changed significantly, but the method of fitting the data to a statistical model has improved in increments. As IMIS (Harris, 1999) fitted the sub-annual maxima to the asymptotic Fisher Tippett type 1 (FT1) distribution, limiting the bottom end of the fitting range to the reduced variate of the lowest annual mean in the sample, typically to y ≈ − 1.2. The reason for this limit was that the bottom tail of wind speed data is limited at V = 0, whereas the bottom tail of the asymptotic FT1 model has no lower limit, and this region of disparity should be excluded from any fit. Thus, the original advantage in using sub-annual maxima was confined to the additional data points from the kth-highest maxima, k = 2, 3, etc, that lie between the highest and lowest annual maxima in the observations, an increase in data of about a factor of 3. It was also noted (Cook, 1982) that the annual rate of independent events was insufficient to achieve convergence to the FT1 asymptote unless the upper tail of the parent CDF was already close to being exponential, leading to the recommendation of dynamic pressure as the variable instead of wind speed in the UK.
This field lay fallow for some years, with observed curvature in the classical FT1 ‘Gumbel’ plot often being attributed to Type 2 or Type 3 behaviour instead of to non-convergence. The introduction of the Generalised Pareto Distribution (GPD) to characterize peak-over-threshold (POT) data, which requires complete convergence for its validity, led to vigorous debate (e.g. Galambos and Macri, 1999; Holmes, 2002; Simiu and Lechner, 2002), and new developments of extreme-value (EV) theory, showed the GPD to be inappropriate for wind data (Harris, 2005). Instead, it was argued (Cook and Harris, 2004, 2008) that asymptotic models should be replaced by exact or penultimate models that avoid the issue of convergence. Accordingly, as XIMIS, Harris (2009) extended the IMIS methodology to accommodate penultimate statistical models.
Implementation of MIS (IMIS or XIMIS) requires continuous data in order to identify individual storm systems, so that the maximum wind speeds extracted from each storm are independent and the multiplication law of probability applies to these events. As the storm maxima are the outcome of discrete independent trials, the resulting distribution of annual maxima is modelled exactly by the Binomial distribution. When the annual rate of storms is large and/or the probability of exceedence is small, i.e. in the upper tail where P 1, the simpler Poisson process model can be used (See Appendix A). A key indicator for the applicability of the Poisson process model is that the time interval between such events should be exponentially distributed (Palutikof et al., 1999), referred to here as the Poisson recurrence model. Figure 1(a) shows the distribution of time between storms extracted by MIS from a 30 year record of hourly mean wind speeds at Boscombe Down, UK, plotted on axes that linearize the exponential distribution. The 5–95% confidence limits shown here, and throughout this paper, were obtained by ‘bootstrapping’ the fitted parameters, using the methodology described in Cook (2004). As the observations fit reasonably well within the confidence limits, it is reasonable to assume that the Poisson recurrence model applies to MIS data.
Often only daily maxima or POT data are available, for which several methods have been proposed. Building on the example of Jensen and Franck (1970), Simiu and Heckert (1996) introduced the concept of ‘n-day maxima’: maximum values from successive periods, each of n day duration, with a minimum separation of n/2 days between events imposed to eliminate correlation. Data by this method, although independent, will always fail the key indicator for Poisson recurrence because each period produces an event, so the separation times must all fall between the fixed limits of n/2 and 2n. An improved method (LM&S), recently proposed by Lombardo et al. (2009) for discontinuous POT data, extracts all maxima that are separated by a specified minimum time interval, but does not set an upper limit to the time between events. Figure 1(b) and (c) show the distributions of time between events for a 2 and 16 day minimum separation, respectively. As these indicate that the time interval between LM&S events is not exponentially distributed, the applicability of the Poisson process as the model for LM&S data relies on the rule-of-thumb limits given in Appendix A and empirical verification.