flood frequency analysis: assumptions and alternatives...distribution family (ahmad et al., 1988)....

Progress in Physical Geography 29 3 (2005) pp 392ndash410

copy 2005 Edward Arnold (Publishers) Ltd 1011910309133305pp454ra

Author for correspondence Current address Corporate Licensing Unit Metropolitan Water SharingDirectorate Office of the Metropolitan Region NSW Department of Infrastructure Planning and NaturalResources (DIPNR) PO Box 651 Level 1 308 High Street Penrith NSW 2751 Australia

I IntroductionFlooding is one of the most pervasive naturalhazards to impact negatively upon theactivities of human beings and requiresvarious responses including construction

(downstream flood defences) forecasting(for warning and evacuation) and land-usemanagement (upstream catchment-scalechanges of land use and runoff characteris-tics) To some degree all of these require

Flood frequency analysis assumptions and alternatives

R Kidson1 and KS Richards2

1Trinity College Cambridge CB2 1TQ UK2Department of Geography University of Cambridge Downing PlaceCambridge CB2 3EN UK

Abstract Flood frequency analysis (FFA) is a form of risk analysis yet a risk analysis of theactivity of FFA itself is rarely undertaken The recent literature of FFA has been characterized by (1) a proliferation of mathematical models lacking theoretical hydrologic justification but usedto extrapolate the return periods of floods beyond the gauged record (2) official mandating ofparticular models which has resulted in (3) research focused on increasingly reductionist andstatistically sophisticated procedures for parameter fitting to these models from the limited gaugeddata These trends have evolved to such a refined state that FFA may be approaching the lsquolimitsof splittingrsquo at the very least the emphasis was shifted early in the history of FFA from predictingand explaining extreme flood events to the more soluble issue of fitting distributions to the bulk ofthe data However recent evidence indicates that the very modelling basis itself may be ripe forrevision Self-similar (power law) models are not only analytically simpler than conventionalmodels but they also offer a plausible theoretical basis in complexity theory Of most significancehowever is the empirical evidence for self-similarity in flood behaviour Self-similarity is difficult todetect in gauged records of limited length however one positive aspect of the application ofstatistics to FFA has been the refinement of techniques for the incorporation of historical andpalaeoflood data It is these data types even over modest timescales such as 100 years which offerthe best promise for testing alternative models of extreme flood behaviour across a wider range ofbasins At stake is the accurate estimation of flood magnitude used widely for design purposes thepower law model produces far more conservative estimates of return period of large floodscompared to conventional models and deserves closer study

Key words extreme floods flood frequency analysis power law self-similar

R Kidson and KS Richards 393

hazard or risk assessment in the form of aflood frequency analysis (FFA) (Dunne andLeopold 1978) This is traditionally mostevident in the design of major engineeringstructures such as dams flood embankmentsand bridges where the accuracy of thesemethods has a profound significance foreconomic investment (Yixing et al 1987) Itis common for a lsquostandard of servicersquo to bedefined for a flood defence structure which isoften to provide protection against an eventwith a specified return period such as the100-year flood Since discharge data for mostcatchments have been collected for periodsof time significantly less than 100 years theestimation of the lsquodesign dischargersquo (designstage or water level) necessarily requires adegree of extrapolation which in turndemands curve-fitting to the existing data

All methods of FFA are thus methods ofextrapolation Extrapolation requires thefitting of a model and here a limitation of FFAis apparent the fitting of any model requiresan a priori assumption about the underlyingdistribution generating flood events Not onlyis this not known for extreme hydrologicalevents beyond the observed record but it isuntestable within human timescales (Klemes1989) Yet a model must be fitted ifpredictions are to be made The literature isdominated by one particular approach tomodelling the use of a range of more-or-lessskewed relatively complex and often theo-retically unjustified probability distributionswhose parameters are estimated from thedata in the observational record However arecent alternative has been to examine theuse of simple power law (PL) models whichcarry theoretical implications about self-similarity in the distribution of flood magni-tudes (Malamud and Turcotte 1999)

There are however some additional toolsfor effectively extending the instrumentedgauging record These include (1) using phys-ically based models including rainfall-runoffmodelling in continuous simulation mode(Beven 1987) (2) combining data fromseveral gauges in a regionalization exercise

(GREHYS 1996) and (3) incorporatinghistorical and palaeoflood information into theinstrumented record (House et al 2002Baker 2003) This review will focus on theimplications of the third of these methodsand in particular will consider the use ofhistorical and palaeoflood data and its implica-tions for the two strategies of curve-fittingnoted above The more recent use of PLdistributions has in part been justified whenpre-instrumental flood evidence suggests thatthe most extreme events are outliers relativeto the conventionally fitted curves

II The mechanics of flood frequency analysisFFA model fitting involves three main stepsdata choice model choice and a parameterestimation procedure (Bobee 1999) Theseintricacies are discussed here for two rea-sons (1) to illustrate the proliferation oftechniques at each step and (2) in order tohighlight the uncertainty consequences ofreliance upon any one technique at each stepYen 2002 proposes a very useful scheme inwhich to consider uncertainty (Table 1) Inthis scheme the modelling process is pre-sented as a hierarchy Progression along themodelling process entails assumptions at eachstep of the process and each of these stepsthus has the capacity to introduce error Ifunquantified this error may cascade down

Table 1 Yenrsquos (2002) hierarchicaluncertainty scheme

Uncertainty Sensitive totype

1 Natural Nonstationary uncertainty conditions

2 Model uncertainty Choice of model3 Parameter Fitting technique

uncertainty goodness-of-fit test4 Data uncertainty Data choice accuracy of

observedgauged data5 Operational Human

uncertainty errorsdecisions

394 Flood frequency analysis assumptions and alternatives

into the model predictions this can have direconsequences in the case of FFA Awarenessof these issues is not often demonstratedin the FFA literature yet challenging someof these assumptions represents the bestopportunity for advancement of the scienceof FFA This paper addresses points (2)ndash(4) ofYenrsquos scheme

1 Data choice Classically FFA is performed on the series ofannual maximum discharge values recordedat a specific river gauging section Howeveran alternative is to use a form of partial seriesthe most common being the peaks overthreshold (POT) method (Hosking andWallis 1987) where every event over a giventhreshold is included in the analysis (Figure 1)POT is now a cornerstone technique for theincorporation of historical and palaeoflood

data since it offers inclusion of the continu-ous instrumented gauged record and the dis-crete historic data within the same modellingframework However in conventional FFA itis common practice to apply both methods inorder to determine the difference that datachoice decisions make for prediction In par-ticular it is known that results are sensitive tochoice of threshold (eg Adamowski et al1998 Tanaka and Takara 2002) with greaterconfidence when the threshold is close to thedischarge for which a return period estimate isrequired (Hosking and Wallis 1986a) Whereit is possible a reliable scheme of investigationwould include a range of thresholds in orderto determine this sensitivity

2 Model choice I A limited number of models have traditionallybeen employed in the analysis of flood peak

Figure 1 A partial duration series (PDS)peaks over threshold (POT) approachapplied to daily discharge records for the 49-year gauged record of the Mae Chaemriver northern Thailand The minimum peak annual discharge over this period is 114 m3s1 (dashed line) and this forms the threshold for inclusion of flow peaks in the PDS (peaks must also be independent a one-month separation of peaks isconventionally used as an eligibility criterion) In the case of these Mae Chaemrecords this yields a PDS with n 78


data for FFA but there are a large number of variants on these The simpler class ofthese conventional FFA models consists of2-parameter functions that can be fittedanalytically such as the log normal andGumbel extreme value type 1 (Gumbel EV1)double-exponential model (see Table 2) Thetwo parameters represent location and shapeand the mean and variance of the samplepopulation are employed (eg the annualseries of discharges for a gauging station) withthe method of moments (see below) The logPearson type III (LP3) and generalizedextreme value (GEV) models belong to aclass of 3-parameter models which cannot befitted analytically The former is based on the gamma distribution In both cases thethree parameters represent location shape

and scale and again when the method ofmoments is used are based broadly on themean variance and skewness of the distribu-tion The 2-parameter models have theadvantage of simplicity and ease of fithowever the 3-parameter models with theadditional scale parameter are regarded as having the flexibility to fit a larger numberof catchmentsrsquo records (NERC 1999)Consideration of scale is also an acknowl-edgement that most annual series aresignificantly skewed

Some lesser-known models includemembers of the gamma distribution family(eg the bivariate gamma distribution Yue2001) the Weibull (Heo et al 2001a 2001b)the extreme value family (eg trivariateextreme value Escalante-Sandoval and

Figure 2 Four FFA models fitted to a synthetically generated record of lsquoannualfloodsrsquo randomly sampled from a log normal distribution and plotted in double-logarithmic space Three conventional models are shown the Gumbel EV1 the two-parameter log normal (LN2) and the log Pearson type III (LP3) All thesedistributions were fitted using the method of moments the LP3 was fitted with theWater Resources Council 1981 skew coefficient Also shown is a power law (PL)model fitted using simple least squares regression to the observed data The threeconventional distributions exhibit a concave-down shape in log-log space contrastingwith the PL which is a straight line


Table 2 Probability density functions (pdfs) andor cumulative density functions(cdfs) for a selection of FFA distributions (adapted from Stedinger et al 1993 andRamachandra Rao and Hamed 2000)

Normal pdf

Log normal (2-parameter) (LN2) pdf

y ln(x)

Pearson type 3 pdf

is the Gamma function

Log Pearson type III (LP3) pdf

Exponential pdf

cdf

Gumble EV1 pdf

cdf

Generalized extreme value (GEV) cdf

Weibull pdf

cdf

Generalized Pareto (GP) pdf

cdfF xx

( )= minus minusminus( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 1

1

f xx

( )=⎛⎝⎜

⎞⎠⎟

minusminus( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

minus1

1

1 1

F xx

k

( ) exp= minus minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

f xk x x

k k

( ) exp=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦

minus

1

⎥⎥⎥

F xx

( ) exp( )= minus minus minus⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪1

1

F xx

( ) exp exp= minus minus minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

f xx x

( ) exp exp= minus minus minus minus minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

1

F x x( ) exp ( )= minus minus minus 1

f x x( ) exp ( )= minus minus⎡⎣ ⎤⎦

f x xx

x( ) ln( )

exp ln( )= minus⎡⎣ ⎤⎦ minus minus⎡⎣ ⎤⎦ minus

1

(( )

( )

f x xx

( ) ( )exp ( )

( )= minus⎡⎣ ⎤⎦

minus minus⎡⎣ ⎤⎦minus

1

f xx

x

y

y

y

( ) expln( )

= minusminus⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

1

2

122

2

f xx

x

x

x

( ) exp= minusminus⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

2

122

2

(continued)


Raynal-Villasenor 1994) and the logisticdistribution family (Ahmad et al 1988) Thegeneralized pareto distribution (GDP) is aspecialized model for POT-type data(Hosking and Wallis 1987) and is based onthe Poisson (binomial) distribution Examplesof the more commonly used models are pro-vided in Table 2 and illustrated in Figure 2

In several instances a national standardi-zation of the approach to FFA is enforcedthis may be to provide the semblance ofobjectivity and protection against legalliability The USA is a notable case where theLP3 has been the official model since 1967(NRC 1988) to which data from all catch-ments are fitted for planning and insurancepurposes There is a correspondingly largebody of research concerning this model (egVogel and McMartin 1991 Wu-Boxian et al1991 Fortin and Bobee 1994 Ouarda andAshkar 1998 Chen et al 2002) By contrastthe UK endorsed the GEV1 distribution(NERC 1999) up until 1999 the official distri-bution in this country is now the generalizedLogistic (GL) There are several instanceswhere a number of alternative models havebeen evaluated for a particular country forexample Kenya (Mutua 1994) Bangladesh(Karim and Chowdhury 1995) Turkey(Bayazit et al 1997) and Australia (Vogel etal 1993) One consequence of the adoptionof a single standard model is the lsquoone size fitsnobodyrsquo scenario where optimal fitting to aspecific catchment is precluded by a (possibly

unsuited) national modelndashthis may result inmediocre accuracy on flood prediction in anygiven basin

Given the confusing range of models avail-able it is perhaps not surprising that therehave also been calls to adopt an internationalstandard for FFA (Bobee et al 1993)However FFA results are obviously influ-enced by model choice and the best means ofquantifying this is through comparing a num-ber of models (Haktanir and Horlacher1993) The proliferation of models is itselfsymptomatic of the weak theoretical basis inhydrology for the application of these FFAmodels Chow 1954 presents perhaps theonly convincing hydrologically plausible theo-retical argument justifying the application of aparticular model for FFA Chow demon-strated mathematically that multiplicativerandom natural events would generate a lognormal distribution Gumbelrsquos extreme valuetheory (EVT Gumbel 1958) and this familyof FFA models have a theoretical basis inclosed-system statistical assumptions thatmay or may not be hydrologically valid Forthe remaining models commonly applied andmost notably the official distributions in the USA and UK theoretical justificationsare weakly articulated if at all and theo-retical considerations are rarely employed as a legitimate criterion to guide model selec-tion in FFA In the case of the officiallymandated models their original selection was based almost exclusively on the basis

Table 2 continued

Generalized logistic (GL) pdf

cdf

Power law (PL) f x Cx( )= minus

F x kx k

( )= + minus minus⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 1

1

11

f x kx

kk

( )= minus minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + minus

minus⎛⎝⎜

⎞⎠⎟1

1 1 1

11

xx kminus⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 2

of empirical goodness-of-fit to the events of record it is implicitly assumed that the best fit will provide the most reliableextrapolation

Classically the appropriateness of a givenmodel is assessed via a goodness-of-fittest (eg Chowdhury et al 1991) and thisis another limitation of FFA different goodness-of-fit tests will favour differentmodels and not all models can be evaluatedby the same test NERC (1999) demonstratea classic example where different goodness-of-fit measures (eg absolute error and leastsquares error) favour alternative models TheFFA analyst therefore must be aware ofsensitivity to choice of goodness-of-fit testand it is useful to employ several of these totest for sensitivity (eg Stedinger et al1993) None of this supports the view thatone of these traditional models deserves to beadopted as a universal choice

3 Parameter estimation procedure Having selected an a priori model the nextstep is to identify the parameters required tofit the model to the selected data This hasusually been achieved using the method ofmoments (MOM) which is based on thestatistical moments (ie mean varianceskew) of the sample data An alternative isthe L-moment method (Hosking 1990) nowhaving wide application in regional analyses(Adamowski 2000) L-moments define thecharacteristics of the sample discharge databased on combinations of the differencebetween two randomly selected events in thedata A third parameter-fitting technique ismaximum likelihood (ML) This is a non-analytical technique where parameters areoptimised via a search through parameterspace hence it is computationally demandingML offers a consistent objective techniquefor parameter fitting where the analysisincludes different types of data and is partic-ularly relevant for FFA where a systematicrecord is combined with categorical data (inthe form of historical POT numbers)(Stedinger and Cohn 1986)

A particular advantage of ML methods isthat they can be applied to probability densityfunctions (pdfs) that are multimodal or other-wise complex (ie without a strong centraltendency or deriving from the combination ofevents drawn from different distributions)ML has the flexibility to deal with informa-tiondata that deviate from statisticalassumptions of normality (eg OrsquoConnellet al 2002) As this characterizes muchhydrological data ML is a valuable toolFurthermore it is an approach that can assistin defining confidence limits on discharge orreturn period estimates However it is sensi-tive to the choice of goodness-of-fit test cho-sen for the optimization and it is not possibleto arrive at numerical solutions for all cases

III Model choice an alternative The PL model (Table 2 and Figure 3) offers asimple alternative to the more complexprobability models discussed above Theevidence supporting PL distributions forextreme natural events is growing withexamples including earthquakes volcaniceruptions landslides avalanches and forestfires (see Turcotte 1994 Scheidegger 1997Birkeland and Landry 2002) Recent work inother fields of hydrology also demonstratesPL relationships for specific ranges of rainfalldata (Gupta and Waymire 1990 Hubert2001) and catchment runoff (de Michele et al 2002) A PL fit implies that dischargeratios are the same (hence lsquoself-similarrsquo) forgiven return period ratios at least for acertain range of discharge magnitudes(scales) eg Q100Q10 equals Q10Q1 wherethe scaling exponent is constant over thisrange Self-similarity offers a plausible theo-retical basis for flood frequency analysis

A prominent application of self-similarityto FFA was presented by Malamud et al(1996) who demonstrated a close fit of thedischargerecurrence interval relationship ofthe extreme 1993 event on the Mississippiriver with a PL model Similar close fits weredemonstrated with historic and palaeoflooddata for the Colorado river This was a



significant outcome as the PL model togetherwith historic (pre-instrumental) flood datapredict substantially larger discharges at agiven recurrence interval If a PL model isindeed the underlying probability model forflood behaviour on these rivers then theestimated discharge of the industry-standard100-year flood requires revision with planningand design decisions needing to be corre-spondingly more conservative (Figure 4)

Improving the theoretical basis and docu-menting additional case studies of PL behav-iour constitute important areas for futureresearch Alila and Mtiraoui 2002 assert thatthe selection of the most plausible distributionfor flood frequency analysis should be basedon hydrological reasoning as opposed to thesole application of the traditional statisticalgoodness-of-fit tests If goodness-of-fit

criteria as applied to short-record annualseries data are not regarded as the best testof the applicability of a flood frequencymodel this leaves few grounds for acceptingany model a priori for a new catchmentndashunless there is a firmly established general-ized theoretical basis A critical outstandingquestion in assessing the plausibility of the PL model is the scale at which PL behaviourapplies flood magnitudes over a period of say10 years may be generated by processes dis-tinct from those which produce the 100-yearor 1000-year floods nonstationarity consider-ations notwithstanding

IV Mixed generating mechanismsIt is notable in Figure 4 that while the PLdistribution fits the extremes well it fails to fit the lower range of annual flood maxima

Figure 3 Power law tail behaviour in daily discharge data For the annual seriesthere is a straight line divergence in plotting position of the observed events with lowprobability relative to the tail of an exponential curve fitted to the same data Alsoshown is the partial duration series (PDS) Use of the PDS tends to remove muchcurvature in space at high exceedance probabilities relative to annual series plots andoften provides more compelling evidence of straight line (ie power law) behaviour inlog-log space


Self-similar behaviour in a wide range of phe-nomena exhibit a phase transition (Schertzeret al 1993) and PL scaling may only bemanifest above this This implies that the PLmay be a model best used for large eventsabove a fairly high threshold In turn this hasthe consequence that PL behaviour may bedifficult to detect in short records if only a

small number of floods have exceeded thephase transition threshold Malamud et al1996 utilized the partial duration series as ameans of overcoming this issue

Many studies of self-similar processes aremultifractal ie exhibit different PL scalingexponents over different scale ranges This issignificant for FFA because different flood

Figure 4 Annual maximum discharges (Q) of the Mississippi River at Keokuk Iowafor the period 1879ndash1995 plotted on log-log axes against return period (T) Theannual series is fitted with a log Pearson type III distribution (LP3) and the partialduration series with a power law model The 1993 flood discharge has a return period of about 100 years according to the power law but a return period in excess of1000 years according to the LP3 model Source after Malamud et al (1996) Reprinted with permission of the authors

generating mechanisms may give rise to dif-ferent scaling exponents over specific rangesThis recalls the approach developed byWaylen and Woo (1982) and Waylen (1985) which focuses on the concept of mixedgenerating mechanisms for floods Theirstudies in Canada demonstrated that floodscaused by winter west-coast rain could beidentified and separately modelled in the floodrecord from those caused by a distinctprocess spring snowmelt and this study iselegant in its simplicity (Figure 5) Recentstudies have further demonstrated the prob-lematic and fallacious nature of attempting tofit a single distribution to flood data generatedby separate identifiable climatic processes(Murphy 2001 Alila and Mtiraoui 2002)The practice is questionable of identifyingmore complex multiparameter pdfs todescribe flood series that may include subsetsgenerated by distinct processes each ofwhich may have a characteristic flooddistribution which is adequately defined bya simple log normal or PL distributionNevertheless attempts have been made tomodel such joint distributions using variantsof the traditional models (eg the Gumbelmixed model of Yue et al 1999 or the multi-variate extreme value with Gumbel marginalsof Escalante-Sandoval 1998)

The mixed generating mechanism conceptis closely related to a form of nonstationarityin the flood record and of course a criticalassumption of all FFA is that of statistical sta-tionarity (Stedinger 2000) and independenceof individual flood events An annual floodseries may include several El NintildeoLa Nintildeaphases with the flood magnitude pdfs forthese different climatic phases being notice-ably different as was classically demon-strated in Australia with the identification oflsquodrought dominated regimesrsquo (DDRs) andlsquoflood dominated regimesrsquo (FDRs) (Warner1987) More recently this pattern has beeninterpreted as a multidecadal nonstationarityassociated with modes in the Inter-decadalPacific Oscillation (Figure 6) (Kiem et al2003) With this recognition it is possible for

FFA studies to be used as evidence forclimatic nonstationarity (Franks 2002)following the temporal partitioning of dataThis of course reduces the sample size forthe fitting of individual pdfs and therebymakes the use of multiparameter extreme-event distributions more difficult by increas-ing the uncertainty of their parameterestimates This suggests that there arepractical reasons (of parsimony) for favouringmultiple simple distributions in addition tothe possibility that there may be bettertheoretical grounds for assuming that theannual flood record is more likely to begenerated by mixed simple distributions thanby single complex ones

V The role of historical and palaeoflood dataPalaeoflood studies have traditionally beenutilized for testing assumptions about thestationarity of the flood record over very long(ie millennial) timescales (particularly theexistence of distinct climatic phases withdifferent flood-generating capabilities Ely1997) Studies of long (Holocene-scale) floodrecords demonstrate distinct temporalclusters of flood events (Wohl et al 1994 Elyet al 1996) reflecting the climatic history ofthe region in question However anotherpossibility is the utilization of palaeofloodstudies over more modest timescales (eg100ndash200 years) for the purpose of testing theself-similar hypothesis in catchments with ashort instrumented record This also repre-sents the timescale of most relevance fordesign structures This approach was utilizedin the following case study from northernThailand (Kidson et al 2005)

1 An example from northern Thailand The Mae Chaem river is a tributary of thePing river to the southwest of Chiang Maijoining the Ping at the town of Hot In itslower reaches the Mae Chaem passesthrough the Ob Luang gorge (Figure 7) andin the gorge there are caves which havetrapped woody debris in extreme floods



Figure 5 (a) Gauging stations on the Fraser River British Columbia and (b) annualflood exceedance probability curves for the numbered stations plotted on Gumbelprobability paper Part (b) shows that flood frequency distributions are linear on thisprobability paper when there is a single generating process (and generating distribu-tion) with the coastal rainfall-generated floods showing a steeper curve than theinland snowmelt floods In a transitional region where the flood record is derived from both sources (and distributions) the curves reflect this combination Source after Waylen (1985) Reprinted with permission of the author


predating the instrumental record Therehave been large gauged floods in recent yearsnotably in August 2001 and it has been possi-ble to survey the water level of this event andto calibrate a 1D hydraulic model to define theroughness coefficient applicable in events ofthis magnitude (Kidson et al unpublisheddata) The highest level flood deposits in thecaves provide four palaeo-stage indicators(PSIs) and the 1D model with a roughnesscoefficient as suggested by the modelling ofrecent large flood events predicts a dischargeof 2420 m3s1 for the water level implied bythese PSIs Dating of this pre-instrumentalevent in the gorge (using historical recordsoral history dendrochronology and radio-carbon dating) suggests that it occurred earlyin the twentieth century and a return periodof 84 years has been assigned to this eventThe details of this investigation are reportedelsewhere (Kidson et al 2005)

If FFA is undertaken for the instrumentalflood record alone and conventional pdfs(eg log Pearson type III Gumbel EV1 and

two-parameter log normal models all fittedwith the method of moments the WRC(WRC 1981) skew coefficient was also usedin the case of the LP3) are used to estimatethe discharge of an event with a return periodof 84 years the estimates are 1005 1012 and1040 m3s1 respectively They thus severelyunderestimate the discharge of the eventassociated with the PSIs in the gorge Bycontrast a PL model fitted by reduced majoraxis (RMA) regression to the gauged recordyields an 84-year discharge estimate of 2479 m3s1 which is a close match to theobserved extreme event (Figure 8) In thisapplication the extreme event is not includedin the model-fitting process this data point isreserved in order to assess the predictivesuccess of the models This represents amethodological departure from conventionalFFA where all data points are used forthe model calibration process leaving none with which to assess the model Hence thepredictive quality of conventionally fittedmodels remains untested

Figure 6 Regional flood frequency index curves for New South Wales Australiashowing the expected values and the 90 confidence interval under El Nintildeo and La Nintildea conditions Source after Kiem et al (2003) Reprinted with permission of the authors


The consequence of this finding in thecase of the Mae Chaem is significant in thatthe Gumbel EV1 model is the official basisfor FFA in Thailand where the maximumrecord lengths tend to be of the order of 50 years However in this catchment itappears to underpredict the magnitude of the 84-year event significantly withinevitable consequences for structuraldesign and planning

2 Implications of threshold exceedanceIt appears that the PL model may be applica-ble for very extreme events but not forevents around the mean annual flood there-fore the important question of appropriatemethods of fitting PL models deserves closerresearch For long records (eg 100 years)the threshold above which PL behaviourbegins may be visible in the gauged recordpermitting a regression relation using these

Figure 7 The Ob Luang gorge lower Mae Chaem river under low-flow conditions


datandashhowever this may not be the case forrecords of short length and other techniquesand supplementary data may be requiredThis increases the importance of historicaland palaeoflood information since this isgenerally available in the form of peaks overthreshold (POT) Often the exact magni-tude of an event is unknown but is it isknown to have exceeded a specific thresholdindeed as Stedinger and Cohn 1986 notelsquohistorical floods are observed because their magnitude exceeds some threshold ofperceptionrsquo

One of the positive consequences of theapplication of sophisticated statistics to FFAhas been the generation and refinement ofmethods which usefully incorporate palae-oflood and historical information into FFAmodels To take the POT example thiscensored form of information where no dataare available on the flows below the thresh-old poses a major statistical challenge (Krolland Stedinger 1996) Inclusion of historical

and palaeoflood information to extend agauging record typically lsquofills inrsquo the ungaugedportion of the historic period with an appro-priate number of replicates of the below-threshold portion of the systematic recordVarious methods of achieving this have beenproposed The simplest is the probabilityweighted moments technique advocated byIACWD (1982) However Stedinger andCohn (1986) demonstrated that a ML-basedtechnique was superior recently Cohn et al(2001) have proposed an expected moments(EM) technique as an alternative whereapplication of ML poses numerical problems(Ouarda et al 1998) All of these techniquesare designed to reduce systematic bias inestimates in the low-probability rangeHowever these are also examples of rela-tively abstruse statistical procedures whichmight be redundant or require revision withina self-similar modelling framework

Hosking and Wallis (1986a 1986b) ques-tioned the utility of incorporating palaeoflood

Figure 8 Four models fitted to the gauged annual floods of the Mae Chaem riverthe reconstructed palaeoflood is also plotted and the success of each of the models inpredicting the palaeoflood (this event was not included in the model fitting process)are graphically illustrated

and historical information in FFA demon-strating that it depends on the number ofmodel parameters and the degree of dischargeerror Frances et al (1994) also qualified itsutility in terms of the relative magnitudes ofthe length of the systematic record thelength of the historic period the perceptionthreshold and the return period for which an estimate is required Some of these qualifi-cations also become less relevant if theextremes represented by palaeofloods areconsidered within a self-similar modellingapproach Several studies have now demon-strated the value of noninstrumental floodinformation in FFA (Stedinger and Cohn1987 Jin and Stedinger 1989 Webb et al1994 Ouarda et al 1998 Williams andArcher 2002) In most cases these studieshave been formulated within an existing para-digm of FFA based on the use of conventionalextreme event distributions It can be seenthat these same data are of especial value inseeking to test self-similar hypotheses in floodbehaviour in a sense palaeoflood and historicdata represent more viable and efficient datasources on which to conduct such tests than ameagre increase in the length of the continu-ous instrumented record which may yield nofloods of sufficient magnitude to add value toa PL model It is possible that the noninstru-mental evidence of censored and thereforeparticularly extreme events may lead to aparadigm shift in which self-similar modelsplay a greater role

VI Discussion and conclusionFFA analyses incorporate assumptions ateach stage of the modelling process Animportant one is the assumption of a highdegree of accuracy in the estimation ofdischarges It is known that considerable erroris introduced both in measurement in theinstrumental record and in modelling ofpalaeoflood discharges Yet this assumptionis the basis for the considerable body ofliterature addressing statistically detailedprocedures focused on model parameterfitting and testing The most fundamental

assumption is that of the conceptual modelof which the self-similar PL approach repre-sents a major alternative to conventionaldistributions

Future research can usefully address theserespective sources of uncertainty at the mostfundamental level challenging basic assump-tions and at the most practical level aiming toreduce the confidence intervals placedaround flood recurrence estimates (NRC1999) The recent history of FFA has seen arange of innovative approaches that reflectthe methods required in historic and palae-oflood studies For example the concept ofthe palaeohydrologic bound (Levish 2002OrsquoConnell et al 2002) has emerged recentlywhich is the time interval during which aparticular discharge is not exceeded ascer-tained through dating the geomorphic surfacesthat have not been inundated This offers amore reliable means of defining upper confi-dence limits on flooding for major structuressuch as reservoirs Also Vogel et al (2001)have introduced the concept of the record-breaking flood ndash an event which exceeds allprevious events and this represents a newapplication of a branch of mathematicsdistinct from extreme value theory Theseapproaches represent a promising avenue inwhich to test alternative modelling frame-works (including the self-similar one) andthere are thus good grounds for suggestingthat the noninstrumental observation of theoccurrence of extreme floods offers the bestpotential for a paradigm shift in FFA

The history of FFA has been distinguishedby the employment of increasingly sophisti-cated statistical techniques to satisfy thedemand for rigorous curve fitting This hasincreased the robustness of estimates and thetreatment of uncertainty ndash but only embed-ded within the broader model assumptionsNational mandates for particular models maybe said to have reduced the tendency to chal-lenge the conventional models and hasinstead facilitated work within this paradigmThese methods have been flexible enough toincorporate the mixture of flood information


that may be available for a single catchmentsite-specific regional historical and palae-oflood information However concomitanttheoretical advance in FFA has been some-what limited for example Singh andStrupczewski (2002) have noted that thefitting of flood frequency models is largely astatistical exercise somewhat divorced fromhydrological input Given the many implicitassumptions in FFA the safest approach liesin testing and citing the sensitivity of a givenanalysis to a range of possible techniques ateach stage of the modelling process

In the sense that FFA for a catchment isultimately information-dependent it is impos-sible to identify which of the many FFA toolsshould be employed in a given instance A state-of-the-art analysis is likely to entail amultidisciplinary approach incorporating the results of both physically based (egrainfallrunoff) modelling with detailed instru-mental gauging data supplemented withregional data and historical and palaeofloodinformation (Stedinger 2000) The caseswhere this full range can be applied in combi-nation are limited Nevertheless in thosecatchments where there is evidence of acensored set of historic and palaeoflood waterlevels which can be converted into dischargesit is likely that flood frequency analyses mayrequire radical revision as new theoreticalconcepts are increasingly employed The self-similar PL model offers particular promise forthe prediction of extreme events Sinceindividual catchments often do not supply asufficient length of instrumented record onwhich to test alternative extreme eventmodels emphasis is placed on the relevance ofhistorical palaeoflood and regional evidenceas a means to overcome the data limitations insingle catchments and ultimately improveextreme flood prediction

ReferencesAdamowski K 2000 Regional analysis of annual max-

imum and partial duration flood data by nonparamet-ric and L-moment methods Journal of Hydrology229 219ndash31

Adamowski K Geng-Chen-Liang and Patry GG1998 Annual maxima and partial duration flood series analysis by parametric and non-parametric methods Hydrological Processes 121685ndash99

Ahmad MI Sinclair CD and Werritty A 1988Log-logistic flood frequency analysis Journal ofHydrology 98 205ndash24

Alila Y and Mtiraoui A 2002 Implications of hetero-geneous flood-frequency distributions on traditionalstream-discharge prediction techniques HydrologicalProcesses 16 1065ndash84

Baker VR 2003 Palaeofloods and extended discharge records In Gregory KJ and Benito Geditors Palaeohydrologyndashunderstanding global changeChichester Wiley 307ndash24

Bayazit M Shaban F and Onoz B 1997Generalized extreme value distribution for flooddischarges Turkish Journal of Engineering andEnvironmental Sciences 21 69ndash73

Beven K 1987 Towards the use of catchment geomor-phology in flood frequency predictions Earth SurfaceProcesses and Landforms 12 69ndash82

Birkeland KW and Landry CC 2002 Power lawsand snow avalanches Geophysical Research Letters29 4941ndash4943

Bobee B 1999 Extreme flood events valuation usingfrequency analysis a critical review Houille Blanche54 100ndash105

Bobee B Cavadias G Ashkar F Bernier J andRasmussen P 1993 Towards a systematic approachto comparing distributions used in flood frequencyanalysis Journal of Hydrology 142 121ndash36

Chen Y Yu H Van-Gelder P and Zhigui S2002 Study of parameter estimation methods forPearson-III distribution in flood frequency analysisIAHS AISH 271 263ndash70

Chow VT 1954 The log-probability Law and itsengineering applicatioin Proceedings AmericanSociety of Civil Engineers 80 Paper 536 1ndash25

Chowdhury JU Stedinger JR andLi-Hsiung-Lu 1991 Goodness-of-fit tests forregional generalized extreme value flood distribu-tions Water Resources Research 27 1765ndash76

Cohn TA Lane WL and Stedinger JR 2001Confidence intervals for expected momentsalgorithm flood quantile estimates Water ResourcesResearch 37 1695ndash706

de Michele C La Barbera P and Rosso R 2002Power law distribution of catastrophic floods inSnorrason A Finnsdottir H and Moss M editorsThe extremes of the extremes extraordinary floods vol271 Wallingford IAHS 277ndash82

Dunne T and Leopold LB 1978 Calculation of floodhazard In Dunne T and Leopold LB Water in envi-ronmental planning San Francisco WH Freeman andCo CA 279ndash391


Ely LL 1997 Response of extreme floods in thesouthwestern United States to climatic variations inthe late Holocene Geomorphology 19 175ndash201

Ely LL Enzel Y Baker VR Kale VS andMishra S 1996 Changes in the magnitude andfrequency of late Holocene monsoon floods on theNarmada River central India Geological Society ofAmerica Bulletin 108 1134ndash48

Escalante-Sandoval C 1998 Multivariate extremevalue distribution with mixed Gumbel marginals Journalof the American Water Resources Association 34 321ndash33

Escalante-Sandoval CA and Raynal-VillasenorJA 1994 A trivariate extreme value distributionapplied to flood frequency analysis Journal ofResearch 99 369ndash75

Fortin V and Bobee B 1994 Nonparametric bootstrap confidence intervals for the log-Pearsontype III distribution In Zanetti P editor Computertechniques in environmental studies V vol II environ-mental systems Billerica MA ComputationalMechanics Publications 351ndash58

Frances F Salas JD and Boes DC 1994 Floodfrequency analysis with systematic and historical orpaleoflood data based on the two-parameter generalextreme value models Water Resources Research 301653ndash64

Franks SW 2002 Identification of a change in climatestate using regional flood data Hydrology and EarthSystem Sciences 6 11ndash16

GREHYS 1996 Presentation and review of somemethods for regional flood frequency analysis Journalof Hydrology 186 63ndash84

Greis NP 1983 Flood frequency analysis a review of1979ndash1984 Reviews of Geophysics and Space Physics21 699ndash706

Gumbel EJ 1958 Statistics of extremes New YorkColumbia University Press 375 pp

Gupta VK and Waymire E 1990 Multiscaling prop-erties of spatial rainfall and river flow distributionsJournal of Geophysical Research 95(D3) 1999ndash2009

Haktanir T and Horlacher HB 1993 Evaluation ofvarious distributions for flood frequency analysisHydrological Sciences Journal 38 15ndash32

Heo JH Salas JD and Boes DC 2001a Regionalflood frequency analysis based on a Weibull modelpart 1 Estimation and asymptotic variances Journalof Hydrology 242 157ndash70

mdash 2001b Regional flood frequency analysis based on aWeibull model part 2 Simulations and applicationsJournal of Hydrology 242 171ndash82

Hosking JRM 1990 L-moments analysis and esti-mation of distributions using linear combinations oforder statistics Journal of the Royal Statistical SocietyB 52 105ndash24

Hosking JRM and Wallis JR 1986a Paleofloodhydrology and flood frequency analysis WaterResources Research 22 543ndash50

mdash 1986b The value of historical data in flood frequencyanalysis Water Resources Research 22 1606ndash12

mdash 1987 Parameter and quantile estimation for thegeneralized Pareto distribution Technometrics 29339ndash49

House PK Webb RH Baker VR andLevish DR 2002 Ancient floods modern hazardsprinciples and applications of palaeoflood hydrologyvol 5 Washington DC American GeophysicalUnion 385 pp

Hubert P 2001 Multifractals as a tool to overcomescale problems in hydrology Hydrological Sciences 46897ndash905

Hurst HE Black RP and Simaika TM 1965Long term storage London Constable 145 pp

IACWD (Interagency Advisory Committee on WaterData) 1982 Guidelines for determining flow fre-quency In Bulletin 17B Reston UA US Departmentof the Interior

Jin M and Stedinger JR 1989 Flood frequencyanalysis with regional and historical informationWater Resources Research 25 925ndash39

Karim MA and Chowdhury JU 1995 A compari-son of four distributions used in flood frequencyanalysis in Bangladesh Hydrological Sciences Journal40 55ndash66

Kidson R Richards KS and Carling PA 2005Reconstructing the 1-in-100 year flood in northernThailand Geomorphology in press

Kiem AS Franks SW and Kuczera G 2003Multi-decadal variability of flood risk GeophysicalResearch Letters 30(2) 1010292002GL01599271ndash74

Klemes V 1989 The improbable probabilities ofextreme floods and droughts In Starosolszky O andMelder OM editors Hydrology of disasters Jamesand James London 43ndash51

Kroll CN and Stedinger JR 1996 Estimation ofmoments and quantiles using censored data WaterResources Research 32 1005ndash12

Levish DR 2002 Paleohydrologic bounds ndash non-exceedance information for flood hazard assessmentIn House PK Webb RH Baker VR and LevishDR editors Ancient floods modern hazards princi-ples and applications of paleoflood hydrologyWashington DC American Geophysical Union175ndash90

Malamud BD and Turcotte DL 1999 Self-organized criticality applied to natural hazardsNatural Hazards 20 93ndash116

Malamud BD Turcotte DL and Barton CC1996 The 1993 Mississippi River flood a one hundredor a one thousand year event Environmental andEngineering Geosciences 2 479ndash86

Murphy PJ 2001 Evaluation of mixed-populationflood-frequency analysis Journal of HydrologicEngineering 6 62ndash70



Mutua FM 1994 The use of the Akaike informationcriterion in the identification of an optimum floodfrequency model Hydrological Sciences Journal 39235ndash44

National Environment Research Council (NERC)1999 Flood Studies Report (in five volumes)Wallingford Institute of Hydrology

National Research Council (NRC) 1988 Committeeon techniques for estimating probabilities of extremefloods estimating probabilities of extreme floods ndashmethods and recommended research Washington DCNational Academy Press

mdash 1999 Improving American river flood frequencyanalyses In Committee on American river flood fre-quency water science and technology board committeeon geoscience environment and resources WashingtonDC National Academy Press 1ndash120

OrsquoConnell DRH Ostenaa DA Levish DRand Klinger RE 2002 Bayesian flood frequencyanalysis with paleohydrologic bound data WaterResources Research 38(5) 101029 2000WR000028

Ouarda TBMJ and Ashkar F 1998 Effect oftrimming on LP III flood quantile estimates Journal ofHydrologic Engineering ASCE 3 33ndash42

Ouarda TBMJ Rasmussen PF Bobee B andBernier J 1998 Use of historical information inhydrologic frequency analysis Revue des Sciences delrsquoEau 11(Special Issue) 41ndash49

Ramachandra Rao A and Hamed KH 2000 Floodfrequency analysis Boca Raton FL CRC Press 350pp

Scheidegger AE 1997 Complexity theory of naturaldisasters boundaries of self-structured domainsNatural Hazards 16 103ndash12

Schertzer D Lovejoy S and Lavallee D 1993Generic multifractal phase transitions and selforganised criticality In Pendang JM and LejeuneA editors Cellular automata prospects in astro-physical applications Singapore World Scientific216ndash27

Singh VP and Strupczewski WG 2002 On thestatus of flood frequency analysis HydrologicalProcesses 16 3737ndash40

Stedinger JR 2000 Flood frequency analysis andstatistical estimation of flood risk In Wohl E E editorInland flood hazards human riparian and aquatic com-munities Cambridge Cambridge University Press334ndash58

Stedinger JR and Cohn TA 1986 Floodfrequency-analysis with historical and paleofloodinformation Water Resources Research 22 785ndash93

mdash 1987 Historical flood frequency data its value anduse In Singh VP editor Regional flood frequencyanalysis Proceedings of the International Symposiumon Flood Frequency and Risk Analyses Baton RougeUSA Baton Rouge LN D Reidel 273ndash86

Stedinger JR Vogel RM and Foufoula-Georgiou E 1993 Frequency analysis of extremeevents In Maidment D editor Handbook ofhydrology New York McGraw-Hill

Tanaka S and Takara K 2002 A study on thresholdselection in POT analysis of extreme floods IAHSAISH 271 299ndash304

Turcotte DL 1994 Fractals and chaos in geology andgeophysics Cambridge Cambridge University Press221 pp

Vogel RM McMahon TA and Chiew FHS1993 Flood flow frequency model selection inAustralia Journal of Hydrology 146 421ndash49

Vogel RM Zafirakou-Koulouris A and MatalasNC 2001 Frequency of record-breaking floods inthe United States Water Resources Research 371723ndash31

Vogel RW and McMartin DE 1991 Probabilityplot goodness-of-fit and skewness estimation proce-dures for the Pearson type 3 distribution WaterResources Research 27 3149ndash58

Warner RF 1987 The impacts of flood- and drought-dominated regimes on channel morphology at PenrithNew South Wales Australia International Associationof Hydrological Sciences (IAHS) Publication 168327ndash38

Waylen PR 1985 Stochastic flood analysis in a region of mixed generating processes Transactions of the Institute of British Geographers NS 10 95ndash108

Waylen PR and Woo MK 1982 Prediction ofannual floods generated by mixed processes WaterResources Research 18 1283ndash86

Webb RH Moss ME and Pratt DA 1994Information content of paleoflood data in flood-frequency analysis EOS Transactions of the AmericanGeophysical Union 75(44) 372

Williams A and Archer D 2002 The use ofhistorical flood information in the English Midlands to improve risk assessment Hydrological SciencesJournal 47 67ndash76

Wohl EE Fuertsch SJ and Baker VR 1994Sedimentary records of late holocene floods along the Fitzroy and Margaret Rivers westernAustralia Australian Journal of Earth Sciences 41273ndash80

WRC 1981 Guidelines for determining flood fre-quency Washington DC US Water ResourcesCouncil

Wu-Boxian Hou-Yu and Ding-Jing 1991 Themethod of lower-bound to estimate the parameters ofa Pearson type III distribution Hydrological SciencesJournal 36 271ndash80

Yixing B Koung TY and Hasfurther VR 1987Evaluation of uncertainty in flood magnitudeestimator on annual expected damage costs of


hydraulic structures Water Resources Research 232023ndash29

Yen B 2002 System and component uncertainties inwater resources In Bogardi JJ and KundzewiczZW editors Risk reliability uncertainty androbustness of water resources systems CambridgeCambridge University Press 133ndash42

Yue S 2001 A bivariate gamma distribution for use inmultivariate flood frequency analysis HydrologicalProcesses 15 1033ndash45

Yue S Ouarda TMBJ Bobee B Legendre Pand Bruneau P 1999 The Gumbel mixed model forflood frequency analysis Journal of Hydrology 22688ndash100


hazard or risk assessment in the form of aflood frequency analysis (FFA) (Dunne andLeopold 1978) This is traditionally mostevident in the design of major engineeringstructures such as dams flood embankmentsand bridges where the accuracy of thesemethods has a profound significance foreconomic investment (Yixing et al 1987) Itis common for a lsquostandard of servicersquo to bedefined for a flood defence structure which isoften to provide protection against an eventwith a specified return period such as the100-year flood Since discharge data for mostcatchments have been collected for periodsof time significantly less than 100 years theestimation of the lsquodesign dischargersquo (designstage or water level) necessarily requires adegree of extrapolation which in turndemands curve-fitting to the existing data

All methods of FFA are thus methods ofextrapolation Extrapolation requires thefitting of a model and here a limitation of FFAis apparent the fitting of any model requiresan a priori assumption about the underlyingdistribution generating flood events Not onlyis this not known for extreme hydrologicalevents beyond the observed record but it isuntestable within human timescales (Klemes1989) Yet a model must be fitted ifpredictions are to be made The literature isdominated by one particular approach tomodelling the use of a range of more-or-lessskewed relatively complex and often theo-retically unjustified probability distributionswhose parameters are estimated from thedata in the observational record However arecent alternative has been to examine theuse of simple power law (PL) models whichcarry theoretical implications about self-similarity in the distribution of flood magni-tudes (Malamud and Turcotte 1999)

There are however some additional toolsfor effectively extending the instrumentedgauging record These include (1) using phys-ically based models including rainfall-runoffmodelling in continuous simulation mode(Beven 1987) (2) combining data fromseveral gauges in a regionalization exercise

(GREHYS 1996) and (3) incorporatinghistorical and palaeoflood information into theinstrumented record (House et al 2002Baker 2003) This review will focus on theimplications of the third of these methodsand in particular will consider the use ofhistorical and palaeoflood data and its implica-tions for the two strategies of curve-fittingnoted above The more recent use of PLdistributions has in part been justified whenpre-instrumental flood evidence suggests thatthe most extreme events are outliers relativeto the conventionally fitted curves

II The mechanics of flood frequency analysisFFA model fitting involves three main stepsdata choice model choice and a parameterestimation procedure (Bobee 1999) Theseintricacies are discussed here for two rea-sons (1) to illustrate the proliferation oftechniques at each step and (2) in order tohighlight the uncertainty consequences ofreliance upon any one technique at each stepYen 2002 proposes a very useful scheme inwhich to consider uncertainty (Table 1) Inthis scheme the modelling process is pre-sented as a hierarchy Progression along themodelling process entails assumptions at eachstep of the process and each of these stepsthus has the capacity to introduce error Ifunquantified this error may cascade down

Table 1 Yenrsquos (2002) hierarchicaluncertainty scheme

Uncertainty Sensitive totype

1 Natural Nonstationary uncertainty conditions

2 Model uncertainty Choice of model3 Parameter Fitting technique

uncertainty goodness-of-fit test4 Data uncertainty Data choice accuracy of

observedgauged data5 Operational Human

uncertainty errorsdecisions














Normal pdf


y ln(x)

Pearson type 3 pdf



Exponential pdf

cdf

Gumble EV1 pdf

cdf


Weibull pdf

cdf


cdfF xx


⎣⎢⎢

⎤

⎦⎥⎥

1 1

1

f xx

( )=⎛⎝⎜

⎞⎠⎟

minusminus( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

minus1

1

1 1

F xx

k


⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

f xk x x

k k

( ) exp=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦

minus

1

⎥⎥⎥

F xx


⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪1

1

F xx


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

f xx x


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

1



f x xx

x( ) ln( )


1

(( )

( )

f x xx

( ) ( )exp ( )

( )= minus⎡⎣ ⎤⎦


1

f xx

x

y

y

y

( ) expln( )

= minusminus⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

1

2

122

2

f xx

x

x

x


⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

2

122

2

(continued)






Table 2 continued


cdf


F x kx k


⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 1

1

11

f x kx

kk


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + minus

minus⎛⎝⎜

⎞⎠⎟1

1 1 1

11

xx kminus⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 2





















































































































































Normal pdf


y ln(x)

Pearson type 3 pdf



Exponential pdf

cdf

Gumble EV1 pdf

cdf


Weibull pdf

cdf


cdfF xx


⎣⎢⎢

⎤

⎦⎥⎥

1 1

1

f xx

( )=⎛⎝⎜

⎞⎠⎟

minusminus( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

minus1

1

1 1

F xx

k


⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

f xk x x

k k

( ) exp=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

minus⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦

minus

1

⎥⎥⎥

F xx


⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪1

1

F xx


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

f xx x


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

1



f x xx

x( ) ln( )


1

(( )

( )

f x xx

( ) ( )exp ( )

( )= minus⎡⎣ ⎤⎦


1

f xx

x

y

y

y

( ) expln( )

= minusminus⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

1

2

122

2

f xx

x

x

x


⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

2

122

2

(continued)






Table 2 continued


cdf


F x kx k


⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 1

1

11

f x kx

kk


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + minus

minus⎛⎝⎜

⎞⎠⎟1

1 1 1

11

xx kminus⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 2













































































































































Table 2 continued


cdf


F x kx k


⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 1

1

11

f x kx

kk


⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + minus

minus⎛⎝⎜

⎞⎠⎟1

1 1 1

11

xx kminus⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

minus

1 2








































































































































flood frequency analysis: assumptions and alternatives...distribution family (ahmad et al., 1988)....

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