ML20094G844

Rev 0 to Review of Enercon Calculation NPP-1-SBO-009,Rev 1

ML20094G844
Person / Time
Site:	Cooper
Issue date:	02/24/1992
From:	NEBRASKA PUBLIC POWER DISTRICT
To:
Shared Package
ML20094G841	List: ML20094G837 ML20094G844 ML20094G850
References
NEDC-92-023, NEDC-92-023-R00, NEDC-92-23, NEDC-92-23-R, NSD920243, NUDOCS 9203030439
Download: ML20094G844 (165)

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,g..,a> a, i t 3 p l J ATTACHMENT 1 TO NSD920243 ' COOPER NUCLEAR STATION [ l NRC-DOCKET NO. 50-298, DPR-46 I i - i e 1 9203030439 920227 PDR ADOCK 0500029B P PDR ...._. _... _ ~,,, _...

'l i N:braska Public Power Di:trict 4 DESl2N CALCULATIONS COVER SHEET k S V 'C d OI E M EACOM C"d C ' Claiculation No. 9 '2

• O 'J 3 Title E9-/9 U M P P 4_ - 5 8 0 - 0 0 9 Supersedes Calc. No.

System / Structure E6 Task identification No. N4 Component ^J 4 Design Change No. N4 C-lec +ro e A/. Classification: } Essential Discipline O Non-Essential

• ASME Stress reports shall be approved by Registered P.E.

NPPD Generated Calculation Non NPPD Generated Calculation Prepared By Date Prepared By EN E44 8 d Date / 9 E tconomaan numet j Date J.gs-9 2. -l Checked By Date NPPD_ Reviewed By M-y Design Venfication By Date NPPD Approval d H [ tE M Date NY-d Approved By Date Calc. Description

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References:

Attachments _:

1. USAR YeI T; TTT ? 0 A.

M PPK-5 8 0- 00 9 - c )} A H. l

2. TECH. SPECS.

NA B. f i i i-i i i Rev. Prepared Checked or Design Approved j No. Revision Description - By/Date Reviewed By/Date Verification /Date By/Date j-(CircitOne) - l.

i NEDC 92-023, Rev. O Sheet 1 of 3 NON-NPPD GENERATED CALCULATION: PREPARED BY: _ENEPCoN DATE_1-7-92 REVIEWED BY S. hh DATE: 7.-21-9 2. v G7 NEDC 92-023 " REVIEW OF ENERCON CALCULATION NPP1-SBO-009, REV. 0" A. PURPOSE This NEDC utilizes existing weather data taken at the CNS site and from the National Severe Storm Forecast Center (NSSFC) to determine the SW and ESW groups for CNS and provide justification that the NUMARC values 87-00 values are not applicable to CNS. B. REOUIREMENTS 1)

Court, Arnold, "Some New Statistical Techniques in Geophysics",

Statistical Laboratory, University of California at Berkeley, circa 1951. This is listed as to this NEDC. 2) Simiu, Emil, and Robert H. Scanlan, Excerpts from Wind Ef fects on Structures, Second Edition, John Wiley & Sons, New York, 1986. This is listed as Attachment 2 to this NEDC. 3) Cooper Nuclear Station Site Specific Wind Speed Data. This is listed as Attachment 3 to this NEDC. 4) Variation of Wind Speed with Elevation (Attachment 4 to this NEDC). 5) Summary of Probability Plot Correlation Coefficient (PPCC) Method, listed as Attachment 5 to this NEDC. 6) NSSFC Program 'TORPLOT' Output for

CNS, listed as to this NEDC.

C. ASSUMPTIONS 1) The data provided by the NSSFC is assumed to be correct and the computer calculations provided are performed correctly. 2) The wind speed data provided to ENERCON by NPPD is Correct.

... ~. ~... -... ~ ~ .. ~. - NEDC 92-023, Rev. O Sheet 2 of 3 !10N-NPPD GENERATED CALCULATION: PREPARED BY: _E}[EE. CON DATE_1-7-92 REVIEWED BY: DATE: 2-71-9 2. v v 3) Court's technique and simiu's technique both employ Type I distributions for description of the C1fa wind ~ speed database, which provides the optimum fit-at most weather stations and is considered reasonable to assume an appropriate method. D. METHODOLOGY 1) This calculation employs extreme value statistical methods to estimate the maximum wind speeds at CNS based on the existing site-specific database. The calculation ~ first follows the extreme value statistical methods developed by Arnold Court and Simiu. 2) Data at the 10-meter elevation will be used for most of the calculations and are considered the reference. basis. Missing data at the 10-meter elevation was completed by I using data at other elevations and correcting I (transpositioning) to the 10-meter elevation. The 10-meter data was conservatively used to. determine the probability of occurrence of winds at _ the 30 meter-elevation without transpositioning the. data to the 30-meter elevation. 3) Simiu's technique-for predicting extreme _. values'of wind-was used as a second check for the Court Method With similar results. 4) In order to verify that the Type I distribution is appropriate-for characterization-of_the CNS wind.. speed database, the' probability pl'ot correlation coefficient (PPCC) method has made used to determine.the best fitting l distribution. This evaluation indicates that the Type-I distribution-is correct. .j E. CONCLUSIONS 1) Court's method resulted in a probability of 6.7803E-8 /yr, which places CNS.in ESW Group 1. 2) Court's method-resulted in a h of 1.012E-3/yr, which 3 places CNS in SW Group 2.

NEDC 92-023, Rev. O Sheet 3 of 3 NON-NPPD GENERATED CALCULATION: PREPARED BY: ENERCoM DATE: 1-7-92 REVIEWED BY: DATE: 2 -?- (- 9 2. 3) Using the 'TORPLOT' data provided by the NSSFC, the probability of tornado occurrepce Q 2.357*10' event per 2.357*10 yr for use in the CNS year, giving a h = 2 severe weather evaluation. 4) The simiu's method resulted in a probability of 3.264E-8 /yr, which places CNS in the ESW Group 1. 5) The Simiu's method resulted in a h of 7.0954E-04 yr", 3 which places CNS is SW Group 2. 6) The PPCC method resulted in a probability of 1.200E-5 /yr, which place CNS in ESW Group 1. 7) The PPCC method resulted in a h of

4. 528E-04, ' which 3

places CNS in SW Group 2. 8) The error analysis, discussed in Section 7, indicates a very high confidence that the extreme-values calculated are conservative. 9) Using Table Sa of NUMARC 87-00, the. combination' of weather groups ESW1 and SW2 indicates that CNS is a 'P1" plant. 10) The methodology used in this NEDC is-' correctly used and the _ results have been correctly calculated. It is concluded from' the values generated in this NEDC, the.CNS minimum allowable EDG target availability is 0.950, i

. _ ~. I *E U SHEET JOB NO NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION, // HOLCOMBf-CLIENT NPPD ORIGINATOR ~E.6,W v, * ' 'g9 2 REVIEWER e--'. .. t o -4PPROVED CALCULATION NO. NPP1-SBO-009 CALCULATION NPP1-SBO-009 COOPER NUCLEAR STATION (CNS) SITE-SPECIFIC WEATHER DATA EVALUATION FOR STATION BLACKOUT (FBO) 9 l l l I r

i i SHEET JOB NO ?!P-119 UATE 1/7/92 PROJECT Cils STATION BLACKOUT ORIGINATORE.__HOLCOMBF,'[ SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD REVIEWER m APPROVEDJ " CALCULATION NO. NPP1-SBO-009 1 REVIEWER'S STATEMDIT This calculation was reviewed in detail and was found to be complete and accurate. The conclusions were fouM to reasonable and justified. Major portions of the review consisted of the following: 4 1. References were reviewed for proper citation. Referenced equations were reviewed and found to be correct regarding their appropriate use and accuracy of transposition into this docunent. 2. The methodology and assumptions were reviewed and found to be both reasonable, apywstiately implemented, and conservative. All numeric calculations were verified. Specifically, the following items are noted: All CIS weather data base monthly extreme wind speed values listed in Table 1 were cross-checked against the original Cis documentation and no errors were found. Hiiting of the data base followed appropriate l l methodology and the calculations were correct. The mean and standard deviation of the set of monthly extreme wind speed values were independently calculated ard found to be correct. - References in this calculation to Court's technique for extreme wird i values (docunentation in Attachnent 1) were reviewed. The criteria for the use of Court's technique for extreme wind values were met, ard the numerical calculations were found to be correct. The conclusions I regarding the CIS -station -blackout weather groups =were deemed to be f accurate. t I 1 ] .. -. ~

SHEET JOB 110 ffP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION j # gI CLIEllT NPPD ORIGINATOR E. HOLCOMIM - REVIEWER A APPROVED _ /, J CALCULATIOli 110. NPP1-SBO-009 REVIEWER'S STATEMDIT - he use of NSSFC tornado data was considered to be consistent with NRC methodoloav in NUR M-1032, and avvivut-iately used in this calculation. References in this calculation to Simiu's technique for extreme wind values (doct 6. tion in Attachment 2) were reviewed. Se criteria for I I the use of Simiu's technicue for predictira extreme wind values were met, ard the resultina numerical calculations and conclusions regardirq the CIS station blackout weather aroups were found to be correct. Re PPCC methodoloay documented in Attachment 5 was reviewod and considered aus us iately applied in this calculation, deicini.catirg that the CIS data is fitted by a type I distribution. Se evaluation determining the level of confidence in the CIS ESW and SW weather group categories is aps us iate. 2e reviewer agrees with the conclusion that the CIS minimum allowable EDG tarcet reliability is 0.950. I i fcui., lh",.nst k <t /~ / 2h2 he l m e e ~

SHEET i JOB NO NP-119 DATE 24ch2 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION g,L CLIENT NPPD ORIGINATOR E. HOLCOMBt7 REVIEWER -W APPROVED CALCULATION NO. NPP1-SBO-009 TABLE OF CONTENTS REVIEWER'S STATEMENT 2 OBJECTIVE 6 CALCULATION OVERVIEW 7

1.0 INTRODUCTION

8 2.0 METHODOLOGY 10 3.0 CNS WIND SPEED DATABASE 11 3.1 Correction of Wind Speed Data to Other Elevations 11 3.2 Editing of Database 12 3.3 Mean and Standard Deviation 13 4.0 Application of Court's Technique-for Extreme Values 14-4.1 Extremely Severe Weather (ESW) Group 16-4.2 Severe Weather (SW) Group 17 4.3 CNS Tornado Data 18 5.0 SIMIU's TECHNIQUE for ESTIMATING MAXIMUM WIND SPEED 20 5.1 ESW Group 21 5.2 SW Group 22 6.0 PPCC METHOD 23 7.0 ERROR ANALYSIS 25 7.1 ESW Group 25 7.2 SW Group 26 8.0 EDG TARGET RELIABILITY 28 9.O

SUMMARY

28-

10.0 REFERENCES

29-r +. ,.r.,--. -s.,e- .r.- ., ~..---.,,. ,.y-.,, .:.i.n m---, ,,r+,,,,f, ,-m,

l SHEET f~ JOB NO NP-119 DATE 44o/fz-PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIEN7 NPPD ORIGINATOR E. HOLCOMBt REVIEWER M APPROVED CALCULATION'NO. NPP1-SBO-009 Summary of Maximum Wind Speed Data TABLE 1 for Cooper Nuclear Station 30 1975-1990 Cooper Nuclear Station - TABLE 2 192 Samples 31 Extreme Value Analysis 35 TABLE 3 Predicted Extreme Wind Based on TABLE 4 Extreme Value Type 1 Distribution 36 TABLE 5 Tornado Damage Scale 37 TABLE 6 Excerpt of Results from 'TORPLOT' Program in the Vicinity of Cooper Nuclear Station 38 + ..y ,_.-,c..- -y-- ,.m--,. +-c.,m, , y yp~ r,-., ~,

SHEET 5 4 JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT 1 SUBJECT SITE-SPECIFIC WEATHER EVALUATIO CLIENT NPPD ORIGINATOR E. HOLCOMB i REVIEWER /b APPROVED CALCULATION NO. NPP1-SBO-009 OBJECTIVE 1 The purpose of this calculation is to evaluate the Cooper Nuclear I Station (CNS) site-specific weather database and estimate the extreme wind values for various return periods. The results will be used to evaluate the CNS weather groupings for determination of the requisite emergency diesel generator (EDG). reliability, according to the criteria of NUMARC 87-00, RG 1.155 and the Station Blackout (SBO) Rule, 10CFR50.63. This calculation supersedes calculation NPP1-SBO-005, since more rigorous techniques will be used to estimate the extreme wind values. j l ,---,w ,y

w h SHEET JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECTSITE-SPECIFICWEATHEREVALUATION[M CLIENT NPPp ORIGINATOR E. HOLCOMB REVIEWER M I? PROVED ~ CALCULATION NO. NPP1 ' O-009 This calculation involves the following CALCULATION OVERVIEW principal steps: 1. A brief introduction is provided. 2. Highlights of the analytical methodology are presented. 3. The CNS wind speed database is presented. Preparatory to application of extreme value statistical methods, the mean and standard deviation of the data are calculated. 4. Court's method is applied to the CNS data. Extreme winds are calculated for various return periods. The CNS station blackout weather groups are determined. Tornado data in the vicinity of CNS are presented. 5. Similar to item 4, Simiu's technique is applied to the CNS data to determine the extreme winds and the SBO weather-groups. 6. Both Simiu's technique. and Court's method employ a Type I distribution to fit the CNS data. The Probability Plot Correlation Coefficient (PPCC) method is used to establish - that a Type I distribution indeed provides the best fit. 7. The level of confidence in the weather group determination is addressed. 8. The CNS emergency diesel generator target reliability is established, based on NUMARC 87-00 methods and the CNS weather group determination. 9. A brief summary is provided. References, tables and the necessary attachments complete the document.

SHEET <f JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMBM REVIEWER <!< '? APPROVED CALCULATION NO..,JPP1-SBO-009

1.0 INTRODUCTION

Using the methods outlined in Chapter 3 of NUMARC 87-00 (Reference 1), weather data, power grid design, availability of emergency alternating current (EAC) power supplies, and diesel generator test data all factor into the required SBO coping duration and the target EDG reliability. Both of these have been determined for CNS. As documented in Reference 2, the required coping duration is 4 hours, and the target EDG reliability is 0.95. Weather data factor into the above determinations. As noted in Reference 2, the CNS severe weather (SW) group and the extremely severe weather (ESW) group were previously determined to be 'SW2' and 'ESW1', respectively. This resulted in an offsite power (OSP) design characteristic group of Pl. The CNS Station Blackout Safety Evaluation Report (Reference 3) was issued in August of 1991. In the SER, the 'SW' and 'ESW' classifications were challenged, the result being that the OSP grouping and the target EDG reliability for CNS were also questioned by the NRC. The staff recommended that: "The licensee should use data provided in the NUMARC 87-00, i.c, severe weather (SW) group "3" and extremely severe weather (ESW) group "3", or provide further justification to demonstrate that the NUMARC values are not applicable to Cooper Nuclear Station. In lieu of the above, the licensee should provide additional plant specific weather data to include the extreme weather conditions in support of 1ts ESW and SW group classifications." In Reference 4, the Nebraska Public Power District (NPPD, or the ' District') committed in its initial response to-the SER to: " revise the existing Cooper Nuclear Station (CNS) plant specific weather calculation to further support our SW and ESW group classifications." Since CNS has a meteorological tower and over 16 years of data, it is appropriate to use the site-specific weather data rather than the NUMARC 87-00 values. l l

SHEET 9 JOB NO. NP-119 DATE 1/7/92 l PROJECT CNS STATION BLACKOUT SUBJECTSITE-SPECIFICWEATHEREVALUATIO l CLIENT NPPD. ORIGINATOR E. HOLCOMB/ REVIEWER /#b APPROVED ~ I CALCULATION NO. NPP1-SBO-009 Using CNS data and methods of extreme value statistics, the weather-data groupings, the OSP design characteristic and the target EDG reliability will be developed in this calculation to meet the above commitment. l l l i \\ ~

SHEET /0 JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOM REVIEWER 0# APPROVED CALCULATION NO. NPP1-SBO-009 2.0 METHODOLOGY This calculation employs extreme value statistical methods to estimate the maximum wind speeds at CNS based on the existing site-specific database. Two primary methods are followed, both of which use a Type I exponential distribution to describe the CNS wind speed data. These methods enable calculation of the frequency of occurrence of high winds. The frequency is then used to determine the CNS station blackout ESW and SW groups mentioned in Section 1. Given a dataset consisting of 'N' extreme values, each one of which has itself been derived from a large number of observations, extreme value statistics provides a way to predict maximum or minimum valtes occurring outside of the recorded time interval for the data of interest. For example, suppose we have a dataset consisting of the maximum wind speeds at a given location over a 15-year period, and that we wish to estimate the peak wind speeds that would occur over a 100-year time period. References 5 and 6 outline methodology for making extrapolations to 100 years and beyond. This calculation first follows the extreme value statistical methods developed by Arnold Court (Reference 5). Court's treatment is included herein as Attachment 1. Methods developed by simiu (Reference 6) are also used, both to supplement Court's treatment and also as a cross-check. Excerpts from Simiu's work are provided in Attachment 2. The method which yields the more conservative extreme wind speeds is used for subsequent analysis, which includes a discussion of the confidence level in the extreme. wind speeds, the return periods, and the CNS weather group determinations. To ensure that a Type I distribution is indeed appropriate for-CNS, the Probability Plot Correlation Coefficient (PPCC) method is used to identify the optimum distribution with which to fit the data. The PPCC method is explained in Attachment 5. Results obtained with the PPCC method are compared to those obtained from Simiu's and Court's procedures. At CNS, a 16-year dataset of monthly extreme wind values has been compiled. From this dataset, the maximum wind speed for longer time periods will be calculated for purposes of the NUMARC 87-00 'SW' and 'ESW' grcup determinations. The CNS dataset is presented in the next section.

a 6 a-a.44;-4. +.L,. ,.a- ._a 2 4 e .E a .2 M SHEET JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMB+ REVIEWER //h APPROVED CALCULATION NO. NPP1-SBO-00,Q 3.0 CNS WIND SPEED DATABASE The CNS Wind speed database used in this calculation spans a 16-year time period from 1975-1990. These data are provided in. For each month, the daily average wind speed and the maximum hourly average are recorded for various elevations above the ground. The maximum hourly average wind speed is the parameter of interest here. This represents the maximum recorded hourly average wind speed for the entire month. For a 30-day. month, this value is the maximum of 30x24 or 720 recordings. Hence, each monthly maximum is an extreme wind speed value which is based on a large number of observations. Data at the 10-meter elevation will be used for most of the calculations herein and are considered the reference basis. Transposition of the data to other elevations will be performed as required, although the data at the 10-meter and 35-feet elevations will be used interchangeably without correction. As used h2 rein,- this is conservative. Using the data in Attachment 3, the yearly maximum hourly average wind speeds for CNS are summarized in Table 1. The maximum for the l 16-year time period is 40.1 mph. 3.1 Correction of Wind Sneed Data to Other Elevations It is possible to use wind speed data recorded at one elevation to determine the wind speed at another elevation. The procedure for doing so is explained in Attachment 4, which has been extracted from-Reference 7. Repeating Eqn. 2.4.1 of Att. 4, in (z/z ) o (1) U(z) U(10), where = in (10/z ) o height above ground (meters), z = o roughness length (meters), and z = l l U wind speed; U(z) and U(10) to be expressed in' = the same units.

M SHEET JOB NO. NP-119 DATE 1/7/92 I PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION g/ CLIENT NPPD ORIGINATOR E. HOLCOMBc. r REVIEWER <M APPROVED l CALCULATION NO. NPP1-SBO-009 Note that '10' meters could be replaced by another elevation since it is merely a reference point. I 3.2 Editina of Database The CNS weather database used herein spans 1975 to 1990.

Hence,

.i there are 16 x 12, or 192 monthly extreme values of wind speed in this database. From the tables in Attachment 3, it can be seen that data were not recorded at the 10-meter elevation for September 1979, nor for February, March and April of 1984. However, the dataset can be completed using the recordings at other elevations along with Equation (1). Using Att. 3 for September of 1979, at 318 f t. (96.93 meters),.Umax = 32.7 mph. From Equation (1), in (10/z ) o U(10) = U(z) In (z/z ) o Let z = 96.93 meters. Choose zo = 0.05 for the roughness length in l open terrain, with confidence that the answer will be correct l within 1 or 2%. (See Attachment 4). Then, the 10.67 meter (35 ft.) wind speed is In(10.67/0.05) U(10) m U(10.67) 32.7 mph = 23.2 mph = In(96.93/0.05) Using a similar procedure for 1984, with z = 100 - meters fro.a Att. 3, yields the other data points: i U(100) U(10) l February 1984 46.0 mph 32.1 mph March 1984 38.0 26.5 mph April 1984 40.0 27.9 mph e -- - e + , +, --e m

SHEET d JOB NO. NP-119 DATE-1/7/92 PROJECT CNS STATION-BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOTrnarah REVIEWER C' _ APPROVED CALCULATION NO. NPP1-SBO-009 From Attachment 3, and including the four edited values, the CNS wind speed database is listed in Table 2 from 1975 to 1990. 3.3 Mean and Standard Deviation The mean of the set of extreme wind values is 24.17 mph, calculated by summing the monthly values and dividing by 192. The mean is denoted by E. The standard deviation of the dataset is

5. 314 - mph, calculated according to N

y E (x,-Tc)2 (2) S.D. ,where = l y, is the 1* sample - and N is the number of samples, 192.-in this 'j ca s e. "' i (a) Using (N-1) n the L- -- -==< of equation (2), the standard deviation is 5.328 mph.

1 i M SHEET JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECTSITE-SPECIFICWEATHEREVALUATION[;Y l CLIENT NPPD ORIGINATOR E. HOLCOMB l REVIEWER D APPROVED CALCULATION NO. NPP1-SBO-009-4.0 APPLICATION of COURT's TECHNIOUE for EXTREME VALUES l In Sections 2.1 through 2.6 cf Attachment ~1, Arnold Court reviews the theory of extreme values. Court notes that both the sample size, N, and the number of observations, n, should be large for the methods to be applicable. For the CNS weather database,.N = 192, i and n = 672 to 744', depending on the' month being recorded.- Hence, according to the criteria given on pages 61 and 62 of Attachment.1, enough wind speed data exist to enable meaningful application of the theory of extreme values. In Section 2.10 of Attachment 1, an application of the extreme-value theory dealing with wind speeds is provided. Following the procedure on 'Worksheet 2', page 73 - of Attachment 1, and using Court's nomenclature, the expected values of extreme ~ wind will~be calculated for CNS. As noted above, N = 192. From Table 2, 24.171354 mph (the mean of the extremes) x = 5.314246 (the standard deviation). and s, = Linearly interpolating in Table III-(p.65) for N = 192 yields Pu 0.5668 (the reduced mean), and a i V = 1.33428 (the standard deviation of the theoretical variate), y Continuing according to Worksheet 2, S, 1/a = = 4.30554 and 73 1/a = 2.44038-The theoretical mode of the sample, u, is l x = u = Tc -(Py 1/a) = 21.73. mph 'l n ..,,,..,,.,i.,.-..,,-.,..~v-,. .,+--,;n...,...n, ,,,.,-,.,,..a.-. .w,--,- .,~ ,n,-,.

a SHEET /I~ JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMBfI REVIEWER

1. 1 APPROVED CALCULATION NO. NPP1-SBO-009 The line of expected extremes can be developed from the theoretical mode.

However, inspection of the sorted wind speed data extremes in Table 2 indicates that it is more conservative to use the actual mode of the dataset. By inspection of Table 2, Ax= 28 mph = u. The line of expected extremes of wind speed 'x' is (3b) x= u + (1/a y) 28 + 4.30554 y (mph). = Here, y is the reduced variate, or the double natural logarithm of probability distribution function p of the wind speed variable 'x', i.e. (4) y= -ln[-In d(x)]. With Court's technique, for maximum values, (5) p(x) = exp[-exp (-a(x Ax)], which amounts to using a Type 1 exponential distribution to describe the CNS wind speed. From equation 2.10 of Att. 1, the return period, T,, is (6a) T, = 1/[1 - p(x)), or (6b) p(x) = 1 - 1/T, Substituting (6b) into (4) yields (7) y = -In[-ln (1 - 1/T,) ] Hence, given the return period, y is known, and the maximum wind speed follows directly from equation (3b). Note also that the return period is related to the probability of occurrence 'p' by (8) T, 1/p(x) = m.______m__.__ ..m_ _ _ _

SHEET 4 JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION [, CLIENT NPPD ORIGINATOR l. HOLCOMB REVIEWER /k'? APPROVED CALCULATION NO. NPP1-SBO-009 and that the probability of nonoccurrence is %(x). (9) q(x) =1 - p(x) = Based on the preceding discussion, the maximum wind speed for CNS can be tabulated for various return periods based on the site specific wind speed data. Return Probability of Reduced Maximum Period Nonoccurrence Variate Wind Speed T, q(x) y x 34 years

• 0.971 3.51147 43.1 mph 40 years 0.975 3.67625 43.8 mph 50 years 0.98 3.90194 44.8 mph 100 years 0.99 4.60015 47.8 mph The problem can also be worked in reverse.

That is, given a maximum wind speed, the return period and hence the expected frequency of occurrence can be determined. This will be done in the next section of this calculation, in conjunction with the NUMARC 87-00 methods for determination of the ESW and SW group. 4.1 Extremelv Severe Weather (ESW) Group The ESW categories are listed in Table 3-1 of NUMARC 87-00. To determine the ESW Group, it is necessary to calculate the annual frequency of storms with sustained winds greater than or equal to 125 mph. An elevation of 30 meters above the ground is representative of transmission line height. Using the logarithmic relationship, a wind speed of 125 mph at 30 meters is equivalent to 103.5 mph at 10 meters, i.e. In (10/0.05) U(10) ,, g U(30) = 0.82826.* 125 mph = in (30/0.05)_ = 103.5 mph It will be conservative to use the 10-meter wind for the evaluation which follows.

• CNS plant design life.

SHEET N JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATIONaf. CLIENT NPPD ORIGINATOR E. HOLCOMB!F REVIEWER 49 APPROVED CALCULATION NO. NPP1-SBO-009 From equation 2.24 of Attachment 1, for large return periods, exp [93 + (x - 32) ( (g/ S,) ] (10) T, = 103.5 mph as developed above. Using the parameters Let x = developed on sheet 13 in egn. (10) above yields T, = 1.7698E8 months = 1.4749E7 years from which p = 1/T, = 6. 7 8 03 E - 8 yr'8 From Table 3-1 of NUMARC 87-00, noting that p << e < 3.3E-4, Court's method indicates that CNS is in ESW Group 1. 4.2 Severe Weather (SW) Groun Section 3.2.1, Part 1C of NUMARC 87-00 outlines the method to determine the estimated frequency 'f' of loss-of-offsite power due to severe weather, i.e. + (1.2

• 102)

(11) f= (1.3

• 10-4) *hi+b*h2
• h

+c*h, 3 4 l Where, for Cooper Nuclear Station, I hi= 30 inches (Annual snowfall for CNS, from Table 3-3 of NUMARC 87-00 12.5 (CNS has multiple rights-of-way) b = 0.0002357 (Tornadoes of 'F2' severity, or greater, h = 2 see Attachment 6 herein and Section 4.3 aelow) 0. (CNS has no vulnerability to salt spray) and c = In the CNS site-specific weather data evaluation, we seek to determine h. As defined in NUMARC 87-00, h is the annual 3 3 expectation of storms for the site with wind velocities between 75 and 124 mph.

SHEET M JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION HOLCOMBh CLIENT NPPD ORIGINATOR E. REVIEWER (h APPROVED CALCULATION NO. NPP1-SBO-009 To evaluate h, select 75 mph as the cutof f. This is conservative, 3 since storms with higher wind speeds are less frequent. It is further conservative to consider the 75 mph wind speed as occurring at 30 meters. Using the logarithmic law as before and transposing to 10 meters, 0.82826

• 75 mph U(10) 0.82826
• U(30)

= = 62.12 mph = Let x = 62.12 mph. Substitution as before in equation (10) yields h 1.1857E4 months = 9.8811E2 years, = from which 4 p 1.012 E-3 yr =h = 3 Substitution into equation (11) gives (1.3E-4)

• 30 + 12.5
• 0.0002357 + 0.012
• 1.012E-3 f

= 6.858E-3 0.00686 = = From Table 3-4 of NUMARC 87-00, Court's method indicates that CNS is in SW Group 2. 4.3 CNS Tornado Data To procure CNS site specific tornado data, the National Severe Storm Forecast Center (NSSFC) was contacted. The NSSFC has provided a computer output listing from Program TORPLOT, which summarizes all reported tornado activity in the site vicinity'from 1950 through 1988. The subject computer output is presented in. The 'TORPLOT' evaluation area is a 2-degree square l centered at CNS, i.e. at Brownville, Nebraska. The output lists data and time of storm occurrence, storm damage class, storm path length and width of touchdown, and other interesting information. 'TORPLOT' wind speed data are instantaneous, ground level winds. The instantaneous wind speed is assumed to apply over the entire evaluation quadrant. To assist in the interpretation of the 'TORPLOT' output, a tornado damage class scale is provided in Table 5. An excerpt-from the 'TORPLOT' output is shown in. Table 6. .e ._,.m ro, n.. m.

SHEET /'I JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMBk REVIEWER

1. 1 APPROVED CALCULATION'NO. NPP1-SBO-009 Appendix A of Reference 9 describes the methods used by the USNRC to develop loss-of-offsite power relationships.

The methodology reported by the NRC is consistent with the 'TORPLOT' program, in which probabilities are assumed to apply uniformly across the 2-degree square, based on the available data. From the 'TORPLOT' output, the frequency of tornadoes of severity - F2 or greater striking in the vicinity of the site is obtained. This frequency is assumed to apply anywhere within the site, i.e. the. tornado is assumed to affect the switchyard or transformers if it strikes at i all. The NUMARC 87-00 evaluation criterion for SW conditions is the probability of tornado occurrence with wind speeds greater than or equal to 113 mph in the site vicinity." Referring to Table 6, and 4 using the conservative evaluation criterion of-113

mph, the

'TORPLOT' database, which contains 68 events, indicates that the probability of tornado occurrence is 2.357*10 events per year, for 4 a mean severe storm return interval of 4243 years. Hence, h 2 2.357*10" yr'8 for use in the CNS severe weather evaluation. (a) Note that power plant transmission systems are designed for wind speeds of 125 mph, based on the National Electric Safety Code. 1

SHEET 20 JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT _ SITE-SPECIFIC WEATHER EVALUATION' CLIENT -NPPD ORIGINATOR E. HOLCOMB-REVIEWER /hD APPROVED CALCULATION NO.-NPP1-SBO-009 5.0 SIMIU's TECHNIOUE for ESTIMATING MAXIMUM WIND SPEED In Reference 6, Simiu discusses a technique for predicting extreme values of Wind. Excerpts from. - Reference 6-are provided in Attachment 2. Repeating equation A1.74 of Ref. 6, and using Simiu's notation, 2 + s(y-0.5772) [ 6/ n, (12) Ox (p) = s where R and s are the mean value and standard deviation, respectively, of the sample of the extreme wind values,-y is the reduced variate, and 0,(p) is the estimated extreme _value-.of-wind for a given probability of. nonoccurrence 'p'. The 'mean and standard deviation of the wind speed database have been' calculated according to equations A1.72 and A1.73 of-Attachment 2 and'are listed in Table 2. Choose a 100-year return period for comparisor} -to the results 'in Section 4. Using the set of monthly extremes,-N.=-12

• 100 = 1200, and y = - In (-ln(1-1/N)] = 7.08966.

Using equation (12) and.the-monthly mean and standard deviation from Table 2 gives Ox (p) = 24.17 + 5.314 (7.08966 - 0.5772) [ 6/-R = 51.15 mph This compares reasonably well with' the 100-year' extreme of 47.8 mph calculated in Section 4,.with Simiu's technique yielding a t more. conservative value by 7% compared to Court's method. ~i .i j

SHEET #I JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT

NPPD, ORIGINATOR E.

HOLCOME-REVIEWER 0' . APPROVED CALCULATION NO. NPP1-SBO-009 5.1 ESW Grouc From page 87 of Attachment 2, for 5 large, note that (13) y = - In[-In(1-1/S)] ~ In Substitution of (13) into (12) and solving for N yields (G,(p) - X)- ff (14) _N = exp + 0.5772 Now, using 6x(P) = 103.5 mph and the values of X and s from Table 2, substitution into egn. (14) yields 3.676 E8 months, or = U 3.063 E7 years. = For comparison to Table 3-1 of NUMARC 87-00, 4 1/ 3.264E-8 yr, e = = with the result that CNS is in ESW Group 1.

SHEET M JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMBk REVIEWER /v APPROVED CALCULATION NO-NDD1-SBO-009 5.2 SW Group Proceeding as in Section 5.1, with O (p) 62.12 mph, equation = x (14) yields (62.12 - 24.17135) n N= exp< + 0.5772 (6-

• 5.314246 w

16912 months 1409.4 years = = 1/A = 7.0954E-04 yr Hence, h 4 = 3 Substituting h into equation (11), as before, 3 f (1.3E-4)

• 30 + 12.5
• 0.0002357 + (1.2E-2)
• 7.0954E-04

= 6.855E-3 0.0069 = = Hence, from Table 3-4 of NUMARC 87-00, CNS is in SW Groun 2. i l

SHEET _ S D JOB NO. NP-119 DATE 1/7/92 PROJECT _CNS STATION BLACKOUT OS SUBJECT SI'1 E-SPECIFIC WEATHER EVALUATION - '~ q l' "LIENT NPPD ORIGINATOR E. HOLCOMB 'EVIEWER P^ APPROVED \\LCULATION NO.JPP1-SBO-009 6.0 PPCC METHOD court's technique and Simiu's method have both employed Type I distributions for desciption of the CNS Wind speed database. These distributions exhibit the same form but have slightly different constants in the functiona, expressions. The constants depend on values used to c h a r a c t e r i r. + t h e data, such as the mode, mean, standard deviation, or location and scale parameters. Based on ' discussion in Section 3.5 of Reference 7, a Type I distribut provides the optimum fit at most weather stations. It is, theret a, considered reasonable to assume that the CNS data would be we.1 fitted by a Type I distribution. However, to ensure that a Type I distribution is indeed sppropriate for characterization of the CNS Wind speed

databass, the probability plot correlation coefficient (PPCC method has been used to determine the best fitting distribution.)

The PPCC method is described in Section A1.6 of Reference 6 and also in Ret'erence 7. Basically, the method examines a Type I distribution and a number of Type II distributions, as defined respectively by equations 2.1.1 and 3.1.2 of Attachment 5. Via a least squares

method, a

correlation coefficient is determined for each distribution, according to equation 3.1.3. The procedure summarized in Attachment 5 was performed for a Type I distribution (GAMMA = INFINITY and equation 3.1.1) and for forty-two Type II distributions (different, finite values of GAMMA from 1 to 1000 in equation 3.1.2 of Att. 5) using computer programs developed by the National Bureau of Standards, which are documented in aoference 8. The extreme value analysis computer program output for the CNS data is listed in Tables 3 and 4. Using Table 3, it is seen that this procedure verifies that the Type I distribution does indeed provido the best fit for the CNS Wind speed database. The maximum value of the PPCC occurs f ar the Type I distribution and is r, = 0. 9 6 4 4 4. Using equation 3.1.1 of Att. 5 and the values in Table 3 for gamma = infinity, the best fitting distribution iF -(v - 21.837) (15) F: (v) = exp --exp' 4.079

SHEET N JOB NO. NP-119 DATE 1/7/92 PROJECT _CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATIOli CLIENT NPPD ORIGINATOR E. HOLCOM M REVIEWER M/ APPROVED CALCULATION NO. NPP1-SBO-QQ9 where 'v' is the wind speed. This distribution is similar to those used previously in this calculation. It yields a 100-yr extreme wind estimate of vim = 50.58 mph for CNS, which is very close to the value yielded by Simiu's method in Section 5. The extreme wind values predicted by the PPCC method are listed in Table 4 for various return periods. Since the ccaputations were made using the CNE monthly extremes, the return periods in Table 4 are also in months. Two extreme wind values are of interest. As before, 103.5 mph and 62.12 mph are used to determine the ESW and SW' categories, respectively. Intepolating in Table 4 for V = 62.12 mph gives a return period of about 26,499 months, or 2208.27 years. Taking the reciprocal of this number yields h 4.528E-04 for = 3 determination of the SW Group. Using equation (11) as before, with this value of h yields 3 f = 6.852E-3 0.00685, = again with the result from NUMARC 87-00 Table 3-4 that CNS is in SE G rout) 2. Regarding the ESW category, an extreme wind value of 103.5 is not listed in Table 4. However, using the much more conservative maximum tabulated value of 78.20 mph yields a return period of 1,000,000 months, or 83333.3 years, i.e. R >> 83333.3 years 4 d f << 1.200 E-5 yr 4 3.3E-4 yr. Hence, from NUMARC 87-00 Table 3-1, the PpCC method also indicates that CNS is in ESW Group 1. l ,,_.,,,e __y.. ,c-..

SIIEET 8 JOB llO. fiP-119 DATE 1/7/92 PROJECT CNS STATIOfi BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMI M REVIEWER APPROVED CALCULATION NO. NPP1-SBO-009 7.0 ERROR ANALYSIS In this section, the level of confidence is evaluated regarding tt a determiraation of the ESW and SW categories for CNS. For a given category, the maximum wind speed is estimated for the minimum return period allowed by the category. The standard deviation of the sampling error of the wind speed estimator is calculated. The upper bound wind speed in the error band in then evaluated by comparison to the allowable ESW and SW wind speeds. 7.1 ESW Group For ESW Group 1, Table 3-1 of NUMARC 87-00 gives d 6. 3.3E-4 yr, or 1/e 3030.30 years llence, take N = 3031 years = 36372 months. From equation A1.74 of Attachment 2, using the method of moments, Sx(p) X + s(y-0.5772) C 6/# where = -In (-in p), and y = 1 - 1/R. p = For 5 = 36,372, and using the mean and standard deviation of the set of monthly extreme values from Table 2, the maximum expected wind speed in 3031 years is S (p) x 65.29 mph = The standard deviation in the estimate of the maximum wind speed is given by b SD (Sx(p))=

1. *+ 1.1396(y-0.5772) # + 1.1(y-0. 5772 ) 2' 6

a T -6 (6 - Cn

S!!EET 2d JOB NO.,NP-119 DATE 1/7/92 PROJECT CNS GTATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION [ CLIENT NPPD ORIGINATOR E. HOLCOMB REVIEWER _;je? APPROVED CALCULATION NO._UPP1-SBO-C09 from equations A1.70 and A1.76 of Attachment 2. With N = 36,372, s = 5.314246, and n = 192 (the sample size), the standard deviation is SD (Ox(p)) = 3.34 mph To a 99.7% confidence level, the 3031-year maximum wind will fall within the band defined by Va., = 0,(p) 13

• SD (0,(p)), or Vg,,s 65.29 + 3
• 3.34 mph 5 75.30 mph.

Since the ESW category 1 minimum allowable return period was used in the above calculation, as opposed to the return periods actually computed using the CNS data, it is apparent that there is considerable margin in the ESW category '1' determination. Further, conservatively correcting to 10 meters, the allowable ESW wind speed is 103.5 mph. Since 75.30 mph is substantially less than the allowable ESW wind speed, there is further conservatism and very high confidence in the conclusion that CNS is in ESW Group 1. 4 7.2 SW Grouo It has been determined in this calculaton that CNS is in SW Group 2. From Table 3-4 of NUMARC 87-00, category SW2 requires that f < 0.0100 Using the previously established values for snowfall and tornado occurrence, solving equation (11) for the maximum allowable frequency of high winds in category SW 2 yields 0.0100 - (30 6 1.3E-4) - 12.5(0.0002357) h 3 0.012 i.e. Waut h < 0.2628 yr'3 This gives 1/h > 3.805 years = 45.66 3 3 months. Conservatively using N = 50 months in Table 4 gives-Vg,,= 37.76 mph. ___.___.__w__.____________---__

N SHEET JOB NO. !JP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATilER EVALUATION - CLIENT NPPD ORIGINATOR E. HOLCOM / REVIEWER /# APPROVED _._ CALCULATIOli NO. NPP1-SBO-009 The maximum allowable wind speed in the SW ovaluation is 75 mph. Conservatively assuming this occurs at 30 meters and correcting to 10 meters gives 62.12 mph for the maximum wind allowed by SW Group 2. Since 37.76 mph is substantially less than 62.12 mph, further quantita*dve analysis is not required,

clearly, there is a significant margin in the SW2 categorization for CNS and very high confidence in this conclusion.

l SHEET M J O B !!O. 11P-119 DATE 1/7/92 PROJECT CNS STATIOli BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION-CLIEllT NPPD ORIGINATOR E. HOLCOM - REVIEWER /)t/ APPROVED CALCULATION 110. NPP1-SBO-009 8.0 EDG TARGET RELIABILITY Using Table 3-Sa of NUMARC 87-00, the combination of weather groups ESW1 and SW2 indicates that Cooper lluclear Station is a 'P1' plant. From Reference 2, CNS is in EAC Group 'C'. For a required SBO coping duration of 4 hours, Table 3-8 of NUMARC 87-00 indicates that the minimum allowable EDG target reliability is 0.950. 9.0

SUMMARY

A site-specific weather data evaluation has been performed for CNS as part of its response to the Station Blackout Rule. The evaluation shows that a Type I exponential distribution provides and excellent fit to the CNS plant specific database for monthly extreme wind values. Three methods, all employing Type I distributions and methods of extreme value statistics, indicate that CNS is in station blackout weather groups ESW1 and SW2, to a high degree of confidence. The resulting conclusion is that CNS is a 'P1' plant with a target EDG reliability of 0.95.

d SHEET JOB llO. NP-119 DATE 1/7/92 PROJECT CNS STATIOtt BLACKOUT SUBJECT FITE-SPECIFIC WEATHER EVALUATION g CLIENT NPPD. ORIGINATOR E. HOLCOMB/W REVIEWER A/ APPROVED CALCULATIO!f NO. NPP1-SBO-009

10.0 REFERENCES

1. Guidelines and Technical Bases for NUMARC Initiatives Addressing Station Blackout at Light Water Reactors, NUMARC 87-00, Revision 1, August 1991. 2. Enercon Services, Inc., " Station Blackout Coping Assessment for the Cooper Nuclear Station", Report No. NPP1-PR-01, Revision 1, dated January 11, 1990. 3. Letter from Paul W. O'Connor, USNRC, to Guy R. Horn, NPPD, " Cooper Nuclear Station-Sp.fety Evaluation of the Response to the Station Blackout Rula (TAC No. 68534)", Docket No. 50-298, dated August 22, 1991. 4. Letter from G. R. Horn, NPPD to the USNRC Document Control

Desk,

" Response to Recommendations on Station Blackout, 10CFR50.63 Cooper Nuclear Station", NLS9100631, dated September 30, 1991. 5.

Court, Arnold, "Some New Statistical Techniques in Geophysics", Statistical Laboratory, University of California at Berkeley, circa 1951.

(Extracted from Advanced Geophysics, Vol. I) 6.

Simiu, Emil, and Robert H.

Scanlan,

} find Effects on Structuren, Second Edition, John Wiley & Sons, New York,1986 7. Simiu, Emil, Changery, Michael J.. and James J.

Filliben,

" Extreme Wind Speeds at 129 Stations in Contiguous U.S.", NBS Building Science Series #118, March 1979. 8.

Simiu, Emil and J.

J.

Filliben,

" Statistical Analysis of Extreme Winds," Technical Note

868, National Bureau of Standards, Washington, D.C.,

1975. 9. Baranowsky, P. W., " Evaluation of Station Blackout Accidents at Nuclear Power Plants", NUREG-1032, June 1988.

SHEET 30 JOD NO. 11P-119 DATE 1/7/92 PROJECT CNS STATIOli BLACKOUT SUBJECT SITE-SPECIFIC WEATilER EVALUATION g,,/ CLIEliT 11 PPD ORIGINATOR E. HOLQQliQf 7 REVIEWER /fh APPROVED CALCULATION NO. NPP1-SBO-009 TABLE 1

SUMMARY

of MAXIMUM WIND SPEED DATA

• for COOPER NUCLEAR STATION yJ;AB WIND SPEED 0 10-METER ELEVATIOl$

1975 30.0 mph 1976 37.0 1977 34.5 1978 40.1 1979 29.1 1980 28.0 1981 31.0 1982 36.0 1983 33.0 1984 32.1 1985 29.1 1986 29.1 1987 25.3 1988 30.3 1989 33.6 1990 27.9 Notes: (a) !.cc Attachment 3 for complete wind speed database. Wind speeds above are nunimum hourly average values far each year. .(b) 10-meter and 35-ft, data in Attachment 3 are both assumed to apply at 10 meters.

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= JS~~ EXTREME VALUE ANALYSIS AF#f .5 Sef-o of 1

1. Y- (8 Tile SAMPLE SIZE N =

192 Tile SAMPLE MEAN = 24.1713638 TIIE SAMPLE STANDARD DEVIATION = 5.3281384 Tile SAMPLE MINIMUM = 5.8000002 TIIE SAMPLE MAXIMUM = 40.0999985 EXTREME VALUE PROBABILITY PLOT LOCATION SCALE TAIL LENGTl! TYPE 2 TAIL LENGTl! CORRELATION ESTIMATE ESTIMATE MEASURE PARAMETER (GAMMA) COEFFICIENT 1.00 0.45931 23.5311928 0.1077400 10.18011 2.00 0.73942 20.3295918 2.2544682 3.39672 3.00 0.83864 16.1745930 5.9701371 2.47043 4.00 0.88058 11.9588499 10.0186853 2.14609 5.00 0.90250 7.7709918 14.1325731 1.98712 6.00 0.91567 3.6100259 18.2566051 1.89429 7.00 0.92437 -0.5315249 22.3780956 1.83394 8.00 0.93051 -4.6594353 26.4944649 1.79175 9.00 0.93505 -8.7775307 30.6056957 1.76069 10.00 0.93855 -12.8884935 34.7124748 1.73691 11.00 0.94131 -16.9941120 38.8155136 1.71814 12.00 0.94355 -21.0955887 42.9154510 1.70297 13.00 0.94540 -25.1938152 47.0127869 1.69045 14.00 0.94695 -29.2894363 51.1079865 1.67996 15.00 0.94827 -33.3830185 55.2014046 1.67103 16.00 0.94940 -37.4748650 59.2932854 1.66335 17.00 0.95039 -41.5653076 63.3838997 1.65667 18.00 0.95126 -45.6545525 67.4733887 1.65082 19.00 0.95203 -49.7427826 71.5619278 1.64564 20.00 0.95271 -53.8302345 75.6496887 1.64102 21.00 0.95333 -57.9168510 79.7367096 1.63689 22.00 0.95388 -62.0028419 83.8230667 1.63316-23.00 0.95438 -66.0882111 87.9088593 1.62979 24.00 0.95483 -70.1732254 91.9942093 1.62672 25.00 0.95525 -74.2578201 96.0791473 1.62391 30.00 0.95689 -94.6753159 116.4984510 1.61287 35.00 0.95803 -115.0871120 136.9117740 1.60516 40.00 0.95887 -135.4954830 157.3213200 1.59947 4$.00 0.95952 -155.9011690 177.7281490 1.59510 50.00 96003 -176.3053890 198.1331790 1.59164 60.00 v.96079 -217.1099550 238.9392700 1.58651 70.00 0.96133 -257.9118960 279.7422490 1.58289 80.00 0.96173 -298.7120360 320.5431520 1.58019 90.00 0.96204 -339.5108340 361.3425600 1.57811 100.00 0.96228 -380.3088380 402.1412960 1.57645 150.00 0.96301 -584.2916260 606.1256100 1.57152 200.00 0.96337 -788.2689820 810.1040650 1.56908 250.00-0.96359 -992.2446290 1014.0803200 1.56763 350.00 0.96384 -1400.1929900 1422.0296600 -1.56666 500.00 0.96402 -2012.1134000 2033.9495800 1.56546 750.00 0.96416 -3031.9772900 3053.8134800 1.56377 l 1000.00 0.96423 -4051.8378900 4073.6762700 1.56330 INFINITY 0.96444 MAX 21.8378620 4.0794487 1.56187 TABLE \\ ,--.--~n --n ..m w-. ..n n

_ -... _ -. _. -. ~ - ~ _ _ _ _ -. -. = h .9G RETURN PERIOD PREDICTED EXTREME WIND M,# f - g gg' y - o p [ (IN MONTilS) BASED ON [g j // EXTREME VALUE TYPE 1 DISTRIBUTION 2.0 23.33 3.0 25.52 4.0 26.92 5.0 27.96 6.0 28.78 7.0 29.47 8.0 30.05 i 9.0 30.56 10.0 31.02 20.0 33.95 30.0 35.64 40.0 36.83 50.0 37.76 60.0 38.51 70.0 39.14 80.0 39.69 90.0 40.17 100.0 40.60 192.0 43.27 200.0 43.44 300.0 45.10 400.0 46.27 500.0 47.19 600.0 47.93 700.0 48 56 800.0 49.10 900.0 49.59 1000.0 50.02 2000.0 52.84 3000.0 54.50 4000.0 55.67 5000.0 56.58 6000.0 57.33 7000.0 57.96 8000.0 58.50 9000.0 58.98 10000,0 59.41 50000.0 65.98 100000.0 68.80 500000.0 75.37-l 1000000.0 78.20 n f4 TABLE 4 ~. ...c.,, ..._,,__,__._.,..-.t r .._..,-,_m,my,y ,-y, r + ,,,,,,m..-

_.._. __ _ m SHEET N JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION CLIENT NPPD ORIGINATOR E. HOLCOMB REVIEWER //M APPROVED CALCULATION ~NO. NPP1-SBO-009 Tablo 5 i Tornado Damage Scale Scale F(wind speed - mph) Damage Pi (miles) Pw (width) - Less than 40 (little or Less than.3 Less than 6 no damage) 0 40-72 Light 0.3 10 6 17 yds 73 112 Modcrate 1.0-3.I 18-55 yas 2 113 157 Considerable 3.2 9.9 56175 yds 3 158 206 Severe 10-31 176-556 yds 4 4 207 260 Devastating 32-99 0.3-0.9 mi 5 261 318 Incredible 100-315 1.0 3.1 mi l l i e

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SHEET /- / 88 42 JOB NO. NP-119 DATE 1/7/92 PROJECT _CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION 4 CLIENT NPPD ORIGINATOR E. HOLCOME P REVIEWER iAm /iu Jt' APPROVED CALCULATION NO. NPP1-SBO-009 Court's Extreme Value Technique l l e ,e- ,,r- .,..n,

Some New Statistical Techniques in Geophysics AftN Of.I) COtlitT .%ttstual Lateatory, Univereoty of Cahfornus, Berleieg. Cahfornne a Coste.wrs l'aee

1. Intrmluttii.n 45 I.I. General.

45

12. Meth.xis.

46 1.3. Functions. 45 1.4. Graphics. 49 1.5. Components. 50 1.6. Separation. 52

2. Extremes.

53 l

21. Intervals.

53 2.2. Frequency. 55 + 2.3. Probel.ility 56 2.4. Risks 58 i 2 5. Theory. 60 2.6. Description, 61 2.7. Parameters. 64 i 2.8. Computaticas 67 2 9. Evaluations. 63 2.10. Applications 71 It 2.11. Cenclusion.. 74

3. Circular Distributions.

76 3.1. Requirement... 75 3.2. Description. 76 s 3.3. Procedure.. 78 s.. QA 3.4. Graphing.... 79 3 3.5. Ilmitations... 81 I,ist of Symbole and Notation. 82 g nererences. k s3 g

1. Inraoover10N 1.1. General j

N O Statistical theories and methode are being applied increasingly in all I fields of science, especially in geophysics. Until the 1930s, the physical N sciences generally used only the rudimentary methods of statistics, pre-ferring, for example, the Gaussian probable error to the analytically stronger and more versatile standard error (or deviation). Statistical 45

Cl i G ) eoni. raw stausricao rn.cumquts m on.orur w M AnNor.o couar Descriptive methods are those which compress many figures into a theorems and methods developed in the preceding half-century were few to represent them adequately for the purpme at hand; these methods 5 employed much more in the biological and social sciences than in the are largely those formerly known as the calculas of obenution They physical involve few assumptions about the nature of the original figures, and [ In the last few decades, however, the physical sciences have adopted a more modern statistica'l outlook. Geophysics in particular has made consider the figures as such, and not as sampics. Descriptive methods l rapid strides in adopting statistical practices, and many techniques have permit computation of means, suodes, medians, ami of variances and hi her moments, as well as of correlations between two or more varinbles. } been developed for the special requirements of its various component 6 Analytical methods use the descriptive techniques to determine how l sciences. Some of these techniques are described in detail in this article, well the obse vations agree with the theoretical model which they are in order to acquaiat a large circle of geophysicists with their potentialitics. From the charatter of the model, m turn, and the r A preliminary discussion of some fundamental aspeds of statistics assumed to follow. descriptive results, the analytical procedures can indis ate the arcu-I which often are overlooked in geophysical applications, and an explana-racy of generalizations foun the data, amt of emopari3ons with otlwr tion of a rediscovered simple method of estimating two nore.al emnpo-nents from a bimodal distribution are given in this section. The article observations Emphasis, in mmt elementary courses in etathairs, un the analytu al i is largely devoted, in Section 2, to a discussion of the lihehhood of occur-ay,e.t.s has obse ured, for immy geophysicists, Imth the hmitatiims aml i rence (return period) of esteeme vahn s, ami the mmt sevent method for the utihty.J she gnorly dcoriiptive methods.4 the rah ulus ut ol.+rva-estimating them, the theory of extreme vaines. The final settiun men-Whrscas description takes the data = they ate, analym a un- [ tions briefly an even newer durclopment, the statistics of tirrufar varia-siders them only as a sample of a population or univer3c. This pare.it i tions f bles, stillin the descriptive stage. population, in turn, is assumed to have certam characteristi 3 who3e Applications, interpretation, and limitations of the technique 3, rather than underlying theory and proof, are stressed. The only statistical numerical values are estimated from the description of the sample-Establishing that the sample does in fact have the attributes of the knowledge presumed of the reader is that of a first course in statistics: parent population is therefore essential to any analysis, yet in least squares computations, characteristics of the normal distribution, this correspondente is not established at all. c.nd simple correlation. deviation can be computed for any set of figures as a valid measure of_ Symbols and notation used in this article are listed at the end of this the amount of dispersion, but only if the Sgures are shown to follow a y article. Most of the symbols are those used in the various original " norma'" distribution can it be assumed timt tao-thirds of ahem isil papers, but some of the nbtation is novel, since statistics has devtloped so rapidly that its notation and symbolism have not yet been fully stand-within one standard deviation from the mean. f Descriptive methods alone may suffice for many gcuphysical appli g erdized. In recent years, the overbar (2) has been accepted to designate cations-more so than in the biological and social sciencts-where a q the mean; in this article, in addition, the tilde (2) denotes ti.e median mass of data is to be reduced to a few characteristic figures (mean s, Q '7 r.nd the circumflex (i) the mode. Grave and acute accents (i and 1) indicate the largest and smallest values, respectively. at des, variances), without any inferer ues about the parent population n or any detaded comparisons with other sets of observations. But statis-g The classical statistical methods of geophysics have been presented tical anclysis of geophysical data must start with a clear expression of the h recently in great detail by Conrad and Pollak [1]. Some more modern populatiota of nhith the data are considered to be a sample, and establish-etctistical concepts, however, are not included these, and may he over-ment that the sample is indeed drawn from such a population. 1 looked by geophysicists. In the following paragraphs cerhin aspects are For many sets of geophysical data,"It is clear that one carmot define 8 discussed which may make more accurate the application of statistics to a population out of which the given sample was drawn at randon2." [2] 4 ( geophysics. Most geophysical data concern measurements of a variable which is con-j . f.f. Methods tinuous in both time and space, and may be relatively uniform over certain ranges of one or both. A single reading of air temperature, or i Statistical methods are of two general categories: descriptive and magnetic intensity, or sea-swell length, may be considered as a sample Wt analytical Both depend in large part on the theory of probability which, of conditions at the spot of observation during a short interval of time, l in the words of Laplace, is merely common sense reduced to figures. i

aous usw cr naricat. TEctINNUM IN GEoPIHs8CB 49 48 artnot.o count "* E" thmetical model, expressed mathematically." euch as a few minutes; c st may be a sample of conditions over a small m either of two ways, one area, a few inches to a few miles in radius. "* '"I

• E' That is, an instantaneous reading is " chosen at random" from all 7

possible similar readings which could have been made at any of an infinite ' * ' ' I finite size; when all frequencies are reduced to perwntages of the aumber of other timer during the interval, or at an infmite number of I " ** places in the vicinity. But when the element is averaged in time it is no this function, denoted by f(x), represents the density of the distribution longer a sample with respect to t.ime: the parent population of a series of of frequency or probabdity, and when plotted on cartesian paper it yields mean da.ly values (temperature, magnetic activity, or sea-swell length) a characteristic curve bell-shaped. for the normal curse. i is composed of all possible values for the vicimty, each averaged m. t.ime. The area under such a curve represents cumulative frequencies or Furthermore, while the mdmdual reading or menu daily value may probabilities; consequently, the cumulafire prdubility funttion is the inte-be a random sample from an mfinite' population, a series of such readings r, gral of the probability density distribution or function: F;r) - ( f(t)dt. is not a random sample, but a stratifed sample: one from each of several distinguishable divisions (e.g., days) or strata of the population. Conse-The graphing of such an integral, if computed from one end of the dis-quently, many analytical procedures, particidarly tests of significance, tribution to the other, yields an ogire, or cumulative frequency or prob-ars not strictly applicable to such data. ability graph; in hydrology, a time-frequency ogive has been called a " duration curve." On cartesian paper the cumulative probabihty ogive 1.3. Functions of a normal distribution is S-shaped or " sigmoid"; special " probability The extensive computations required for statistical description or mper" (Section 1.4) transforms. this curve into a straight line,. antlysis are laborious if done by hand, but can be done rapidly on modern Each fortn of frequency function, the density distribution and its computing machines. Recent improvements in such machines, in fact, integral, has separate uses. In general, the density distribution is used. hava permitted great simplifications in the routine computations, in that to graph the theoretical function for comparison with a graph of observed, ~ involved calculations can be done more rapidly than simpler calculations values, while its inte;;ral, the cumulative probability function, is used for chich require additional manipulation. Unfortunately, these advances numerical comparison of the agreement between theory and observations, are rarely reflected in elementary textbooks, which describe methods and for discussion and conclusions sJter correspondence is establishedA, epplicable to manual computation, perhaps aided by an adding machine. While a density distribution eurve can he approximated from area values. C For example, combination of observations into classes is desirable and theoretical ordinates can be compared numerically with obe.crved when a large mass of data is to be summarized manually, but imposes frequencies, such procedures are not s.s correct as the proper use of the rpme loss in accuracy as the price for convenience. With modern two functions. hk mochines, individual observations can be squared and the results added M WPE3 N in less time than is required to select class limits, assemble data into 1 classes, and perform the computations. Consequently, the classic rules For any cumulative rirobability function, whose ogive plotted on g as to the number and size of classes no longer are very important. g, g gg g Quantitative data or measurements, however, already are grouped on which the ogive becomes a straight line. Such papers were firi,t h by classes, defined by the umt of measurement even though the vari-d Q cugineers, and they are used chiefly in that field. "Tla> ugh j able measured as itself continuous. Any further grouping usually is mathematicians look with disfavor on the use of graphical methods in o madvisable. the evaluation of statistical parameters, engineers find them very con N Likewise, although the standard deviation is deh.ned bas.ically as the venient and time saving, especially if the accuracy required is not too squase root of the mean of the squares of all deraations from the mean 3***'. g3y y I (root mean square),in practice it is obtained most readily by the " vari-Graph paper for the normal probability paper was designed and intro-able squared" method: the square root of the ditTerence between the duced in 1914 by IIasen [4] without comment, and explained in 1916 by $ mean of the squares of all the ongmas ulwvations and the square of the his coworker Whipple 15], who also presented a logarithmic normal paper mean of the observatwns. Indmdual departures from the mean need previously suggested by IIasen; revision of this paper has recently been not be computed at all. u

50 susota couny w,m. n w arriranesi m imiuiu m m.oru u w.3 r, I proposed by Kottler [0) As soon as the statistical theory of ex.reme bles whnh aie not uniform but include subvariahks of ihticient base vclues (Sections 2.5 et seq.) was introduced into the Umted S ates, t har actes is tics. Powell 17] designed a probability paper for its fimetion (Section 18). l'o r e xa m ple, lan.dal.a g 1101 has shmm tlut olm i mi t hu mal Other probability papers include the " Probit" and "Logit" graphs of giadwntsin the car th's crust f all mto two group % passil.!y for seduntntary Berkson and Gumbel's new "Itange" paper 15]. and metamorphic r ot k s, respectively. Similaily, in mi<ldte latit ude. The chief virtue of any probability paper is that a set of data whiih the t rupopaux n.uy 1 e either high and.ohl (tropi als o, bnsundiot.o plots along a straight line on it can be assumed to be drawn from a (ohl (polar), so that a f requency distnbution of d.uly tiopopruse height population whose distribution is that on which the paper is ba3ed A determinations Inis two definite modes further tulvantnge is t hat such a st raight line, whetimr drawn by insper-A g.nti,d motimd hir hmling two ma mal romponens m an y <h inho-tion or fitted mathematically, can be in.ed to obtain estimutm of other amn, amuming noihing ahimt them estept their esi,.temr, win gnesented vtlues, such as the expected frequeury of a given vnhic or Ihe value wit h a by l'earson llIj in the hrat el his famous "Contributiow, to t!.c Mathe-given probability of occurrtnce. matic:d Themy of 1: volution" befoic ti c lluyal % wty on Nm e mbre Ilowever, probability paper cannot be n3ed alone to determine how 16, 1893. It requires solution of a complete ninth dtgrte equation well data follow the assurned distribution, e.g., to test for " normality," involving the first five moments of the given thstribution. because a straight line cannot be litted to plotted points by inspection: Pearson applied this method not only to markedly skrwt d ihatnhu-the paper is not linear, and slight departures from a straight line are tions, in which the presence of two components is indicated strongly, but mcgnified at both ends. " A Log-Probability Chart should be used only to some which are quite symmetrical (although not normals to funi tom-to represent an exact ogive by a straight line but not to judge how the ponents with identical means but differing standard devotions. lii3 d;ta fit it. It is impossible to achieve any reliable judgment by mere general method applies even when one of the cominments 34 nimainvi, in;pection of such a graph." [6] i e., the given distribution is the di!Terence between normal ours. To plot a set of values on any probability paper they must be arranccd To Is!geworth's [121 suggestion for simphfying ax,umptions, Pia 3on _in order of magnitude and their cumulative rank established. The 113} retorted that the " process is not so laborious that it need be thscarded smallest value is No.1, the next-smallest No. 2, etc.; if the smailcst for rough methe # approximation based upon diopping the funda. occurs twice, it has Nos. I and 2, and the next. smallest is No. 3, etc. mental nonie and guessing suitable solutions " llouever, t brher 11 ll,, Alternatively, the largest value may be No.1. considered the general solution "a very laborious opera tion," and Ilowever, there has been little agreement on how to plot these cumula-developed simplo solutions for two special cases: (1) where means ate tive ranks on probability paper, If the ranks are divided by the number assumed for the two components and (2) where the s ariantes of the two x components are assumed to be equal. g of observations, N, then the last one is unity, which is at infinity on le grrph paper. Compromises have been suggested, by which either i or 1 Charlier's development, pubhshed in English in a journal of the Q is cubtracted irom the rank before division by N, or division by 2N; these thiiversity of Lund (Sweden), attracted little attention, and no mention 1~ either omit an observation or distort the original data [9]. of it appears in his later textbook nor ducs it seem to have been used by g Certain theoretical considerations indicate advantages in dividing anyoneel,c. Of the two methods, the first, involving assumption of the % cach cumulative rank by N +

• for plotting; in addition to providing means of the two components, is far simpler than the second, whichR 8

more realistic frequency values, this method permits all olmen vations to sequires computation of the fourth moment of the gis en distriimtion and he plotted on graph paper. This procedure is used in analysis by the solution of a cubie equation. { theory of extreme values (Section 2.81 llowever, Charlier devoted little sp. e to the first metho.1 and x. expanded on the second, terming it the " abridged method for dissecting L5. Components frequency curves." Since the cubic equation involved is actually one s I Typical of the subordination of descriptive methods to analytical step in the general solution,"hence it is no loss of time to begin with this procedures is the neglect of a very simple and useful technique for esti-epprosimate method." lie felt that assuming equal variances for two } mating two normal components in any frequency distribution. Many components "is of a more general character" than assuming values for measurtments, in geophysics as well as other sciences, involve varia-their means: "Especially in biology it is a fairly probable supposition

m e-SoitE N EW BTATtsTICAL TECHNIQUES IN GEUPHT63c3 53 52 ARNOLD CoUMT given distribution. The best pair of means generally has maximum that two types found together in nature possess raurly equal stand-ordinates agieeing well with the observed values, due regard being given ard deviations. We may then use this method to separate the two to the cositnhution each component makes to the oil.er's peak. Suth agreement can be made as clo:,e as dc3 ired by assuming vaines of components." lie admitted that "this abridged method is appliiuhle only when the inaximum ordinates ja and s, in adihtion to the means 3f and 3f., there are a priori reasons for the assumption that the two components Then have nearly equal standard deviations. There are many problems where and ,, = A d y r, y. no such reasons exist," such as those involving several sets of errors to a (1.5) ei = Nd v'2r y. g of a different type and magnitude. In effect, this short cut to Charlier's procedure replurs the e,tandard reading, each set bem. In geophysics, equal variances may be present in some cases, but in deviation and skewness of the. original distribution by a subjective general the first method, of assumed means, is the most applicable. evaluation which may M mo.? e!Iective for some thstn!,utions, but is not Both methods, and one further simplification, are presented in the next of as general applicability in finding two normal components. paragraph, without the theoretical basis or development and in mosc Assuming the two presumed components to have equal vasiances, condensed and modern notation [15L instead of assuming values for their means, led Charlier to a cubic equa-g;g.i inwelving the difietente between the variances of the given distribu-f.6. b,eparation tion and the assumed components: Jn obviously bimodal dbtributions, and many unimodal ones with pronounced " humps" or " shelves," means 31i and 3f for two supposed Their departures irorn the mean M of components may be apparent. where z = ei: -,* and,. is the fourth moment of the d6tnhotion. Thee the given distribution, discriminant of this c:.bic, (1.1) M - Mi = mi. and M: - 31 = mi III) C' " I*/216) (13. San * + E*) give the variances of the two components: where as = rd,'is the skewness and E = (rs/,*) - 3 the excess, almost ,,' = a' - 2mimd3 - (m */3 + rs/3mi) always is pos;tive, indicating only one real root: (1,2) ei' = e' - 2mimd3 - (m */3 - ed3mi) = 0 4082,5 (fM6742an* + y - WWs76:5h) (1.8) where a* and r, are the variance and third moment of the given distribu. The total areas or frequencies of each component depend only on where v = V13.5a * + E'. g a N tion. Except for almost symmetrical and very flat-topped dettibutions, the assumed means: i8 positive, so that z wdl be negative, and e.' < e'. But if - >,8, 7 (1.3) N - Nmd(m + mi) and N - Nmd(mi + mi) then an' as negative, and there is no actual solution, indicating that tl.e 1 i ates assumption of equal vasiances is unwarranted. If the assumption is v Finally, from a table of the normal frequency distribut,on ordm. Q justified, and ei s real, the means are: (Section 1.5), 4(i), the ordinates of each component at any distance i g (in i units) from the mean may be found, since , #' " # ~ "' ~ # - ("* M - O F - ' (1#) (1.4) ys = (Ndai)4(l) and y = (Ndes)+(t) , Af - M + m = M - (rd6) + V({r )* - s The' larger cornponent always corresponds to the smaller departure The areas N and #3 of the two components are foamd from equation 1.3 *D from the mean, whichin turn is miif reis positive, meif negative. Should as before. impossible means be assumed for the two components, ei' or an' will be ,,.. ExTatuss N a ive, Indicat.mg no real solut. ion. E.f. Intertals g negat. liowever, the method of assumed means does not give a unique solu-Extremes of any distribution of observations are of interest becam,e tion: usually trial of several pairs of means is required to find one set yielding two components which, added together, closely approximate the they afford a roughindication of the range of the variable: extremes which ---v-

boH E N EW STATISTICAI. TECHNIQUE 8 IN GEoPIM6tes 55 51 AnNot.o count In particular, most hydrologic analyses use the relative frequency aml have occurred may be expected to occur again. In geophysics, extreme. apparent return periods of annual floods (maximum stream discharge), cre of greater importance than in many other sciemes, because many ignoring the menmd-highest floods of each year although some of them questions of engineering design hinge on the most extreme value to be expected. Dams must be constructed to withstand the maximum flood may be greater than the largest floods of other years. To ree tify this apparent fault, other analyses use all floods exceeding 11 e base value enticipated in the lifetime of the structure, skyscrapers must be designed (" partial duration series"), so that "the recurience intervat is the average with the most severe earthquake in mind, chimneys abould be able to interval between floods of a given size negardiens of their relationship to endure the strongest vind, communications circuits should operate during the most severe magnetic and electrical disturbances, and piens the year or any other period of time." [16] It is less than the recurrence interval computed on the annual basis, although "for sarge floods the two must f.e located and constructed to withstand the heavicat unticipated appenas h numerical equality " surf. In' all such problems, specified calculated sisks may be taken if the ^y 4,, i likelihood of occurrence of these extremes can be e.timated within known If the occurrence or recurrence of an event depends on so many limits of accuracy. The basis for such estimates of risk, and methods for independent factors that it may be considered to follow the laws of t their calculation, are explained first in this section. Then follows a dis, chance, its relative frequency usually is assumed to be the same as the cussion of the most recent method of estimating the most extreme value to be expected in a given period, the statistical theory of extreme values. Frobability of occurrence in any one trial. This espaivalence, which appears intuitively sound to the engineer, is questioned by the rnathe-By definition: matician, and has encountered much statistical discussion. An event which happens 11 times in N trials has a relative fre-It is the subject of an early theorem, acclaimed as une of the f nnula-t i quency of occurrence of H/N, and an apparent return permd of tions of probability theory, proven by James Bernoulli in his Jrs con-jecfandi (published posthumously in 1713): ~ b The apparent return period, or reciprocal of the relative frequency, is As the number of trials increases, the probabihty approaches U therefore the average mterval between recurrences of the event in the unity that the relative frequency of occurrence will differ by less particular series of trials. Despite the rigor of this definition, it has not than any desired amount from the true probability of occurrence. j been fully appreciated, and there even have been some attempts to This theorem does not say that the relative frequency itself approaches g prove it. Distinctions have been drawn, in hydrology, Letween two kinds of the true probability as a limit, although Riet 117] proposed such a state-ment as the basic definition of probability, from which Bernoulli*n % . return periods: the "exceedance interval" and " recurrence interval,n theorem wouhl be an immediate consequence. In recent years these I respectively the average periods between exceedances and recurrences fundamental assumptions of probability theory have been the subject of % of an event. These distinctions may be justified in dealing with discrete renewed discussion (181 variables, such as number of points on a throw of two dice, but they grow k In most geophysical problems, the true probability is unknown ami meaningless for continuous variables as the unit of measurement becomes 8 must be inferred from the relative frequency. "Bernoulli, himself, in cmaller. The distinction is part of the earlier empirical approach to the establishing his theory, had in mind the approximate evaluation of 2 problems, which has been superseded by the recent advances outlined unknown probabilities from repeated experiments," Uspensky [19] 4 in this article. Pointed out, quoting Bernoulli as saying: "If somebody for many pre-Events for which relative frequencies and return periods are estimeted ceding years had observed the weather and noticed how many times it N exe defined in one of two ways: by time or by magnitude. Events 8 was fair or rainy,... by these very observations he would be able to de6ned by time are the largest (or smaIIest) individual values during a discover the ratio of cases which in the future might favor the occurrence N given interval, such as a month, year, or solar cycle. Events defined by or failure of the same event under similar circumstances." magnitude are those values which exceed some predetermined base, such While the relative frequency based on very many occurrenets pio-as a temperature of 100'F or an earthquake intensit, of 6.0; the time unit vides a reasonable estimate of the true probability of occurrence, the is usually much smaller than that used for the first type.

e L_ 57 56 ARNot.D CoUR'r soggE F3EW STAT 18 TICAR TECnNIQuE3 IN oEoFinsics relative frequency in a few occurrences is not at all reliable. l'robability although it can be developed as a corollary of the oldest problem in the estimates usually are made in terms of two limits which are expected. theory of piobability. In this problem,300 years ago, Pascal found that with some given degree of cc,a6dence, to include the true value; for a while the probability of a double six on any one throw of two dice is ifs given relative frequency, the grea a the degree of confidence, the wider and its return period is therefore 36 throws, there is better than a 50 50 the interval in which the true probability is estimated to be. The limits chance of obtaining at least one double six in only 25 throws. of the estimate converge sharply as the number of trials on which it is in general, the probability that an event zr, whose probaleility of based increases; this is shown by Table I, for 95% confidence, based on a occurrence in a single trial is p = 1 - g and whose return period is diagram by Clopper and Pearson 120] w?tich has been reproduced widely ti.erefore f - 1/p, will not occur in any of N trials is (notation as in 1.ist [21); a similar table is presented by Sur. decor [22) without explanation. of Symbols, page 82): Tamax L Limits of estimate of true p ebability with 95% confidence from (3,1} p(1, < 2,) - 9" - (1 - p)=. (1 - 1/ fp relative frequency based at samples of varying size. Coru equently, ti.e probability of et least one occurrcnce in N m.d4 im Number of Trids - Sample Size (2.3) P(1, 2 zr) = 1 - V" = 1 - (1 - I/i)" g Ret freq. 10 20 30' m 100 1000 gp M ~ .00 .00 to.31 .00 to.17.00 to.12.00 to.07 .00 to.04 .00 to.01 Simil.dy, the probability of occurrence for (Ac firal time on lac Nih .10 .00 to.45.01 to.32.03 to.27.05 to.22.07 to.17 .08 to.12 trialis the compound probability of non-occurrence in N - I trials and ,20 .02 to.57.05 to.44 .07 to.39.10 to.3 4.12 to.30.17 to.22 .30 .06 to.66.12 to 55.15 to.50.18 to. 45.21 to.40 .27 to.33 of occurrence in one trial; 40 .11 to.75.18 to.64.22 to.60.26 to.55.30 to.50.37 to.43 (2.3) P[(i _, < z )(x, 2 z )) = pg"-'== 'g"4 - k" = (f - IP '/t" g\\ r r .50 .17 to.82.27 to.73.31 to,69.35 to.65 .40 to.60 ,47 to.53 This probability is greatest on the first trial ud de ases with each suc-Table I shows, for example, that a relative frequency of 0.20 based cessive trial because the probability of occurrence on the prece fing trials on 10 trials (2 occurrences in 10 years) may arise from true probabilities increases. In Pascal's dice problem, the probability of a double six for * : - enywhere between 0.02 and 0.57. For the aame relative frequency the first time on the Nth trial (equation 2.3) decreases, while that for a N observed in 50 f *als the corresponding limits are 0.10 to 0.31. Ilased on double six in at least one of N trials (equation 2.2) increases, a.s follow: 1000 trials the limita are only 0.17 to 0.22. Estimates of the true prob-N: 1 2 3 4 5 10 15 20 25 30 36 . r.bilities based on the rather small samples used in geophysics ljave very P(z,2 zr):.028.027].026.026.025.022.011).016.014.012.010 wide confidence intervals-so wide as to vitiate many computations based P(1,2 zr):.028.055. 081.107.132.246.315.431.506.471.638 g on them. Probably the most valuable contribution of the theory of ex_tgtnf A fourth iclationship, extensively used in some probability problems, h vplues discussed in detail later in this section, is that it provides an but rarely of direct interei.a in geophysics, gives the probability of exactly % k a l estimate of the true probability of occurrence of extreme values based, H cccurrences in N trials: I not on one extreme alone, but on all the values. An estimated relative I14) UIII" 12r) = UI = INI/UI(N - U) IIP"f"~" h frequency or return period obtained by this method,.s outlined in "INI/NI(N - N)IIIi ~ II"~"/ " Bection 2.9,is the closest obtainable approximation to the true probability The factorial terms are the binomial coefficient, usually written (") but N or return period. fermerly written as,C, or C"; they represent the number of combina- % f.3. Probability tions of N objects taken H at a time. For no occurrences, H - o and l Return periods, observed or estimated, are used extensively in various the coefficient becames unity, so equation 2.4 reduces to equation 2.1; for br:nches of geophysics, especially 'in hydrology for flood analysis. exactly one occurrence, H = 1 and the coefficient becomes simply N, so N:vertheless, the significance of the return period is not well known, equation 2.4 is N times equation 2.3: the probability of exactly one .3_ --,m,... .g

59 arrusrimi. texusiunts as utururauw m ui;ra.n 58 nun couirr 1* actical upplic tion of these findmg3 can he snude re.uhly in tea ms of calculated risks. The probability (equations 2.1 and 2.5) that an occurrence in N trials is N times as great as the probability of occurrence event zr, whe seturn period is f, iedi not occur in any of N = f/r for the first time on the Vth trial. The significance of these equations, especially equations 11 and 2.3 trials is also the probahCity that in each of these trials the vaiiable 1 becomes clearer if the number of trials N is expressed as a fraction of the wW be less than the value Ir. This in turn may be comidered as the true return period f by the substitution N = f/r, where r is any positive con:idence that a structure, designed to withstand a masimum esent number. This substitution permits the evaluation of the equations as f Factor r I y which desired bretime N met 1,e multipieed to diam Letz 11. increases without hmit, since by the definition of e, the base of natural design return period T, for various calculated risks U (equation 2 8). logarithms, the hmit of (1 - a/f)* as f increases is e-* Thus the probability that an event zr, whose return period is f, trdt not cerar mentated r4 U A32 m.4m.333 m.250 m m uso with,m N = f/r trials,is (from equation 2.1), '2 a.27 1.71 2.22 2.73 3 OG 3.73 4 74 9 75 19 7*s Desered Me, N 10 1.05 1.49 2 ot 2.52 2.85 3 52 4 52 9 r,2 19 57 (2.5) P(1, < zr) = [(f - 1)/fjf*

,+

I'" r-** Likewise, the probability that 2r willoccurfor thefrst time on the N - f/t trial is (from equation 2.3), whose return period is f, trill not failin a shorter period f/r. Thus the confidence is 50% that a bridge designed to withstand a 100-year !!ood, (2.6) P[(1,_, < 27)(x, 2 2,)]. (f _ g ym,rijf A. o but whicia will fail in the slightly larger 101-year flood, will not be washed f.4. Risks out in'less than about 70 years; the confidence that it wit! not be washed out in 100 years is only 37 percent-the risk of such faih re is conw guently These equations illuminate the nature of the intervals between recur-rences of zr in a very long series of trials, of which the average interval f 63 percent. Conversely, for any desired lifetime N = f/r, and a calculatol rid of failure U within a lesser interval, the design return period T can is by definition the return period. The median T is the period with a 50% probability of at lesat one cccurrence (Pasesd's original problem), I P(b 2 rr) e-t', - }. A, f increases,1/r approaches log 2 determined by substituting for N in equation 2.2 and solving for r: s - - 0.69315, so that the median is a little more than I of the average, U = 1,k. 2 M = 1 - U -,M)r.,, .g _,-o,

r...

t f23) i.e., 7 A 03f. The mode, f, or most frequent interval between recur-rences, is always 0: there is more chance that an extreme value will recur (2.8) r - log (1 - 1/T,)r'/ log (1 - U),,_, j -1/ log (1 - U) in Table 11 for various calculated risks U and on the next trial following an occurrence (interval 0) than that it will fr , recur for the first time on any specific trial thereafter, but this proba-for lifetimes N of 2 and 10 (trials, e.g. years) as calculated irom the exact N . bility for any specific trial approaches 0 as f increases without hmit. I rst portion of equation 2.3, as well as the limiting values from the second ) When r - 1, that is N - f, the probability by equations 11 and 2.5 'These limiting values are approached so rapidly that they may for various occurrences of an event x, during a very long perimi equalling dicient accuracy for any desired lifetimes greater than g its average return period f approach: This table indicates, for example, that a tower which is to 0 or 15. last 50 yents, with a risk of only 10% of failure due to strong winds g 0 occurrences .1/c = 0.36785 before that time, shoidd be designed for the strongest wind expected m o .1/e = 0.3G788 1 occurrence. s 2 or more occurrences.. - 0.26124 T, = 50 X 0.49 - 475 years > 3 1.00000 Tables II and I show dif. :nt aspects of the same fundamental fact: This that the intervals between recurrences of an event, are variable. fact, though known intuitively and demonstrsbie as a corollary of a i Consequently, the probability that the event rr will occur at least problem solved more than 300 years ago, has not 1,:en used extensively % once in an infinitely long series is 0.63212, not much less than the value One of the few investigations of the problem, 0.638 given above for occurrences of a double six in 36 thro.vs of two dice. in numerical estimates. Actually, the limiting values can be used for practical purposes whenever by Thornas 123], used a different version of equation 2.4 (for the proba-f exceeds 10 or 15, as shown in Table II.

a t 1 1 i 60 annot.n couar sout sa srarisman, munsp N WuN N Lility of exactly H occurrences in N trials), considered as a general expres-to many cases in which some of the conditions are met only approxi-1 sion of which others such as equations 3.1,2.2, and 2.3 are special cases. mately,. particular, it may be used for extremes of distrd,,utions whic By this more indirect method, conchisions analagous to thome presented 2 are lim' I at either end, as long as the limits are well beyond t he region o here were reached, and the resulting tables are reprodmcil in a recent observation. Temperature has a delinite lower hmit (ahmolute zero) am textbook [21). possibly an upper limit, but since these are far removed from the vu urs 2.5. Theory observed on carth, extremes of air temperature (or water, or rocks) inay t be analyzed by the theory. Simil$rly, rainfall amounts usul flood stages ) Use of Table II implies accurate estimation of the magnitude of x, caa he analyzed if tho smallest values in each set are still well above zero. the " design extreme" whose return period Ta is obtained from the table the highest th,od stage of each year in a perennial river can be analyze j for the desired lifetime N and calculated risk II. Each estimation ~ but not the highest stage in a dry wash which may have no water at a ever,imhiccualhdimitations of Table I as long as it is haJJon, how. l ~~~ only for several vam in a row-j the observed relative frequency of the extreme in question. Improve- _The fundamental theorem of the theory of extieme values a m 4 j ment in the estimate can be achieved only by increasing the size of the j sample from which the relative frequency is determined, or by weighting in a set of N independent extremes zu,23. 2,. z,,each or correcting the estimate in some way. king the extreme of one of N seta of n observations ca':h of an The most obvious weightmg procedure is to consider all the observed unlimited, exponentially-distributed variable, as both A and n extremes matead of only the extremes equalling or erceeding the required grow large the cumulative piobability that any one of these A 3 vdue. In effect, this process increases the sample size synthetically, and exttemes will l>e less (greater, for smallest values) than any cho-men quantity, r, approaches the louble exponential expttwinn f thus narrows the confidence hmata of the estimate. The various empirical I weighting procedures proposed in the last few decades have been replaced (2.9) 9(#) " N#) " **E! t in recent yeara by a newer method, with theoretical foundations: the q statistical theory of extreme values. la the exponent, the - sign applies for largest extremes, the + sign % M trm exp** is another way of writing "e to the power": j ] From foundations laid during the previous 15 years, the statistical distribution of the extreme values in a sample was developed during the ,,p g,),,, This expressiou gives the probability. of nonoccurrence _ [ s-g,) of the event x in a single trial, and thus affords a way of deter-T j j 1930s by pr. EmiU. Gumbey25}. (De fundamentals of the theory are d O.e probabihty of occurrence p g Mx) used in j summarized by hendall[2tl].) After applying the theory to auch widely i Sections 2.3 and 2.4. Consequently, the return pedod of extremes equal 3 diverse things as the ages of the oldest inhabitants of each region and the tervals between radioactive emiaions, Gumbel adapted it to flood to or exceedmg e is ta N r j rndysis and introduced it in this form (27] shortly after coming to the (2.10)

f. " 1/II ~ Md

{ United States in 1940. Introduction of the expre33 ion for MxL equation 2.9, into equatmn y %e theory attracted widespread intescat, and was adapted by others [7,231 for hydrological computations, and applied to breaking strength 2.10 yields a ma t unwichly expression, so that is practice the probability g 1 of non-occiurence, 4 (r). is obtained first, and then the return pernel is i i [291 problems, the determination of gust loads on alacraft [301, and to frnd. climatic evaluations [31]; additional refinements were made 1,y Gumi.cl 132}. LO U**'i W " g V ] The theory applies to the largest (or smallest) values in cath oil The mam.cr su which this probabihty of non-occurrence, +(s), varses independent setaM E nt ependenlobactntions_rnch. drawn from the 1 i l with z is shown by differentiation: sams population. This parent population must be dist4ibuted according x to some exponential law (~as is the normal di.tribution so that it is I~ 4'(x) = a - e W'-L 4(x) "I ' d unlimited but tends to sero as 15e'v~ariabN l'ndeases or)decicases, the i O Further differentiation shows that the densit_y_of probabih.ty (Sec-distribution also must possess all moments. iion 1.5) is a maximum at x - 5 i e. that 2 is theoretically the most I Mdle based on these premises, in practice the theory may t,e applied frequent value (nM e) of the set of extremes l>eing considered. Graph-l l

id bTATibrtCAl. Tt;ctsNaquts IN GLot*tlielC6 So H P. N BC W AHNoLD COUHy i ing reveals the density function (equation 2.11) to be a gene ally bell-shaped curve, roughly similar to the normal curve but skewed markedly (to the right for largest values, to the left for smallest values), so that the mean is different from the mode (it is greater for largest values, less f__,_... - for smallest values). The skewness of the density of probability curve shows that there is } n greater likelihood of very great extremes than of very small ones, i.e., ~_ L than of extremes which are closest to the mean of the parent values. Although derivation of the theory of extreme values is far beyond the scope of this article, some intuitive basis for it can be mentioned. \\ - in any fair-sized cample drawn from a normal distribution, or from one of the same general unimodal, unlimited type, it is almost certain --'_"' ~ ~ that there will be at least one value as much as one standard deviation greater than the mean. On the other hand, since the distribution from which the extrernes are drawn has no limits, a few such samples _ wili epn,tain_vg!ues gregter thartthe mean b_y.mgre than ti ree standard devi-t , _ations. Consequently, when the extremes of each of many such sampics I I at: considered as a group, they are found to range from around one stand-rid deviation above the mean of the original distribution up to a few very large values, but to be concentrated close to the lower end of this n r nga. ds s The skewness of the density distribution of the extreme value func-Ts ' tion is shown in Fig.1, which also illustrates the relation between a set I of extremes and the observations from which it is drawn. The large 6% m,,, histogran. :o which a normal curve has been Stted, shows the frequency !=ta,itr i h of occurrence of the highest temperature of each summer day (June- ' ""*'3"~ { ) July-August) at War,hington, D. C., during 74 years-a total of 6,803 - Jg, , daily observations [33]. -~ g In the lower right a solid histogram shows the frequency of occurrence

't' ~ W*4 of the highest temperature in each of the 74 summers, with an extreme I/ -

,,,,,,,.,-n.- velue probability density curve fitted to it. Since the daily values are ""J #** by 5'F class intervals, the scale for the annual values has been multiplied Frem.encies of Mshe.t temperatur=* of e*ch ""*"' d*7 y,a t. by 5 to make the two curves comparable. w hington, D. C. 74 y..,. Os72-usCu=@-^"8"* f One gnoral of Fig.1 is that even a small set of extreme values must 1 are not titted too acil by a es n Fi5 .UPresep[k r3etively large number of 1.ctual observations,,since each The highest daily tem , to the right, but not as m ach N x value in the set of N extremes is itself the extreme of a large number, n, normal curve-they are and thus subject to of readings: here N = 74, n = 92, since this example involves the as would be required if they were in extremes of each of 74 sets of observations each containing 92 obser. the theory of extreme values ,em, the ldghest temperature a e n g valions. The theory of extreme valuespsumes both N and n to be, applies only to sumin er months, once in f.1ay and large and in general it should not be applied if either is less than 20, of the year came ou m Ed preferabiy 30 or even 50. four times in September. ~~

65 suME NEW STATISTICAI. TECHNI4yg, g3 ocornwics M ARNot.D counv Iteauced n>eans and =tandard di*t""* 'I '*d"*d ****"**~ f.7. Parameters T st: 111. .hl3hc dtillitY digitibuti9a.(equation 2.11) of extreme values, the Sangle Iteduced Std-S*'" P ', Reduced d 1* I Y' "7 N ipflec3 ion _ pointe, where curvature changes from convex upward around 'd* d*' g the mode to concave in the tails, are at z 10.962:la; in the normal curve, the inflection points are at ia. Thus 1/a is som'edat ]~ ][gh 8' ] 78 so .Sts5 3 M7 85 '. 8 '** analagous to a, in that it indicates the degree of dispersion of the various is 5157 1 0318 8' 5 g 37 5581 1 19E7 cztremes about their mode; consequently, "a" itself is a measure of con-37 .5181 1.0411 52 .c;ntratioa about the mode. 18 .5202 1.0493 53 5 7 3 658 83 .5583 1.1998 This measure of concentration, a, and theoretical mode, f, of any set is .5220 I g9 ,gg3 3 ;oog 52 I g g9, ,,tts, gg .333c g oo7 ]{j sr. 5508 1696 9: SSW 3 M13 of extremes depend in theory on the density distribution f(x) of the m l{ l entire set of values and on its integral, the cumulative probability ,j i n7s 57 .55:1 8 17ns 9 ' [,,$ 9 .5s 38 98 5592 tm 25 5p 1E5 co u21 1.1747 95 .5593 3 2038 (2.12) a = n f(x) and F(f) = 1 - 1/n IEl 552 3.3759 DG 5595 1MH .53M Since these theoretical definitions require kno rledge of the density 2G 4332 1.1004 n27 s.1770 97 .559s 1.20t3 c3 '5530

a. 7s2 98 5598 1 2055 distribution of the population from which the set of extreme valm has 27 been drawn, and in general the only knowledge of this population is N~

i to a ct 5533 1.1793 99 MM 3* hh 3g 53e2 1.1124 65 5535 1 8803 IU t'20 d:rivable from the sample, these definitions cannot be used in practice. Jnatead, these two values are estimated by the theory of least squares 31 5371 1.1159 66 5538 I.1814 from the data of the sample (as explained in Section 2.8), using two 32 .5360 1.1193 67 .5540 1.18 theoretical quantities: 33 .52ss 1.122s 68 . 2 43 11 8 m Sc2 1 3593 L. (2.13) a = a,/s, and J = 2 T s,(b/a,) 2 1 To .5548 1.185s 250 sess 1.2 292 4 71 1.1863 1 (N3 '.5550 3a .5410 72 5552 1 1873 3m .5m9 i 24786 IIere 2 is the mean and s, the standard deviation of the set of extremes. ,g while the mean p, and standard deviation a,, of a theoretical variate is 1,,24 5 1.1 63 73 5555 3 888I d; pend only on the sample size N, and thus can be tabulated for ready as .S t30 1.t388 74 .5557 IN 'nn : m So NN l'[8 use. Table III gives their values for every integer of N from 15 to 100, 40 .5436 1.1413 75 529 {, 3g m2, i mo 1 ,and for selected greater sample sizes; linear interpolation is adeguate_ 41 .5842 1.1436 76 g , hen N > 100 since as N increases both quantities approach limiting 42 .5HS 1 H55 73 3.1923 w vtlues asymptotically. Tahle III was computed by Dr. Gumbe! [31]. j3

8f3 '{[$

[ges h 3 g 79 5567 1.1930 750 5738 8 "* Decause the double exponential form of the basic equation (2.9) 5:c3 1.1519 60 .5569 1.1938 O . A.58 0 6 5658 1.1538 31 .5570 1.1945 tom .5745 imposes difficulties in computation and analysis, it is reduced to linear $573 1.1557 92 [9 f:rm by taking the double (" iterated natural") logarithm of both sides; 47 gog. .5772 28255 g g3$ I 1557a 1.1987 I 5 a n:w variate, y - log [- log 4(x)], is called the reduced rariate: 8 5 9 -(2.11) F - ia (z - f) where, as before, the upper sign is used for extremes of maximums, the g I lower for those of ininimums. This equation gives the crpected extreme Solved for x, this equation becomes (2.15) z-t y/a ""I return period f corresponds to the probability given by y. With the definitions of Eq 2.13 introduced, this expression becomes In this form, the results of application of the theory of estreme values (2.10 x-2 (s,/,)(y - p,) to a set of extremes can be compared with results given by earlier, more

-~_ 67 SoM FEw Frnt3 TIC A1. TECHNIQUES IN GLorH ulCa 66 ARNote CoUltr expected extreme x whose probability of not being cyualled or exceede (e4"ation 2.9) is +(x) - exp (-c'), and therefore whose return period ' **I f rrt u. A " general formula for hydrologic ficquency is (equation 2.10) T, = 1/11 - 4(x)], can be computed if the mean i Pp e to all analyses of the probabilities or return periods For example, and standard deviation a, of N extremes are available.

• • reme values, has recently been proposed by Chow 1341 With the extreme expected to occur (on the average over a long period) once in

,non a tered to conform to the remainder of this article, it is: ~ (2.17) x = 2 + Ks, ~ ~ nv rsely, the expected return period f, curresponding to any given where x is the departure of an individual observation (flood) f I mean the series, a,is the standard deviation of x (i.e., of the series) extreme value z can be obtained from equations 2.10,2.13, and 2.11, but hich depends upon the law of the resulting expression is cumbersome, and the determmation is easier occurrence" of t par cular e ent' The only difference bet erent law of occurren is a their de n of co pt on o f.8 Computations Certain computations based on the theory of Extrerae mes can be a in sorne cases is quite laborious and requires extensive tables. 13

• • ',Ing equ tion 2.17 by 2, Chow obtained an expression for the "y-mean made directly from a set of extremes (obtaining the mear c snd standard g

8 u d im ) in terms of K and deviation s ) 6.; the use of Tables III or IV, and eque;or+ 2.10 or 217. For complete analysis of a set of extremes, however, and in particular to the coefficient of var atto - (2 18) z/i = 1 + K(s./2) determine how well the set follows the theory, it is more convenient to graph the data, using a special extreme probability paper. This form he considered more useful than the first (2~17) i" #"*P"""E On this paper, one of the coordinates is linear, for the oboerved t vr.rious formulas. From equation 2.lG, the " frequency factor" for the theory of extreine extremes (denoted by x), while the other is double loganthmic, for 4-(x) In the original which is (equation 2.9) a double exponential expression. values is: version of this paper 17,27), the double-logarithmic coondinate was the Y~ abscissa;in a revised version l31] the coordinate 3 are revened so that the (2.19') ~ N ~ E")/#" Since y is the double logarithm (" iterated natural logarithm") of the observed values, denoted by z, are plotted along the abscissa as is cus-To facihtate pro ability, and p, and ay depend only on the sample size, K can be tomary, and the double-logarithmic scale is the ordinate. # ? ated readily, as in Table IV. With the values in this table, the Y" MN "NE' linear scale for the reduced variate y, and at the nght a msilogarithmic Taniz IV. Values or K - f (y, g,3f,, fo,,,,ious Prohaf>ilities 4fe) sent various samNe sizes N. scale for the return period T. 1 I Extreme probability paper is identical in function and use to other i ability papers (Section 1.4), and observations are plotted on it, by %

1. ~ 3 - 8 /7' IAch extreme is plotted at an abscissa correspond-rank and magnitude.

ing to its value and at An ordinate, on the double-logarithmic scale, N 0.999 0.990 0.980 0 060 0 0.000 0.800 AII such points, correspomling to its curnulative rank divided by N + 1. 6.265 4.005 3.321 2 631 2.Jto 1.7o3 0.967 are then connected by short straight lines, producing a zig-zag line which O 2.517 2.302 1.625 0.939 25 5. 3.728 3.088 2.444 2 235 1.575 0.PS8 2 3 a straight line. 2[188 'lYts straight line is simply equation 2.14 or 2.15, which was fittgj to } 30 5.727 3.653 3.026 2.393 1[466 the observations by a method of least squares: the estimates of a and J 40 5.573 3.554 2.943 2.326 126 s I 0'820 50 3.473 - 3.491 2.839 2.283 2.086 (equation 2.13) actually minimize the sum of the diagonal distance ~ the line to each plotted poir s.* presenting one of the observed extremes. 5 359 3.413 2.824 2.230 2,038 1.430 0.797 ] Ordinary least squares pre jure minimises the sums of either the 8 '898 I 401 0.779 200 s.'130 3.'2 13 2.698 2.129 1.944 1.362 0.755

1 A4NoLD COURT soM E h t;W wrr sisTlers. TLeHNIQUns IN U CorH Y sles ID horizontal or vertical departures, but this method provides a best fit, obscrvations are concentrated in the lower part of the diagram: the independent of whether x or y is considered as the independent variable. median (frequency.500 or return period 2) is less than a third of the way This "line of expected extremes" is expressed customarily by equa-up the figure. liccause the largest and next-to-largest values in this tion 2.15, since in practice specific values of x are determined for various probabilities as represented by y, such as O and 5. This procedure, ,g -* however, a,mphes no dependence of z on y: they are mutually lependent. . 'g,p l To indicate how well the line fits the observations, a confidence band ~'" e cc.n be drawn on both eides of it. Generally, the limits of this band are chosen so that there is a probability of 0.G8268 (correspondmg to is of i'" the normal distribution) that the extreme correspondir.g to any frequency 6(z) will fall within the band. For frequencies from 0.15 to 0.85, the . Eo / width of this band is obtained by dividing a centain theoietical value, [ \\ [_.'"

• iere called h, by a s/N, so that the limits of the band (sometimes called

./ ' g control curves) are, b' equation 2.10, j [ y ooo .o (2.20) z ~ t t Ka, t h/a VN where the first double sign is + for largest values, - for smallest values, [ and the second gives, respectively, the upper and lower limits of the _.f. f, .,,,,,,e...... - [ b:nd. Values of A for various frequencies are: [ m Freq. 4'(z):.150.200.300.400.500.600.700.800.850 ~ ~ ' e' - N 8 A: 1.255 1.243 1.268 1.337 1.443 1.598 1.835 2.241 2 'i85 -+

,,o For frequencies greater than 0.85, the width of the 0.08200 confidence

=> , m.a band is calculated for the largest and next-to-largest extremes: Q F w. 2. Mhe.t temperatures cf each summer at Washington, D. C. (1871- ( (2.21) 4,., = 11.1407/a 4,',_ - 10.7592/a[(N - 1)/N] 1945) plotted on extreme probabihty graph and Etted by Ime of espected entremes, On either side of the line of expected extremes, intervals as obtained with confideace b*"d "dd"d-by dividing the tabular values above by a VN are plotted at the corre-k 'sponding frequencies; the values computed from equation 2.21 are laid particular example are equal.mvau (a not un ommon curience in off similarly at the frequencies of the largest and next-to-largest observed some sets of extrernes), Se conMenee band badens inarW kr de g last value. In Figure 2 the confidence band has not been extended past I values, but symmetrically about the line and not about the points repre-senting those observed extremes. Two lines are drawn connecting the the largest observed value, as may be done. points so plotted, forming a characteristic horn-shaped figure; technically.

19. Bdiu Uons ths two lines should be drawn smoothly, with a french curve, but in 1

prcetice short straight lines are adequate. For frequencies greater than if about two-thinis of the observed extremes as plotted on the extreme g probability paper fall within the confidence band, the extremes may be th:t of the largest observed extreme, the confidence band is extended parallel to the line of extremes at the same width as for the largest value. considered to be represented ' adequately by the theory of extreme values. g Figure 2 shows, for the same data represented by the solid histogram Usually tha hrgest few valuce will show the greatest departures from the ) of Fig.1, the sig-zag plot of the 74 observed extremes, their "line of line, but uniss one of them is well outside the confidence band it la not czpected extremes," and the confidence band centered on this line. The subject to serious question. g scales and grid of Fig. 2 are skeletonized from extreme probability grapb The probability ps that the greatest extreme z, of the sample will ~ paper. Since the ordinate cf this paper is doubly locarithmic, most of the depart, by an eenount equal to or less than a (ita actual departure), from

II song scw srATwricA1. TECHNI4ggg gN oEoFHY6fCS 70 rnNot.o couar f the established that the variable in question does fall within the acope o exuemes (or by theory, a complete analysis, using a confidence band on extreme prob-qua i e21 2.17) s ability paper, is desirable before any conclusions are draw n. p 20 P2 = exp (-e--4) - exP (-c*2) 830 A PP C"'i""" II Values of aA, the " relative departure," for various probabilities are: ml applicatios u o{ the theory of extreme valties, in geo-bl lu n A The i Probability, ps: 0.0100 0.1000 0.3000 0.5000 0 E827 0 7500 0.9000 h ics as e h al y is either the return period of some specified Rel. departure, a a: 0.0136 0.1342 0.4200 0.74291.14071.2940 2.2511 riu t When the actual departure a of the largest extreme from its expected extreme value, or else the converse, the greatest extreme to be expec e value is multiplied by "a" (equation 2.13), this table permits estimation within some upccitied period. Either of these (luestions can be answere. satisfactorily, together with the conMence hmits of the answers l of the probability that the largest extreme of the given set could have As demonstrated in Sections 2.3 and 2.4, the return ty.rual _ u pc such a departure. Another method of determining the reliability of the largest extreme, averace of all the intervals between recurregces of an event yg along se e, ~ ie if it deviates markedly from the expected value, is to omit it from an but half of the intervals will be less than about.7 of this average, an< The probability that an event will not most probable interval is zero. entire new computation of 2, s,, and the line of expected extremes, and occur until the end of its return period is only 0.37, which u also t e then determine its relative departure frcm the new line for' evaluation by robability that it will occur exactly one time before the end of the Periot. Confidence limits of the return period also can be expreaxed m another the above table. When the most extreme value of a set of extremes is very different Instead of a single value, the return period can be, dicated by the m from its expected value, which is beeed on it and all the others in the set, way. it may be so as the result of chance: there is always a probability of 0.01 interval within which there is a given probability P, that the extreme zr The limits of this mterval are 3 that the 100-year value will occur on the next trial (i.e. year). But such (whose return period is T) will occur. 8 departure Warrants investigation ci the original data for possible errors and T/6, where e** - ed - Pr. This gives, for various values of Pr. .68269 .750 .900 .95t50 g In observation, recording, or transcription. 2[500 Pr. 105 3.129 3.909 U.503 21.485 g,. When the two or three most extreme values depart markedly from 0: I- '522 1/6: .873 .657 .475 319 256 .105 .0465 ths expected values, or when many of the observations plot outside the ccafidence band, the observations simply may not follow the theory of will occur for the Thus the probability is.68 that the extreme value ze extreme values, for any of several reasons:

a. The eet of extremes in question may not be independent.

first time in at least.327. and in no more than 3.13fk% The first of the two queet.ons concerning extremes, that of the return

b. The individual extremes may not be comparable, i.e., may not be period i. for a specified extreme value z, is difficult to answer directly.

cxtremes of samples from the same population. For example,- annual wit.1 extremes at a weather station where the anemometer height or Combination of equations 2.10,2.13, and 2.14 gives expositre has changed markedly through the years do' not follow the p, = g j{g __,,I, g _,,p gg, f (, _ g)(,,/s.)lll (2.23) g thtcry; nor do maximum ennual river stages (heights) if the channel width increases irregularly with the height. Fortunately, as x increases, this converges toward 4 (2 24) p* r. -,,,p gg, f (, _ g)(,,fsy - e. Q

c. The original population, from v.hich independent samples are pre-sumed te have been drawn with each sample yielding a separate extreme, In both these equations, the + appues to largest values, the - to small-may notbe unimodal and unlimited. Maximum relative humidity values would not follow the theory (except in very arid areas) because the upper Thus, with the mean 2 and standard deviation s. of the set of s

"t. d 'xtremes, and the values of g and,,in Table 111. f. can be calculate. { limit (100%) is within the range of the observations. tysually t is simpler, however, to obtain it graphically: it is read on the Lack of correspondence between observation and theory does not riod scale, at the right of the extreme probability paper, opposite discredit the theory:it merely shows that the theory of extreme values t d oi t of intersection of the line of expected extremes with the des, ire cannot be used to analyze the observations. Thus, unless it has been he

80" ^" " ^ ' 72 ARNot.D COURT ~ tuet valus of 2, as given on the abscissa scale at the bottom of the sheet. (Equations 2.23 and 2.24 indicate the nature of the relatiorahip between g. .. a i. .oe i s.3 *. s o.o tha return period scale on the right side of the extreme probat.ility 33 p:per, the frequency scale in the body of the paper, and the reduced h 82JS*' ?.

n. ro.'$$

v. verinte scale along the lef t side; all three scales are indicated in Fig. 2.) .o

• ,,],]

d-r The secoml_quenion_ concerningl extremes, that of the nrobable ' ' " * " * * * ~ * * - a/ta o - 1xtreme with a riven return period fuis much simpler:it is answered by 5l .s ns equLtion 2.10 and Table III, or equation 2.17 and Table IV, using 2 and s. s..... m either case. Or the probable extreme can be read directly on the ,,. ;$.,o,,,. 3.3,,. re. ose extreme probability paper:it is the abscissa at which the line of expected ,.rra. extremes Intersects the appropriate return period Imc. ,[ . m ::::,.;: :-,:,. Once the expected cxtremes, xi and x, for any two return periods, 1 ,n...... ?, " *o$' T and T, are determined, the expected extremes x, for any of her ret arn ..3,,,,. rw om x=

• =

85 period T. een be determined: is.s >* r t r.s no s r.aon rs.sse. .n.orre c oo g (2.25) rr = xi + [x - xi][(y, - yi)/(y, - yi)] 3 '""[is.sers ... rs.sosi ~...i se s sr.iers,.r. core ,n ssrs n, a

== - In this equat.h n, the last fract. ion mvolv.mg only the reduced variates ,,,,,,,,,,,,,,,,,,, c,,,,,,,,,,,,,,,,,, afn ai. i, wi [ #s[ g ,y,, (y) d: pends only on the lengths of the two periods Ti and T, and is called ~

== ".6 0 .m ..a Zr. For two convenient periods of to and 100 trials (years), values of . m .ros ' * *s4 r

• s s t

.s46 .s6s .s94 4s4 ..os _ ..r e e 3, Tact.m V. Facter (Zr) by which di5erence betw een 100-year and 10. year en tremes 1,'; '"***'"'d*.'"*" must be multiplied to give excene over 10-year value of extreme to be expected in F.e.='- a. e- ' ". a... I"/ia "1 " *

• T years.

g g s s v i,.-.= 0 =. E. n -M. 4 1 z. a.m.m r. rawv.. T Zr T Zr T Zr d s. .m, 4 n.. 15 .10018 60 .78118 140 1.14399 n .so. un 1 20 .30634 70 .84717 150 1.17306 y %g es.ro m. ,, s.e s 25 .40352 80 .90451 200 1.29607 ,, m 20 .48257 90 .95497 300 1.46941 ~ ~~ 35 .54924 100 1.00000 400 1.59150 '~ g 40 .60632 110 1.04080 500 1.68666 ~* '" 45 .65759 120 1.07812 750 1.85937

• • [

ote i,as 50 .70287 130 1.1123G 1000 1.98186

• $ ms,

,,,o spego irastest an in tao. uantn. aan es. manusenser rannariosi no o 's wratura eu,* tao Zr for various other return periods T, are given in Table V, which can e m se be used to determine the expected extreme for those periods: uo \\ Example of computation form for evaluatiris extremes by the methods 1 r = zie + Zr(r - zie) Fio. 3. i (2.20) r di=,=>amed in Sections 2.7 to 2.10. Only those portions of the workebeet neces.ary to x a h Most of the computations discussed in th.is and preceding Sections are ., particular question need be used. (Taken from (311.) arranged in logical on!er on a "Worksheet 2," reproduced as Fig. 3. "Werksheet 1," printed on the reverse of the original of this form, pro-vides space for arranging the extremes in order, computing their mean and standard deviation, and their cumulative frequencies and plotting

75 brrridTICAl. Ti.CHNIQUE3 IN OLorHY6scs soM E N EW 74 ARNoLb COURT by any one of a very large a,et of extreme values, obtained as specified. po:itions. These twoworksheets, and the form of the extreme probabilP y Observed extremes are then fitted to this function by an ingenious least P2Per used with them, were developed from Gumbel's original work by equares procedure, involving in addition several approximations based tha Climatology Unit, Environmental Protection Section, Itesearch and on the assumption that the sample of observed extremes is so large that Development Bronch, Office of The Quartermaster General; they are dis-limiting (asymptotic) values can be used. This procedure is essentially sirnitar to that u ed for the " normal" cussed in a report of this Unit [31], from which Fig. 3 is taken. This exanple involves winds, rather than the temperatures of Figs. I and 2, distribution, and many other statistical and mathematical " laws," in Ar,is ofter the to present a different application of the theory and method. which observed data are fitted to a theoretical function. case in many other fields, the theor3tical function has been found to f.11. Conclusion apply to aamples which depart markedly from the original premines This Section has shown that a combination of classic probability (small in number, not wholly independent, not unlimited, etc.). In sume theory and the very recent theory of extreme values permits accurate cases, however, other samples which appaiently should follow the theory andysis and evaluation of the extremes of many geophysical elements. equally well do not do so, for some reason wnich may not be apparent Tha highest temperature, strongest wind, severest earthquake, greatest Ilitherto, many distributions of extreme values, falling within the magnetic disturbance, or worst flood which has occurred or been exceeded scope of the theory of extreme values, have been analyzed by other caly 5 times in 50 years has a relative frequency of 5 in 50 or 0.10, but the Chief of these has been the logarithmic normal distribution; methods. best estimate, with 95% confidence, is that its true probability is some-that is, the logarithms of the individual extremes have been considered to whcro between 0.05 and 0.22. Thus its return period is not necessarily be normally distributed (Section 1.4). Some of the earlier hydrologic 10 years, but is somewhere between 4.5 and 20 years. analyses used a logarithmic transformation, and more recently the break-When all the 50 observations are considered, instead of only the 5 which have equalled ing strengths and analogous properties (e.g., wr.ter penetrability) have or exceeded the value in question, then the theory of extreme values leen evaluated by using logarithms. b[ provides a reasonably accurate method of estimating the return period-As yet, no simple method has been proposal to determine whether an or tha expected extreme for any given return period. actual set of observed data are fitted better by one theoretical function i. t Even after the return period is established, however, the chances are Familiarity with the logarithmic normal procedure, and than another. the complexity of the extreme value theory in its earlier stages, has two out of three that the value in question will occur within a shorter s It is hoped that the j interval, and are also two out of three that it will occur in at least 0 32 caused many investigators to prefer the former. end no more than 3.13 times the return period. For engineering and exposition of the theory of extreme values in this section will enable ] simila appi; cations, the design return period Ts can be determined (See-geophysicists and others to determine for themselves whether the newer I tien 2.4) for any desired lifetime N and calculated risk U of failure in theory cannot be used to greater advantage in analyzing any problem % less than Ts: I"* I'I" E * * ****' I (2.27) Ts a -N/ log (1 - U) - rN

3. Cmcut.An OISTRIBITr*. N8 g

o Table 11 provides a simple way of determining Ts for most riska U actu_ clly used. Once this design return period is established, the expected extreme Circular variables are those which vary continuouely through a!! corresponding to it (zr) can be obtained by the theory of extreme values. angles of a circle, in contrast to the more familiar linear variables, which This is done most simply by equation 2.17 (x, - 2 + Ks.) and Table IV, may hsve no limits or be limited on one or both ends. ILIore so than any } for K; this requires only the mean 2 and standard deviation a, of N other sc4nce to which statisticais applied, geophysics has many problems N l (xtremes, provided that extremes of the type in question are known to involving circular varables: many elements (e.g., winds, tidal forces, M magnetic fields) vary around the compass, and almost all geophysical follow the theory reasonably well. elements vary continuously with time through a day, a lunar month, Fundamentally, the theory of extreme values involves the develop-liitherto, such data have been analyzed inent on strictly theoretical grounds of a function (equation 2.9) for the a solar (27-day) cycle, or a year. probability that a given extreme value will not be equalled or exceeded i ~ ~ -.

anNos.o counv souE NEw symsvicAt, TECHNIQuEd IN oEortlpicts U 76 either as though they were linear, or as trigonometric functions, espe-In the equation of the circular normal distribution, the degree of con-cially through the use of Fourier series, in which several sine or cosine centration of the variable at one time or direction is mdicated by a terms of different amplitudes and periods are adde i to approximate the parameter, denoted by k. This parameter is O for a uniform circular original data. distribution, and has no upper limi., although values of i exceeding 3 As long as a circular vaiiable does not extemi completely aroumi the indicate so great a concentration within a narrow sector that the disto-circle, it can be linearized for statistical analysis without great error. bution may be coneidered as linear. Thus 1, a measure of concentration Ocean swells reaching a beach have a total variation in direction of about around the mean, is in many ways analogous to the reciprocal of the half a circle, and all days of snowfall in temperate latitudes occur in about standard deviation, of the linear normal distribution, a, ce,is a measure m half a year. In such cases, statistical analysis based on the norma'l dis-cf dispersion around the mean; k in analogous to "a" in the theory of tribution, or any other linear distribution, is adequate:it may be con-extreme values (Section 17). sidered that the variable has no limita on either side. Ifowever, when The density of probability of the circular normal distnbution is: cli directions, hours, or months are represented in the distribution of the I variable, the linear approach cannot be justified: there is no more logie (3.1) +(a, 4 = gg) e*-- in considering the day to start and end with midnight than at noon or 7 2.u., and changes in the limits can affect any analysis seriously. where a is the angle measured from the mean, and the denorninator Approximation of a circular variable by a Fourier series avoids the involves an incomplete Dessel function of the first kind of zero order for g difficulty of artificial limits, but introduces another artificiality: the a Pure imaginary argument, and has real values. k periods of the vanous terms usutlly have no physical basis. What, for This function is completely epecified by the two parameters, a, the cxample, is the significance of a half-yearly term in a series approximating angular departure from the mean, and k, the measure of concentration the annual course of air temperature or geomagnetic intensity? At best, about the mean. In turn, k may be estimated by the method of maxi-Q comparison of two circular variables by Fourier series can indicate the mum likelihood from the obeervations themselves:it,s umquely deter-i phase retardation, i.e., the amount by which the peak of the curve lags mined by the length of the vector mean a of the data ('l,able VI). ~l he behind some point, such as the solstices for temperature. Furthermore vector mean is simply the vector eum of the data divided by the total h Fourier analysis cannot be applied readily to spatial variables, i.e., those number of units, not observations. h involving directions such as wind. T.tx vi. values or the parameter a of the circular normal d*talat*"a i it. Descriptior ""**P"*8"'"'**d***""'"** h wo mi x2 m m E M m During the last year, a c~rcular normal probability function has been described by Gumbel [35,36]; when developed it will permit circular d viriables to be analyzed m the same way that linear variables now are o 'ono o2o ojo.oco .oso .noo .120 .140 . loo .381 g 201 221 .212 .262 .283 .303 .324 .345 .366 .387 0 discussed with the aid of the linear " normal curve." The circular nor- [2 $409 .43o .451 .473 .495 .516 .539 .561 .584 6M % m-1 distribution has the same theoretical basis as the linear noimal one: it assumes a large number of random " errors," or departurer from the .3 4 29 ms2 .07G .700 .724 .748 .772 .797 .823 s mean, with the frequency of the departures varying inversely with their j af2 1 12 l.

1. c2 1 8 1 1 475 m gnitude.

i-x A crude experiment illustrating the theory of the circular normal dis-1.51S 1.557 1.600 1.645 1.691 1.739 1.79o 1.ss2 1.896 1.954 ) tribution is provided.by a tiltable roulette whes!. When horizontal, the - .' 6 7 2.014 2.077 2.144 2.214 2.289 2.300 2.4t4 2.547 2.646 2.754 x t frequencies of the numbers on which the ball alights is uniform around .s 2.s71 3.000 3.143 3.301 s.47s s.sso 3.911 4.177 's the wheel. The more it is tilted, the more the frequencies concentrate I 1: ward the numbera at the bottom, regardless of their value. When the For observations which have magnitude as well as d.irection (such as wheel is inclined by 30' or 40*, the distribution is confined to the two or wind speed by directions or flood stages by dates), the division is by the three numbers at the bottom. total number of units (miles per hour, or feet) rather than by the total

i 78 Annot.o couar a e i,wrisuca recuniquEs in cuornrsics 79 number of observations; there is no distinction for data which are merely pens to fall clo3e to the center of one of the original clas6e. (.ectors) can frequencies of occurrence (such as number of hours of wind from each ,ng,t oinc,y,,tional series be compared numerically with the expected direction or number of people dying per month). frequencies. Furthermore, s. yet no criterion has been developed for the goodness of ht of observations to theory (such as is provided by the 3.3. Procedure chi. square test in linear normal theory, or the confidence hand m the In fitting a circular normal curve to observed data, the first step is to theory of extreme valucat compute the resultant direction (time or date is considered as a direction) cnd length, which together form the rector mean. 13asically, two methods 3 4. Graphig for such computation are available: graphical and trigonometric. Each Comparison of observations and theory can be made most readily has several variants. and satisfactorily by graphing both the dsta and the theoretical density In the graphical method, vectors representing all the observations of of probability. From a carefully drawn graph of the probability density, each class are added, on plain or ruled paper or on a circular plotting the expected recquency for each of the original classes (sectors) can be board. blagnitude of the resultant vector, from the start of the first est mated for comparison with the observed frequencies. This elimina tes to the end of the last, is measured with a scale, and its direction deter-the need to regroup the data for comparison with the probability values mined by a protractor. Alternatively, the vectors may be plotted on a given in the area tables. polar diagram and their components parallel to two perpendicular axes Buch estimates will be most accurate, and any graph.ical repreanta. measured by a scale. From the algebraic sums of each component, the tion or comparison of circular variables more meaningful, if equivalent s g resultant is found as in the first method. polar paper is used instead of the customary polar coordinate graph simper. In the trigonometric method, components of each vector are ebtained On equivalent. polar paper, concentric circles are drawn at distances from, by multiplying it. by the appropriate sine and cosine values; after addi-the center corresponding to the sguare root of the indicated number s, - tion, the two components are then used to determine the direction of the instead of the numbers themselves as on the customary paper. Thus, on s recultant by a tangent formula, and its magnitude either from a sine or equivalent polar paper, for each sector the crea is directly proportional ecdne relation or from the root mean square. to thefrequency which it represents. From the vector mean, the proper value of k is found from Table VI, The same results may be obtained on conventional clar coordm. ate I and the equation of the function may be written directly. Or, the paper by using the square roots of the observed and theoretic observed and theoretical frequencies for each class interval (sector) may cies. be compared, numerically or graphically. gives the square roots, rather than the actual values, of the radius vectors g For a numerical comparison cf theoretical and observed frequencies, for unit-area circular normal distributions with various k values. This g% .the observations must be regrouped into sectors so that one will be table is condensed from a more extensive one {36), which itself required a centered on the resultant direction. For example, if the resultant of a complex computational procedure. Table VII gives values for 10* series of monthly observations turns out to be 86* (1 Jan. being 0* and intervals, but satisfactory curves can be plotted by using ordm, ates at 360*), the data originally grouped as 0-30*, 30-60*, 60-90*, etc., must intervals of W* or 30*. Q be grouped into the following sectors: 11 -Il',41-71*,71-10l*, etc. The To obtain a curve for comparison with one plotted from the aquare numher of observations falling within these new classes can then be com-roots of n observations grouped into w equal sectors (includmg any with no pared with the theoretical expectancies, as obtained from the appropriate observations), the tabular values must be n.ultiplied by Vn' 7io; when area table, and multiplied by the number of observations. 06vations are expressed as percentages, no multiplication of the tabular In the present stage of development of the circular normal theory, values is required. In either case, however, square roots of the observed y such numerical comparison is not very practical cr fruitful. Unless the values snust bo used, until equivalent polar paper becornes available. origmal data were reported to much greater ace nacy than the classes Although the principle of equivalent polar paper is obvious, it does used (such as directions to the nearest degree or time to the nearest minute not seem to have been applied to any great extent m geophysics, or in or day of the year), no basis is as yet available for regrouping them int graphic presentation generally. Yet a sector is a truer representat,on of i the new classes based on the resultant. Only in case the resultant hap-observations which may have fallen anywhere within it than the conven-

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82 ARNOLD COUlt? BUME NEw STATISTICAL, ygettNtQUES IN OWHMM circular case remains to be seen, as well as many other applications third and fourth moments (3.6) analogous to those of the linear normal curve. Obviously, there are 3.1415926535 a6) opportunities for many people to develop this new branch of statistics, { [,*gI',[,f,', p.53;w cucular d t*a]21 of such potential value to geophys,cs. sundard deviation (3 6); ** " 'h** i At present the circular normal theory afford.s only (1) a measure of the concentration of a circular variable about its resultant and (2) a normal function to which the observations can be compared qualita-2 * **'I"I'I" I ' tively. Yet its development is of benefit to geophysics simply by point-fauvdu of za f a - **** *' # "*' ' ' I*'T} ing out that " average" times or datea should be computed vectorially, as roedian of all values of (2.3) are " average" directions, and that circular variablea cannot be analyzed 1 : mode of dl,gues of : (2.4) "{*df *"[,,",',',g,,g a y,va,m (2.31 cdequately by the linear methods of classical statistics. t Ints I di List or Srpoou axo Norsrior. z' I'** ... (2 3) (Bection in which first usage is made shown in parenthemes) approximately equal (2.4) .n. parameter of distribution of extreme values (2 5) -. ' approaches as a Umit M o 6 factor de8ning interval of occurrence of extreme value (2.10; log natural logarithm (2.4) hts K excess of distaibution -../,* - 3 (1.6) Itsrsamnesa base of natrrallogarithme - 2.7182818281 (2.4) e F(z) cumulative probability function (1.3)

1. Conrad, V.,

id Pollak,1,. W. (1950). Methods in ClimatoIogy. liar,ard f(z) probability density function - F'(z) (1.3) Univ. Press. Cambridge, Mass,459 pp. ed correlation coefficiente. f. H number of occurrences of an event (2.1)

2. Panofsky,11. A-WE D, D

& factor de8ning confidence band of ext.4me values (2.8) BufL Am. Jfeteorot see. 30,326-327.

3. !(alinske, A. A.

(1940). On the logan. hmic-pro - Transact,.1 =I-1.(&) Bessel function of first kind of zero order for pure imaginary argement (3.2) t K frequency factor in frequency analyses (2.7)- g,,p,.Irn.27,709-711. g 4.11asec, A. (IU14). Storage to be provided in un ding reservoire for munici- & parameter of circular normal distribution (3.2) - M{ mean of a bimodal distribution; #s, # means of componenta (1.6) p g vater supply. Paper No.130s Tra Card Kay. 77,1539-1669. departures of component means from common mean (1.6)

5. W1 ipple, G. C.. ' (1916). The element o n manitdion. J.Frankha fest se, me a

N sise of sample: number of observations in bimodal distribution (1.6); number 133,37-59,205-227. . of trials (2.1); number of observed extremes (2.5); desired lifetime (2.4)

6. Kottler, F.. (1950). The distribut. ion of paru e sizes J FranAhn fase. 260, 4 ptide number of values in each sample from which extreme is taken (2.5) 333-356, 419-441. (1951).. The goodness o n

an g p probability of occurrence (2.3) .izes. abid, 251, 499-514; 617-641. C8 1 7, Powell. R. W. (1943). A semple met eg mg og Sood frequency.,,,;,', g% g, probability of non-occurrence, p (2.3) ratio of return period to number of trials - T/N (2.3); of design return Eng.13,' 105-107. Discuenion: W. E. gg r .adysing Good W period to desired lifetime - Ts/N (2.4) - pp.185; 11 J. Gumbel, Ezeeedance or recurrence g s, standard deviation of z (2.71 discharge, p. 438. T retum period of an event - I/p (2.1); 2's - design return period (2.4)

8. Berkson, J. Codex Dwk Co. WoM, I1")

per No. 32,450, logistic ~ o 4,455, range ruling. Cf. O i normalised deviate of a variable = (2 - z)/, (1.8) ,,1; g; paper no. 32,451, normal ruling no. Gumbel, E. J. (1947). fIbe distribu 3g,,g, 3,g, 3g, 4 number of sectors of circular distribution (3.4) is z.a variable; an extreme value (2.5) 384-412. }' -y ordinate (1.6); reduced variate of extreme value function (2.7);in - theo- ,, g;,, ball, D. F. (1946). Assigninent og g o,neien to a cornpletely orded met : N & dinummeon by I - retical mesa (2.7) h g sa,aple data. Transmet. Ass. Os*P ys I Zr factor *. sblain extreme espected in T years (2.10) E. J. Gumbel and B. F. Kimbell, (1947). 36sd. 28, 961-953-' d gathermd N ' difference between varianess of bimodal distribution and of components -

10. Iandsberg, H. E. - (1944). Note f I),seussion:H.CecilSpleer s

e8-e8 (1.6) - A angle of circular distribution measured from mean & (3.2) "radients. Transact A** G'*P - and H. Landsberg (1947). ' se: #34 the mathematical theory 4 enlution. a r /,8 = skewness (1.6) as 11' Pearson, K. (1894). Contributions a A, departure of extreme value from expected f2.8) FAIL Transaci. R*F. 38c L*"d** # 8 cK ~

84 AltNOt.D COtHrr 8 The Application of the Statistical Theory of Extreme Valuce

12. Edgeworth, F. Y.

(1899). On the representation of statistics by mathematical

30. Prem,1 I

formulae (Part II). J. Roy. Stat. Soc. 82: 125-140. 18 tems. National Adnsory Committee for Aeronautics, Wash-I*E *** D. C^* Technical Note 1926,43 pp.

13. Pearson, K.

(1901). On some applications of the theory of chance to racial t differentiation. PhiL Mag. 6th ser 1,110-124. 31 Anonymous (1951 Evaluation ef Climatic Extremes. Research and Develop-Pla g Division,05ies of The Quartermaster General.

14. Charlier, C. V. L.

(1905). Researches into the theory of probability. Lunds ment ranc, 'y ~ C' Environmental Protection Section Report No.175. D Univ. 3redriff, ny foljd, afdelningen 2, Vol.1, No. 5,51 pp. W"' *8 (3945). SimpliSed plotting of 8 Stistical observations. Treas-IS. Court, A. (1949). Separating frequency distributions into two normal com-

32. Gur ponents. Scienes 110, 500-501.

3 a op ps. Un. 26, 69-82; (1945). Studies on the extremes of statistical I" tea. Yearbook Am. Phil. Soc. 1944, 140-141;(1945). Floods estimated by

16. Langbein, W. B.

(1949). Annual floods and the partial-duration flood series. "*tLod. Eng.- News Racerg 134, 97-101. (1946). Forecaansa Trossaci. Am. Geophys. Un. 30,879-881. Discussion: Ven To Chow and W. B. [* gg,135, 96. (1918). The Statistical Forecast of noods. Columbus, Langhein (1950). 31,939-941. ()hio Water Itesources lloard,21 pp.

17. Riets,11. L (1927). Afathematical Statistics. Carus 41athematical Efono-graph No.3, Af athematical Association of America, Open Court Pub. Co., Chicago,

33. 7,och* flichmond, T.

(1949). The Climatie llandbuuk for Washm.aton, D. L. i U g Dept.cf Cmomerce, Weather Bureau, Technical Paper No. 8. Waabmston, .Ilinois,181 pp. C, 35

18. von hiises, R.

(1941). Probability and statistics. A nn. Math. Stat. 12, yy (1951). A general formula for hydrologie frequency analysis. 191-205. Doob, J. L Probability as measure. ibid. 206-214. Also dise. by A both, 2!5-217. Transact. Am. GeeP vs. Un. 32,231-237.

19. Uspensky, J. V.

(1937). Introduction to Stathematical Probabihty. SicGraw-

35. @ M E. 1 (1950). 'Ihe cyclical normal distribution (at stract). Ans Math. Stat. 21,143. (1952). 'Ihe circular normal distribution: apptwations.

Ilill, New York,411 pp. (ref. on pp. 103-107).

20. Clopper, C. J. and Pearson, E. B.

(1934). The use of confidence or fiducial j" u 3,,,,, ;3 p,,,,, se. Cumbel E. J., Greenwood, J. A., and Durand. David. (1952L The circular limits illustrated in the case of the binomial. Biometrata 26,404-413. "*'*al distribution: Theory and tables. A nn. Masi. Stat., in prema.

21. Eiscnhart, C., Ilmstay, Ef. W. and Wallis, W. A. (Editors). (1947). (Selected)

1. J 0 928). Graphie studies in climatology. II. The polar form of Ef1 31 Techniques of Statistical Analysis. McGraw-Ilill, New York, 473 pp. (ref. on I"8 arna in the plotting of the annual climate cycle. Ifair C*li/ Pdl 8'*8-pp.332-333). David, F. N.

(1951). Probability Theory for Statistical afeth-ods. Cambridse Univ. Press, Iondon, p. 78. Dixon, F. N.. and &f assey, F.J. Jr., 2,387 M (1951). Introduction M Statistical Analysis. LicGraw Ilill, New York,322-3.

22. Snedecor, G. W.

(1946). Statistical Methods. 3rd ed.. The Collegiate Press, Ames, Iowa, 485 pp.

23. Thomas, II. A., Jr. (1948). ' Frequency of minor floods.

J. Boston Soc. Cie. i Eng. 35, 425-442. ' Reprinted as No. 466 of Puhtications from the Graduate School of Engineering, liarvard Univ., Cambridge, Ef ass.

24. Linsley, R. K., Jr., Kohler, M. A. and Paulhus, J. H.

(1950). Applied Ilydrol-ogy. AfeGraw. Hill, New York,689 pp. (ref. on pp. 548-550). , 8S. Gumbel, E.. J. (1935). ' Les valeure extrbnes des distributions statistiques. Anm Inst. Henri Fanearn S, 115-158. L

26. Kendall, 51. G.

1947). Tho' Advanced Theory of Statistics, Vol;]l,~3rd,'ed.

j Charles Griffin and Co., landon,457 pp. (ref. on pp. 217-224).

27. Gumbel, E. J. - (1941). The return period of flood flows.

Ann. Math. Stat. 12 163-190. (1941). Probability-is.terpretation of the observed return-periods b 3 of floods. Transaci. Am.Geophys. Us. 22,836-849, Discussion: Ralph W. Powell, SS, 849-850. (1942). Statistical control. curves for Sood-discharges. ibid. y ) 38,489-500, Discussion: Bradford F. Kimball, SS, 501-509. (1943). On the 3-pietting of Sc4 M 'ges. ibid. 34, 699-716, Diseumion: B. F. Kimball and k R. 8. Goodridge $8,716-719. (1942). The frequency distribution of extreme hsf I w valees in & Ueal data. bee. Asn. Nedesret. Soc. 23."05-105. (1943). g [) Statistical' analysis In bydreingy, Pres. Ass. Sec Cie. Enes. 89,995-10G6. 1

28. Potter, W. D.

(1949). Simpliacation of the Gumbel biethod for Comput%g g Probability Curves. U. S. Dept. of Agriculture, Soil Conservation Service (SCS-TP-78), Washington, D. C.,22 pp. 0 N

29. Epstein, B.

(1948).~ 3tatistical aspects of fracture pr'oblems-J. Appl Paps. 19,140-147. 1

SHEET 2~/*f 3/ JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATIONg CLIENT NPPD ORIGINATOR E. HOLCOMB+ REVIEWER /'/c/M._ b APPROVED CALCULATION NO. NPP1-SBO-009 Excerpts from Wind Effects on Structures by Emil Simiu \\ l

I A/PPf-58cd cof f% d' .-z WIND EFFECTS ON STRUCTURES AN INTRODUCTION TO WIND ENGINEERING SECOND EDITION 4 Emil Simiu Center for Building Technology National Bureau of Standards Gaithersburg, Maryland Robert H. Scanlan Depanment of Civil Engineering The Johns Hopkins University Baltimore. Maryland (Emeritus Professor, Princeton University) A Wiley-Interscience Publication JOHN WILEY & SCNS New York / Chichester / Brisbane i Toronto / Singapore l l

2-Y Affi-~, 58$ - vo 9 f / FSTDIATION OF EXTRDIE WIND SPEEDS 85 in which sheltering effects by small-scale obstacles are present, the data may be adjusted by using a procedure presented in [3-4]. A situation commonly encountered in practice is one in which, while the anemometer may not have been moved, the roughness of the terrain surrounding } the anemometer has changed significantly over the years as a result of extensive j land development. In such situations, the adjustment of the data to a common r~ighness may pose insurmountable problems, unless detailed information on the phases of the land development is available. Anemometer elevation and location changes are listed for most U.S. weather stations in Local Climatological Data Summaries [3-3]. 3.2 ESTIMATION OF EXTREME WIND SPEEDS IN WELL-BEHAVED CLIMATES Infrequent winds (e.g., hurricanes) that are meteorologically distinct from and considerably stronger than the usual annual extremes are referred to herein as extraordinary winds. Climates in which extraordinarv winds may not be ex-pected to occur are referred to as well behaved. In such climates it is reasonable to assume that each of the data in a series of the largest annual wind speeds 1 contributes to the description of the probabilistic behavior of the extreme winds. i A statistical analysis of such a series can therefore be expected to yield useful predictions oflong-term wind extremes. Thus, in a well-behaved climate, at any given station a random variable i may be defined, which consists of the largest yearly wind speed. If the station I is one for which wind records over a number of consecutive years are available, then the cumulative distribution function (CDF) of this random variable may be estimated to characterize the probabilistic behavior of the largest annual I wind speeds. The basic design wind speed is then defined as the speed corre-j sponding to a specified value p of the CDF or, equivalently, to a specified mean recurrence interval N.* A wind corresponding to an N-year mean recurrence i interval is commonly referred to as the R-year wind. This section is devoted to the question of estimating (a) the CDF of the largest 2 r 1 & annual speeds and (b) errors inherent in the wind speed predictions. Such errors I 1 .h include, in addition to those associated with the quality of the data (see Sect. 3.1), i .A modeling errors and sampling errors. Modeling errors are due to an inadequate e .g choice of the probabilistic model itself. Sampling errors are a consequence of I e the limited size of the samples from which the distribution parameters are "4 V estimated and become, in theory, vanishingly small as the sample size increases

3 indefinitely.

a ~ \\}. 1 ./.'3.2.1 Probabilistic Modeling of Largest Yearly Wind Speeds l, l l Several probability distributions have been proposed to model extreme wind f g, behavior. These include: the Type I distribution of the largest value in n ? ' :e ..s. l n i.

• Recall that R = 1/(1 -p)(see Appendix A1. Eq. A1.45L qQ h

W o m .J

.2 - $ WPf-SB gf - c e ? $$ do ESTLMATION OF EXTREME WIND SPEEDS 87 eters of the distribution and, hence, the value of the variate corresponding to a. I given mean recurrence interval.* However, inherent.in these estimates are sampling errors. A measure of the magnitude of the latter can be obtained by calculating confidence intervals for the quantity being estimated, that is, intervals of which it can be stated-with a specified confidence that the statement is correct-that they contain the true, unknown value of that quantity. Techniques t that can be used to estimate the R-year wind, and confidence intervals for the i i R-year wind, are discussed in some detail in Sect. A1.6. One of these techniques is l presented and illustrated below. s Using the approximation -In[-In(1-1/R)] min R, it follows from Eq. A1.74 (which is based on the method of moments) that the estimated value 4 0 ofthe -year wind vg is 3 l Og =i+0.78(1n R-0.577)s (3.2.1) t 'vhere i and s are, respectively, the sample mean and the sample standard I deviation of the largest yearly wind speeds for the period of record. As pteviously noted, inherent in the estimates of up are sampling errors. It a follows from Eqs. A1.76 and A1.70(which are based on the method of moments) that the standard deviation of the sampling errors in the estimation of vg can 2 s be written as SD(0 )=0.78[1.64+ 1.46(In R-0.577)+ 1.1(In R -0.5 3 n e where n is the sample size. h n Example d At Great Falls, Montana, the largest yearly fastest-mile wind speeds at 10 m above ground during the period 1944-1977 (sample size n = 34) were [3-9]:

e 57,65,62,58,64,65,59,65,59,60,64,65,73,60,67,50,74

g e

60,66,55,51,60,55,60,51,51,62,51,54,52,59,50,52,49 is . (mph). The sample mean and the sample standard deviation for these data are i=59 mph and s=6.41 mph. From Eqs. 3.2.1 and 3.2.2 it follows that for f al R= 50 years and R = 1,000 years, n 0oa 76: aph SD(0 o)s3.7 mph 3 3 .gf 0 ooom91 mph SD(D ooo)m6.4 mph i i Y ~ Ifit is assumed that the largest yearly wind speeds are dercribed by a Rayleigh Wdistribution,t the R-year wind, denoted by tj, can be obtained from Eq. A1.65 ue D n-f ' 'In Appendix Al this value is denoted by G (p), where p = 1 -1/R and 8 is the mean recurrence 2 D interval. In this chapter the notation Gy(1 - t/R)= q is used. l sI $Qttit is recalled that the Weibe distribution with tail length parameter y= 2 is commonly referred on w to as the Rayleigh distributic Note that of all Weibull distributions with y;> 2. the Rayleigh dis- . ibution is the closest to the ' ype i distribution (i.e., it has the longest tail). tr

is i c J. i$iss.s $, 4 4 o.+ n u 4 a -Q .ke i K ' '1 - ----J = e '5 i _7 ) f. %-y c a I S'/ -k ' 3,. k dir 1 -g.,,. ,.gy,; e "'s;

\\

4. %a' w)?[

,-.l_, m., a. _. IQ' %, \\. L ~L e.. = 3 N_...- .7 s t [y 5 >N 's '70, i .:s% N ~'g/ s . __f ig- ,,n) - Q ~ < i _., ' \\. '- \\. ~

g.

q m= - +, \\ ! q,o. g l ~s7.. - r. a J (ir d.. w _,~....'=......_.M.....,,o so- , 'l; ~ '. + _ =,, -,ur,,- ,, c ... t_., ..a.. g '. /

2. Lhe r eat.<p.t.non betw a kid e.ed c.aemer. a..ee.penba..

d,10' 11. i,, 3 C.wis.a an ei.....f wand.p d contour. he sn.was.ea.w. r.ea.n..e l11. _ _, i -- - _ _ g - - ' ; - -.t AI.*k.s..d e s. j { c-t h FIGURE 3.2.1. Map of basic design wind speeds. Reproduced with permission from American National Standard A58.1 Building Code Requirementsfor Minimum Design Loads in Buildings i} and Other Structures, copyri;;ht 1982 by the American National Standards Institute. Copies of 4 this standard may be purchased from the American National Standards Institute at 1430 k llroadway, New York, NY 10018. g( l 1 S% gasa r M.:.- hr .o

NPPf-58c4 cio'1 f% d' 2-z WIND EFFECTS ON STRUCTURES AN INTRODUCTION TO WIND ENGINEERING SECOND EDITION Emil Simiu Center for Building Technology National Bureau of Standards Gaithersburg, Maryland - Robert H. Scanlan Department of Civil Engineering The Johns Hopkins University Baltimore, Maryland (Emeritus Professor Princeton University) A Wiley-Interscience Publication JOHN WILEY & SONS New York / Chichester / Brisbane ! Toronto / Singapore

/dP' f - Sao - h o 'P ,-3 N ;,Y 1 i I i 3 Copynght @ 1986 by John Wiley & Sons. inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copynght Act without the permission of the copynght owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons. Inc. Library of Congress Cataloging in Publication Data Simiu, Emil. Wind criects on structures. "A Wiley interscience publication " includes bibliographical references and index. 1. Wind. pressure. 2. Buildings-Aerodynamics. 3. Structural dynamics. 1. Scanlan. Robert H. II. Title. TA654.5.S55 1985 624.l*76 85-10598 ISBN 0-47186613-X Printed in the United States of Amenca 10987654321 a --n. e-4,- -,..y ,g.

AfM-SB$'- v o 9 $W-ESTIMATION OF EXTREME ND SPEEDS 85 in which sheltermg effects by small-scale obstacles are present, the data may be adjusted by using a procedure presented in (3-4]. A situation commonly encountered in practice is one in which, while the anemometer may not have been moved, the roughness of the terrain surrounding j the anemometer has changed significantly over the years as a result of extensive j land development. In such situations, the adjustment of the data to a common roughness may pose insurmountable problems, unless detailed information on the phases of the land development is available. Anemometer elevation and location changes are listed for most U.S. weather stations in Local Climatological Data Summaries [3-3]. 3.2 ESTIMATION OF EXTREME WIND SPEEDS IN WELL-BEHAVED CLIMATES Infrequent winds (e.g., hurricanes) that are meteorologically distinct from and considerably stronger than the usual annual extremes are referred to herein as extraordinary winds. Climates in which extraordinary winds may not be ex-pected to occur are referred to as well behaved. In such climates it is reasonable to assume that each of the data in a series of the largest annual wind speeds contributes to the description of the probabilistic behavior of the extreme winds. A statistical analysis of such a series can therefore be expected to yield useful g ~ " predictions oflong-term wind extremes. Thus, in a well-behaved climate, at any given station a random variable may be defined, which consists of the largest yearly wind speed. If the station I is one for which wind records over a number of consecutive years are available, f then the cumulative distribution function (CDF) of this random variable may be estimated te characterize the probabilistic behavior of the largest annual I wind speeds. The basic design wind speed is then defined as the speed corre-sponding to a specified value p of the CDF or, equivalently, to a specified mean ) recurrence interval N.* A wind corresponding to an N-year mean recurrence i interval is commonly referred to as the R-year wind. I 2 This section is devoted to the question of estimating (a) the CDF of the largest I 1 & annual speeds and (b) errors inherent in the wind speed predictions. Such errors I a include,in addition to those associated with the quality of tne data (see Sect. 3.1), j 1 .A modeling errors and sampling errors. Modeling errors are due to an inadequate f choice of the probabilistic model itself. Sampling errors are a consequence of e L the limited size of the samples from which the distribution parameters are ..s @ estimated and become, in theory, vanishingly small as the sample size increases e, indefinitely. a ' W 3.2.1 ]' -cf. Probabilistic Modeling of Largest Yearly Wind Speeds ll k Several probability distributions have been proposed to model extreme wind g g' g behavior. These include: the Type I distribution of the large n n Ni Recall that S = 1/(1 -p)(see Appendix Al, Eq. A1.45). n +

2-C' 4'P/'1 - SBa!- o c y 5 ',p ' v 86 EXTREME WIND CLIMATOLOGY the Type 11 distribution of the largest values (Eq. Al.42), and the Weibull distribu-tion (Eq. A1.65). Extreme wind speeds inferred from any given sample of wind speed data depend on the type of distribution on which the inferences are based. For large mean recurrence intervals (R > 50 years, say) estimates based on the assumption that a Type II distribution is valid are higher than corresponding estimates obtained by using a Type I distribution, while estimates based on a Weibull distribution with taillength parameter y3 2 are lower.* According to [3-5], extreme winds in well behaved climates may be assumed l to be best modeled by a Type II distribution with p=0 and y=9. However, subsequent research has shown that this assumption is not borne out by analyses r of extreme wind speed data [3-6,3-7,3-8]. In [3-6],37 year-series of 5 minute i' largest yearly speeds measured at stations with well-behaved climates were subjected to the probability plot correlation coeflicient test (see Sect. A1.6) to determine the taillength parameter of the best fitting distribution of the largest values. Of these series,72% were best fit by Type I distributions or by Type II distributions with y= 13 twhich differ insignificantly from the Type I distribu-tion); 11% by Type II distributions with 7 G y < 13; and 17% by Type II distribu-tion with 2Gy <7. Virtually the same percentages were obtained in [3-7] from the analysis of sets of 37 data generated by the Monte Carlo simulation from a population with a Type I distribution. On the other hand, the analysis of sets l generated by Monte Carlo simulation from a Type II distribution with tail i length parameter y=9 led to percentages differing significantly from those p H h corresponding to the actual wind speed data. On the basis of these results it can be confidently stated that in well-behaved climates extreme wind speeds are c! modeled more realistically by the Type I than by the Type II distribution with L y=9. This conclusion was reinforced by studies reported in [3-8], in which i h techniques similar to those of [3-7] were used in conjunction with wind speed data at one hundred United States weather stations obtained from [3-9]. e As indicated earlier, the Type I distribution results in lower estimates of the f extreme wind speeds than the Type II distribution with y=9. An interesting i result obtained in [3-8] is that at most stations in the United States even the ]; J Type I distribution appears to be an unduly severe model of the wind speeds i corresponding to large mean recurrence intervals; at these stations a better } fit to the data is obtained by Weibull distributions with 732. Thus, structural L, reliability calculations based on the assumption that the Type I distribution jqj holds are in most cases likely to be conservative [3-10]. ~ ! ji 3.2.2 Estimation of and Confidence Intervals for the S-year Wind: i Numerical Example i' l It is shown in Sect. A1.6 that, given a set of data with a Type I extreme value j j underlying distribution, several techniques can be used to estimate the param-4 l i.

• The differences between speeds estimated on the basis of Type 11 distributions and the T l

distribution increase as y decreases. Di!Terences between speeds based on the Type i distributio f and Weibull distnbutions merease as y increases.

2-$ W/Y .r8 $ - c c ? $N :lV ESTTMATION OF EXTREME WIND SPEEDS 87 eters of the distribution and, hence, the value of the variate corresponding to a i given mean recurrence interval.* However, inherent in these estimates are sampling errors. A measure of the magnitude of the latter can be obtained by J: calculating confidence intervals for the quantity being estimated. that is, intervals of which it can be stated-with a specified confidence that the statement is correct-that they contain the true, unknown value of that quantity. Techniques that can be used to estimate the R-year wind, and confidence intervals for the R-year wind, are discussed in some detail in Sect. A1.6. One of these techniques is t presented and illustrated below. j Using the approximation -In[-In(1-1/R)]=In R, it follows from Eq. A1.74 (which is based on the method of moments) that the estimated value Og of the R-year wind vp is ) 0;= f +0.78(In R-0.577)s (3.2.1) I I where i and s are, respectively, the sample mean and the sample standard deviation of the largest year;y wind speeds for the period of record. i As previously noted, inherent in the estimates of og are sampling errors. It follows from Eqs. A1.76 and A1.70(which are based on the method of moments) a that the standard deviation of the sampling -~r., in the estimation of vg can a s be written as 11 SD(0 )20.78[1.64 + 1.46(In R-0.577)+ 1.1(In R-0.577)23 t/2 _3 e 3 ,[I (3.2.2) n e where n is the sample size. h h Example d At Great Falls, Montana, the largest yearly fastest-mile wind speeds at 10 m above ground during the period 1944-1977 (sample size n = 34) were [3-9]:

e 57,65,62,58,64,65,59,65,59,60,64,65,73,60,67,50,74 g

60,66,55,51,60,55,60,51,51,62,51,54,52,59,56,52,49

e IS (mph). The sample mean and the sample standard deviation for these data are

} i=59 mph and s=6.41 mph. From Eqs. 3.2.1 and 3.2.2 it follows that for N = 50 years and N = 1,000 years,

n W'

0oa 76 mph SD(0 o)23.7 mph 5 3 0 ooo291 mph SD(0 ooo)m6.4 mph + 1 Y _ Ifit is assumed that the largest ye trly wind speeds are describ Wdistribution,t the R-year wind, denoted by v, can be obtained from \\,e 1 8 3 ^ Y interval. In this chapter the notation Gx(t - t/R)= c; is used.T s t e mean recurrence 75g'"7to as the Rayl i h di, tit is recalled that the Weibull distribution with tailleng g 1 _ tribution is the closest to the Type I distnbution ti.e., it has the longe eg ak \\ l',

2-7 k'PPf-SBe' - 0o'? f" y' J> e'? 88 EXTRE51E WIND CLINIATOLOGY (with y = 2) as follows: (3.2.3) v5 ~ f +0.463 [(in S)u2 -0.886] where i and s are defined as in Eq. 3.2.1. In the case of Great Falls, i = 59 mph and s = 6.41 mph, so that ufo = 74 mph and ufooo = 83 mph, versus v3o = 76 mp and viooo=91 mph, as estimated in the preceding example by assuming the validity of the Type I distribution. As indicated previously, in engineering calculations it is prudent to assume the validity of the Type 1 distribution (Eq. 3.2.1), rather than using Eq. 3.2.3. This conservative approach was in developing the map of basic design wind speeds (i.e., fastest-mile wind sp at 10 m above ground in open terrain, with a 50-year mean recurrence interval) included in the American National Standard A58.1 [2-49] (Fig. 3.2.1). As shown in Sect. A1.6, the probabilities that up is contained in the intervals SD(Dy),0;i2SD(0 ), and 0;i3SD(Og) are approximately 68%,95%, and 99%, respectively. These intervals are referred to as the 68%,95%, and 99% Og 3 confidence intervals for vs, and are shown for the 34-year Great Falls sample in line (1) of Table 3.2.1. It is also shown in Sect. A1.6 that the width of the confidence interv 2g be reduced if a more efficient estimator is used; however, the intervals cannot be narrower than those obtained by using the Crambr-Rao (C.R.) lower bound M, (Eq. A1.77). For the Great Falls sample, the confidence intervals based on t 4 latter are shown in line (2) of Table 3.2.1. It is seen that the differenc fm the results of lines (1) and (2) of Table 3.2.1 are small. This is co the conclusion of Sect. A1.6 that the efficiency of the method of moments4 (Eq. 3.2.1) is generally adequate for structural design purposes. j L It is noted that, in Table 3.2.1, the errors in the estimation of the 50-year 9 wind are of the order of 10% at the 95% confidence level. Since the wind pres-sures are proportional to the wind speeds (see Chapter 4), the corresponding i. errors in the estimation of the pressures are of the order of 20%. r u f e Confidence Intervals for the S-year Wind at Great Falls is 5 TABLE 3.2.1. Confidence level 68 % 95 % 99 % k interval, R (years) 50 1000 50 1000 50 1000 Mean recurrence E U (1) Estimated by ^ 76 3.7 91 6.4 76 i 7.4 91t12.8 76t il.1 91 19.2 method of moments I C.R. lower bound 76 3.1 91 i 5.0 76 i 6.2 91 t 10.0 76t9.3 91 tl5.0 j (2) Estimated usmg i E. ' ~ ~ ' ' ' ^ - - - - - - - - - j

ig is i i i .~., 1 b hMdi:l%kd j.0, f 3, 1_ ,p' ;9 i! ' 4 i yN w_, Q f - Q p l' v,- p -w 4'f s' x?ar ~ - J :.d.W4 1 A,,l > g-W - q',.'..: ^ /N 't 4,xg%s z ~ + s --j m-- f i . g.-4gR 4:\\l sj- ._ q - f f,k, ..,C,,, y- .- --- -.--,\\ \\ p, -J. = = - \\ .j .r, _ _ N x Nxg. 3 r akylt_ x x -.- F _ _.._ __ g w%x 3./ \\ J a, s a& -[. x 4 O~s,- Ng's k %gM4]- \\ ,,o - l 4,_ v.,. ='n.2_1, \\. y a y --O s_ ~ l *' e0 r_ "gh ' ^' \\ M A- -~. - .e 7 '3 i __ \\ 'IT ino. i ., +/ p .~~~~r-a 7 L x _3 ID I "_"_'."*p... t. -J ',.-..i 3 e.: L. ,, 4= e o.p n q m -.. . c .a.........,-...m--e... F p . 7-2.t 6 m,.. w.., ,c ..a ,.eoo2. 1"oi ;: n .c ....e. w. _+- -2 l t. .no 5 FIGUR E 3.2.1. Map of basic design wind speeds. Reproduced with permission from American National Standard A58.i Building Code Requirementsfor Minimum Design Leads in Buildings s\\ } \\ o r Other Structures, copyright 1982 by the American National Standards institute. Copies of q this standard may be purchased from the American National Standards Institute at 1430 g 11 roadway, New York, NY 10018. bs ( l i S% y ^ b;w &&.

s. -..).

r- =.

.2 - 9 A W f ' 58 $-vo P [M 90 EXTRE!.1E WIND CLIMATOLOGY An alternative approach to.accountmg for sampling errors, which applies the theorem of total probability, is suggested in [3-51]. 3.2.3 Methods for Estimating the Extreme Speeds at Locations with Insufficient Largest Yearly Wind Speed Data There are about one hundred U.S. weather stations for vhich reliable and l relatively long wind speed records are available D.e., records over periods of, I say,20 years or more). Some of these stations cover areas of tens of thousands of square miles, over which-for meteorological reasons or owing to topographic effects-the extreme wind climate is not necessarily uniform. There arises therefore in practice the problem of estimating extreme wind speeds at various locations where long-term records of the largest yearly wind speed data do not exist. Esthnates of Extreme Wind Speeds in a Marine Environment. Reference 3-1I lists three methods that are in principle available to carry out such estimates for marine environments where the extreme speeds are associated with extra-ql tropical storms. The first method makes use of climatological information on h! various parameters of the storm and of physical models relating those param-eters to the surface wind speeds. It is shown in Sect. 3.3 that such a method can be applied to estimate extreme wind speeds in hurricane-prone regions. How-ever, as noted in [3-11], owing to the complexity of the surface wind patterns p j in extratropical storms, the usefulness of this method appears to be uncertain 1 in regions where such storms are dominant. A second method listed in [3-11] is the use of objective analysis schemes.

j These consist of

(a) an initial guess at the surface wind on a regular grid,(b) an automated procedure for screening wind reports from ships to eliminate l ;

erroneous readings, and (c) a procedure for correcting the initial guess on the l p basis of the usable set of ship reports, which involves relations among the J surface wind speeds, sea-level pressures, and air and sea temperatures. Details i [j on objective analysis schemes and of errors currently inherent in such schemes '.j l (which may range from 10% to 30%) are given in [3-11]. 4 The third method listed in [3-11] is referred to as direct kinematic analysis. j The method, which involves subjective judgment by experienced analysts, con-4 sists of synthesizing discrete meteorological observations to obtain a continuous field represented in terms of streamlines and isotachs. Objective or kinnnatic h, l l, analyses applied to a sufficient number of strong storms make it possible to y, j provide estimates of extreme winds that may occur at any one location. As lg 7 indicated in [3-11], one of the major difficulties in conducting such analyses is that much of the vast store of existing data is currently not accessible in u readily usable form. hI Estimation of Extreme Wind Speeds from Short-Term Records. A practical i procedure for estimating extreme wind speeds at locations where long-term data are not available is described in [3-12]. The method, whose applicability i I 1 j6 I h

-d-so MMI-58$ -ov 9 fW. 9 i ESTIMATION OF EXTREME WIND SPEEDS 91 was tested for a large number of U.S. weather stations. makes it possible to infer the probabilistic behavior of extreme winds from data consisting of the largest monthly wind speeds recorded over a period of three years or longer. Estimates based on the monthly speeds, denoted by Op,,,, are obtained by re-writing Eq. A1.74 as follows: Og,,,, = X,,, + 0.78[In(12S)- 0.577]s,,, (3.2.4) where X,,, and s,,, are, respectively, the sample mean and the sample standard deviation of the largest monthly wind speed data, and R=mean recurrence interval in years. The standard deviation of the sampling error in the estimation of 6,,,, is 5 obtained from Eqs. A1.76 and A1.70 as 2 ) ~ SD(0s.,,,) = 0.78 { 1.64 + 1.46[In(12S)-0.577] + 1.1[In(12 )-0.577]2} i2 _8 i. E { (3.2.5) where n,,, = sample size. 1 l Example t l At Great Falls, the sample mean and the sample standard deviation of the largest monthly fastest-mile wind speeds at 10 m above ground for the period 3 September 1968 through August 1971* (sample size n,,,= 36) are X,,,=42 mph, 3 s,=6.96 mph. From Eqs. 3.2.4 and 3.2.5, the estimates for N=50 years and N=1000 years are: / / 1 0 o.,,,= 74 mph SD(0 o,,,,)= 6.23 mph 3 3 0:00o.,,, = 90 mph' SD(0 ooo,,,,)= 8.85 mph It is seen that th; estimated speeds based on the set of 36 largest monthly data s are only slightly lower than those obtained from the set of 34 largest yearly speeds (0 o = 76 mph and D oco =91 mph; see Sect. 3.2.2); however, the sampling s 3 i errors are larger. I Similar calculations carried out for 67 sets of records taken at 36 stations are reported in [3-12], where it was found that the differences 0 o.,,,-0 o, s 3 5 where Oso is the 50 year wind speed estimated from long-term largest yearly- .c data, were less than SD(Oso.,,,) in 66% of the cases and less than twice the value o ofSD(0 o,,,,)in 95% of the cases.This remarkable result. confirmed by additional 3 s culations reported in [3-13], indicates that the estimates based on largest s a

g. m nthly wind speeds recorded over three years or more provide a useful n

3,_ description of the extreme wind speeds in regions with a well-behaved wind climate. 1; Inferences concerning the probabilistic model of the extreme wind climate y fu .a

y

• For the actual data, see the Local Climatological Data summaries for the years 1968-1971.

a .I

li .2 - // N W F .S89 -o09 5 92 EXTRESTE WIND CLihlATOLOGY have also been attempted from data consisting of largest daily wind speeds [3-12), or of wind speeds measured at 1-hour intervals [3-14]. One problem i that arises in this respect is that data accorded on two successive days are generally strongly correlated. Nevertheless, as shown in [3-14], in practice such correlation has a negligible efTect on the statistical estimates, and the assumption of statistical independence among the data can therefore be used. However, a second and more serious problem is that the daily (or hourly) data reflect a large number of events (e.g., morning breezes) that are altogether unrelated meteoro-logically to the storms associated with the extreme winds. These events can be viewed as noise that obscures the information relevant to the description of the extreme wind climate. Indeed, it was verified in [3-12] that estimates of extreme winds based on daily data differ significantly from estimates obtained for long-term records of largest yearly speeds. This conclusion is a fortiori true for -{ inferences based on hourly data. I i' 3.3 ESTIMATION OF EXTREME WIND SPEEDS IN HURRICANE-PRONE REGIONS We now consider the prediction of extreme winds in climates characterized ,d by the occurrence of hurricanes. It was suggested in Sect. 3.2 that in a well-1 behaved wind climate each of the data in a series of the largest yearly speeds contributes to the description of the probabilistic behavior of the extreme winds. h[ However, in a hurricane-prone region most of the speeds in a series of the largest yearly winds are considerably lower than the extreme speeds associated with hurricanes; they may therefore be irrelevant from a structural safety point L of view. This situation is illustrated by the plot of Fig. 3.3.1, which shows the 5-min largest speeds recorded at Corpus Christi, Texas between 1912 and 1948 [3-6]. It may then be argued that in hurricane-prone regions the series of the largest yearly speeds cannot provide useful statistical information on winds of interest to the structural designer, much in the same way as the population of a L first-grade classroom-which might include a teacher-is of little use in a statistical study of the height of adults. That this is the case is suggested below. l j The abscissa in Fig. 3.3.1 represents the reduced variate -In i 1 - -I V il f [ y=-In 1 l i \\ N/_ l l where R is the mean recurrence interval. In virtue of Eqs. A1.43 and A1.45, a Type I extreme value cumulative distribution function would be represented in Fig. 3.3.1 by a straight line, the intercept and slope of which would be equal to the distribution parameters y and o, respectively. To the extent that the I 'l population oflargest yearly speeds would be described by a Type I distribution, t the actual data would then fit, approximately, a straight line. In Fig. 3.3.1 this j

}

is roughly the case as far as the winds ofless than hurricane force are concerned. However, if-as in Fig. 3.3.1-the hurricane-force winds are included in the i .~__m

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J-/3 NPP1 ,SB f-ooy ?WlV 522 ELEN1ENTS OF PROBABILITY THEORY AND APPLICATIONS It is noted that if X and Y are independent, then Corr (X, Y)=0.This follows cur-immediately from Eqs. A1.26, A1.18, and A1.24. However, the relation whi Corr (X, Y)=0 does not necessarily imply the independence of X and Y [Al-4]. stre yea Th< A1.5 PROBABILITY DISTRIBUTIONS COMMONLY 'JSED IN Cor WIND ENGINEERING ~ oth-var: The Geometric Distribution dur Consider an experiment of the type known as Bernoulli trials in which (a) the the only possible outcomes are the occurrence and the nonoccurrence of an event A, p(n. c (b) the probabilityp of event A in any one trial is constant, and (c) the outcomes out-of the trials are independent of each other. I Let the random variable N be equal to the number of the trial in which the r event A occurs for the first time. The probability p(n) that event A will first occur on the nth trial is equal to the probability that event A will not occur IfE on each of the first n - 1 trials and will occur on the nth trial. Since the probability i van of nonoccurrence of event A in one trial is 1 -p (Eq. A1.2) and since the n trials

oce, are independent, it follows from the multiplication rulo (Eq. A1.8) oft 1

p(n) =(1 -p )*- 'A (n = 1, 2, 3,...) (A1..iO) k witi This probability distribution is known as the geometric distribution with param-f- [Al eterp. p l ave: The probability P(n) that event A will occur at least once in n trials can be i L of a found in the following manner. The probability that event A will not occur in n add i trials is (1 -p)". The probability that it will occur at least once is therefore The t P(n) = 1 -(1 -p)" (A1.31) hou The expected value of N is, by virtue of Eq. A1.22,in which Eq. A1.30 is used-Non = f n(1 -p)"-'A (A1.32) Con n=1 Cont The sum of this series can be shown to be com R = 1/p ( A1.33) The quantity is referud to as the return period, or the mean recurrence interval, of event A. whe: Examples respi g;on 1. For a die, the probability that a "four" occurs isp = t. If the total number thep of trials is large, it may be expected that, in the long run, a "four" will appear on is m the average once in N = 1/t = 6 trials. varn 2. A structure is designed so that the stresses in its members will attain the N allowable stress under the action of extreme winds with a (0-year mean re. appl m_.__

J-/'/ kWf-58 $ - c o 7 $W;iO PROllAlllLIW DISTRIBUTIONS USED IN WIND ENGINEERING 523 currence interval. The probability of occurrence in any one year of winds for which N = 50 is p = 1/N =0.02 (Eq. A1.33). The probability that the allowable stress will be attained at least once in n years is given by Eq. A1.31. For n= 25 years P(25) = 1 -(1 -0.02)2s 20.396: for n = 50 years. P(50)a0.63. The Poisson Distribution Consider a class of events, each of which may occur independently of the others and with equallikelihood at any time of an interval OctG T A random variable is defined, which consists of the number N of events that will occur during an atbitrary time interval r = '2 -it (f > 0, t < t 6 T). Let p(n, r) denote t i 2 the probability that n events will occur during the interval T. Ifit is assumed that p(n, r) is not influenced by the occurrence of any number of events at times outside this interval, it can be proved [Al-4] that (Art -*' p(n,r)= n! e (n = 0,1, 2, 3,...) (A1.34) li Eqs. A1.24 and A1.25 are used, it is found that the expected value and the variance of n are both equal to Ar. Since it is the expected number of events occurring during time T, the parameter i is called the average rare of arrival of the process and represents the cycted number of events per unit of time. The applicability of Poisson's distriomm m y be illustrated in connection with the question of the incidence of telephone cads in a telephone exchange [Al-5]. Consi :.:r an interval of, say, a quarter of an hour, during which the average rate of arrival of calls is constant. During any subinterval, the incidence of a number n of calls is as likely as during any other equal subinten'al. In addition, it may be assumed that individual calls are independent of each other. Therefore. Eq. A1.34 applies to any time interval lying within the quarter of an hour. Normal and Lognormal Distributions Consider a random variable X which consists of a sum of small, independent contributions X, X,..., X,,. It can be: proved [Al-1] that, under very general i 2 conditions,if n is large the probability density function of X is 1 -(x ,)2g 5 f(x)= _ -exp ) (A1.35)

J_2na, la,

/ 2 I, 2 where p,= E(X) and a = Var (X) are the mean value and the variance of X, respectively. This statement is known as the central limit theorem. The distribu-tion represented by Eq. A1.35 is called normal or Gaussian. It can be shown that } the probability distribution of a linear function of a normally distributed variable i is normal. Also, the sum of two or more independent normally distributed i variables is normally distributed. j Normal distributions are used in a wide variety of physical and engineering i applications, for example, the description of errors in measurements. At the iE

l 2-d~ NP/Y-Sad - o07 59 f Pl $24 ELD 1ENTS Of PROllABILilY TilFOlW AND Al'PLICATIOM same time, it should be c:.refully noted that many phenomena may not be value and normally distributed, for example, the extreme wind speeds occurring at any given geographicallocation. If the distnbution of the variable Z = log X is normal, the distribution of the variable X is said to be lognormal. Type I and Type 11 Distributions of the Largest Values. Mean Recurrence The et Intervals larqcst rc the genen Let the variable X be the maximum of n independent random variables Y, Y.... Y, [Al 6]. Since the inequality X G x implies 1;G x for all f(f = i 1, 2,..., n), it follows that F(X G.*:) = Prob (Y G x, Y 4 x..... Y.5 x) ( Al.36a) i 2 = Fy,(x)Fy,(x)... Fy,(x) ( A1.36b) where y, c where, to obtain Eq. A1.36b from Eq. A1.36a, the generalized form of Eq. Al.8 ete was used. The probabilities Fr (y) are referred to as the underlying (or the initial) distributions of t'te variables li. The latter are said to constitute the p Equati. i parent population from.ihich the largest values X have bem extracted. In Point func the particultr case in which all the variables Y, have the same probability distribution Fy(y), Eq. A1.36 becomes ""Y Efdis

ype Fx(x)=[Fr(x)]"

(A1.37) In the case m which they are unlimited to the right, the initial variables Y d whereas ft are said to have distributions of the c::ponential type if their cumulative distribu-f tion functions converge (with incicasing y) toward unity at least as fast as an i exponential function; the initial variables Y are said to be of the Cauchy type if it is convt lim [1 - F(y)]y" = A ( A > 0: k > 0) ( A1.38) by p and : r-w As the number n becomes very large, the distributions Fx(x) of the hrgest d and for th values approach limits known as the Type / and the Type 11 distri%tions 3 according as the initial distributions are of the exponential and of the Cauchy 'M type, respectively [Al-4, Al-7]. The cumulative distribution function for w. ype I distribution of the largest Hy Frcm i Let the ra values (also referred to as the Type 1 Extreme l'alue distribution, or the Gumbel ge distribution) is given loca i the wind 1-m<x<m 'I 1 -p. By s F,(x)= exp{ -exp[-(x - p)/a]} J-m < p < r (A1.39) jjgj 0<a < m in Eq. A1.39, y and a are referred to as the location and the scale parameter, i< i Thus, the respectively.' It can be shown, using Eqs. A1.24 and A1.25, that the mean Ib to the valt I ' As shown in Eqs. Al.40 and Al.41, these parameters are not the expectation and the standard t deviauor. of X. g

i j ? - / (, } A'P/'t 7 38 gf-vo 9 l fplV l PROII AlllLITY DISTRIBl#10NS USED IN WIND ENGINEERING 513 I value and the standard deviation of X are E(X) = p + 0.5772a ( A l.40) SDfX)=1a ( Al.41) d6 The cumulative distribution function for the Type 11 distriburton of the largest values talso referred to as the Tvpc // Extreme l'a'ue distribution, or the generalized Frecher distribution)is p<x<w E ~ Fn(x)= cxp(-[(x-p)/a]-7)q 0 - ( A1.42) l y>0 I where p a. and / are the location, the scale, and the shape (or taillength) param-eter, respectively [Al 8]. In the particular case p =0, Eq. A1.42 is referred to as the Fr6chet tas opposed to generalirrd Fr6chet) distribution. 1 j Equations Al.39 and A1.42 may be inverted to yield the so-called percent point function, that is, the value x of the randont variable that corresponds to any given value of the cumulative distribution function. In the case of the Type i distribution x(F )= p-a in(-In F ) (A1.43) i i whereas for the Type 11 distribution x(Fn) = + a(-In Fn}- U7 (A1.44) It is convenient to denote the cumulative distribution function value F or Fu i by p and x(F ) or x(Fn) by Gx(p). Then, for the Type i distribution i Gr(p)=p-a In(-in p) (A1.43a) and for the Type 11 distribution Gx(p) = p + a(-In p)- U7 (A1.44a) From the definition of p and Eq. A1.2 it follows that Prob (X > x)= 1 -p. Let the random variable X represent the extreme annual wind speed at some given location. Each year may then be viewed as a trial in which the event that the wind speed X will exceed some value x has the probability of occurrence I 1 -p. By virtue of Eq. A1.33, the mean recurrence interval of this event is l R=1-p (A1.45a) Thus, the wind speed x corresponding to a mean recurrence interval R is equal to the value of the percent point function of X corresponding to p=1 (A1.456) N -l -. - - -...

.2-/7 f NP/1-ssf-w 7 ( lW,W 526 ELDIENTS OF PROllAlllLi'lY T11EORY AND APPL.lCATIONS been itelations Between Type I and Type 11 Extreme Value Distributions variat. Let the Type 11 distribution be written as tion a Fn(y)= exp(-[(y-un)/a ]'} ( A1.46) Type n as an (in the present context it is convenient to denote the location and scale param. eter of the Type 11 distribution by pn and o, respectively). If the transformation Let 2 n (Al.47) has a y-pn = exp x tion ft is applied to Eq. Al.46. the expression obtained is a Type I distribution with parameters where (Al.48) p =ln an 1 ( A1.49) o-7 It is now shown [Al-12] that as y approaches infmity, a Type 11 distribution Thec approaches a Type I distribution. Consider the distribution of the standardized variate ( A1.50) Z = ' ~ le(X) sca Ther. where loc (X) and scale (X) are measures oflocation and scale, respectively, of the distribution of X. Examples of measures oflocation of a random variable X are its expected value E(X) and its median Gx(0.5). Examples of measures of scale of a random variable X are its standard deviation SD(X),its imerquartile is, by difference 6 o= Gx(0.75)-Gr(0.25), and its 95% difference 6,3 = Gx(0.975) From 3 - G x(0.025). The percent point function Gz(p)is given by Gz(p)= Gx(p)-loc (X) (0 < p < 1) (A1.51) or, iP-scale (X) With no loss of generality, a reduced variate with p =0 and a = 1 may be used in the demonstration. Substituting Eq. A1.44a with p =0 and a= 1 into Eq. A1.51 and choosing, for simplicity, loc (X)= Gx(0.5) and scale (X)=do, Co tion, i [-In( p)]- "7 -[-In(0.5)]- ilt [-In(0.975)]- "' -[-In(0.025)]- "7 As y-m, this expression becomes indeterminate. However, application of or,m: L' Hospital's rule yields, after simplification, (0 < p < !) ( Al.53) Gz(p)= - In[ - Inl0.975)] - { - In[ - In(0.025)] } As can be seen from bqs. A1.43a, the terms in the numerator and denominator of Eq. Al.53 are, respectively, the percent point function, the median, and the it is n 95% difference of the reduced variate for the Type I distribution. It has thus any gm

.2-/C AY,Pf .SB $ - 0 0 9 ^ f g,'/> 'f PRollAillt.ITY DISTRitstTTIONS 1: SED IN WIND ENGINEERING 527 been demonstrated that, as y approaches infinity, a standardized Type 11 variate approaches a standardized Type i variate and, hence, a Type 11 distribu-tion asymptotically approacher a Type i distribution. Type i Distributions: Mode of the Largest Value from a Sample of Size n as an Aporoximation of the Percent Point Function 6x[l/(1 -n)) Let Z be the largest of a set of n values of a random variable X, each of which has a Type 1 Extreme Value distribution (Eq. A1.39). The cumulative distribu-tion function of this largest value is F,(:) = [F (:)]" = cxp[-- n exp(- w)J (A1.54) i where -x<:<x

-p w=

-x<y<x ( A1.55) 0 0<a<x The corresponding p.obability density function is f,(:) = 1-n exp[- nc "- w) ( A1.56) 0 The root of the equation df,(:) I - - w][ne "- 1] = 0 (A1.57) dr "p n exp[-nc " is, by definition, the mode

• of the largest of the set of n values considered.

From Eq. A1.57 it follows immediately 1 c"=- (Al.58) n or, if Eq. A1.55 is used. mode (Z)= -a in I (A1.59) n Consider now the initial random variable X. Since X has a Type I distribu-tion, its percent point function is Gr(p)= p -o In(-In p) (A1.43a) or, making use of Eq. A1.45 in which is the mean recurrence interval f1-1) 1 -.1 V Gr l - I= -a in -In l (A1.60) \\ N) N).

• lt is recalled that the mode of a vanable X is the value of that vanable most likely to occur in any pven tnal(Sect. Al.41

?-/7 NA#f-sB gt - c o'/ $W j',' 528 ELD 1ENTS OF PilOllAlllLi1Y TilEOltY AND APPLICATIONS In the particular case in which S = n Gy (1 - - mp-o in ~ l - 13 -In 1-- ( A1.61) nj n, in Eq. A 61. G y(1 - 1/n)is the value of X corresponding to the mean recurrence interval n. it can be verified that for n sulliciently large, say, n > 10, ' I -In I l-IV :xin th (A1.62) ~ / la l i \\ nj_ nj (For example, for n = 20, the right and left members of Eq. A1.63 are equal to -2.970 and - 2.996, respectively. For n =40, they are equal to - 3.676 and - 3.689, respectively.) It follows therefore that Gx (1 - l := p -a in 1 = mode (Z)( A1.63) n n Equation Al.63 shows that if X is a random variable with a Type i distribution, the mode of the largest value in a sample of n values of X is very nearly equal ] to the value of the random variable corresponding to the mean recurrence mterval n [Al 9]. An interesting experimental verification of this statement is provided by the p data of[Al-11], which cover a period of 37 years. For example, for the first five sets of [Al-11], the values of the largest of the maximum yearly wind speeds recorded in 37 years, r.., and the values of the estimated 37 year wind, r33, m are (in mph) s a 4 1 l-Cairo Alpena Tatoosh Isl. Williston Richmot.d + (111.) (Mich.) (Wash.) (N.D.) (Virginial i I e., 51 50 84 50 48 m p t -( r3, 52 51 81 52 50 'C. -( 4 r The probability that the largest of a set of n values of the random variable X i r with a Type 1 distribution is contained in a given interval can be easily calculated d using Eq. A1.54. For example, from a 37-year record of the largest annual wind 4 speeds at Richmond, Virginia the values of and a were estimated to be 16.8 I j mph and 3.78 mph, respectively [Al-ll]. Using these values, the probability f ( that the largest wind speed Z= V., in a set of n= 37 largest annual speeds is j( m contained, say,in the interval V (1 0.24) = 50 t 12 can be estimated as follows: I 3 u 3 v' f33(:)d:= F (62)-F,(38)=0.95 ( A1.63a) i P(386 Z 462)= a 33 3 '( tl 3s I I: I J E.

,2.2o M&f-- Sapi-co ? $W,N 5'ROllAlllLifY TilEORY AND STATISTICAL, D ATA 529 l foint Extrerne Value Distributions 1 'v 'he joint Type i Extreme Value probability distribution of two correlated ariables X, Y has the expression ~ f ( ~ Um' x - p,' + exp I - m -y - y, (Al.64a) '!) F rlx. y)=exP - exPI -m t \\ 0 \\ G ' J *' r shere ,j m = (1 - Fxt)- U2 (Al.64b) - g

nd the correlation ccefficient Fxr>0 [Al 23]. It can be verified that for

'robabilities of interest in structural reliability calculations (c.g., Fxr(x, y) > 0.99) .nd for values pxr GO.7, say, Fxr(x, y)a Fx(x)F,(y) (A1.64c) .here F (x) and Fr(y) are the Type i Extreme Value distributions of X and Y, t espectively, that is, it may be assumed that X and Y are statistically inde-3endent. Die Weibull Distribution the Weibull cumulative distribution function is fx-p s' F(x) = 1 - exp -1 (A1.65) _ ( a The expected value and the standard deviation of the variate (x-p)/a are, espectively, F(1/y + 1) and { F(2/y + 1)-[F(1/y + 1)]2) na, where r is the gamma ' unction, and are listed here for various values of y [Al 8]. f 1.2 1.6 2.0 2.2 2.6 3.0 3.2 3.6 4.0 6.0 Expected Value 0.9407 0.8966 0.8862 0.8856 0 8882 0.8930 0 8957 0.9011 0.9064 0.9264 standard Deviation 0.7872 0.5737 0.4632 0.4249 0.3670 0.3245 0 3072 0.2780 0.2543 0.1850 For ya3.6, the shape of the Weibull distribution is similar to that of the aormal distribution. The Weibull distribution witt parameter y = 2 is common!y referred to as the Rayleigh ('istribution. A1.6 PROBAlllLITY TilEORY AND STATISTICAL DATA Goodness of Fit Data obtained-or that may be obtained-from actual cbservations may be viewed as observed values of random variables. The behavior of the data is then assumed to be described by models governing the behavior of random variables, that is, by such mathematical models as are used in probability theory.

} .2 - H NFPf-58 ef-wf !)/ jV 530 ELD 1ENTS OF PROllAlllLITY TilEORY AND APPLICATIONS in practical applications two important problems must be dealt with. First, from the nature of the phenomenon being investigated (or on the basis of I observationsi, an inference must be made on the probability distribution that i' i will adequately describe the behavior of the data. Second, the data must be i used for drawing inferences on the parameters of the distribution or on some of i its characteristics, for example, the mean or the standard deviation. l In practice, given a set of observed data, or a data sample,it is hypothesized that its behavior can be modeled by means of some probability distribution believed to be appropriate. This hypothesis must then be tested. Tests incorpor-ate quantitative measures of the degree of agreement, or goodness offir, between "f h the data and the hypothetical distribution or, conversely, of the degree to which the data deviate from that distribution, if the measure of this deviation is L i appropriately small, then the hypothesis will be accepted. and vice versa. h~ Associated with the testing of a hypothesis is a level ofsignificance, that repre-sents the probability of rejecting the hypothesis when it is in fact true. Tests f commonly used in applications, including the well known x test, are discussed, 4l for example, in [Al 1] and [Al-4] Brief mention is made of the probability plot correlation coeflicient test [Al-10] that has been used in the study of the i behavior of extreme winds [Al-11, Al-12]. The probability plot correlation coefficient is defined as [(X,-i)[Af,(D)- Af(D)] (A1.66) y r= o (((X,- X)2 [(Af,(D)- Af(D))23u2 in which i =[ X,/n, Af(D)=[ Af,(D)/n, n is the sample size, and D is the prob-l l ability distribution being tested. The quantities X, are obtained by a rearrange- ! 7 ment of the data set: X is the smallest, X: the second smallest,...,X, the g' i l-th smallest of the observations in the set. The quantities Af,(D) are obtained as 4 follows. Given a random variable X with probability distribution D and given a sample size n, it is possible from probabilistic considerations to derive mathe-matically the distributions of the smallest, second smallest, and, in general, the i th smallest values of X in that sample.The quantities Af,(D) are the medians of each of these distributions. If the data were generated by the distribution D, then, aside from a location and scale factor, X, will be approximately equal to the theoretical values Af (D) 4 y for all i so that the plot of X, versus Af,(D)(referred to as probability plot) will be approximately linear. This linearity will, in turn, result in a near unity value of r. Thus, the better fit of the distribution D to the data the closer ro o will be to unity. To test whether the behavior of a given set of extreme data is better described r by a Type I distribution or by a Type II distribution with some unknown value ( '7 4 of the taillength parameter y, the probability plot correlation coefficient rois '! ;k I computed for a large number of extreme value distributions, defined by various values of y suitably spaced from y= 1 to y= m (it is recalled that y = m corre-sponds to a Type I distribution). The variable in these distributions is written If in standardized form so that for any given set of data the coeflicients to depend lp d 4 r .d i

Tf,t 2 ~ 2.2 4Tff-S8 p'- co 9 ~ [R',l. l'HOUAlllLITY Tile 0llY AND STATISTICAL DATA 531 8 solely upon. that is, are independent of the location and scale parameters y and a on whih. therefore no prior assumptions need to be made (Al ll]. The distributi in that best tits the data is that which corresponds to the largest of the calculat d values of r. o Estimation of Ditribution Parameters From the data of a sample it is, in principle, possible to make inferences on the parameters of the distribution that describes the behavior of the population from which the data are extracted for on characteristics of the distribution, e.g., the meani. An estimator may be defined as a function 3(X, X. .X.) of i 2 the sample values such that i is a reasonable approximation to the unknown value 2 of the distribution parameter (or characteristici.The particular numerical e value assumed by an estimator in a given case is referred to as an estimate. As a function of random variables. i(X, X,..., X,) is itself a random variable. i 2 This is muso ::d by the following example, consider the observed sequence of 14 outcomes of an experiment consisting of the tossing of a coin: 11 T T T 11 T 1111 T li 1111 T 11 (A1.67a) The random numbers associated with this experiruent are the numbers zero and one, which are assigned to the outcome heads and to the outcome tails, respectively. The data sample correspondmg~ to the observed outcomes is then: 0,1,1,1,0,1,0,0,1,0,0,0,1,0 (A1.67b) This sample is assumed to be extracted from an infinite population that, in ~T the case of an ideally fair coin, will have a mean value, denoted in this case by 2, equal to {. A reasonable estimator for the mean a is the sample mean s* i=I T X, ( A1.68) a n s) where n is the sample size (number of observations) and X, are the observed data. In the case of the sample consisting of all 14 observations in A1.67b, i=4. If the samples consisting of the first seven and of the last seven observa-l tions in A1.67b are usca,i= 4 and i=i, respectively, As a random variable, an estimator 3 will have a certain probability distribu. l tion with a nonzero dispersion about the true value s. Thus, given a sample of statistical data,it is not possible to calculate the true value a of th: parameter i sought. Rather, confidence intervals can be estimated of which it can be stated, with a specified confidence level q (level of significance 1-q), that they will contain the unknown value 2. l In order that the confidence interval corresponding to a given confidence level q be as narrow as possible, it is desirable that the estimator used be egicient. l 'The symbol ' is used to denote estimated value.

2 ~,2 3 AW'f-SB pf-vo f' 21V s 532 LLDIENTS Ol? PilOllAlllLITY TilEORY AND APPLICATIONS Of two different possib!: estimators si and $ of the same parameter 2. the 2 estimator i is said to be more etlicient if E[(si -21 ]< E[(5 -21']. 2 2 i Details on procedures for estimating distribution parameters can be found. for example,in [Al 1] and [A14)(see also [A1 17] and [Al 22]).The question of parameter estimation for the Type i Extreme Value distribution-which is widely used in the study of extreme wind speeds-will be examined subs quently in this appendix. Before proceeding to this topic it is useful to discuss first the simulation of the behavior of a Type i Extreme Value distribution by means of numerical techniques commonly referred to as hionte Carlo methods. N1onte Carlo h1ethods. Simulation of a Type i Extreme Value Process As defined in [Al 13], hionte Carlo methods comprise that branch of experi-mental mathematics that is concerned with experiments on random numbers. " The simulation of the phenomenon of interest is achieved by subjecting avail. able sequences of random numbers to appropriate transformations. The new sequences thus obtained may be viewed as data, the sample statistics of which ' are representative of the statistical properties of the phenomenon concerned. Examples of engineering applications of hionte Cario methods can be found in [Al-4) and [A1 14]. The simulation of the behavior of a random variable with a given distribu- { tion is a simple application of hionte Carlo techniques that is now discussed. [ lt is assumed that the distribution is Extreme Value Type I with given param-1 eters y and a (Eq. A1.39). Consider a sequence of n uniformly distributed random numbers 0<li<1 [ ~ (i= 1,2,..., n) such as are listed in [A1 14] or as may be generated by procedures j i discussed in [Al 2], [Al-13], or [Al 14]. These numbers are viewed as prob- ) abilities of occurrence of the data X(li) obtained by the following transforma-i tion (Eq. A1.43): X(li)=p-a in(-In 1;) ( A1.69) i From the sample of size n X(li)(l= 1,2,..., n),it is possible to obtain estimates .t of p and a (i.e., the distribution parameters) and of Gx(p)(the percent point j Y function corresponding to any given value of p, see Eq. A1.43a). Since, as was 17 previously indicated, the estimates are random variables, the estimates will difier,in general, from the known parameters and percent point function of the y underlying distribution. The procedure just described can be repeated a large number M of times. Then M sets of values p,6, and Ox(p) and corresponding { histograms can be obtained. From those sets it is possible to calculate summary 14 statistics (such as the mean, the variance, the standard deviation) for, d, and 0x(p). } i A hionte Carlo study cf the behavior of a random variable with a Type I k t distribution conducted for the purpose of predicting extreme wind speeds was first reported in [Al-15]. A similar study, subsequently conducted by the writers, is now summarized. The parameter values of Eq. Al.69 used in this i study were p = 36.8 and a = 3.78 (these values in mph represent estimates of ( i A p.

.2.2 '/ "W1.mp-on F#, a PROilAlllLirY TilEORY A.ND STATISTICAL DATA 533 Type i distributions found in [Al ll) to best fit the annual extreme wind speeds recorded in Richmond. Virginia between 1912 and 1948). Two sets of 100 samples each were generated. the size of the samples being n =25 for the first set and n = 50 for the second. The main results of the study are listed in Table Al.l. For example: G (0.98)= 51.57 (calculated from the underlying distribu-t tion with parameters p = 36.8. c = 3.78); the mean of the 100 estimates dx(0.98) based on the samples of size n = 25 is Mean[dx(0.98)] = 52.58: the standard devi-ation of these estimates is 5[d (0.98)] = 3.46: the largest of the estimated d (0.98) 3 1 is max [dx(0.98)] = (l + 16.6/100) x {Mcan[dx(0.98)] } = Mean[dx(0.98)]+ 2.5s[dg(0.98)]. A histogram of the estimates dx(0.999) for the 100 samples of size n = 50 is shown in Fig. A1.5. The results of Table Al.1 were obtained by fitting a Type 1 Extreme Value distribution to the data samples generated from sequences of random numbers by Eq. Al.69. Ilowever. it is conceivable that because of the random character of the sampling, the behavior of some of the samples would be better described by Type 11 Extreme Value distributions rather than by a Type I distribution. To verify whether this is indeed the case. the probability plot correlation coefficient test was applied to each of the samples. The results obtained, which are independent of the parameters and cr of the underlying distribution are shown in Table Al.2. As shown in Sect.3.2 percentages such as those of Table A1.2 can be com-pared to similar percentages obtained from the analysis of measured extreme wind speed data in an attempt to draw inferences on the applicability of the Type i distribution to the modeling of extreme wind behavior in certain types of climate. For details on such inferences, see [Al-21]. TABLE Al.l. Monte Carlo Simulation of a Type ! Extreme Value Process y a Gr(0.98) Gy(0.99) G (0.999) t Original t Underlying) Distribution 36.80 1.78 51.67 54.24 62.97 Mesn' n = 25 36.90 .t.01 52.58 55.38 64.64 n-50 36.80 3.89 51.92 54.63 63.60 Standard Deviation' n=25 0.86 0.81 3.46 4.00 5.85 n = 50 0.65 0.50 2.14 2.49 3.61 Maximum Deviation Below n = 25 5.90 52.00 19.60 21.30 25.70 Mean*(Percent of Meant n = 50 3.80 32.00 10.20 11.00 13.60 Maximum Deviation Above n = 25 6.60 64.00 16.60 19.00 25.50 Mcan* (Pereent of Mcan) n-50 4.00 32.00 10.70 12.00 14.70 Maximum Deviation Below n = 25 2.50 2.60 3.00 3.00 2.80 Mean*(Standard Deviationst n = 50 2.20 2.50 2.50 2.50 2.40 Maximum Deviation Above n = 25 2.80 3.20 2.50 2.60 2.80 Mean*(Standard Deviationst n = 50 2.30 2.50 2.60 2.60 2.60

• Estimated from 100 samples of size n.

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.i o I 3 siano.ra asvi iion FIGURE Al.5. Histogram of estimated values Ox (0.999) for 100 samples of size n = 50. TABLE Al.2. Percentage of Samples from a Population with a Type i Distrit,ution that are Best Fit by Type I and Type !! Distributions Sample Size Extreme Value Distribution n=25 n=50 s 'l Type i or Type II l (y>13) 57 77 1 7 6 y < !3 13-12 I 26y<7 30 11 Estimators for the Type i Extreme Value Distribution A classical method of approaching the problem of estimation is the method of momems. In this method it is assumed that the distribution parameters can be obtained by replacing the expectation and the mean square value of the random variable X by the corresponding statistics of the sample. In the case of the Type i distribution, using Eqs. A1.40 and A1.41e l J.d (A1.70) 5 n y = f -0.57728 ( A1.71) 4 l e 4 ~~

.2 - R f,,, ATPf- $8pl-oo 1 / W,l PROBAlllLITY TilEORY AND STATISTICAL DATA 535 where i and s are the sample mean and the sample standard deviation. respec-tively, that is, i=b T X, (A1.72) n~ ~g ~ lI2 - T (X,.i)2 g A g,73) s=.n - From the estimators ( A1.70) and (A1.71) the following estimator of G (p) can t be constructed: Sy(p)= i + 3(y-0.5772)d6/n (A1.74) where y= -In(-in p) (A1.75) Under the assumption that the random variables,i and s defined by Eqs. A1.72 and A1.73 are, asymptotically, normally distributed, it can be shown [Al 7. pp.10.174, and 228] that for large samples of size n SD[dx(p)] = _6 + 1.1396(y-0.5772)]* + 1.1(y-0.5772)2 3 (A1.76) A more efficient estimator of Gx(p) has been developed by Lieblein on the basis of the method oforder statistics [Al-7, Al 16, A1-17]. A method frequently used in applications is based on least squarcsfitting of a straight line to the data on probability paper. This method is used in the computer program of[Al 11). A simplified approximate version of this method is presented in [Al 7, pp. 34, 227. and 228]. For a discussion of other estimation methods used for the Type i distribution. for example, the maximum likelihood method, the reader is referred to [Al-7] and [Al-18]. It can be shown that the standard deviation of any estimator of a parameter is larger than, or at least equal to, a theoretically specified standard deviation known as the Cramir-Rao lower bound. In the case of the percent point function of a Type I distribution, the Cramer-Rao lower bound is A SDea[dx(p)]=(0.60793y +0.51404y+ 1.10866)u2 A (A1.77) 2 Jn where y is given by Eq. A1.75 [Al 19] For n=25 and n=50, the ratio (1/a)SDen[dx(p)] is now compared with the-ratios (1/a)SD[dx(p)], where SD[dx(p)] denotes the standard deviation of the percent point function esti-mated by the method of moments, by Lieblein's method of order statistics, and by the method ofleast squares fitting. In Table A1.3 the quantit :s of line (1) were calculated by Eq. A1.76. The i quantities ofline (2) were obtained from [Al-16, p.131] (through multiplication of corresponding quantities given for n = 10 by J10/25 and,/10/50 or of quanti-ties given for n = 20 by,/20/25 and,/20/50). The quantities of line (3) were J

O. Ratios (1/alSD[SatP)] and (1/a)SDc,[S (p)] 3 TAULE Al.3. " = Si * ~~ n = 25 Estimation Metimd R 20 50 100 100:1 20 50 100 luut 0.65 1.02 1.13 1.45 0 46 0 72 0 hu I03 Moments (2) - SD[Sx(p)] Order Statistics (Liebicin) 0.62 0.90 1.32 0 43 0 64 0 93 (1) I 6 0.92 1 06 1.55 0.57 0 66 096 037 0.70 0.81 1.16 0 40 0 49 0 57 0.82 (3) Least Squares (1/a)SDc,[G (p)](Cram 6r-Rao Lower Bound) (4) 2 f m s u i % b % 1 gL 1 1 e D L ,,,- ;. itpe.% ewue. x-an er. w o w s.,c.: :.>

c

. ~ _ _.

.2 ,2 & AWf-- 58 pf - w 9 l9f,0 PRO!!Allli.lTY TilEORY AND STA flSTICAL DATA 537 obtamed from Table Al.l* (as shown in [Al-ll], these quantities are inde-pendent of the parameters y and a used in the calculationsl. Finally, the quanti-ties ofline (4) were calculated by Eq. Ai.77. Assume that Gy(p)is normally distributed. The approximate statement can then be made that the interval dy(p) SD[dx(p)] will contain the true unknown parameter Gx(p)in about 68% of the cases. This interval (referred to as the 68% confidence interval) is said to correspond to the 68% confidence Icrel. For ti.- interval dy(p), 2SD[dx(p)] the percentage rises to 95%, while for the interval Gy(p) 3SD[Gx(p)] it rises to over 99% (99.7%). As noted above. these per-centages should be viewed as only approximate: however, the approximation is satisfactory for reasonable sample sizes such as are used in the analysis of wind speed data. Estimation.\\1cthods and Reliability of Extreme Wind Speed Pndictions it is of interest to examine the efTect of the estimation methods upon the reli-ability of predictions of extreme wind speeds corresponding to mean recurrence intervals used in structural engineering calculations.t Consider, for example, the case n=25. The 68% confidence interval for the 100 year wind, xico= Gy(0.99), is Sx(0.99) SD[dx(p)]. If the most reliable method of estimation of Table A1.3--the order statistics method-is used, then the interval is dx(0.99) 0.90a ~ dx(0.99) 0.7s (Eq. A1.70). If, on the other hand, the least reliable method of Table A1.3-the method of moments-is used, then the estimated interval is dr(0.99) 0.885. Numerous analyses of wind records show that the ratios s/i are of the order of 0.07 to 0.15 [Al ll, Al-15]. Then the 68% confidence intervals ob-tained by the method of order statistics and by the method of moments are (using the ratio sli =0.12) dy(0.99)[1 +0.061] and dx(0.99)[1 + 0.077], respec-tively.The difference between the respective reliabilities of the estimates of the values of X corresponding to p = 0.99 (or, in virtue of Eq. Al.45, to a mean recur-rence interval N = 100 years) is seen to be quite small, that is, of the order of 2%. Results of similar calculations carried out for p=0.95, p=0.99, p =0.999: n = 25, n = 50: and s/X = 0.12, are shown in Table A1.4. The difference:. between the reliabilities of the various procedures can be verified to be negligible also l} ( for s/f =0.07 and s/f =0.15. It is seen from Table A1.4 thrt any of the methods listed will provide an acceptable estimate of the order of magnitude of the 68% confidence limits. The width of the 95% confidence limits is approximately twice the width of the 68% limits: for example, for = 20 and n = 25, the nondimensionalized 95% confidence limit estimated by the method of moments is 1 0.098. The dif-

• The standard deviation of S (p)in line (3)is an estimate based on a finite sample. In accordance t

with the convention adopted herem, the notation s rather than SD should therefore be used for the quannues ofline O). This was not done m Table Al.3 for the sak: of clarity. tOf two different possible esumators si and d of the same quanuty z. the estimator 3, is said to i be more reliable than s if(assummg the estimators to be unbiasedi SD(i )<SD(i )[Al 16]. i i ~ ~

~~~ v.U 6x% Confidence Intervals

• Based on Various Estimation hiethods and on the Cramer Rao Lower ihmnd, TABI.E AI.4.

n = 50 n=25 0.95 098 0.99 0999 095 093 0.99 0 999 R 20 50 100 1000 20 50 100 10t N) p ..=_ hiethod of bioments i 1 0.049 110.073 I 1 0.077 1 0.085 1 +0035 I10052 110055 110 tka hfethod of Order I 1 0.047 I 1 0.06! I10078 I10033 1 0 044 110056 Statistics (Lieblein) Least Squares hiethod I10066 I 1 0.072 110091 110047 110051 110065 Crambr-Rao lower liOD*3 I f D 055 110068 110068 110031 110036 l iO O30 110 n49 Bound "Nondimensionalized with respect to $s[I-1/S)]. \\ ( s Q(, i M ~ U N L i a 4 M gg .g G g a g

E ~ 2-h A/PPf-SB / -M 9 pg jh' REFERENCES 539 ferences between estimates based on various procedures are seen to remain acceptably small for the 95% confidence limits as well. It has previously been shown (Eq. A1.64) that if X is a random variable with a Type i distribution, it is possible to view the largest value in a sample of n values of X as an estimator of the value of X corresponding to a mean recurrence interval n. While this estimator has the obvious advantage of extreme simplicity, its reliability is relatively poor. This can be shown by the following example. If Eq. A1.76 is used to estimate the 95% confidence interval for the 37 year wind speed at Richmond, Virginia (ii=36.8 mph, d = 3.78 mph: see (Al 11]), the interval obtained is (50 5) mph. Using the largest value in a set of 37 values as an estimator of the 37 year wind, the estimated 95% confidence limit interval obtained is (50 12) mph (see Eq. A1.63a), that is, an interval more than twice as wide as the interval estimated by the method of moments. 'l x REFERENCES Al 1 H. Cramer. The Elements of Probabdity Theory. Wiley. New York.1955. Al.2 A. G. Mihram. Simulation. Academic. New York.1972. Al 3 R. von Mises. Probabdity. Statistics and Truth, Allen and Unwin. London, and Macmillan. New York.1957. A l-4 J. R. Bennmin and A. C. Cornell. Probability. Statistics and Decision for Cleil Engsneers. McGraw-Hill, New York,1970. Al 5 T. C. Fry, Probabdity and its Engineering Uses. Van Nostrand, Princeton,1965. Al 6 B. Epstem. " Elements of the Theory of Extreme Values," Technometrics,2.1 (Feb.1960L 27-41. A17 E. J. Gumbel, Startstics of Extremes. Columbia Univ. Press. New York,1958. Al 8 N. L. Johnson and S. Kotz. Contsnuous Unitariate Distrsbutions, Vol.1, Wiley, New York, 1970. Al 9 E. Simiu and B. R. Ellingwood. " Code Calibration of Extreme Wind Return Periods." Technical Note. J. Struct. Dir ASCE,103, No. ST3 (March 1977L 725-729. Al.10 J. J. Filliben,"The Probability Plot Correlation Coefficient Test For Normality." Techno-metrics,17.1 (Feb.1975L ti1-117. Al ll E. Simiu and J. J. Filliben. Statistical Analysis of Extreme Winds. Technical Note 868. National Bureau of Standards, Washington, D.C,1975. Al.12 E. Simiu and J. J. Filliben, " Probability Distributions of Extreme Wind Speeds." J. Struct. Dir, ASCE 102. No. ST9, Proc. Paper 12381 (Sept.197611861-1877. Al 13 J. M. Hammersley and D. C, Han, 'nb, Afonte Carlo Afethods. Methuen. London, and Wiley, New York,1965. Al 14 J. H. Mize and J. G. Cox, Essentials of Simulation. Prentice-Hall. Englewood Cliffs. NJ., 1968. Al 15 P. Duchene-Marullaz. " Etude des vitesses maximales annuelles du vent," Cahiers du Centre Scients)1gue et Technsque du Batiment, No. I31. Cahier i Ll8. Paris.1972. Al-16 J. Lieblein. A New Method of Analyzing Extreme Value Data. National Bureau of Standards Report No. 2190, Washington D.C.,1953. Al 17 J. Lieblein. Egicient Afethods of Extreme Value Afethodology, Report No. NBSIR 74-M)2. National Bureau of Standards, Washington, D.C 1974. Al 18 J. Tiago de Oliveira. " Statistics for Gumbel and Frechet Distributions" in Structural Safety and Reliabdity, A. Freudenthal (Ed.k Pergamon. Oxford and New York,1972, pp. 91 -105.

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.58pf-uo9 840 ELD 1ENTS OF PROBABILITY TilEORY AND APPLICATIONS hj f F. Downton. "Lmear Estimates of Parameters m the Extreme Value Distribution." Tech, A1 19 nomearses. 8.1 IFeb.1966L 3-17. A1 20 G.1. Schueller and 11. Panggabean. "Probabilistic determmation of design wmd velocity in Germany.' in Proc. /nst. Cir. Eng 61. Part 2 (1976L 673-683. A121 E. Sinaiu.J. Bi6try and J.J. Filliben. -Sampling Errors in the Estimation of Estreme Winds." J. Struct. Dir. ASCE.104 il978t 491-501. Al.22 1.1. Gringotten. " Envelopes for Ordered Observations A ied to Meteorological Ex. tremes.' J. Geophys. Res 68 (1976L 815 -826. Al.23 N. L. Johnson and S. KotL Distributwns an Statistscs: Contsnuous Multirarsate Distrsbu. tions. Wiley. New York.1972. A. W. Marshall and I. Olkin. "Domams of Attraction of Multivariate Extreme Value A1 24 Distributions." The Annals of Probability,11 (1983L 168-177. 1 6 4 4 g. 1 f - .y. 3 4 m$I }. ie .I ..h. fg 6 I l .--..-.-,..--__.._.-.--.-..-...-.4 I.

SHEET 7~/ d /9 JOB NO. NP-119 DATE 1/7/92 PROJECT CNS STATION BLACKOUT SUBJECT SITE-SPECIFIC WEATHER EVALUATION, g,# CLIENT NPPD ORIGINATOR E. HOLCOMBl/ REVIEWER l A eh__,#' APPROVED CALCULATION NO. NPP1-SBO-009 l l Cooper Nuclear Station Site-Specific Wind Speed Data i i ( I l l 9 e e-- ,r ,s-, -,,--.n.-_ .x ,m.,-,,w-- e, w-, n. -n.-

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s Table 3. 3-19. Monthly wind speed statistics, Cooper Nuclear Station, January-December 1975. Wind Speed (mph) 35 ft Wind Speed 318 ft Wind speed flourly llourly Month Mean Maximum Mean Maximum Jan. 7 18 15 37 Feb. 7 16 14 32 Mar. 3 29 16 45 Apr. 12 28 10 39 May 10 27 16 35 June 7 18 13 45 July 7 23 11 29 Aug. 9 23 15 34 Sep. 8 20 11 25 Oct. 9 27 13 32 Nov. 11 30 14 38 Dec. 9 27 13 36 Annual 8 30 14 45 4 4 4 <^ 4. 3-39 4 ___m_______m.__

N II' ~ N AL.CO Q N VIR O N M ONT A L. OClO NOOO ], 3~3 , ll, 2' Table 3.3-19 Monthly wind upeed statistics, Cooper Nuclear Station, January-December 1976. Wind Speca tmpn) 35 ft Wind Soeed 318 ft Wind Soeed Hourly Hourly Month Mean Maximum Mean Maximum __ January 11 28 15 34 February 11 29 16 45 March 11 35 15 44 April 11 37-16 44 May 9 20 13 34 June 10 34 14 42 July 8 19 11 26 August 8 22 12 28 September 8 20 11 28 October 8 23 10 28 November 10 28 12 36 December 10 28 13 35 Annual 10 p 37 11 SNY Yd O ~ h 3-32

l /W?Y{- 50/ cci N ALCO ENVIRONMENTAL BCIENCEB FN p V \\ Table 3.3-19. Monthly wind speed statistics, Cooper Nuclear Station, January - December 1977. Wind Speen tmph) 318 tt Wind Speed 35 ft Wind Soeed Hourly Hourly Month Mean Maximum Mean Maximum January 9.5 26.0 12.9 40.0 February 10.4 28.0 14.2 42.0 March 12.4 32.0 16.8 37.0 April 10.1 34.0 14.5 47.0 May 9.9 25.0 14.2 36.0 June 8.9 25.0 13.5 36.0 July 9.3 24.0 14.8 41.7 August 8.2 28.4 11.9 29.6 september 7.2 19.0 13.5 38.4 October 8.4 21.4 15.1 39.1 November 11.2 34.5 15.3 45.1 December 11.4 26.6 15.9 34.9 Annual 9.7 34.5 14.4 47.0 e e^ h 3-32

t Table-3-1. Summary of meteorological data measured at Coooer Nuclear Station, January - December 1978. l t 4 Meteorological Data I> j-35 ft Temperature 35 f t 318 ft N Mean Mean Abs. Abs. Mean Max. Mean Max. F 'Mean Min. Man. Min. Man. Direction Speed Speed Direction Speed Speed Precip. M Honth (F) (F) (F) (F) (F) Prevailing (sph) (eph) Prevailing (eph) (eph) Tot a l (in) 2 January 14.1 6.3 21.8 -9.8 44.4 HNW 4.2 27.5 NNW 13.3 37.9 0.08 m February 17.2 9.8 23.6 -15.8 42.5 Ntt 8.7 33.2 N 11.5 37.7 0.47 Z j March 35.2 26.5 43.5 -11.3 80.4 N 8.3 25.5 NNW 12.7 33.9 0.11 g 0 N April 53.0 45.1 60.9 30.5 78.8 ESE 11.6 25.8 5 16.6 36.1 3.04 Z s May 61.1 53.4 69.1 -38.1 86.1 E 8.6 31.2 SE 13.3 39.6 3.60 m ' June 73.1 63.7 82.4 50.4 98.5 SSE 9.9 40.1 S 13.6 52.4 2.86 2 g., I July 76.6 67.9 85.2 60.8 96.0 S 6.7 21.7 5 13.0 30.5 5.11 p i i August 75.4 65.4 86.1 51.8 95.4 SSE 6.6 21.2 SSC 11.1 36.6 1.12 01 0 September 71.0 61.3 81.4 47.1 95.4 S 6.4 19.9 S 12.3 28.4 6.44 m 2 l ' October 54.1 42.9 67.0 32.7 86.1 N,5 6.8 18.2 NNW,S 12.0 29.1 0.62 O M November 40.9 33.8 49.2 14.0 77.5 N,5 6.2 17.7 N,SSW 9.8 23.7 1.36 UI 4 4Qh December 27.2 20.2 35.3 4.9 47.8 mi 7.9 28.0 N 12.5 30.6 0.23 vN g l 's s s 9 3 l -V r Dj A. I t n, = t 8

Sufistiary of meteorological data measured at the Cooper Iloclear Station. January-December lable 3-1. 1979. ileteorological D3ta 35 ft 318 ft 35 f t tenverature Mean Mean ABS AUS Direction Itean Max Direction itean Itax Precip. Mean flin Max !!in Max Pre-Speed Speed Pre-Speed Speed Total. !!anth (F) (F) (F) (F) (F) vailing (mph) (mph) vailing (mph) (mph) (ic) 1> January 12.6 5.0 19.5 -8.9 40.0 riffW

6. 3 21.5 lits 13.1 35.0 0.70

-4 February 17.0

8. 0 25.0

-16.0 42.4 Il 6.0 25.0 titM 12.6 34.0 0.02 0 Z l

tiarch 38.8 31.0 47.5 16.2 73.3 IlrM

7. 4 29.1 tiffW 18.1 41.5 3.22 y

April 50.5 41.0 59.8 23.5 75.9 ESE 4.0 10.8 ESE,SE 14.3 32.3 1.61 5 0 flay 62.4 '52.0 72.6 38.5 85.9 5 3.8 15.4 S 15.6 43.2 1.40 Z E June 72.1 62.0 82.1 45.2 94.0 S 3.2 13.4 S 12.5 38.6 2.06 4 . July

74. 5 67.0 85.0 55.0

94. 4 SE 2.7 10.8 SSE 11.0 29.7 4.40 h

m Augus} ' 74.0 65.0 83.4 38.2

94. 1 E!!E 2.6 5.8 5

15.1 28.0 3.20 g 3'3 S 13.4 32.7 1.20 , September 67.8 56.0 79.9 40.1 88.3 a a m October 55.9 44.0 67.8 30.6 85.4 fifM

5. 9 27.0 ritM 14.1 32.8 3.99 m

flovember 39.0 31.0 47.9 20.2 68.8 S.WIM

8. 0 19.9 fM 13.5 26.9 1.55 f

g 4v 'I December 33.1 24.0 42.9 -0.1 61.6 it!M 8.1 24.7 firm 14.4 34.0 0.19 m i y Annual 50.0 40.7 59.6 -16.0 94.4 ti!M

5. 7 29.1 titM,5 14.0 43.2 23.54 5

4 i 2 u a = llo Data Available E <A w,1 r,. m u. h L 4 6pA. e. 4L +

t j i I it i Table 3-1. Susanary of meteorolo'llcal data' measured at the Cooper Fluclear Station Brownville. j i j. Hebraska, January-December 1980. i [ 31s~rruard 25:Triss ~ 75:tt Tieperiiwi j Meae Es ~Mean lien Meen Ilires Abs es Precipa tettee 7 5 peed Speed Prevailleg Speed Speed : Prevelling Mese Mes set e fees sete letet i g-Month ' leph) Imph) Strectlee leph) leph) Directlee (C) (C) (C) (C) (C) lie.] q .13.1 )$ 7.2 24 -3.8 1.4 -F.3 13.5 -23.8 0.66 Jameery 6 fehrwery 11.7 - 30 Imai-m* 6.5 25 susu-m* -4.5 -8.2 -9.8 18.6 -22.3 s.IF t 4 - March 15.2 ~ 31 0.2 28 2.8 6.7 -2.7 19.8 -19.9 1.92 i ) April 13.6 34 2.5 23 II.4 16.7 5.0 30.4 -1.6 1.80 l May-12.8 - 31 ESE-55E* 5.7 le less-s* *h II.2 22.8 II.4 38.4 3.3 1.49 7 i' July 12.7 36 4.6 17 26.9 33.0 28.2 39.8 14.7 9.82 Jeee 13.2 31 5.7 15 23.5 29.5 17.5 39.7 32.1 0.84 i 3 Aetest - 13.1 29 SE-5* 5.1 11 55E-5* 24.8 38.5 19.7 M.3 14.8 2.70 f 5eptember 13.2 41 5.5 26 39.6 25.8 13.2 34.1 5.7 e.44 L actater 11.3 M 6.2 26 9.2 15.5 3.4 24.4 -5.e 1.39 r moved er 12.7 . 30 seef-u* 8.5 24 ment-N.55E-5* 6.2 12.8 8.7 26.5 -10.3 8.29 I

• ~'

Deceder 12.4 30 9.8 25 -1.4 3.8 -5.6 16.5 -21.8 8.64 i 4 a s 1 Anemel 13.8 41 Imel-N.55E-5 6.7 28 asaf-u.55E-5 11.3 16.4 5.7 39.5 -23.8 14.22 1 3 i

• ere. lisse direcisen is 9:ees ter each geerter er the yeart Jeemery-nerch. Apell-June. July-September, actober-Secember.

[ i-pig 615 et the led eeta et 35-f t leeel was rece.ered eerse, this guerter. l 4-i i-t t i i, I i $q h i 5 1 Qesp

1. 4 g

\\ l e t A { t I n i j D1 ( i l-1 % 4 H i r t t a b i. I

.=. ,W 7'f-5 B '-cof 9 y$ >{lI/ >* 5 l l i TABLE 4-1 $UMMARY OF METEOROLOGICAL DATA MEASURED AT THE COOPER ttVCLEAR STATION. co0WfivitLE. flEBRASO. JANUARY 1981-DECEMBER 1981. JAN FE8 MR APR MY M 318.rt utna Mean Speed (aiph) 11 17 14 16 14 13 mante m 5 peed (mph) 28 39 33 38 31 37 Otrection of Mastmum Speed NNW N $5W $5W I $5V Date of Manimum 5 peed (8I 6 10 28.29 3 22 13 Prevailing Otrection IM.N 15E.15W 35.rt vind mean Speed (men) 8 11 9 11 9 9 Museum Speed tmph) 22 30 30 31 23 30 Otreetton of Mantmum Speed NIN N WSW $5W $$W.$$!.$ $5W Date of Mastmum 5 peed (a) 6 10 31 3 3.16.21 13 Prevailing Olrection NW.N 13[.15W 35.Ft Ambient Temperature Mean(C) 1.9 0.5 6.3 1%1 15.6 22.6 Departure fran (b) hermal ( C) 1.9 0.2 2.2

3. 0'

-2.0 0.0 Date of Hastmum 24 25 30. 26 ' 27.9 33.4 Mastmum ( C) 18.4 19.0 23.1 30.6 29 8 Minimum ( C) 15.8 25.9 7.5 1.6

• 1.4 11.9 Date of Minista 17 11 8

6 11 1 preetettatten total (in.) 0.22 0.00 0.94 1.68 2.37 1.75 Departure from, ) Horiaal(in.) 0.66 1.05 1.30 1.33 2.30 -4.31 Ratn Days 1 0 7 7 8 12 maatsuu in a 5tngle Day (in.) 0.22 0.63 0.46 0.93 0.59 Date 31 4 12 18 15 Maa tats: 18 a Single Mour (in.) 0.11 0.21 0.46. 0.17 0.23 Date 31 4 12 17.18 25 I I'I Prevailing direction is cerived frors the Quarterly joint frequency tables and is reported for the cuarterly period only. The cuarterly pertoos used aret Jan. Mar. Apr-Jun. Jul.5ep, and Oct.Dec. fb The climatological normals were derived fran NOAA climatological data for Auburn, Nebraska. C Rain cays are defined as a day in which 0.01 in. of rain or rain equivalent of froaen precipitation has fallen. l 1 ? ,5 4 4-2

f MP/- say< - w t

Y if f

I off TABLE 4 1 (CONT.) JULY AUG ttp OCT NOV Ott Annual 318-Ft Wind Mean Speed (mph) 10 9 13' 15 14 12 13 mantas Speed (spn) 28 24 28 37 36 37 39 Directles of Hastmum Speed 55E NW NW W' N W N Date of Maatmum Speed (a) 24 7 26 17 19 3 10 Feb Prevailing Otrettion 3$g,$$g $$g,$$y $3g.gsg 35.Ft Wind Mean Speed (mph) 7 6 7 9 9 8 9 Maxima speed (mph) 21 17 20 26 25 28 31 Directton of Mattmum 5 peed WSW NW,55W, W NW WNW WW,NW. .W_ 55W $W Dat.' of Mastmum Speed (,) 17 7.14 26 17 18,19 3 3 Apr Prevelling Direction 55E-55W $[.5 SE 5 M.Ft Assient Temeerature Mean ( C) 23.7 21.6 18.5 10.9

5. 9.

-3.4 10.6 Departure fromIbI -1.4 -2.7 -0.8 -2.9 0.6 2.3 1.0 Matteam ( C)) floreal ( C 35.2 31.7 31.9 25.3 18.4 13.7 35.2 Date of Maatsum 14 30 29 5 17 7 14 Jul Min 6mm ( C) 14.2 12.1 2.0 -4. 5 -7.5 -28.6 -28.6 Dats of Mtataum 28 11 18 23 21 19 19 Dec prectottation Total (in.) 4.77 4.87 3.15 1.84 1.58-0.43 23.60 Departure fraa(b) riornal(in.lg) 0.66 0.39 -0.92 -0.68 0.4 el -0.62 -11.70 Rain Dars 10 11 3 6 2 2 69 Maximum in a Single Day (in.) 2.88 1.67 1.17 1.13 1.44 0.27 1.67 Date 26 5 7 3 1 16 5 Au9 Maxima in a singleHour(in.)

t. 4 5 0.73 0.75 0.41 0.35 0.05 0.75 Date 23 5

.7 3 1 16.27 7 Sep ?

AWf 38 d-ce f 9 }W,lY .2-t0 TABLE 4-1 SUMitARY OF METEOROLOGICAL DATA 11EASURED AT THE COOPER llVCLEAR STATI0fle BROUllVILLE e flEBRASKA e JAflUARY 1982-DECEMBER 1982. iLA" U1 21 E El US Jje.ft vtad mean $ peed (eph) 14 11 13 Il 12 9 maaiman speco teen) 39 21 3g a6 35 31 Strectien of Pasimus Speed WW N SW W 1 1 Date of Masiews $pese 22 23 30 2 9 14 Prevalital Strection' w.m. 112.!!W 75.ft wine

• • en Speed (mehl 9

10 11 12 9 I maatam spees leek) 32 34 36 28 27 - Otrectten of.*asimus Speet WW 5W.*:nt.n.tnN W W 15W WW Cate of maatmum spece 22 12.23.24 30 2 9.10 14 Peeve 11 tnt Ottection'

• W.%N!

I I*IIW 35-F t Amn t eat f emoeret ure Peen (C) 9.8 1.7 3.3 1.9 17.5 19.8 Cesarture free nomal (C')b 6.0 3.0 0.s 2.2 0.1' 2.8 P.astnum (C) 5.9 21.2 10.7 26.8 ' 28.4 33.0 Date of massm a 27 !?

12. 30 2

4 29 Minim m (C) 23.1 .!!.0 14.5 4.9 6.4 8.1 Cate of minim a 10 6 6 6 7 1 Pm t et t et t on total (In.) 0.69 0.27 1.05 0.96 6.96 .2.41 Departure fras nomal (tn.)h* 0.19 0.78 .l.19' 2.C5 2.29 -43.61 Rata Days' ? 6 10 5 18 6 l nanta m in a l 5talle Day (In.) 0.41 0.11 0.25 OM 2.64 1.28 l l Date. 22 1; 19 28 to 8 nastown in a Single Mour (in.) 0.19 0.03 0.14 0.11-0.88 0.84 Date 22 17 19

5. 28 20 '

8

• Prevotitet etrection is eertved from the everterly ans annual jotat freewncy tables and ts reportes for the guarterly ans annual pertoes only. The svarterly pertoos uses are: Jan. Mar. Apr Jun..lui.5ep. and Ott*Dec.

" The citaatological normals vers serives from noAA citaateletical case for Auourn. neerasta. 8ata says are eef tnee as a day in unten 0.01 in. of rain er rain eovivalent of frozen p.ectattation mas f allen. I- ,RS 4-2

= _ _. _ - _ _ _ _.. l. /1./Y 5 S(lk - (( *l f'

Y N

/VS$ TABLE 4-1 (CONT.) au ss m s.1 s3. m = ~ ai 319.Ft viad maan Spees (apa) to 9 11 13 13 11 tt

• ssiem teees (mont 28 24 ft 33 a0 28 45 Direction of Pasimum Soeed th!

W$W f;g

• 1NW NW prd Date of passam loved to 4

28 19 I! 18 ! Apr

• cevailtat Direction'

$$t.1sv 15t.15V 11t.51W M.ft viad mean Spees (men) 7

• 5 6

6 7 7 8 wasimum sotto (men) 19 19 16 33 24 36 cirection of *asimum Speed 15W W5W 1 W

• NW SW NW case of mesimum noeed 1

4 28 11 Il 13 2 Apr Peevailing Direet ton' st.5 $$t*15d 15t*$3V 35.f1 AJnoient te.peratvet mean (C) 25.4 22.3 18.1 12.3 3.3 0.0 9.9 ceparture f ras hermal (C*)* 0.3 .t.0 1.2 .l.5 2.0 1.1 .l.7 massa m (C) 36.3 31.5 29.7 21.5 18.8 16.3 36.3 Cate et *ssiam 3 3 1 5 9 1 3 Jul minimum (C) 14.8 12.2 2.8 3.3 11.1 13.6 28.1 Cate of Pinimum 31 11 21 21 24 21 10 Jan Pref tfit $ tion total (in.) 1.71 7.47 0.93 0.88 0.79 3.32 27.44 Departure f ree b hermal (in.l 2.40 2.99 3.34 1.64 0.37. 2.27 7.86 tala cars' 8 15 4 6 4 8 17 masimum in a Statie car (in.) '.h J.G 0.50 0.38 0.47 1.31 f.64 Cate 6 12 6 28 11 27 to May masium in a $lptie nour (tn.) 0.45 1.19 0.2$ 0.16 0.13 0.51 1.19 Cate 6 12 6 8 11 1 12 Aet i l I l 4-3 l l

Table 3-1. Sunwary of *wteorological Data Neasureo at the Cooper Nuclear Station, January 1983 throu9 e December 1993 f Jan Feb Har Apr May Jee July Aug Sep Oct Nov Dee. Annual 318.Ft Wind Hean Speed (eph) 13.2 12.5 15.5 15.8 13.9 11.8 13.1 10.4 14.8 '3.0 14.1 14.9 13.6 Hasteue Speed (eph) 39.0 29.0 33.0 44.0 36.0 33.0 30.0 25.0 3C.0 3C.0 41.0 33.0 44.0 Ofrection of Haaloue Speed NNW NNW SSE,$E NNW SE 5 55W,5W 5 NNW M NNW NW NNW Date of Harlsue Speed 11 2 4 2 1 12 3 20 20 13 9 24 2Apr 35.Ft Wind Mean Speed (eph) 9.0 7.5 9.7 10.5 10.0 7.9 7.8 5.9 8.F 7.5 9.1 10.1 8.7 Naaloue Speed (sph) 30.0 24.0 24.0 33.0 28.0 27.0 22.0 16.0 23.0 21.0 27.0 29.0 33.3 Direction of Naslooe Speed NNW NhW MnW N SW 5 SW -e-nNW -e-N Date of Ha:Iscn Speed 2 27 2 6 12 3 15 20 27 9 24 2Apr 35.Ft Ambient Temperature .e- -e- -o-45.7 58.9 71.2 80.0 81.3 69.4 Sa.1 40.7 12.1 56.7 w i s N Hean (or) 78.5 86.0 89.5 100.5 104.0 94.5 88.0 15.0 35.0 104.0 Date of Haslaus N/A N/A N/A 26 27 30 22 17 9 2 2 4 IFAug -e- -e- -e-Manimo,(of) -e- -e- -e-28.5 39.0 44.5 62.0 61.5 33.5 34.5 13.5 -11.5 4 Date of Rinleue N/A N/A N/A 18 8 1 25 12 23 13 29 22 22Dec Hinimus-{ofi precipitation lolal (in.) 0.18 0.68 1.03 1. 04) 1.34 2.87 0.19 0.64 3.10 0.75 3.54 0.14 IS.52 h kain Days (a) 5 2 6 8 9 8 2 4 6 4 5 4 63 + ,Q g Single Day (In.) 0.07 0.67 0.40 0.31 0.29 0.76 0.17 0.48 1.82 0.57 1.20 0.10 1.82 Haslemurs in a 6 26 1 26 12 13.18 17 13 23 19 21 9 20 195ep Date Single llour (In.) 0.02 0.13 0.17 0.21 0.12 0.37 0.10 0.29 1.0a 0.13 0.55 0.05 1.tr) Manteue in a S w 29 1 15 12 10 17 13 23 19 21 3 20 195ep gN M Date L aRain days are defined as a day in which 0.01 in. of rain or rain equivalent of froaen precipitation has fallen. .e-Indicates missing data; N/A Indicales Not Available. Note: a ---a w

T Table 3-1. Summary of Meteorological Data Heasured at the Cooper Nuclear Station, January 1984 through December 1984 Jan Teb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual 100-e Wind 13.6 15.9 14.6 18.3 13.0 13.0 11.4 10.9 13.0 12.6 14.9 14.3 13.s MeanSpeed(eph) 51.0 46.0 38.0 40.0 37.0 43.0 25.7 26.0 30.4 32.0 34.5 30.4 51.0 Direction of Maslaue Speed juni. NW W W,NW W 5 i 5 5 W NW N tmW Ma Isue Speed (eph) Date of Manteue Speed 29 5 7 29 25 7 8 31 7 18 10 24 29Jan 60-m Wind (a) -e- .-e- -e- -e-11.5 12.3 9.9 9.0 11.3 10.6 13.1 12.8 II.3 Manteum Speed (eph) -e- -s- -m- -s-37.0 40.0 22.9 22.8 30.6 29.5 31.8 29.3 40.0 Mean Speed (eph) -e- -e- -e- -e-luni 5 5 550 5,55W W hNW N 5 Date of Maslaue Speed N/A N/A N/A N/A 25 7 9 31 7 IS 10 24 IJun Direction of Maalaus Speed 10-e Wind

1. 8.5 8.4 6.8 5.8 7.6 7.0 9.0 9.0 7.6
2. -e 748 e 7,f 27.0 27.5 16.0 16.8 23.7 23.5 25.7

'Z.4 21.5 Mean Speed (eph) 6.5 -e- -s-MaalausSpeed(eph) 18.0 -m-3L i m2 -e- -s- -e- -e-N 5 5 SSE 5 W Mnw 5 ' Date of Manteue Speed 3 N/A N/A. M/A 25 7 14 31 7 18 10 2'. 7Jun .. Direction of Manteve Speed w k 10-0 Ambient Temperature -4.6 2.0 0.2 9.1 15.9 23.2 24.8 25.1 18.1 12.7 6.0 -0.1 11.0 Manteue (oc) 10.3 18.3 12.2 26.4 28.5 31.6 36.7 38.2 35.6 26.4 20.8 20.7 3a.2 SaloC) Date of Manteus 29 22 25 . 26 18 26 8 28 . 6 3 14 28 20Aug

• A

-k' Mlaisum (oC) -23.3 -16.7 -11.4 0.8 3.5 10.0 16.7 12.8 -3.0 -r - -4.7 -15.9 -23.3 Date of Minleue 20 8 6 8 3 29 23 29 E 28 6 20Jan N V _N3 10-mDewPojat 'feeperaturelal ' N' -m- -e- -e- -e-8.1' 15.6 16.0 15.9 9.5 6.5 -2.1 -6.5 1./ 3 -e- -e- -e- -e-20.0 23.1 21.9 23.1 19.9 16.4 11.5 13.1 23.1 Mean(oc) l Date of Maxleue ' N/A N/A N/A N/A 24 14 10 6 23 27 9 20 14 J.m 6 .Manimus (DC) WJ -e- -e- -e- -e- -4.5 6.7 9.4 4.2 -5.8 -4.2 -14.1 -24.5 -24.5 1 l ~ Date of Minleue N/A N/A' N/A N/A 9 2 7 30 29 23 15 6 6tlec Minleum (oc) l A q%c v4 f y %* ff 0* fg gg ~

A............................................. s. ....__.m.... 1 Table 3-1. Summary of Meteorological Data,Hessured at the Cooper Nuclear Station for January 1,1985 ttroush December 31, 1945 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual 100-m Wind 15.1' 12.6 13.8 14.4 13.8 13.9 11.0 11.6 14.8 12.8 12.5 15.1 13.4 Hastmum Speed (mph) 36.0 27.5 36.0 40.5 36.5 37.1 25.1 34.7 29.5' 35.8 34.2 33.8 40.5 tfean Speed (mph) Direction of Maximum Speed NNW NNW W S WSW SSE WNW N NW S MNW NW S 25 23 4 19 11 23 11 5 23 7 6 '17 19Apr Date of Nazimus Speed 60-m Wind 13.8 11.3 13.0 13.6 11.9 12.0 9.1 9.6 12.7 10.9 11.4 13.5 11.9 Hasinum Speed (sph) 34.4 25.9 36.9 36.7 32.4 31.3 22.1 - 31.3 28.9 33.6 32.7 31.1 36.9 j Hean Speed (mph) Direction of Maximum Speed NNW NNW 5. S,SSE WSW SSE NNW(2) N S S NNW NW S 25 23 26 19 11 23 4 .5 19 7 6 17 26 Mar Date of Maximum speed 10-m Wind 10.4 8.0 19.1 '10.0 8.5 8.4 6.0 6.4 8.4 6.9 7.5 14 8.2 '28.2. 21.0 29.1 28.0 26.6 21.0 17.7 19.7 22.1 26.6 25.5 25.5 29.1 Mean Speed (mph) [ Direction of Maximum Speci HNW NNW S SSW(2) USW SE ItMW N NW NW(2) NW NW S - Hasimus Speed (mph). 25 16 26 18 11 23 4' 5 23 4 6 17 25 Mar Date of Maximus Speed 10-m Ambient Temperature Mean (Degree C)-. -6.5 -3.6 7.9 13.5 19.0 21.0 24.8 21.4 17.7 12.7 0.7 -6.0 10.3 9.9 13,4 22.7 30.0 30.2 35.2 35.5 34.1 ~ 33.1 24.5 20.4 7.5 35.5 6 28 26 38 25 8' 9 31 2 16 18 30 9 July -Haminum (Degree C) Date of Mazimus' -3.5 7.9 8.5 14.5 ..-11.4 2.8 .-0.5 -13.6 -21.6 -25.4 'llinimum (Degree C)" -25.6.-23.0 -3.1~ 1 18 13 3 26 30 1 30 18 19Jan g 19. 6. 4 Date of Minimum A h_ 10-m Dev reint Tesecrature ~-11.8 -9.1 -1.1 5.3 ,10.3 12.3

16.9 17.2 12.8 5.9

-4.1 =-10.6 3.8 -1.0-9.2-12.0' 16.6 --21.5 23.6 22.9 25.5 -23.7 17.3 15.3 1.7 25.5 \\ N-Mean (Degree C)-. 18 21.. '3 29 30 - 24 - 12 9 1 18 IS 30 9Aug l Hamtmum (Degree C) Minimum (Degree C) -29.8 -27.4 -11.9 -10.1 -2.0 -2.2 7.5 10.3 -2.0- -5.9 -16.7 -25.6 ~-23.8 Date of Manimum 31 1 4 8 2. 17 4 to a 30 28 -27 14 '313an m

Date of Minimum.

~ A i i o \\!.y N w a _msm_ .--di m.

s" Table 3-1. Summary of Heteorological Data Heesured at the Cooper Nuclear Station for January 1, 1986 through December 31, 1996 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual 100-m Wind Hesa speed (eph) 15.5 12.3 15.8 16.7 12.7 11.6 12.1 11.1 12.2 12.2 13.5 11.1 13.1 Haminum Speed (eph) 37.1 35.6 34.2 38.0 34.9 28.2 27.3 27.3 32.9 26.8 36.7 28.9 38.0 Isirection of Maximum Speed NW WNW NNW WNW S NNE S S,SSE SW N N WNW WNW Date of tenimus Speed 4 25 5 14 4 29 5 12 28 31 25 29 Ir. Apr 60-m Wind Hean Speed (mph) 13.6-11.0 14.4 14.8 10.9 10.0 10.4 9.1 10.5 10.3 11.9 9.5 11.4 Haminum 3 peed (sph) 35.1 31.8 35.8 34.9 34.2 23.0 26.8 24.6 27.7 24.4 33.1 22.8 35.8 Direction of Hazimus Speed NW NNW S W 8 PHE S SSE WSW N N NNW S Date of'Hautaus Speed 4 20 24 14 4 29 5 12 28 31 25 29 24Har Y 10-m Wind Non Speed (mph) 9.5 7.7 10.1 10.5 7.6 6.9 7.0 5.9 7.1 6.8 8.1 6.6 7.8 y Hastmum Speed (sph) 29,1 25.3 26.8 29.1 27.3 17.2 20.1 17.9 19.7 18.3 25.1 18.3 29.1 Direction of Hazimus Speed NW NtfW NNW,S W S SSW S $$E S NNW W NNW W 4 20 5,24 14 4 26 5 18 28 11 7 29 14Apr .( ., Date of Hazimus Speed' 10-m Ambient Temperature 25.8 21.1 20.1 12.8 2.3 -0.2 12.0 man (Degree C) 0.5 -2'.7 8.3 '13.0 18.1 24.5 Haminum (Degree C) 16.3 16.2 31.4 29.5 29,1 34.6 35.0 31.4 30.8 25.1 17.7 9.3 35.0 Date of Heminum 20 26 29 24 31 28 24 25 26 7 21 14 24 July Minimus (Degree C) --18.9 -22.0 -10.8 -2.0 7.2 13.4 14.8 7.9-7.1 1.0 -16.4 -12.1 -22.0 Date of Minimum 27 12 7 14 19 12 21 28 8 14 11 10 2Feb h y ( i 10-m Dew Point Temperature 'J \\ b-Haan (Degree C) -7.3 -6.2 - 0. 4 4.8 . 8.9 15.2 18.5 14.8 14.0 7.3 - -3.7 -3.9 5.2 It.eximus (Degree C) 2.7 8.5 13.1 17.2 17.5 23.1 23.4 21.6. 21.0 18.0 11.1 4.1 23.4 Q \\ ( Date of Hamimue 31 2 31 29 9 29 30 17 24 2 7 7 30 July t i Hinimum (De8ree C) -26.2 -23.8 -17.8 -?.0 -2.9 7.5 11.7 2.0 2.4 -5.3 -20.0 -17.8 -26.2 Date of Minimum 26 12 7 14 1 2 20 28 7 13 13 10 26Jan' )R i k % i l 4 t. \\' 6 Ne I l

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06/21/91 14:03 r+PD ru m - w-o. -e- * ,Vrf{ ~.'>& ~ C C Y fy/lU MI Tehle 31. Sonnery of Meteorelestcel Date Measured et the Cooper Meelear 8tetton for Jenuery 1. 1989 throu6h December 31, 1998 Jan Teb Her Apr Hey Jun Jul Aug Sep Oct Pav Dee Annual 100*e Wihtf Wen Speed (mph) 13.7 14.4 16.4 14.6 13.8 13.3 10.8 12.6 13.0 12.7 15.4 13.6 13.9 1 Hestense Speed (mph) 36.2 40.3 35.6 34.7 34.1 24.'A 28.3 29.7 34.9 31.6 32.6 30.9 40.3 Direetion et Natena Speed WW NW NW MW 8 8 3 N W NW W MW FW Debe ej Mittssse 8p9ed il 14 12 3 7 21 1$ 22 19 22 3 14 14Feb .F 't 80-m Wind Heen speed (eph) 12.3 12.6 14.4 12.8 14.0 11.4 9.3 10.4 11.9 11.0 14.1 12.0 12.2 Hautaus Speed (mph) 43.6 ' 37.2 36.0 32.3 32.5 24.2 26.4 24.2 12.2 31.1 33.2 32.1 %1.2 6 reekten et M4ximum speed NW ltpW 88W MNW 3 3 w N W NW W NW NW Debe et Hemlansa Speed ,.,.12 14 27 3 7 7 4 22 le 22 14 14Feb p. 3 10 m Hind Weh $#ed (til 9.0 ' 8.9 10.3 8.8 9.7 7.6 6.2 7.0 7.8 7.3 10.1 8.4 8.4 Mae&'aue 8 peed (e @ ) '24.8 28.3 30.3 27.3 24.2 18.6 19.4 19.4 24.7 23.6 24.0 26.6 30.3 Direenten of Hentaswan Speed le6t NW E8W M WMW 8 Mf RNW W NW W ltNW SSW ~ Deke et Havianas treed 12 14 27 26 9 13 8 22 19 21 14 27Har .y 10.m Ambient teamerat.ure Heen (Dettee C) -4.7 -3.3 4.6 11.2 20.4 23.1 24.8 23.3 10.2 10.7 6.0 0.4 11.4 m aimum (Do6 tee c). 13.2 21.1 27.6 $8.0 31.3 al.4 36.6 37.9 34.0 26.0 21.2 18.3 38.4 Date of Mattmas tl 26 22 8 18 23 31 13 17 14 15 2 ISJun Hintown (Dettee b) ,*14.3 -23.1 10.6 -0.3 9.8 8.1 13.4 6.6 $.S 3.3 -8.2 -1J.9 23.1 Date of Hinimum - 26 11 14 16 17 10 1 24 24 28 16 13 11Feb .p ..t 16-= be* 9eint tammazature Heen (Dettee C) -9.3 -9.6 -4.3 -0.2 8.2 21.6 14.4 15.$ 10.8 0.7 -1.6 7.2 2.5 Massanas (De6te6 C) 7.4 1.8 13.1 11,$ 17.2 21.2 23.9 24.0 19.$ 12.1 ' 15.3 5.7 24.0 Date of Hastense 29 71 24 2 31 22 14 17 18 22 15 26 17Au6 Hinimum (De6:44 C) -24.9 -26.0 -19.2 -14.2 4.3 -1.9 9.8 0.9 -0.1 10.0 -12.6 -21.6 26.0 i Debe et Hintaum 4 11 14 18 9 9 21 28 6 30 28 14 11Feb n. I 1 + 3-2

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== TeMe 3-1. senery of meteorotesket sete measured me the omsoor inseteer sameten for Jersmery 1,1990 IBresch -31,1998 Jet Fett stor Apr mey Jean Jid Ass Sugp Oct mer Dec Asemat b 4 n e-1M-e 1 find y mean speed (eph) 14.3 13A 14.8 14.8 13.T M.1 12.6 11.8 11.3 15.9 T.2 13.T 12.9 marismus speed (mph) 34.4 33.2 35.T 38A 31.3 3RJ 26.T 26.2 26.F 39.9 21.9 31A 3F.4 l Dete/skaar of Mustinus speed 2T/8 13/ 9 13/21 M/2 18/1T 19/5 19/2p 3/1 27/5 1T/19 30/M 3/7 cn1T I 40-e vind -{ nous speed (agdt) 12.6 11.9 12.5 13.9 12.3 12.f" 11.3 9.6 9.6 11.T 12.6 11 4 11.9 b Munisass Speed (medt! 33.6 32.1 33.1 31.3 31.2 41.4 25.6 21.5 25.1 34.9 33.5 29.6 41A Dete/mour of sentisaan Speed 27/8 13/10 13/21 23/1T 18/1T 19/4 T/1T 1r/M 22/t3 17/19 20/17 3/T Josir 10-e viruf Maart Speed (asAs) 8.5 8.9 5.4 9.8 8.6 5.9 T.4 6.1 6.2 9.0 a.5 a.3 7.9 metime Speed (up) ZT.2 Zsa 21.6 24.7 22.8 27.9 1s.1 M.2 19.5 so.e 26.6 23.1 30.e -j Dete/dans of Maaim e Speed 2T/s 24/3 11/M 1/12 T/16 19/4 T/14 1T/M 22/13 17/20 21/11 3/7 oct1T t m Ambient Tumerates e h namn (Segree c) 1.9 0.8 6.0 11.3 15.5 26.4 25.2 26.6 21.3 Et s.4 -3.s 12.6 naala m (segree C2 18.1 22.2 24A 29.1 29.3 34.8 Jf.t 35.6 36.2 32.0 26.T 16.6 39.5 bete of Reafsmus M 12 12 25 T 28 4 31 6 5 1 11 Jut 4 Minimman (Oegree C) -9A -17.2 -6.9 -3.9 4.9 8.0 11.7 12.5 3.9 -2.8 -6.0 -22A -22A Dete of minim m 1-1T r.i 12 1 4 M T 23 28 2s 22 SEC22 10-e Dee Poirve Teumerature steen (Degree C) -5.2 - T.5 -9A 1.2 7.8 15.5 M.S E5 10A 4.4 0.9 -9.3 4.9 Realsam (segree c) 11.3 2.5 MA 15.0 E3 25.6 21.2 23.4 21.9 15.5 13.9 5.1 23A S-k Date oflamalems 16 7 10 22 15 13 1 28 1 2 29 M AIEBB minissa (Segree C) -15.9 -20.6 -17.5 - 15.5 -T.1 1.9 6.3 74 -2.9 -8.8 -10 4 -38.0 -35.0 Q

1. ate of minimum 12 17 3

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SHEET Y~/ d JOB NO. NP-119 DATE 1/7/92' PROJECT CNS STATION BLACKOUT HOLCOMB/ /g'/,., SUBJECT SITE-SPECIFIC WEATHER EVALUATION ff, CLIENT NPPD ORIGINATOR E. REVIEWER /M~ /*L J APPROVED CALCULATTON NO. NPP1-SBO-009 4 Variation of Wind Speed with Elevation Now: Tne informadon se. Attachment 4 has been extracted from the following document: Simiu, Emil, Changery, Michael J. ami James J. Filliben. " Extreme Wind Speeds at 129 Stations in the Contiguous U.S.". NBS Building Science Series #11B, March 1979. J

y~2 p?/'/- 5 8/ - e c Y (3 T ;[ r'(/ Nashville, TN (1963 & 1972) Abilene, TX (1971) Amarillo, TX (1972) Brownsville, TX (1963) Corpus Christi, TX (1955, 1961 e 1970) Port Arthur, TX (1972) Salt Lake City, UT (1968) Burlington, VT (1968) Lynchburg, VA (1962 & 1967) 2.3 ROUGRNESS CONDITIONS AT AIRPORT STATIONS In an attempt to ensure that the terrain roughness conditions are stations unif orm among all the sets of data being analyzed, only airport have been considered herein. In principle, it may be assumed that at such stations open exposure conditions prevail. Nevertheless the mere-fact that wind speed measurements are taken at an airport station does not necessarily ensure that the wind climatological conditions reflected by these measurements are identical, from the standpoint of the terrain exposure, to those prevailing at a different airport. For example, it is noted in Reference 2 that the estimated 50-year wind at Chicago Kidway The-Airport is about 15 mph less than at the Chicago O'llare airport. probable reason for this difference is that the terrain around the Chicago tddway Airport is relatively heavily built-up. Similar consid-erations might explain to some extent the difference between the esti-mated 50 year winds at the Washington National Airport and the Baltimore-Washington International Airport, which are estimated in this report to be 66 mph and 75 mph respectively. Thus, in interpreting airport data for the purpose of developing wind maps, it is appropriate to take into account the possibility that, at the airport of concern, the terrain exposure conditions might differ somewhat from those defined as "open' (e.g., in Ref erence 3). 2.4 VARIATION OF WIND SPEED WITH HEIGHT ABOVE GROUND To ensure the micrometeorological homogeneity of the data at any given station it is necessary to reduce all the wind speeds recorded at that station to a common elevation. The elevation chosen for this pur-pose is 10m above ground. The mean vind profile near the ground in homogeneous terrain is given by the well-known logarithmic law, which may be written in the form: In 'L-z U(z) = U(10) (2.4.1) in IS. o 7

Y-3 sypf-seg n 1 5W j?V where z = height abovc ground and z = roughness length, both expressed o in meters. In open terrain, z may vary from, say, 0.03m to 0.10m. In o this report the reduction of the data to an elevation of 10m is based on the assumption z = 0.05m. It can be verified that the errors inherent o in the assumption z = 0.05m - when in f act the values z = 0.03m or o o = 0.10m were correct -- are small (of the order o.1% or 2%). z o An approximation to Eq. 2.4.1 is given by the power law ( *'* ) U(z) = ( ) U(10) where, for open terrain conditions, it is generally assumed a = 1/7 (3). It is noted that Eq. 2.4.1, and therefore its approximate equivalent given by Eq. 2.4.2, is valid f or mean wind speeds averaged over a rela-tively long time interval, e.g., one hour. The question thus arises of expressing the variation with height of the f ast.r:st-mile wind speed, which is averaged over a relatively short time (30 to 90s or so). To obtain an approximate expression for the fastest-mile wind pro-file, note that it may be assumed, approximately, Upk ~ Ufm 1 (2.4.3) Upk - U ,T pk = peak vind speed, U, = f astest-mile speed, and U = hourly where U f mean speed (see, e.g., Reference 4, p. 62). The expression for Upk can, in open terrain, be written as ___ 1/2 Upk (z) = U(z) + 3 u'2 (2.4.4) 1/2 '2 where u = r.m.s of longitudinal. velocity fluctuations, and 1/2 u'2 U (10) (2.4.5) = in.10,

• o where z is expressed in meters (see Reference 4, pp. 45 and 54).

o It can be verified by us..ng Equations 2.4.1, 2.4.3, 2.4.4 and 2.4.5 that, within the anenometer elevation range of interest in this report, it is possible to write approximately 8

WM 9p737.- gsc/n oV .M j[9 fa(10). U(10) (1,z-10 0.02) (2.4.6) U,(z) U(z) 10 f where z is expressed in meters. The errors inherent in Equation 2.4.6 are of the order of -1 to 3Z,the higher errors being on the conservative side (i.e., yielding slightly higher f astest-atle values at 10m above ground than would be obtained by a more " exact" expression). Eq. 2.4.6 has been employed to obtain the corrected speeds at 10m above ground in this report. 9

l l SHEET I' l' cr/ I JOB NO. NP-119 DATE 1/7/91 / PROJECT CNS STATION-BLACKOUT gt/ SUBJECT SITE-SPECIFIC WEATHER EVALUATION rk' g CLIENT NPPD ORIGINATOR E. HOLCOM'81' REVIEWER /44_/ N / m APPROVED CALCULATION NO. NPP1-SBO-009 Summary of Probability Plot Correlation Coefficient (PPCC) Method

\\ t k [~' d AT/'t 280 -c ei ~ f _i _i--. _i

==-=... 43 3300564emaa. a. I I I I I I 19 7604437 8 a a 1 3 1 8 1 $7 471F04F = t I B I I I S4.894S322 a I a 1 8 a 8 8 a a f 94.4433878 I i 1 aa 8 8 I et.4 34 8 886ea l>. 8 aa i I aa t i 1 e6.etS0076 aa 8 I I I I I es.3FS&338 a as a 4 na i I I I as an I 48 4944545 = an i t i I a 1 39 4874434 A a I a a l 8 8 8 4 3s.s3s3e es..s,.-, i i. _i_--..i------..i -i.3 ivsei .o. ioon i.e.esset a.sv, sis. 3...rer3s EnfRfut VALWE TYPt i E EaPONE$eTldA TYPf4 3. STATISTICAL ANALYSIS 1 3.1 OBJECTIVE OF STATISTICAL PROCEDURE Probabilistic considerations, as well as available empirical evidence suggest that the asymptotic probability distributions of the largest values with unlimited upper tail are an appropriate model for the behavior of the largest yearly wind speed. There are two such distributions, known as the Type I and Type 11 distributions of the largest values, whose cumulative distributions functions, F (v) and y Fry (v),-respectively, are of the form Fy(v) = exp [-exp ( V ")]; - < v < *; a (3', 4 /1 -<p<; o<a<- 11 1 4 1 l

f-3 A77'f-iBsl~ccT $9/ lit and Fyr (v) exp [-(v - u) ~ T); y <y --<u<=;O<a<=; y>0 (3.1.2) in which u, o, and y are location, scale, and tail length parameters, respectively. Actually, the Type I distribution may be shown to be a Type II distribution with y = = (see Reference 4, p. 422); however, it is convenient to refer-to it separately. The data were analyzed using -- with minor modifications -- a com-puter program listed in Reference 5. For convenience, the main fea-tures of the procedure used in the analysis of the data are summarized in this section. The procedure consiuts of three distinct stages. In the first stage the value of y (Eq. 3.1.2) is determined which yields the closest fit to the observed data set (recall that y = = corresponds to an extreme value type I distribution). The " closest fit" criterion used in this stage is the so-called maximum probability plot correlation coafficient criterion. The probability plot correlation coefficient is defined as I(X1 - X) [M (D) - M(D)] (3,1,3) 1 rD = Corr (X,M) = {I(X1 - Y)2 ggg (g) _ 3(p)j2) 1/2 i in which X '= TJC /n; M(D) = IM (D)/n; n= sample size; and D = probability 1 g distribution tested. The quantities Xg are obtained by a rearrangement of the data set: X is the smallest; X the second smallest; and X i 2 i the ith smallest of the observations in the set. The quantities M (D) 1 are obtained as follows. Given a random variable X with probability distribution D and given an integer sample size n, it is possible from probabilistic considerations to derive mathematically the distributions of the smallest, second smallest, and generally the ith smallest values of X in a sample of size n. There are various quantities that can be utilized to measure the location of the distribution of the ith smallest value Xg (e.g., the mean, the median, or the mode). It is convenient to use the median as a measure of location in Eq. 3.1.3 - these medians of the distribution of the ith smallest value being denoted by M (D). g If the data set _was generated by the. distribution D, then aside from a location and scale factor, Xt will be approximately equal to M (D) for g all 1, and so the plot of Xi versus M (D) [ referred to as probability i plot] will be approximately linear. This linearity will, in turn, result in a near unity value ir vD. Thus, the better the fit of the distribu-tion, D, to the data, the closer r D will be to unity, k l 12 } l l

E \\ r- + AuY/-SBg/ cai j' ?^$/(hV { The procedure just described makes use of 46 extreme value Type 11 distributions defined by various values of Y from 1-25 in steps of 1, from 25-50 in steps of 5, from 50-100 in steps of 10, from 100-500 in steps of 50, from 500-1,000 in steps of 250, and Y = a. For any given data set, 46 probability plot correlation coef ficients are computed corresponding to these distributions, and the distribution with the maximum probability plot correlation coefficient is chosen as the one which best fits the data (see, for example, computer output for Dallas, Texas, Section 4). The final result from this first sta8e is a value, T of y correspanding to the estimated best fitting distribution.

opt, The second stage in the procedure consists of estimating the loca-tion and scale parameters, y and o, respectively, in Eqs. 3.1.1 and 3.1.2 for the observed data set and for the determined optimal value, as determined in stage 1.

Estimates of the location and scale y$$o,wdirectlyfromthebasicprobabilityplotapproach. fo If a least-squares line is fit to the probability plot corresponding to yopts then the computed intercept and slope of the fitted line serve as esti-mates for the unknown location and scale parameters, p and o. In serms are as follows:t(D), these estimated location and scale values, 3 and 6, of the Xi and M i - 3)(M (D) - M )) (3.1.4) g, I(X i I(M (D) - M(D) i D = X - 6 M(D) (3.1.5) The third and final stage in the procedure determines the predicted wind speed v, for various intervals N of interest. The estimate for N i vN s = G + 6Gq (1 - ) (3.1.6) y opt in which yo e = the optimal value of y (as determined in stage 1); G and6aretSeestimatesofthelocationandscaleparameters,pando in Eqs. 3.1.1 and 3.1.2 (as determined in stage 2); and G y,E[stribution. (p) = the g percentage point function of the best fitting extreme value If y 4 = (i.e., if a member of the extreme value type 11 family pro-videsNhebest fit), then (p) = (-In p)-I/Y (3.1.7) G I i 13 i, h 1

p g-AW128/~MY 7% y 'l If y = = (i.e., if the extreme value type I distribution provides the t ofit), then best G (P) = -In(-In p) (3.1.8) Xg In effect, the procedure described in this section is an automated e.quivalent of probability paper plotting in which 46 types of probability paper, corresponding to 46 extreme value distributions, would be used and in which fitting would be carried out on the basis of the least-squares method, rather than by eye. 3.2 J'ROBABILITY PLOT 3 A majority of the Type I probability plots generated by the computer from the data taken at the 129 stations fit a straight line reasonably well (see, e.g., plot included in computer output for Ely, Nevada, Section 4). However, in a number of case.5 the fit was relatively poor. A discussion of various rer. sons leading tu a poor fit is presented in Section 3.5. To provide an idea of various types of deviations from a Type I distribution, probability plots were included in Section 4 for the following stations: Indianapolis, Indiana; Des Moines, Iowa; Topeka, Kansas; Wichita, Kansas; Boston, Massachusetts; Nantucket, Massachusetts; Detroit, Michigan; Grand Rapids, Michigan; Minneapolis, Minnesota; Missoula, Montana; Omaha, Nebraska; Valentine, Nebraska; Ely, Nevada; Albuquerque, New Mexico; Albany, New York; Abilene, Texas; and North Head, Washington. 3.3 ESTIMATION OF SAMPLING ERRORS As indicated in Section 1, the computer output of Section 7 includes estimates of the standard deviation of the sampling errors, i.e., errors that are a consequence of the limited size of the data sample from which the Type I distribution parameters are estimated. Two such esti-mates were used. One estimate is based on the method of moments and has the following expression given by Gumbel in Reference 6 (pp.10,174 and 228): g' SD(v ) * + 1.1396(y-0.5772) + 1.1(y-0.5772) ]1/2 6 (3.3.1) N 6 W g t i in which SD(v ) = the (estimated) standard deviation the sampling error N in the ertimation of the N year wind l y = -In ( -In (1 - 1)] (3.3.2) N 8 = the estimated value of the scale parameter; and n = the sample size. 14

i-i - s:(, l . isyff-SBc/- C0 '/ iSV W l A lower bound for the estimated sampling error is giv/.(n by the - following expression: 4 l_- SD v ) = (0.60793y2 + 0.514y+ 1.10866)l/2 where the notations are the Same as in Equation 3.3.1. Equation ~3.3.3-is commonly referred to as the Cramer-Rao lower bound (7). i ~f. l 3.4

SUMMARY

OF RESULTS The results-of the analysis are summarized in Tab'le 3.4.1, in which the following notations are used: i n = sample size X = sample mean-s = sample standard deviation i v,,, = sample maximum yopt = value of optimal tail length parameter (see section 3.1) { v = estimated extreme wind corresponding to a n year return n period, based on Type I distribution i; ppec = probability plot correlation coefficient.(see Section 3.1) for Type I distribution d l v = estimated 50 year wind speed i 50 j SD(v "E '## 'I'* 50 wind speed. I i f ~i ( .15 I i I 1

3* f AW/- S'Br/ - M f 3.5 TYPE I VERSUS TYPE II DISTRIBUTION Of the 129 stations listed in Table 3.4.1, 15 stations (marked with the superscript (c) in Table 3.4.1 and listed in Appendix 1] have been

M j noted to have largest yearly speed records that may not provide a reli-P

+ able basis for predicting extreme winds. The remaining 114 stations may [ I be divided into three categories characterized by the value of the optimal tail length parameter Yope, as shown in Table 3.5.1. p p j i Table 3.5.1 Classification of Stations According to Value of Y opt Category Range of Y Number of Stations Percentage opt 1 I 131 Y 89 78% II 71Yopt < 13 11 10% p < 7 14 12% III 21Y op The sample size for the stations of Table 3.5.1 varies between n=10 and n=45. It is noted that the percentages of Table 3.5.1 are in qualitative agreement with those found from the analysis reported in Reference 8, in which all sample sizes were n = 37 This tends to confirm the hypo-thesis advanced in Reference 8 to the effect that, for stations in well-behaved wind climates, the best fit of a Type II (rather than Type I) distribution to a set of extreme wind data might be attributed to a sampling error in the estimation of the tail length parameter. This hypothesis does not exclude the possibility that stations exist for which a Type II distribution might provide an appropriate'dascription of the wind climate; however, according to the results of both Reference 8 and Tahle 3.5.1, the number of such stations, if they exist,-is very likely to be small. Thus, it appears justified to ascume, as in Refer-ence 8, that the Type I distribution of the largest values provides, in generaYa better 'descriptio~n~iff tE widCclimate than Tpp'e II d'istri- ~ butions with small values of the tail length _ parameter (say, 21 Y I 12). 3.6 LARGEST WIND SPEED IN A SAMPLE OF SIZE N AND THE N-YEAR WIND it is shown in Reference 9 (see also Reference 4, p. 423) that, if a variate X has a Type I distribution, the mode of the largest value in a a sample of n values of X is very nearly equal to the value of the variate corresponding to the mean return period n (recall that the mode of a variate X is the value of that variable most likely to occur in any given trial). It can be seen from Table .5.1 that, for most set: for whichY 19 large, the ratio v,,xN is indeed close to unity. opg n I 4 9 y

SHEET 5 l v i 5b JOB NO. NP-119 DATE 1/7/92 SUBJECTSITE-SPECIFICWEATHEREVALUATIO PROJECT CNS STATION BLACKOUT CLIENT NPPD ORIGINATOR E. HOLCOMB1 REVIEWERt/Z~ '# m M APPROVED CALCULATION NO. NPP1-SBO-009 NSSFC Program 'TORPLOT' Output for CNS

C-A NPPf - SB pf - c o ? $W ;l9 NATIONAL SEVERE STORMS FORECAST CENTER TORNADO DATA The enclosed tornado listing provides information on all reported tornadoes in the area indicated since 1950. The various entries, and tables are explained below. If you have additional questions, please writo or call the National Severe Storms Forecast Center, Room 1728, 601 E. 12th St., Kansas City, MO. 64106, phone (816) 426-3367. The item-by-item listing shows the year, month, date and time of occurrence of each tornado in Central Standard Time. The columns labeled SEQ and SEG indicate the sequence number and segment number of each tornado. Sequence numbers are assigned chronologically within each state. The first tornado in 1973 in Ohio is given sequence number 1 for the state of Ohio that year. Many tornadoes have lengthy paths that cross county or state lines. Some change direction quickly. In such cases the tracks are broken into segments that arc denoted by segment numbers. A tornado with 3 segments has the same sequence number, but a different segment number, for each separate segment. The statistics in the tables are based only on the initial touchdown points. The Latitude and Longitude of the beginning and ending points of each tornado are shown followed by the overall length and width. Deaths and injuries for each segment are listed, followed by Damage Class. Damage Class numbers range from 1 to 9 and provide an estimate of the damage according to the table (#1) below. The columns labeled FPP provide the Fujita-Pearson scale estimates of Force, Path Length and Path Width. All three scales are logarithmic with values ranging from " " for the smallest category to +5 for the largest. The following table (#2) shows the range in each scale. The Path Length and the Path Width values represent estimates as to the actual amount of ground contact for each tornado. For instance, if a tornado had an overall length of 45 miles but made actual ground contact only 60 percent of the time the Path Length scale value would be a 3. The AZRAN column indicates the azimuth and range from the center point. 129/83 indicates the tornado touchdown was 129 degrees (southesst) at 83 nautical miles from the center point. A circular plot of tornado touchdown points is enclosed. The city of interest is at the center of the plot, north is at the top, east at the right, etc. Each digit represents the number of touchdowns in a small square area, about 2 miles on a side. Thus, what might be plotted as 21 actually represents 2 touchdowns in one square and 1 touchdown in the adjacent square.

4-3 Nfff-SB$ - oo '1 lW ph The four frequency tables provide detailed information about the I time of day, time of year and length and width characteristics of tornadoes in the area of interest. The Path Width vs Path Length table is computed from the F1 and Pw data. Also, the mean path length and mean-path' areas are computed from the P1 and Pw data. When the length and width scale values are converted back to length and width figures the minimum values in sach range are used. For example, a-P1 value of 3 is converted to a length of 10 miles in the calculation. The monthly and hourly distribution tables indicate the favored times of day and year for tornadoes in each area.- Monthly and hourly percentages are shown on the hourly distribution *,able, t l Mean times are shown for each month and for the entire year. These times should be interpreted and used in conjunction with the hourly percentages in examining the diurnal trend of tornadoes. All times in these tables are Central Standard Time. The latitude and longitude of the center point used by the search' program is listed at the upper right of the Hourly Distribution Table. These figures are in degrees and hundredths. The map scale used in the circular plot is compatible with the WSR 57 l radar map, 125 nautical mile range. Table #1 (Damage Class) 1 Less than S'50 2 $50 to $500 3 $500 to $5,000 4 $5,000 to $50,000 5 S50,000 to--S500,000 6-S500,000 to S5 million 7_ $5 Million.to S50 Million-8 $50 Million-to-S500 Million Table #2 (FPP Scale) l Scale F (mph) -Damage P1_(miles) Pw (width). Less than.40= (little or .Less than .3-Less than:6 l l no damage) l 0 40-72 Light 0.3-10 6-17' yds 1 73-112 Moderate 1.0-3.1-18-55 yds. 2 113-157 Considerable 3'.2-9.9 56-175 yds _- 3 158-206 Severe 10-31 176-556 yds-4 207-260 l Devastating 32-99 'O.3-0.9 mi-5 -261-318 Incredible 100-315 1.0-3.1 mi E x-w-v -y ,---.---..v.,. p.,m--- c-,.._......w-eswe, +- .iese,....--. ww.i--v-.y n,-- -,--,-~,.,, w,,,,. 4-v-a.-,+, ...--,,me--

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