FLOW-DuRATION CURVES.

I: NEW INTERPRETATION AND CONFIDENCE INTERVALS

By Richard M. Vogel,J Member, ASCE, and Neil M. Fennessey,2

Associate Member, ASCE

ABSTRACT: A flow-duration curve (FDC) is simply the complement of the cu-mulative distribution function of daily, weekly, monthly ( or some other time intervalof) streamflow. Applications of FDCs include, but are not limmited to, hydropower

planning, water-quality management, river and reservoir sedimentation studies,habitat suitability, and low-flow augmentation. Although FDCs have a long andrich history in the field of hydrology, they are sometimes criticized because, tra-ditionally, their interpretation depends on the particular period of record on whichthey are based. If one considers n individual FDCs, each corresponding to one ofthe individual n years of record, then one may treat those n annual FDCs in muchthe same way one treats a sequence of annual maximum or annual minimumstreamflows. This new annual-based interpretation enables confidence intervals andrecurrence intervals to be associated with FDCs in a nonparametric framework.

INTRODUCTION

"It is a capital mistake to theorize before one has data," Sir Arthur

Conan Doyle.

A flow-duration curve (FDC) represents the relationship between the

magnitude and frequency of daily, weekly, monthly (or some other time

interval of) streamflow for a particular river basin, providing an estimate

of the percentage of time a given streamflow was equaled or exceeded over

a historical period. An FDC provides a simple, yet comprehensive, graphical

view of the overall historical variability associated with streamflow in a river

basin.

An FDC is the complement of the cumulative distribution function ( cdf)

of daily streamflow. Each value of discharge Q has a corresponding ex-

ceedance probability p, and an FDC is simply a plot of Qp, the pth quantile

or percentile of daily streamflow versus exceedance probability p, where p

is defined by

p = 1 -P{Q :s; q} (la)

p = 1- FQ(q) (lb)

The quantile Qp is a function of the observed streamflows, and since this

function depends upon empirical observations, it is often termed the em-

pirical quantile function. Statisticians term the complement of the cdf the

"survival" distribution function. The 'term survival results from the fact

that most applications involve survival data that arise in various fields

I Assoc. Prof. , Dept. of Civ. and Envir. Engrg. , Tufts Univ. , Me

2Res. Assoc., Dept. of Civ. and Envir. Engrg., Tufts Univ., Medford, MA.

Note. Discussion open until January 1, 1995, To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The

manuscript for this paper was submitted for review and possible publication on

January 28, 1993. This paper is part of the Journal of Water Resources Planning and

Management, Vol. 120, No.4, July/August, 1994. @ASCE, ISSN 0733-9496/94/0004-

0485/$2.00 + $.25 per page. Paper No.5474.

485

such as medicine, manufacturing, and demography [see Anderson and Vaeth(1988) for a review of the literature].

Brief History of Application of Flow-Duration CurvesA sequel to the present paper (Vogel and Fennessey, work in progress)

summarizes, in detail, the many applications of FDCs. The first use of anFDC is attributed to Clemens Herschel in about 1880 (Foster 1934). Thewidespread use of FDCs during the first half of this century is evidencedby many studies that sought to develop FDCs for particular regions of theU .S. For example, Mitchell (1957) developed procedures for estimatingFDCs at gaged, partially gaged, and ungaged sites in Illinois. Cross andBernhagen (1949) summarized FDCs in Ohio, and Saville et al. (1933)summarized FDCs in North Carolina. In the U .S. , regional FDC procedureshave been developed for ungaged sites in Illinois, New Hampshire, andMassachusetts by Singh (1971), Dingman (1978) and Fennessey and Vogel(1990), respectively. See Fennessey and Vogel (1990) for a review of otherrecent regional FDC models.

Mitchell (1957), Searcy (1959), and the Institute of Hydrology ("Low"1980) provide comprehensive manuals on the construction, interpretation,and application of FDCs. Interestingly, most of the important work relatedto the construction, analysis, and interpretation of FDCs predates the com-mon application of computers (e.g., Foster (1934); Beard (1943); Mitchell(1957); Searcy (1959); Hoyle (1963)].

Since the advent of computer technology, few articles on FDCs have beenwritten, yet, ironically, many recent advances, due to computer technology,can be exploited along with FDC concepts as is shown in the present paper .Fienberg (1979) found that in general, there has been a prolonged declinein the relative use of graphical devices for displaying statistical informationever since the advent of computer technology. Although many of the articleson FDCs were written during the first half of this century , current textbooksstill contain discussions pertaining to this important tool [see, for example,Warnick (1984); Gordon et al. (1992)].

Streamflow-duration curves have been advocated for use in hydrologicstudies such as hydropower, water-supply, and irrigation planning [see,Chow (1964); Warnick (1984)]. Mitchell (1957) and Searcy (1959) describeadditional applications to waste-Ioad allocation and other water-quality man-agement problems. Male and Ogawa (1984) show how FDCs can be usedto illustrate and evaluate the trade-offs among the variables involved in theselection of a wastewater-treatment-plant capacity. The U .S. Bureau ofReclamation (Strand and Pemberton 1982) use FDCs in river and reservoirsedimentation studies that examine the frequency of suspended sedimentloads and determine the long-term average suspended sediment yield for agiven site. The U.S. Fish and Wildlife Service (Gordon et al. 1992) useFDCs in their "Instream Flow Incremental Methodology" for determiningthe suitability of habitats to streamflow of different magnitudes and fre-quencies. Alaouze (1989) describes the use of FDCs for determining theoptimal allocation of water withdrawals from reservoirs, where each with-drawal is to have a unique reliability.

Some Caveats Associated with Flow-Duration CurvesFig. 1 displays an example of an FDC along with the probability density

function (pdf) of average daily streamflow for the Acheron River in Aus-tralia for the period 1947-1987. Also depicted are the mean, median, and

486

Probability Mass Function, fQ(q)0- 0.00 0.02 0.04 0.06 0.08 0.10 0.12

50'i ~ fQ(q)O;-;:::: 40 p{Q)ql

S Meanro 30~ Median~I/) Mode20"0'w~ 10w00 .

,0 0O 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Flow Duration Curve, PiQ)q~

FIG. 1. Comparison of FDC with Probability Density Function of Observed DailyStreamflows (m3/s) for Acheron River, 1947-1987

modal daily streamflows. Note that only 34.2% of the daily flows exceededthe mean over the period of record. This is not unusual, and this resultemphasizes how misleading it can be to use the mean as a measure of centraltendency for highly skewed data such as daily stream flow. Daily streamflowsare so highly skewed that ordinary product moment ratios such as the coef-ficient of variation and skewness are remarkably biased and should be avoided,even for samples with tens of thousands of flow observations (Vogel andFennessey 1993).

Although FDCs are appealing for depicting the hydrologic response of ariver basin, they can be misleading because the autocorrelation structure ofstreamflow series is effectively removed from the plot. To clarify this point,Fig. 2 uses a single graphical image to compare theFDC with the hydrographof the Acheron River. It should always be understood when viewing anFDC, that streamflow behaves the way it is illustrated in the hydrographappearing as a dotted line in the background. We recommend plotting FDCswith the complete hydrograph in the background, perhaps using light shad-ing, to reinforce the serial structure of all flow sequences. One could alsoplot a correlogram, which is a plot of the lag-k serial correlation versus lagk, to expose the significant serial structure associated with daily streamflow.

Although FDCs have a long and rich history in hydrology, they are some-times criticized because, traditionally, their interpretation depends on theparticular period of record on which they are based. If one considers nindividual FDCs, each corresponding to one of the individual n years ofrecord, then one may treat those n annual FDCs in much the same way onetreats a sequence of annual maximum or annual minimum streamflows.Viewed in that context, the FDC becomes a generalization of the distributionof daily streamflow where the distribution of annual maximum flood flowsand annual minimum low flows are simply special cases drawn from eitherend of the complete annual-based FDC. This new annual-based interpre-tation of FDCs provides a general approach to streamflow frequency analysisthat allows us to derive confidence intervals, recurrence intervals, and quan-tile-estimation procedures for FDCs in a nonparametric framework.

487

Year1947 1957 1967 1977 1987

-FDC~ 100.0

~0

~S 10.0

roa>~ 1.0

o+.J(/)

0.10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Exceedance Probability, PIQ)q~

FIG. 2. Comparison of Hydrograph of Acheron River with Its FDC

TRADITIONAL FLOW-DURATION CURVE IS AN OGIVE

Prior to the advent of computer technology, Searcy (1959) and othersrecommended constructing FDCs by separating observed streamflow into20-30 well-distributed class intervals, and defining the FDC as the cumu-lative histogram of streamflow based on those class intervals. Searcy (1959)provides explicit guidelines for the construction of class intervals to be usedwith his procedure [see Table 1 in Searcy (1959)]. Searcy's approach pro-duces what statisticians term a ogive, which is a plot of the cumulativefrequency corresponding to each class interval versus the upper limit of eachclass interval where straight lines connect consecutive points. An ogive is agrouped data analog of a graph of the empirical cumulative distributionfunction. Ogives are useful for representing selected percentiles or quantilesof a distribution or for constructing box and whisker plots. However, if oneis interested in obtaining an accurate computerized description of FDCs andt~eir a~soci~ted confidence intervals, t~e more efficient and smoother quan-tlle-estlmatlon procedures to be descrIbed are recommended.

NONPARAMETRIC QUANTILE-ESTIMATION PROCEDURES

Consider the construction of an FDC or empirical quantile function fromn observations of streamflow qi' where i = 1, ..., n. If the streamflowsare ranked, then the set of order statistics q(i)' where i = 1, ..., n, results

where q(l) is the largest and q(n) is the smallest observation.Even before the era of computers, quantiles and associated FDCs could

be estimated from one or two order statistics. For example, the simplestempirical quantile function, or quantile estimator, is obtained from a single

order statistic using

Qp,l = q(i) if i = [(n + l)p] (2a)

Qp.l = q(i+l) if i < [(n + l)p] (2b)

where the quantity in brackets [(n + l)p] denotes the integer componentof (n + l)p that is always less than or equal to (n + l)p. We recommend

488

setting the smallest possible observation q(n+l) equal to zero, the naturalminimum for streamflow. If the observations are not bounded above, theestimator Qp.l is undefined for values ofp that lead to i = [(n + l)p] =0, since q(O) = 00. Essentially, Qp.l is equivalent to plotting the orderedobservations q(i) versus an estimate of their plotting positions Pi' where Pi= i/(n + 1) is an estimate of the exceedance probability p in (1) known asthe Weibull plotting position. The Weibull plotting position provides anunbiased estimate of 1 -FQ(q), regardless of the underlying probabilitydistribution from which streamflows arise.

The main drawback to the simple quantile estimator Qp.l, is that due tothe variability of individual order statistics, it is often an inefficient estimator .An efficient quanti le estimator is one with low bias, variance, and meansquare error. The lack of efficiency associated with Qp,l is particularly sig-nificant for small samples (n < 100) and for values of p near zero or unity,even for large samples.

One way to improve the efficiency of Qp,l is to reduce its variability byfonning a weighted average of two or more adjacent order statistics usingan appropriate weighting function, Such quantile estimators, based on alinear combination of the order statistics are termed L-estimators, analogousto L-moment estimators of distributional parameters recently advanced byHosking (1990) and summarized by Stedinger et al, (1993) for hydrologicapplications.

For example, Parzen (1979) introduced a simple quantile estimator basedon the weighted average of two adjacent order statistics. One such weightedestimator, which is slightly smoother than Qp,1 is

Qp,2 = ~1 -6)q(i) + 6q(i+l) (3)

where i = [(n + l)p] [and the brackets are defined as in (2)]; and 6 =«n + l)p -i). The estimator Qp,2 is undefined for values of p that leadto i = [(n + l)p] = 0. .

In a comparison of 10 alternative quantile estimators, Parrish (1990) foundthat (3), with i = [np] and 6 = (np -i + 0.5), perfonned slightly betterthan (3), with i = [(n + l)p] and 6 = «n + l)p -i); however, theseestimators of i and 6 only yield improvements for small samples. Again,one sets the smallest possible observation q(n+l) equal to zero; the naturalminimum for streamflow.

The estimators Qp.l and Qp,2 are probably adequate for constructing FDCswhen thousands of daily streamflows are available; however, if one wishesto construct a series of annual FDCs, each of which is only based on 365highly correlated daily streamflow observations, it may be wise to use moreefficient nonparametric quantile estimators. Similarly, if one wishes to es-timate the empirical quantile function associated with a particular quantileestimator, as we do later on, then one requires a reasonably efficient non-parametric quantile estimator for independent samples with n ranging from10 to 100.

Harrell and Davis (1982), Kaigh and Lachenbruch (1982), Yang (1985),and Sheather and Marron (1990) introduced quantile estimators with smallermean squared error than either Qp,1 or Qp,2 for a wide range of distributionsand sample sizes. Harrell and Davis (1982) introduced a distribution-freequantile estimator based on linear combinations of all n order statistics,with significantly lower variance than the estimators Qp,l and Qp,2. Theirestimator is derived from the fact that the cumulative probability FQ[q(i)]associated with each ranked streamflow q(i) follows a beta distribution [see

489

Loucks et a" (1981)] Hence, the expected value of the ith order statistic

is given by

E[q",] = ~ I:. qF(q)'-.[1 -F(q)]"-' dF(q) (4a)

I ('E[q",] = w:n -i :;:---ij J, F-.(q)q'-'(1 -q)"-' dq (4b)

where B[a, b] denotes the beta function. Now taking i ~ (n + I)p, regardlessof whether or not i is an integer, E[q(,,] converges to F-'(p) as n ~ 00,

which leads to the Harrell and Davis estimator

Q,3 = i A,q", (5a),-,

with the weights " estimated from

Ii ""

, ~ -,."" q'"+"p-'(1 -q)("+"(.-P'-' dq, B[(n + I)p, (n + I)(I -p)] "-,,,"

(5b)

" ~I.,Jp(n + I), (1- p)(n + I)] -l"-,,m[p(n + I), (1- p)(n + I)]

(5c)

where IJa, b] ~ the incomplete beta function and B[a, b] ~ the betafunction Yang (1985), and Sheather and Marron (1990) describe Q.'i' asan analytical form of the bootstrap estimate of the mean of the i = [(n +I)p]th order statistic In other words, if one were to use simulation to obtainthe bootstrap estimate of E[q",], for i ~ (n + I)p, one would obtain the

estimate Qp, Eq (5a) is the exact analytical version of the computationallyintensive bootstrap estimate of E[q",] that requires simulated resamplingEfron (1982) provides an introduction to the bootstrap method

As long as n 2 100, Harrell and Davis (1982) suggest estimating the

weights A, using numerical integration with two intervals between (i -1)/n and i/n Otherwise they suggest estimating the incomplete beta function

exactly For this purpose one caneither use the efficient algorithm suggestedby Majumdar and Bhattacharjee (1973) or the approximations given by

Abramowitz and Stegun (1972)IJa, bJ = 1 -"'(X'/'1) if (a + b -I)(I -x) $ Q8 (6a)

IJa, b] = "'(y) if (a + b -I)(I -x) 2 Q8 (6b)

wherex' = (a + b -I)(I -x)(3 -x) -(I -x)(b -I) (7a)

'1 ~ 2b (7b)

y ~ 3 ( w, ( I -~) -w, ( I -~) ] (¥ + ~) -." (7c)

w, ~ (bx)"' (7d)

w, = [a(1 -x)]"' (7e)

490

."d"")j"q'.looPIZs,"wh,reZ~."."d"d""m.I,."dom"ri.b"

K.jgh ."d L.,h,"bru," (1982) j""od",d ."mh" q,."'j" ,"jm."cwhj,hj,",00."L."jm.mM'h"ry'jmj",poo"rtj"ooQ" Th,K.jgh."d Lo,h,"bru,h ,"jm.oo' j" rem,d .g'",,",jred ,.mpl, q"mj" ."d j,ob"j",dby."rngj"g.".ppoopri.""b,.mp"q"mj"0",",Ip'"'jb"rub"mp"'offi"d""g'hkTh,j,,"jm.00,j,

Q" ~.f'wq (8)

(j- 1)(" -1)-~ (9)", -(;) "

, ~ [(k + 1),) (9b)

(" ) - (-"- ) (9"k (" -k)!k!

K.jgh."dllij.",1(1987)'howhow(8),."b'wmp",d,ffj,j,mlyo".p,"w.lwmp"""j"g.'jmp"ruw",,"fmmw. ffk ~ " Q"red"",

ooQ,c Th,p'rnm""k.W!"moo,h'h,q".mjk'"jm.c'm.fI"'.!""" k prud"mg 'moo'h" ,",m.," " Qr Th, ,"m~ " k 'h. mm,m.re,'h, m,." 'q".rud "ru, req"j", k",w"dg' " 'h, '"""yi"g dj",jbmjocSj"~pru,j~ru",ro"","j"g'h,rub".mp""irek."mi.,dwi'h'h,",jm,",Q".ru,",".i"b"."',i""Sh,.h"."dM,n,"""0)."dP"ri'h(I.")ro"""h.Q",,"",m,db"",'h."Q,j",m.fI,.mp"~::~";:;,,~;::~:~:;~;"d;:::"Q{O'j;'i~";h:"7~:'.,':::::i~.~:':f,;,~:Y;re "

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'h,w.""reru"~,'j"rn',,,byA'.mow,kj"98;"Goo(1991"."dmh,"fo,",jm"j"8'h'p'fof."",,'m.,jmomnoo,now,y."g,,98","'Sh"'h"."'M.n""(I990)"mm.rire'h,prup,"j,""k'ru"q".mj","jm"m,Sh",h","dM.no",how,h"fo,'""",mp"'Q"i"q,;,."m".k,ru",";m"mwj,h.G.".i."k,ru"Th,y"wro""d'h"',m,1 q".mi" ~';m"m, h,", ,"'p",j"gly ,;miloc p,""m."~ 00 Q"K,m" "';mw" wm.i" , k,m" fu,,'io" wh"h i, ,",mj.fly .,m~'hmg

fu""joo.".'",",",h".,"",ki"Q"Sh,"h"."dM."0"(I990)'h"w'h".,ympoo'i'"'y(lm"ry"'g,""

.JI"o"p'rnm",;,q".mi","im"o"d""op,dOO""h.",.ppru,im.,,'h".m".mpl;"gpru""j"Th.y'how'h",","fo,,m"'""mp"'wj,h" ""g;"g "om ;0'" 100 'h, ,"im.oo" Qv ,", Q" ,." p,"o,m "mm'

!1fi::'~:', ::Q::~~":Q;:"\;~:~7, ::' ~~~f;;,?:;~~ir,~~:f~~;Ti1~~,m;rely gi'," 'h, "" 'h" 'hoy ,,"o,m,d 00ly m.,g;""" b"'" 'h." 'h,"mp"'";m.ill";"Sh",h"."dM,",""m.JI,.mp"M"moC."o,rud

ies. Nevertheless, the estimators Qp.3 and Qp.. provide markedly smootherestimates of the quantile function than the simpler estimators for smallsamples, and bootstrap and jackknife estimates of the variance of theseestimators are available and perform well. One cannot obtain bootstrap orjackknife estimates of the variance of Qp.l since the quantile function isdiscontinuous.

Each of the above quantile estimators has advantages in particular set-tings; these situations are described in the following sections within thecontext of estimating FDCs.

PERIOD-OF-RECORO FLOW-DURATION CURVE

Previous investigators, for example, Searcy (1959), Mitchell (1957), Beard(1943), and Foster (1934), have suggested constructing FDCs as ogives.Those studies, which were published prior to the advent of computers,advocated plotting the empirical cumulative histogram or ogive since onlyapproximately 30 points needed to be plotted. When the complete periodof record is used to construct an FDC, the quantile estimators describedhere are based on 365n daily streamflows or between 3,650 and 36,500observations for records ranging from 10 to 100 years, respectively! Withso many observations, one need not fit a curve through the points, sincethe points, themselves, could be used to create a curve. Another equivalentapproach is to plot a hundred or so points and use spline curve-fittingprocedures to draw a smooth curve through the points. Furthermore, withsuch large samples, the differences among the estimators Qp I, Qp 2, Qp 3,and Qp.. are negligible, and the simplest estimators Qp.l or Qp.2 suffice. .

Comparison or Quantile EstimatorsFig. 3 uses lognormal probability paper to compare the period-of-record

(1947-1987) FDC of daily streamflow constructed using an empirical cu-mulative histogram [Searcy's (1959) recommended approach] with the es-timator Qp.1 for the Acheron River. Lognormal probability paper is con-structed by plotting the logarithms of Qp versus the inverse of the standard

100.0~ Qp,l

~- ~~, OgiveO 10.0 "

~ , ro

Q) 1.0

~+J(/J

0.112 51020305070609095 99

P~Q)q~ X 100

FIG. 3. Period-of-Record FDC (m3/s) Based On Searcy's (1959) Ogive MethodCompared with Estimator Qp,1 for Acheron River

492

normal cdf Zp = <I>-I(p). See Stedinger et al. (1993) for a review of pro-cedures for constructing probability plots.

The estimator Qp.l yields a slightly smoother and more representativeFDC than the piecewise linear empirical cumulative histogram advocatedby Searcy (1959) and others, even for a large sample such as this one. Onecould argue that both curves in Fig. 3 are almost equivalent, but that theestimator Qp.l has the advantage of being easily implemented on a computerand leads to significantly smoother quantile functions than the traditionalogive for small samples.

Fig. 4 uses lognormal probability paper to compare the period-of-recordFDC for the Acheron River using the estimators Qp,l, Qp.2, and Qp.3 basedon the complete 40-year period-of-record 1947-1987 in Fig. 4(a) and basedon only the single year 1987 in Fig. 4(b). Here one observes that the threequantile estimators are almost indistinguishable for the 40-year sample of

(a)100.0

1947-1987--- -, """

10.0 "

q .',-

1.0 Q -"'p,l '\

-Qp,2--Qp.s

0.112 510203050 70809095 99

P~Q)q~ X 100

(b)100

1987

~"-

,.q 10 "

"'"

Q ,p.l -Q p.2 Qp.s

112 510203050 70809095 99

P~Q)q~ x 100

FIG. 4. Comparison of Period-of-Record FDC for Acheron River: (a) Based onComplete Record 1947-1987; and (b) Based on Single Year 1987

493

~

(a)

--Period-of-Record100.0 -.-Mean Annual

-Median Annual

10.0

q '::., 1.0

0.112 510203050 70809095 99

P~Q>q~ X 100

100.0 (b)--Period-of-Record--Mean Annual-Median Annu81

10.0 ~

q

1.0 -,\

0.112 51020305070809095 99

P~Q>q~ X 100

FIG. 5. Comparison of Period-of-Record FDC with Mean and Median Annual FDCs:(a) Moss Brook; and (b) Acheron River

except for exceedance probabilities above about 0.8 (Iow-flows), in whichcase the period-of-record FDC is always significantly lower than either themean or median annual FDC. Similar results were found at other sites inMassachusetts. The significant differences between the period-of-record FDCand either the mean or median annual FDC occurs because the period-of-record FDC is highly sensitive to the hydrologic extremes associated withthe particular period of record chosen, whereas the mean and median annualFDCs are not nearly as sensitive. This effect is explored in Fig. 6.

Fig. 6 compares the median annual FDC at Moss Brook with the period-of-record FDC using two different periods of record. Fig. 6 illustrates howsensitive the lower tail of an FDC can be to the chosen period of record.The period of record 1950-1981 contains the 1960s' drought that was moresevere than any drought experienced over the 1917-1949 period, hence theFDC for these two periods are significantly different. Fig. 6 makes it obvious

495

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',".'-.'-'.~N ","-,...) , ---',"0'-0".,oN""""')

0"'2 5 " 2"050 "'0 '0" "

PjQ)q~ X 100

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why hydrologim ."oft," "I,cr.", '0 ,~ p"iodoC"w,d FDCc 'h,i,im"p"'.'io"d,p,"d,wh,.,ily,po",h",I,cr,dp,,iodofrew,dA"",.1FDQ pro.id, .wl,'io" 10 lhi, di"mm. Th, m,di." ."",,1 FDC "p"~""'h,di",ib,'io"ofd,ily"".mnowi",1ypi"r"m,di."hypo'h"i','y",."di"im,'P""'i""i'"m,f"cr,dby'h'o",~,'io"of.b""m.llyw""dryp"iod,d,ri"g'h,p"iOOofrew,d

CON"OENCE INTERVALS 'OR auAN"LES a,FLOWoURATlaN CURVES

Wh,"0",foc""0".p."iwl.,q'.mi",,p"""'i"of."FDC,h,FDC d~, ""' by i",1f "PO~ 'h, ,oc""i",y .,=i,"d wi,h .p."iwl"q'.mi","im."F",'bi'p,'PO"w"fid'"rei"",...,ho"db,w".rucr,d.00",h"ru,q'.mi"roq'."'ify'h",p'ct,d,"re".imy."w,i."d wi'h ,.im.,i"g ,.,h q'.mi" A"",h" impo".m .d,.m,g' .,~i."dwi,h'h""""li"",p""'io"ofFDC,i,'h.'"o"p.'.m"ri,wofid,oreim,~"'.""'ilyw",",ct,dToo"k"ow"dg,""prored,re, "i. fo, ,omp"'i"g wofid,oc, i""~.. fo, p"iodofrew,d FDCcWh," "", ,"im.", ." FDC ., ." ,"g,g,d ,i'c regi""'1 regre"i"" prored,re"rewm"im"",,di"whi,h,.~F,"",,~y'"dVog"(lm),hoWhow,om,d,"reim,~..""",,ppro,im."dl"'hi,~ctio"W'd,~,i", ,imp" "oop.,.m",i, p,oc,d"", f" ,0",'rucri"g wm,d,"re i""~..,bo,'",hq,.m;"."oc;."dw;,h'h,m,di."""",.IFDC

W,d"i",Q"(;).,.","im."of'h'p'hq',",i"of.",mnowb",d°",h,365d,ily,",.mnoW,;"y",;,,;"go",of,h,,";m.rn,cQ,"Q"",Q""d,~ri"'d,.rl;,c G;""°""",I"i"F;g4{b"w,,,wmm,"d'h,m'ofQ,"h're;hoW"'"Q"""ffirecTh,"yo."of,","mnowd.'.yicld"o"im.re.ofQ,(;)fo';~C "Th,",.I""ofQ"(;)",,",,',du.c~dom-"f~wh;ch°""~,";m.,,,l00(l- .)%oonfido,re,,""" q-1e Qr To -fo fbe upp" ~d -, =.--;...,... "" ~, ,.. "' .., .h= ~.~- = he ~ "' the--Q,"),i w 1,.C..~~p",""ogthe~U~""Q,.,

'~1 -.)% =6do= "f,"" ., ~ u [Q~L),~

Qp(U)], where Qp(L) and Qp(U) denote the lower and upper limits of that

interval computed from

Qp(L) = (1 -9)Qp(i) + eQp(i + 1);

with i = [(n + 1)a/2] and e = (n + 1)a/2 -i (lOa)

Qp(U) = (1 -e)Qp(i) + 9Qp(i + 1);

with i = [(n + 1)(1 -a/2)] and 9 = (n + 1)(1 -a/2) -i (lOb)

In Fig. 7(a) we use (10) to illustrate 90% confidence intervals for eachquantile associated with the median annual FDC (solid line) for the QuahogRiver at West Brimfield, Massachusetts. Here, the solid line Qp is estimated

(a)

-- -,

lOO 0 :-::-= -::-:-:-::~ ----

100 , ,q "

-- -- --10 ---Median Annual FDC--90% Confidence Interval

112 510203050 70809095 99

p~Q>ql X 100

(b)

1 00 0 :-::-:::-::-:c::-:~ ----

100 ' , , q --

10 -

-Median Annual FDC

--90% Confidence Interval

112 51020305070809095 99

p~Q>ql X 100

FIG. 7. 90% Confidence Intervals Associated with Quantiles of Median Annual

FDC for Quabog River Using: (a) Qp,2; and (b) Qp,3

497

using the median of the sample of n values Qp( i) .These confidence intervalswere constructed by setting a = 0.10 in (10) and repeating these compu-tations 365 times to obtain values of Qp(U) and Qp(L) corresponding to p= i/366 for i = 1, ...365. Smoother confidence intervals are obtained byusing the estimator Qp.3 instead of Qp.2' which is assumed in (10). Forexample, Fig. 7(b )illustrates the 90% confidence intervals for each quantileof the median annual FDC at the Quabog River when Qp.3 is used insteadof Q 2.

The nonparametric confidence intervals illustrated in Fig. 7 and describedby (10) have a precise interpretation that corresponds to each individualquantile Qp associated with the median annual FDC. For a particular quan-tile Qp, the confidence intervals Qp(L) and Qp(U) represent the randominterval within which one would expect the true annual median quanti le Qpto fall 100(1 -a)% of the time.

GENERALIZED NONPARAMETRIC HYDROLOGICFREQUENCY ANAL YSIS

Given the annual interpretation of FDCs illustrated in Figs. 5- 7, onemay now think of each individual annual FDC in much the same way onethinks of each annual maximum flood flow or each annual minimum lowflow drawn from a streamflow record of length n years. On~ may envisioneach annual FDC as a continuum bounded by two end points that are theannual maximum and annual minimum daily streamflow for that particularyear. Then, similar to our assumption of independence among the n annualminimum and n annual maximum streamflows, one envisions n independentannual FDCs.

Another advantage of defining and estimating annual-based FDCs is thatit allows us to define an annual probability of exceedance, which we term£, (or probability of nonexceedance, which we term v) associated with eachannual-based FDC 0! quantile function Qp. The definition of an annualprobability of exceedance or nonexceedance associated with each annualFDC allows one to define an annual FDC with a specified average recurrenceinterval T, where T = 1/£ for high flows, and T = l/v for low flows. Forexample, one could easily define and estimate the annual FDC with anexceedance probability of £ = 5%, which would be the hypothetical annualFDC that is exceeded on average, once every T = 1/0.05 = 20 years,assuming independence among years. Unfortunately, one would never ac-tually observe a T-year FDC. Nevertheless, if one wishes to understand thefrequency of daily streamflow during an unusual but hypothetical year, aT-year FDC provides such information. This concept is analogous to theuSe of T-year design hydrographs and hyetographs used in flood studies.One could never observe a T-year hydrograph or a T-year hyetograph.

To formalize this concept, we first recall that Qp(i) is the estimate of the .pth quantile of streamflow based on the 365 daily streamflows in year i,using one of the estimators Qp.l' Qp,2' or Qp.3' described earlier. The nyears of stream flow data yield n estimates of Qp(i) for i = 1, ...n. Nowsuppose we wish to estimate the annual FDC with a prespecified annualprobability of exceedance £, or probability of nonexceedance v. The threenonparametric quantile estimators can be used again to define the £th andvth quantiles of Qp(i), which we term Qp,. and QI""' respectively. We rec-ommend the use of Qp.3 for this purpose, since there are only n values ofQp(i) for each value of p, and Qp.3 leads to much smoother estimates thanany of the alternative quantile function estimators.

498

The annual FDCs described by Qp.£ and Qp." are identical when £ = v= 0.5 or, equivalently, T = 1/£ = l/v = 2 years. In this case both FDCscorrespond to the median annual FDC, which we term Qp.O.5. In generalQp.£ lies above Qp.O.5 and Qp." lies below Qp.O.5.

In Fig. 8 we apply the generalized annual FDCs described to the QuabogRiver, Using the 77-year record of available daily streamflows, we firstestimated 77individual annual quantile functions Qp(i), i = 1, , , , 77, usingthe estimator Qp,2' Next, in Figs. 8(a) and 8(b), the estimators Qp,2 and Qp.3are used, respectively, to estimate the median annual quantile function Qp.O.5(depicted using solid lines) along with the generalized quantile functionsQp,£ for T = 1/£ = 20 and 50 years, and Qp", for T = l/v = 20 and 50years (depicted using dashed and dotted lines), For example, the dashedand dotted lines above each solid line represent the annual FDC that isexceeded on average once every T = 20 and 50 years, respectively.

We illustrate the generalized annual FDCs in Fig, 8 using lognormalprobability paper, although we make no distributional assumptions, Thegeneralized FDCs depicted in Fig, 8(b) are slightly smoother than the cor-

(a)

,~ '.-..'. 0;;..1000 0;,:.."' -.""'..0;;.':.;..:.;.:;.: , 100 ---

q .~-10 -T = 2 Years ::-..~.::-..::- ..--T = 20 Years 1

T = 50 Years1

1 2 5 10 2030 50 7060 90 95 99

PIQ)q~ X 100

(b)

~.1000 ""'....'",.

100q .'..-:- --.~..~. ..

10 T = 2 Years ...,,-:-:,,-:- --.., --T = 20 Years .."..,.

.,., T = 50 Years

1

1 2 5 10 2030 50 7060 90 95 99

PIQ)q~ X 100

FIG. 8, Generalized Annual FDCs for Quabog River Using: (a) Qp,2; and (b) Qp.3

499

responding curves depicted in Fig. 8(a). This is because the estimator Qp.3provides a smoother estimate of the generalized quantile functions than theestimator Qp.2. This effect is particularly apparent in the tails of the gen-eralized quanti le functions. In general, we recommend the use of the esti-mator Qp.3 for constructing generalized quantile functions, as shown in Fig.8('1J).

Since there are only n = 77 annual values of Qp(i) available for estimatingthe annual FDCs Qp.. and Qp.v, the selection of an appropriate quantileestimator is extremely important. In most hydrologic applications for whichthese procedures are appropriate, n is in the range 10 s; n s; 100. Thenonparametric quantile estimators presented are limited to applications inwhich both E and v are greater than l/n, or, equivalently, T is less than n.

FLOW-DURATION CURVES IN FLOOD FLOW AND LOW-FLOWFREQUENCY ANAL YSIS

The nonparametric quantile-estimation procedures provide a generalizedalternative for estimating the magnitude and frequency of the completecontinuum of daily streamflow ranging from the T -year annual minimumlow flow to the T -year annual maximum flood flow. In this section, wedescribe how an annual FDC can be used to estimate extreme design events.Beard (1943) first suggested the use of FDCs in flood frequency analysis.

In a given year, an unbiased estimate of the expected probability ofexceedance associated with the largest observed average daily streamflowq\.I), is p = l/(n + 1) = 1/366 = 0.002732. Similarly, an unbiased estimateof the expected probability of exceedance associated with the smallest ob-served average daily streamflowq(n)isp = n/(n + 1) = 365/366 = 0.99726.Hence, the distribution of annual maximum average daily flood flows isgiven by Q~.. with p = 0.002732, and the distribution of annual minimumaverage dally low flows is given by Qp.v with p = 0.99726. In Fig. 8, weplot, using filled diamonds, the log Pearson type 3 estimators of the T =2-year, 20-year, and 50-year annual maximum flood flows. Similarly, weplot using filled diamonds, the log Pearson type 3 (LP3) estimators of theT = 2-year, 20-year, and 50-year annual minimum low flows. We use thestandard method-of-moment estimators in log space for estimating quantilesof the LP3 distribution as described in "Guidelines" (1982) and in mosthydrology textbooks.

Fig. 9 uses lognormal probability paper to illustrate the goodness of fitof a LP3 distribution to the annual maximum flood flows and annual min-imum low flows of the Quabog River. The LP3 distribution provides a goodapproximation to the distribution of annual minimum low flows and only afair approximation to the distribution of annual maximum flood flows atthis site. This is why there is good agreement in Fig. 8 between the non-parametric and parametric LP3 procedures for the annual minimum lowflows, and poor agreement between the nonparametric and parametric LP3procedures for the annual maximum flood flows. Since Vogel and Kroll(1989) showed that the LP3 distribution provides an excellent approximationto the distribution of annual minimum 7-day low flows at 23 sites in Mas-sachusetts, the agreement between parametric and nonparametric proce-dures for the annual minimum low flows is not surprising. The smoother~ene;ralized nonp:arametric quantile fun~tions based on !he estimator Qp.3ID FIg. 8(b) provIde better agreement with the parametnc LP3 proceduresthan the generalized quanti le functions based on the estimator Qp.2 in Fig.8(a). This is to be expected since the estimator Qp,3 uses all the observations

500

..~ =77 years ...

1000Annual Ma;ximum Flows /' q 100 ~ Annual ¥in1mum Flows

10 'L '~-- .Observations

-Fitted Log Pearson type m1

12 510203050 70809095 99

P~Q)q~ X 100

FIG. 9. LPJ Distribution Fit to Annual Maximum and Annual Minimum Stream-flows at Quabog River

to obtain each value of Qp.. and Qp"" unlike Qp,2, which only uses twoadjacent observations.

The comparison in Fig. 8 does not validate either procedure; however ,it does document the correspondence between the classical parametric an-nual maximum flood-flow and annual minimum low-flow computations, andthe generalized nonparametric procedures recommended. Future researchis required to ascertain the efficiency of the nonparametric procedures rel-ative to the parametric alternativ:es. Even if the nonparametric proceduresare less efficient, which they are likely to be, the nonparametric proceduresprovide a wealth of information regarding the frequency and magnitude ofstreamflow, in excess of the traditional parametric low-flow and flood-flowprocedures.

CONCLUSIONS

The present study introduces a variety of nonparametric quantile-esti-mation procedures useful for estimating and interpreting FDCs. The tra-ditional period-of-record FDC can be interpreted as representing the mag-nitude and frequency of daily streamflow during the period of record or, inthe limit, over some long period of time. Experiments indicate that thelower tail of such FDCs are highly sensitive to the particular period of recordused. This fact led us to consider the alternative of representing and inter-preting FDCs on an annual basis. We introduced the median annual FDCthat represents the exceedance probability of daily streamflow in a median,or typical, but hypothetical year. The median annual FDC is not influencedby ~he occurrence of extreme low-flow periods or extreme fl,?ods over t~eperIod of record, yet it still captures the frequency and magnItude of dally~treamflow in a typical year. The use of annual FDCs ~Iso allo,!,ed us.toIntroduce simple nonparametric procedures for computing confidence 10-tervals and for estimating a hypothet.ical T -year FD.C:., Annual FDCs provide an alternative to the traditional approach of es-

tImating a period-of-record FDC. A period-of-recor.d FDC. repres~n~s theexceedance probability of streamflow over a long perIod of tIme. This mter-pretation can be quite useful, as long as the period of record used to construct

501

the FDC is long enough to provide the "limiting" distribution of streamflowor if the period of record corresponds to a particular planning period ordesign life. As an alternative, the median annual FDC represents the ex-ceedance probability of streamflow in a typical year .

Engineers often wish to estimate quantiles of daily streamflow for use inhydrologic design and planning. Such studies typically define a design eventusing the concept of average recurrence intervals. For example, storm sewersmay be designed for the SO-year peak annual flood flow and waste-Ioadallocations may be based upon the 7-day 10-year low-flow event. The presentstudy shows how FDCs can be constructed so as to provide a generalizeddescription of hydrologic frequency analysis using average recurrence in-tervals. In Fig. 8, generalized FDCs were constructed to illustrate how onecould estimate a hypothetical FDC with a specified annual probability ofexceedance £ or nonexceedance v ( or corresponding average return periodT = 1/£ or T = l/v). Such FDCs provide a description of the frequencyand magnitude of the entire continuum of daily streamflows ranging fromthe T -year annual maximum flood flow to the T -year annual minimum lowflow.

The annual FDC procedures introduced are appealing because they gen-eralize both fiood-flow and low-flow events in addition to all events inbetween. Yet FDCs are characterized by complex shapes requiring muchmore complex probability distributions than for the usual fiood-flow andlow-flow applications. The procedures developed are entirely nonparamet-ric, unlike the parametric procedures usually recommended in fiood-flowand low-flow frequency analysis. Nonparametric quantile-estimation pro-cedures are chosen to simplify the analysis and to alleviate the need for"goodness-of-fit" procedures that typically have very low power [see Vogeland McMartin (1991)]. Nevertheless, studies should be undertaken to de-termine the efficiency of the proposed nonparametric quantile-estimationprocedures relative to the usually recommended parametric procedures,such as the LP3 distribution, recommended by the Interagency AdvisoryCommittee on Water Data ("Guidelines" 1982).

ACKNOWLEDGMENTS

The writers gratefully acknowledge the financial support of the U.S.Geological Survey who sponsored this research under the terms of contractnumber 14-08-0001-G2071 titled "The Application of Streamflow DurationCurves in Water Resources Engineering." In addition, the writers acknowl-edge the partial financial support of the U .S. Environmental ProtectionAgency Assistance Agreement CR820301 to the Tufts University Centerfor Environmental Management. The writers are also indebted to Wilbert0. Thomas, Gary Tasker, William Kirby and Thomas A. McMahon fortheir comments on an early version of this manuscript.

The views and conclusions contained in this document are those of thewriters and should not be interpreted as necessarily representing the officialpolicies, either expressed or implied of the U .S. Government.

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