NIST BUILDING SCIENCE SERIES 174
Extreme Wind Distribution Tails: A ‘Peaks Over Threshold’ Approach
E.Simiu1andN. A.Heckert2 1BuildingandFireResearchLaboratory NationalInstituteofStandardsandTechnology Gaithersburg,MD 20899-0001 and DepartmentofCivilEngineering The JohnsHopkinsUniversity Baltimore,MD 21218 2ComputingandAppliedMathematicsLaboratory NationalInstituteofStandardsandTechnology Gaithersburg,MD 20899-0001 Sponsoredinpartby: NationalScienceFoundation Arlington,VA 22230
March1995
U.S.DepartmentofCommerce RonaldH. Brown,Secretary TechnologyAdministration Mary L.Good,Under Secretaryfor Technology NationalInstituteofStandardsand Technology AratiPrabhakar,Director National Institute of Standards and Technology Building Science Series 174 Natl. Inst. Stand. Technol. Bldg. Sci. Ser. 174,72 pages (Mar. !995) 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 w LIST OF F’IGURES...... 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 DEHAAN 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..5Influence of Data Errors on Analysis Results ...... 19
6. LOAD FACTORS FOR WIND-SENSITIVE STRUCTURES ...... 23
7. CONCLUSIONS ...... $...... ,,...... $$...... O 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 & 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–, 1OOO– and 100 000-ye= speeds versus threshold and number of threshold exceedances (based on four-day interval samples) ...... 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 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 ofexceedances (de Haan, 1990) ...... 19
Figure 7. Plots of parameter 6 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-, 1OOO- and 100 ()()()-year speeds versus threshold and number of threshold exceedances (based on 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 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 eailier phases of this project, some of which 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 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 leadto 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 Fr6chet 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 Bi&try, 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, tobe 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 wind- sensitive 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 viewof the theory’s basic assumption that the threshold is high, excessive loweringof t.e threshold 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
1 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 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 fact that exceedances of a sufficiently high threshold are rare events to which the Poisson 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 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>O, c=O and c Given the mean E(Y) and standard deviation s(Y) of the variate Y, a = % E(Y)(l + [E(Y)/s(Y)]2) (2) c = %{1–[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-@/aG]} +x<@ (4) Fw(x) = exp(-[(pW - x)/oW]~), Xspw (5) For the Gumbel distribution, the relations between distribution parameters and the mean E(X) and standard deviation s(X) are UG = (61f2/n)s(X) (6) p~ = E(X) – 0.57722(6112/n)s(X) (7) For the Weibull distribution, Dw = s(x)/{r(l+2/T)-[r(l+l/7) ]2}1/2 (8) Pw = E(X) + a. r(l + 1/7) (9) where I’is the gamma function (Johnson and Kotz, 1972). For example, for E(X) = 50, s(x) = 6.25, and~= 2, Ow= 13.49 andpw= 61.96. The tail length parameter ~ is related to the parameter c in the GPD distributions as follows: 7 = -l/c (lo) (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 ScanIan, 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 –l/(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 - l/(AR) (11) Using eq (1) -II. = I – l/(,uL) 1 – [1 + cy/a] (12) Therefore y =–a[l – (AR)c]/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/~,=, 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) . ~ ‘~1 {lWG%-l,J - log(%-k,.))’ r-l,2 (15) k i=o The estimators of c and a are then 1 6==~(1)+1- .--———.----_— (16) 2{1 - (%(1))2/(%(2))) a ==~(l)/pl (17) 1 ;>() PI = (18) { 1(1–;) ;<0 The standard deviation of the asymptotically normal estimator of c is s.d.(:)=[(l+;2)/k]li2 620 (19a) 8(1-2;) (5-116)(1-26) s.d.(&) = {[(1-E)2(l-2&)[4 - ----— + —-—----- ]/k)li2 :<0 (19b) (1-36) (1-36)(1-46) (de Haan, 1994). 5 4. WIND SPEED DATA 4.1 Uncorrelated Samples Obtained from LargestDaily Data Records. Setx 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. Inmost samples anumber 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. 1 The influence of these corrections on the results of the analyses is discussed in Section 5.5. The length of the records ranged from 1,5to 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; 9.8 m 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.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 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 periodis four to eight days.) Pick the largest value in each period. If the maxima of two adjacent 1 The following discrepancies between records of yearly maximum fastest miles and daily maximum fastest miles were found: Cheyenne (6.1 melevation), 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.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 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 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) 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 four- day 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. 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, Fr6chet, 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 (PPCC) (Filliben, 1975). As an example, Figure 1 shows the PPCC plots for the Albany, New York four-day interval samples. For this sample the mean and standard deviation were E(X)=1O.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.l 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 PPCC differed only in the fourth or even fifth significant figure from the PPCC 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 stationwe calculated the PPCC’S by assuming that the shape parameter ywas 1,2,3, ...50. For samples of data based on eight-day intervals the mean and median of the PPCC’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 Y=Q). 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<-Ys20, and 7>21 are shown in Table 1. Also shown in Table 1 are the numbers of observed samples basedon 8-day intervals for which the analysis yielded 7<12, 13<-Y<20, and 7>21. The results of Table 1 would suggest that a reverse Weibull distribution with 7x12 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. 9 NORMAL PFIOBABILilY PLOT V DOUBLE EXPONENTIAL PROBABILITY PLOT V LOGNORMAL PROBABILIN PLOTV 60 , J 50- 40- 30- 20- lo- 0 1 1 1 1 1 1 1 -4 -3-2-101234 -lo -5 0 5 10 -4 -3 -2 :101234 CORRELATION COEFFICIENT = 0.990863 CORRELATION COEFFICIENT = 0.977449 CORRELATION COEFFICIENT = 0.994978 EXTREME VALUE TYPE 1 PROBABILITY PLOT V WEIBULL PROBABILITY PLOT V NEGATIVEWEIBULL PROBABILITY PLOT V 60 60- GAMMA = 2 GAMMA = 8 50- 50- 50- 44- 40- 40- t-J 0 30- 30- 30- 20- 20- 20- lo- lo- lo- 0 1 1 1 # 1 I 1 1 , 0 I 1 1 I 1 0 1 1 -2.10123456 78 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 -0.4 CORRELATION COEFFICIENT = 0.99108 CORRELATION COEFFICIENT = 0.996035 CORRELATION COEFFICIENT = 0.997976 EV2 PROBABILIN PLOT V 60 GAMMA =50 50 40 30 20 10 ; ,,[, FIGURE 1. Probability plots for four-day interval sample, o 0,9 0.95 1 1.05 1.1 1.15 1 2 Albany, New York. CORRELATION COEFFICIENT = 0.988443 Table 1. Comparison of Results for Simulated and Observed Samples @2 13<7<20 7z21 Simulated samples, ~= 8 44 0 0 Simulated samples, 7=12 26 15 3 Simulated samples, 7=16 7 22 15 Observed samples* 25 10 9 *stations for which 7z21 wara: Graen Bay, Greensboro,FIuron,Lansing,Louiavilla,Macon, Moline, Portland, OR, San Diago. Thosa for which 13S7S20 were: Binghamton,Fort Smith,Fort Wayne, Grenville, Milwaukaa,Minneapolis,Springfield,Topaka,Tuceon,Yuma. 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 MU, the next higher thresholds we considered were ~a-l, %,X-2, ....%..-24. 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 6 f 2s.d, (;)). 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 (; 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-O 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 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 t.~q,/Siq2)/ Z l/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 ~iq, Siq 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 ABILENE.TX (15 YEARS) ALBANY.NY (19 YEARS) ALBUQUERQUE.NM (19 YEARS) BILUNGS.~ (26 YEARS) 0,5 ,-, 0.2- -’, . /’ ,. /’, ~”,, , -. . ~., . t .’, ,. --, 0.5 , 0 . ..- ,,, .J. ./ * -/’, ,, , 1- . ‘- ,,,,,”-.,’. 0 ,,.-,-. ‘. ,’, ,’ *, ‘. @.26 ‘, ‘., ..* . . . J. ‘. ~-0.5 ,, . 4. ‘, ‘-, $ ‘. ,.-‘-.”. ,4 -!*. 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Lg N E E . . ,0 1, . ,, .0.26 - -1 !, ,- w,, 4 .1 .25 1 n,, 1’ ,, 1, 1, .3 , ,, 1* ,.., !, 1: .1.6 ,: I .35 J:,1 , T ) , < .1.5 r , r 41 263532292623201; 4229363330272421 1! dPH .1 2835322926232017 tiPH 59 66 53 50 47 44413636 IPH 18 26 64 117 235 412 693 993 IZ 16 42 64 133 264 461 745 10441% 8 29 61 141 279 460 795 1071 13C 20 30 60 82 1S1 170 251 337 46 CONCORD.NH (26 YEARS) DAYTON_CORR.OH (17 YEARS) DENVER.CO (15 YEARS) DULUTH.MN (15 YEARS) 0,5> , 0,5 f’. ,. , l“.., ,“. . , ‘,,, \ o- 0 ,“, \ ‘, 0 , c) , ,“~ .- . ~, ! -. ‘ /’, ~. $ J ,, $ ,,1 ...! ,~ g ; ,, ,.’ ,, ~i~~ . n ,. ,, ,,J -, !.:* ., -0.5 ‘. JI, , , .1 ,,, , ‘. -0,5- , #“ ., ●, ,,. . ‘, # ,, . -, ,,’, -,, $ ,,, -.. , ,, ,, J’.’ ,. ,. ,, ,, % ,,, ,. -1- ‘ ,, ., - ,, ,,, m~-.’ .1 ,, ., .IJ,I,, I # r {:,,,,,,,,,,,,,,,,,,,,,,,, h -1 , , , 1- ~ 29 36 33 30 27 24 21 18 15 4744413635 32 29 26 2: lPH 33 25 22 aPH 44 41 23 35 32 29 26 23 20 bPH t7 41 106 193 404 666 965 1393+71 16 36 65 92 152 254 402 605 83 17 56 79 139 274 431 537 747 SS$ 20 47 72 131 206 302 580 790 98 FIGURE 2. Plots of parameter ~ and 95% confidence bounds versus threshold and number of threshold exceedances (based on four-day interval samples). ---- . -z-~! ‘.-. t-g .’ -. -. , ,- . .. . -’ -N ~ . .’.’ . .,,, %% ‘---- .g . -N .-. , R: #- . . :s .’ , ‘. . ? 38 --- -.’ -. ..[. , i %!% 3 NvvH3a 13 MINNEAPOUS.CORR.MN (15 YEARS) MSSOUIA_CORR.MT (25 YEARS) MOUNE.IL (15 YEARS) OKLAHOMA.CITY.OK (16 YEARS) a5 . , l,- ‘, ,, . . 0,5 ‘. ~. .~ . . *, 0.25 ‘-- . . .-, , ‘, . ,, ,’ ,.$ 0.26 \ ‘, ~-. ,.1 ‘,. . ‘.. . ,- . ,--- . l“~~, :0 , ,. : . . . ‘-,-: .,-.,, , . . . ~ o- ‘, .,,, “.* : -. / . r. . : ,. “,1 ~, s “-, ; + ,.- ‘, !, , ,, -025 ./1, ,’!1 ,, ,. .,$,.’ ‘r ,, .0.26 ,. ,, ‘ ,. -- ... ,,~, * .1- It ,. ,,,’ 1, ~, ,’. I ,, ,’ ,) , ,’ u J,,!, ,, ‘. -0.5 1 .,,, -0,6 ‘,, . , ,, ? . : * ,, \ ,, I I > , , 11,,,,,,,,,,,,,,,,,,,,,,,( .ls~ 41 33 35 32 29 26 23 20 17 MPH 46 42 29 35 33 30 27 24 21 MPH 47444135353229 262 !PH 4345423936 33 30 27 24 MPH 18 31 73 138 237 401 661 850 1021 17 34 55 144 159 375 540 10291243 17 27 45 6% 109 172 273 400 84 21 42 74 120 234 378 586 763 943 POCATELLO.ID (21 YEARS) PORTbiND_COR.ME (20 YEARS) PORTIAND_CORR.OR (23 YEARS) PUEBLO_CORR.CO (17 YEARS) , 0.5 0.5 . 0.5- t , ,, ,, 0 . . 0.25, ,, ,,, ,,. , ,“. ., 0.26- “ ‘, ,-. ,’. I . . $ ,~ ‘ ,, . ..- ‘,, , -’ , . . . . ,.’ c1 . . ,/, ,5 1.., ~.O.6 ~,,1 J,, ,, ,- ,,. z , ., , ‘, ~ 4-, ,--- 1, ‘,. , \ ,, , ! & , . . -1 ‘., ,,, , .. .0.5 - ,’, , w ‘,, ,,, I .1.5 I 1 -1.5 ~ r > , , , , a r , , , 44413+13532292623 20 MPH 47444123353223 26Z MPH 56 53 50 47 44 4’I 33 35 32 MPH 21 56 59 179 403 618 877 11221301 16 22 28 56 119 180 325 511 78 18 31 44 79 122 215 303 435 855 RED_SLUFF.CA (21 YEARS) RICHMOND_CORR.VA (16 YEARS) SALT_LAKE_CITY.UT (26 YEARS) SAN_DIEGO.CA (17 YEARS) 0.5 . .,, /. . , 0 .,, ?. ‘.’- .* ‘.. . . ,. .’ ,- .,,’.- ‘.. -,’, ,,.’ ‘, -,’.. ,, ,- .,-~ ,-, . ,, . 0.0,5 . . ‘.,1 ,*,, . t, ‘. ,“ . ‘-’, , . . -., ,, ,\- ,. $ ,, , ,, / ,“. -. . ,, t ,. . . +,’ 1,, n .1 ,, 1, ‘,’ ‘- ‘,, . . . ,J, + ,, ‘.’ 1, ,, * ,’, ● ~,,’ ‘. ,, , -1.5 ,, $, :, ‘, I ? ~, -1.5. > -2 r, r 45 42 39 35 33 30 27 24 21 MPH 4 31 28 25 22 19 16 13 10 4 4+ 3835322926232( UPH 16 24 67 91 136 221 393 395 S13 3 57 128 261 418 617 944 11521276 30 52 92 134 3%0 614 867 1183 13$ FIGURE 2 (continued) TOLEDO.OH (25 YEARS) SHERIDAN_CORR.WY (15 YEARS) SPOKANE.WA (24 YEARS) SPRINGFIELD.MO (15 YEARS) \ ., , ,- -, ‘. $“, , .,, , , # ,’. . 0 .,’. ‘ , t’, , ,. . . . . 0 -. ~, “., , ‘. $ ,,,~~.-, , ,*W ,-- .-,, , , , ..-, ,Jt *. ..-. . . ,. $26 ,,, ,, J,-- . ‘-’, . ‘.. ,,-, ,. . . ,, ... ., z ‘, -. .’.. , ‘, ,o.~ . ..’.,. -. ; ~.” , .I ‘. ‘, -0.5 P, ‘, ,, . , , ‘,, - ., + ‘, , , ,, ,. ,, ‘, , ,, ‘, ,,, , ‘, y ‘. - m -0,75 . ‘, “ -1,6 ‘. , ‘J, ‘1 ‘, ‘, r ) , -1 ) ( 85 42 39 36’ 33 30 27 24 21 MPH MPH ~;; 333532292623 2 MPH & 36 33 30 27 24 21 18 15 MPH B 45422936333027 21 18 39 63 115 235 326 502 649 1162 16 26 47 91 144 252 367 642 75 59 121 210 351 523 855 11 23 37 63 140 257 467 670 878 1111 WJh14.AZ (22 YEARS) TOPEKA.KS (15 YEARS) TUCSON.AZ (15 YEARS) WINNEMUCCA.NV (19 YEARS) , . 1- 0.5 . , , WI , . “ ~, .,’,,’ . ‘, ., ,. “. . ,,, ,. 8 . 1-, -- ~,. ‘. .,-, ‘, , ,. ..- , ,. --,,..-,.. ,-. ‘,’ ., , . . \ ‘1,, ‘, ,’, . 4 , ,(, . ‘,, \ ,,, ,,, % ,-, ,--, ., . ,tt~ ‘. ,,. , -, .~%. , ,,, ... ,., ,’ , $$ . ‘. ,. ‘,, .0.5 ‘, , ‘,’, ,, ‘, ., ,fl, , ,, ,, , . q ,,~, ‘, ,, * \ $’1 ‘f , 1. % , ,, \ ‘,‘ I,,,,,,,,,,,,,,,,,,,,,,,fi 1 , n, , f 1 f -1, 1642293633202724 2 MPH 42 39 36 33 30 27 24 21 If MPH 4229363330272421 1 I!PH 42293633302724 2111 IPH 17 27 44 92 177 366 467 744 9 20 23 57 135 270 523 IOW 1472 20( 21 43 70 123 224 373 690 772 @ 17 29 66 157 263 444 676 875 10 FIGURE 2 (continued) 0.5 * ‘1 0.4 i i 0.2 0.1 0.0~ 1 0.1: ---- ,/. ------. ‘..”- *---.-.------‘,---- o 100 200 300 400 500 600 700 800 900 1000 FIGURE 3. Point estimates of tail length parameters and 95% confidence bounds versus number of exceedances (de Haan, 1990). q=lfor each station was so chosen that at least 16 data points exceed that threshold. ) 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 is negative, perhaps c=-O.2 or c=-O.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 5.3 Estimation of Tail 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 6 f 2s.d. (;)). 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 -0.1 -0.2 -0.3 -0.4 -0.5 I 1 I I I I I I I I I I I I I I I I I I -0.6 I I o 5 10 15 20 25 THRESHOLD FIGURE 4. Mean of tail length parameter estimates weighted over 44 four–day interval samples versus order of threshold. -0.1 -0.2 / -0.3 -0.4 \ -0.5 -0.6 “0.7 -0.8 -0.9 I I I I I I I I I I I I I 1 I I I I I I I I I 1 0 5 10 15 20 25 THRESHOLD FIGURE 5. Mean of tail length parameter estimates weighted over 115 largest yearly data samples versus order of threshold. 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 lOO, 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. 1000 ;f $ ,, IL .: ,-~ ,,t; 900 \ I , ::: , : ;’ I::;,! t ;,, ! ~ , ;4;” , ,, 800 “t 1“f! ‘, ..,,,-.~. . 700 ..------.------600 p‘y\ ““ -.. .’( 500 [11i)J ““~d +00 Iv ‘“”- { ---- 300 i .< ., *,.,,*., I 200 ; t ., ,11; :*: .., !’ ...”. . . . . Y,...... ---- .--” .------100: ~’”,i --- ~------p’ , OJ, , , b o 100 Qoo 300 400 500 600 700 800 900 1000’ 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 L.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 .“/. -lM I ,. ‘., ‘.. . ‘, ,/ ‘, , , ,’ ,‘, . ‘. ., ,, t, . G . , , . . { -. 2 0 w T 3 NVVH311 ‘ 3 NVVH% 21 CHEYENNE.WY DAYTON.OH FORT_WAYNEIN 100- 90 IOQ I — 100,0W 204 — ioo,ooa — Iw,ooa - -1,000 - -1,600 1,000 --1oo - - 164 160 90 ‘A 90- 150 80 80 1 70 II ,\ %0+ E 70 g 60 /’,f \ . [. ~,’’,’, , 50 \ ,., . !/, 70- 50 60 ,,, :,,. 40 ,/ ‘., ‘,,,, d\ 30 60 ~ 0 , 1 # 50 1 20 59 56 53 50 47 44 41 38 35 MPH 1 44 41’ 38353229262 JPH 46 45 42 39 36 33 30 27 24 MPH 20 30 60 82 131 170 251 337 461 7 37 65 92 152 254 402 606 82 i? 25 35 60 39 140 222 266 564 LANDER.WY LOUISVILLE.KY MILWAUKEE.WI MI NNEAPOUS.MN 150 I — 100.06-2 104- 200. — too,06Q — 1 W,ou 140- - -I,OQO – -1,3+0 90- \ --104 - - 164 :~ 130- 150- 120- 80- 110- 3 N gloQ - IOQ - E N 60- 90- .= ..- . ..: 50- 80. 50- 70- 40- 60~ MPH o~ MPH 30 ~ OJ, ,,, ,, , 39 36 33 30 27 24 21 MPH :~~3532zJ 262320 17 21 29 41 63 101 155 200 272 376 20 36 60 145 265 420 580 776 1025 MPH 19 31 75 110 218 356 564 645 1067 73 f26 237 401 661 850 1021 MISSOULA.MT PORTIAND.ME PORTLAND.OR RICHMOND.VA w. — 100,000 Iw] 200. —100,000 — 134,000 60- - -~,ooo — 100,000 - -I,lxlo - -1,000 --100 i20- - -l,OQO --to Q 80- --100 --100 110- i50 54- 70- \ 1oO- ,/ \ \ 90...... - 60 - K ~80- . X144 z40- . \ 70- , 50 - 60- .1 50. 30- 40 - 50- . 40- 20J, ,IIL III! , , 45 42 39 36 33 30 27 24 21 MPH 20- , ( 0. I I 20-, ,,,,, 40 37 24 31 28 25 22 19 1( IPH 5 42 29 36 33 30 27 24 21 MPH , 17 24 55 *O4 169 375 540 1029134 24 31 23 25 22 19 16 13 10 MPH 19 24 70 162 302 463 751 1041 14( 0 33 46 92 165 275 413 694 971 23 57 128 251 418 617 944 11521276 FIGURE 8. Point estimates of 100-, 1000- and 100 OOO–yr speeds versus threshold and number of threshold exceedances (based on 4-day interval samples, uncorrected data) . 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 1,31/235.8=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 PPCC 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 1.311227.9=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 1,31/2 times the wind speed with a 50–year mean recurrence interval? For Denver, 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 likely to be somewhat lower, that is, the distribution tail is likely to be somewhat shorter than that corresponding to c=-O.2), and X50=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. Assuming X50=26.8 m/s (60 mph), it would follow from eqs (13) and (14) &=2.5 m/s (5.6 mph). The estimated 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 recurrence interval of the nominal ultimate wind speed 1.31f2x~0=1.31/2x26.8=30.6m/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.3 31n this instance code writers attended to the apparent insufficiency of the nominal ultimate design speedby 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 s~ ~ = 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=2.14, rather than just 1.3. The mean recurrence interval for the 40.7m/s (91mph) (i.e., 1.311280 = 2.14112X62) 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 unaffectedly 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 RE)?ERENCES 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. Bingharn,N.H. (1990), “Discussion of the Paper by Davison and Smith,” Journal of the Roval Statistical Societv, Series B, 52, p. 431. Castillo, E. (1988), Extreme Value Theory in En~ineerinr, 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,” Statistic Neerlandica 44, pp. 45-68. de Haan, L. (1994), “Extreme Value Statistics,” in Extreme Value Theorv and Applications, Vol. 1 (J.Galambos, J. Lechner andE. Simiu, eds.), Kluwer Academic Publishers, Dordrecht and Boston, 1994. Ellingwood, B. et al. (1980), Development of a Probabilitv-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,” Technometrics, 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 Theorv and Armlications, Vol. 1 (J.Galambos, J. Lechner andE. 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 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 the Estimation 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) and histograms of four-day interval uncorrelated wind speeds (bottom) 29 . t [ I I t ~“ $?” 8n N N . 30 ALBuQuERQUE.NM - DAILY MAXIMA BILLINQS.MT - DAILY MAXIMA 1000 2000 I 90Q 804 t60+ 700 60+ 500 1000 444 300 5D0 200 100 0 o 80 0 13 26 39 62 65 0 10 20 30 40 50 60 70 ALBUC2UERQUE.NM -4 DAY NA7ERVAL BILLINGS.MT -4 DAY IN7ERVAL 40Q I 1 300 20Q 160 204 100 100 50 0 0 0 13 26 39 52 65 0 10 20 30 40 50 60 70 80 BI NGHAMTON,NY - DAILY MA%IMA BCiSE.lD - DAILY MAMMA ‘“”~ 1534 1 ‘1500 I 1Olw 1 06+ 500 500 0 0 0 13 26 w 39 52 65 0 5 10 15 20 25 30 35 40 45 50 66 N BINGHAMTON.NY -4 DAY INTERVAL BOISE.ID .4 DAY INTERVAL 3CQ 3“ ~ 254 20Q 150 104 50 0 o 13 26 39 52 65 0 5 10 15 20 25 30 35 40 46 50 55 BURLINGTON.W - DAILY MAXW6A CHEYENNE.CORR.WY - DAILY MAXIMA 900 ‘“w~ 800 700 600 500 1000 464 300 500 200 164 0 0 0 15 30 45 60 75 0 5 10 15 20 26 30 35 40 45 50 55 w u BURLINGTON.VT -4 DAY INTERVAL 1 I 250 1 I 0 15 30 45 60 75 0 6 10 15 20 25 30 35 40 45 50 56 CONCORD.NH . DAILY MAMMA DAYTON CORR.OH - DAILY MAXIMA 2004 ‘“”~ 1500 1500 i I 1006 1000 500 5W 0 o 0 5 10 15 20 25 30 35 40 45 60 55 0 13 26 39 52 66 CONCORD.NH -4 DAY lNIERVAL DAYTON.CORR.OH -4 DAY INTERVAL 364 4“~ 1 I 256 3W 2C4 - 2W 150 – 100 — 100 50- r 0 0 0 5 10 15 20 25 30 35 40 45 50 55 0 13 26 29 52 65 DENVER.CO - DAJLY MAWdA DULUTH.MN - DAILY MAXJMA Boo I 1 ‘ow~ 0 13 26 39 52 65 0 10 20 30 40 50 60 LA u-l DENVER.CO -4 DAY INTERVAL 200 103 !!0 0 o 13 26 39 52 65 0 10 20 30 40 50 60 FOR’.SMITH.AK - DAILY MAXMA FOIT_WAYNE_CORR.lN . DAILY MAXIMA i O&l 900 -1 n 902 i II SOQ + -I i-h 800 i 111111 700 -j 11111 704 1 Al Ill 600 + 111111 604 500 i II HIII 400 -j 111111 4W ‘w-l All Ill 3W 200 + 1111111 2s4 100-i 1111111 100 1 1 I 1 1 1 I 1 t I 1 1 ! I I 1 1 1 — 0 I # 0 o 13 26 29 62 65 0 13 26 39 52 65 FOtT_SMITH.AK -4 DAY lNIERVAL FoRT_WAYNE_CORR,lN -4 DAY INTERVAL 260 204 160 100 50 0 o *3 26 29 52 66 0 13 26 39 52 65 GREEN.BAY.WI - DAILY MAXIMA GREENSBORO.NC - DAILY MAMMA 900 I I 800 700 i 500 I 600 504 io3a - 400 304 500- 200 % 100 0 o I I o 13 26 39 52 65 0 13 26 29 62 65 GREEN_BAY.Wi -4 DAY INTERVAL GREENSBORO,NC -4 DAY INTERVAL i I 150 Iw 59 b o t3 26 39 62 66 0 13 26 28 52 65 GREENVILLE_CORR.SC . DAILY MAXIMA 2000 1504 1530 100Q 103U 500 50Q 0 o 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 GREENVILLE_CORR.SC .4 DAY INTERVAL HELENA-MT .4 DAY INTERVAL ‘w~ ““~ 300 30G 200 2CG 100 lW 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 s I -3 G $_ -3 I I I-. I L- - . I I 1 I I I I I 8 i? I I A I I ,-. I I I I I L.- 8 . . I I I I I I I I I .3:3 :g8: :8” . * .- 40 MACON.GA - DAILY MAXIMA 1000 1 90Q 800 1500 700 600 500 1000 400 304 5W 200 100 0 o 0 io 20 30 40 50 60 0 13 26 39 52 65 MACON.GA -4 DAY INTERVAL MILWAUKEE_CORR.Wl -4 DAY INTERVAL 200 1 3w~ 260 200 m 100 60 0 0 10 20 30 40 50 60 0 i3 26 29 52 65 MINNEAPOLIS_CORR.MN - DAILY MAXIMA MISSOULA_CORR.Ml - DAILY MLXIMA 1000 I n Wa - ‘“”~ J 800 — 1500— 709 — 600 — 500 — 1000- 4s0 - 3CQ — 600 - 200 — 104 — o 0 I i I I 0 13 26 S9 52 66 Jo 40 20 30 40 50 60 70 MINNEAPOLIS_CORR.MN .4 DAY INTERVAL MISSOULA_CORR.MT .4 DAY INTERVAL 20a 400 150 300 100 200 50 100 0 0 o --d13 26 39 52 65 0 10 20 30 40 50 60 70 MOLINE.IL - DS2LY MAXIMA OKLAHOMA_CITY.OK - DA2LY MAXIMA ““~ 1500 I 1000- 56Q - 0 I I I 1 I I P o 10 20 30 40 50 60 70 0 10 20 30 40 50 60 w MOLINEJL -4 DAY INTERVAL OKLAHOMA_C~Y.OK .4 DAY INTERVAL 1 I 250 1 0 10 20 30 40 60 60 70 0 10 20 30 40 50 60 d I I 8 I -=+’ I I [ r I e 1 L I , , I I I I I I os . 44 PoRTLAND.CORR,OR - DAILY MAXIMA PUEBLO_CORR,CO - DAILY MAXIMA 1064 ‘“”~ 90Q 8@3 764 600 1000 50Q 4W 3W 540 240 100 0 0 0 15 30 45 60 75 0 10 20 30 40 50 60 70 PORTLAND.CORR.OR -4 DAY INTERVAL PUEBLO_CORR.CO -4 DAY INTERVAL 250 250 264 200 150 160 Iw 100 50 64 0 0 0 16 30 45 60 75 0 10 20 30 40 50 60 70 RED.BLUFF.CA - DAJLY MAMMA RICHMOND_CORR.VA . DAILY MAXIMA 9&3 ma ‘“”~ 700 1500 1 I 600 500 434 300 200 too o 0 10 20 30 40 50 60 0 10 20 30 40 60 RED_ BLUFF,CA -4 DAY INTERVAL RICHMOND_CORR.VA -4 DAV lhlERVAL 40D 300 250 3D0 200 200 150 100 100 50 0 0 o 10 20 30 ho 50 60 0 10 20 30 40 50 : -8 8 I I I r . I ‘d [ . I I I 47 SHERIDAN_CORR,WY DAILY MAXMA SPOKANE.WA - DAILY MAXIMA 80Q 704 ‘“”~ 604 1500 500 409 1000 300 2@l 500 100 0 0 0 13 26 30 52 65 0 10 20 30 40 50 60 SHERIDAN_CORR,WV .4 DAY lNIERVAL SPOKANE.WA -4 DAY IN7ERVAL -i I 4“~ 250 i I 200 150 100 50 0 o 13 26 39 52 65 0 10 20 30 40 50 60 I 49 I I L s I I % 1’ I I I I . I I . I [ I I I I I I I . . I I L 1 , . , I I [ I 1 I I , I , I I I 1 I J 1 \, I I I I I I I I APPENDIX 2 Plots of parameter & and 95% confidence bounds versus 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) -- ...... -> ...... -.. ------.....1 .. -. ,, ;’.-. -, , ‘- ., ,. ,’:,~ ,, ,, !; ,,’ 1..,,, . ..’ ,;; , ,: ● ,! , , , ,’ W-! t., , ,: i $ . . . -1.6 * i -., 1’● ..! T I t .2 -1 .3 J 4646444240 d8’36 J4’i’” 2 30 MPH dl’ds’h’ dl 39 37 35 33 31 20 27 25 MPH $654526048444442402436 34 MPH d14947454341393736 33 MPH 13 17 17 27 29 34 34 34 34 34 1210182329303434343434 34 12 16 16 19 24 26 27 35 36 39 39 39 13 13 16 21 25 29 23 37 39 39 FORT SMITN, AR (31 YEARS) LITTLE ROCK, AR (39 YEARS) FRESNO, CA (37 YEARS) RED BLUFF, CA (4z YEARS) 1- 1- ------0.5- -. . . . . ,-. . . . 0- 5 -.> -. I (J z < J-, <.1- “., , ~ n ~ x *.. ~-: ~ ,, -2- -5 !, -1.5- ,’ ., ,’ 1-. .---, . , . --l 1 -2” ;; -3- -lo T 8745$41 39 37 ?!5 33 MPM , 434644424036363432 IP3i 36 34 32 30 26 26 24 22 20 18 16 MPtl 57656351494746434139 37 MPH 12 14 27 29 29 31 31 12 13 16 24 28 30 33 35 37 12 77 26 30 34 31 37 37 37 3? 37 12121518212735364042 42 SACRAMENTO, CA (39 YEARS) SAN DIEGO, CA (4S YEARS) DENVER, CO (33 YEARS) GRAND JUNCTION, CO (33 YEARS) . . -. . . -, . ‘.. . .- -. . . . - . . - - . . 0 . 0 ...... --. . . . ,. . ‘. ---- ..- v . Z $. ‘, : .1 . ‘..,, ‘w”,, ,’,/ A 1.., ,“ - ‘,.‘, k%’ ~ \ ‘,,, k ,,,, . ,1 , . . . ,! , ,, -2- ,, -2 ., * ., -2- r -3- r -3 464442403830343230 MPH ~ Mp” -3 J 37 36 33 31 29 27 25 23 21 19 17 m+ O 4346 4442 40 36 36 34 32 30 526043464442403636 3432 MPH 12 18 21 26 30 37 39 39 39 12 19 32 37 42 45 46 46 46 43 4( 2172224323333333333 33 14 26 2S 28 32 33 33 33 33 33 33 PUEBLO, CO (43 YEARS) HARTFORD,CT(44 YEARS) WASHINGTON, OC (39 YEARS) ATLANTIl GA (42 YEARS) 1- ‘.. - - . . . . . - .-. , ,. o- . . . ------. ---,. ------0- .- ..-, *.-. ,. *----- ...... - -1- I ., ,. ,.. . v ‘.,., ---, ------..’.. 0 .1 . ,,- ,.’ ,.‘..”. * .2- .- ,. . ~ ● ,.” ---- < -. . < . . ..’ * . . . ~ -, ; 2 . : .3 . , ‘, 0 .2. - y ‘, * --) , y ‘% ... . :: -4 ‘. 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MS (29 YEARS) COLUMBIA, MO (35 YEARS) KANSASCITY,MO(51 YEARS) ST.LOUIS, MO (21 YEARS) 1 1 1 1 . ‘. --- . 0 0 ------. 0.5 ...... -. -. -’..-.. -. . 0 -1 ..-----. -... . ---- ‘-, ‘. -.”.. -1 . 0 (2-2 : (2 1. , “. ‘, ; * -2 . z < .,,. < 9.5 < <-2 z ,,’, ~ Ig .3 \ t. , -. .,,, E N 0 ,. f. -4 , -1 ,.’ , ‘.-,’ -4 -2 -5 T. ---- -1,5 -. -5 . -6 , 1 ‘ ** -2 1 -6 J ( .3 t -7 ~ 14’h’4’1 ’39 31 35 33 31 24r 5’4’3’2’60’46 46 44 42 40 24 36 34 3 S755536 $404746434139 3735 45 42 41 39 27 36 33 31 29 27 25 MFW 131720232929292924 131821242629322234 36353 12141920252223424446 5051 15 16 20 20 20 21 21 21 21 21 21 SPRINGFIELD, MO (44 VSARS) BILLINGS, MT (49 YEARS) GREAT FALLS, MT (44 YEARS) HAVRE, MT (27 YEARS) 1 . ---- . -. ‘.. ‘. . . ..- -’ ..-. ‘. . ●✎✍✎ . . . 0 ------0 -. ,-...... -. . . . .-. . ‘. . . . c1 0 . . . . z . ,. .-,. z --- . < ● < -., ~ -1 -. s,’. . ~a -1 ~ . : y . o h x . . . . -2- ,* . -2 ,, ,, . . ..’ ,-. ,, ., 1 r -3- I 1 i149474542413937 353337 -3 W5i 63 61 59 d7 55 53 51 49 i7 46”” m 1361595755535146 474542 ~6’3’4’d2’d 0 43’44’& ‘42’JO’ MPH 318273436424244444444 16771927263242434746 g 3162126283134404244 U 15 t6 19 21 21 24 27 27 27 HELENA, MT(4s YEARS) MISSOULA, MT (43 YEARS) NORTH PLAllE, NE (31 YEARS) OMAH& NE (51 YEARS) f 1 . . . . --- . . . . --- . ‘~ 0.5 -- . . . . . ‘. 0.6- .-. .-’. . -., , ‘., . 0 -.,, . 0 v z z w < ● m .. ~ o- :.0.5 l., . ~ ,, .,1 S’ n J-, ; a -1 Y .0.5 - ,,., ‘,, ,. , , -1.5 ., ‘, .2 -5 , r -2 h 57665351494743424139 373 ml DU-W*4Z4UVJ **34W *PH 646260 S6565452504644 44 WH 85634626043434442 MPH 1$16283033414346464340 4 12 15 2335 40 43 42 43 42 43 13 13 17 20 22 25 26 31 31 31 31 21516222634404f46 VALENTINE, NE (27 YEARs) ELY, NV (4S YEARS) LASVEGAS,NV(20 YEARS) RENO, NV (4S VEARS) 1 ------0. -- . . . . 0,! -.. :~ ... . .- .1- ( .-. -. c1 u u .1 -, z ~ .‘.. - z < ~ .2- $-0, e ~ ,, h ~ : ~ . . . . ,“,. . : ,’ .-----, E ,, 0 .2 i * , ‘ ‘,,’, -3- , ,“ .1 ., ... ,, ,, ‘ ,-, ,, ,, ,, .‘!”)v. ,, -2- ,, ,, 1, -.. ‘! -3 4- ,.. -1.5 . !, ., ,- ., -5 J -2 -1 r 63615957555351494743 42 MP16 .4 -3-’ 55 53 51 49 47 45 43 41 39 37 35 IPH 54525046444442403836 34 MP+I 62605056545250444444 42 MPH 12 14 16 18 20 22 24 25 26 27 27 121721303643444949 4949 13 16 17 19 20 20 20 20 20 20 20 12161721253235414445 45 WINNEMUCCA, NV (38 YEARS) ALBUQUERQUE, NM (52 YEARS) ,.. . l-l 1- 1- ‘. -- .-.,- --- 0,6- -. 0.6- -’. .- . ..- .4* 0- ---- . -. ...-, --. 7,, - ---- *-. . ------.-. ,.., . . . 0 ,.. , ‘. , ,. u -. 0-1- , . 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V.,,, ~ ,, . ,, ,, ,, > ‘.4 ,, ,, ,, .1 ~. ‘n . -4 ,, ,, ,, -5 ,, , ,, ,, -1,6 -5 ,, *, .6 -..1 ,., +111 -it. l!r!!l. ,ll! lrrlrl > -? -2 56 56 54 82 60 44 46 44 42 40 36i36 MPH 62 SO 46 44 44 i2’40 36’&’$4’$2’2! b’&’&rh’&’&’d6’d4’d2’dO’;6< MPH k’h’41 39 37 32’3!3 3% 24 27< MFil 12141923293541454545 45a46 f4242a3s 394245424542 45* 121314*4202326262629 29 141624263743472050 60 BISMARCK, ND (40 YEARS) FARGO, ND (45 YEARS) WILL3STON, ND (18 YEARS) CLEVSLAND, OH (35 YEARS) 1 1 . ‘.. - ...... - 0.5- ---- 0 -...... -. 0 ... - ...... ------...... 0. ------.---’ . . v .1 . u .! $ “ . ~ ,, ,-* -, . ,,, , ,, .1 f0,6 ,, ‘..1’! -, ; .2 -. g - b Q ---- ?’-yw > -1- , ●, ,; ~., -. ,, -2 ,, ,’. .3 , ,, .- -1.5. ‘, ‘ ,’ ‘. , J-.1 .4 ? .2 r- -2 J 1 H!.957d5rJ35194745 k’i’r 1 3$ WH k 59 67 55 d3 Eh 49 dl’.b’k WH 50464644 ;:4~36 i452d0434544 h“’’’”40 36 36 14 f72127343437404040~ 4( 15 18 20 27 32 36 37 39 4f 44 15 16 18 13 132025272930243535 35 COLUMBUS, OH (30 YEARS) DAYTON, OH (41 YEARS) TOLEDO, OH (45 YEARS) OKLAHOMA CITY, OK (30 YEARS) 1 t 0.5 -’. . . 0.5------. m ---- . . o ---- o- -. ‘. 0 -. ------. 1 ‘. .-. . . o- ‘. . . -1. ,, ;..,5 ~ !-.,l ,, 0’ * ,. J?- . ‘. * ,, .,! J, ‘ w % -1- -* ‘ ‘, ,, .,!,’$ ,,’, ,, ,, ,------2- -, ‘, -, * ,.‘, ... -1.5- .-r , --f -3 , , r -3 ~ -2 ~ 04443444240233634 ml 57666361404745424139 37 WI+ 5351494744424+3937 3! 34526043434442403636 3464Pli 51619262629303030 13161925283023364041 41 -l_____131621252934294244 4! 13 16 19 27 29 30 30 30 30 30 30 TULSA, OK (35 YEARS) PORTLAND, OR (38 YEARS) HARRISBURG, PA (38 YEARS) PHILADELPHIA, PA (33 YSARS) 1 1 1> . . -...... -. ------. 0.s . 0 .. ---- . . . -.. ‘. . . . . -- , 0 . . . . . -. .-. . o ,. . , 0 ,, ,,. 3 .’..,, ● <-0.5 . ,. ~ -J?- ., * ,, ~ ,, 0 ..’ *-.’ -1 ,’ .-,, )’% -. ,, -2 -2 -, ‘t ,- .1.6 -4 . . . . -3 > -2 -5 1 -6 ~ !9 47 M 42 41 39 37 35’d3’&’{9r2; WI 45250464644424036 36 MPH 19 47 45 42 47 39 37 35 33 31 MPH 50434444424033363432 30 MPH 15 21 23 37 30 32 34 25 35 35 %6 35 4 14 15 20 29 29 30 24 27 38 13 13 20 21 26 20 35 37 38 36 13 16 22 25 29 30 33 33 33 33 33 SLOCK ISLAND, RI (31 YEARS) GREENVILLE, SC (42 YEARS) PmSSURGH, PA (18 YEARS) SCRANTON, PA (33 YEARS) . ..-...... ---- -. -.. . -. ”.. . ..-. . ..- -. . -. ..- -.--..,,.. y- . . . ., $-. , %-, ”,- ,,,$ , 7 , ,, ,, ; ,, ,, .2 . . 1, .-. . ,. # MPH 4 MPH &’d0’3’8’ d4’5’4’5’2’dO’i6’ J6’4$’42 MPH b“ 52’60 44 &“’& ’ion” ‘d6’d4’i2’2’O’& ?4’i4’d2’44’d6’36 2’4’32 2’O’;O’i i6’i4’42’io ’d6 12 15 19 21 27 31 31 31 31 31 31 1212141822 ;2826~ 12 15 16 17 Is 18 18 18 18 11 12172230323322233362 X CNA~ANOOGA,TN (35YEARS) HURON, SO (49 YEARS) RAPIO CITY, SO (42 YEARS) 1 1 --- 0 -. ‘~ ‘. .. ---- ..- ..-. -.. ----- ,., 0- -1 -- -.. 0 ...... -...... --”- -. ---- ,’ ! ---- ,, .2’, , ~-l- .. ‘-. ~ -3 “,,; :, ~ , ,.. . ~ z ~ f ., ,. w -4 .$ ,’. + n .2- ,, ,, $- 0 .$ ‘lJ’lJ ,;. ... .----, ,, . . . ,., .1 ‘,,’, ,, -5 ,, :,, f , ,.’ ,, $, .**’ -2 ‘% ~-, -3- t, ,,’!. 1 --- ,, .4 ;,t.- -7 .-l ‘- , -4J f I -8 IJ .2 J c.2’do 46 i6’i4 42 44’26 36 24r MP61 !i2’5b’46 44 44’& ’40 26 36 2’4’32 30 MPH 6’7’65 62’3’1’2’0 67’~5’C13 6’1’i4’47’i6 MPH 6i’;2’6b’&7 56’6’4’d2’50 46 46 U 42 MPH 12 12 13 15 20 21 26 24 24 35 1214192124263131332233 22 1215172123272440413547 49 1418243536374142424242 42 AMARILLO, TX (24 YEARS) MEMPHIS, TN (21 YEARS) NASHVILLE, TN (34 YEARS) -. . ‘~ . . . . 0.5 *- -. .--, ---- . . . ----- . 0.5 ! . . --- I -. ---- ,. ‘.- . , ‘, 0 -. ~ . ---, ,-, . % ,, --.,, -..8”, .. ”., -----. -4.5 ‘--) $0 -.. . c. . . -1 .! ., .’-,). .,/ ‘, ‘.,’ I -1 r -2 ) ‘ -1J 3’4’1’;s 42’ mli i3’i6’i4’42 io’38 36 34’;2’30 MPH 55 53 51 49’47’45 42’41 39 37( MP14 57’i5’4’3’d1’4’9’i7’ w 42 40 28’36 34 32 30 28 26 21 2 16 19 2S 26 31 32 33 34 34 12 13 17 20 21 21 2t 21 21 21 2 12 15 16 22 26 32 34 24 24 34 15 20 21 24 28 22 35 26 36 36 AUSTIN, TX (37 YEARS) DALLAS, TX (32 Y2ARS) EL PASO, TX (32 YEARS) SAN ANTONIO, TX (36 YEARS) 10. 1 1- ‘. 6 . . . . , . -. 0 . . . . . 0.6- ‘. . ●..” *... ‘-. . . 0 . 0 4, *---- -1 ,,, , 2 ,,, . . . . < -5 -, i.; \ ‘. ... :,$ g g . . . -,!, f -2 P 0 . . ., ,t~ , -10 f“y-J -1- ,’ “., ,, ,’ .2 ,, -15 , ,’ .1.s .-.1 i .20 r r -4 r, -2 I k 44 42 40 38 36 34’32 2t’2’6’2i o ./3’’’’’’’”454442402’8 363432 67 55 6dS1’49 dl 43 & 41 39 37 35 MPH 44’J4’&’4’240262634’ 32d02’6MFll 420262336363737373737 2 16 19 27 31 31 32 32 32 32 15 16 16 26 30 32 32 32 32 32 32 32 13182225282536333636 3S SALT LAKE CITY, UT (46 YEARS) BURLINGTON, VT (40 YEARS) LVNCHSURQ, VA (44 YEARS) 1 --- .- 0,6 . --. 0- -.. . . ------.. . . . -. , .. ,’.”. . .,, ----- ‘.. . . . 0 -1- \ .-. . .\. . m (J ‘. z . ,-- -\. - N . . . ..1’ , ,, . < ., ... <-0.5 /. r ,,, .2 ~ . ‘., * ~ > ,“$ n . ,* -., , ,. I-, J’.,. . ,$1 .1 -, -2- k ‘,-. q ,’ . 3 ,’ 1’, ,: ‘, , ,, , ,, ., ‘, $ -1.6 ‘. , -4. . -2 ~ -2 > , 1 -5- 1- 54526043434442403836 MP64 49474042413937352231 29 MP61 424436362432302626 MIW 42 41 36 37 35 33 31 29 27 25 23 MPH 121820293026424344 43 12151824323537404040 40 121623302442444444 ls 217.227303333623362 22 NOR7H HEAD, WA (41 YEARS) QUILLAYU7E, WA (21 YSARS) SEATILE, WA (20 YEARS) SPOKANE, WA (47 YEARS) 1 1 1 0 ------, .- --- !. -.. . ----. ‘. . . . -.. -1 ------0.5 -, 0 , ., , ,’ ‘. $..- -2 -. c1 . z .3 . -4 , . . <0 ., ~ n &---.. l-u -4 F -0.5 .--, , -2 . . . ., ●✍✎ i !.. . -1 > -3, I 7 -3 ‘ , n 70 68 66 64 62 60 56 MP65 34 32 30 20 26 24 22 20 18 16 *4 42 40 23 36 34 32 30 28 26 24 22 Umi 5149474243413037353331 29 MPH 13 15 17 20 36 41 41 41 13 15 21 21 21 21 21 21 21 21 21 12 17 17 19 20 20 20 20 20 20 20 :U$4202533374244434747 4747 TATOOSH ISLAND, WA (54 YSARS) GREEN BAY, WI (36 YEARS) MADISON, WI (41 YEARS) MILWAUKEE, WI (42 YEARS) 1 1 ...... - .0-.. ● ✎ ✎ ✎ ------. ,. ..- . . ✎ ✎ -. ‘-. , 0 o -.. ‘------. . . . ,.. . .-. ,’ .,.. ,, -. -. ,’. . ,. ,* \ -., * ,t,’-”~ , ., W-m-i G*.t . , ,, ‘,,, , =’s .- ,,,-’ , . -2 ,,, -z , -l!; . $. . -2 J .3 .3 )0’d8’;6’dk’(2’dO’iO’i6’d4’d2’ 2b7 MPH d6’3k’5’4’ i2’6’0r4b’&’&’d d MPH ?Z’d6’i4’;Z’5b’d4’k’ 4i’i2’k’h MPH h’ i4’5’2rdo’JZ’i4’44’dZ’k’ 12 17 19 34 40 44 46 52 63 54 54 12 13 17 21 24 27 26 32 3: 3( 13 13 14 t? 17 20 27 32 36 39 41 14 30 23 26 33 29 41 42 42 :: CHEYENNE, WY (46 YEARS) LANDER, WY (42 YEARS) SHERIDAN, WY (44 YEARS) 1 1 ‘~ ------‘-. . --- 0 -. -. 0 . . ------. ‘“. -. ,“* . ,- -1 ..-, f.,.. ,. 1, .-, h , ~ ., ..,. % > ,. -3 . , ‘. ,,’-. , ,’ . ,. ,, *; .3 b. -4 .--1 ,4 f #“ .sJ 4 1 -5 -1 d4’&’d0’ 5’6’5’6’;4’i2’iO 4k’44 44* ml 5’63 61 59 67 5’5’3’3 5’1 44 4? ml 33 61 54 67 55 53 61 49 47 45< MPH 12162531364041424443 44 21415212230333436 39 13162730342535394144 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 ABILENE.TX (15 Y124RS) ALBUQUERQUE.NM (19 YEARS) BILLINGS.MT (26 YEARS) 110 m- 400 — 160,000 — lGO,om — 100,000 - -1,040 - -1,000 - -*,000 foo h --1oo --1oo --100 60 334 \ 70 ‘\ \ 1,, ~ g 240 ./, ,, ,, 60 \ ‘-. “, 100 --. \ 30 -.-~-, : 4 / . . zo~ zo~ o l-r, , 43 40 37 34 31 26 25 22 19 MPH 42393633302734 2jq6 r4PH ‘L.L- ao~5249464340373437 2 !Ptl 5047444120363229 26 MPH 16 52 96 191 3LW 536 741 925 1045 16 41 65 136 247 428 659 912 116 19 30 69 39 199 315 455 690 90 16 40 63 121 203 326 566 823 1129 BINGHAMTON.NY (20 YEARS] BOISE.ID (2s YEARS) BURUNGTON.VT (19 YEARS) CHEYENNE_CORR.WY (15 YEARS) 200 70 90 — 1to,ooo — 100, OW — 100,000 — Ioo,ow - -IJOO - -1,000 - -1,000 - -1,000 - - 124 --102 --1oo --*OO 24 60 l\ 30, -! ,’.- u gso . x76. .. ‘,- “. -. ‘. >--- s 70. \-, . . . . - ‘. 64 \ 40 * \ 44. k \ 0 r, , 9-I 30 -1 m. -1 , , , , , 1 30 35 32 29 26 23 20 1 IPH ! 29 36 33 30 27 24 21 11 !PH 9565350474441333k MPH 8 26 54 117 235 412 693 993 72 F 42 64 133 254 461 74S 4044 1% o 30 60 82 131 170 251 337 461 DENVER.CO (15 YEARS) DULUTH.MN (15 YEARS) 140. 110- — 134,000 — 100,000 i30 - - -I,LXIO - -1,000 --140 100. --400 WO - 30- 110. 1oo- e4- 90- \ E E70. , 30- ,\ .\ 60- . 60- . 50- 54- 40. 40 ~ 40J, , , , 8 t 30 ~ ~p” .~ r4PH 47 44 41 33 35 32 29 26 23 MPH 33 35 32 29 26 23 20 MPH 16 36 65 92 162 254 402 603 832 77 56 79 139 274 4% 537 747 369 20 47 72 131 206 362 560 790 902 FORT.SMITH.AK (17 YEARS) FORT_WAYNE_CORR.lN (15 YEARS) GREEN_BAY.M (15 YEARS) GREENSBORO.NC (19 YEARS) 120 T 400 2CQ — 100,000 — 164,000 —100,000 — 16Q,000 - -1,000 - -1,000 - -1.000 - -1,000 120- I --100 100 --100 --IA I 110. 150 30Q *OO - 90- a ,6\ Xloo ,\ gao- E 200 Ii /\ 70- I d,’,’, k . 60- ,- \ \ ● .’ ‘,\’\ -- E4 ., ,t.-t. im \ -.=, *,,1 50- ‘“+ s, 40- 0 , r 30- , < o ~ ;:~ 353229262320 ~ iPH BJ45 42 39 36 33 30 27 W MPH 4126353229262320 1’ IPH B 35 26 23 20 17 14 MPH 62 i03 167 251 45S 667 31 725356089140222266 53 22 36 66 127 223 360 604 817 10 !0 29 61 117 202 234 547 770 *043 GREENVILLE_COR R.SC (19 YEARS) HELENA.MT (26 YEARS) HURON.SD (15 YEARS) LANDER_CORR.WY (15 YEARS) 324- iw — 100,000 — Im,ooa - -1,000 - -1,600 - -1,000 150 --100 --100 1 --100 90- 250- 140 :F 120 60- 200 120 E 70- gmcl %10 -\ ;\ 1 -- 164- , 1’ \l\ , 60- ioo ~, \,\ .A, -...., 90 ‘ bd~\ l\l-.’ ,1,\//\ \ --- ,. , . >---/. 2 >.- .,. v.. ,t ?,. ,/, , w , , cw - 50 i,L d?l “ , ,., 70- \\. , : -- o~ ~p” 40~ o~ 60- 3431232622 4345422936339027 Z #PH 46 45 42 39 36 33 30 27 24 i!PH 54 514645423936333 MPH 20 46 106 236 419 672 1051 13201497 i6 27 60 126 204 365 572 875 12 16 22 43 93 125 224 232 536 811 20 23 4* 63 101 155 200 272 % LANSING.KH (15 YEARS) LOUISVILLE_CORR. KY (15 YEARS) MACON.GA (16 YEARS) MILWAUKEE_CORR.Wl (17 WARS) 140- 104- 90- — 100,000 — too,ooo — 10.3,0m %20 - - -I,oao 90 --100 120- 30- 110- 60 100- 70- 70 ;\ } =90- K. ~ , \l \ x x $i l\ ~ 30- .~ 60- . ““, \ ,,-, 50- . . 70- ‘ .\ .- 50 . 50- -.-’ ---. ~,\ . . . . ~:% 5n - , 60- . . \ 40 - 40’, , r- o~ 30 ~ .~ 46’42293633302720 2 ,lPH 29 36 33 30 27 24 21 16 1! !4PH 26 26 22 19 16 MPH 33 30 27 24 21 MPH 20 37 M 07 158 256 377 603 81 20 36 80 145 266 420 533 776 IW 17 25 29 71 121 267 416 607 930 19 31 75 %0 218 356 560 645 1067 MINNEAPOIJS_CORR.MN (15 YEARS) MSSOULA_CORR.MT (25 YEARS) MC3LINE.IL (15 YEARS) OKLAHOMA.CIN.OK (16 YEARS) 20Q - 200 12Q -! 70- — 16Q,00Q — 100,000 — 100,OLW — Ico,om – 1,640 - -1,000 - -1,000 - -<,000 . . 164 --104 110 - - 100 --104 66 154 164 104 / 66- I /’ .- .\ ,, w -. ,1 u Elm E1OO x“ * %55 ,-, l\ f ,- \ ~- fl -- . 4\ II ~/ J,.-\ \/A\ ..- .X --..\- 60 . -\ . . ...>-- -=. I ‘/, ,%, ..- 70- -’ ..-.1 b 66 50 /1’, , /\ ~ 4 L {./, ~ -’,””. \ ~ k . \ ,’*\, 64 # % ‘.\, , . I o~ 0 r, r 54~ 40 J,,, , 36 35 32 29 26 23 20 77 MPH 15 42 39 36 33 30 27 24 21 iiPH 47444136 ~ 45 42 39 36 33 30 27 24 MPH 18 31 73 i38 2S7 401 661 850 1021 7 34 55 104 139 375 640 1029134 17 27 46 63 fW 172 273 400 d 42 74 120 224 378 566 763 943 POCATELLO.ID (21 YEARS) PORTblND_COR.ME (20 YEARS) PORTLAND_CORR.OR (23 YEARS) PUEBLO_CORR.CO (17 YEARS) 140- 140- — 100,000 — 100,004 W 130- - -1,040 124- - -1,600 I --100 .- 164 I 12Q - 120- 75 110- 110- 70 100- 1o4- 90- 90- ~ E ~ 65 3a- 60- $ 70- 70- -. 60 50- 6@- 60- 54- E-5 40- 40- 20’ , , , 30- , , , , r , o~ 60~ # 41 24 35 32 29 26 23 20 MPH 44 44 26 35 32 20 26 23 20 MPH 2S 35 32 29 26 23 MPH 66 53 50 47 44 41 33 35 32 MPH 56 69 179 403 618 877 1122 i301 21 56 69 179 403 618 877 11221301 16 22 26 56 119 190 325 611 769 18 31 44 79 122 215 303 435 655 RED_BLUFF.CA (21 YEARS) RICHMOND_CORR.VA (16 YEARS) SALT_LAKE_CITV.UT (26 YEARS) SAN_DIEGO.CA (17 YEARS) 90- 70 Iw - 500 1 — 140,004 — 100,004 — 160,000 — 1LM,ooa - -1,000 - -1,000 - -1,040 - -1.600 --100 - - ,100 96. --1oo I’m \ - - Oo- 34- 70- 56. . 70- E J \ E z . . - ., . 60- 40- 64- 50- 20. . 40- ‘“w’’-+- 40 ~ 20- r 36~ o~ 4542393633302724 21 ilPH 34 31 26 25 22 19 16 13 10 MPH 26 35 32 29 26 23 20 MPH 19 16 13 11 MPH 16 24 67 91 136 221 393 595 81: 23 57 126 251 418 617 %t4 11521276 20 52 92 164 330 614 667 11831391 21 36 ~ 1;; ~~ 604 952 1290 14: SHERIDAN_CORR.WY (17 YEARS) SPOKANEWA (2S YEARS) SPRINGHELD.MO (24 YEARS) TOLEDO.OH (15 YEARS) 364 69 — Ioo,ow — IO0,06Q A - -1,640 – -1.000 - -1,000 90- - - I’w --1C4 90- 250 70 80- 80- 200 ,\ 70- % 70- s %150 g60 60- , \ 60- Iw 50 - f\ :\ 50 .--. .> - --- 60- 50 . 40 - ,~ 40~ .~ 0 40 ~ IPH 35 32 29 26 23 2{ MPH 2936333027242118 1 IIPH 33 30 27 24 21 MPH 16 26 47 91 144 252 367 5.32 7! 18 51 69 $21 210 351 523 855 11 23 37 08 440 257 467 670 878 11 8 29 63 116 235 325 502 849 1162 TOPEKA.KS (2S YEARS) TUCSON.AZ (15 YEARS) WINNEMUCCA.NV (15 YEARS) YUMA.AZ (19 YEARS) 100,000 100 — Ioo,oco 20Q — 100,000 -I,cao - -1,000 - -1,640 --100 90 --100 --100 — 350 300 - 3Q \ 70 \ E x gloo 60 . . 50 100 r 60 “L-L 40 o 1 f o~ 20 > , , r a 1 , 42 29’ ’36 33 30 27 24 21 11 MPH 293633302724211 flPH t542393633302724 2’ MPH 12 29 36 33 30 27 24 21 18 MPH 21 48 78 *2a 224 373 590 772 94 *7 39 66 157 253 444 675 875 10 17 27 44 92 777 365 437 744 97 20 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 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 71 *U.S.sowmmmm ~ OFFICE:1995-386-627/37909