The 3в•'Parameter Lognormal Distribution and ... - Wiley Online Library

VOL.

6, NO.

2

WATER

RESOURCES

RESEARCH

APRIL

1970

The3-Parameter Lognormal Distribution andIts Applications in Hydrology B.

P.

SANGAL

AND

ASIT

K.

BISWAS

Inland Wat,ers Branch and Policy and Planning Branch Department of Energy, Mines and Resources,Ottawa, Canada Abstract. The 3-parameter lognormal distribution is a general skew distribution in which the logarithm of any linear function of a given variable is normally distributed.The distribution is applied to the frequency analysisof floods,annual fio.ws,and monthly flows, and a comparisonwith other commonlyused methodssuggeststhat it can be successfullyused for this purpose.A procedurefor its applicationhas been suggested.usingonly the median, the mean, and the standard deviation of the original data. The Gumbel distribution is a special case,and any straight line on the Gumbel probability paper can be transformedinto a straight line on the lognormalprobabilitypaperby the 3-p•rameter lognormMdistribution.The sample of 10 stations used in this study all exhibited negative skewnessfor the logarithms of the data and therefore the lognormal distribution that assumesthis skewnessequal to zero has generally given high estimates.

INTRODUCTION

THEORY

The 3-parameter lognormal distribution representsthe normal distribution of log (x -- a) where the logarithm is to the base e. The variable x is the randomvariable, a is a parameter, flows is much more skewed than the distribuand (x -- a) is the reduced variable. This is tion of annual flows. Sincemany of the statisti- a general distribution in which the logarithm of cal conclusions are based on the normal distriany linear function of x is normally distributed. bution, it has often been found advantageous The parameter a has to be evaluated in terms

Few hydrologic populations can be representedby the normal distribution,and the degree of skewnessdependsupon the type of the data. For example,the distribution of daily

of the statistical

to transform a skew distribution into the normal

measures of the variable

x. It

one. Theoretically it is always possibleto de- can be positive, zero, or negative. The significance of these signs will be discussedherein. termine a function that would yield such a Matalas [1963] derived an expressionfor detransformation[Hald, 1962], eventhoughsome termining the parameter a which involved the transformationsmay be quite involved. One of the most important and useful of such coefficient of skewness of the variable x. If the data are limited, as is the usual case in hytransformations is the logarithmic transformation. It has been observed that whereas the drology, the computation of the coefficientof skewnessmay be subject to error. Its compuoriginal data show considerableskewness,their when the samlogarithmsare nearly normally distributed,and tation is, therefore,discouraged hence the original data are said to follow the ple size is less than 100 events [Interagency lognormal distribution. The distribution has Committee,1966]. Although Matalas and Benbeen extensivelyused in many areas of science son [19.68] attempted to show that the skew and engineering.Chow [1954] made a compre- coefficient,when computed even from smaller hensivestudy of this distributionand theoreti- samples,may be reliable, the parameter a can cMly derived its parameters.The objectiveof be determined in the following manner without this paper is to present the theory and some using this coefficient. mathematical characteristicsof the more geneFor the random variable x, the mean/x• and ral 3-parameterlognormaldistributionand to variance a•' are related to the parameter a, show someof its applicationsin hydrology. and to the mean tx• and variance • of the 5O5

506

SANGAL AND BISWAS

transformedvariable y, equal to log (x - a) which is normally distributed [Matalas, 1967]. Thus

g• = a q- e('•'/2+"•)

data can be analyzedwith this assumption(appendix). From equationsI and 2

(1)

and

(7) and

•

=

-

ß

(2)

0'z2 =

'0¾ 2

(S)

Let • representthe median of y. Sincey is By substitutingthe value of e• from equation normally distributed,•y -- •y. If •. denotes 3 and by eliminating %'"from equations7 and the median of x, then (•. -- a) denotesthe 8, we get median of (x -- a). Therefore • -- log (•

2

-- a). Thus

e•'=

•--

a

By usingthis value of e•' and by eliminating

e•"

from equationsi and 2, we get

2a•(•-

a-- •'x-- 2(g• -- •'•)

(3)

This equation can be conveniently used in determiningthe parameter a. Its dimensionless form is

L)

2

Y--f•) er= f•-- 2(1

+ a•(•.• + rf - 5• • +

+ 2a(2• • - r•f+ rf•f

(9)

- •

r•f-

+ •rf

•L •)

= 0

which can be used in regional studies.

(4)

This is the basic equation that relates the

parametera to the meang•, the median•, and the standard deviation ,•

of the variable x.

This is a cubic equation in terms of a. It must have one real and two imaginary roots that would be proved later. If a -- alga, • -- ••, and • -- ,•/g•, then equation 4 is transformedto

CI-IARACTERISTICS

TI-IE

DISTRIBUTION

The characteristicsof the 3-parameter lognormal distribution are analogousto those of the lognormal distribution. In the former case the distribution applies to the reducedvariable (x -- a), whereasin the latter caseit appliesto the variable x. For the lognormal distribution Chow [1954] derived the following basic relation between the coefficient of variation

(5)

This is in dimensionlessform. In the special caseof the logoreal distribution when a -- 0 and thus a -- 0, the equation gives

(•)

and the

,

g• = 3v•q- v•a

+ 2.(2 - ½,• - ½ - ½•)

½ = 1/(• + ,•)'/•

OF

coefficient of skewness:

2d(• - ½)+ d(, • + f - 5 + 4½)

+•%a -- 1 +•a = 0

(10)

(11)

where g• and v• are the coefficientof skewness and the coefficient of variation

of x. For the 3-

parameter lognormal distribution the relation becomes 3

g(z-a) -- 3V(z-a)-[-V(x-a)

(12)

where g(x-a) and V(x-a) are the coefficient of Thus if x is lognormallydistributed,equation6 skewness and the coefficient of variation of the a). has to be satisfied. This is a stringent condi- reduced variable (xThe coefficient of skewness and the standard tion, andl field data, especiallyhydrologicdata, deviation of the (x -- a) seriesare the same as will rarely satisfythis conditionexactly. If ,• is assumedto be small, an approximate those of the x seriessince the parameter a has but simple expressionfor a can be obtained no effect on these measures. But the mean of by neglectinghigher powers of ,•. This as- the (x -- a) series differs from that of the x •mption is not unrealisticand mosthydrologic seriesby an amount equal to a. Thereforefor.

3-Parameter Lognorma! 3-parameter lognormal distribution

gx- 3V(x-a)-•-•)(x-a) s

(13)

It gives a cubic equation in

507

a reductionin the mean of the x series,a would be positivein this case.The case(b) represents a condition of lognormal distribution and a would be equal to zero. In case (c) the effect of a would be to make the coefficient of varia-

3

v(x-a) q- 3v(x_a)-- g• = 0

(14)

Equation 14 has three roots. According to the Descartes rule of signsit has only one positive root. The other two roots are either negative or imaginary. Since the equation doesnot have a v(x_a) 2 term it cannot have negative roots and

therefore

the

other

two

roots must

be

imaginary. This is also evident from the fact that v(•_•) must be positive. Since there is only one positive value of v(•_• to satisfy equation 14, there can be only one real value of a, which proves the earlier statement that a can have only one real value. In any given seriesof x, three casesare possible

tion of the (x -- a) serieslessthan that of the x series.Since it would require an increasein the mean of the x series,a would be negative in this case.Thus the sign of a can be predetermined from the plot of the original data on the lognormal probability paper. If the plot is curved upward, a would be positive. If the plot is a straight line, a would be zero. If the plot is curved downward,a would be negative. In the caseof lognormaldistribution the three measuresof locationare given by [Hald, 1962] Mode

log x = log •j -- 2.3026 S,og•,

(18)

log x = log •j

(19)

logx = log// q- 1.1513Slog/'

(20)

Median

(a) g• > 3v•q-v, 3

(15)

(b) g• = 3v•q- v•3

(16)

(c) g• < 3v•q-v•s

(17)

Mean

and

In case (a) the serieswould plot curved upward on the lognormalprobability paper. The coefficientsof skewnessof the x and the log x serieswould both be positive. In case (b) the series would plot a straight line on the lognormal probability paper. The coefficientof skewnessof the x serieswould be positive, and the coefficient of skewnessof the log x series would be zero. This is the required condition of the lognormal distribution. In case (c) the serieswould plot curved downward on the lognorma] probability paper and the coefficientof skewnessof the log x serieswould be negative, but the coefficient

For the 3-parameter lognormal distribution these equationsbecome Mode

(x - a) = log•j -- 2.3026 Median

log (x -- a) = log }

(22)

Mean

log (x -- a)

of skewness of the x series

can either be positive, zero, or negative. However, only the serieswith positive skewnesscan be treated by the 3-parameter lognormaldistribution. The caseof g• -- 0 with v• -- 0 is unimportant, sincethe variable x is constant. For these three cases of positive skewness, the role played by the parameter a will now be examined.In case(a) the role wouldbe to make the coefficientof variation of the (x -- a) series greater than that of the x series so that equation 13 is satisfied.Since it would require

(21)

log// q- 1.1513 Slog In these equationsthe logarithms are to the base 10. ThSni [1969] has given a more accurate method of estimatingthe mean of a lognormal distribution. However, if the sample size is 30 or more, equation 20 is sufficiently accurate. ESTIMATION

OF

PARAMETERS

The parameters used in the present study have beenestimatedby the followingformulas'

508

SANGAL

AND

BISWAS

areas in squaremiles, and the number of years of record. The original streamflow data are available on request to the authors.

Mean

(x)= Z x

(24)

The

where n is the number of items in the series.

Standard deviation

cient

ß Coefficient

(2s)

of skewness

z

- (x>)

g•= (n- 1)(n-2)s• a

the median

measures that

have

of skewness. These measures were com-

puted for the original data as well as for their logarithms to the base 10. For the 3-parameter lognormal distribution a was computed from equation 9. The median was computed as the mean of the middle

l•rge, the determinationof the medianis simple. However, with hydrologicd•ta • b•sic difficulty is the s•mple size since this is usually sm•11. For • sm•11s•mple the •rithmetic interpolation would not yield reliable results; therefore the graphical interpolation is necessary.The dat• lying between30% and 70% probabilitiesm•y be plotted corresponding to their probabilities on •rithmetic paper to • l•rge scale, and the median m•y be estimated from • smooth curve through these points. Generally the curve between 40% •nd 60% probabilitiesis almost a straightline. If it is •ssumedtMt the frequency is uniformly distributedover this interval, then the arithmetic mean of the d•ta lying between this interwl would give • fairly accuratev•lue Therefore

statistical

fifth

of the data. For

re-

(26) gional considerationsthe ratio of the median to

The median represents the value of the v•ri•ble •t 50% probability. If the s•mple is

of the median.

different

been usedin the analysisof the data by various methods include the mean, the standard deviation, the coefficientof variation, and the coeffi-

can be

the mean (M) and the ratio of the parameter a to the mean (R) were also computed.Having computed a, the different measures were computed for the log (x -- a) series.All these results are listed in Table 2. The data analyzed include the annual flood flows of all the 10 sta-

tions, and the annual flows and the April flows of station 2GD-1 only. The last two serieswere includedto demonstratethe applicability of the 3-parameter lognormal distribution to the annual flows and to the monthly flows. The analysis of April flows is presentedas an example TABLE

Water Survey of Canada Station No.

8MF-5 2FC-1

to avoid the high floods,the so-called0utliers. Table 1 givesthe 10 test stations,their Water Su•ey of Canad•inventorynumbers,drainage

Stations

Drainage Area, mi 2

Fraser River at Hope,

Years of Record

83,700

57

1,570

52

Columbia

SaugeenRiver near Port Elgin, Ontario

2FC-2

Saugeen River near Walkerton, Ontario

850

52

2GD-1

Thames

519

51

82

51

River

near

Ealing, Ontario

E•PLES

The usefulnessof any theoretical method is dependenton its practicaluse.To test the •pplicability of the 3-parameter logno•l distribution to flow frequency•nalysis,it was decidedto compareit with severalother methods that have been in commonuse. A group of 10 long-term stations was selected where flows were not •ffected by artificial re•l•tion. All data were considered,and no effort was made

Ten Test

Location

British

computed •s equ•l to the mean of the middle

fifth of the d•ta, and unlessthe d•t• showappreciable curvature in the 40% to 60% interval this method of median computation c•n be •dopted for the s•ke of uniformity.

1.

2HB-1

Credit

River

Cataract,

2HL-1

near

Ontario

Moira River near

1,040

51

524

46

Foxboro, Ontario 2CE-2

Aux Sables River

near

Massey, Ontario

4L J-1

Missinaibi River near Mattice,

5PB-14

3,450

46

1,880

46

5,240

45

Ontario

Turtle River near Mine Centre, Ontario

5QA-1

English River near Sioux Lookout, Ontario

509

3-Parameter Lognormal TABLE

Standard

Station

No.

Series

Mean

Deviation

2.

Statistical

Coefiqcient of Variation

Measures

Coefiqcient of Skewness

Median

with

Ratio to Mean (M)

311,000 (0.980)

Parameter

a with

Ratio to Mean (R)

--6,460

(--.020)

8MF-5

x log x log (x -- a)

317,000 5.49310 5.50219

62,700 0.08558 0.08378

0.198 0.01558 0.01523

0.685 --0.19541 --0.19859

2FC-1

x log x log (x -- a)

17,160 4.20208 4.82617

6,410 0.17521 0.04106

0.374 0.04170

0.00851

0.387 -0.49986 --0.08171

2FC-2

x log x log (x -- a)

10,060 3.96274 4.02177

4,500 0.18860 0.16508

0.447 0.04759 0.04105

1.101 --0.00478 0.13454

9,090 (0.903)

--1,230 (--.123)

2GD-1

x log x log (x -- a)

6,620 3.76091 4.01654

3,610 0.23892 0.13083

0.545 0.06353 0.03257

1.781 --0.52635 0.37617

5,980

--4,260

2HB-1

x

2HL-1

827

441

log x log (x -- a)

2.85784 3.01873

0.23925 0.16163

0.08372 0.05354

--0.56127 0.12390

x log x

7,280 3.83898 4.51166

2,220 0.14932 0.02986

0.306 0.03890

--0.009 --0.87706

0.00662

--1.12765

3,770

1,320

log (x -- a) 2CE-2

x

log x log (x -- a) 4L J-1

x

3.54861 3.86173 31,690

0.16317 0.07674 9,350

0.534

0.349

0.04598 0.01987 0.295

1.318

0.555

16,850

(0.982)

(0.904)

732 (0.885)

--50,150

(--.644)

--292 (--.353)

__

7,200 (0.990)

(--2.923)

--

--25,280

__

(--3.475)

--

3,650 (0.968)

--3,620

31,370 (0.990)

--105,700

(--.959)

--0.69850 --0.03036 0.006

(--3.336)

log x log (x -- a)

4.48009 5.13701

0.14118 0.02977

0.03151 0.00579

--9.71110 --1.45705

x log x log (x -- a)

4,400 3.59122 3.78939

2,150 0.21907 0.14024

0.489 0.06100

4,020 (0.914)

0.03701

0.811 --0.15953 0.15723

5QA-I

x log x log (x -- a)

10,320 3.96336 3.99697

5,140 0.21457 0.19822

0.498 0.05414 0.04959

1.159 --0.17820 --0.06861

8,950 (0.867)

--661 (--.064)

2GD-1

x*

460 (0.970)

--230 (--.485)

897 (.858)

--307 (--.294)

5PB-14

474

139

log x log (x -- a)

2.65753 2.83961

0.12940 0.08493

0.04869 0.02991

--0 20293 0.05552

x]' log x log (x -- a)

1,045 2.94973 3.09247

597 0.25501 0.18378

0.571 0.08645 0.05943

1.052 --0.26476 0.14828

x*

Annual

xt

April flow in cfs.

0.293

0.495

__

__

-2,090

(--.474)

flow in els.

All other x's are flood flows in cfs.

meter lognormal distribution, are now in common use, and the proceduresfor their applicaThe followingfive methodsof frequencyanal- tions have been fairly standardized.However, ysiswere appliedto the floodseries: (1) Gum- a brief descriptionof the proceduresusedin this bel distribution; (2) Pearsontype 3 distribu- study is given below. The parameters of the Gumbel distribution tion; (3) log-Pearson type 3 distribution;(4) lognormal distribution; and (5) 3-parameter were obtainedby the method of maximum likelognormaldistribution.To the annualseriesand lihood [Panchang and Aggarwal, 1962; Panthe monthly seriesthe followingthree methods chang, 1967]. In the Pearson type 3 distribuwere applied: normal distribution,lognormal tion, the skewnesswas computedby equation distribution,and 3-parameterlognormaldistri- 26 and no adjustment for the length of record bution. All these methods, except the 3-para- was made. The Pearsontype 3 coordinateswere

although the method is applicable to other

months also.

510

S•,NG•,•, AND BISWAS

computedusingthe approximateformula given by Beard [1965]

where K is the Pearson type 3 coordinate; g• is the coefficientof skewnessof x; and T is the standard normal

deviate.

assumedmedian being equal to 99% of the mean. Plots for both these stationswere very nearly straight lines on the normal probability paper.This methodhelpedin fitting the 3-parameter lognormaldistribution without causing any significantchange in the distribution.

For the normal distributionthe discharges were obtained

from

The log-Pearsontype 3 distributionis of speß= + (30) cial significancesince it has been recommended where (x) and s• are the mean and the standard as a basemethod for analyzingflood flow frequenciesfor the U.S. federal agencies[Benson, deviation of the x series. Table 3 givesthe estimatedmode,median, 1968]. Bensonhas discussed its application.The suggestedprocedurewas used, except that the Pearsontype 3 coordinateswere computedby TABLE 3. Measures of Location Estimated from Various Distributions

equation 27.

For the lognormaldistribution,the logarithms of the dischargesat the selectedprobabilities were obtained

from

log X --- (log X) + T,•log x

Station

Distribu-

No.

tion

8MF-5

G LN 3LN G LN 3LN G LN 3LN G LN 3LN

(28)

where (log x) and S•o•• are the mean and the 2FC-1 standard deviation of the logarithmic series. The procedurefor the 3-parameterlognormal 2FC-2 distribution was similar to the one used for the

lognormal distribution. In this method the lo-

garithmsof the dischargesare given by

log (x -- a)

= (log (x -- a)) -• Ts log(x--a,

2GD-1

288,000 308,000 320,000 299,000 311,000 317,000 300,000 311,000 317,000 14,100 16,130 17,300 13,530 15,930 17,280 16,270 16,860 17,160 8,080 9,300 10,000 6,840 8,780 9,940 7,870 9,280 10,070 5,090 6,050 6,600 4,260 5,770 6,710 5,230 6,130 6,610 635 532 617

754 721 752

822 839 827

2HL-1

G LN 3LN G LN 3LN G LN 3LN G LN 3LN G LN 3LN

6,170 6,130 7,050 3,150 3,070 3,440 27,060 27,180 30,730 3,420 3,030 3,460 8,070 7,200 7,400

6,950 6,900 7,200 3,570 3,540 3,660 30,280 30,210 31,380 4,020 3,900 4,070 9,450 9,190 9,270

7,400 7,320 7,280 3,820 3,800 3,770 32,150 31,850 31,700 4,370 4,430 4,400 10,250 10,380 10,360

2CE-2 Equation 9, used to determine a, is sensitive to the conditionof the medianbeing very close 4LJ-1 to the mean. In this situation one of the follow-

ing two methodsmay be used' 5PB- 14

be a straight line. (2) Assume a value of the median close to

5QA-1

but lessthan the mean,say 99% of the mean,

2GD-i*

and compute a. Two examplesof this method are the analyses of flood flows of stations

LN 3LN

435 435

474 461

2GD-i$

LN 3LN

647 727

897 930

2HL-1

and 4LJ-1.

Both these basins are af-

fected by appreciablenatural lake storage.In both casesthe computedmedian exceededthe mean. For a positively skewed distribution, the

median

should be less than

the

mean.

Therefore the analyses were made with the

Mean

G LN 3LN

series.

(1) Plot the data on the normalprobability paper. In mostcasesthis plot wouldvery nearly

Median

2HB-1

(29)

where (log (x -- a)) and Slog(•-a) are the mean and the standarddeviationof the log (x -- a)

Mode

*

Annual

•

April flow in cfs.

flow in cfs.

All other data are flood flows.

G, LN, 3LN,

Gumbel; lognormal; 3-parameter lognormal.

495 475

1,060 1,050

3-Parameter Lognormal

and mean from the Gumbel, the lognormal,and the 3-parameter lognormal distributionsfor the flood flows. It

also shows these measures for

the annual flows and for the April flows of station 2GD-1 correspondingto the last two distributions.In the Gumbel distribution,the mode, the median, and the mean occur at 1.58 year, 2 year, and 2.33 year recurrenceintervals, respectively.In the lognormaland the 3-parameter lognormal distributions these measures have beencomputedfrom equations18 to 23. Table 4 lists the resultsof the computations of flow magnitudesat the selectedprobabilities of 20%, 10%, 2%, 1%, 0.1%, and 0.01%, which for annual data respectivelycorrespondto 5, 10, 50, 100, 1000, and 10,000-yearrecurrenceintervals. Computationsat small probabilitiesare chosen to illustrate the basic differences in vari-

ousmethodswhich do not becomequite evident at higher probabilities. Figure 1 is a probability plot of flood flows of station 2GD-1 on lognormalprobability paper. The probability for each item was computed as m/(n + 1), where m is the order number of the descendingseriesand n is the total number of events.The transformedline represents the distribution

of the reduced variable

(x -- a), which must be a straight line. The fitted curve through the plotted points represents the curve obtained from the transformed

line by algebraicallyadding a to the values of the transformedline at someselectedprobabilities. These points are denotedby crosses.This curve representsthe distribution of the variable x. The position of the 1937 flood is of interest sinceits cumulativeprobability is 0.1%; that is, it is a 100.0-yearflood. Figures 2 and 3 show the analysesof the annual flows and the April flows of the same station.The methodof fitting the curvethrough the plotted points was the same as that used for the flood flows.

511

of the logarithms equal to zero, generally gave high flow estimates.For the same reason, no attempt was made to evaluate the log-Gumbel distribution

that assumes the skewness of 1.139

for the logarithms of the data. An examination of the results presentedby Benson [1968] also suggestedthat the log-Gumbelestimatesare unrealistic and should be avoided.

There is no standard method of comparing the merits of one distribution

with another. In

the central part of the frequency curve, where plotting positionsare more reliable, all distributions have given almostthe sameresults,which are closeto the data values. However, at small probabilitieswhere no basis for comparisonis available, the divergenceis quite substantial. Generally at these probabilitiesthe 3-parameter lognormaldistribution has occupiedthe middle position. In the analysesof the annual and April flows of station 2GD-1, the lognormal distribution has given the highestestimatesfollowedby the 3-parameter lognormaldistribution.The normal distribution, as expected,gave the lowest estimates.

In all casesgoodresultswere obtainedby the 3-parameter lognormal distribution. Table 3 shows that the theoretical

median and the mean

as obtainedfrom this distributionare quite close to the median and the mean as computedfrom the original data. It proves the validity of the method.

The 3-parameter lognormal distribution is a general skew distribution. The Gumbel distribution, which has a fixed skewness,is a special case, and any straight line on the Gumbel probability paper can be transformed into a straight line on the lognormalprobability paper by the 3-parameter lognormal distribution. It may be noted that this flexibility is not available with the lognormal distribution. The Gumbel

andthe lognormaldistributionsare identicalonly

when the coefficientof variation of the original RESULTS AND DISCUSSION data is 0.364. This is a very limiting condition Sincethe streamflowrecordsusedin the pres- and would rarely be encountered in practice. ent study are of reasonablelength, there is a However, with the 3-parameter lognormal disgood possibility that the parameters of the tribution, the coefficientof variation of the reduced variable (x -- a) is automatically advarious distributionshave been computedwith justed to 0.364, irrespectiveof the coefficientof somedegreeof accuracy. In all casesthe skewness of the logarithmsof variation of the original data. the data was negative.That is why the logThe Pearsontype 3 distributionsand the lognormal distribution, which assumesthe skewness Pearsontype 3 distributionsrequire the compu-

512

SANGAL

TABLE

4.

AND

BISWAS

Computed Flows for SelectedProbabilitiesby Various Distributions Cumulative Probability, %

Station

No.

Distribution

20

10

2

1

0.1

0.01

8MF-5

G P LP LN 3LN

372,000 367,000 368,000 367,000 367,000

414,000 401,000 399,000 401,000 401,000

506,000 468,000 457,000 467,000 466,000

546,000 494,000 478,000 492,000 491,000

675,000 574,000 542,000 572,000 570,000

804,000 649,000 598,000 648,000 645,000

2FC-1

G P LP LN 3LN

22,390 22,400 22,490 22,370 22,420

26,530 25,600 26,010 26,710 25,500

35,660 31,630 29,980 36,470 31,230

39,510 33,880 35,090 40,700 33,350

52,260 40,600 42,080 55,400 39,600

64,980 46,570 47,820

2FC-2

G P LP LN 3LN

13,070 13,400 13,230 13,230 13,250

15,560 16,060 16,010 16,020 15,880

21,050 21,690 22,370 22,390 21,720

23,380 23,980 25,160 25,200 24,220

31,050 31,730 35,010 35,120 32,800

38,710 38,510 45,960 46,170 42,000

2GD-1

G P LP LN 3LN

8,990 8,920 9,240 9,160 9,130

10,950 11,320 11,230 11,670 11,020

15,240 16,850 15,220 17,850 15,020

17,060 19,240 16,750 20,730 16,670

23,060 27,370 21,290 31,560 22,090

29,050 35,780 25,160 44,640 27,560

2HB-1

G P LP LN 3LN

1,120 1,140 1,160 1,150 1,140

1,360 1,410 1,400 1,460 1,390

1,900 2,010 •,890 2,240 1,950

2,120 2,250 2,070 2,600 2,190

2,870 3,060 2,600 3,960 3,010

3,610 3,860 3,040 5,600 3,880

2I-IL-1

G P LP LN

3LN

9,350 9,150 9,250 9,220 9,140

10,940 10,120 10,250 10,730 10,190

14,440 11,830 11,840 13,990 12,130

15,920 12,430 12,330 15,360 12,830

20,810 14,110 13,480 19,970 14,890

25,690 15,500 14,170 24,800 16,670

2CE-2

G P LP LN 3LN

4,880 4,830 4,880 4,850 4,830

5,740 5,510 5,520 5,730 5,510

7,640 6,850 6,620 7,650 6,840

8,450 7,360 6,990 8,480 7,360

11,100 8,900 7,960 11,290 8,940

13,750 10,320 8,640 14,310 10,420

4L J-1

G P LP LN 3LN

40,240 39,560 39,890 39,720 39,520

46,840 43,680 44,340 45,820 43,970

61,350 50,930 51,830 58,890 52,100

67,490 53,480 54,310 69,340 55,070

87,760 60,670 60,630 82,480 63,710

108,000 66,600 64,990 101,200 71,190

5PB-14

G P LP LN 3LN

5,900 6,070 5,990 5,970 6,000

7,140 7,270 7,380 7,450 7,230

9,880 9,690 10,530 11,000 9,870-

11,040 10,640 11,890 12,610 10,960

14,860 13,630 16,560 18,540 14,620

18,670 16,450 21,520 25,480

G P LP LN 3LN

13,720 14,090 13,980 13,930 13,920

16,550 17,180 17,140 17,320 17,170

22,770 23,740 24,170 25,360 24,700

25,400 26,420 27,180 29,000 28,050

34,090 35,100 37,380 42,300 40,030

42,770 43,600 48,040 57,750 53,590

5QA4

71,430 45,110

18,380

513

S-Parameter Lognormal TABLE 4.

(Continued)

Cumulative Probability, % Station

No.

Distribution

2GD-I*

2O

NORMAL LN 3LN

2GD-1 t

591 584 585

NORMAL LN 3LN

*

Annual

t

April flow in cfs.

10

2

652

760

666 658 1,810 1,890 1,820

1,550 1,460 1,460

I

0.1

798

838 803 2,270 2,980 2,640

909 859 2,430 3,490 3,000

0.01

904

991

1,140 1,040 2,890 5,470 4,270

1,380 1,200 3,270 7,910 5,670

flow in cfs.

All other data are floods in cfs.

G, P, LP, LN, 3LN,

Gumbel; Pearsontype 3; log-Pearsontype 3; lognormal; 3-parameter lognormal.

tation of the coefiqcient of skewness. When com-

applicability of the method is provided by the straight line plot of the reducedvariable (x -a) on the lognormalprobability paper.

puted from limited data this measure,whether of the original data or of their logarithms,may not be reliable.

CONC•LUSIONS

Thus it is suggestedthat the 3-parameter

1. The 3-parameter lognormal distribution is a general skew distribution and can be suc-

lognormaldistribution affords a mean of adequately analyzing the hydrologicdata. Its applicationis simple.It can be easilyappliedwith

cessfullyused for analyzinghydrologicdata. 2. The different methodsof flow frequency analysisyield differentresults,especiallyat low probabilities.

or without the use of computers. The method

is directly applicableif the skewhess can be reliably computed.An immediate check on the 99

98

95

90

80

70

60

50

40

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40[ [ • • [ I I I I I I I I • [ I • TRANSFORMED LINE,•,,.,.•.,,,• •,.x •j' •'FITTED CURVE

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51YEARS OFRECORD

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CUMULATIVE PROBABILITY, PERCENT I0

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9• 9•

Fig. 1. Distribution of flood flows of Thames River near Ealing, station 2GD-1.

514

SANGAL AND BISWAS 98

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