Journal of Wind Engineering

and Industrial Aerodynamics 89 (2001) 225–227

Response to discussion byN.J. Cook and R.I. Harris

J.D. Holmes*, W.W. Moriarty

Department of Mechanical Engineering, Monash University, Clayton, Vic. 3800, Australia

Accepted 8 September 2000

The authors are appreciative of the interest shown by the discussers in our paper,and the useful analysis of the mean excess of the Weibull distribution included in thediscussion.

It is interesting, and significant from a practical engineering point of view, that thepredictions from the approach advocated by Cook and Harris (C and H) (theirFig. 1) are within 1m/s of those obtained from our approach for return periods up to400 years (a reduced variate of about 6). At higher return periods, the C and Hpredictions become progressively greater, and give unlimited predictions. Predictionsto about 1000 years are significant for design, as this return period represents about a5% risk of exceedence in a 50-year building lifetime. If a lower return period (say 50years) is used for nominal design wind loads, the appropriate wind-load factor willbe determined by the assumed prediction line, i.e., the assumed probabilitydistribution function, as discussed by Simiu and Heckert [5].

The prediction curves for high-return periods for the two approaches can bewritten as follows:

ðH and MÞ VR ¼ C ÿDRÿk ðEq: ð8Þ of the original paperÞ

ðC and HÞ VR ¼ ðAþ B loge RÞ1=w:

Note that there are three parameters to be determined in both cases. The significantdifference is in the divergence of the two distributions at high-return periods asdiscussed above.

Cook and Harris’s arguments are based on the ab initio assumption that the parentdistributions for winds for all ‘‘naturally occurring wind engineering variables’’are of the exponential type (C & H’s Eq. 3) of which the Weibull distribution isan example, and consequently the observed distribution of storm maxima is an

*Corresponding author. Tel.: +61-3-9584-5885; fax: +61-3-9585-3815.

E-mail address: john.holmes@eng.monash.edu.au (J.D. Holmes).

0167-6105/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 6 7 - 6 1 0 5 ( 0 0 ) 0 0 0 6 4 - 7

unconverged extreme value Type I distribution. It is certainly true that the Type I(generalised extreme value distribution with shape factor, k, equal to 0) is the correctasymptotic extreme value distribution for parents of this type, as shown by Gumbeland others. However, we dispute that the parent distribution of any wind engineeringvariable can be unlimited in its upper tail. Invariably there are insufficient data toproperly define the tail, and physical limits will sooner or later come into play.

C and H later state that ‘‘the Weibull distribution is well established as the parentfor wind speeds’’. The Weibull distribution is certainly very useful for defining thecentral part of a complete population of synoptic wind speeds, and for interpolationwithin the probabilities of exceedence corresponding to the recorded data, but it willundoubtedly be conservative when extrapolated to low probabilities. It is alsounlikely to be appropriate to the wind speeds generated by isolated downburst gusts,such as those we have analysed from Moree in our paper.

Since the Types I–III asymptotic extreme value distributions are part of a singlefamily of G.E.V. distributions with a variable shape factor, k, as shown by Jenkinson[7], there seems to be no reason why the correct value of k should be ‘forced’ to be 0rather than say 0.1 or 0.2. The latter values have the advantage that they givebounded distributions.

The question of what is the correct upper limit of wind speeds for particular typesof storms certainly needs more consideration, but we disagree with the discussersthat such wind speeds are so high as to be of no practical consequence. For example,reasonable estimates of the maximum downdraft velocity in microbursts (the type ofwind storm analysed in the Moree data), can be made from thermodynamicarguments (e.g. Spillane [21]). The world-wide recordings of wind speeds fromground stations give an indication of the magnitude of the maximum possible windspeeds that can be generated by the earth’s atmosphere. For example, gust wind

Fig. 1. Generalized extreme value distribution with k ¼ ÿ0:2, 0 and 0.2.

J.D. Holmes, W.W. Moriarty / J. Wind Eng. Ind. Aerodyn. 89 (2001) 225–227226

speeds exceeding 100m/s have never been recorded in the severest tropical cyclone,to the authors’ knowledge. Some knowledge of these expected upper limits isimportant for setting reasonable load factors for design, as discussed above, and forthe prediction of insurance losses.

We agree with the discussers that the G.P.D. is a limiting distribution of theexcesses, as the G.E.V. is the limiting distribution for the maxima in a specified‘epoch’ such as one year. This fact plus the inevitable sampling errors, no matterwhat fitting method for the G.E.V. distribution is adopted, can lead to significanterrors in the estimation of all parameters, including the shape factor, k. Many of thedifficulties involved in fitting the distribution were discussed in Sections 4 and 5 ofour paper, and sampling errors were discussed by Hosking et al. [8]. The majorsource of sampling error is the short lengths of available wind records. Refinementssuch as adding sub-annual maxima, removing bias and pre-conditioning the data (asproposed by the discussers) may improve the fitting of the G.E.V. to the availableobservations, but will not address the possibility that the available extreme data isunrepresentative of longer term occurrences.

The assumption of a fixed value of one parameter (i.e., setting k equal to zero) forthe asymptotic distribution as advocated by the discussers, does not seem to be asolution to these problems. The method we have advocated in the paper, fromamongst the several available, has two major advantages for making predictions ofextreme wind speeds:

* ease of application;* use of data from all available storms; and* no prior assumptions on the ‘correct’ parent distribution, or shape factor in the

asymptotic generalized extreme value distribution, are made.

If indeed the Weibull distribution was the correct one for the parent distributionof the Moree data, and assuming values of w of 1.47 (C and H’s value), and c of0.1 (m/s)ÿ1, then the discussers’ Eq. (14) would predict values of k between 1.0 and1.6, for the thresholds, u, actually used of 20 to 34m/s. These values of shape factorare far in excess of the value of 0.16 obtained by us (Table I of the paper). In fact, forthe range of thresholds used in our analysis, in order for Eq. (14) to produce anaverage shape factor of 0.16, a scale factor (1/c) of about 45 m/s would be necessary!This value of course exceeds the largest extreme gust recorded of 36.5m/s.

The point of extreme value analysis is to avoid having to make any assumptionsabout the parent distribution, otherwise predictions could be made by the fitted parentdistribution. It is certainly clear that in the example of Moree wind data we have usedin our paper, that the Weibull distribution is not the correct underlying distribution.

Finally, an error was made in the plotting of Fig. 1 of our paper. A replacementfigure is attached.

Reference

[21] K.T. Spillane, On the microburst family. The Third Australian Wind Engineering Workshop,

Brisbane, Queensland, July 8–9, 1993.

J.D. Holmes, W.W. Moriarty / J. Wind Eng. Ind. Aerodyn. 89 (2001) 225–227 227