NIST BUILDING SCIENCE SERIES 174

Extreme Wind Distribution Tails:A ‘Peaks Over Threshold’ Approach

E.Simiu1andN.A.Heckert2

1BuildingandFireResearchLaboratoryNationalInstituteofStandardsandTechnologyGaithersburg,MD 20899-0001

and

DepartmentofCivilEngineeringTheJohnsHopkinsUniversityBaltimore,MD 21218

2ComputingandAppliedMathematicsLaboratoryNationalInstituteofStandardsandTechnologyGaithersburg,MD 20899-0001

Sponsoredinpartby:

NationalScienceFoundationArlington,VA 22230

March1995

U.S.DepartmentofCommerceRonaldH.Brown,Secretary

TechnologyAdministrationMary L.Good,Under Secretaryfor Technology

NationalInstituteofStandardsandTechnologyAratiPrabhakar,Director

National Institute of Standards and Technology Building Science Series 174Natl. Inst. Stand. Technol. Bldg. Sci. Ser. 174,72 pages (Mar. !995)

CODEN: NBSSES

U.S. GOVERNMENT PRINTING OFFICEWASHINGTON: 1995

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402–9325

TABLE OF CONTENTS

wLIST OF F’IGURES.......................................................... “

ABSTRACT ..........................................................~......

1. INTRODUCTION ...........................................................

2. ~PEAKS OVER THRESHOLD’ APPROACH .......................................

2.1 Generalized Pareto Distribution. ................................2.2 Gumbel and Reverse Weibull Distributions ........................2.3 Mean Recurrence Intervals of Variate X as Functions of GPD

Parameters and Exceedance Rate ..................................

3. DESCRIPTION OF DEHAAN ESTIMATION METHOD ...............................

4. WIND SPEED DATA ........................................................

4.1 Uncorrelated Samples Obtained from Largest Daily Data Records. ..4.2 Largest Yearly Data Samples .....................................

5. ANALYSES AND RESULTS. ..................................................

5.1

5.2

5.3

5.4.

5..5

Analyses of Uncorrelated Data Sets by the Probability PlotCorrelation Coefficient Method ..................................Estimation of Tail Length Parameter by ‘Peaks over Threshold’Analyses of Uncorrelated Data Samples ...........................Estimation of Tail Length Parameter by ‘Peaks over Threshold’Analyses of Largest Yearly Data Samples .........................Estimation of Speeds with Specified Mean Recurrence Intervalsby ‘Peaks over Threshold’ Analyses of Uncorrelated Data Samples.Influence of Data Errors on Analysis Results ....................

6. LOAD FACTORS FOR WIND-SENSITIVE STRUCTURES .............................

7. CONCLUSIONS ........................$.............,,......$$...........O

REFERENCES ........................................................-.....””

APPENDIX 1 --

APPENDIX 2 —

APPENDIX 3 —

Histograms of daily largest wind speeds and histogramsof four-day interval uncorrelated wind speeds. ..............

Plots of parameter & and 95% confidence bounds versusthreshold and number of threshold exceedances (based onsamples of largest yearly wind speeds) ......................

Plots of point estimates for 100–, 1OOO– and 100 000-ye=speeds versus threshold and number of threshold exceedances(based on four-day interval samples) ........................

Iv

vi

1

3

33

4

5

7

78

9

9

11

16

1919

23

25

27

29

53

65

iii

-

APPENDIX 4 -- Instructions for accessing data sets and attendant programs. 71

LIST OF FIGURESI

Figure 1. Probability plots for four-day interval sample, Albany,New York ....................................................... 10

Figure 2. Plots of parameter & and 95% confidence bounds versusthreshold and number of threshold exceedances (based onfour-day interval samples) ..................................... 12

Figure 3. Point estimates of tail length parameters and 95% confidencebounds, versus number of exceedances (de Haan, 1990) ........... 16

Figure 4. Mean of tail length parameter estimates weighted over 44four-day interval samples versus order of threshold ............ 17

Figure 5. Mean of tail length parameter estimates weighted over 115largest yearly data samples versus order of threshold ........... 18

Figure 6. Quantiles: point estimates and 95% confidence bounds versusnumber ofexceedances (de Haan, 1990) ........................... 19

Figure 7. Plots of parameter 6 and 95% confidence bounds versusthreshold and number of threshold exceedances (based onfour-day interval samples, uncorrected data) .................... 21

Figure 8. Point estimates for 100-, 1OOO- and 100 ()()()-year speedsversus threshold and number of threshold exceedances (basedon four-day interval samples, uncorrected data). ,.,............ 22

lFigures in Appendices are not included in this list.

iv

E. Simiu acknowledges with thanks partial support by the National ScienceFoundation (Grant No. CMS-9411642 to Department of Civil Engineering, The JohnsHopkins 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 ofStandards and Technology, who suggested the collection of large samples of dailydata 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 theNational Institute of Standards and Technology for advice on testing thehypothesis that a universal reverse Weibull tail length parameter exists; andDrs. J.L. Gross and J.L. Lechner of the National Institute of Standards andTechnology for valuable collaboration on eailier phases of this project, some ofwhich are reviewed in this report.

COVER PICTURE:

Honor6 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 appropriateextreme wind speed model by performing statistical analyses based on the ‘peaksover threshold’ approach. We use the de Haan method, which was found in previousstudies to perform about as well or better than the Pickands and Cumulative MeanExceedance methods, and has the advantage of providing estimates of confidencebounds. The data are taken principally from records of the largest daily windspeeds obtained over periods of 15 to 26 years at 44 U.S. weather stations inareas not subjected to mature hurricane winds. From these records we createsamples with reduced mutual correlation among the data. In our opinion, theanalyses provide persuasive evidence that extreme wind speeds are describedpredominantly by reverse Weibull distributions, which unlike the Gumbeldistribution have finite upper tail and leadto reasonable estimates of wind loadfactors. Instructions are provided for accessing the data and attendantprograms.

Key words: Building technology; building (codes); climatology; extreme valuetheory; load factors; structural engineering; structural reliability; thresholdmethods; wind (meteorology).

vi

1. INTRODUCTION

A fundamental theorem in extreme value theory states that sufficiently largevalues of independent and identically distributed variates are described by oneof three extreme value distributions: the Fr6chet distribution (with infiniteupper tail), the Gumbel distribution (whose upper tail is also infinite, butshorter than the Frechet distribution’s), and the reverse (negative) Weibulldistribution, whose upper tail is finite (Castillo, 1988).

The wind loading provisions of the American National Standard ANSI A58.1-1972were based on the assumption that a Frechet distribution best fits extreme windspeeds blowing from any direction in regions not subjected to mature hurricanewinds. An extensive study concluded, however, that the Gumbel distribution is amore appropriate model (Simiu, Filliben and Bi&try, 1976). It is a physical factthat extreme winds are bounded. Their probabilistic model should reflect thisfact. To the extent that an extreme value distribution would be a reasonablemodel of extreme wind behavior, one would intuitively expect the best fittingdistribution to have finite tail, that is, tobe a reverse Weibull distribution.

In addition to the certainty that wind speeds are bounded, there is at least oneother indication — albeit indirect — that the Gumbel model might be aninappropriate model of extreme wind behavior. Estimated safety indices for wind-sensitive structures based on the Gumbel model imply unrealistically high failureprobabilities (Ellingwood et al., 1980). This may be due, at least in part, tothe 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 isan appropriate extreme wind speed model by performing statistical analyses basedon the ‘peaks over threshold’ approach. This approach enables the analyst to useall the data exceeding a sufficiently high threshold, and is more effective thanthe classical approach, which uses only the largest value in each of a number ofbasic comparable sets called epochs (typically, for extreme wind analysis anepoch consists of one year). To illustrate this point, consider, for example, twosuccessive years in which the respective largest wind speeds are 30 m/s and 45m/s. Assume that in the second year winds with speeds of 31 m/s, 37 m\s, 41 m/sand 44 m/s were also recorded (at dates separated by sufficiently long intervalsto view the data as independent). For the purposes of threshold theory, for a 30m/s threshold the two years would supply six data points. The classical theorywould make use of only two data points. It may be argued that, by choosing asomewhat lower threshold, the number of data points used in the analysis couldbe considerably larger than six in our example. However, in viewof the theory’sbasic assumption that the threshold is high, excessive loweringof t.e thresholdwould introduce a strong bias. Simulations reported by Gross et al. (1994)suggest that, in samples taken from normal or extreme value populations, optimalresults are obtained if the threshold is chosen so that the number of exceedancesis of the order of ten per year.

Given a sample of data exceeding a sufficiently high threshold, the analyst usingthe ‘peaks over threshold’ approach must choose an appropriate estimation method.In this report we use the estimation method proposed by de Haan (1994). Ourchoice is based on two reasons. First, Monte Carlo simulations suggest that the

1

de Haan method performs about as well or better. than two available alternativemethods, the Pickands method and the Cumulative Mean Exceedance method (Gross etal, 1994). Second, the de Haan method has the advantage of providing estimatesof confidence bounds.

Data used in this report are taken principally from records of the largest dailywind speeds obtained over periods of 15 to 26 years at 44 U.S. weather stationsin areas not subjected to mature hurricane winds. A storm system usually affectsa given location for longer than one day, so that wind speed data recorded on twoor even more consecutive days are not necessarily independent. We describe thedata samples and our procedure for creating, from the samples of largest dailyspeeds, samples with reduced mutual correlation among the data, In addition tosamples of daily data we describe and analyze 115 samples consisting only oflargest yearly speeds recorded over periods of 18 to 54 years at locations notsubjected to mature hurricane winds. To our knowledge no tornado winds haveaffected any of our data. All the data samples used in our analyses are availablein an anonymous data file. Instructions for accessing that file and attendantprograms are available in Appendix 4.

In our opinion, the results presented in this report provide persuasive evidencethat extreme wind speeds of extratropical origin and excluding tornadoes aredescribed predominantly by reverse Weibull distributions. This result is initself useful from a structural engineering viewpoint, and we discuss itspotential implications for the estimation of load factors for wind-sensitivestructures. Our analyses also suggest that estimates of extreme wind speeds basedon the reverse Weibull model may not be obtainable with sufficient confidencefrom the information and by the method used in this report. This suggests theneed for (i) data samples based on longer recording periods and (ii) moreefficient 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 Haanmethod is reviewed briefly in Section 3. The data used in the analyses aredescribed in Section4. Section5 includes the main results of our investigation,Section 6 discusses the load factors issue. Section 7 presents our conclusions.

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 factthat exceedances of a sufficiently high threshold are rare events to which thePoisson distribution applies. The expression for the GPD is

G(y) = Prob[YS y] = l-([l+(cy/a)]-l/c) a>O, (l+(cy/a))>O (1)

Equation (1) canbe used to represent the conditional cumulative distribution ofthe excess Y = X - u of the variate X over the threshold u, given X > u for usufficiently large (Pickands, 1975). The cases c>O, c=O and c

2.3 Mean Recurrence Intervals of Variate X as Functions of GPD Parameters andExceedance 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 beexceeded in any one year (see, e.g. , Simiu and ScanIan, 1986, p. 525). In thissection we give expressions that allow the estimation from the GPD of the valueof the variate corresponding to any percentage point 1 –l/(AR), where A is themean crossing rate of the threshold u per year (i.e., the average number of datapoints above the threshold u per year). Set

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

Using eq (1)

1 – [1 + cy/a]-II. = I – l/(,uL) (12)

Therefore

y =–a[l – (AR)c]/c

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

XR= y + U

(13)

(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 thresholdu represents the (k+l)-th highest data point(s). We have A = k/~,=, where ~r~denotes the length of the record in years. The highest, second, ..,, k-th, (k+l)-th highest variates are denoted by &,n, ~.l,n, ~-(k+l),n,~.km~=u, respectively.Compute the quantities

~(r) . ~ ‘~1k i=o

The estimators of c

{lWG%-l,J - log(%-k,.))’ r-l,2

and a are then

16==~(1)+1- .--———.----_—

2{1 - (%(1))2/(%(2)))

a ==~(l)/pl

{

1 ;>()

PI =1(1–;) ;

4. WIND SPEED DATA

4.1 Uncorrelated Samples Obtained from LargestDaily Data Records. Setx of dailyfastest mile wind speeds for winds blowing from any direction were obtained fromthe National Climatic Data Center, National Oceanic and AtmosphericAdministration. Inmost samples anumber of daily fastest miles were missing. Thespeeds on days with missing fastest mile data were estimated from speeds recordedon the respective days at 3-hour intervals, using observations of the approximaterelation between these speeds and daily fastest mile speeds. Wind speeds soestimated 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 usedin the analyses. For fourteen of these samples corrections based on the largestyearly records were effected. 1 The influence of these corrections on the resultsof the analyses is discussed in Section 5.5. The length of the records rangedfrom 1,5to 26 years, the average length being about 18.5 years.

The anemometer elevations were changed during the period of record at thefollowing stations: Duluth (16.2 m to 10/15/75; 6.4 m thereafter), Dayton (6.1m to 2/4/64; 6.7 m/s thereafter), Missoula (6.1 m to 6/24/82; 9.8 m thereafter),Oklahoma City (16.8 m to 10/21/65; 6.1 m thereafter), Portland, OR (7.6 m to3/1/73; 6.1 m thereafter), San Diego (6.4 m to 8/13/69, 6.1 m thereafter),Toledo, OH (6.lmto 11/1/68; 10m thereafter) and Winnemucca (10.4mto4/22/66;6.1 m thereafter). For these stations the daily data were corrected to correspondto a common 10 m elevation using the logarithmic law for open terrain. For allother stations the anemometer elevations did not change during the period ofrecord and (except for Denver, where the data were also corrected to correspondto a 10 m elevation), the original recorded data were used, that is, no elevationcorrection was effected.

From samples of largest daily wind speeds we obtained as follows samples thathave reduced mutual dependence among the data. Partition the sample of dailymaxima into small periods of size equal to or larger than the duration of typicalstorms in days. (A reasonable choice of the length of the periodis four to eightdays.) Pick the largest value in each period. If the maxima of two adjacent

1 The following discrepancies between records of yearly maximum fastestmiles and daily maximum fastest miles were found: Cheyenne (6.1 melevation), 69vs. 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), 51vs. 55 (8/27/65); Greenville (6.lm), 57vs. 52 (7/15/66); Lander (6.lm), 49vs.61 (4/12/68); Louisville (6.1 m), 57 vs. 61 (2/15/67); Milwaukee (6.1 m), 52 vs.56 (6/16/73); Minneapolis (6.4m), 52 2VS. 56 (7/10/66); Missoula (6.lm), 67vs.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 thecorrections were effected by replacing the recorded daily maximum by the yearlymaximum recorded on the same day, The yearly maxima were checked for correctnessagainst the original charts — see Section 4.2.

7

periods are less than half a period apart, replace the smaller of the two maximaby 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 whichadjacent data are one period apart on the average and never less than half aperiod apart. We show below the daily maximum fastest miles at Boise, Idaho inthe first six eight–day periods of the year 1965. The periods are separated byvertical bars. The data selected by the procedure just described are in boldtype. In the sixth period we underlined the period maximum (26), discarded andreplaced by the next largest value (18) because of the proximity to the largermaximum (31) of the adjacent period.

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

15,12,12, 7,15,12,29,10 I 7,10,15,20,20,17,24,31 I 26,9,16,14,18,16,14,121

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 anuncorrelated data sample based on eight–day (four-day) intervalsor, for short, an eight–day (four–day) interval sample. An assessment was madeof differences between results of analyses based on four–day and eight–dayinterval samples at the same station. Since in all cases the differences wereinsignificant, we present in this report only results based on four–day intervalsamples.

Appendix 1 contains histograms of the full samples of daily data and of the four-day interval samples obtained from them for each of the 44 stations. Owing to thesmall scale of the graphs, in some cases high wind data are not perceptible onthe daily data histograms; however, they can be seen clearly on the four-dayinterval data histograms. A comparison between the histograms of the full dailydata samples and the histograms of the four–day interval samples shows that ourselection procedure considerably reduces the number of lowest wind speed data.The selection procedure also results in a shifting of the highest ordinate of thehistogram toward higher wind speeds.

4,2 Largest Yearly Data Samples. Also available were samples of largest yearlyfastest miles for winds blowing from any direction, recorded over periods of 18to 54 years at 115 U.S. stations not subjected to mature hurricane winds. Thedata in those samples were obtained and checked against original charts by M.J.Changery, Chief, Applied Climatology Branch, National Climatic Center, NationalOceanic 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 ananonymous data file. Instructions for accessing that file and attendant programsfor creating sets of uncorrelated data are available in Appendix 4.

5. ANALYSES AND RESULTS

5.1 Analysis of Uncorrelated Data Sets by the Probability Plot CorrelationCoefficient Method. Before applying the ‘peaks over threshold’ approach, weestimated the best-fitting distributions for the four-day samples from among aset of seven distributions or families of distributions (normal, doubleexponential, lognormal, Gumbel, Fr6chet, Weibull, and reverse Weibull). Thisanalysis was viewed as a tentative step toward understanding the probabilisticstructure of the populations from which the threshold exceedances were taken. Theestimation of the best fitting distribution was based on the probability plotcorrelation coefficient (PPCC) (Filliben, 1975). As an example, Figure 1 showsthe PPCC plots for the Albany, New York four-day interval samples. For thissample the mean and standard deviation were E(X)=1O.5 m/s (23.5 mph) ands.d.(X)=3.14 m/s (7.03 mph); for the full sample of daily data E(X)=7.8 m/s (17.5mph) and s.d. (X)=3.l m/s (6.98 mph). The analyses were repeated for the eight–daysamples, and the results were found to differ insignificantly from those basedon the four-day samples.

The reverse Weibull distribution was found to best fit the data in the majorityof the cases. Even in the cases where other distributions fitted the data better,the reverse Weibull was typically very close to being the best fittingdistribution, that is, its PPCC differed only in the fourth or even fifthsignificant figure from the PPCC of the best fitting distribution. We thereforere–analyzed the eight-day interval samples by assuming that the populations forall stations have reverse Weibull distributions with a single, site-independentvalue of the tail length parameter, and site-dependent location and scaleparameters. For each stationwe calculated the PPCC’S by assuming that the shapeparameter ywas 1,2,3, ...50. For samples of data based on eight-day intervals themean and median of the PPCC’S, taken over all the stations, were largest for 7=11and 7=13, respectively. If it were true that a reverse Weibull distribution witha single tail length parameter characterized the extreme winds at all sites, thenour analyses would indicate that the value of that parameter is 7=12.

The assumption that there exists a universal tail length parameter for extremewind distributions is implicit in current practice, except that it is applied tothe Gumbel distribution (for which Y=Q). To see whether that assumption istenable if applied to the reverse Weibull distribution with 7=12, 44 samplescorresponding to 18-year record lengths based on 8-day intervals were generatedfrom reverse Weibull populations with (1) 7=8, (2) 7=12, and (3) 7=16. The numberof simulated samples for which the best fitting reverse Weibull distribution hadshape parameters with 7

NORMAL PFIOBABILilY PLOT V

60 ,J

50-

40-

30-

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lo-

0 1 1 1 1 1 1 1

-4 -3-2-101234

CORRELATION COEFFICIENT = 0.990863

EXTREME VALUE TYPE 1 PROBABILITY PLOT V

50-

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0 1 1 1 # 1 I 1 1 ,

-2.10123456 78

CORRELATION COEFFICIENT = 0.99108

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0,9 0.95 1 1.05 1.1 1.15 1

CORRELATION COEFFICIENT = 0.988443

DOUBLE EXPONENTIAL PROBABILITY PLOT V LOGNORMAL PROBABILIN PLOTV

-lo -5 0 5 10 -4 -3 -2 :101234CORRELATION COEFFICIENT = 0.977449 CORRELATION COEFFICIENT = 0.994978

WEIBULL PROBABILITY PLOT V

60GAMMA = 2

50-

40-

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lo-

0 I 1 1 I 1

NEGATIVEWEIBULL PROBABILITY PLOT V

60-GAMMA = 8

50-

40-

30-

20-

lo-

0 1 1

0 0.5 1 1.5 2 2.5 3 -1.3 -1.2 -1.1 .; -0.9 -0.8 -0’.7 -0.6 -0.5 -0.4CORRELATION COEFFICIENT = 0.996035 CORRELATION COEFFICIENT = 0.997976

FIGURE 1. Probability plots for four-day interval sample,2 Albany, New York.

Table 1. Comparison of Results for Simulated and Observed Samples

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q=lforthreshold.

o 100 200 300 400 500 600 700 800 900 1000

Point estimates of tail length parameters and 95% confidence boundsversus number of exceedances (de Haan, 1990).

each station was so chosen that at least 16 data points exceed that) The plot of Cwa is shown in Fig. 4 and, in our-opinion, tends to

confirm the view that, -at most if not all stations, the estima~ed value of c isnegative, perhaps c=-O.2 or c=-O.25. We note that, as suggested by Monte Carlosimulations (Gross et al. , 1994), for sample sizes not exceeding about 10 percentof the total number of data, the bias in the estimation of c is about -0.05, thatis, sufficiently small not to invalidate our”judgement that, predominantly,C

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THRESHOLD

Mean of tail length parameter estimates weighted over 44 four–dayinterval samples versus order of threshold.

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15 20 25

THRESHOLD

parameter estimates weighted over 115 largestversus order of threshold.

5,4 Estimation of Speeds with Specified Mean Recurrence Intervals by ‘Peaks overThreshold’ Analyses of Uncorrelated Data Samples. Appendix 3 contains plots of

point estimates of the extreme winds with mean recurrence intervals of lOO, 1,000and 100,000 years. The estimates were based on four-day interval samples at eachof the 44 stations. They are plotted against the threshold speed and the numberof exceedances of the threshold, as in Fig. 2.

We reproduce in Fig. 6 an example of a plot where the quantile fluctuatesstrongly as a function of threshold (de Haan, 1990). De Haan comments: “if onewould be forced to give a point estimate ’a value of 5.1 m. .. would not beunreasonable.” The comment is indicative of the spirit in which results based onthe ‘peaks over thresholds’ method must be interpreted in cases where fairlylarge fluctuations are present, as is the case for Fig. 6 and many of the plotsin Appendix 3. We do not attempt in this report to estimate extreme wind speedsfor various mean recurrence intervals. Rather, having found that the tail lengthparameter c of the GPD is negative for the majority of the stations, we assessin the next section the potential implications of this finding for the estimationof load factors.

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5.5 Influence of Data Errors on Analysis Results. The footnote

bounds

to Section L.1lists errors in the recorded daily data found at fourteen stations, and therespective corrected values. Figures 2 and 4 and Appendix 3 are based on thecorrected 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 lengthparameters and speeds with various mean recurrence intervals for twelve of thesestations and are based on the uncorrected data. Comparisons between these plotsand their counterparts in Fig. 2 and Appendix 3 show that estimates.of speedscorresponding to various mean recurrence intervals are in some cases affectedfairly significantly by the errors in the data. This is true to a much lesserextent for the tail length parameters. The plot of the weighted mean over all 44stations, computed from results obtained by using the uncorrected data, wasindistinguishable from Fig. 4.

20

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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 loadis an extreme load with specified probability of being exceeded during a basictime interval. In the United States that interval is usually 50 years. A basicdesign load with a 50–year mean recurrence interval has a probability of almosttwo thirds of being exceeded during a 50-year period.

A structure or element thereof is expected to withstand loads substantially inexcess of a 50- or 100-year wind load without loss of integrity. The wind loadbeyond which loss of integrity can be expected is referred to as the nominalultimate wind load. The nominal ultimate strength provided for by the designeris based on a nominal ultimate wind load equal to the design wind load times awind load factor. This statement is valid for the simple case where wind is thedominant load. It needs to be modified if load combinations are considered, butfor clarity we refer here only to this case.

The load factor should be selected so that the probability of occurrence of thenominal ultimate load is acceptably small. This probabilistic concept isimportant from an economic or insurance point of view. To the extent thatevacuation or similar measures cannot be counted on to prevent loss of life, itis also important from a safety point of view.

A probabilistic approach has proven helpful in a number of cases, particularlyfor relative assessments of alternative design provisions, for example for mobilehomes. However, in most cases the difficulties of obtaining wind load factors byprobabilistic methods have proven to be substantial if not prohibitive. For thisreason code writers have largely relied on wind load factors implicit intraditional 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 verylarge number of applications the wind load is proportional to the square of thewind speed, so that a basic design wind speed and a nominal ultimate wind speedmay be defined that are proportional to the square root of the basic design windload and the square root of the nominal ultimate wind load, respectively. Forexample, for Lander, Wyoming, the American Society of Civil Engineers StandardA7-93 (1993) specifies a basic design 50-year design speed of 35.8 m/s (80 mphfastest-mile) at 10 m (33 ft)elevation. The corresponding nominal ultimate windspeed would then be 1,31/235.8=40.8 m/s (91.2 mph).

Reliance on traditional code values is part of the process sometimes referred toas “calibration against existing practice,” Traditional codes were generallyadequate for many types of structures, but questions remain on whether safetymargins implicit in those codes may be applied to modern structures, which candiffer substantially from their predecessors in their materials anddesign/construction techniques. For this reason an assessment of wind loadfactors used in codes and standards would be desirable. For example, one wouldwish to answer the question: what is the approximate mean recurrence interval ofthe nominal ultimate wind speed?

The answer to this question depends strongly upon the probability distributionassumed to best fit the extreme wind speeds. For example, a PPCC analysis oflargest yearly fastest-mile speeds recorded at Denver between 1951-1977, basedon the assumption that the best fitting distribution is Gumbel, yielded a 27.9m/s (62 mph) estimate of the 50-year wind speed at 10 m above ground. The

23

corresponding nominal ultimate wind speed would be 1.311227.9=31.8 m/s (71.0mph), to which there would correspond, under the Gumbel assumption, a meanrecurrence interval of about 500 years (Simiu, Changery and Filliben, 1979). Iftaken at face value this would be an alarmingly short recurrence interval, sinceit would entail an unacceptably large probability of exceedance of the nominalultimate 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 appropriatemodel is a reverse Weibull distribution, rather than a Gumbel distribution, wewish to answer the question: what is the mean recurrence interval correspondingto 1,31/2 times the wind speed with a 50–year mean recurrence interval? ForDenver, if one had to estimate the tail length parameter ; 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=–O.2 (a conservative choice: according to the plots ; is likelyto be somewhat lower, that is, the distribution tail is likely to be somewhatshorter than that corresponding to c=-O.2), and X50=26.8 m/s (60 mph). For athreshold of 16.5 m/s (37 mph) –- a value that is roughly consistent with thesechoices, see Denver plot, Fig. 2 -- we have A=139/15=9.27/year. Assuming X50=26.8m/s (60 mph), it would follow from eqs (13) and (14) &=2.5 m/s (5.6 mph). Theestimated maximum possible wind speed corresponding to the parameters 6=-0.2 and~=2.5 m/s (5.6 mph) is obtained by letting R + ~ in eqs (13, 14). Its value is~aX=u-ti/~ = 16.5+2.5/0.2 = 29.0 m/s (65 mph). The estimated mean recurrenceinterval of the nominal ultimate wind speed 1.31f2xis therefore infinity (i.e.,

~0=1.31/2x26.8=30.6m/s(68 mph)such a wind speed is estimated to never occur). This

estimate is of course subject to sampling errors: the actual maximum possiblewind speed may be higher than 29.0 m/s (65 mph), and the mean recurrence intervalof the 30.6 m/s (68 mph) speed may in fact be finite, though likely much longerthan 500 years. In spite of the uncertainty inherent in our estimates, our resultsuggests that a load factor of 1.3 –– specified in the ASCE A7–93 Standard on thebasis of practical experience –- is in fact adequate from a probabilistic pointof view. This is contrary to what would be concluded if the analysis were basedon the assumption that the Gumbel distribution holds.3

31n this instance code writers attended to the apparent insufficiency of thenominal ultimate design speedby substantially inflating the value of the 50-yearwind 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 thesampling error for the estimated 50–year wind is s~ ~ = 1,3 m/s (3.0 mph) (Simiu,Changery and Filliben, 1979), the specified 35.8 m/s”(80 mph) value differs fromthe estimated 27.9 m/s (62 mph) value by almost six standard deviations. One mayview the actual load factor implicit in the ASCE A7–93 Standard as being equalto (1.3)(80/62)2=2.14, rather than just 1.3. The mean recurrence interval for the40.7m/s (91mph) (i.e., 1.311280= 2.14112X62) nominal ultimate load inherent inthe ASCE A7-93 Standard provisions for Denver, based on the Gumbel model, wouldbe about 80,000 years (Simiu, Changery and Filliben, 1979), which is much moreacceptable than 500 years.

24

7. CONCLUSIONS

In this report we presented estimates of the tail length parameters of extremewind distributions for non-tornadic winds blowing from any direction in regionsunaffectedly mature hurricanes. In our opinion, these estimates support the viewthat the reverse Weibull distribution is an appropriate probabilistic model inmost if not all cases. They also suggest that load factors for wind sensitivestructures specified by current standards provide for reasonable safety marginsagainst wind loads, and that the adoption of the Gumbel model likely results inan unrealistic assessment of structural reliability under wind loads.

However, owing to fluctuations of our estimates with the threshold value, it isdifficult to provide reliable quantitative estimates of the tail lengthparameters. This difficulty is even more pronounced for quantile estimates.We tentatively ascribe these difficulties to the relatively small size of oursamples (15 to 26 years). It would therefore be desirable to assemble data forlonger records than those used in this report. In addition, more efficientestimation methods should be developed. Efforts to develop such methods arecurrently in progress (Coles, 1994).

25

RE)?ERENCES

American National Standard A58.1-1972 (1972), American National StandardsInstitute, New York.

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

Bingharn,N.H. (1990), “Discussion of the Paper by Davison and Smith,” Journal ofthe Roval Statistical Societv, Series B, 52, p. 431.

Castillo, E. (1988), Extreme Value Theory in En~ineerinr, Academic Press, NewYork.

Coles, S. (1994), private communication.

Davison, A.C., and Smith, R.L. (1990), “Models of Exceedances Over HighThresholds,” Journal of the Royal Statistical Society, Series B, 52, pp. 339-442.

de Haan, L, (1990), “Fighting the Arch-enemy with Mathematics,” StatisticNeerlandica 44, pp. 45-68.

de Haan, L. (1994), “Extreme Value Statistics,” in Extreme Value Theorv andApplications, Vol. 1 (J.Galambos, J. Lechner andE. Simiu, eds.), Kluwer AcademicPublishers, Dordrecht and Boston, 1994.

Ellingwood, B. et al. (1980), Development of a Probabilitv-Based Load Criterionfor American National Standard A58, NBS Special Publication 577, National Bureauof Standards (U.S.), Washington, DC.

Filliben, J.J., “The Probability Plot Correlation Plot Test for Normality,”Technometrics, 17, pp. 111-117.

Gross, J., Heckert, A. Lechner, J., and Simiu, E., (1994), “Novel Extreme ValueProcedures: Application to Extreme Wind Data,” in Extreme Value Theorv andArmlications, Vol. 1 (J.Galambos, J. Lechner andE. Simiu, eds.), Kluwer AcademicPublishers, Dordrecht and Boston, 1994.

Gross, J.L., Heckert, N.A., Lechner, J.A. and Simiu, E. (1995), “A Study ofOptimal Extreme Wind Estimation Procedures,” Proceedings. Ninth InternationalConference on Wind Engineering, New Delhi.

Hosking, J.R.M. and Wallis, J.R. (1987), “Parameter and Quantile Estimation forthe Generalized Pareto Distribution,” Technometrics, 29, pp. 339-349.

Johnson, N.L. and Kotz, S. (1972), Distributions in Statistics: ContinuousMultivariate Distributions, Wiley, New York.

Pickands, J. (1975), “Statistical Inference Using Order Statistics,” Annals ofStatistics, 3, 119-131.

Simiu, E., Changery, M.J. and Filliben, J,J. (1979), Extreme Wind Speeds at 129Stations in the Contimous 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 theEstimation of Extreme Wind Speeds, ” Journal of the Structural Division. ASCE,104, pp. 491-501.

Simiu, E., and ScanIan, R.H. (1986), Wind Effects on Structures, Second Edition,Wiley-Interscience, New York, 1986.

Smith, R. L. (1989), Extreme Value Theorv, in Handbook of Armlicable Mathematics,Supplement edited byW. 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) andhistograms of four-day interval uncorrelated wind

speeds (bottom)

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II

I.,-

8.

II

II

II

II

I.

.3:3:g

8::8”

*.-

.

40

MACON.GA - DAILY MAXIMA

1000 1

90Q

800

700

600

500

400

304

200

100

0

0 io 20 30 40 50 60

MACON.GA -4 DAY INTERVAL

3w~260

200

m

100

60

0

0 10 20 30 40 50 60

1500

1000

5W

o

0 13 26 39 52 65

MILWAUKEE_CORR.Wl -4 DAY INTERVAL

200 1

0 i3 26 29 52 65

MINNEAPOLIS_CORR.MN - DAILY MAXIMA

1000

I n

Wa -

800 —

709 —

600 —

500 —

4s0 -

3CQ —

200 —

104 —

o

0

20a

150

100

50

0

13

I26 S9 52 66

--do 13

MISSOULA_CORR.Ml - DAILY MLXIMA

‘“”~J

1500—

1000-

600 -

0Jo 40

MINNEAPOLIS_CORR.MN .4 DAY INTERVAL

26 39 52 65

i20

I I

30 40 50 60 70

MISSOULA_CORR.MT .4 DAY INTERVAL

400

300

200

100

0

0 10 20 30 40 50 60 70

Pw

MOLINE.IL - DS2LY MAXIMA

1000-

56Q -

0 I I I 1 I I

o 10 20 30 40 50 60 70

OKLAHOMA_CITY.OK - DA2LY MAXIMA

““~

1500 I

MOLINEJL -4 DAY INTERVAL

1

0 10 20 30 40 50 60

OKLAHOMA_C~Y.OK .4 DAY INTERVAL

I250

1

0 10 20 30 40 60 60 70 0 10 20 30 40 50 60

d

II

[

so.

e

II

8

I

-=+’

rI

1L

I

I,

I,

II

II

44

PoRTLAND.CORR,OR - DAILY MAXIMA

‘“”~

1000

540

0

0 15 30 45 60 75

250

264

150

Iw

50

0

PORTLAND.CORR.OR -4 DAY INTERVAL

0 16 30 45 60 75

PUEBLO_CORR,CO - DAILY MAXIMA

1064

90Q

8@3

764

600

50Q

4W

3W

240

100

0

0 10 20 30 40 50 60 70

PUEBLO_CORR.CO -4 DAY INTERVAL

250

200

160

100

64

0

0 10 20 30 40 50 60 70

RED.BLUFF.CA - DAJLY MAMMA

‘“”~

1500 1 I

40D

3D0

200

100

0

0 10 20 30 40 50 60

RED_ BLUFF,CA -4 DAY INTERVAL

o 10 20 30 ho 50 60

RICHMOND_CORR.VA . DAILY MAXIMA

9&3

ma

700

600

500

434

300

200

too

o

0 10 20 30 40 60

RICHMOND_CORR.VA -4 DAV lhlERVAL

300

250

200

150

100

50

0

0 10 20 30 40 50

:

-8

8

II

rI

..

I[

II

I.

47

‘d

SHERIDAN_CORR,WY DAILY MAXMA

80Q

704

604

500

409

300

2@l

100

0

0 13 26 30 52 65

SHERIDAN_CORR,WV .4 DAY lNIERVAL

-i

250

i

I

I200

150

100

50

0

SPOKANE.WA - DAILY MAXIMA

‘“”~

1500

1000

500

0

0 10 20 30 40 50 60

SPOKANE.WA -4 DAY IN7ERVAL

4“~

o 13 26 39 52 65 0 10 20 30 40 50 60

49

I

s

I%

I

I 1’I

I

I..

II

I[

II

II

II

I..

II

L1

,I

I[

I,

1I

I,

I,

I.

IIL.

II

I1

J

\,

1

I

II

II

II

I

APPENDIX 2

Plots of parameter & and 95% confidence boundsversus threshold and number of threshold exceedances

(based on samples of largest yearly wind speeds)

53

BIRMINGHAM, AL (34 YEARS)

-- . . . .. ..->

MONTGOMERY, AL (34 YEARs) TUCSON, AZ (39 YEARS) W64A, AZ (39 YSARS)

-.. ----- -------.

.. . . . .

W-!.....

..1

-,

., ,.,, ,,

,,’ 1..,,,,;;. ..’● ,!t., ,

~,, ;’.-.

, ‘-,’:,!; ,

, , ,$

,:,’,:

. . .

i-1.6i 1’

*-.,● ..!

.2dl’ds’h’

I

dl 39 37 35 33 31 20 27 25 MPH1210182329303434343434 34

T4646444240 d8’36 J4’i’” 2 30 MPH13 17 17 27 29 34 34 34 34 34

.3 Jd14947454341393736 3313 13 16 21 25 29 23 37 39 39

-1 t$654526048444442402436 34 MPH12 16 16 19 24 26 27 35 36 39 39 39

MPH

MPH

MPH

FORT SMITN, AR (31 YEARS) LITTLE ROCK, AR (39 YEARS)

1-

0.5-,-.

~-: ~

-1.5- ,’1-.

., .

-2” ;;43464442403636343212 13 16 24 28 30 33 35 37

FRESNO, CA (37 YEARS) RED BLUFF, CA (4z YEARS)

1-

-. . . .0-

-.>

~ x

-.

(Jz< J-,

PUEBLO, CO (43 YEARS)

1-

.-. ,0-

-----.- ..-,

0 .1

- y

,.. .~ ,,-,.”< . . ..’ *~ -, ;0 .2. ‘, *

::

-2 -‘. (

-4’d7’&43’4i’d9’5’7’6’5’d3’3’I’ d9’i7f14151022202238424246 42

MACON,GA(33YEARS)

1

0

-2

-2

1

0

-2

-4

.“.. ----

------

W-h.-...,.,. ,.,1

-,/. ,$-.,,

,-,,!,

,,

,-,)13’k’i4’d2’4b 38 26’;4’;2’20’;8 MPH14151820232720313223 33

MOLINE, IL (44 YEARS)

-----. ------- ----

,’ .,,-1,,, ,“‘.-J

;0’d8’d6’ 54’62’iO’~’~ 44 42 40121717203229333842 4242

Wii

HARTFORD,CT(44 YEARS) WASHINGTON, OC (39 YEARS)

,.-. ---,.*.-.

--).,-- .,.

,.’ ‘..”.----.

i7’43’i2’ il’d9’37’d 52331t

141821343642UUU

BOISE, ID (46 YEARS)

1-

.. .

0- ------

~ y

.- .,

0.1- ‘, ,-.z< -, J ‘,

-.

~...,’~ $ . .,. .

-144ri7 i2’42 4* 39’37 35 33 33 23 a121S 2329343637373737 373

EVANSVLLE, IN (44 VEARS)

WH

MP24

-.. --,

y’-hf-....----.,-*,,*,t~”,

,, $,,, , ,“,,,, ,’,,,,!,

.,

)11494742424139373533 3129 MPH13 16 20 2729 24 35 40 44 4444 U

FORT WAYNE, IN (46 YEARS) INDIANAPOLIS, IN (36 YEARS) BURLINGTON, 1A (23 YEARS)

~=

DES MOINES, 1A (37 YEARs)

1 1

0,6

0

uz-4

.-. ,. . . . .

-.. ..

.. ’.. .0 .1 .-’*z<a~

o .2 1...,

-3-

-4- , r656361444745424139 3161019232622223434 3

------- -.0

-----

: ~

-.,,. -..

..-, --.--~

0 .1 .

z . .

z~0 .2

-3-

-4’ r:: 57 65 63 61 49 4T 45

19 20 32 36 41 44 43

------ .-‘-..

o-

-’:~ ,--2 .

,

. . .

,. . ..-

6 63 51 49 47 46 42 41 39 3721213141620202020 20

-4J , ,S4626068666462504346 U

r

12 14 16 23 39 22 37 40 41 41 41WI 6P61

WH

MP61

MPH

WICH~A, KS (41 YEARS) K3RTLAND,ME(43 YEARS)

‘~

BALTIMORE,MD(39 YEARS)

0.5- ------

~-o: -+.1 ,- - . .’ ,

$- ,

---- -. .-.. . . . . . . .

. . .0.5- ●

.------ .

!!!..:. *x , ,“, . .;’ I

‘..1- ‘-.-,’ .’

-1 .s -

-2-63614947454341393735 3312 13 15 16 24 30 26 42 42 45 42

..-

k‘. -.. .-

-.,.

,.

. . .-,’

!: ..’.,;;;t. r‘I:L

636169676553614947 45424112 12 16 21 25 26 31 36 40 41 41 41

.3 J,’,, r514947434241393736333112 20 23 28 30 36 37 39 39 39 39

-4 J56565462504646444240 3612 13 16 23 27 30 35 36 39 39 39

Ii%l

BOSTON, MA (50 YEARS) NANTUCKET, MA (23 YEARSI DETROIT, Ml (46 YEARS) GRAND RAPIDS, Ml (29 YEARS)

1. . . . .

--- ----0 ----- ----

1t

0.5

0

vz<~-0,6

1%

.1

-1 .!

-2

‘. . . ..-...-

-

----.

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0.5‘...

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..-.

‘)+/‘. ‘. ., ,’,

J,,.-,,!-f

46’5’ 2’io ’i3’J4’J4’ *42403W17192122222323 Z

----..-. ------ ---

a

:L121314141818212524 2

.1

-,-l.!

-i462 S04645 &“ ’JO’’’’’”417 f7262934%42zzz

MP66MPH

WH

Wli

SAULT STE MARIE, Ml (47 YEARS) MINNEAPOLIS,MN (42YEARS)LANSING, Ml (36 YEARS)

1

0.5

0

0z<

wm

SPRINGFIELD, MO (44 VSARS)

-----.

‘.

x

. . .

,-.. . . .-. .. .

..-.,

,-.

1i149474542413937 353337

r

318273436424244444444

HELENA, MT(4s YEARS)

. . . . ..

Y‘.

.-’. .

-.,, .

.. l.,,,.,1J-, ;

.257665351494743424139 3731$16283033414346464340 4

VALENTINE, NE (27 YEARs)

W5i

ml

0.

.1-c1z<~ .2-

:

-3-

4-

---------- -- . . . .

h.,’ .-----,

, ,“,, ,,,, ,,,, ,,-..,..

-5 J63615957555351494743 42 MP1612 14 16 18 20 22 24 25 26 27 27

BILLINGS, MT (49 YEARS)

. .‘.

. ..-0

: y

--------.

-.c1 ‘. .z

. . .

<. ,. .-,.

~ -1 ●-. s,’..

~ ...

-2-

-3-63 61 59 d7 55 53 51 49 i7 46””16771927263242434746 g

MISSOULA, MT (43 YEARS)

‘~

0.6-

0z<~ o-

S’

.0.5 -

.-.-., ,‘., .

a ●..,,,‘,,,,. ,

.,‘,

DU-W*4Z4UVJ **34W12 15 2335 40 43 42 43 42 43

ELY, NV (4S YEARS)

1

0,!

(u~

$-0, e

E.1

-1.5

-2

-.. .... .-.-.

‘!”)v.

‘.. -

,“,.,,, ‘ ‘,,’,‘ ,-,

. .‘!

..,

.,

-1 r55 53 51 49 47 45 43 41 39 37 35121721303643444949 4949

m

*PH

IPH

GREAT FALLS, MT (44 YEARS)

-’ ..-..

,* .,, .,,.,

I 11361595755535146 4745423162126283134404244 U

NORTH PLAllE, NE (31 YEARS)

f

. . .

-5 , r646260 S6565452504644 4413 13 17 20 22 25 26 31 31 31 31

LASVEGAS,NV(20 YEARS)

:~

u .1z

~

.0 .2

-3

.4

: ~

-. -,~ ,,. . .

54525046444442403836 3413 16 17 19 20 20 20 20 20 20 20

WH

1

0

0z<a -1~o

-2

-3

1

0.5

0

vz

:.0.5~

n

-1

-1.5

-2

MP+I

HAVRE, MT (27 YEARS)

‘...●✎✍✎

h.......---

. . .

,,. ..’

~6’3’4’d2’d 0 43’44’& ‘42’JO’15 t6 19 21 21 24 27 27 27

OMAH& NE (51 YEARS)

------

-- . . . . .

h8563462604343444221516222634404f46

RENO, NV (4S VEARS)

MPH

MPH

i *., ...,,

-2-,,1,!,,-

-3-’62605056545250444444 42 MPH12161721253235414445 45

WINNEMUCCA, NV (38 YEARS) ALBUQUERQUE, NM (52 YEARS)

,.. . l-l

0,6- -.

.--.

------ . ..

~“i::*..,-..--.-,,

,’ ----- , .

-1.6-

1-

0.6- -’..4*

*-. ..-.

0u

~ -y

‘.-.

2 .‘,

-..‘.-..

‘.. ,.

.. .

.,

. . . .‘.

‘.”..

‘n‘.

.-.

.-..,.

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~..

----- ---.,0 0.6

W.... .“$-,,,,,V ,-.--.,,--.,,. ..,-1 0,,,,,,,,,,,,,,,,-..1

,,,,,,,,,,,,,,,,*,,.,

.1

-4 ,-1,6

-5

+111 -it. l!r!!l. ,ll! lrrlrl56 56 54 82 60 44 46 44 42 40 36i36 MPH

>

12141923293541454545 45a4662 SO 46 44 44 i2’40 36’&’$4’$2’2!f4242a3s 394245424542 45*

-2k’h’41 39 37 32’3!3 3% 24 27<141624263743472050 60

b’&’&rh’&’&’d6’d4’d2’dO’;6<121314*4202326262629 29

MPH

1

0

v .1$

; .2

.3

.4

1

m o0

-3

1

0

-2

-3

BISMARCK, ND (40 YEARS)

‘.. -

-.. .

Q.-----,,“

. ,,, ,

‘..1’!

,;,,, ,,.-

?H!.957d5rJ35194745 k’i’r 1 3$14 f72127343437404040~ 4(

COLUMBUS, OH (30 YEARS)

-------- ---- -.

J?-,,,,,,,....‘,

.-r--f

, , r0444344424023363451619262629303030

WH

ml

1

0

.1

-2

.2

FARGO, ND (45 YEARS) WILL3STON, ND (18 YEARS)

.. . . . . . . -

>

. . . ... -. . . . . .. . . . .

.---’ . ..

. -,

-.

----

k 59 67 55 d3 Eh 49 dl’.b’kr-

15 18 20 27 32 36 37 39 4f 44

DAYTON, OH (41 YEARS)

TULSA, OK (35 YEARS)

o-

-1.

-2-

------------ .

. .

%

.

,,!-.,l

. ,, ‘..,! J,

.,!,’$ ,,,-----

-,

-3 ~57666361404745424139 3713161925283023364041 41

. ..

)’%‘. -- .

. . .-. .-.

.,

● ,..,

-, ‘t

,-.

>!9 47 M 42 41 39 37 35’d3’&’{9r2;15 21 23 37 30 32 34 25 35 35 %6 35

WI

0.s

o

0

3

------ -.. . . .

~

.

,. . ,,.,,.’. ,.,. ,,

*-.’

-6 ~50434444424033363432 3013 16 22 25 29 30 33 33 33 33 33

MPH

GREENVILLE, SC (42 YEARS)SCRANTON, PA (33 YEARS) SLOCK ISLAND, RI (31 YEARS)PmSSURGH, PA (18 YEARS)

. . . . .-. ”..

. ..-. . . . ---- -.

y- . .$-. ,

-.. . ..-. .

7.-.--..,,... ..- -. -. ..-

%.,.-, ”,-

,,,$ ,

,,,

;

.-. .,,,,1,,.

?4’i4’d2’44’d6’36 2’4’32 2’O’;O’i12172230323322233362 X

,,. ..2

4 &’d0’3’8’ d4’5’4’5’2’dO’i6’ J6’4$’42#

12 15 19 21 27 31 31 31 31 31 31b“ 52’60 44 &“’& ’ion”1212141822 ;2826~i6’i4’42’io ’d6 ‘d6’d4’i2’2’O’&

12 15 16 17 Is 18 18 18 18 11

MPH

MP61

MP14

MPHMPH

CNA~ANOOGA,TN (35YEARS)

‘~

HURON, SO (49 YEARS) RAPIO CITY, SO (42 YEARS)

1

0

~

--- -... ----

-..-1

,’ !.2’, ,

~ -3 “,,; :,

z f .,w -40 .$

1

0-

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‘...-

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..-.. . . . . . -. ---- ,,‘%,.....,’. ,.,, $-,;..1 ‘,,’, . . ., ,.’ ,,~-, ,,t,

,,.-l

----- ,.,-- . . . ...--”-0

+.----,

,,,,,.,

:,, f.**’

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I,,$,.4 ,,’!.;,t.-

-7 ‘- ,

-2

.2 Jc.2’do 46 i6’i4 42 44’26 36 24r12 12 13 15 20 21 26 24 24 35

-4J f!i2’5b’46 44 44’& ’40 26 36 2’4’32 30 MPH1214192124263131332233 22

-8 J6i’;2’6b’&7 56’6’4’d2’50 46 46 U 42 MPH1418243536374142424242 42

I

6’7’65 62’3’1’2’0 67’~5’C13 6’1’i4’47’i6 MPH1215172123272440413547 49

AMARILLO, TX (24 YEARS)NASHVILLE, TN (34 YEARS)

‘~

MEMPHIS, TN (21 YEARS)

-. .. . . . -. .--,

%

-----,., ‘,

,-,,, --.,,

0.5 !0.5 . . *- . . .

‘--)----.-..--- ,.-.----.8”,

-.

---- .--- I -.

‘.-0~

$0c-4.5

. . .., .’-,).

.,/ ‘, ‘.,’

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-1 .!

-155 53 51 49’47’45 42’41 39 37(15 20 21 24 28 22 35 26 36 36

r3’4’1’;s 57’i5’4’3’d1’4’9’i7’42’2 16 19 2S 26 31 32 33 34 34

-1J Ii3’i6’i4’42 io’38 36 34’;2’30 MPH12 15 16 22 26 32 34 24 24 34

-2 ) ‘w 42 40 28’36 34 32 30 28 26 2112 13 17 20 21 21 2t 21 21 21 2

mli

AUSTIN, TX (37 YEARS) DALLAS, TX (32 Y2ARS) EL PASO, TX (32 YEARS) SAN ANTONIO, TX (36 YEARS)

10. 1

0

-1

-2

.2

1-

‘.0.6- ,

*...

i.; \

‘-.. . .

. . . .‘. ...

g . . .0 . .

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6

00

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g

-10

-15

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f“y-J*----

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-2 I44’J4’&’4’240262634’ 32d02’6MFll13182225282536333636 3S

.20 -4 r,67 55 6dS1’49 dl 43 & 41 39 37 35 MPH15 16 16 26 30 32 32 32 32 32 32 32

k 44 42 40 38 36 34’32 2t’2’6’2i420262336363737373737

ro ./3’’’’’’’”454442402’8 363432 r2 16 19 27 31 31 32 32 32 32

SALT LAKE CITY, UT (46 YEARS) BURLINGTON, VT (40 YEARS) LVNCHSURQ, VA (44 YEARS)

1

0,6

0(Jz<

-..,.,,

-1- \

-\. -,, .

.2.,

-2-

-4.

-5-42 41 36 37 35 33 31 29 27 25 23

1-

ls 217.227303333623362 22

--. ------ -. .

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.

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q

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,$1 ‘,-.,’ 1’,‘, , ,,

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mN

-,

$

.-2 ~

54526043434442403836 MP64121820293026424344 43

-2 > ,49474042413937352231 29 MP6112151824323537404040 40

1424436362432302626 MIW121623302442444444

MPH

MPH

NOR7H HEAD, WA (41 YEARS) QUILLAYU7E, WA (21 YSARS)

1

SEATILE, WA (20 YEARS) SPOKANE, WA (47 YEARS)

1

0.5

c1z-4n 70 68 66

-3, I 764 62 60 56 MP65

13 15 17 20 36 41 41 4134 32 30 20 26 24 22 20 18 16 *413 15 21 21 21 21 21 21 21 21 21

-3 ‘ ,42 40 23 36 34 32 30 28 26 24 2212 17 17 19 20 20 20 20 20 20 20

Umi

TATOOSH ISLAND, WA (54 YSARS) GREEN BAY, WI (36 YEARS)

1

. . .- .0-..

\

. .‘-. ,-..

. .,.. .,’ .,.. .-.

,,

-.

.,,

$..

)0’d8’;6’dk’(2’dO’iO’i6’d4’d2’ 2b712 17 19 34 40 44 46 52 63 54 54

CHEYENNE, WY (46 YEARS)

‘~

-------- -.

>

.‘“. -.

. ,-f.,..

1, .-,h ,

,.

,4

.sJf #“

d4’&’d0’ 5’6’5’6’;4’i2’iO 4k’44 44*12162531364041424443 44

. . . . . .● ✎ ✎ ✎

✎ ✎

=’s.-

MPH-2 J

d6’3k’5’4’ i2’6’0r4b’&’&’d d12 13 17 21 24 27 26 32 3: 3(

1

0

.3

4ml

LANDER, WY (42 YEARS)

~ ., ..,.

, ‘., ,’,,b.

15’63 61 59 67 5’5’3’3 5’1 44 4?21415212230333436 39

0

-2

MPH.3

1

0

-1

-3

-4

-5ml

MADISON, WI (41 YEARS)

1

-------- . ,. ..-

W-m-i-., *

, ,,. ,,,-’,,,-l!;

?Z’d6’i4’;Z’5b’d4’k’ 4i’i2’k’h13 13 14 t? 17 20 27 32 36 39 41

SHERIDAN, WY (44 YEARS)

--‘-. .

%

-.. ------ --

,“* ...-, ,.

,. .,,’-..

*;.--1

-133 61 54 67 55 53 61 49 47 45<13162730342535394144

o

-z

MPH.3

MPH

MILWAUKEE, WI (42 YEARS)

-.‘------

. .G -.,’. . ,.,*,t,’-”~ ,*.t .‘,,, ,

,.

h’ i4’5’2rdo’JZ’i4’44’dZ’k’14 30 23 26 33 29 41 42 42 ::

APPENDIX 3

Plots of point estimatesfor 100-year, 1000-year and 100 000-year speeds

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

65

ABILENE.TX (15 Y124RS)

110 — 160,000

h

- -1,040foo --1oo

zo~43 40 37 34 31 26 25 22 19 MPH16 52 96 191 3LW 536 741 925 1045

200

64

0

BINGHAMTON.NY (20 YEARS]

— 1to,ooo- -IJOO- - 124

k\\“.‘. >--- s-, . . . . -r, , 9-I1 30 35 32 29 26 23 20 18 26 54 117 235 412 693 993 72

IPH

r4PH

zo~42393633302734 2jq616 41 65 136 247 428 659 912 116

70

60

gso

40

30

BOISE.ID (2s YEARS)

— 100, OW- -1,000

--102

\-1! 29 36 33 30 27 24 21 11F 42 64 133 254 461 74S 4044 1%

r4PH

!PH

ALBUQUERQUE.NM (19 YEARS)

m- — lGO,om- -1,000

: 4

--1oo

60

\70

‘\~

\ 1,,

./, ,,,,

60\

‘-.

30

ao~5249464340373437 219 30 69 39 199 315 455 690 90

BURUNGTON.VT (19 YEARS)

— 100,000- -1,000--1oo

*

l\

-!,’.-... ‘,-

-.

‘.

\

DENVER.CO (15 YEARS)

140.— 134,000

i30 - - -I,LXIO

--140WO -

110.

1oo-

90-

E30- ,\

60- .

54-

40

!Ptl

400

334

g 240

100

BILLINGS.MT (26 YEARS)

— 100,000- -*,000--100

‘L.L-/“,--. \-.-~-, . .90

24

30,

ux76.

70.

44.

m.

CHEYENNE_CORR.WY (15 YEARS)

— Ioo,ow- -1,000--*OO

o l-r, ,5047444120363229 26 MPH16 40 63 121 203 326 566 823 1129

-1 , , , , ,9565350474441333k MPHo 30 60 82 131 170 251 337 461

DULUTH.MN (15 YEARS)

110-— 100,000- -1,000

100. --400

30-

e4-

\E70. ,

60-.\

.

50-

40.

~

40J, , , ,47 44 41

8 t33 35 32 29 26 23 MPH

30 ~ ~p” .~

16 36 65 92 162 254 402 603 832 77 56 79 139 274 4% 537 747 36933 35 32 29 26 23 20 MPH

20 47 72 131 206 362 560 790 902

2CQ

150

aXloo

E4

0

FORT.SMITH.AK (17 YEARS)

— 100,000- -1,000--100

I

Ii‘“+

,;:~ 353229262320 ~

62 i03 167 251 45S 667 31iPH

GREENVILLE_COR R.SC (19 YEARS)

:F

o~ ~p”343123262220 46 106 236 419 672 1051 13201497

LANSING.KH (15 YEARS)

90- — 100,000

30-

70-

$i

50-

5n - ,.

40’, , r-46’42293633302720 220 37 M 07 158 256 377 603 81

,lPH

FORT_WAYNE_CORR.lN (15 YEARS)

— 164,000- -1,000

J100

,6\,\/\

I d,’,’,k .

\ ● .’ ‘,\’\., ,t.-t.-.=, *,,1s,

rB 45 42 39 36 33 30 27 W725356089140222266 53

HELENA.MT (26 YEARS)

90-

60-

E 70-

60-

cw -

- -1,000--100

-\

--

40~4345422936339027 Zi6 27 60 126 204 365 572 875 12

LOUISVILLE_CORR. KY (15 YEARS)

o~29 36 33 30 27 24 21 16 1!20 36 80 145 266 420 533 776 IW

MPH

#PH

!4PH

GREEN_BAY.M (15 YEARS)

120 T —100,000

120- I- -1,000

--100

110.

*OO -

90-

gao-

70-

60- ,---

50-

40-

30- , <4126353229262320 1’22 36 66 127 223 360 604 817 10

HURON.SD (15 YEARS)

324- — 100,000- -1,600

250---100

200

gmcl

ioo

i,L

1’~, \,\

.A, -....,--- . >---/. 2 >.-

50

o~46 45 42 39 36 33 30 27 2416 22 43 93 125 224 232 536 811

MACON.GA (16 YEARS)

140- — too,ooo

%20 -

120-

110-

100-

=90-

x30-

70-

50- -.-’ ---. ~,\

60-.

.

40 -

400

30Q

E 200

im

oIPH

i!PH

GREENSBORO.NC (19 YEARS)

— 16Q,000- -1.000

--IA

\

\

~B 35 26 23 20 17 14 MPH!0 29 61 117 202 234 547 770 *043

LANDER_CORR.WY (15 YEARS)

iw — Im,ooa

150 1 - -1,000--100140

120

120

%10

‘ bd~

;\ 1164- ,

\l\ ,

90 \ l\l-.’ ,1,\//\ \,. ,

.,. v.. ,t ?,. ,/, ,w , , d?l “

70-, ,.,

\\. , : --

60-54 51464542393633320 23 4* 63 101 155 200 272 %

MILWAUKEE_CORR.Wl (17 WARS)

104- — 10.3,0m- -I,oao

90

~:%

--100

60

70 ;\ }K. ~ \x

, \l

.~ l\ ~60- . ““, \ ,,-,

. . ‘ .\ .-

50 . . . .

\

MPH

30 ~26 26 22 19 16 MPH

.~

17 25 29 71 121 267 416 607 93033 30 27 24 21 MPH

19 31 75 %0 218 356 560 645 1067

MINNEAPOIJS_CORR.MN (15 YEARS)

20Q -— 16Q,00Q

– 1,640

164

~ k

. .

154

Elml\

.~/ J,.-\..- .X

. -\66 ..-

o~36 35 32 29 26 23 20 77 MPH

18 31 73 i38 2S7 401 661 850 1021

POCATELLO.ID (21 YEARS)

140-— 100,000

130-I

- -1,040

--10012Q -

110-

100-

90-

~3a-

$70-

50--.

60-

40-

20’ , , ,# 41 24 35 32 29 26 23 20 MPH

56 69 179 403 618 877 1122 i301

RED_BLUFF.CA (21 YEARS)

90-1 — 140,004- -1,000

--100Oo-

70-.

E J \

60-. . - .,

50-.

40 ~4542393633302724 2116 24 67 91 136 221 393 595 81:

ilPH

200

164

E1OO

50

0

MSSOULA_CORR.MT (25 YEARS)

— 100,000- -1,000--104

L

\4\

--..\-. . ...>-- -=.

r, r15 42 39 36 33 30 27 24 217 34 55 104 139 375 640 1029134

iiPH

PORTblND_COR.ME (20 YEARS)

140-— 100,004

124-

I

- -1,600.- 164

120-

110-

1o4-

90-E

60-

70-

6@-

54-

40-

30- , , , , r ,44 44 26 35 32 20 26 23 20 MPH21 56 69 179 403 618 877 11221301

RICHMOND_CORR.VA (16 YEARS)

70— 100,004- -1,000- - ,100

56.

E40- .

20.

20- r34 31 26 25 22 19 16 13 10 MPH23 57 126 251 418 617 %t4 11521276

MC3LINE.IL (15 YEARS)

12Q -!— 100,OLW- -1,000

110

# %

- - 100

104

wux“ *

~- flf

\/A\II

I ‘/, ,%,70- -’ ..-.1 /1’, ,

b

{./, ~ /\-’,””. \

64. \ ,’*\,

‘.\, ,.

54~4744413617 27 46 63 fW 172 273 400 d

PORTLAND_CORR.OR (23 YEARS)

o~2S 35 32 29 26 23 MPH

16 22 26 56 119 190 325 611 769

SALT_LAKE_CITV.UT (26 YEARS)

Iw - — 160,000

\

- -1,040

96. --1oo

34-

70-

z

64-

40-

36~26 35 32 29 26 23 20 MPH

20 52 92 164 330 614 667 11831391

OKLAHOMA.CIN.OK (16 YEARS)

70- — Ico,om- -

SHERIDAN_CORR.WY (17 YEARS) SPOKANEWA (2S YEARS)

90-

80-

% 70-

60-

60-

A - -1,640

40~16 26 47 91 144 252 367 5.32 7!

TOPEKA.KS (2S YEARS)

300

100

r —100,000--I,cao--100“L-L

o 1 f42 29’ ’36 33 30 27 24 21 1121 48 78 *2a 224 373 590 772 94

IPH

MPH

90-

80-

70-

s

60-

50 -

40 -

.~35 32 29 26 23 2{

18 51 69 $21 210 351 523 855 11

TUCSON.AZ (15 YEARS)

o~293633302724211

*7 39 66 157 253 444 675 875 10

SPRINGHELD.MO (24 YEARS)

364 — Ioo,ow– -1.000

250

200

%150

Iw

50

0MPH

100

90

3Q

70Ex

60

50

40

20flPH

- - I’w

,~

\

:\f\

.--. .>- ---

2936333027242118 123 37 08 440 257 467 670 878 11

WINNEMUCCA.NV (15 YEARS)

— Ioo,oco- -1,000--100

\\

. .

> , , rt542393633302724 2’17 27 44 92 777 365 437 744 97

IIPH

MPH

69

70

g60

50

40

20Q

350

gloo

60

a

TOLEDO.OH (15 YEARS)

— IO0,06Q- -1,000

--1C4

,\

,

.

~33 30 27 24 21 MPH

8 29 63 116 235 325 502 849 1162

YUMA.AZ (19 YEARS)

— 100,000- -1,640--100

1 ,12 29 36 33 30 27 24 21 18 MPH20 28 57 135 270 523 *0031472 2UOI

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 anonymousenter 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 (this copies a specific file; example: get boise.idboise.id)

>quit

71*U.S.sowmmmm~ OFFICE:1995-386-627/37909