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NIST BUILDING SCIENCE SERIES 174

Extreme Wind Distribution Tails:

A Teaks Over Threshold' Approach

E. Simiu and N. A. Heckert
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NIST BUILDING SCIENCE SERIES 174

Extreme Wind Distribution Tails:

A 'Peaks Over Threshold' Approach

E. Simiu' and N. A. Heckert^

'Building and Fire Research Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

and

Department of Civil Engineering

The Johns Hopkins University

Baltimore, MD 21218

^Computing and Applied Mathematics Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

Sponsored in part by:

National Science Foundation

Arlington, VA 22230

March 1995

U.S. Department of Commerce

Ronald H. Brown, Secretary

Technology Administration

Mary L. Good, Under Secretary for Technology

National Institute of Standards and Technology

Arati Prabhakar, Director
National Institute of Standards and Technology Building Science Series 174

Natl. Inst. Stand. Technol. BIdg. Sci. Ser. 174, 72 pages (Mar. 1995)

CODEN: NBSSES

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1995

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

ABSTRACT ; vi

1. INTRODUCTION 1

2. 'PEAKS OVER THRESHOLD' APPROACH 3

2.1 Generalized Pareto Distribution 3

2.2 Gumbel and Reverse Weibull Distributions 3 2.3 Mean Recurrence Intervals of Variate X as Functions of GPD

Parameters and Exceedance Rate 4

3. DESCRIPTION OF DE HAAN ESTIMATION METHOD 5

4. WIND SPEED DATA 7

4.1 Uncorrelated Samples Obtained from Largest Daily Data Records... 7

4.2 Largest Yearly Data Samples 8

5. ANALYSES AND RESULTS 9

5.1 Analyses of Uncorrelated Data Sets by the Probability Plot

Correlation Coefficient Method 9

5.2 Estimation of Tail Length Parameter by 'Peaks over Threshold'

Analyses of Uncorrelated Data Samples 11

5.3 Estimation of Tail Length Parameter by 'Peaks over Threshold'

Analyses of Largest Yearly Data Samples 16

5.4 Estimation of Speeds with Specified Mean Recurrence Intervals

by 'Peaks over Threshold' Analyses of Uncorrelated Data Samples. 19

5.5 Influence of Data Errors on Analysis Results 19

6. LOAD FACTORS FOR WIND-SENSITIVE STRUCTURES 23

7. CONCLUSIONS 25

REFERENCES 27

APPENDIX 1 — Histograms of daily largest wind speeds and histograms

of four-day interval uncorrelated wind speeds 29

APPENDIX 2 — Plots of parameter c and 95% confidence bounds versus

threshold and number of threshold exceedances (based on samples of largest yearly wind speeds) 53

APPENDIX 3 — Plots of point estimates for 100- , 1000- and 100 000-year

speeds versus threshold and number of threshold exceedances

(based on four-day interval samples) 65

iii
APPENDIX 4 —

Page

Instructions for accessing data sets and attendant programs . 71

LIST OF FIGURES^

Figure 1. Probability plots for four-day interval sample, Albany,

New York 10

Figure 2. Plots of parameter c and 95% confidence bounds versus

threshold and number of threshold exceedances (based on four-day interval samples) 12

Figure 3. Point estimates of tail length parameters and 95% confidence

bounds, versus number of exceedances (de Haan, 1990) 16

Figure 4. Mean of tail length parameter estimates weighted over 44

four-day interval samples versus order of threshold 17

Figure 5. Mean of tail length parameter estimates weighted over 115

largest yearly data samples versus order of threshold 18

Figure 6. Quantiles: point estimates and 95% confidence bounds versus

number of exceedances (de Haan, 1990) 19

Figure 7. Plots of parameter c and 95% confidence bounds versus

threshold and number of threshold exceedances (based on four-day interval samples, uncorrected data) 21

Figure 8. Point estimates for 100- , 1000- and 100 000-year speeds

versus threshold and number of threshold exceedances (based

on four-day interval samples, uncorrected data) 22

^Figures in Appendices are not included in this list.

iv
ACKNOWLEDGMENTS

E. Simiu acknowledges with thanks partial support by the National Science

Foundation (Grant No. CMS-9411642 to Department of Civil Engineering, The Johns

Hopkins University), and the helpful interaction on this project with R.B.

Corotis of the University of Colorado at Boulder. The writers wish to thank S.D.

Leigh of the Center for Computing and Applied Mathematics, National Institute of

Standards and Technology, who suggested the collection of large samples of daily

data and the plotting format adopted in this report. Thanks are also due to Dr.

S. Coles of Lancaster University for helpful criticism; Dr. J.J. Filliben of the

National Institute of Standards and Technology for advice on testing the

hypothesis that a universal reverse Weibull tail length parameter exists; and

Drs . J.L. Gross and J.L. Lechner of the National Institute of Standards and

Technology for valuable collaboration on earlier phases of this project, some of

which are reviewed in this report

.

COVER PICTURE:

Honore Daumier (1808-1879)

"UN LEGER COUP DE VENT" (A light gust)

Lithograph (newsprint)

In the Collection of the Corcoran Gallery of Art,

Gift of Dr. Armand Hammer

V
ABSTRACT

We seek to ascertain whether the reverse Weibull distribution is an appropriate

extreme wind speed model by performing statistical analyses based on the 'peaks over threshold' approach. We use the de Haan method, which was found in previous

studies to perform about as well or better than the Pickands and Cumulative Mean

Exceedance methods, and has the advantage of providing estimates of confidence

bounds. The data are taken principally from records of the largest daily wind

speeds obtained over periods of 15 to 26 years at 44 U.S. weather stations in areas not subjected to mature hurricane winds. From these records we create

samples with reduced mutual correlation among the data. In our opinion, the

analyses provide persuasive evidence that extreme wind speeds are described

predominantly by reverse Weibull distributions, which unlike the Gumbel

distribution have finite upper tail and lead to reasonable estimates of wind load

factors. Instructions are provided for accessing the data and attendant

programs

.

Key words: Building technology; building (codes); climatology; extreme value

theory; load factors; structural engineering; structural reliability; threshold

methods; wind (meteorology).

vi
1. INTRODUCTION

A fundamental theorem in extreme value theory states that sufficiently large

values of independent and identically distributed variates are described by one of three extreme value distributions: the Frechet distribution (with infinite

upper tail), the Gumbel distribution (whose upper tail is also infinite, but

shorter than the Frechet distribution's), and the reverse (negative) Weibull

distribution, whose upper tail is finite (Castillo, 1988).

The wind loading provisions of the American National Standard ANSI A58. 1-1972

were based on the assumption that a Frechet distribution best fits extreme wind

speeds blowing from any direction in regions not subjected to mature hurricane

winds. An extensive study concluded, however, that the Gumbel distribution is a more appropriate model (Simiu, Filliben and Bietry, 1976). It is a physical fact

that extreme winds are bounded. Their probabilistic model should reflect this

fact. To the extent that an extreme value distribution would be a reasonable

model of extreme wind behavior, one would intuitively expect the best fitting

distribution to have finite tail, that is, to be a reverse Weibull distribution.

In addition to the certainty that wind speeds are bounded, there is at least one other indication — albeit indirect — that the Gumbel model might be an inappropriate model of extreme wind behavior. Estimated safety indices for windsensitive structures based on the Gumbel model imply unrealistically high failure

probabilities (Ellingwood et al . , 1980). This may be due, at least in part, to

the use in those estimates of a distribution with an unrealistically long —

infinite — upper tail.

In this report we seek to ascertain whether the reverse Weibull distribution is an appropriate extreme wind speed model by performing statistical analyses based

on the 'peaks over threshold' approach. This approach enables the analyst to use

all the data exceeding a sufficiently high threshold, and is more effective than

the classical approach, which uses only the largest value in each of a number of

basic comparable sets called epochs (typically, for extreme wind analysis an epoch consists of one year). To illustrate this point, consider, for example, two

successive years in which the respective largest wind speeds are 30 m/s and 45

m/s. Assume that in the second year winds with speeds of 31 m/s, 37 m/s, 41 m/s

and 44 m/s were also recorded (at dates separated by sufficiently long intervals

to view the data as independent). For the purposes of threshold theory, for a 30

m/s threshold the two years would supply six data points. The classical theory

would make use of only two data points. It may be argued that, by choosing a somewhat lower threshold, the number of data points used in the analysis could

be considerably larger than six in our example. However, in view of Che theory's

basic assumption that the threshold is high, excessive lo^jeriiYg, oi tVve t.\\xes\vo\d

would introduce a strong bias. Simulations reported by Gross et al . (1994)

suggest that, in samples taken from normal or extreme value populations, optimal

results are obtained if the threshold is chosen so that the number of exceedances

is of the order of ten per year.

Given a sample of data exceeding a sufficiently high threshold, the analyst using

the 'peaks over threshold' approach must choose an appropriate estimation method.

In this report we use the estimation method proposed by de Haan (1994). Our

choice is based on two reasons. First, Monte Carlo simulations suggest that the
de Haan method performs about as well or better, than two available alternative

methods, the Pickands method and the Cumulative Mean Exceedance method (Gross et

al , 1994). Second, the de Haan method has the advantage of providing estimates

of confidence bounds

.

Data used in this report are taken principally from records of the largest daily

wind speeds obtained over periods of 15 to 26 years at 44 U.S. weather stations

in areas not subjected to mature hurricane winds. A storm system usually affects

a given location for longer than one day, so that wind speed data recorded on two or even more consecutive days are not necessarily independent. We describe the

data samples and our procedure for creating, from the samples of largest daily

speeds, samples with reduced mutual correlation among the data. In addition to

samples of daily data we describe and analyze 115 samples consisting only of

largest yearly speeds recorded over periods of 18 to 54 years at locations not

subjected to mature hurricane winds. To our knowledge no tornado winds have

affected any of our data. All the data samples used in our analyses are available

in an anonymous data file. Instructions for accessing that file and attendant

programs are available in Appendix 4.

In our opinion, the results presented in this report provide persuasive evidence

that extreme wind speeds of extratropical origin and excluding tornadoes are described predominantly by reverse Weibull distributions. This result is in

itself useful from a structural engineering viewpoint, and we discuss its

potential implications for the estimation of load factors for wind-sensitive

structures. Our analyses also suggest that estimates of extreme wind speeds based

on the reverse Weibull model may not be obtainable with sufficient confidence

from the information and by the method used in this report. This suggests the

need for (i) data samples based on longer recording periods and (ii) more efficient estimation methods.

The report is organized as follows. Basic theoretical results pertaining to the

'peaks over threshold' approach are briefly reviewed in Section 2. The de Haan

method is reviewed briefly in Section 3. The data used in the analyses are described in Section 4. Section 5 includes the main results of our investigation.

Section 6 discusses the load factors issue. Section 7 presents our conclusions.

2
2. 'PEAKS OVER THRESHOLD' APPROACH

2.1 Generalized Pareto Distribution. The Generalized Pareto Distribution (GPD)

is an asymptotic distribution whose use in extreme value theory rests on the fact

that exceedances of a sufficiently high threshold are rare events to which the

Foisson distribution applies. The expression for the GPD is

G(y) = Prob[Y < y] = l-{ [l+(cy/a) j'^/^} a>0 , (l+(cy/a))>0 (1)

Equation (1) can be used to represent the conditional cumulative distribution of

the excess Y = X - u of the variate X over the threshold u, given X > u for u sufficiently large (Pickands, 1975). The cases c>0 , c=0 and c<0 correspond

respectively to Frechet (Type II Extreme Value) , Gumbel (Type I Extreme Value)

,

and reverse Weibull (Type III Extreme Largest Values) domains of attraction. For

c=0 the expression between braces is understood in a limiting sense as the

exponential exp(-y/a) (Castillo, 1988, p. 215).

Given the mean E(Y) and standard deviation s(Y) of the variate Y,

a = h E(Y){1 + [E(Y)/s(Y)]2} (2)

c = h{l-[E(Y)/s(Y)]2} (3)

(Hosking and Wallis, 1987).

2.2 Gumbel and Reverse Weibull Distributions. We recall that the expressions for

the Gumbel and reverse Weibull distributions for maxima are, respectively,

Fg(x) = exp{-exp[-(x-/iG)/aG] ) -«<x<co (4)

Fw(x) = exp{-[(/ZM - x)/a„]'^), x < /x„ (5)

For the Gumbel distribution, the relations between distribution parameters and

the mean E(X) and standard deviation s(X) are

ae = (6i/V^)s(X)

fiQ = E(X) - 0.57722(6^/V7r)s(X)

(6)

(7)

For the Weibull distribution.

aw = s(X)/{r(l+2/7)-[r(l+l/7)]2}i/2 (8)

//w = E(X) + a„ r(l + 1/7) (9)

where F is the gamma function (Johnson and Kotz , 1972). For example, for E(X) =

50, s(X) = 6.25, and 7 = 2, a„ = 13.49 and /i„ = 61.96. The tail length parameter

7 is related to the parameter c in the GPD distributions as follows:

7 = -1/c (10)

(Smith, 1989).

3
2.3 Mean Recurrence Intervals of Variate X as Functions of GPD Parameters and

Exceedance Rate. The mean recurrence interval R of a given wind speed, in years,

is defined as the inverse of the probability that that wind speed will be

exceeded in any one year (see, e.g., Simiu and Scanlan, 1986, p. 525). In this

section we give expressions that allow the estimation from the GPD of the value

of the variate corresponding to any percentage point 1 -1/(AR) , where A is the

mean crossing rate of the threshold u per year (i.e. , the average number of data

points above the threshold u per year) . Set

Prob(Y < y) = 1 - 1/(AR) (11)

Using eq (1)

1 - [1 + cy/a]-i/= = 1 - 1/(AR) (12)

Therefore

y = -a[l - (AR)=]/c (13)

(Davison and Smith, 1990). The value being sought is

Xr = y + u (14)

where u is the threshold used in the estimation of c and a.

4
3. DESCRIPTION OF DE HAAN ESTIMATION METHOD

Let the number of data above the threshold be denoted by k, so that the threshold u represents the (k+l)-th highest data point(s) . We have A = k/riy^^, where nyj-s

denotes the length of the record in years. The highest, second k-th, (k+1)-

th highest variates are denoted by „, X^-i n> ^n-d^+D ^-k.n^'-^. respectively.

Compute the quantitites

1 k-1

M„(r) „ 2 {logCVi,,) - log(X„_i,,J}- r=1.2 (15)

k 1=0

The estimators of c and a are then

1 c = Mn^^^ + 1 (16)

2{1 - (M,(i))V(Mn^'^)}

a = uM^^^Vpi (17)

Pi =

' 1 a>o

•\ (18)

l(l-c) c<0

The standard deviation of the asymptotically normal estimator of c is

s.d. (6) = [(l+c2)/k]i/2 a>0 (19a)

8(l-2c) (5-llc)(l-2c)

s.d.(c) = { [(1-6)2(1-26) [4 + ]/k)^/2 (19b)

(1-36) (1-36) (1-46)

(de Haan, 1994)

.

5

T

4. WIND SPEED DATA

4.1 Uncorrelated Samples Obtained from Largest Dally Data Records. Sets of daily

fastest mile wind speeds for winds blowing from any direction were obtained from

the National Climatic Data Center, National Oceanic and Atmospheric

Administration. In most samples a number of daily fastest miles were missing. The

speeds on days with missing fastest mile data were estimated from speeds recorded

on the respective days at 3-hour intervals, using observations of the approximate

relation between these speeds and daily fastest mile speeds. Wind speeds so estimated exceeded 15.6 m/s (35 mph) only at the following stations and dates:

Boise (4/26/87, 16.1 m/s); Portland, OR (11/14/81, 19.7 m/s). Salt Lake City

(2/1/87, 16.1 m/s) and Toledo (2/6/86, 18.8 m/s). Forty-four samples were used

in the analyses. For fourteen of these samples corrections based on the largest

yearly records were effected.^ The influence of these corrections on the results

of the analyses is discussed in Section 5.5. The length of the records ranged

from 15 to 26 years, the average length being about 18.5 years.

The anemometer elevations were changed during the period of record at the

following stations: Duluth (16.2 m to 10/15/75; 6.4 m thereafter), Dayton (6.1

m to 2/4/64; 6.7 m/s thereafter), Missoula (6.1 m to 6/24/82; 9.8m thereafter),

Oklahoma City (16.8 m to 10/21/65; 6.1 m thereafter), Portland, OR (7.6 m to

3/1/73; 6.1 m thereafter), San Diego (6.4 m to 8/13/69, 6.1 m thereafter),

Toledo, OH (6.1 m to 11/1/68; 10 m thereafter) and Winnemucca (10.4 m to 4/22/66;

6.1m thereafter). For these stations the daily data were corrected to correspond

to a common 10 m elevation using the logarithmic law for open terrain. For all

other stations the anemometer elevations did not change during the period of

record and (except for Denver, where the data were also corrected to correspond

to a 10 m elevation), the original recorded data were used, that is, no elevation

correction was effected.

From samples of largest daily wind speeds we obtained as follows samples that

have reduced mutual dependence among the data. Partition the sample of daily

maxima into small periods of size equal to or larger than the duration of typical

storms in days. (A reasonable choice of the length of the period is four to eight

days.) Pick the largest value in each period. If the maxima of two adjacent

^ The following discrepancies between records of yearly maximum fastest

miles and daily maximum fastest miles were found: Cheyenne (6.1m elevation), 69 vs. 75 (72/3/6), 61 vs. 66 (1/18/75), 65 vs. 70 (6/14/76); Dayton (10 m) , 44 vs. 49 (6/27/66), 61 vs. 86 (2/16/67), 52 vs. 67 (5/14/70); Fort Wayne (6.1 m), 51 vs. 55 (8/27/65); Greenville (6.1m), 57 vs. 52 (7/15/66); Lander (6.1m), 49 vs. 61 (4/12/68); Louisville (6.1 m), 57 vs. 61 (2/15/67); Milwaukee (6.1 ra), 52 vs.

56 (6/16/73); Minneapolis (6.4 m), 52 2vs . 56 (7/10/66); Missoula (6.1m), 67 vs.

61 (7/31/83); Portland, ME (6.1 m) , 38 vs. 57 (12/24/70); Portland, OR (10 m)

,

48 vs. 42 (12/11/69); Pueblo (6.1 m), 65 vs. 70 (5/12/75); Richmond (6.1 m), 47 vs. 42 (7/11/73); Sheridan (6.1 m), 52 vs. 57 (3/2/74). For these stations the

corrections were effected by replacing the recorded daily maximum by the yearly

maximum recorded on the same day. The yearly maxima were checked for correctness

against the original charts — see Section 4.2.

7
periods are less than half a period apart, replace the smaller of the two maxima

by the next smaller value in the respective period which is at least half a period apart from the larger maximum. A data sample is thus obtained in which

adjacent data are one period apart on the average and never less than half a period apart. We show below the daily maximum fastest miles at Boise, Idaho in

the first six eight-day periods of the year 1965. The periods are separated by

vertical bars. The data selected by the procedure just described are in bold

type. In the sixth period we underlined the period maximum (26), discarded and

replaced by the next largest value (18) because of the proximity to the larger

maximum (31) of the adjacent period.

23,32,35,20,26,24,24,14 | 13,16, 5,11, 5,12,12, 7 | 6, 6, 9, 9,11,12,25,26 |

15,12,12, 7,15,12,29,10 | 7,10,15,20,20,17,24,31 | 26 , 9 , 16 , 14 , 18 , 16 , 14 , 12

|

In spite of our selection procedure, small correlations among data might subsist.

Nevertheless, we refer to a sample obtained by the selection procedure just

described as an uncorrelated data sample based on eight-day (four-day) intervals

or, for short, an eight-day (four-day) interval sample. An assessment was made

of differences between results of analyses based on four-day and eight-day

interval samples at the same station. Since in all cases the differences were

insignificant, we present in this report only results based on four—day interval

samples

.

Appendix 1 contains histograms of the full samples of daily data and of the fourday interval samples obtained from them for each of the 44 stations. Owing to the

small scale of the graphs, in some cases high wind data are not perceptible on

the daily data histograms; however, they can be seen clearly on the four-day

interval data histograms. A comparison between the histograms of the full daily

data samples and the histograms of the four-day interval samples shows that our selection procedure considerably reduces the number of lowest wind speed data.

The selection procedure also results in a shifting of the highest ordinate of the

histogram toward higher wind speeds.

4.2 Largest Yearly Data Samples. Also available were samples of largest yearly

fastest miles for winds blowing from any direction, recorded over periods of 18

to 54 years at 115 U.S. stations not subjected to mature hurricane winds. The

data in those samples were obtained and checked against original charts by M.J.

Changery, Chief, Applied Climatology Branch, National Climatic Center, National

Oceanic and Atmospheric Administration (letter to E. Simiu of December 20, 1988)

and are an update of the information included in Simiu, Changery and Filliben

(1979) .

As noted earlier, all the data samples used in our analyses are available in an anonymous data file. Instructions for accessing that file and attendant programs

for creating sets of uncorrelated data are available in Appendix 4.

8
5. ANALYSES AND RESULTS

5.1 Analysis of Uncorrelated Data Sets by the Probability Plot Correlation

Coefficient Method. Before applying the 'peaks over threshold' approach, we estimated the best-fitting distributions for the four-day samples from among a set of seven distributions or families of distributions (normal, double

exponential, lognormal , Gumbel, Frechet, Weibull, and reverse Weibull) . This

analysis was viewed as a tentative step toward understanding the probabilistic

structure of the populations from which the threshold exceedances were taken. The

estimation of the best fitting distribution was based on the probability plot

correlation coefficient (FFCC) (Filliben, 1975). As an example. Figure 1 shows

the FFCC plots for the Albany, New York four-day interval samples. For this

sample the mean and standard deviation were E(X)=10.5 m/s (23.5 mph) and

s.d.(X)=3.14 m/s (7.03 mph); for the full sample of daily data E(X)=7.8 m/s (17.5

mph) and s.d.(X)=3.1 m/s (6.98 mph). The analyses were repeated for the eight-day

samples, and the results were found to differ insignificantly from those based

on the four-day samples.

The reverse Weibull distribution was found to best fit the data in the majority

of the cases. Even in the cases where other distributions fitted the data better,

the reverse Weibull was typically very close to being the best fitting

distribution, that is, its FFCC differed only in the fourth or even fifth

significant figure from the FFCC of the best fitting distribution. We therefore

re-analyzed the eight-day interval samples by assuming that the populations for

all stations have reverse Weibull distributions with a single, site—independent

value of the tail length parameter, and site-dependent location and scale

parameters. For each station we calculated the FFCC's by assuming that the shape

parameter 7 was 1,2,3, .. .50. For samples of data based on eight-day intervals the

mean and median of the FFCC's, taken over all the stations, were largest for 7=11

and 7=13, respectively. If it were true that a reverse Weibull distribution with

a single tail length parameter characterized the extreme winds at all sites, then

our analyses would indicate that the value of that parameter is 7=12.

The assumption that there exists a universal tail length parameter for extreme

wind distributions is implicit in current practice, except that it is applied to

the Gumbel distribution (for which 7=00) . To see whether that assumption is

tenable if applied to the reverse Weibull distribution with 7=12, 44 samples

corresponding to 18—year record lengths based on 8-day intervals were generated

from reverse Weibull populations with (1) 7=8, (2) 7=12, and (3) 7=16. The number

of simulated samples for which the best fitting reverse Weibull distribution had

shape parameters with 7<12 , 13<7<20, and 7>21 are shown in Table 1. Also shown

in Table 1 are the numbers of observed samples based on 8-day intervals for which

the analysis yielded 7<12, 13<7<20, and 7>21. The results of Table 1 would

suggest that a reverse Weibull distribution with 7=12 is an appropriate model for

the populations of extreme winds representing data based on 8—day intervals

,

except for the larger number of samples with 7>20 among the observed samples than

among the simulated samples. We interpret this larger number as reflecting the

relatively frequent presence of outliers among the observed samples. This may

suggest that, because wind speed populations are mixed (in addition to extremes

they include ordinary winds whose meteorological structure may differ from that

of the extremes) , a sample taken from such a population is likely not to be a sound basis for inferences on extremes. It is therefore desirable to "let the

tails speak for themselves." The application of the GPD-based 'peaks over threshold' approach is an attempt to do just this.
1

I-l>c

T31 uOuo

(4-1 npo u

1-1 o a >H

>^

4J (U r-l iH

•iHo .o u

p-l < M

10
Table 1. Comparison of Results for Simulated and Observed Samples

7<12 13<7<20 7>21

Simulated samples, 7= 8 44 0 0

Simulated samples, 7=12 26 15 3

Simulated samples. 7-16 7 22 15

Observed samples* 25 10 9

*Statlons for which ^>Z1 were: Green Bay, Greensboro, Huron, Lansing, Louisville, Macon, Moline, Portland,

OR, San Diego. Those for which 13<7<20 were: Binghamton, Fort Smith, Fort Wayne, Grenville,

Milwaukee, Minneapolis, Springfield, Topeka, Tucson, Yuraa.

5.2 Estimation of Tail Length Parameter by 'Peaks Over Threshold' Analyses of

Uncorrelated Data Samples. We applied the de Haan estimation method (Section 3)

to the four-day interval samples using, for each sample, a highest threshold such

that the number of its exceedances be equal to, or larger than and as close as possible to, 16; any higher threshold was deemed to result in data sets too small

to yield useful statistics. Denoting a sample's maximum threshold by u^iax. the

next higher thresholds we considered were u^ayr^> ^nax~2 , . . . ,Uj^^y-2^ . The

estimated values (point estimates) of c are shown for each station in the plots

of Fig. 2. Also shown on the plots are 95 percent confidence bounds (i.e., lines

corresponding to c ± 2s.d.(c)). On the horizontal coordinate axis of each plot

we indicate the thresholds, in miles per hour, and the size of the data samples

(i.e., the number of exceedances) for each threshold.

Figure 3 is an example of a similar plot (c versus number of threshold

exceedances) presented for a different type of extreme value problem by de Haan

(1990). Like the plots of Fig. 2, this plot exhibits fairly strong fluctuations

in the region of the highest thresholds where the sample size is relatively

small. In the region of the smaller thresholds the 95 percent confidence bounds

become narrower — a result of the increasing sample size — but a bias sets in,

which is due to the inclusion in the data samples of data not properly belonging

to the tails. In de Haan's judgment, "it looks from the graph as if the value c=0

is not a bad choice in this case."

We propose to apply this type of qualitative judgment to the plots of Fig. 2. For

example, it would appear that, for Abilene, c<0 , perhaps c=-0.25. The plots of

Fig. 2 indicate that c<0 for most, though not all, stations. This in itself is an interesting result, insofar as it would indicate that in most cases extreme

wind distribution tails are indeed finite.

Let us again assume for a moment that extreme wind speeds in regions not

subjected to mature hurricanes are described by a reverse Weibull distribution

with site-dependent location and scale parameters and site-independent tail

length parameter c. The weighted mean of c may be written as a function of

threshold order q as (Gross et al . , 1995):

44 44

Cwq = { 2 iiq/Siq^}/ S 1/Siq2 (20)

i=l i=l

where the index q = 1,2, .

, .25 is the order of the highest, second highest, .

. ,25-

the highest threshold for the 44 samples being analyzed, and c^g, s^g are the

estimated value of c and the estimated standard deviation of c for station i,

based on the threshold of order q. (Recall that the threshold corresponding to

11
3NWH3a ° ONWHBQ ° 3 NWH3a

12
I

j

13
o V) T- to tnotn»-in T-moini-mNtn

oNWHaa oNvvHaa ONWHaa

' o d ' o o

0 NWHaa 0 NvvHaa o NvvHaa

14
T3c

•Hcou

CMwoM

15
0 100 200 300 400 500 600 700 800 900 lOOO

FIGURE 3. Point estimates of tail length parameters and 95% confidence bounds

versus number of exceedances (de Haan, 1990)

.

q = 1 for each station was so chosen that at least 16 data points exceed that

threshold.) The plot of c„q is shown in Fig. 4 and, in our opinion, tends to

confirm the view that, at most if not all stations, the estimated value of c is

negative, perhaps c~-0.2 or c~-0.25. We note that, as suggested by Monte Carlo

simulations (Gross et al .

, 1994), for sample sizes not exceeding about 10 percent

of the total number of data, the bias in the estimation of c is about -0.05, that

is, sufficiently small not to invalidate our judgement that, predominantly,

c<0.2

5.3 Estimation of Tall Length Parameter by 'Peaks Over Threshold' Analyses of

Largest Yearly Data Samples. Appendix 2 includes point estimates of c for each

of the 115 stations for which largest yearly speeds were available. Also shown

on the plots are 95 percent confidence bounds (i.e., lines corresponding to c ±

2s.d.(c)). The estimates are plotted against the threshold speed and the number

of exceedances of the threshold, as in Fig. 2. For these plots the larger samples

are not likely to be affected by bias, since the lowest wind speed in those

samples is itself a largest yearly wind, and hence it will be within or close to

the distribution tail. Though the plots are not always easy to interpret, in our opinion they confirm the view that at most stations c is negative. Figure 5,

which shows the weighted average of the estimated tail length parameter for the

115 data samples (eq (20)), lends further credence to this view.

^We note a typographical error in Table 5, p. 147 of Gross et al . (1994).

In the last line of the Table the population value of the parameter c should be - 0.275 (as in line 7 of p. 142), rather than -0.50.

16
in

CMoCM oX

0)

LU

CCI

-o1 uo

<r

<!•U>On

<u •u -a

•l-l (U

l-lO

0)

m

4-> 0)

cdB

•i-i 4-) W <w

(U o

}-l J-l

(1> (U •U T)

Q) S-i e o

Vj

w

n3(X «

S-i x:

•p ?>

CO

r-l r-l

cx, r-l e

•H CO

to cn

i-lCO

O >

5-4 c 0)

CO 4J

CD

C s

in

EdM

CMd

COdI d

dI d 0 dO 30Va3AV

17
inCMoCM

in

in

OX

(/) UJ

CEX

Ipw

0)bO

i-i in

r-H i-l>o

x>

bO r-Ho

<U& CO

(1) CO

(U +J 4J e I4-I •H 0

4J

CO }-l (U (U •0 U >-l 0

•p

CO s

CO

}-i to (U> ^ CO 4J (1) tJOi-H

CD e

iH Cd

CO

I-l •H «d

cd 4J

•U CO CHo >^

iH

C

U

cfl cd

<U (U M

CMd

COd 9

(O

d

CO 9 0 do aovuaAv

18
5.4 Estimation of Speeds with Specified Mean Recurrence Intervals by 'Peaks over Threshold' Analyses of Uncorrelated Data Samples. Appendix 3 contains plots of

point estimates of the extreme winds with mean recurrence intervals of 100, 1,000

and 100,000 years. The estimates were based on four-day interval samples at each

of the 44 stations. They are plotted against the threshold speed and the number

of exceedances of the threshold, as in Fig. 2.

We reproduce in Fig. 6 an example of a plot where the quantile fluctuates

strongly as a function of threshold (de Haan, 1990). De Haan comments: "if one would be forced to give a point estimate a value of 5.1 m. . . would not be

unreasonable." The comment is indicative of the spirit in which results based on

the 'peaks over thresholds' method must be interpreted in cases where fairly

large fluctuations are present, as is the case for Fig. 6 and many of the plots

in Appendix 3. We do not attempt in this report to estimate extreme wind speeds

for various mean recurrence intervals. Rather, having found that the tail length

parameter c of the GPD is negative for the majority of the stations, we assess

in the next section the potential implications of this finding for the estimation

of load factors.

j 300 ^

J 200:

100:

100 200 300 400 500 600 700 800 900 lOM

FIGURE 6. Quantiles: point estimates and 95% confidence bounds

versus number of exceedances (de Haan, 1990).

5.5 Influence of Data Errors on Analysis Results. The footnote to Section 4.1

lists errors in the recorded daily data found at fourteen stations, and the

respective corrected values. Figures 2 and 4 and Appendix 3 are based on the

corrected data sets (these are marked in Fig. 2 and Appendix 3 by the suffix

19
"corr"). Figures 7 and 8 include, respectively, plots of estimated tail length

parameters and speeds with various mean recurrence intervals for twelve of these

stations and are based on the uncorrected data. Comparisons between these plots

and their counterparts in Fig. 2 and Appendix 3 show that estimates of speeds

corresponding to various mean recurrence intervals are in some cases affected

fairly significantly by the errors in the data. This is true to a much lesser

extent for the tail length parameters. The plot of the weighted mean over all 44

stations, computed from results obtained by using the uncorrected data, was indistinguishable from Fig. 4.

20
o

x:

V)

(Uu

4-> twO

>-l <U J3 s ° s ° 0 NWHBQ ° S " S ° 0 NWHaa °

SQ ° a c

,-

V CO •u

CO T3

r—1 o •d

0)

(U o u 0) X. S-i u !-l o w oc w

>-l a;> M

1—1 T3c e

to

O cn

r-l

OJ CO o > c

0) Q) T) U •i-l C

i-t co >^ o CO T31 in 5-1 On

pu fo

CO c

«o o

5-1 T3

0) 0) •u tn 0) CO E

XI

cd

S-i CO

M

a a>oc o CO T) M 0) •U (U o O

r-(

X

0)

w

OS

21
o

-100,(

-

1,00( 100

1 i

;

Y'

V'

. \ \

dX

22
6. LOAD FACTORS FOR WIND-SENSITIVE STRUCTURES

Extreme wind loads used in design include nominal basic design wind loads (e.g.,

the 50-yr wind load) and nominal ultimate wind loads. A basic design wind load

is an extreme load with specified probability of being exceeded during a basic

time interval. In the United States that interval is usually 50 years. A basic

design load with a 50-year mean recurrence interval has a probability of almost

two thirds of being exceeded during a 50-year period.

A structure or element thereof is expected to withstand loads substantially in excess of a 50- or 100-year wind load without loss of integrity. The wind load

beyond which loss of integrity can be expected is referred to as the nominal

ultimate wind load. The nominal ultimate strength provided for by the designer

is based on a nominal ultimate wind load equal to the design wind load times a wind load factor. This statement is valid for the simple case where wind is the

dominant load. It needs to be modified if load combinations are considered, but

for clarity we refer here only to this case.

The load factor should be selected so that the probability of occurrence of the

nominal ultimate load is acceptably small. This probabilistic concept is

important from an economic or insurance point of view. To the extent that

evacuation or similar measures cannot be counted on to prevent loss of life, it

is also important from a safety point of view.

A probabilistic approach has proven helpful in a number of cases, particularly

for relative assessments of alternative design provisions, for example for mobile

homes. However, in most cases the difficulties of obtaining wind load factors by

probabilistic methods have proven to be substantial if not prohibitive. For this

reason code writers have largely relied on wind load factors implicit in

traditional codes and standards. For example, the American Society of Civil

Engineers Standard A7-93 (1993) specifies a wind load factor of 1.3. In a very

large number of applications the wind load is proportional to the square of the

wind speed, so that a basic design wind speed and a nominal ultimate wind speed

may be defined that are proportional to the square root of the basic design wind

load and the square root of the nominal ultimate wind load, respectively. For

example, for Lander, Wyoming, the American Society of Civil Engineers Standard

A7—93 (1993) specifies a basic design 50-year design speed of 35.8 m/s (80 mph

fastest-mile) at 10 m (33 ft) elevation. The corresponding nominal ultimate wind

speed would then be 1.3^/235 8=40.8 m/s (91.2 mph).

Reliance on traditional code values is part of the process sometimes referred to as "calibration against existing practice." Traditional codes were generally

adequate for many types of structures, but questions remain on whether safety

margins implicit in those codes may be applied to modern structures, which can differ substantially from their predecessors in their materials and

design/construction techniques. For this reason an assessment of wind load

factors used in codes and standards would be desirable. For example, one would

wish to answer the question: what is the approximate mean recurrence interval of

the nominal ultimate wind speed?

The answer to this question depends strongly upon the probability distribution

assumed to best fit the extreme wind speeds. For example, a FFCC analysis of

largest yearly fastest-mile speeds recorded at Denver between 1951-1977, based

on the assumption that the best fitting distribution is Gumbel, yielded a 27.9

m/s (62 mph) estimate of the 50-year wind speed at 10 m above ground. The

23
corresponding nominal ultimate wind speed would be 1 . 3-'-/^27 . 9=31 . 8 m/s (71.0

mph) , to which there would correspond, under the Gumbel assumption, a mean

recurrence interval of about 500 years (Simiu, Changery and Filliben, 1979). If

taken at face value this would be an alarmingly short recurrence interval, since

it would entail an unacceptably large probability of exceedance of the nominal

ultimate wind load during the life of the structure.

However, the 500 years mean recurrence interval is based on the Gumbel model.

Since our results support the assumption that, predominantly, the appropriate

model is a reverse Weibull distribution, rather than a Gumbel distribution, we wish to answer the question: what is the mean recurrence interval corresponding

to 1.3^^^ times the wind speed with a 50-year mean recurrence interval? For

Denver, if one had to estimate the tail length parameter c from the plots of Fig.

2 and Appendix 2 , and the 50-year speed from the plot of Appendix 3 , one might

choose, say, c=-0.2 (a conservative choice: according to the plots c is likely

to be somewhat lower, that is, the distribution tail is likely to be somewhat

shorter than that corresponding to c=-0.2), and X5o=26.8 m/s (60 mph). For a threshold of 16.5 m/s (37 mph) — a value that is roughly consistent with these

choices, see Denver plot. Fig. 2 — we have A=139/15=9 . 27/year . Asstaming X5o=26.8

m/s (60 mph), it would follow from eqs (13) and (14) a=2 . 5 m/s (5.6 mph). The

estimated maximum possible wind speed corresponding to the parameters c=—0.2 and

a=2.5 m/s (5.6 mph) is obtained by letting R «» in eqs (13, 14). Its value is

Xn,ax=u—a/c = 16.5+2.5/0.2 = 29.0 m/s (65 mph). The estimated mean recurrence

interval of the nominal ultimate wind speed 1 . 3^''^X5o=l . 3^'^x26 . 8=30 . 6 m/s (68 mph)

is therefore infinity (i.e. , such a wind speed is estimated to never occur). This

estimate is of course subject to sampling errors: the actual maximum possible

wind speed may be higher than 29.0 m/s (65 mph) , and the mean recurrence interval

of the 30.6 m/s (68 mph) speed may in fact be finite, though likely much longer

than 500 years. In spite of the uncertainty inherent in our estimates, our result

suggests that a load factor of 1.3 — specified in the ASCE A7-93 Standard on the

basis of practical experience — is in fact adequate from a probabilistic point

of view. This is contrary to what would be concluded if the analysis were based

on the assumption that the Gumbel distribution holds.

•^In this instance code writers attended to the apparent insufficiency of the

nominal ultimate design speed by substantially inflating the value of the 50-year

wind speed: the ASCE A7-93 Standard specifies for Denver a 35.8 m/s (80 mph) 50-

year speed at 10 m above ground. Since the estimated standard deviation of the

sampling error for the estimated 50—year wind is Sg g = 1.3 m/s (3.0 mph) (Simiu,

Changery and Filliben, 1979), the specified 35.8 m/s (80 mph) value differs from

the estimated 27.9 m/s (62 mph) value by almost six standard deviations. One may

view the actual load factor implicit in the ASCE A7-93 Standard as being equal

to (1 . 3) (80/62)^=2 . 14 , rather than just 1.3. The mean recurrence interval for the

40.7 m/s (91 mph) (i.e., 1.3^/^80 = 2.14^/2x62) nominal ultimate load inherent in

the ASCE A7-93 Standard provisions for Denver, based on the Gumbel model, would

be about 80,000 years (Simiu, Changery and Filliben, 1979), which is much more acceptable than 500 years.

24
7. CONCLUSIONS

In this report we presented estimates of the tail length parameters of extreme

wind distributions for non-tornadic winds blowing from any direction in regions

unaffected by mature hurricanes. In our opinion, these estimates support the view

that the reverse Weibull distribution is an appropriate probabilistic model in

most if not all cases. They also suggest that load factors for wind sensitive

structures specified by current standards provide for reasonable safety margins

against wind loads, and that the adoption of the Gumbel model likely results in an unrealistic assessment of structural reliability under wind loads.

However, owing to fluctuations of our estimates with the threshold value, it is

difficult to provide reliable quantitative estimates of the tail length

parameters. This difficulty is even more pronounced for quantile estimates.

We tentatively ascribe these difficulties to the relatively small size of our samples (15 to 26 years). It would therefore be desirable to assemble data for

longer records than those used in this report. In addition, more efficient

estimation methods should be developed. Efforts to develop such methods are currently in progress (Coles, 1994).

25

REFERENCES

American National Standard A58. 1-1972 (1972), American National Standards

Institute, New York.

ASCE Standard A7-93 (1993), American Society of Civil Engineers, New York.

Bingham, N.H. (1990), "Discussion of the Paper by Davison and Smith," Journal of

the Royal Statistical Society . Series B, 52, p. 431.

Castillo, E. (1988), Extreme Value Theory in Engineering . Academic Press, New

York.

Coles, S. (1994), private communication.

Davison, A.C., and Smith, R.L. (1990), "Models of Exceedances Over High

Thresholds," Journal of the Royal Statistical Society . Series B, 52, pp. 339-

442.

de Haan, L. (1990), "Fighting the Arch-enemy with Mathematics," Statistica

Neerlandica 44, pp. 45-68.

de Haan, L. (1994), "Extreme Value Statistics," in Extreme Value Theory and

Applications . Vol. 1 (J.Galambos, J. LechnerandE. Simiu, eds . ) , Kluwer Academic

Publishers, Dordrecht and Boston, 1994.

Ellingwood, B. et al . (1980), Development of a Probability-Based Load Criterion

for American National Standard A58 . NBS Special Publication 577, National Bureau

of Standards (U.S.), Washington, DC.

Filliben, J.J., "The Probability Plot Correlation Plot Test for Normality,"

Technome tries . 17, pp. 111-117.

Gross, J., Heckert, A. Lechner, J., and Simiu, E. , (1994), "Novel Extreme Value

Procedures: Application to Extreme Wind Data," in Extreme Value Theory and

Applications . Vol. 1 (J.Galambos, J. Lechner and E. Simiu, eds.), Kluwer Academic

Publishers, Dordrecht and Boston, 1994.

Gross, J.L., Heckert, N.A. , Lechner, J. A. and Simiu, E. (1995), "A Study of

Optimal Extreme Wind Estimation Procedures," Proceedings. Ninth International

Conference on Wind Engineering . New Delhi.

Hosking, J.R.M. and Wallis, J.R. (1987), "Parameter and Quantile Estimation for

the Generalized Pareto Distribution," Technometrics . 29, pp. 339-349.

Johnson, N.L. and Kotz, S. (1972), Distributions in Statistics: Continuous

Multivariate Distributions . Wiley, New York.

Pickands, J. (1975), "Statistical Inference Using Order Statistics," Annals of

Statistics . 3, 119-131.

Simiu, E., Changery, M.J. and Filliben, J.J. (1979), Extreme Wind Speeds at 129

Stations in the Contiguous United States . NBS Building Science Series 118,

National Bureau of Standards (U.S.), Washington, DC.

27
Simiu, E. , Filliben, J.J. and Bietry, J., (1978) "Sampling Errors in the

Estimation of Extreme Wind Speeds," Journal o£ the Structural Division. ASCE . 104, pp. 491-501.

Simiu, E. , and Scanlan, R.H. (1986), Wind Effects on Structures . Second Edition,

Wiley-Interscience , New York, 1986.

Smith, R. L. (1989), Extreme Value Theory , in Handbook of Applicable Mathematics . Supplement edited by W. Ledermann, E. Lloyd, S. Vajda, and C. Alexander, pp. 437—

472, John Wiley and Sons, New York.

28
APPENDIX 1

Histograms of daily largest wind speeds (top) and

histograms of four-day interval uncorrelated wind

speeds (bottom)

29
30 i ' L
31
L

32
£

I

'

I

'

I

'

I

' I

o

?

o

-

" 1 -8 i

— S 2:O1-

(5z

*^ :

15

o

' 1 ' 1 ' 1

' S S S 8

> Ul O lO N ^ ^

— o

o

'

1 ' 1 ' 1 ' 1

' 1

1 S § S 1 s

°

33
34
35
36
37
38
39
- 3

40
41
§oo§§§SSS^ 0)(0(^u>io^rteMT- s s s

CM ^ T42
43
g 8 g g ° gSgSS

44
[

45
46
47
T.

48
49
50
t:

51

APPENDIX 2

Plots of parameter c and 95% confidence bounds

versus threshold and ntomber of threshold exceedances

(based on samples of largest yearly wind speeds)

1 53
54
i i

T^csie^^ ^ o m ^ o n y

0 NVVH3Q 0 NVVHaa 0 NVVH3a

55
0 NVVHSa 3 NvvHaa

0 NVVHSa

0 NVVHBQ

56
i i i

—I , r—I 1

1

I I h—'1 ' 1 I

' r—• 1 ' 1 I ' I—'

I— r—^>—I——I

'—I>—

I

ONVVHiQ ONVVHiQ " 0 NVVHSQ

1 1

ONvvHaa ONVVHaa oNvvHaa

57
» o

58
59
0 NVVHSa 3 NVVH30

I I

0 NVVHaa 3 NVVH30

60
61
lU>

lU

3 NVVHaa ' 0 NVVHBO

0 NVVHaa 0 NVVHia

0 NVVHaa

-I - "•" I I w • i O T- f\4 O

3 NVVHia

o »- WW

"WW w w

.«>WW

T «? 0 NVVHaa

ONVVHBQ

62
63

APPENDIX 3

Plots of point estimates

for 100-year, 1000-year and 100 000-year speeds

versus threshold and number of threshold exceedances

(based on four- day interval data samples)

65
ax ux ax

<

UJ>

<

ux ux

*" ax

66
ux ux ux

67
o

o

=>

1 ^ III

1

1

,

1 1 I

<r< «

/it § s s s s s

ax

68
O O Q O O

CO r> <o m ^

ux S S S 8

8QOOOOOQ Q> to *o tf> ^ n QOOOQ otnouio UX dX

69

APPENDIX 4

Instructions for accessing data sets and attendant programs

NOTE. Only corrected data are included in the data files.

ftp enh.nist.gov (or: ftp 129.6.16.1)

>user anonymous

enter password >guest

>cd emil/datasets (to access data)

>cd emil/programs (to access programs)

>prompt off

>mget * (this copies all the data files)

>dir (this lists the available files)

>get <enh name> <local name> (this copies a specific file; example: get boise.id

boise . id)

>quit

71

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