Modeling Hydrologic Change: Statistical Methods - Chapter 5 doc

Specifically, separate frequency analyses can be performed at a large number of sites within a region and the value of the randomvariableX for a selected exceedance probability determine

Statistical Frequency Analysis

5.1 INTRODUCTION

Univariate frequency analysis is widely used for analyzing hydrologic data, includingrainfall characteristics, peak discharge series, and low flow records It is primarilyused to estimate exceedance probabilities and variable magnitudes A basic assumption

of frequency analysis is that the vector of data was measured from a temporally orspatially homogeneous system If measured data are significantly nonhomogeneous,the estimated probabilities or magnitudes will be inaccurate Thus, changes such asclimate or watershed alterations render the data unfit for frequency analysis and othermodeling methods

If changes to the physical processes that influence the data are suspected, thedata vector should be subjected to statistical tests to decide whether the nonsta-tionarity is significant If the change had a significant effect on the measured data,

it may be necessary to adjust the data before subjecting it to frequency analysis.Thus, the detection of the effects of change, the identification of the nature of anychange detected, and the appropriate adjustment of the data are prerequisite stepsrequired before a frequency model can be used to make probability or magnitudeestimates

5.2 FREQUENCY ANALYSIS AND SYNTHESIS

Design problems such as the delineation of flood profiles require estimates ofdischarge rates A number of methods of estimating peak discharge rates are available.They fall into two basic groups, one used at sites where gaged stream-flow recordsare available (gaged) and the other at sites where such records are not available(ungaged)

Statistical frequency analysis is the most common procedure for the analysis offlood data at a gaged location It is a general procedure that can be applied to anytype of data Because it is so widely used with flood data, the method is sometimesdesignated flood frequency analysis However, statistical frequency analysis can also

be applied to other hydrologic variables such as rainfall data for the development

of intensity-duration-frequency curves and low-flow discharges for use in waterquality control The variable could also be the mean annual rainfall, the peakdischarge, the 7-day low flow, or a water quality parameter Therefore, the topic will

be treated in both general and specific terms

5

5.2.1 P OPULATION VERSUS S AMPLE

In frequency modeling, it is important to distinguish between the population and thesample Frequency modeling is a statistical method that deals with a single randomvariable and thus is classified as a univariate method The goal of univariate predic-tion is to make estimates of probabilities or magnitudes of random variables A firststep is to identify the population The objective of univariate data analysis is to usesample information to determine the appropriate population density function, withthe probability density function (PDF) being the univariate model from which proba-bility statements can be made The input requirements for frequency modelinginclude a data series and a probability distribution assumed to describe the occurrence

of the random variable The data series could include the largest instantaneous peakdischarge to occur each year of the record The probability distribution could be thenormal distribution Analysis is the process of using the sample information toestimate the population The population consists of a mathematical model that is afunction of one or more parameters For example, the normal distribution is a function

of two parameters: the mean µ and standard deviation σ In addition to identifyingthe correct PDF, it is necessary to quantify the parameters of the PDF The populationconsists of both the probability distribution function and the parameters

A frequently used procedure called the method of moments equates tics of the sample (e.g., sample moments) to characteristics of the population (e.g.,population parameters) It is important to note that estimates of probability andmagnitudes are made using the assumed population and not the data sample; thesample is used only in identifying and verifying the population

characteris-5.2.2 A NALYSIS VERSUS S YNTHESIS

As with many hydrologic methods that have statistical bases, the terms analysis and

synthesis apply to the statistical frequency method Frequency analysis is “breakingdown” data in a way that leads to a mathematical or graphical model of the rela-tionship between flood magnitude and its probability of occurrence Conversely,synthesis refers to the estimation of (1) a value of the random variableX for someselected exceedance probability or (2) the exceedance probability for a selected value

of the random variableX In other words, analysis is the derivation of a model thatcan represent the relation between a random variable and its likelihood of occurrence,while synthesis is using the resulting relation for purposes of estimation

It is important to point out that frequency analysis may actually be part of amore elaborate problem of synthesis Specifically, separate frequency analyses can

be performed at a large number of sites within a region and the value of the randomvariableX for a selected exceedance probability determined for each site; these valuescan then be used to develop a regression model using the random variableX as thecriterion or dependent variable As an example, regression equations that relatepeak discharges of a selected exceedance probability for a number of sites to water-shed characteristics are widely used in hydrologic design This process is called

regionalization These equations are derived by (1) making a frequency analysis ofannual maximum discharges at a number (n) of stream gage stations in a region;

(2) selecting the value of the peak discharge from each of the n frequency curvesfor a selected exceedance probability, say the l00-year flood; and (3) developingthe regression equation relating the n values of peak discharge to watershed char-acteristics for the same n watersheds.

5.2.3 P ROBABILITY P APER

Frequency analysis is a common task in hydrologic studies A frequency analysisusually produces a graph of the value of a single hydrologic variable versus theprobability of its occurrence The computed graph represents the best estimate ofthe statistical population from which the sample of data was drawn

Since frequency analyses are often presented graphically, a special type of graphpaper, which is called probability paper, is required The paper has two axes Theordinate is used to plot the value of the random variable, that is, the magnitude, andthe probability of its occurrence is given on the abscissa The probability scale willvary depending on the probability distribution used In hydrology, the normal andGumbel extreme-value distributions are the two PDFs used most frequently to definethe probability scale Figure 5.1 is on normal probability paper The probability scalerepresents the cumulative normal distribution The scale at the top of the graph isthe exceedance probability, that is, the probability that the random variable will beequaled or exceeded in one time period It varies from 99.99% to 0.01% The lowerscale is the nonexceedance probability, which is the probability that the correspond-ing value of the random variable will not be exceeded in any one time period Thisscale extends from 0.01% to 99.99% The ordinate of probability paper is used for

FIGURE 5.1 Frequency curve for a normal population with µ = 5 and σ = 1.

the random variable, such as peak discharge The example shown in Figure 5.1 has

an arithmetic scale Lognormal probability paper is also available, with the scale forthe random variable in logarithmic form Gumbel and log-Gumbel papers can also

be obtained and used to describe the probabilistic behavior of random variables thatfollow these probability distributions

A frequency curve provides a probabilistic description of the likelihood ofoccurrence or nonoccurrence of a variable Figure 5.1 shows a frequency curve, withthe value of the random variableX versus its probability of occurrence The upperprobability scale gives the probability that X will be exceeded in one time period,while the lower probability scale gives the probability that X will not be exceeded.For the frequency curve of Figure 5.1, the probability that X will be greater than 7

in one time period is 0.023 and the probability that X will not be greater than 7 inone time period is 0.977

Although a unique probability plotting paper could be developed for each ability distribution, papers for the normal and extreme value distributions are themost frequently used The probability paper is presented as a cumulative distributionfunction If the sample of data is from the distribution function used to scale theprobability paper, the data will follow the pattern of the population line when properlyplotted on the paper If the data do not follow the population line, then (1) the sample

prob-is from a different population or (2) sampling variation produced a nonrepresentativesample In most cases, the former reason is assumed, especially when the samplesize is reasonably large

5.2.4 M ATHEMATICAL M ODEL

As an alternative to a graphical solution using probability paper, a frequency analysismay be conducted using a mathematical model A model that is commonly used inhydrology for normal, lognormal, and log-Pearson Type III analyses has the form

(5.1)

in which X is the value of the random variable having mean and standard deviation

S, and K is a frequency factor Depending on the underlying population, the specificvalue of K reflects the probability of occurrence of the value X Equation 5.1 can

be rearranged to solve for K when X, , and S are known and an estimate of theprobability of X occurring is necessary:

(5.2)

In summary, Equation 5.1 is used when the probability is known and an estimation

of the magnitude is needed, while Equation 5.2 is used when the magnitude is knownand the probability is needed

5.2.5 P ROCEDURE

In a broad sense, frequency analysis can be divided into two phases: deriving thepopulation curve and plotting the data to evaluate the goodness of fit The followingprocedure is often used to derive the frequency curve to represent the population:

1 Hypothesize the underlying density function

2 Obtain a sample and compute the sample moments

3 Equate the sample moments and the parameters of the proposed densityfunction

4 Construct a frequency curve that represents the underlying population

This procedure is referred to as method-of-moments estimation because the samplemoments are used to provide numerical values for the parameters of the assumedpopulation The computed frequency curve representing the population can then beused to estimate magnitudes for a given return period or probabilities for specifiedvalues of the random variable Both the graphical frequency curve and the mathe-matical model of Equation 5.1 are the population

It is important to recognize that it is not necessary to plot the data points inorder to make probability statements about the random variable While the four stepslisted above lead to an estimate of the population frequency curve, the data should

be plotted to ensure that the population curve is a good representation of the data.The plotting of the data is a somewhat separate part of a frequency analysis; itspurpose is to assess the quality of the fit rather than act as a part of the estimationprocess

2 1

3

For use in frequency analyses where the skew is used, Equation 5.3c represents astandardized value of the skew Equations 5.3 can also be used when the data aretransformed by taking the logarithms In this case, the log transformation should bedone before computing the moments.

5.2.7 P LOTTING P OSITION F ORMULAS

It is important to note that it is not necessary to plot the data before probabilitystatements can be made using the frequency curve; however, the data should beplotted to determine how well they agree with the fitted curve of the assumedpopulation A rank-order method is used to plot the data This involves orderingthe data from the largest event to the smallest event, assigning a rank of 1 to thelargest event and a rank of n to the smallest event, and using the rank (i)of theevent to obtain a probability plotting position; numerous plotting position formulasare available Bulletin l7B (Interagency Advisory Committee on Water Data, 1982)provides the following generalized equation for computing plotting position prob-abilities:

(5.4)

where a and b are constants that depend on the probability distribution An example

is a = b =0 for the uniform distribution Numerous formulas have been proposed,including the following:

n

i n

i=2 − = −12

0 5

P i n

+

0 4

0 2

The Hazen formula gives smaller probabilities for all ranks than the Weibull and

Cunnane formulas The probabilities for the Cunnane formula are more dispersed

than either of the others For a sample size of 99, the same trends exist as for n= 9

5.2.8 R ETURN P ERIOD

The concept of return period is used to describe the likelihood of flood magnitudes

The return period is the reciprocal of the exceedance probability, that is, p= 1/T

Just as a 25-year rainfall has a probability of 0.04 of occurring in any one year, a

25-year flood has a probability of 0.04 of occurring in any one year It is incorrect

to believe that a year event will not occur again for another 25 years Two

25-year events can occur in consecutive 25-years Then again, a period of 100 25-years may

pass before a second 25-year event occurs

Does a 25-year rainfall cause a 25-year flood magnitude? Some hydrologic

models make this assumption; however, it is unlikely to be the case in actuality It

is a reasonable assumption for modeling because models are based on the average

of expectation or on-the-average behavior In actuality, a 25-year flood magnitude

will not occur if a 25-year rainfall occurs on a dry watershed Similarly, a 50-year

flood could occur from a 25-year rainfall if the watershed was saturated Modeling

often assumes that a T-year rainfall on a watershed that exists in a T-year hydrologic

condition will produce a T-year flood

5.3 POPULATION MODELS

Step 1 of the frequency analysis procedure indicates that it is necessary to select a

model to represent the population Any probability distribution can serve as the

model, but the lognormal and log-Pearson Type III distributions are the most widely

used in hydrologic analysis They are introduced subsequent sections, along with

the normal distribution or basic model

n= 9 n= 99 Rank p w p h p c Rank p w p h p c

5.3.1 N ORMAL D ISTRIBUTION

Commercially available normal probability paper is commonly used in hydrology

Following the general procedure outlined above, the specific steps used to develop

a curve for a normal population are as follows:

1 Assume that the random variable has a normal distribution with population

parameters µ and σ

2 Compute the sample moments and S (the skew is not needed).

3 For normal distribution, the parameters and sample moments are related

by µ = and σ = S.

4 A curve is fitted as a straight line with plotted at an exceedance

probability of 0.8413 and at an exceedance probability of 0.1587

The frequency curve of Figure 5.1 is an example for a normal distribution with a

mean of 5 and a standard deviation of 1 It is important to note that the curve passes

through the two points: and It also passes through

the point defined by the mean and a probability of 0.5 Two other points that could

removed from the mean has the advantage that inaccuracies in the line drawn to

represent the population will be smaller than when using more interior points

The sample values should then be plotted (see Section 5.2.7) to decide whether

the measured values closely approximate the population If the data provide a

reasonable fit to the line, one can assume that the underlying population is the normal

distribution and the sample mean and standard deviation are reasonable estimates

of the location and scale parameters, respectively A poor fit indicates that the normal

distribution is not appropriate, that the sample statistics are not good estimators of

the population parameters, or both

When using a frequency curve, it is common to discuss the likelihood of events

in terms of exceedance frequency, exceedance probability, or the return period (T)

related to the exceedance probability (p) by p = l/T, or T = l/p Thus, an event with

an exceedance probability of 0.01 should be expected to occur 1 time in 100 In

many cases, a time unit is attached to the return period For example, if the data

represent annual floods at a location, the basic time unit is 1 year The return period

for an event with an exceedance probability of 0.01 would be the l00-year event

(i.e., T = 1/0.01 = 100); similarly, the 25-year event has an exceedance probability

of 0.04 (i.e., p = 1/25 = 0.04) It is important to emphasize that two T-year events

will not necessarily occur exactly T years apart They can occur in successive years

or may be spaced three times T years apart On average, the events will be spaced

T years apart Thus, in a long period, say 10,000 years, we would expect 10,000/T

events to occur In any single 10,000-year period, we may observe more or fewer

occurrences than the mean (10,000/T).

Estimation with normal frequency curve — For normal distribution, estimation

may involve finding a probability corresponding to a specified value of the random

variable or finding the value of the random variable for a given probability Both

problems can be solved using graphical analysis or the mathematical models of

X X

(X−S)(X+S)

(X−S, 0 8413 ) (X+S, 0 1587 )(X+ 2 0 0228S, ) (X− 2S, 0 9772 )

Equations 5.1 and 5.2 A graphical analysis estimation involves simply entering theprobability and finding the corresponding value of the random variable or enteringthe value of the random variable and finding the corresponding exceedance proba-bility In both cases, the fitted line (population) is used The accuracy of the estimatedvalue will be influenced by the accuracy used in drawing the line or graph.

Example 5.1

Figure 5.2 shows a frequency histogram for the data in Table 5.1 The sample consists

of 58 annual maximum instantaneous discharges, with a mean of 8620 ft3/sec, astandard deviation of 4128 ft3/sec, and a standardized skew of 1.14 In spite of thelarge skew, the normal frequency curve was fitted using the procedure of the pre-ceding section Figure 5.3 shows the cumulative normal distribution using the samplemean and the standard deviation as estimates of the location and scale parameters

The population line was drawn by plotting X + S = 12,748 at p = 15.87% and –

S = 4492 at p = 84.13%, using the upper scale for the probabilities The data were

plotted using the Weibull plotting position formula (Equation 5.5a) The data do notprovide a reasonable fit to the population; they show a significant skew with an

FIGURE 5.2 Frequency histograms of the annual maximum flood series (solid line) and

logarithms (dashed line) based on mean (and for logarithms) and standard deviations (S x and

for logarithms S y): Piscataquis River near Dover-Foxcroft, Maine.

X

TABLE 5.1 Frequency Analysis of Peak Discharge Data:

Piscataquis River

Rank

Weibull Probability

Random Variable

Logarithm of Variable

TABLE 5.1 Frequency Analysis of Peak Discharge Data:

Piscataquis River (Continued)

Rank

Weibull Probability

Random Variable

Logarithm of Variable

especially poor fit to the tails of the distribution (i.e., high and low exceedanceprobabilities) Because of the poor fit, the line shown in Figure 5.3 should not beused to make probability statements about the future occurrences of floods; forexample, the normal distribution (i.e., the line) suggests a 1% chance flood magnitude

of slightly more than 18,000 ft3/sec However, if a line was drawn subjectivelythrough the trend of the points, the flood would be considerably larger, say about23,000 ft3/sec

The 100-year flood is estimated by entering with a probability of 1% and findingthe corresponding flood magnitude Probabilities can be also estimated For example,

if a levee system at this site would be overtopped at a magnitude of 16,000 ft3/sec,the curve indicates a corresponding probability of about 4%, which is the 25-yearflood

To estimate probabilities or flood magnitudes using the mathematical model,

Equation 5.1 becomes X = + zS because the frequency factor K of Equation 5.1 becomes the standard normal deviate z for a normal distribution, where values of z

are from Appendix Table A.1 To find the value of the random variable X, estimates

of and S must be known and the value of z obtained from Appendix Table A.1

for any probability To find the probability for a given value of the random variable

X, Equation 5.2 is used to solve for the frequency factor z (which is K in Equation 5.2);

the probability is then obtained from Table A.1 using the computed value of z For example, the value of z from Table A.1 for a probability of 0.01 (i.e., the 100-year

event) is 2.327; thus, the flood magnitude is:

which agrees with the value obtained from the graphical analysis For a discharge

of 16,000 ft3/sec, the corresponding z value is:

Appendix Table A.1 indicates the probability is 0.0377, which agrees with thegraphical estimate of about 4%

5.3.2 L OGNORMAL D ISTRIBUTION

When a poor fit to observed data is obtained, a different distribution function should

be considered For example, when the data demonstrate a concave, upward curve,

as in Figure 5.3, it is reasonable to try a lognormal distribution or an extreme valuedistribution It may be preferable to fit with a distribution that requires an estimate

of the skew coefficient, such as a log-Pearson Type III distribution However, sampleestimates of the skew coefficient may be inaccurate for small samples

The same procedure used for fitting the normal distribution can be used to fitthe lognormal distribution The underlying population is assumed to be lognormal

The data must first be transformed to logarithms, Y = log X This transformation

creates a new random variable Y The mean and standard deviation of the logarithms

are computed and used as the parameters of the population; it is important to recognizethat the logarithm of the mean does not equal the mean of the logarithms, which isalso true for the standard deviation Thus, the logarithms of the mean and standarddeviation should not be used as parameters; the mean and standard deviation of thelogarithms should be computed and used as the parameters Either natural or base-

10 logarithms may be used, although the latter is more common in hydrology Thepopulation line is defined by plotting the straight line on arithmetic probability paperbetween the points ( +S y, 0.1587) and ( − S y , 0.8413), where and S y are themean and standard deviation of the logarithms, respectively In plotting the data,either the logarithms can be plotted on an arithmetic scale or the untransformed datacan be plotted on a logarithmic scale

When using a frequency curve for a lognormal distribution, the value of the

random variable Y and the moments of the logarithms ( and S y) are related by theequation:

(5.6)

in which z is the value of the standardized normal variate; values of z and

corre-sponding probabilities can be found in Table A.1 Equation 5.6 can be used toestimate either flood magnitudes for a given exceedance probability or an exceedanceprobability for a specific discharge To find a discharge for a specific exceedance

probability the standard normal deviate z is obtained from Table A.1 and used in

Equation 5.6 to compute the discharge To find the exceedance probability for a

given discharge Y, Equation 5.6 is rearranged by solving for z With values of Y, , and S y , a value of z is computed and used with Table A.1 to compute the exceedance probability Of course, the same values of both Y and the probability can be obtained

directly from the frequency curve

Example 5.2

The peak discharge data for the Piscataquis River were transformed by taking thelogarithm of each of the 58 values The moments of the logarithms are as follows:

= 3.8894, S y = 0.20308, and g = −0.07 Figure 5.4 is a histogram of the logarithms.

In comparison to the histogram of Figure 5.2, the logarithms of the sample data areless skewed While a skew of −0.07 would usually be rounded to −0.1, it is sufficientlyclose to 0 such that the discharges can be represented with a lognormal distribution.The frequency curve is shown in Figure 5.5 To plot the lognormal population curve,the following two points were used: − S y = 3.686 at p = 84.13% and + S y =

4.092 at p = 15.87% The data points were plotted on Figure 5.5 using the Weibullformula and show a much closer agreement with the population line in comparison

to the points for the normal distribution in Figure 5.3 It is reasonable to assumethat the measured peak discharge rates can be represented by a lognormal distributionand that the future flood behavior of the watershed can be described statisticallyusing a lognormal distribution

If one were interested in the probability that a flood of 20,000 ft3/sec would beexceeded in a given year, the logarithm of 20,000 (4.301) would be entered on thedischarge axis and followed to the assumed population line Reading the exceedanceprobability corresponding to that point on the frequency curve, a flood of 20,000

ft3/sec has a 1.7% chance of being equaled or exceeded in any one year It can also

FIGURE 5.4 Histogram of logarithms of annual maximum series: Piscataquis River.

FIGURE 5.5 Frequency curve for the logarithms of the annual maximum discharge.

be interpreted that over the span of 1,000 years, a flood discharge of 20,000 ft3/secwould be exceeded in 17 of those years; it is important to understand that this is anaverage In any period of 1,000 years, a value of 20,000 ft3/sec may be exceeded orreached less frequently than in 17 of the 1,000 years, but on average 17 exceedanceswould occur in 1,000 years.

The probability of a discharge of 20,000 ft3/sec can also be estimated matically The standard normal deviate is

mathe-(5.7a)

The value of z is entered into Table A.1, which yields a probability of 0.9786 Since

the exceedance probability is of interest, this is subtracted from 1, which yields avalue of 0.0214 This corresponds to a 47-year flood The difference between themathematical estimate of 2.1% and the graphical estimate of 1.7% is due to the error

in the graph The computed value of 2.1% should be used

The frequency curve can also be used to estimate flood magnitudes for selectedprobabilities Flood magnitude is found by entering the figure with the exceedanceprobability, moving vertically to the frequency curve, and finally moving horizontally

to flood magnitude For example, the 100-year flood for the Piscataquis River can

be found by starting with an exceedance probability of 1%, moving to the curve ofFigure 5.5 and then to the ordinate, which indicates a logarithm of about 4.3586 or

a discharge of 22,800 ft3/sec Discharges for other exceedance probabilities can befound that way or by using the mathematical model of Equation 5.1 In addition tothe graphical estimate, Equation 5.6 can be used to obtain a more exact estimate

For an exceedance probability of 0.01, a z value of 2.327 is obtained from Table A.1.

Thus, the logarithm is

(5.7b)Taking the antilogarithm yields a discharge of 23,013 ft3/sec

5.3.3 L OG -P EARSON T YPE III D ISTRIBUTION

Normal and lognormal frequency analyses were introduced because they are easy

to understand and have a variety of uses The statistical distribution most commonlyused in hydrology in the United States is the log-Pearson Type III (LP3) because itwas recommended by the U.S Water Resources Council in Bulletin 17B (InteragencyAdvisory Committee on Water Data, 1982) The Pearson Type III is a PDF It iswidely accepted because it is easy to apply when the parameters are estimated usingthe method of moments and it usually provides a good fit to measured data LP3analysis requires a logarithmic transformation of data; specifically, the commonlogarithms are used as the variates, and the Pearson Type III distribution is used asthe PDF

While LP3 analyses have been made for most stream gage sites in the UnitedStates and can be obtained from the U.S Geological Survey, a brief description ofthe analysis procedure is provided here Although the method of analysis presentedhere will follow the procedure recommended by the Water Resources Council, youshould consult Bulletin 17B when performing an analysis because a number ofoptions and adjustments cited in the bulletin are not discussed here This sectiononly provides sufficient detail so that a basic frequency analysis can be made andproperly interpreted.

Bulletin l7B provides details for the analysis of three types of data: a systematicgage record, regional data, and historic information for the site The systematicrecord consists of the annual maximum flood record It is not necessary for therecord to be continuous as long as the missing part of the record is not the result offlood experience, such as the destruction of the stream gage during a large flood.Regional information includes a generalized skew coefficient, a weighting procedurefor handling independent estimates, and a means of correlating a short systematicrecord with a longer systematic record from a nearby stream gaging station Historicrecords, such as high-water marks and newspaper accounts of flooding that occurredbefore installation of the gage, can be used to augment information at the site.The procedure for fitting an LP3 curve with a measured systematic record issimilar to the procedure used for the normal and lognormal analyses described earlier.The steps of the Bulletin 17B procedure for analyzing a systematic record based on

a method-of-moments analysis are as follows:

1 Create a series that consists of the logarithms Y i of the annual maximum

flood series x i

2 Using Equations 5.3 compute the sample mean, , standard deviation,

S y , and standardized skew, g s , of the logarithms created in step 1

3 For selected values of the exceedance probability (p), obtain values of the standardized variate K from Appendix Table A.5 (round the skew to the

nearest tenth)

4 Determine the values of the LP3 curve for the exceedance probabilitiesselected in Step 3 using the equation

(5.8)

in which y is the logarithmic value of the LP3 curve.

5 Use the antilogarithms of the Y j values to plot the LP3 frequency curve.After determining the LP3 population curve, the data can be plotted to determineadequacy of the curve The Weibull plotting position is commonly used Confidencelimits can also be placed on the curve; a procedure for computing confidenceintervals is discussed in Bulletin 17B In step 3, it is necessary to select two or morepoints to compute and plot the LP3 curve If the absolute value of the skew is small,the line will be nearly straight and only a few points are necessary to draw itaccurately When the absolute value of the skew is large, more points must be used

Y

Y= +Y KS y

because of the greater curvature When selecting exceedance probabilities to computethe LP3 curve, it is common to include 0.5, 0.2, 0.1, 0.04, 0.02, 0.01, and 0.002because these correspond to return periods that are usually of interest.

Example 5.3

Data for the Back Creek near Jones Springs, West Virginia (USGS gaging station016140), are given in Table 5.2 Based on the 38 years of record (1929–1931 and1939–1973), the mean, standard deviation, and skew of the common logarithms are3.722, 0.2804, and −0.731, respectively; the skew will be rounded to −0.7 Table 5.3

shows the K values of Equation 5.8 for selected values of the exceedance probability (p); these values were obtained from Table A.5 using p and the sample skew of −0.7.Equation 5.8 was used to compute the logarithms of the LP3 discharges for theselected exceedance probabilities (Table 5.3)

The logarithms of the discharges were then plotted versus the exceedance abilities, as shown in Figure 5.6 The rank of each event is also shown in Table 5.2and was used to compute the exceedance probability using the Weibull plottingposition formula (Equation 5.4a) The logarithms of the measured data were plottedversus the exceedance probability (Figure 5.6) The data show a reasonable fit to thefrequency curve, although the fit is not especially good for the few highest and thelowest measured discharges; the points with exceedance probabilities between 80%and 92% suggest a poor fit Given that we can only make subjective assessments of

prob-FIGURE 5.6 Log-Pearson Type III frequency curve for Back Creek near Jones Spring, West

Virginia, with station skew ( ) and weighted skew (- - -).

TABLE 5.2 Annual Maximum Floods for Back Creek

Source: Interagency Advisory Committee on Water Data,

Guidelines for Determining Flood-Flow Frequency, Bulletin 17B, U.S Geological Survey, Office of Water Data Coordina- tion, Reston, VA, 1982.