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Consolidation of analysis methods for sub-annual extreme wind speeds

2013; Wiley; Volume: 21; Issue: 2 Linguagem: Inglês

10.1002/met.1355

1469-8080

Nicholas Cook,

Energy Load and Power Forecasting

Meteorological ApplicationsVolume 21, Issue 2 p. 403-414 RESEARCH ARTICLEFree Access Consolidation of analysis methods for sub-annual extreme wind speeds Nicholas J. Cook, Corresponding Author Nicholas J. Cook RWDI, Dunstable, UKN. J. Cook, RWDI, Unit 4, Lawrence Way, Dunstable, Bedfordshire LU6 1BD, UK. E-mail: [email protected]; [email protected]Search for more papers by this author Nicholas J. Cook, Corresponding Author Nicholas J. Cook RWDI, Dunstable, UKN. J. Cook, RWDI, Unit 4, Lawrence Way, Dunstable, Bedfordshire LU6 1BD, UK. E-mail: [email protected]; [email protected]Search for more papers by this author First published: 08 January 2013 https://doi.org/10.1002/met.1355Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This paper consolidates recent advances in methodologies in extreme-value analysis of wind speeds by using sub-annual maxima in conjunction with exact and penultimate extreme-value models. By avoiding asymptotic models and the associated issues of asymptotic convergence, the consolidated methodology is able to extend analysis further into the lower tail, greatly increasing the statistical confidence. The standard error in design predictions of dynamic pressure is reduced to less than a third of the corresponding standard error from annual maxima. The methodology is demonstrated by re-analysing wind speed data from previously published studies at sites in simple and in various mixed-mechanism climates. Copyright © 2013 Royal Meteorological Society 1. Introduction 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. Figure 1Open in figure viewerPowerPoint Distribution of time between events for MIS and LM&S methods 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. 1.2. Extreme-value models 1.2.1. Exact distribution of extremes The Binomial gives the exact distribution, Φ, of the maximum, ◯, of r values drawn from a parent distribution, P, of independent events, x, as: (1) from which it is clear that the form of the extreme distribution depends on the form of the parent for all finite values of r. When assessing annual maxima, r is the annual rate of all independent events and represents the upper limit to the rate of events that can be extracted from the wind record by MIS, LM&S or similar methods. 1.2.2. Parents of the exponential type The parent distribution, P(x), of any variate, x, can always be expressed as: (2) Fisher and Tippett showed that the form of (1) converges towards one to three possible types as x→∞. When h(x)→∞ more rapidly than ln(x), Φ(◯) converges towards Type 1. In this case, P(x) is called a parent of the exponential type. Here, h(x) is a slowly increasing function of x which penultimately behaves like xw as x→∞ and ultimately behaves like x. For example, consider h(x) = x2, in which case w = 2: with very large values of x, say x = 100, 101, 102…, h(x) = 10 000, 10 201, 10 404…≈ 10 000 + 200(x − 100), i.e. h(x) behaves like a linear function of x. The Type 1 distribution is given penultimately (Cook and Harris, 2004, 2008) by: (3) and asymptotically as n→∞ to: (4) The FT1 distribution is unlimited in the upper tail and has an exponential asymptote. The standard reduced variate, 〈y〉 for the expectation 〈h(x)〉, the ensemble mean from an infinite number of random trials, is from Equation : (5) Hence, when observations of h(x) are plotted as abscissa against 〈− ln(−ln(Φ))〉 as ordinate, a straight line is expected, with an intercept of ln(n). The conventional 'Gumbel plot', in which observations of ◯ are plotted as abscissa against 〈− ln(−ln(Φ))〉 as ordinate, will be a straight line when w = 1, a concave-upwards curve when w < 1, a concave-downwards curve when w > 1, and the intercept is Uw. When h(x)→∞ less rapidly than ln(x), Φ(◯) converges towards Type 2. This is also unlimited in the upper tail, increasing faster than Type 1. Type 2 distributions appear on Gumbel plots as concave-upwards curves, hence mimic Type 1 with w < 1. When h(x)→L, a finite upper limit, Φ(◯) converges towards Type 3. This appears on a Gumbel plot as a concave-downwards curve, straightening as it approaches the limit, L, but in the region of the mode (where most of the observations lie) mimics Type 1 with w > 1. When observations on a Gumbel plot form a curve, it is not possible to distinguish between Type 1 with w ≠ 1 and the corresponding Type 2 or Type 3 behaviour without a priori knowledge of the form of h(x). In the case of wind speeds, observations in single-mechanism climates are invariably very well represented by the Weibull distribution: (6) and in mixed climates by the disjoint sum of two or more Weibull distributions. This is a distribution of the exponential type and leads, via the penultimate FT1 distribution, to the Type 1 distribution as the asymptote for extreme wind speeds. 1.2.3. Penultimate distributions of extremes Recently, attention has been focussed on avoiding the issues of asymptotic convergence in the upper tail by discarding asymptotic models in favour of exact, or penultimate, models. Note that here, and later in this paper, the term 'exact' refers specifically to the relationship between the extreme and the corresponding parent. In all real-world data there is always uncertainty associated with the finite sample size, so this does not imply that 'exact' models provide 'exact' values. Cook and Harris (2004, 2008) proposed Equation as the penultimate model for wind speeds because parent wind speed data conform closely to the Weibull distribution, where h(x) = (V/C)w. This model is exact for Weibull parents and is the penultimate distribution for all parents of the exponential type. In developing XIMIS by extending MIS/IMIS to accommodate penultimate models, Harris (2009) re-examined the role of the Poisson process model: (7) This is the form of the extremes of independent events that follow the Poisson recurrence model at an average rate, r, for any form of the parent, P. For the derivation of Equation see Cook et al. (2003). More generally, Equation provides a very good approximation to Equation within the rule-of-thumb limits given in Appendix A. Although Equation is independent of any model, it uniquely links the extreme to the parent, so selecting a model for one also selects the model for the other. For parents of the exponential type, P = 1 − exp(−h(x)), Equation becomes (3), the penultimate FT1 model of Cook and Harris (2004, 2008). The standard reduced variate evaluates from (7) as: (8) Where the events do not follow a Poisson process, Harris (2009) demonstrated that significant differences between (8) and the series expansion of the exact expression (1) are confined to the lower tail, i.e. as P→0, and are equivalent in size to the error in the Cauchy approximation in the derivation of (3) (see Cook and Harris, 2004 and Appendix A, here). Figure 2 compares the Poisson reduced variate 〈y〉 from Equation with the asymptotic FT1 for various values of rate, r. The upper tails are convergent, but the lower tails are different: the Poisson process model is limited at y = − ln(r) whereas the asymptotic FT1 is unlimited in the lower tail (effectively − ln(r)→− ∞). As MIS data extend well into the lower tail, this limit has important implications for the fitting range. Figure 2Open in figure viewerPowerPoint Poisson reduced variate (ordinate) compared with the asymptotic FT1 reduced variate (abscissa) for various rates, r 2. Estimating the mean reduced variate from the order statistics 2.1. The order statistics The order statistics of a sample are obtained by ranking the values in ascending or descending order of value. Conventional EVA uses the rank from the bottom in ascending order, usually denoted by m for the mth smallest value. However, some of the relevant expressions are simpler when the data are ranked in descending order of value, with rank denoted by ν for the νth largest value, or when probability P is replaced by its complement, Q = 1 − P. (See Harris, 1999, 2009) The expectation, 〈ym〉, of the mth out of N ranked values of any function, y(P), is the best unbiased estimator of ym and is evaluated from the Binomial by: (9) It is important here to note that Equation is universally applicable to all functions of P, including probability itself, i.e. including y(P) = P. This case evaluates to the Weibull (1939) estimator: (10) Although Gumbel (1958) advocated the use of this estimator, Gumbel noted that it produces a mean bias in estimates of the variate of all non-linear distributions. As all probability distributions in nature tend to be non-linear and follow an S-shaped curve, so all estimates for the variate using Equation will be biased. A thorough discussion of this issue is given by Cook (2011, 2012). In most analysis and design applications, the aim is not to obtain the best unbiased estimate of probability for a given observational value, but is to obtain the best unbiased estimate of the variate for a datum (design) probability: in this case, the best estimate of the reduced variate, y. Whether or not (10) can be evaluated directly for y depends of the form of y(P). 2.2. The Fisher Tippett type 1 reduced variate The expectation of the FT1 mean reduced variate, 〈ym〉, for any rank is given by inserting the FT1 function for y = − ln(−ln(Φ)), from (5), into (9). A closed form solution does not exist, and evaluation of (9) requires numerical integration (Harris, 1999) or a Monte-Carlo (bootstrapping) approach (Cook, 2004), with the latter also able to evaluate confidence limits. 2.3. The Poisson reduced variate The expression for the mean reduced variate, ym, for any rank is given by inserting the Poisson model, y = − ln(1 − Φ)− ln(r), from (8), into (9). A closed form solution does exist, as given by Harris (2009) for the XIMIS method: (11) where ψ is the digamma function. Evaluation of ψ(x) is usually done by exploiting its recurrence relationship, ψ(x + 1) = ψ(x)+ 1/x, starting with the value ψ(1) = − γ = − 0.577215665. Evaluation of (11) is the difference of two summations: one of length N and one of length N − m. Hence (11) simplifies to the single summation of length m: (12) an expression also given by Gumbel (1958, p. 117) for the case of r = 1. Implementation of (12) is computationally efficient when 〈y〉 is required for all ranks, since 〈y1〉 = 1/N − ln(r) for the first rank, then each successive rank is given from the previous by one division and one add operation. Provided this is done in at least 32-bit double precision arithmetic, values can be obtained for very large N without accumulating significant rounding errors. The issue with POT data is that the threshold excludes the lower tail, so the observations are left-censored. The population of events above the threshold is not the population required to evaluate 〈ym〉Poisson in (11) or (12). If the length of the record in years is denoted by R, the unknown population of events becomes N = R × r. Since ψ(x + 1)→ln(x) as x→∞, then from (11) (Harris, 2008, equation .22): (13) where ν is the rank in descending order (νth largest value). Equation is a more rigorous confirmation of the insensitivity of MIS to the annual rate than that given by Cook and Harris (2004, appendix B). Equation can be used for POT data, where N is unknown in value but is large. The digamma function, ψ(ν), is evaluated first for the largest value, ψ(1) = − γ, then for the 2nd largest, etc., using the recurrence relation sequentially until the smallest data value is reached. 3. Implementation of MIS and LM&S methods 3.1. Maximum annual rate of independent maxima and the 'relevant parent' It is convenient at this point to distinguish between the annual rate of independent maxima extracted from the data record, re, and the maximum annual rate of all independent maxima inherent in the record, ri. The ratio re/ri can be interpreted as a measure of the efficiency of the extraction methodology in maximizing the population of values for analysis. For the temperate climate of the UK, specifically for the case of Boscombe Down which is examined later, Harris (2008) found the correlation time scale to be T = 22.15 h, or approximately 1 day. Given that the interest is in maxima of independent events, the shortest time between such maxima is t = 44.3 h, since at least one event minimum must exist between any two consecutive maxima. Hence the maximum annual rate of independent maxima for the temperate UK climate is ri = 198.6 ≈ 200. A method that could extract all these independent values from each year of a record would provide what Harris (2008) calls the 'relevant parent'. It follows that if the MIS or LM&S method extracts independent events at an annual rate re then these events are directly related to the relevant parent by Equation , where Φ represents the extracted maxima, P represents the parent, and r = ri/re. Hence, providing that ri is known, or can be independently estimated, it is possible to estimate the relevant parent by inverting (1). An early attempt to formulate a model for this parent was made by Brooks et al. (1946, 1950). They suggested that the wind vector be resolved into orthogonal x and y components relative to a set of arbitrarily oriented Cartesian axes, each component Normally distributed and mutually independent. As noted by Davenport (1968), this results in a parent distribution which is Rayleigh in form, i.e. Weibull with w = 2. As noted in Section , above, all parents of the exponential type behave penultimately like xw, hence they exhibit Weibull-distribution equivalence in the upper tail. Wind speeds are observed to be very well represented by the Weibull distribution, Equation , or the disjoint sum of several, but the shape parameter, w, can take a wide range of values depending on the wind mechanism. This is the basis of the penultimate extreme model proposed by Cook and Harris (2004, 2008). 3.2. Boscombe Down, UK 3.2.1. MIS analysis Boscombe Down has been used as an exemplar of the UK wind climate in two recent studies: Cook and Harris (2004) fitted their penultimate model for y > − 1.3, assuming a single climate mechanism. The observations lay well within 5–95% confidence limits, but some lay outside the more onerous 37–63% confidence limits, and, Cook and Harris (2008) used the Jenkinson-Lamb index to separate the hourly mean observations into cyclonic and anti-cyclonic sets for separate analysis. The distribution of the hourly parent, irrespective of direction, was a very good fit to the disjoint two-mechanism Weibull model and the extremes to a joint two-mechanism penultimate FT1 model. When the standard MIS method is run on hourly mean wind speed data from Boscombe Down, the annual rate of events recovered is re = 147, representing an extraction efficiency of around 75%. As the method extracts maxima, the data are therefore expected to have diverged slightly from the 'relevant parent', in accordance with Equation , and towards the FT1 asymptote. However, from reference to Figure 2, observations plotted using the Poisson ordinate are expected to lie close to the FT1 model down to y ≈ − 4. Figure 3 shows the dynamic pressure for all ranks plotted against the mean reduced variate for the FT1 ordinate (small solid circles) and for the Poisson ordinate (large open circles). The bold curve is the fit to the penultimate FT1 model, assuming a single mechanism climate, and the chained curves are the 5–95% confidence limits on the data. The penultimate FT1 model was fitted for 〈y〉Poisson > − 4, with 1665 observations contributing to the fit. All model fitting in this paper was made using the multi-parameter non-linear optimizer 'Solver' in MS Excel™ spreadsheets by optimizing the parameters of the model to achieve the least mean square error between 〈y〉Poisson and the model y. The resulting shape factor wq = 0.989 for dynamic pressure gives almost a straight line, and corresponds to wV = 1.98 for wind speed which is typical of wV ≈ 2 for the UK climate. As expected, the data remain close to the fitted curve down to 〈y〉Poisson = − 4, much further into the tail than the y = − 1.8 limit suggested by Harris (2009), but is limited at y = − ln(147) = − 5.0. On the other hand, the data plotted using the 〈y〉FT1 ordinate have diverged significantly above the penultimate FT1 model by 〈y〉FT1 = − 3 as the lower tail asymptotically approaches the q = 0 axis. The fitted model line intersects the q = 0 axis at y = − 5.41, giving another estimate for the rate of independent maxima, ri = 224. Comparisons of estimates for the annual rate of independent events, ri, should be made in terms of − ln(ri) as this is represents a linear shift of the variate, the so-called 'Poisson shift'—on which basis, the correlation time scale estimated by Harris (2008) gives − ln(ri) = 5.3, while Figure 3 gives − ln(ri) = 5.4, matching to within 2%. Figure 3Open in figure viewerPowerPoint MIS dynamic pressures for Boscombe Down, UK, using FT1 and Poisson ordinates Figure 4 shows the corresponding relevant parent recovered using r = 200/147 in Equation which, plotted using the Poisson ordinate, gives an excellent match to the penultimate FT1 model for the full range of the observations. The bold model curve and chained confidence limits are from the MIS fit in Figure 3. The extended range includes sufficient observations to attempt a direct fit for the six parameters of the joint two-mechanism penultimate FT1 model, shown by the dashed curve, which is comparable to the fit from the cyclonic/anti-cyclonic separated data in Cook and Harris (2004). The two-mechanism mixed climate model is explored later in Section Figure 4Open in figure viewerPowerPoint Poisson parent of MIS dynamic pressures at Boscombe Down, UK 3.2.2. LM&S analysis Daily maxima were abstracted from the hourly dataset, then independent maxima selected by the LM&S method for separations of n = 2, 4, 8 and 16 days. The shortest separation interval, n = 2 days, gives re = 103, representing an extraction efficiency of about 50%. The longest interval, n = 16 days, gives re = 12, an extraction efficiency of only about 6%. Figure 5 shows the dynamic pressure for the median rank of each integer wind speed plotted against the mean reduced variate for the FT1 distribution (small solid symbols) and for the Poisson process model (large open symbols). The fit to the penultimate FT1 model from the MIS analysis of Figure 3 is included for comparison as the bold curve, and the chained curves are the 5–95% confidence limits for this. The trend for the FT1 ordinate is for the observations in the lower tail to converge gradually towards the asymptotic FT1 model, from above, as the interval increases—but note that the observations for n = 16 days lie below the model. As the interval, n, increases, the rate of events extracted, re, decreases rapidly, so that the observations plotted using the Poisson ordinate follow the trend in Figure 2. The observations on the Poisson ordinate begin to move away from the MIS model curve before they approach y = − ln(re), and the smaller the value of re the earlier this deviation begins. The deviation becomes significant at a position approximately ln(ri/re) above y = − ln(re), i.e. at y = − 2ln(re)+ ln(ri), and this is proposed as a reasonable rule-of-thumb for setting the lower fitting limit for LM&S data. Figure 5Open in figure viewerPowerPoint LM&S dynamic pressures for Boscombe Down, UK, using FT1 and Poisson ordinates Figure 6 shows the relevant parents recovered using r = ri/re in Equation , for comparison with Figure 4. The parents lie close to the fitted MIS model curve, with n = 2 giving the best match, and with a trend for increasing slope (increasing dispersion, C) as n increases. This action removes the deviation in the lower tail and substantially increases the fitting range to y = − 4 for all values of n used here. Figures 5 and 6 empirically demonstrate the validity of the Poisson process model for LM&S data in terms of the wind speed values, despite the poor match for recurrence interval in Figure 1. Figure 6Open in figure viewerPowerPoint Poisson parents of LM&S dynamic pressures at Boscombe Down, UK 3.2.3. Discussion of Boscombe Down results Table lists the penultimate FT1 parameters when fitted for y > − 2ln(re)+ ln(ri) using the Poisson ordinate. The two right-hand columns give the predicted 50 year return period dynamic pressure and the percentage difference from the datum MIS value. The LM&S predictions become increasing less accurate as n increases, underestimating by 17% for n = 16 days. It is apparent that any additional confidence that the events are uncorrelated is negated by the increase in sampling variance as the fitted population falls with increasing n. The aim should always be to use shortest interval that achieves independence. Table 1. Penultimate FT1 parameters fitted to observations of dynamic pressure at Boscombe Down, UK Standard MIS and LM&S Lower limit: y = − ln(re)+ 1.5 N fitted w (Pa) U (Pa) C (Pa) q50 (Pa) Δ(q50) MIS 1665 0.989 198.6 36.0 343.9 0.0 LM&S n = 2 787 0.956 198.6 29.4 330.6 − 3.9 n = 4 443 0.991 198.0 40.4 370.8 4.6 n = 8 201 0.873 194.5 18.3 272.4 − 10.3 n = 16 96 0.761 193.0 8.1 224.5 − 16.8 Table gives the corresponding penultimate FT1 parameters for the 'relevant parents' fitted over the full available range, i.e. for y > − 5. Each case includes two to three times more events in the fit than above and this gives a large improvement in the statistical accuracy of the parameters. The LM&S predictions gradually overestimate as n increases, but by only 1.7% for n = 16 days. There is a small, but consistent trend to greater slope (greater dispersion, C) in the LM&S curves with increasing n which is attributable to the corresponding increase in sampling variance. The ranking process subsumes the sampling variance into the variance of the observations but the mean is unchanged, so the dispersion increases and the mode decreases in value. This trend is consistent in Table and is the reason that the observations for n = 16 days lie below the model on the FT1 ordinate in Figure 5. Table 2. Penultimate FT1 parameters fitted to relevant parent of dynamic pressure at Boscombe Down, UK Relevant parent MIS and LM&S Lower limit: y = − 5 N fitted w (Pa) U (Pa) C (Pa) q50 (Pa) Δ (q50) MIS 4476 0.989 198.6 36.0 343.9 0.0 LM&S n = 2 3157 0.980 200.2 34.9 344.9 0.3 n = 4 1750 0.969 198.0 34.9 347.8 1.1 n = 8 796 0.975 195.7 36.3 348.3 1.3 n = 16 375 1.002 193.5 40.2 349.6 1.7 Table gives the penultimate FT1 parameters for the joint two-mechanism fit to the MIS 'relevant parent' shown in Figure 4. The top half of the table gives the parameters for dynamic pressure, while the bottom half gives the values converted to wind speed in knots for comparison with the fit in Cook and Harris (2008). The results are quite closely comparable, given that the observations in Cook and Harris (2008) were separated before analysis and each set fitted for three unknown parameters, while the observations in this paper were fitted together, for six unknown parameters. The difference between the single and joint models in Figure 4 is small in comparison with the range of the confidence limits. Until recently, the UK strong wind climate has been regarded as 'simple', i.e. dominated by Atlantic depressions, but Cook and Harris (2008) showed that analysing separated cyclonic and anti-cyclonic components accounted for the observed deviations of the hourly parent and annual extreme distributions from the single-climate models. It remains debatable whether this represents a true mixed climate, or is simply two aspects of a sing