,zt5ll

SHAH ABDUL LATIF UNIVERSITYKHAIRPUR

STATISTTCAL STUDY OF FLOOD FRf,QUENCYANALYSIS OF INDUS RIVtrR AT VARIOUS BARRACES. IN SINDH

By

- ABDUL GHANI MD,MON

THESIS SUBMITTED TOTHE FACULTY OF SCIENCES

In Fulfillment OfThe Requiroments For The Degree Of

DOCTOR OF PHILOSOPHY IN STATISTICS

TII310.5491t

MEM

a

-_'-\

Decemb€r 20(M.

F--

CERTIFICATE

This is certifigd that lhe r€search *ork embodied in lhe

thesis entitled astodsrrca, Sntdy ol Floot Freqaency Anobr'tr ofIndus River At ,/aious Ba ages tn Sindl, i was carded out by

Prof. Abdul Ghani Memon undcr my guidanc€ and supervision.

His res€arch work is original and his th€sis is worthy ofpres€ntalion to shah Abdul Latif univeNity, Khairpur, fofawarding Degree ofDoctor ofphilosophy ( ph.D) in tlrc disciDlin€

of Statistics.

Prof: Dr. Noo! Mustafa Shaikh

Supervisor

To my lat€ frther,late mother, wife snd childrcn.

ACKNOWLf,DGEMENT

The aulhor wishes to express his deep appreciation and graritude tohis supewisor Prof€ssor Dr. Noor Mustafa Sheikh, Ex- Dcrn Faoulty ofNatural Sci€nc€s, University of si lh, Ja$shoro, for lus €ncouragement,

valuable guidance and keen interest sholn in aocomplishing this r€searoh

The author is gmteffrl to lhe offic€s of Chief Engine€r Guddu, Sukkutand Kotn banages for providing lhe Data of p€ak flood discharges al the

above barrrges atrd also proudltre valuable bformariotr.

The aulhor appreciates the help md encouragem€nt ofr€r€d byhof€ssor Dr. A.R. Malik, th€ Vic€ Chanoelor, Shah Abd' Latifu venity,Khairpur, in lcconplishitrS this gigadic res€arch work.

Th€ au$or also wishes to 6ai* I\4r. Irshad Ali Junejo, cl€rk/Computer Opemter on ddly wag€s in tFing the manuscript on computerusing equation writer softwar€.

The author is also deligtned rnd thanlfil to Prof€ssor Dr. Fillibin,from uK, the author of Data Plot Software for sending lhe manusoript andCD of lh€ software fi€€ of cosr wilh intediors io help ald encouragc thesci€ntists and rcsearchers lo promotc r€s€arch for the bclt€rnent of thehunan kinds by usirs his softwarc.

The author is rlunkftl to Sycd Maqsood za, Assistanl Professor,Dcpartment of Slatistics, Shah Abdul latif Udvcrsity, Khairpu, for help inediting and printing the manusc.ipt.

In the end I m thankfil lo ny fEnily m€mbeF panicularly ny €ld€rson Engr.AMul Qad€er Mcnon for givitrg encourag€ment, noml support,inspiration and their prayeB to enable m€ lo completc this inponantresearch wo* in spite ofmany hardships.

ABSTRACT

Estimation ofPeak flood flow frequ€ncy is an imponant elenenl 'n

d€ plaming and desi8n ofwater resouro€ projects.

Thc preseni study d€als wilh fnding lhe efr€cls ofconstuction of

barrages and idedtificalion of app.opnat€ probability dislribution fiuctiotrs

toge$er with an efficient n€thod of €stination for predictiotr of floods, for

next 100 ycars , in dver Indus.

For this purpos! thc daaa of Amual Maximun flood at thee places,

oamely: Guddu, S*kur and Kotsr are slatistically analyzed. Tb€ suilabiliry

of tlle data for trend (lndepedenc€, Srationariry), homogpneousness and

nomality is tested. The exist€rc€ ofoudieG has also been check€d.

nood frequenci€s are presenl€d in gaphical, tabular as well as

marhematical fom. The effectr of construction of banages al a site and

dom srream de exanined usios descriptive Siatistics, fr€quency

distributions, and Weibull Plotting and L.S mod€ls.

Ia has been obseNed that all da(a sets are posilively skewed,

m'tltinodal ud platy kunic. All fie av.rages and drsp€rsions ar€ larg€st ai

G ldu and smallest at Kotri, and incr$s€ aner construclion ofbarrag€s.

Coefroient ofvariation (CJ incr€as€s, coefiicient ofSkewness (CJ

d€deases and Co€fEcient of Kurtosis (c!) boom€s negative (shape

chds€s sone tines) spccialy aner Guddu BarraSe. Frequency of v€ry low

and very high floods increases; Lealt Squar€ (L.S) model is accepted and is

better than Gumbels model for predication of floods at €ach barrage.

- The €xpecled flood magnitudes at ditrerent retuB periods me

comp t€d using six distsibutiors, nmcly : Gutnbel, G€nenliz.d Extrem€

- value (GEv), Normal,l,og Nornal (2) (LN (2)), Pearson (3) (P (3)) and

t s Pe'son (3) (LP (3)).

- The applicability ofdifrerent probability distributions lo th€ data

_ s€ri€s from difre.ent barr.ges and fieir sup€riority over o$ers is €xanined

uling S-K tcst, z'-tesl, PPCC, RMiE, C,, C" , Cr , ed L mom€nt ratios.

- The Suitability ofeach distribution along with illre€ nethods of

esdmanon of their pameiers, tranely: MOM, MLM, PWM. is then rankcd

- for each barrage, using S.E's as wel.

_ It is found thal LP (3) is supqior for applic{tioa followen by Gunh€l

and LN(2).

The res'nts obtained in $is study are, m Senerd, comparabl€ wlth

those obrained by oth€r researchers in different oomtries, and in Pahstan to

- thos€ by Nixon v/ho analyzcd lhe data of nl€r lldus at Sukkur and Kotri,

usrng lhe data for 4l yetrs ard 76 years

As the floods 30% hider tlan lhe designed capacity of balrages have

- alr€ady passed tkough them aDd ar€ also exp€cled lo pass durhg nen 100

years, so therc is Decd to lak€ apFopriate sleps to save the struchle of- barrages fi om any danages and to slorc the €xcess wat€r by corsEucling

cany over dams.

Trblc OfConterts

Chapter Title Prge

Table OfconlentsList OfTablcaUsi OfFiSure

Stat€medt (X Th€ Pmblem Of Floo&,Modcli|rg Ard Their Eldnlliotr

iv

3

,

I

6

7

7

8

10

l8

l3

l9t9

20

2l

22

23

25

26

Llt2ll

t5I6

Revie* Of Llteraturc

Tcsting Of D.tr And Dc€ciiFirc St ti.ri63.1

3.4

3.5

s@p. Of R*dch And Motivetion

M€dDdolog/

Hwothesh To B€ T€ded

Or8Aiz.tion OfThc Study

3.2.2 Hinqigt@ OfD.r!

3.1.1 Tct For In.LFnd.@ Ald Si|tioMity

3 3 2 T6t for tlomSarty Ard S!.lioMily

D.siptiva St ristic! ofDara

ll

4 Th€ ElTec$ Of Co[st ction Of Brrnge! Usrng

Frequ€ncy DistributioN, Ske*ne.^3 And KurrGis

4lTbc Etr .rs ofcotutrucdon OfB@ad u3ina

P.r.. sge lr€qu€ncy DidnbltioN Ald Hi$ogr@

Co.mcie.ts Of StdN$ Ard Knttoris

4.3.1 lnports* OfCqC, and Cr

4 3.2 Rdrhr ud Diseon

50

5l

52

tl

57

71

19

30

80

80

3l

a2

3l

3l

5 Pr.dictiotr Of noo& By weibol PlottirgAnd Gunbcl An.Mictl Mclhod5 I l.troducton

5 2 Probtbiliry Plotr, Plotli.S losnioN,P.obsbilrty ?aps dd Weihul llollitg5.2. I Probabilily tlpd sd Prcb.bilirv Plot

G!fr b.ls Ardtaicrl M.thod

t.2.2 PlofiingPolitions

523 W€ibul Plotdng

s3 Gumbels Afrlylicd Mcthod

5 3.1 M.thotlology

5.1.2 GDb.l'3 Eq!.tion

5 3.3 Proceddo

5.3,4 Veriffmtion

51.5 Gff$el P@habilitY PaPd

5.3.6 confddE Lituc For Predicton of fl@ds Bv

cubbd Ardyli.d M.tbod

5 4 Th. Eflet OfcoNlruction OfBd.Ad On

Oi P..t Di$hd8.! By Grnb€l s Mclhod

5.4 I At Sulhrr Bd.A.542 ArKorf,B5ng.

84

85

5.2

62

543 At Ditrsdn Banlg.r

Codprlisor Of Prediction Of Floods By Weibul PhningAtrd Mod€ri For L€rst Squir€ And cunb.| Me6od!

6,1 nood PrcdidioN by Wcihrll plofli.g ,

61.2 ta$ Sql@ M.thod (LS Mcilod)

Mod.L for pr.di.rio. offoods

6 2.1 Mod€ls by L.S tud c@bd'3 M.rhod

6.2.2 CompEi.on OfPrcdiction Ol Floods By L S tudGurnlEl Method With W.iboll Plottir8

88

6.1

ll7117

lt7119

l19

120

126

127

7 M.rho& OfPrrrd.t rArdQu tilc EliimrriooOf Probabilit Diltribulions? | Inrroduction 138

7.2 Ptublbilily D3rriburion Fu.cri@ 138

7l choic.ofProbrr, iyDicntlioE 139

73.1 DBqipriwAbiliry 140

7.3.2 Prcdicriv.Alility l4O

7.4 Choi@OfBsriDtors t4O

7.5 Pu.ndaEsfturio. l4l75.1 M.thod orMonente (MOIo 142

7.5.2 Mdhod of Mdiim Likc$ood (MIn4 742

7 s.3 M.tLd ofP.otr! ny Wd8h.d Ml)'Mts (?wlv0 l4l?.5.4 LMoMrs r43

?.6 Qwtil. Esti@tioi 146

?.7 Pro!.rii6 Ot lhd Erri,nll6 t46

lrt3

t50

t52

153

15,

154

155

156

156

156

158

E

7.9 ConfddE hrdv.l.7.lo Robultnds OfD/E

Te3tirg Thc coodr.i! Of trtt Of Prob.blity

Dtu|'ibut'rotr!

8.2 Tsting th. Cio6.$ of 6l

8.2.1 Clt-S$eT.ra8.2.2 Tb. soimov- Kdmg@v T.n8,2 3 Prcbalility Plor Cor.lllion CHfncidn (PPCC)

8.2.4 Ret M@ Squ. Emr

8.3.2 A.D-Test

9 Pr€dkt'or OfFlood! By ftdrg Enr.nVlrue ( EVI) Ard G.r.rrlir.d EdEr. Viue (Gcv) Diltribulio$

9.2.1 Th.Emd vdF Trpel (Ev l) Di.tobd6 160

159

160

163

166

t63

168

170

t1l

t76

179

9.2.4 Role of Stowrx Kut6i! dd L moF@i Rdi6 166

9.2.2 GEV Didrihotiotr

92.3 Goodnd offt lcstt

9.3 lt$Il'tudnii.Nio9.1.4 Gud!tuBdr.g.

9.3B SlkkurB.r g.

9.3.C Kotri Bar.gc

l0 Prdicdon (Xnood! By Finiog Nonr.lArd Log-Nomrl (2) Dilt burioB

|0.2.| Nor@l D\tnibld@

10.2.2 Ttu-P.r.@16t amrEd (!N (2) Ddrihdioi10.2.3 Goodn $ ofFir Tcsr!

10.2.4 Rotc of St m.!r! (Irrdb ed LMorEnt Rdios

10.2.5 Nornrliy Te<.

l0 3 Ral$ nn Disq$ion

| 0.3. A Guddu Be.g.l0.l.B Su*lorBM!a.103. C KoLiBr.g.

l0 4 FidiIgt

Predicdod Ol Flood! By Fttdng Pc.Fon (3)

Ad Lo&PerEor (3) Dittrib|riiotrs

11.1 Introduclion ztstl.2 Mcfioddo8 2t7

ll.2,l P@(3)[tltriburiotr 2r7

11.2.2 L.g-Pdsn (3) Dlrriburion 721

11.2,3 Goode$ ofFit T6i! 225

r1.2.4 RoL ofst wtr6s, Kun6i! ed Ll|e.rr Rlrior 225

11.3 R.alttrndDilq|!.ion 225

ll.3,Acud&Ed.g. 225

ll.3.B Sukt{Bs.gc 22g

I1.3. CKoliiDqng. 213

fl.4 Findin$ 237

11.5 Conclu.ior 139

136

t88

t33

t9lt93

193

194

194

199

203

207

2tb

lt

t2 RANKING OF DTSTRIEUTIONS' AND

COMP f,tgON OF MDTSODS OF ESTIMATION

122 Conpdi$nOlDtultiburim

12 2 I Rdting bY PPCC dd RMSE

12.2.2 compdisn bY x: leltandS-Ktdt

12.2.1 Conp.den bY C' Q , h

12.3 Comp&ilon of BtinutoE

SUMTT'ARY OT RDSTJLTS AND RECOMMDNT'ATIONS

13.1 Teding OfData Ard D46prire Sl4isri6

132 FGqucncy DstdbutioN. C And Ct , w€ibul Plotling

13.3 CompesorofDdnbtdou

13 4 ConPUi$n Of htiMtors

13.5 Coipeilo! ofRdlt! About Dhributiom ofthi! Siudv

*nh rh@ of orhd R.g.ehm13.6 R.mndendalio.dsuggdDns

245

245

245

241

244

250

l3251

252

254

255

251

:4' )5E

conpurd Prdgr.n For The Pdm.td Esrimion Of

c{nbcl Diltribliio! By MLM M.rhod ln C+ l5gu8e 261

E$ple OfIidi.S tN (2) Dfiit$tion To Amu.l P.d(

nood Dan At Guddu Bet€p By MLM Method 263

Emplc Ollining GEv Dslribulioi To Ansual P.tk

Fl@d Di$hsa. D.t. A. Gudd! Bdtg.By PWM Mdnod 27t

EMple Of Ftuina ? (3) Distdbution ro The Amul PetL

Flood Di!.!trg! Du! At Guddu B.rdgc Bv MOM 216

&p€ndn E Mod.l! for lh. Gunh.l GEv' Nomd, LN (2), P (l)a'dtx (l) 279

Appc.dix F Gnph3 OfFitling Dbrdburioir On Thc Dd6 OfP.lk Fl@ds

Ar C{ddu Br.agq SuthrBengc Ard KoEi

B&.gcByMOM 281

- LIST OF TABLES

- Trbl€s Tille Pagc

121 Fr€$dcY Of!"k n@d! At Suklor 5

- 12.2 C.Pdtv, Exp'd'd AND Actual rl@d P4|<!

rllThc BuEg6 5

- 2 \ Sunnnary On TIE U* OfP'ob'tjlilv Dhtibution3

AnonA Countn€s

' ",

Goodt*Of-Iit Tdt Re$lt! F oollood

Htdrologv Sludv For TnG Dsribudd of

A,rud Mdnum watd lAd s€n€s l4

2 3 Goodnq$_Of'Fir les R'$lls Frcm Flood

- Hlimlogv Sndv For Tn' Detiibottoa Of

Amal Mdi'lM Dehrei' Scds 15

- ".0

Sumarv Of Coodms-Of-Fit T€srs ?'rfom'd

Bv Polss And Ann'd ( 1986) In Banglld6h l5

t2l Psk Diehoge AT G!'ldu Sutlu Ard

Kotri B@g' (ooo ost' tr., ?aildonina o! Dala AT suklot And Koli

Bdlgdln D1f'r6'i Ir'rio& 29

- ,.,tt R41r oF \I-w Tdt For lnd.pcde@ And

surio.dilv OfDat' Of P'!k D*hdg'8 Al

Tnc Bed8.3

3 3 2.r Rdlts OF M@-l{hiL'v T'n For Dctt OF

P€lk nood! B€fore Ald lfrd Btr6gr' l|*sArd T4l St.tistic

' I I I Reuls Ofcnubbs And B4l' (lanl (G_B) TdL'

For Thc D!t! OfPak Di!ch"g6 At Glddu Sdraga 32

1112 RNh! Of Gnrbbs And Bdk ( 1972) (C-B) ldt

- For lh€ Dat. OfP€at ll@ds Ar Suklu B@s'

For Diftiql Pcdo{ts

I I 3.3 Re$tlts Ofcrubbs Aid B@k (19?2) (GB) lc!t'

For Th€ Dtl! Of P4L Floo& At Kotn Bdag'

ForDtr r.ilPdio{ts 1l

3.,1I Dddiptiv. Shiitticsol FloodPalc (0m cuc)AT Sulhi BMrgp

1 4 2 DeeriFiv. Slstisdca OF llood P.,lG (000' Cweo)

AT l(oti Bd.ac l5

3 4 I DsiFiw stdidiB oF rt@d Pels (000' cus)Fd Thc BrEs6

42 | P@ntag. lr€qudcv Ddnbution OF n@d P"kAT Sullur Bdage The Efr.cls OF Balges

(S ror tud Oud,tu) t3

4.22 Pficeltlgc Fr.queJ Ddnbui@ ofnood Pdr3

Ar Kori Bdage Tbe Efrccts O! Sul*ur Ba!g' Ald

fte €f..ts OfKo$i And Glddu Buragd To!'tbq 59

42.1 Pe!@l|g. Frcque4, Disrriburi@ Of flood Pal6

Ar Koto Erdg. Th6 Efdt3 ofsutlor And Kotri

BM.g€s Togrrhd Attd Thc Gff@t ofclddu Bamge 60

4 t I Stwne3t And Kunosi. OF Flood Petl AT Sukrur

Barnge For Dtrq@t Pdiods 6l

43 2 Sk€Ma and Kurr6i3 OF Flood Pat! AT Kolri

Bm.gc For Dif€rsil Pdio& 61

4.1 I SkMsr Atd Kurtosh OF Flood Pcak! AT All

Bdng6 idDitr66t!@o& 62

5.1 I Cal@ltlioD For Gin$.| kob.blj'rv PtF and

W.ibullPlot Gs.ltu 8e.g.0962199) 39

5 | 2 Cdalalion lor Cnmbd Problbility ?ap6 And

w€ih Pbt ro. Slll<u (B) (1901 - lelo) eo

_ 513 Cal.d.tiof, For Tn C6pno.Glnb.lProb.bility

Pap6 And W.ibull Plol Kotri Bsd.g€ O90l- 1999) 92

- 51.6.I Conndene I d.l lu Thc Dltr OfDischargc

At Cnddu Bung. 0 962,1999) Gmbd tu lyiicd M€rnod 94

_ 5362 Condd@ Irtdtls For !.rl Flood D$h.€.At su*klr Baftge (1901-1999) Gunbl€ Aral}lic.l M.thod 94

_ 5 3.6.3 Conndde Inr.dd For Tt. D.t! OfD&hracAr Sulh! a.foe Sutlo. Br|ge (r$r l9l | ) Ginnble

_ A.rlrlicd M.thod 95

5 I 6.4 Co.idence Int Nal For Thc D!t! OfDschlge

_ At Sut&! Afrd Snktw Bd!8c ( l9r2- 1999) Gunbel

A!.lyiicrl Mdhod 95

_ 5 3 6 5 Coifid€re lir.dsl Fo. Th. D!t! OfDisch{e.

At Sul&rr Bcfore Cuddu Ed!s. (l g0l - 1961) C!ftble

- An lyii..lM.rhod 96

5 3.6.6 Confd€@ Inrdsl FsThc D.n Of Dlcneg.

- A1sutkur An r Grddu Bmsgc (l 2-1999) clnbl€Alaltlrcal M.thod 96

- 536? Confd@ I ctud For Thc Prcdiclion OfFlood

At Kot i B.mg. (l9ol -199) cunbl6 Ar.lydot MedDd 97

- s r 6.3 Conffd€n€ Intdal lcir The D!1t OfDischlrg.

Ar Kotri Barage Before Koiri (t 901t 955) Gumbt.

- Aralyricd M.rhod 91

5 I 6 9 Confde.c€ Intd.l For Tbe Dai! OfD$Idge

- lt1 Kotri Bamge An.r Koiri Biras€ 095c 1999) Cmbel

Anlyricd M.thod 98

- 536.10 Con6d.@ lnGv.l !d thc D.tt OfD'$lD.gc

Ar Kotri Bd.sc B€forc suktur Bmsge 0 901- l9l2) cunbl.

- Aialfc.l Mcthod 98

- t 3 6.11 Conid6@ lntdal For ThG Dds OfDish'rg€

At Kotn An6 so}lu B@g' (1932-199) Gombel

- Analttical Mlrnod

5 3 612 Conid€ne Intdd lor Th. D!t' OfDrshag'

- Ar (otli Bn.g. BeIorc Cdddu(1901_1961) Glnbel

An lt-ticil Mclho<l

- 53611 Co.ndqr Intwd Fd th' Dltt ofD$hltac

At Xoln Bs!8' Anq Glddu t l9o2_ l99a) Gonbd

- Analtical Mcthod 1oo

t 4 I T- Y.s Flood (ooo'S Cus) Ar Suktur Bdlge

- cdnb.l M.rhod lol

5 4 2 T-Y@s Flood (ooo's cu$6 ) At Ko$i BenS'

- Gmbd Mcthod

5 4 I T-Ym Flood ( ooo 3 Ors ) At lrifeut BMlg6

- Gumb€l M.thod €dl D !) l0l

6 I I 1 Models For Predictio. Of lloods At Suklur Bst's€ 128

- 6 | | 2 Modeb For Prcdjdior Offloo& tu knti Be'ge 129

6.1 l3 Modele For Pr.di.rr@ Offloo& Ar Difiircd Btrag4 ll0 '

- 621 Conpsisn Of ft.didion Of Floods By W'ibu['

L s And Cumbet Suthf(B) O9ol-I99) l3l

- 622 Cotlleisn OF Pr.didion OF noods AT Sukknr B@g'

Before Sukklr Bd!8c Bv W'ibuls Mcthod Tr6d Lin'

- (L. s.Melhod) tud Cumbel Mdhod 090I n 9l I ) l3l

6.2,3 Conpti$n OF Predicrron OS Ftood! AT s!&u BdaS'

- eno Suklor Bdras' Bt w'ibull Mdnod Trcnd Lin'

(L.S Melhod) And cum!.lMcthod 0912n99) t32

' 6 2 4 Coopeison OF Prcdiction OF n@& AT Suklu Bdage

B.forc Gud.tu Bdngc By Wobult M'thod, Trcnd Li@

- (L.S Mahod) AND Glnb.l Mahod o$l-l%1) ttz

6 2.5 ConpDson OI ft€dicdon OF Floo& At Sukkd BeaS'

- Aid c{dd! Bd!s' Bv w'ih Mahod' Trdd Lin'

S M.rliod) And Gunb'l M'rhod0962'199) lll

_ 626 Conpdisn OF Pr.dici@ OF Floodt AT Korri Brma.By W.ital M.th.<I, Tr€nd Lin. (L.S M.thod) Ald Glnh.lM.thod (1901-1999) lll

6 2 7 Conpdisn OF Pr€didion Flood! AT Korri Bar.ge

Beforc Suktu Earmg. By W.ib!l! MeUo4 T@d Li@

(L.S.Manod) AND Cmbd Medtod09ol-l9r l) D4

- 6 2.8 Conpdi$n OF P..dicrion OF noo& AT Kotti Bd.ac

A.n.r Suklur Aamg. Bt wcibuf i Metho4 Trcnd I irc

_ (t .s.Mclhod) And Glnh€l M.thod (19321999) 134

6 2.9 Conprilon Ol Pr€diction O! !le& AT Korri Barage

_ B.for. Korri B&Rge By W.it{l'3 Medo4 T@d Un.(L.S.M.rhod) Ald G@bd Mcthod (1901-1e55) r3J

_ 6 2.10 Conpqien OF Pftdiction OF noo& AT Korri Bo.g.AnGr Kotd Bmlge By W.ibull Maho4 TMd Lin

_ (L.S.Mcrhod) Andcrnb.l Mdhod0956-199e) 135

6 2. I I Compeilon OF PEdi.lion OF Fl6& AT Kord Bd.gc

- BaforcOldd! Bdds. By Wcibul's M.$od Trcrd

Lii. (L.S M.rhod) AND Cut l Mc6od O90l- 196t ) 136 .

- 62.12 Codp.'isa OF Pr€diclio! OF noods AT Kot.i Bdr€cAad Glddu Bgmse By W€ibut's M€rhod, TMd Lin

^ (L S.M.lhod) AND Cruobel M.thod (1962-1999) 136

62.13 Compdisn OF Prcdiclion OF Flood, AT Cflddu BulgcBy W.jhol's M.thod, T€d Litu (L.S.Md[od) And

cumb.l Mdnod (1962-199) r37

^ 7 5.4 I M6 Floq And Moddl R.tio! For 3 Dsta Sci! ro Th.

InduRivd r45

- 7 5 4 2 EstimtB ofccf6ci€nr for PWM'3 For AI Probdiliy

Dbtiburions t45

- 1 .2 4 I L - MoM(. And Rdid For Thc Dir! Of ThcBdlsr! ln TIE lt{hB Ris l4t

- 9.1A B$ic St.ti3ti6 (Guddu B'n'!G) 1962_199 r80

9.18 Balic Slltbtjcs (Sullor),190!1999 130

- 9 | C Bsic St litti6(KorriBltog!) lgol-1999 l3l

9.t A P.6tEld O! Gurnb'l Ard Od Dlrrihdrid In

O0O Cus (Guddu Bd!g') 1962n999 r8l '

. 92.R PsBh.t6 OF Gumbel And G'v Distibudon ln

0OO CuFr (Sukhr Ed'g') l9O!199 182

92.C P.mct r! OF Gumbd Ald OEV Diliribrnion In

- ooo Cut4e (KoltiBdtSe) lgol 1999 132

9.1 A Quantile E3tidt6 Ald Th'it S'El 0n PaMlh'$t

Bv Gunbel Ald CEV Distdbutions Ai Guddu Baitase'

1962-t999 133

9 ! B Quudc Edjnes A'd thdr s E 3 0n P!@dts)

Bv Csdb.l\ Ad GEv Diffibllidr ( sul}ur B@'g' )

lgol-t999 134

9.3 C Qumtilc hdnat.s And Thcir s E s (In Prdhc!4)

Bv Gunb.l Atd Gd Fo! Kori B'n'g'' 19Ol199 185

- 94A TBls oFcrodBoFFn( G!ddrDF!8') 1962-1999 l?0

94.8 T61s OI G.odre$ O! Fit (Sukku Bdrg') 1901n99 112

' gq1 Tet! OF Coodn ss OF rtu (AT (otd rgol-1999 l7t

9.5.A Obstd Atd Suggd.d Vtlues OF C' Ci ! lnd l'

(Gqddo) 1962_1eee

9 5 B Obw.d And E4.<r.d Vj6 Of C" CL rr Atd r'

(Sukkut 19011999

9 5 C vatu.s ofc., Cr lr .nd ! (Kotd) 1901- 1999

170

1?5

lO 2.A P'met€B OF Nomal And L.g-No.m!t

Diltrit{tiou (cuddr) 0n 000'E Cus) le62'199

l02B PUa|acB OfNotuul^Dd rr8-Non4l (2)

Distriburion3 In OOO Cus@! (Sltlor Bdag.) 190l' 1999 210

- lO 2.C PlmclcB Of Nod.d Atd Log-Noml

Disiiblnom Gofr) l90t-1999 2ll

- lo3 A Qu&rtit. E imi6 Ard Th.t S E\ 0. PudhG)

By Nomal Atd l4g-Nom.i Dbldbutiod (G!ddu)

1962-199 212

- lol.B Qurntilc Ertm6 Ard Ttxlt S E 3 ( I! ParotlE!tu )

By Nornil And r-B-Nom.i LN(2), (sukLu BM.8c)

- l9ol-199 213

lo.3.C QusiiL Bdirot s And Tbat S.B'3 (It Pdslhcsd)

- ByNomd Dsnihnid ( xotri ) 1901"1999 214

104 A Rrlt6 OF Qlotile Esidlg ( rrg-Nordl VS Nomal )

- cuddu 1962-t99 196

l04B Rltio! OF Q@liL Esimdd ( Log_Nonnil v3 Nom'd )

- AT Sukkur 200

lOa.C PdqllS.Inc|asInQu.r il. Erlibll6 OF to8_Nonul

r.l Nomal Diniftuti@ ( (olii ) l9ol _1999 For Ditr€r€.t

Retum Paiods 2M

- lo5 A R.tios OfS Et OF Ql4lnilc Elti@t sForDitrcr.nt

Renn Pdiods ( tu Grddu ), 1962_1999 197

- r0.5. B R.lid OF S.E sofq6|n. Estmdc (Sul&t)

1901-199 201

_ lo5c RltiosoFsEsolQuMailoEsdtutdlorDiffeienrR.tum Pdiods Kotri, :901.1999 205

210

l06Bl06c

l0?Bl0? c

l0 3.8

10.8c

II

r12B

ll.2 c

198

202

245

202

n6

l99

T.sts OF Cood&s OF Fit ( Coddu ) 1962-199

Tqls oF Goodn $ Or Fir (Sukku) 1901-199

Telr OF C@dre Or Fir ( (otri ) l9or-r99Norn lityT4( Cud{tu ) 1962-1999

Nomdliry Tst! ( Sulhn )NomdlnyTqrs ( Kotri) 190l-1999

V.tu6 O! C. , Cr And LM@ r( Cflddu )t962-1999

V.!ue OF C. , Cr And LMoGrt Ratios ( Sullu )1901,1999

valus OF Ca, cb t And r. ( Ko$ ) 1901-1999

Sur@ry OF Th€ nsft OI P@iu Snldid

rddd4 oF P (3) AId Ll (3) Disflburioro,

cflddu (In 000'S Cuso 1962199

P&uclc6 OF P (3) An LP (3) Drlihnid!(su*u.) 0n 000's oi6, lr0l-leee

Para'neleff OF P (3) AND Lp (3) Distnlutions (Kord)

Qudtile E dh.t s And TI&i. S.Et (In Pu.dhs)ByP 0) And LP (l) (Gutldu) 1962-199

Qulil€ E.timat$ And Itdr s.E's Gn rafulh.scd

By P (3) Ald Ll (3) (suk*ur) l90l-r999

Qurile Eninal6 Ard Th.ir s Et (b Pdnh6.6)

By P (3) And LP (3) (Kori) t90t-1999.

Priid oF Qudrilc E rimrd (G!ddu) 1962n999

R ios oF Q!.'tilc Esnnwc (Suktuo l9ol n 99P..rios OI QlstiL Esirri6 (Xoiri) l90l-199

203

206

2t6

240

240

241

ll.t B

742

24J

ll.t.c

lt 4B

lt 4c

226

230

xt4

1t.5. B

ll.5c

il.6B

ll6.c

11.7 B

r| 7.C

t22.l.A

12.2 | B

t2.2.2

12.7.7

12.3.4

r2.3.8

t2.l c

Rriq ol s B'i ( afl&il ) 19611999

Rdior OF S.E! (Surh!) l90l-199

Ratid OF S.E! (ktn) 1901-1999

T6tg O! Coo.ln* oI Fi (Goddu) 1962n999

T6ts OF Coodncs O! Fh (Suklur) 1901- 1999

T€s! oF cioodc6 oF rn 6olri) l9orl99Vd@ OF CD Cb tr dd q (Cuddu) 1962-1999

valu* oF cs Ci and t Mon6t tutid (Sultur)

l90I-1999

vd6 OF C. Cb ! rrd L (Koui) 1901-199

RNnling oflh. Dstdbulioi, !y PPCC

Ls.king ofrha Diddblni@r by RMSI

Acc.ptrbility of the DrtibutioB !trd thin orda of Prcfe@s

Showii8 wh.ihd in Ddrihdion @ ,.6!&.1dy lrd rh.ir

Qudlil6 ald 0Fir S.E.'r at Guddu (B)

Qulmi|4 .'d lkir s.E.! !I Sltlu (B)

Qudil.3 |nd d*ir s E 3!(otd(B)

228

231

235

224

23)

2]6

223

236

245

246

246

241

248

249

249

LIST OF FIGURES

Grrphs Tillc Plae

t 2.1 tistoflgramofP.at Ftood! ar C{ddu Bd[!p l1

J 2.2 }irstorig@ OfP6k Floo& Ar Suktu Brr.4p 33

7.2) Hisrodgm of Par Fbodr Ar Stllar B.ngc Ana

consruclion Ofcldd! Be.g. 39 ,3.2.4 rftbrig@ Of Pat Fl@d! Ar SuLlurBdag.B€forc

ConsioctioaofcudduB!.ug€ 40

3 2 5 lfttodgrm OfP!.t Floodr At Sukkur Ba!g! AAd

Coctdctron ofsDl&lr BM!g. 4l

!.2 6 tlifiorigrd Ofr..k Flood! Al SuLlor Bdage B€fo..

Consttuction OfCDddu BM|ge 42

321 EislorigmofP€at Flood! At Koin Bdage 4t

L2 8 tlstongrdn OfPak Floo& At Koiii B$nAe Before

CoEtnction Of Korri Bung.

1.2.9 ttidorigm OfP.rt Flood. Ar Kot i Br@gp AIiaconsru.rion Of Korri B@9.

I 2. l0 rnstongrm OfPdt Floo& Al Koti B.mgc Bef6.

Corslflctior Of Sukj(ur Bdlg.3 2.I I gisrorigre Of!.ik noods Al Kotri Bm!g. Afts

Coddcrion Of Sukhr Bmgci 212 Histoigrd OfPdl !loo& At Kod B@ge aff.r

CoEtoction Of Guddu Batrlgc

J 213 Hisrorigm Oflerl llood. Al Kord Barngt Beforc

consiruciron Of Glddq 81168.

4.2 \ ltrstogrrm & Fr.qu.ncy Cle of nood Fdr al Glddu

B&69. (1962-1999)

4 22 Hinoglu & Fr€quacy Co@ ofnood p6*! rl Su*kur

Bdag. (1901-199)

45

4.2 5

421

424

429

42.10

4-2.12

4.2.13

541

542

12

1t

105

ttls:togan & Frcqu"cy cs' of Ftood P"k st SuLhr

Barage An€. Clddu fB) I 19621 999)

ltbtoltan & t rcqucscv Cud' of Flood P'!k3 'l susnr

BstS€ B€tor€ Glddu (E' ( l90l-196))

Histosr.m & Fcquctuv Cud' ofltood P'ak at sumr

Burlg€ Aner Suklur 8M!g' I I912'1999)

Ui*osrd & Frequ4v Curu' offl@d p"l! at slkxur

BM!8€ B€forc Suklur Badtgc I l9Ol-r9ll)

Hrstosdn & Frcqudcv CuN! ofll@d poks al Kotn

BMtge t I90l_1999)

HitloBten & Frcqu"cv CuFc ofrbod p'!ks at Ko'n

Bs!8c And Cuddu Bsagc ( 1962- 1999)

Histogs & Frequ'ncv Cure ofFl@d p"k8 al Koh

Bdsge Bcforc Grd'lu B$tg' ( l90t- 196l)

t!3logtrm & FrcqudK-v CuFc olFl@d pdk xoln

Bmge A.nd SuL*u B&!8c ( | 912- | 9s9)

rftbgnn & Fr.quoc, CuM ofnood pak! d Kq

AlrEgF B.foc Sutlu BdEgG t l90l - l9l l )

}llstosrD & Fr.qu'nc-v Cuffi of Flood p'irr tl Kotn

B@s. AAd Kotti BtdS' ( 1956199)

fli$osan & Frcqucncv CuNe of rbod F5kt !t Kom

Bdase B€foe Kori Bdts' ( l90l'1955)

Conitldcc I Gdth for P"* nood !! cuddu aMtSe

rlq62-1499) Gumb'l Attlvticti LS& Weibul Melhnd

ConnddEe lntedrk for Pcdk flood d Sukklr Bsrg€

(lool-l@ol c!tr$el And)4rcd Ls& wcibull Melhoo

confrddd lnt.Bik lor Pdk nood 'r

Sukkur Brras€

Befote Sukt$ {S)(l9Ol_l9ll' Clnbel Anslt'lic'l l s

conid.n@ Intd'b for Pcd( lood d Sukkur B8ntS€

And Suklor (B) (1912'1999) Goitb'l Analgicd Ls

& W€ibull M.thod

65

61

7l

T!

106

107

5 4.5 Corfido@ Iltdak for P.!t [@d ,r Su]lor 8e.8.B.foE Cuddu (B) (1901-1961) Glnbd Atrrlyricd, L.S

& w.ibutl Mahod r08

s.4 6 Conndd@ Intw.ls for Pdl ll@d d S!*hr BeaA.

Aner Guddu (B) (1962199) Guntb.l A$btic.l, L.s

& WabuU Mcthod 109

5.4.1 Confid@ lttdds f,r Pcr food tt Kotri Bdrg.0962-199)G!sbel Anrlyrkrl, L.S & W.ibull Mahod ll0

5.4 8 Confddco Ini€ods for PerI {ood at Koti Bdo.€(o|n (B) (l9ol-1e05) ct]nb.l Aslrli€j, L S &

WcibullM.tlbd lll5 4.9 Co.fdo@ l.r.Mk forP.r& 8ood.r Kotri An6

Kotri (B) 09561999) cflnb.l ADlydc.l, L.S &

w€ibullMethod 112

5 4 l0 Conid6@ lntd.ls for P.l} llood rl Kotfl Bdorc

sukhj (B) O90l'19D cnnd.l Adt-ticil, Ls &weiblllMahod 113

5.4.I I Cnnfdcoc.Iindds for ?qk flood !l Kofii Anrsul&w (B) (1932-1999) clmbel Aelytiqt L S &

WeibdlM.lhod ll45.4.12 Cotrida@I qds for Pel lood.l Kotn B.foc

Guddu (B) (1901-l%D c{n$.I An'briol L.s &Weibull M.thod ll5

5.4 13 conffddc. Int€tuab tur Pdl ll@d 3r Kot i Ancr

clddu(B) (l962lee9) Glmbcl Aidrtic.l, L.s &

w.iblllM.ttod 116

Chapter I

INTRODUCTION

l.l. Shrc|nent Of The Probl€rn Of Ftoodq Modeting And Their Estinatiotr

Paldsh is bsicdly m ryiolrunt @unrry &d poss6s th€ ldasr ondBuousirrig.rion sysr€m in rh. wo.ld cov..ing approxin0tely t4 milion b@tad oaeea Th.nrigarion sysl€m is abour l0O y.m otd lnd a tarye ponron of Mter is tost intr.nspotudon due to s@paee. Pilistan has big rive6 wn€.€ diehalscs in summ€rsdon vary froh l5o tiouland a!e! lo l2OO tnourmd @secs Md ciuse kemendous

lo$ ro dr€ hlnan liv.q crops and propedy

Floods e on. ol rh. natunl h&srds with which numditv has tived and

sttuSgle<l in al! 6e @rd.d history Ftoods eer quite fr€quatly i. dnosl atl

countie of th. world Ar av.rag. snual dmlg€ ssciate{r *ilh floods dount toovq I billion dollds li P*ilrm d.mag6 wonh billiotu of tup6 have oeur.<r io

rhc 1950,1955,1956,r9?1,t9?6,1978,t988,199 md t994, bsid6 los of human tite

R@mly, durina rh. yd 1992, p.tis!6 hs bq sbjdrcd ro two tujor atsl@phicddts; first i. Jult and ,Auausti dd s@nd ii S.prmbq In ,uty Dd Algust provi.@or Sindt dperie.*d dc@tioMtty h@ry 6ins rh tu$d inundarion of larsc ar.. InS€Fember, ther. w.E €xcption.tty hid fl@ds in Jhetnn dd Ch.nab avers Thc

other triburary rivers of Indus al$ pm$d high flows sihultan€ouslr, rhe amularivdeffecl of w h ich Senerated sp€r R ood conditions rr varioue controt poims The sir ual ion

resull.d in qide sprad <lama8! to thc piivat. propeny and dmaee ro lh. rrigaion,D.ainage and Inlrartuclur€ inctudins Road6, Schools md H€trh feitities The irritisl

estimates ndieted thar 1he damlg* in th. order of Rs 60 billion equivdt€nl to US

$ 24 biuion look pla@ in rhal panicut.r yd Dunns fte yd t994 addirionll

flood/rain danags wonh Rs lo bilions (ep,oximately) w.re ats ecumd.

D,€veloped @untdq likc Unir.d State, Chmy, Frde. Chin4 eic\ hav€

bd r&ins aote nood probl@ in th.ir rivq! tikc Misisipp', ColoBdo. Dynnb,

Rnyinq Yelow, .rc., in y€r t998 wrious Arn ricd aM Europtu count es and Chimfa@d sedous floods probt€m ofih€ c€.tury

!loo& in th. Indus bdin ee gd.rally qusd by the oau,,l evcnk lik. healyrainfall in lhe !pp.r otchEcnts ofrir€G o. ,.jnfall sp.rieposed on !now,m.tl. H€r7rainf,ll in m@nsoon s.en is @s.d by lropic.t d.prcions Aqerslty fomcd in rh.BayofB€ngrl, shcofwhich nnve innialty in. Norlh _w*tcrty ed lhcn in a

Northely dii.crion to ihe f@r hills oflhe Weslrn Himalayas wiih consequent

orograpnicenhancem.ntof raiofath Flows.xcadiigbankfilcapeitisof Ueivscluonels r€sults in inuDdation ofldge rers, fajlurc ofwucr @nv€yMcc sy$fl, and

dandae lo vital riv.r 31Nct!r4' EnginerinA d6ign r.4uiB 6 int.gntcd srudy ofsfdy, .mci.ncy

!fld @nohy 011n. prcjer u.dd $udy. Th. sf.ry sp@r r€qnir6 not only the

efety olfie srtudurc nlclfbul atso ofrh. tife hd prcp.rty ofp@ple conc.m.dwnich mighr be cndanger€d by its fairur.. Thus larg€ ba@s€s nu$ be dcsisied ro

wihstand extr€ncly high nows qp€cl.d 1o p!s! within rh€i, expecr.d tif. tim.The banage.t cuddu, @mpl€led in t962, ws dcaigned ro pe a nuimuo nood

disoh{ae of Ll hillion os But in 19?6 rh.bs6gec.@udsndimum pNr. tllto pos nood dis€hra. of I 2 mitlim @s which ws 0.1 nillio. us albv. itsdBigned qpacily rnd llckily the baragc ws not dMag€d

The baiilgc al Sutkur, @optelcd in 1912, wa designed lo pss a

maxnun nood dbch.rS€ of I I million cu!.6. Lat€r on i1s apaciry was rcduced to09 milliotrs cuses. in l94l,byctGing l0 spans ofbMrg€. In 19?6, tb. bdrag.@. under nainun prGsurc ro pas nood itilc.lurg. of I 2 ni ion o@, vhich

B 0 I millio. @3e aiovc ns '€d6ign.d

qp&iry. olhd supo nmds p.$.d thousn

rhe b@s., s&ly in 1942, r9B, €tc. Tn€ bs!8. .t Korri, 6npt.r.d in | 955, hB ihe

desiged dpacity ofo 875 dillioi .uses It crDc und€. ndihum pr.ssuE in 19j6, topas ihe flood dilchrrg. of 0.982 million cuscs whicn ws als !bov. ns d€si8n€d

I. ord.r to rclG !h. Foblm ofsddy, n ir n@s.ry lo cstiolt wirh

praision thc hiShdt po$ibl.llood lil@ly to rrivc ar Clddn, Suktur lnd Kotri, at

lest foi rhc n€ l00y.s. Thair mlld b.po$ibl.lo sgSBr to th!dtbonti6bnenEd for rdodcling the bm.ga @rdingly !d .tso to .doF olhd n@u.eviz: construction of snall dac for lrooge ofwlr.r ro stop rh. ttow of hieh p€ak

fl<jod lioto rhe bamg6 beyofld iheir .aplcily ofdiehsgesThe nood ftqu.ncy nod.ting pobtd retat€s rol

G) Cboie of mod.l rt?e (AM,PD, or olh.r),

(b) Choi€ ofdtrribution 1o b. !r.d in ih. chlM modcl,

(c) Choi@ ofn€rhod ofpatuncrcl lld qurn$te dlinulioo.

Two s.parat. asp€cts ofsch choi.e de inpondt Th6e de rh. descriptive

and pred,crivc propeniB ofrhe choFn m.thod The de$rip$r. propeny 'e'aE ro rh.

r.qui@l th.r chos. nodel din ihltion mln b. sr.tuicdly linilr ro rhe prop.niqof rq! n@d si* ed th€ Fcdictiv. pmp.ny r.lar6 to thc r.quiM6t rhd qu..rit..srinarer re robun wnn satl bis md st.nddd dor

A choi@ musr bc nad€ bersc.n flood.etirores b!$d on (r) sl-site dala rlonc

or (b) akne pbs r.giooal data.

(a) Flood .slim.ta may be basd on !t -sit d.ra ato.e it(i) th. -3i1. .@rd is mpionslly tong:

(ii) 1h* a€ ro rcgjoDl d.la Nlihblci(iii) the rcgion is Ery bcierog€n@us, i., Cy>0.4.

Single rt -siic r@ord @n providc linit.d quality .!rimt4 ov€r a liniled rangd

or telrm psiods But lhe bias will (probabt) b. Lfger. Thtu i! rhc linitatior of ue ofal-rl. dah alorc. Floo<l6tinar6 tuy profrlbly b. h.!cd on joi.t u$ of ar -sn. Md..sjonar dala. provid.d a @..bly hooo8.@us fled rcgion en b. idatifi.d. In

rhls @nldt a honogc@us r€gion is ! @ll@lion of olchmcnts who* nood 6biislics

arc honog€neols. Il doq nol imply ih.t .ll catchmenB ir h .re in a confrn d

Thc Btitu{on of p€l flood fow frcquaEy qith unifomitieq dd@mpdbility, requne o.e apprcpdtc Fobability di.riblrion tuncrion (ifposiblc)

to be used dl tmdad. It ds Equircs & eft.idt nahod for esrin.lion ofparameteF of rh. Probrbiliry distribulion, to b. usd

l. ptrcti@, tbe to. prcb ilitv distrib{tion ofth' dait ofAnnull Msxrmlm

(AM) or P.rial Dualon (PD) !r . site o' ' r'sion B unrdoM Howdq fot Dr'ct'car

puooses! sinpl. disdburions !r. oftd Dsd to chdsderiz rh' r'lation b€lwdn nood

;snnnd€s tnd lhen ft€quencie The p'rforman@ of distributioN is 4duat'd usng

difdor staiislicd t6ts Thc P.rfoM@ of m'thods of ednarion is daluaLd bv fie

S E s of 6tim 4. Io. dmplq in U S A sd Auslrclit 148-P@Mn-3 (LP3)

dinrihutio. is u$d md Mdhod ofMon nts (MOr'! it prcldr'd

In Bangladest\ Gumbl., Ing Nomsl dnd cdeBti*d Expon'nthl Vdiate

distribuijont e n*41 In Palisl4, Gdd"liz'd kgisric dislribution alld cEv

distnbutioo have bs r@nmcnd.d for tud'ling 0@d nequ'icv dda or nvs

Indus at Jinnah baflse ed fo. Ihclum dvd al kiul' r6p@liv€lv I 4s I

Thc @rcepr of flood fr€qu€ncv anatvsis a! 4 aid in dcc'sion dakiog h !

usetul onc Hows, it is a lcclnique *hich nlv 'rsnv

b€ nieu!'d Mosl d'si8i flood

caimat6 involve $aridical qlr.pol.rron ofsm knd. tlE ddScB ofwnid should be

SCOPE Oi RESEARCII AND MOTWA.TTO

Fo! lh. !4r $.ntv v.m @omout inpon!@ l$ bdn giv@ oi *tt'r

iesour@s .nd hydrNlic eiginc.dng trcjdtt In Paki$ar, Indus tiv'r is a' 'nPonint

and bi8861 3o!re ofiftigElion, bd ufodundclv no mmprch'nsive $idinc

r€s.eh wo* rcgeding $. nui.l cnd&t.ri3tic of Indw nEr hld b@ wncd

our in the plt, P.niot&lv i. Sindh B.if,g d .Srrolrunt @unrrt n is nd€s$ry lo

coll@t thc dats of ilood pqks ad to snalve it fot ddclopi'g $me r'l'iionships

so@i.g th. $uvial cftdact.rislics ofrhc Indu! nver'

tfrhc nuvial chandcnsti* sc known wc ould prcdicl wirh onsidotble

a$urao@ ihe nunbd of tloods within a sPecili.d nnse ol nasnitDde which &uld

prcb.bty bc dp4ted to o4r dudng 4v long p'iod of rin'

h b€@6 appMt froo th. stldv of 0@d r@tds tlt 4 tic gs ot\

highd p..k di$hdges e. bcinS r€@rdld no.. ft'qu€dtv I indicrr'd in $e

rhbh r,2.r FREQuENcy oF PEAK rl,ooDs ar suKK u ^%L

Ntu: * irdicdE signiient ch.rB6 ii d. % fr.qu*y of fi@& ud4 tdE r MpL periodSou!€e Fl@d Commi$ion R.port (1973) ed D.ta @lLdcd fto6 otr* of Chi.fEngine.

Frem thc abow t!bt. r signifiqt inqq!. ir qp@tcd in tlE prcblbiliiy ofhighd pak! for longd pcriods

Ro$E ror hl8b.r pal dircn.rgdli N* blnds ![ich pre spsd ofooodr wd6 oG rlE lcnd rcslrinS in rh.

incras. of pdt .iva tlow.

ii Morc ..pid Motrin thc upps c.rchneft ofrhc dd sFi.n.iii To@.li.l 6ic in th. erchlM$ ofthe ldbubry nEd.

Thus qelhe( propheh predicr lik.lihood of. psk oa l5 0 lac and I I 0 lm

Guddu ad SukLur bamg6 cp.diEly l20 l.

I

T.b|.1,2.2 Crp.ciry! Erpected $d Acrrd nood Pelt! (niuion cu!..r)

l.t0 150 1.20(r976)

1.50 120(1976)

0375 0 982 (rcs6)

SouG Flood Comi*ion R?od (19?8).

l90l-1976 l90l-1999

,'o50 58 163 7l 17,

53.9 45

2l 216 24 24

80 l0 tt2 l5 l590 1 9.2 l3 t3.100 8.ll0 2 2.6

t2.0 I t3

Hencc, it hds be@me nes!ry to nt diffqdt probrbihy distributions, using

dir.rqt o.Oods of c.ridrion, for for@dng *irh ndil]rm a@.&y lh. lit tihood

of flood &d usins D.scriptivc Slatislics C@mcidt of Skwn.ss, Kurlosis, HisroBrafrs

Rc8r6ion and Co@l'rjon Analt{i! for llsing thc cfer ot @nsttuclion ofbara8€s (ifany) at sit or down srreans aswell ss p.edicanon of0oods.

I.3 OtsJECTIVES

Th. pimr.y objEriE of fr€qudcy &at sis is ro Elrre fi. msnitudc ofdrreme €v.nts to rh€if fftquency ofoc.ur€n@ tbrorgh the usc ofprobabihydisr.ibdions Orh.. objdlive ofthh srudy e:

' To for@.r 1h. @M@ of doods lik ly ro dive ar rh. barng6 @d6 study

by LS, Glmb.l a.d W.ibutt ploui.s m.thods

ii To .sss rh. impacr ofoBttuctio. ofbarf36 (ifdy) .t sit€ o. olhs b.rag6

iii. To idcndry I prcbaliliry di$nbldon frcdon which is apprcpnaE fo. ftcqudcya.aly sis of anoull m.x ihun nood disch&ge u sing coodn.ss of ft l.sls.

iv. To id.ndry ftc .ifici.'n m.iho<t for erimrion ot the pdeass of fin€d

v To @mrEr€ rh€ r€sults wilh thosc obrain d in prdious sludies for s6e baragdby vlrious SciatisrYEnainqs

vi. To iuggBr th. r€vis.d d6i8n.<t dishtge for $" bM.g€s or ro sgger loGovchm.nt ro @ rtuc1 rhe Rsfroin dd sndl drms to slonge the exc6s

I.' METEODOI,OGY

Conparilon of desoptive stslilli.s&d vsrioN probabilitv distributions,

baed uPoi lhe dalt 6[e&d fiom l90l ro 1999, involv* th€ following sl.ps:

ir T€sring lhc tnd@4ddcq Eortog..dty ad sbionDtv ofd.ra using sr{i$idrl

,6u ( w.W ud M-w r6rs)

ii Testins ihe presence of ourlid us'i8 Cdbbs dd Bdk (G-B) t*riii. Iindina M.!n, M€dj&, Modc, Stind&d Deviation, Qurlil€s Frequtrcv

Distribution., C@fficidl of v&i!tio4 Co.frcidr! of Sicwn.s &d Kuno$

Outliqr (D&iPliE St lisi,cs), ctc, ro b. B.d ror d.luatins tbe 'fel3 or

conslru.Jiof, of o.e barrage on th. othd bdrag€s if .nv' dd lor d.Lrnent lhc

sn{p€ of distdbulio n.

iv Drawing HistotiSraN, tliltoextB and liequdcv cud€s

v UsinA ploning ponridro fot p.cdiclion bv Snphic n€thod, OLS 6nd

vi Fn ns P.obrbitity dtrtibuliont (C!ob.l, Non l, b8- Notn l P (3). CEvand

LP (3) on lhc lhfte dals *t3, by tn@ n€thods of€limation (MOM, MLM 'nd

vii Ranking of wrious distnbltion!, u!in8 Eoodnes of fit tats like

Snimov - Kolnorgov t6r, Chi-squs. te!t, PPCC, RMSE dd Lmomenl

- I.5 IIYPOTIIESIS TO BE TESTED

I Tbe independo@ md st.liondity oadal!.

2. Tn Homg@ity of d.t .

3 Th. pr*@ of oldid in ih. dtl!4. Thc posilivc skM6s &d Nomtlitt ofrbe distdbutions. (u6i.g F equ€ncf

distributions, Fr€qumcy Cufl*, Co-€flici€nt ofsreMcs .nd Co_€fiiciml

5

6.

Th. efeE of con$tuction of bmg.s on the oc.M6@ of highc p"t tl@ds

(using so$43 of Centrat Tenddcy, Ditp.rsioi, CGeftrci.nt of veiairoi and

Dr.dictioN by GumU probalility dhtiibutio4 wobu[ and L.S m'thod!)

The afra3 of Tift &il Di.dle bav6 lh. betia6 on tlE o'cll|rt@ oI

hjghd p..t flood! usins wi@! problbilhv diddbuti@

1.6

7 th€ e|!.ct3 of con$rucrion of . b|mg. !t thc olhd bdtgs on lh. o@@@

of bishd p..k floods (usin8 % frcq!6oic0

8. th€ approprial.o*s of probability dittdbutio.s dd oelhods of.stimtlion in

reprMtinS thc shrislicat cheiclsi3lics of pe€t floods !t $rc. bMag6 o'

hdw nv.r in Sind\ usins S.E '.t, X: - tes! Simimv- Kolnogtovc tel PrcC

dd RMSE.

OrginizNtion of the study

Thk slldy is o.eaniad into thind chapts. Cb.Pl.r on€, n a

srlt.nat of th. Drcbt@ of noo& rnd their dtimtio,L to8'rh'r *ith lhe

obidivd, ndhodolo6| &d hyPoth64 to b. L*td Ch. cr r*o, is lit4ttue

fd4 8iling dcilk of cidid ( lidilr ) trudiB @ndud.d in sindlr P*tuL4

Bdghd*h dd oth; o'nrie, . ChlPtd tbin€dr ill. I.d orc, $nMi4lh€ rcsultt ol raarch, l@mn€ndations .nd di@tiotu for tunh'r r4arch

Thc d.t.ih of *ork don.. svstc@ticatlv, i. ods !rc Avm in the

scnd. of $udie a follo*:

Schene of Studi€.

ffiaop'd**fl "oas$addeg'did(otjditd! l{d}.dolop, rt/Ddhar b b' 6d)

( PdiNs Srtrdi€ roi Rid irr Sldh' Paridtur crc)

ffiiuiPeomroronr( Kddig'@ foi $ind du * )

(sratioflnry, !d.9.irb@tlom!.€iv ind pmE oI On i'6)

{ li{.! I{clitt Mod., s D cv . fsl]*)

fl6t ol coNi*ror or Brd&s

Gieosd$,%F.qrfft Dl$iburioE Gticrft'll4)

@!n P,inic'd of rbod3

( Pobdilit Plor!, tDd.ls bv L s lti Gln$" Mdn'4 coI6&G tl{'Mk

ror lr *B)

- r..r, s-K E. P?ac, RMSE Nodriv )

Fii;qPoblttlrt6l*r]hb.rlsut"giffi-'M,M dd PwM d the &h as

( ClFb.l CEv. Nonol , LN(?) P(r) ad LP (3) )

r Er.'@ of Enids, l: sx rcgi PPCC RMSe)

c.mtusioc R4omErdnim !i Ditdim for Fdrhd R.&eh

Chrpt€r 2

Review OI Lirer|ture

ln mmy @utri.s, a unilom l@hniqu€ for dd@inirg nood-flov ftequenci*

G followed in order lo a$ure unifomiry, compstibilitt &d onp&abilirics in ftood

csiiDaiions by vdous adnn€nr sd p.ivat. orCaniations Ditre.enr Stalisticians and

htdrcloSiis have propo*d and us€d wiour distribuiion and erimarion ndhods, totpr€diction of flood of rivqs.

(l) In usA dd Auslnlia rrs-Pdsn Typ.-3 (LP3) distdbutio. is led in ftood

6.qumcy a.alysb t63l md [s01. Th.F @untri.s e v.ry lra. dd @nrain

r%ions ofdif.rdr hydrolog] such !s did, hunid md @ld; y€t om dist.iblrion

tunclion is used firougbout rh€ counlry

(2) li tlK and scvcnl oln€. 6unid.s, lh€ c€nerdtiz.d Extrd€ Votu. (cEV)

ddnbution tu u*d l47l .

(3) The nd $i.ntific snrdy of dal. of Inds rivcr, rr Suktur, E ffi.d ooi by

Nixon had .alculd.d 7 7 lac cuss !s lfi. md a.nu.l llood for 4j ycss an<t

I 1 5 l8c cus@s for 76 y€d, at Sukkur. Firiing P€arson Typ€ I skev curve on lh€

dara for 43 y@8, NixoD .ored Oal the uppd linit is 8.51 tac oecs and lowsInn is 3.63 lac @s Bur $ine rh. dara for 76 ys, rhe uppd lioir is t2 4

lac d*c dd lhrt rhe fl@<ts ofm.gnirude of r2.0, I I O ed lO.O tac cusecs ar.exp{tqr 10 o@ur once tbrie md w€o in6 in s cdrury, $p@tiwly 1201.

(a) The iist scicitific study orthc datr of t.dus nv€r, at Rorri from r 90 I lo t 943

and fton l90l to 1976, v6 6ri.dour by Nixon llal. pNn Typet st.worue va fi(.d and n ws not.d thd r fled of7 tac and 6 tac tu3c or mor.may b€ expcct€d wnh . Fequ€ncy oft*s1hd 5 in IOOO ys6 and tcss rhan 2 in

r entury { 4l y.ds ) dd thal lowd nood of 1.9 t& auscs dd highdt nood

oll0 5lac cuses hwe no chs.@ of o@r€.@ (76 y.r,

(?)

(8)

o)

(6)

(e)

(2)

(r)

In Banglad6h, difer€nr orgsi2ltions €mploy diE r€ni il@d Fequ.ncy

disdbutioi tunctio.s in the Mlysis, desgr.nd pl4ming elldiq Th.

B.ngliddh wd6 Dcvclop@'n B@d (BWDB) dd s*r.l oder orSstadoc

@ploy Evl distriburion, which b aLo loown s Gnnbe| dislribuion. Son.

onsullcncy fims use lo8-nordrl (Llt dbtnbution Th. Mrstd plon

orsoiaion (MPO) ha mdoy.d LP €) di$nbution ror di$hag. dau aid

Feden Tt?e 3 (Pl) dislribuion lor war€r l.vcl dda. In ihc Flood Action Pbn

(FAr), contultdts a. usins cEv dirlibut on 1311.

In lakjstan , Gudb.l &d GEV disldbntioni e odPrcd for Jllcbn nva al

Raul is lifi.d for Ihc dlta ofriv.r Indns al Jinnah bamgc, l4tl, His

rNhs are not rcliobl., s he hs Eind qq.d My mc0Fde of sliFttion not

Bhuuo H.B. I24l *orked oo fluvial chu&teistica ofthe Ind$ Nvq ftom

l90l-1985.

Shaikh N M ard Bhuuo H.B ltl had us.d Fd14 I, UI B!@ Nnd cumbel

o es on th. dlta offloods poks oflnd$ tivd.t Guddu for 24 yea6

( 1062. I985) Th.y sg8en.d @d.ring or rhc b.r6gF

Sn ith N M. and Bhu11o H.B l9l &alrad llE ttlta of hdns nvq al Suklar,

l9ol-1985, usingFdt€tI,lII, H@ndd Gunbel rh€y ohefr€d thal tbeE ee

$re etr crs of @rnodion of Suurar bd.g! $ EI a Guddu badagc on tl'e

dah at Sukklr. They lddked tn th@ i! ! Ecd for fittin8 be11er di$ributions

ro susg.$ r€modelina oflh€ hn.g.

Sunniry of fi. worli dor. bt $n. @.rcb.B i3 p6.nt d t.n:

Dmshtodi lr9l u$d the TR-20 @nDurs prog@ ro tinulate 0oods. H.

u*d obedcd or sioul.i.d ninalll dd. &d vclid wtdhcd nodeb to

6timar. fte flood fr.quocis.Ala.rd.rl2l dissscd rhe prcbld of.srintting rhc r.l.tiorenip h.lw4n (h.

mSnnudc and Fcqucncy of iarc lloods llt us of moncfl 6rio diagnms war

8le db.used by th.m

(3) Kiihyl3sJ disused ih€ radon o@r€@ of@ fioods dd rh. e of. Poisson disr.iblrion in flood frcqucrcy datysis.

(a) vog.l et d 16?1 0rdysed flood dlra Fom 383 sit6 in rhe south estnU. S to explor€ lhe $n.bitiri of ditr rcnr din.ibution lo nod.t noodarequcncics L-nomeni r io di.Brams v€.. lsd by rhefr for s.lerion of

(5) lrr(J and woo 1681 Btimared dnud oood probabitirica usins louri.r s.ries.(6) Smnh 16ll r.pr6dl.d bsin sl. ii fl6d pol disrriburio4 uinS a

Lo8- Nomrl modet

(?) Bou8hlon Jl2l deletoped and thcn a thr.cpemtd distrib{tion to th. mualflood dara 6om ?8 etctftnts in 6t.m Ausrrati. Ee dddred nontinsr

. rehlionsnip betw*. the tiequacy telol sd a double logrithfric tu.cion ofthc r@@@ inr.ryal.

. (3)l Anhed d at J r I cohparcd rhe Log-Logislic disrnburio n toOEV,LN(t),andp

(3) disrribudons by usina dara from S@d.nd ir M foud lo perfom b.rre,rhln orh€r distjbutions lnd hene ws r@nhetuled for tudbc analy,is

(e) Sh&k! &d Fjo 156l $cd loseithdic k&sfo,narion (porer) ror rransfominghyd.ologic d.ra betorc anatrzinS lh.n

(10) JaiD a.d Singh I32i @npdcd rhe SMEMAX r.r,sfomrrion dd ir\dodilicd v4ioro ro UE porer rnnsfom ion by usiog flood {trra

(ll) Majumdar md saehiey {4rl conducted e a$lr{ic{r dd a simurationstudy by Nuhirg th. popltarion distribution lo bc EVI (Z) dirrribudon.

Somc disrribulions werc 6fi.d to the Sererare<t d!ra. I1l€ .uiho6 found thalFosrs's type III dislribotion gave q goo<t s Giimrc s Evl (2) disrnbution

(r2) Lowcry sd Nlsh l3rl .nd rain and Siish ljzl @nparcd a .umber ornethodsofturiis Evl(2) dindbldoi ro srnpl€ dtrta. Th. Medod of MoMrs (MoM)was found lo bc dosl &curate, ncxt lo ih. Mdioun Likelih@d Method

0dn4. MOM ws also foud to b€ vinudty unbised ,nd rhe limplet io

(lr) kes I3El inv.srigaied r.d u*d ceiso,ed dara in ln. csr,naroi ofclmbcldislriblrioi p&n€rds for AM flood flows, lnd pacnt d rn cquarions lo bc

t

usd for ML aslinard H. sls trenl.d .qulrioN ro 6dmt. lar8€ splcSE's.

(t4) t dw.h( ct al lltl d.!€lop.d .ciod,on orpdadEtd of Grsb.l's

' distnbution by P.ob.bitity w.ighl.d Moftnt{Pwr'O. Tbc PwM .stiml6wde @nde<t to the MOM BliNt6 md rhe MLM gtimltca. TbQ FWM

eslinat6 w€re shoM to be comorablc lo other 6riddt6.(i5) Laum.ir ed BuB.s Fol r..on6dd thal Gmbcl\ mbe set to infinny.

.arhcr lhsn ro spl. ldgrh, for .slinadng th. pai.nEt G oflh. disdbudon.

Thc population noMt! l,4 ..phed by rheir smplc 6linlld.(16) Ches sd Mmc U6l ur.d Ev ! (2)dtttiburion ,'d tn€ lndd Flood n thod to

daiv.. rcgional flmd ft.qu.icy tum for s.llwlt.ch.dt in soulh.m

(l?) Shiih I6U d*eloped a fahily oflrathtiql dhrnbudotu and €sinatots ior

€xtr.n. values basd on . 6r€d numbd of th. ldg€sr Mud evcnts. H.

illusr.t.d his nethod by d lopliotion lo th. s lev.ls in v.ni..(18) Phicn [5rl daiEd qF*ioN for vtdM a.d@vui.@ of.srituroc

of rhe Evo) disrdblrion.

(19) Cro md C!|m. {221 iN6ligrLd lhc lstuli* of hjstoricd 8rd

t.l.ologicd fl@ds in qudlil. aimtion. Tney foutd thal in typo I c.Nred-

dara MLM eslindor ws a rob$t modcl for Gumb€l di$rihltion

(20) Mrtlica.nd w.llis (43) usd P (3) th.y obs*ed rhat MLM.ttinates

v.rc lds bied and nld !n lLr S E '3 tn n MoM.srinard.

(21) Sinsh .rd Dirs (32) had found PwM a.mo€nt a Moi4 for P {l).(22) Cu..!@lrsl had F@ .d s s@llot did of the is. orsl*ti.s a

dhnbudon for a r€3ion or d ry.H. ob*o€d thal @NdlioMl 8@dnd of

6t tdts sch 6 Cni-Sqw. $d K-S t6& m oflittle valu. in this @nlql. Th.jt

rdiew is orceted in Table 2. L He @ntun.led EVI di6tdbution.

i

EV1 EV2 EVI GEV lJ{ P! I.PJ

2a 3 21 t7 22 5

2 l6 \2 ll

l8 l l6 l t1 3

lo l 8 l 7 2

IABLE 2.1 Sumn.ry on th. u. of Prcbnbilitv dittribudon! .nong coutrtri.!

TABLE 2-2 Godtrdloa-ftt T6l R6!ll! fbd nmd EydmloE/ S.trdt (1992)

(21) R€,hns and Alm.d I52l had aPpl,€d Goodn s of ft tes:ts, in Bdsl.d€sh, on

EVI, EV2, EVI,LN ,P3 and LPI drsltibution. Thcir rdts 4 p@ted in lh.

following thr€e Tabls .

loi rh.ItirrribltioN of,{M w.t r lrv.l S..id.

Not Tn! r$L shM ft. prolrbility of qc.cdee of tltc i6t Slatici6 aod thehien.$ v.tu. r.l.t.s lo fi. bd fi1. K>-.ol or <.01 or <.01 n4$ thd eith.r EV2 orEvl do rot provide a Bood fit.

EVt EV2 EV3 LN EI LPJ

KSCQ

0.9800109

0.9910 288

K<01 0 9860.288

0.99930 288

0 9980 129

KSco

0.9380056

0.9360.214

K<01 0 9?9 0.9910.317

KSCQ

o992 0 976o.234

K<.01 0.9910.234

0.9750480

KSCQ

0.9000.079 o.zt1

I<< 0r 09850.217

0 9640.340

KSCQ 0.083

0.86?0 317

0.35303t7

o9t6 03730025

TABLE 2,3 Goodn.leof-Et T6t R.ruhs frem Flood Eydlolog Sludy (1992) Forth. Dalrib{rbm of AM tti*hrrg. S.ri6

TABLE 2.a Sunnry ofcood..!&of-fit T6|3 P.rfom.d by Rdrnin .nd Ahned(1986) in E.rylrd.ln

Th.y ddir.d lh! no ddnn. @ncbsion @ld b. drM fton lh. l.sls. How evd $eyrc@mncnd EVI dbfibuliotr

(24) Jnaw.l C.nsut I34l srudi.d thc flood t€qua.ies in the Chao Phrar? rier by

md3 of r oinciddt {@d li.{uocy mod.l. And cdibr.rion, ln. nodel is

appli.d to detmirc th. .fi4$ of r nood @nt ol di@ion .h8Nl which

diEris flood dis.leg. sr Prl Xar on rhe w6l bank of th. Che Phrlya nrc..

Thi! m.aur. b found b r.ducc ih. lows flood lw.l frcqu.ncy in rhe uppd

F

EVI Ett LN P3 LPf,KSco

0.8r00.197

0.9360.234

0.9190.234

0.84202t4

0.312

KSco

o6170.564

K>-0 0l 0919003?

0.9460134

0.8530.037

0.442

KSco

0.99r0 273

0.9910 480

K<0.01 09910.480

0.9910480

0.9?l0ll7

KSco

0.477o t24

K>4 0l K<0.01 0.8830.028

0?890 028 0.022

KSco

0.988015?

K>-0.01 0.93a0.nt

0.9340lll

0 9300.|ll

0955

Note: Sm a in lh. Trbl.2.2

R6ulr. rt 5 % drnn .nc.ld.lLN EVI P! LP't303

l30z

t30

ll

t2l

II22 I

352

852

r@ch efiiciently. For the lowr rqcl\ it is found not apable enoug)' lo redu@

rhe fl@d l.vcl ficqDfty efeivct.

(25) M. AbdB S!b!. lasl pdfom.d fi. Egioi.l flood E€quacy adysat i. thte

et.gofi*, i..., lhc d€tminuon of lhc probrbility di$ribuuoN of th. mnual

flood pcaks of Thaildd, th. divi$on of flood rcgions, dd th€ salysh of flood

fr€quency lor lh. divided llood reAion!.

lour rob&bility distdburionE nMcly Gumbcl, lN (2), r (3) 4d LP (l) w€rc

sel@r.d ss $e crndidrcy for thc dilt butions ofil| amtl fl@d Pat!. BN*d

upon X: - r*i K- S t6t md orhd otuidqarions, lhe Cumb.l dislribudon w6

eld|.d for rhc dislribulion ofoNal [ood p..r s6Th. regr*ion halysit sho{ very Sood coft.ldion betw thc mdn dnudl

il@d .nd th. drrirug. arB for all dividcd oood tegions wnh ihc rtnge of

@relstion coeftici..t betwd 0.8 to 0 92.

(26) Thambihjan Jivljinjn 162l @nsider.! thrc stlnsticd dslJibDions, namele

LP (3) Joh.$n'3 SB ed lmsfonn d gm distribltioN in ordd to in€ti8alc

rh.ir lppli€biliry ro htdroloSt with rcgq& lo lh.ir flenbiliry tnd difiidlti6 in

drimting th. pldcl.G.Il w3 fodd that the LP (l) distlibutions usins nErhod of dix€d noncnk for il!p&andcr csdmadon b spoio.lor thc.ppl'cation i. hydrologv t'lowd.r, cue

should b. tak.n in using th€ LP (3) dhlribulion in dood Maly3iq !s in host of

th. es.i il Lndr lo hav. an uppd bound Thc tresfomed ganna ddriburion

va folnd to bc betq ino lh. Ss dilinbllon 'n

bolh mud slr6 flow rnd

(27) Anador C. Ar8d. 6n.d l3l ffw diff.r. problbility di$riburioB, omdy,

No@|, LN (r), clnbel, r (3) and LP (3) disnibutioB. thc $itabilhy of

h.sfomation and tb. lra.sform.d daia is .rMincd uling th. w-tst and thc

Pa$on resb for nomlity Probibility dislribniiois fitled to rhe data series

too dif.rdr stdiotu \N@ ehin d by u.ing K-s t6t. Ihe suitlbility rlsrh.n r&k d on 3 nadon to dalion bai!

(28) Y&8 Chos-Te t69l ob$std thlt U. di$rBr8€ ltd Y!l.r lev'ts in an

intdlidAl '6 d€p.td upon both lorn lWfr 6oh Oc b&!in ud thc s *ar€r

ld.l5 Thscfor., in orda to 85s ihc ds.lwge &d *crd l4cl! with vaio6

rctum p6ods, it 's

n@ssry to trudv th. joinr prob$ititv and timinS belwo

high $a w.l.r ldels dd h*vy rrin stoms in the sludv ar.! Sin@ tbese two

tacto6 arc not iod.p.ndenl €ven6, thc lssoe.t has bccn bed on

Blth€m.lically nrcdehg t nunb.r ol historic.lly @rd.d svc. slom evq$

(twhooB) in de tludy sa Fiv. pdfs wldcn e comtronlv o$d in the

frcqudcr.nlysis of fldd P.!k, nDdv Notud r-N (l)' P(3) LP(l) dd

Oumbd dinrihnliom, w. @mps.d for th.ir good6 of it to thc tuitum

watcr ldeb and di$hs8.s obl.in d lion hvddulc onPubtiotu at sl@lcd

points along rhe Tdsui rivd

(29) Iahiodditr chowdhury pol @nparcd six cmdidatc dhtibuiions mmelv

Cunbcl, LN (21. LN (lr, CEV, P tll dd LP (lr disLibulion! in fi. di't otnver

ii Bogladesh. The goodngs-of 6t of ondidare dBuibltio.s r. comDdrdl bv

Ning PPCC dd RMSE. On llE bsb of PPCC and RMSE CEV TN(2)

disrributioB @ siml.r. dF slltini..l b.hlti(lm ofbo6 AM Mrcr ld€l and

diehlrge data enpl6 in Besl.d6b Gumbd, LN (3) and LP (3) dtttiburions

sn ro b€ not $ilrbl4 for tequocf edvsis of AM wslcr l4cls in

B&gl!d6h. The P 0) dislributions !.d to be not suirrbt. for AM di$bd8'

dBra. The performane of ih. OEv distribuion is fte bst follow.d bv LN (2)

..d !(3) dtribulionc in th. ce of AM wtd tdel dah Th. p.rformM@ of

rh. GEV disrnbndon b th. bdr abo in th. {@ of AM dischlt8c drt. follow€d

by LN (2), clnb€l and LN (3) dittlibutlon in d.$oding ordd or rh'ir

H.kton l2sbl 6rr.n nie dbttibulioc virh rhe d.i. ftom 45 unr'gulared

sllans ir Turk y who @nclud.d that LN (2) and Evl (2) di6tributioB wr€

sup.!io. ro orhd distributions.

(31) Oioz ud Bay&ir I4Sbl usd s.v.n distribuilons on th. dall of tot l of l819 sne

- yds from t9 stalioc in thc qorld md found thar CEV distrib iot wa

$plrior ro olhs dittribulions.

'rlI

o0)

11

Chapt€rf,

TESTING OT DATA AND I'ISCRTPTIVE STATISTICS

3.1 Inlroduction

k is gcodal praclice that the dsta to b. us.d lor Statistical Araiysis is re$cd for

Indcpddd@, StalioMiry and for fie de&crion of ounrqs.

A diehb oo for AM 0mds cmor b. cho$n sldy on rhe b6is of $6r.rrcdargun.rts. Th. chdcr.ridiq ol obseErd food dal. nusl be tletmin€d in r suitlble

fshion so thal some distribDrio.s @ be excludcd if $e 6doh spl6 fron rhem do

not have chardc|.ristics in @nmoo !r'iih ob3ded nood dala. Thus Hislorigram! arc

drawn to s* thc b€hdvior of ocqrdc. oi floods, b€for€ dd aAe con$tudion of lh.berdgcs and th. Chraci.ri$ic ( M@, Mcdian, S D, C.V, . )oferchoflh€dltas.lsde 6lcul.r.d to 6nd th. .tr{|s of @nsiruction of bdlge.

L2 D.a..ndHistorigr.ms

3.2.1 Datlrnd Sourc.

Dat. on p€ak disoharces at Cuddu, Sukhr and Korn bans€s on Indu6 river,

used in rbo prcscnt study, hrve hd. @llcdcd fiom $e ofiies of Chicf Engineers

Clddu barhg., Sukku banasc and Kori b!ftrg., rcsp@rivety. Dda is shown i'r

rabb I 2 | In ord.r ro fiod the cf@c of @nntucrion of bsrg€s\ the dak s tot

Sukku banac is panitioned inlo four ubs.rs dd rhlr for (otri bmag. is pmnion.d

into six $brrs ai shoM in Table I 2 2. h rhould b. nor€d lnat th€ us ofd AM sri.smy i.volv€ som. loss of informarion. For *ample, rne *@nd o. tni.d pek wirhin ry.dr hay b€ g.eai€r rha. lhe tuimum flow ln oih€f yem bd yel th.y sle ignored

r'bwcver. the AM nodel is slalislially moE effici..t thd the P D Dod.l Cunnlie Il51

l3

3.2.2 Elao.ia|:D ofltrltHidodgnn|!of 13 !..!ofd r@tuminGtryh!No. 3.2.1

Iii. inbmrrid lhdr food Fr. oDrd.cd to6 l{4odgr|irpft.ortcd b d|c forn ofllhld .nl clun| |. !nd!:

ro 1.2.13.

Ouddu >l l 3E 4 194?8,85,88

>t.l 99 t958,73,76,78,8658

Kolri > 0.8 99 1955,56,?3,94

&XcrilEL

ar suklu (Bl

surhI (B) Koti (B) c'ddu (B)

It90l t9t2

I

t955 1962 t99,

0

sl|.ru (8) a{ddu (B)lll90t 1932 tg62

Ir99t

Fr@&015ADoE .lurr. ii.lic{. rht

(i) No $F Fbod oconrld d Kor.i hn gr or Sorhtr b&8. hd@ tt*@nnrudlon of S'nhr bsn8.

(ii) Maitun.tr.r on ilE o.srtud..!d tlqElcy of o@rc of dp< Eood!

.l Surrur brrr.g. i. du to tlF @EludiM of G!dd! blrna. rn d Korri

b.rng. n dE b dr aEsnxrid of bofi Kdri !d Grd.tu bir€€.

3.3 T€ling of DrtrTF bdc lgnpioo i! drliliql udydr of food t quftr r rlE

Ind.padaE.rd Sulbriity of rL drt! sh. rhid.rutrptior i! th.t tlF d.l!itnon|%!'tous, i.c, n h! om A06 dE r.m didihdior

Frcqucicy .n lysis of hydrclogic dlra .cquir.s rial rhe dala be homoad@us

and indep.nddt. The rdricrion of honog@ity asures that alt the obsdsrioN ...from thc samc populaliof, (i.e , a s&€m ga8ing srarion hs not b€q nov€d, a w.t.Bhed

has not b.@ne ulbanired, or .o slrucrurcs havo b.er placed on rbe strem or its najortribut.rj.t The rstriction of ind.pdden@ !s$r.s that a hydrclogic ryent ,uch rs a

singl. l!rg. stom do6 nor eoter th. d.ra *t nore ihu on@. For .Mptc ! sin8t.

stom systen my produ@ two or 6orc 1a.8. ru@If pets o' y one of which (hcrargBt should olcr lhe dsta sd. Iunha tio.lh. pr.diclion ofrhe ft.quocy oftuturc.v.nG thc Grncrion ofhono&ncjty r.quires rh.r th. dala on hmd bc rcpr€sdrrsrivc oftuurc nows (i.., th.re will be no ns stdcturB, divdiotu, land u€ chd8cs, ac, in

rh. .ac of strem flow dals)

It is uslally Nsu€d rhal atl ihe peak naeritudes in rh. AM sri€s are murually

indep.ndent in tb€ S taiistiql sene this .ssunption is unauy j Ntitied

3.3.1 Tsts tror l.d.p.nd.m. And St!(ioDrirtW.ld-Wolfoyitz (W-\D i6t is ued 10 r.st for Indcpmd@ of a drr. s ed |o

ldl whalhr lr.ndr dist in ir lal

(i) IL Dsta is lid€p€nd.nt dd Srrtiooary.

Hr D.ta is nor Indepad.fl.nd Sraio.ary.

(ii) The Signin@ce lev€l is sel al o= s%

(iii) If rh. .len€fls oftbe Safrpt€ xi, r , ,.., t N se indepanldi, lhen rh. r.srStatistic R giv€n by

^= L 4 x,,t + r\ xt

aollows a nomll dinribuion with nd i..d d;, wh€ret = VDl. ,/, isthenh6ondt ofrhe smple about the origi..

..) -." \dl =!i:-t- i' + l.{ -al rt +arl r' "r- ^'/- /v-l (x l)('v 2)

- ("1 -", )tv-l

(iv) rh. t6r srdnic U- (n-li )/d, islpproddr.lydrEllydbdbur.d

wilh ild 0 ad vri.icc L

lf tlE lel vde U k l.3s thrn lh. critic.l !du. i 5vo l.v.lofsi8nific.re w..cc?l tl|c htTottc* of in&Fdd@ ad ndionuity dh@is lh. hrpothBi!

(t) R6ol.! .trd DL.uIo!TIE t@lB e 3hoM in T$1. 3.!. L I

As thc tca! Sbrilric u k ld thn rhc qiticd !d@ !t 5% ldd of.igni6occ, E !c..pt dt hypodErL of Ld.Frdqp .d $|tiowity . Thus

Iidu! ric d!r. d ach ba!g. @ @ldrdcd to b. irdcFdort Dd !|iri@ry.i... rrcd .xint

l.J.t Tdl! for [.orog.!.lty A.d Strtb.rtilyMal{N - WnITNEY (M-W) U TIST it !r.d to t !t for honbg.nchy dds.tiowily ofr dlll !a l58l

W.stn !.? null Hyporiatu s

(i) fio:TlE l{o sd6 or!i4 nr sd ,: (r, <,,r ) .onE tin id6tic.l

Er: Tha tN e6d6 e mr ila icrl.

(ii) Thc aignifi@oc lcvcl h !.r !t o - O05

(iii) To ..rry out th. t4!i *! .mgc .ll n, +n I ob€qv.rioln of tn @mbir.d

splcs in thc odd of ifuering tugninrd. .rd Bi8. thc ruk,|,2,....,n'}nr to thq!. In ce oftiB w aign tlE !wrg. of .d nnkjRr= S@ofmt!lssigrEd to obr. |lb in

'lnpLR1= sufr of nnl3 i. em9L 2.

u, =laa+ala+rll2l-R'

we cho$ tne sd$l!.r oftbe vdue fouid for U! rnd U,, ddoLd bv U

!n this t.sl two 3mplcs of sia nr .nd nr ( nrs t, ) @ onpeed

(iv) The 1€st sratislid b u, which b@rne3

U L,,

As nrand ni de both gr.al.. thd 8l Z i! approximtely st3nd.rd nomal

(v) rhe c'iiicd r.sion is lz l> z!d'

(O Nult hyrotheb s rcj{t.d ifth. crlollt.d ldue ofz ir gEldthd the 8iv.n

value in the table

(b) Rdrltt Dd Di*o$iotr

Th. K s e sltM in T3bl.3.3.2.I

Ar lhe NuI hypoth6b t.e.9tcd ar t% lcrcl of !i8ni6@ w Nm tl[l 'ihe M smpl6. bcfor. sd tnd @nltruciion of bstg.1 oft fion d.dic.l

populati06.

4=1n,4+ry1,4+rl1 z1-4

3.3.3 T6t for oldiNThe pres.@ of outlicr! in th. dlta .Ni.s dimdlti* wbo litting d

did.ibulion to rhe datt r4w lnd higl Ourlid. ar€ both possibl. and havd

ditrer€nt efe6c on the aialysi3, Thus it b@nca nessary lo p..aorm Oullic6

l6t befor6 doinS dy Stali$icrl .ndFi!.

GRUBAS AND EEC.I( (G"!) tf,IlT to D.r.c' Oudh" l'l

(.) St pt

(l) li ll|it l.n dF qultitic. xt.nd rL e cdol'tcil

x"=aeltvx,s)''"=*p(rFr,")if-(" =-l.6220l+6.28446JV'/' -249835w'n I O491436lVI/r -OOl79II/V'

N= SmPlc !i2; dd 3 rr th. nan .td S D of dr itllt l loSaidn of thc sPlc'

rdp.cli*ly Kx ! crlcd G'B ndidrc V.}l6 of X{ e trlul|tld for vsi@r spLti'a !trd dgtrific|nc. bvclt, bv Cobt6 rd B.d( I a I

(ii) SdPL v.lu€ grdd thu ri o. cdiddld to b' hir! Oudi4' whl' $o!'

let thd & @ druidcd to b. lry OuOd

(b) Rdllt rtd DLdio.TlE 6It! u. .hwn itr T.bL. tij.l to 33JJ

Thc t!bl6 dloeth.t th6. i! rc Oullia in r[ ('rlt !'i! sc?t oic (i '982

lrc q$"!)'

i. yd 1956) in $. d.lr d ofKob b.rrrg. bcfo€ @ndtuclio' otcuddu bMlsc

I! so|.r.l, ordi.ta c qclud.d nln $c sdv' bql if ii i! cedcd t! r@ thll

aM nood con. ft@ ts vctv drfiddt $b'popuLtio!' rnt ddkt! tun ba i"i"d if

th. mplc i! ro b. rcgld.d s 6don ud uibi.!'d

3.4 Dd.riptiv. St t'uih3 ofDrlrD*dtlirc Sriliniq of flood P..r! rr Sulh! rd Kotti Bmgd for wi@t

Fiodt b.f@ ud.nd @rdd.lio ofbor.gs "xl

fq tt[e bcbgr! e lhos in

TrbLr 3.,1.1 to t,aj.T$h 3.4J ro. Srt b' trn!. (5 t n) .!m lt!l:

(i) M.n > Mcdln > ModG, v dic.ibxiou - t!' di'l,ihltioo @ +tv

Jc*.d (la &.quc.cv fo. vqv hid nood!)

(ii) All th. mcer.s of @mr.t r.nddcy !.d disFRion ar Suktlr ircr.e !n.r@nsh.rio. of Sukku ud crddu b&ns6. % iicit@ in 11!@6 ofdhp.Biotr k Surd ftrn in n a$c of enrrt radacy which r..!hs ii rh.

Mi.imum uluc and Dr d&1!e *hit. Mui!|M !d!. ed D, iM.e aft.,coisrtuclion ofbangB.

TrbL 3.a.2 lor Kotn b.m8. ( 7 rab) lnom rhrt:

(iri)

(D M6 >M.dio > Mod., for dt ihc distdbuiois\ o.epr for rhc d& for BcforcSlrkw(B).

All rh. dilrdburiom .e +ry st.e.d for rh. d$. bd@ Sut*u bd.E AlrhosSh

Md < M.did< Modc yct S r - 0.2?? be!!$ ditrcl.n€ b.rren rhsn is

(ii) .Z il|clt@in tlla$cof dtpflio> % inc|q* ii l|l@r6orenrrtt6d.ncy (6 lr Suklor (B) ). na.c, C.v. itut!&

(iii) Q,, Minimum vdu. Dd D, da:!e |n.r @nrtucrion ofrhe bsna6.Q', Q D., Mdimm lde !d D, ircrce lnq @nntudjd of $e bmsB ( .s|t Sukkur (B) , qc.pt rh. Mrrimm v.tu. 0. 982 hjlio. whi.h is & OurtiaT.bL 3.a3 for rhE hnt6 ( t *r. ) ltM ln.r:

(i) All$.nlc&cof enrdd r.ndary (q4FD,!nd dLp.dion (d@pr C v)for d a ofpsl dis.hlgB !r Su*to. (B) lic b.rqq rhos u Kolti ..d codd!bM!a*. Thi! ihows Ih.r p.rt di!.hjg6 ,r Gd.tu (B) e hignd rhesukkur(B). d Xord (B), in o.dd.

(ii) Th. dille@ bdvq rh. rrtle of$6.3rldrric,r Kori ed Sutt!, bh|a6.rc r.rg.r the lho$ b€{w@ Sukror ed C!dd! b.r|gs ( i... rhd. ie .ttd of

M.3n > M.die > Modcj fr d.i. !1 lhE bM|g6.(iii)

5 lindingg

(i) IIi origr.n.

G) A.ll the $per floo& o@ricd lnd th. coosiructio! of&y b3dase

(b) Mdinun €trect n aiy banr8. is du. lo ft! @Nlruction ofa b.rage in up

stt@, i.6. of Guddu bdr8. at Sul(ur banla. and of Suklor bMag. it Kolri

(ii) T6titrs of Drr,

(!) 4ll tle data sers !l eh bMagc aE ind.p@ddt ond slalioDtry, i.. rend

disrs (w-w rest).

(b) Th€ dau belor€ and ai.r rh. @n$ructio. of.ach baase @m.! fo6id€.rical population, i.e. thc ddr aE homos.nous (M-w ldt).

(c ) Tbd. is only on oullier in thc ddr s.t .l Kotri (B) before thc @nstruclion ol

Ouddu (B), i.e. in 1956.

(iii) De.ipriv.Sr.rtris

(!) MalPM.did>Mode fordlrhcdlt &tslr.!l b.rages, q..pt forrh. dair dKotri (B) befo'e sukkur (B).

r lte dislribution! !rc +ly skc*cd.

Ol

F.€qlenry of Fl@ds dccr.&r r! th.ir mgritude i..M*.(b) All th. ma$.e oa C..tr.l T..dacy !d DbPssion ofrhc dlrt at Sukku.

dd Kolri barnsg in6.e lns th. @nllo.tion ofb.nrge (i . rh..G rccrals or@nno.rioi ofb.n s6).

(c) th. dkpqdotu inctt& nm rhM Nd.g6 s.h rh!. C.V d$ ircr.g.ncr $c onslruction ofb.mg6

(d) Thc nagnilude oflowcr food p.rt! d@.Ns and ofhishd fl@d p@ts

incrcas aftd lhe consttuction ofblrag6.

(.) Alt the 'Esr.s or Ccdtnl Todqcy dd DtupaTion for thc dlu Ni Sulftu (B)

li. b€tPq the Korri (B) and rl Gtddu (B). TIE dif.|uB b.trd rhc

vdud of tn@ lt.lirti6 fo. th. dalr !t .iy rrc bar|ad dcrEnd lpon the

dbiancB bct*an th6n.

3.6 Concluion

L All rhe di$riblrioru .a poslr\dy Sk.w.d.

2. Tn &ta ar. sitablc for $atbdc.l ai.lysh b.ing i.d.p6ddt, 3tlrionary and

homogd@s

3. T1@ is only o@ oldia in th. d|lt s.t rt Kor.i (B) bcfor. lh..oNrdcrion orGrddu (B).nd n i! r.rdrEd i. d.ridicd ut3i..

4. nE dispdiom ircBlc norc rh.n .vcngq, tt @gnitud. of 0ood p€rs

dcc@sa aod othigh.r ilood pqki in@!a !ftr fi6 coNrtuction! orbea8.!5 Histodgrms ud C.V indicor. rhar thc ctr cts of m$rucrior of a banagc on

olh.r borEgd d.p.trd lpon the dbtll|cc b.tw.€n tlE!.

T

PI D

rtl.!.tl Pok Dtuh|i!6.t Coddu, Soltllr, rnd Koai E|nr. (00o crs

t90l 564 4t1

t902 243462

t55

t905 162

1906 504 J91

IEJT )621908 645 412

1909 514 156

t9l0 561

l9|l 149

t9t2 584 178

t9ll 519 186

l9l4 997 500

1915 454 108

t9l6 588 331

t9l7 592 505

L9l8 421 356

l9t9 423 492

9)n 5tt 4t7

1921t922 436 415

t92l 5ll 413

1924 106 552645 451

'1926 415

t921 625 415

t92a 4n 121

1929 702 550

t90 657

l9ll 390

t912 572 501

19ll 627 608

t9l4 491

r9l5 6i13 .t3l

l916 500 372

1937 t39l913 515

t939 485 396454

126

1942 510

774 t50

t944 694

1945 661 515

504 42r

t941 356 286

1943 tt565s 622

1950 661

l95l lr3t952 594 481

1953 608 577

lg54 426 lt4t955 8ql 804

1956 998 947

t957 707 561

1953 t091 164

r959 679

1960 684 502

l96l 648 446

t962 441 432 301

1961 548 525 i22t964 ?ll 710 516

t965 410

t966 6ll 665 6t91967 673 657 519

1963 651 586 t6l681 655 551

t970 726 260

l97l 6ll 582

t912 408 j15 UIo

t97l t034 ln6 | 811

\914 124 t06 | 164

r915 too2 t05t I 49O

t200 l2oo 791

\917 541 522 t491973 ll56 ll16 122

1919 57A 501 J0l1980 652 616 281

l93l 129 631 ll6t9a2I93l 159 755

644 t7lr985 137 t361936 |73 lt66 501

1987 t43 116 l5l

1933 I l6.l llt9 661t989 9t5 9t0 109t990 589 551 t07t99l 606 561 t00t992 1086 t064 689l99l 626 564 420

711 757 425t95 988 986 7991s96 588 551 4t21991 187 E3l t2lt98 667 629 295t999 4t9 189 220

.Trbl.3.l.! P.nidotrirs olD.tr.t Suktur And Koti E.rrrs.r Ir Ditt (nl P.riodS.No.

t90l- t999 99

2 1962-t999 t8J l90t-t961

1932-t999 68

5 l90l-l9ll llS No.

l90l-1999

2 t90l-1955 55

3 1956-r999

l90l-l9ll5 t932-1999 63

19621999 l8l90t-1961 6l

S No. GUDDU BARIAGE PERIOD

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Chapter 4

Tt. Effcc.' of CoBtruc.io! of Brrsg.s od Th. D.L ot Ftood P.akUsing F..qu.rcy Dturibuaio , Slawn4 .!d Kurtosr3

Ti. .fi€ts ofco.sruction ofbsBges on lh. 'snlpe and tpo of dislnbudore' 6 Nell 6maSnitude of lhe lood pql$ dl &y b6Ege !rc studi.d Sblisically lsitg Coefiicienls oi

s*ewne$ ud Kut6is HidoErms, P€renIaae fr.{u.ncy disliibutioft lnd Flequency cunos

!r. also used for invetgaling lhe inpa.l of o.. or dors bdrag6 on lhe dda of floods De.ls al E

biraAe on lhe river Indus in t don@d strcd posrlion.

Tne dkdburion oa nood naCnitud6 in ihe AM eri6 alwrys hd soe valus lo de len

or thc nodo Tb6. lalter valu$ ftfl*t lh6 p(!$c. ol mu.! ndidun llood roi yed havinB

Floods in sdi rid bm g€nd.lt h!v. huch hiSh& C" ihd thde ofhum'd m6.H.nce. loose. E@'ds ( ufotunarely nor on.. *aihbt.) e tqu@d tor such ac Al$ the

losr AM llood v.lu6 nay be moc of! d8lracrion lhs vtle lo lhe Eridtion $hefre. serioG

cdsid.tulion should bo giv@ lo @on.g O. lo*r AM vd!*. Sim thn would 16. v.ry

te\ ll@d vrlu.\ ar sh slriron rlHnpeidDe lo 4 rcSronal strmatun methods'nsu(h

wilhin a ountry wnich hrv€ difiqcot Qr/ Q v.u6 T relalio.s son. ctrchmcnl lyp6

nny have sle€per €w€s (I.€ei C, rhM othdt &d lhis dillerdc. may t€ a ruolion or

cltchment cheacleislics olhe. lhu aGa slo.oi sin€ sralcr qustilc d@urdcy dn be mh'eved by

groupi.a oalchde.rs i 6 hono8enous arolp3, .Iforrs should be msde in uy t ood cstinarion

scbem. ro check od rqioral honog.n ity' A sn.ll mowt ol rc8)oml heterogdsly is

rol€oble and in such 66 re3jonal oood esidation s.henes slill p.rtom bener $u a! sile

wtil. mdry lh6Elic.l foms of problbiliry distribulioB r@vide a saGfactory fil lo

histogtus within fie oh*Fcd @gc olp.d( valu6, su.n disttibulioN nry dillq in lhc shape

of thcn t!b. The ditincoon bel$@ di(tq..l $rp6 of Q- T relarions is of Cdl Dractrcd

sitnrfid@ it o{ €Ilediv. ramn.nddioN e to b. madc by Itood hyd@logsr5 The

histosu alone, bed uurlt on 50 or L*r ob*o.lio.s, is ulbl. lo oid lhe t6t oa

discrininarins b€tween durnburions rlils. Thercfore, collecrive evidedce lbout disLibutionil

behoriour of 0oods neds ro b. srldicd. Thi! b.ha our is r€fl4ted in rh. higher dineBionl.$monols q, C, ad Cr dd in lh. crlrd. ralu6 ol sdp'6. sucb qumririd arc di*6cd in

Apd rron 8@ind of fil 9p. t .E, infomdi@ rbout d4rnbuuo ryp. should b.

inf@blc rom th. dindi6l6 nomori Ct, C, !d CL, m6urd fron AM dal4 Vdio6dislribuljoN mly 6. diilinsunh.d b, rh. rddiorl'iF whici dis bdwg cv dd q odbel{eo c" dd q 161 4h di$nbudond lom. Sonc or ths disfibulio.s @rc ptoD@d

innialy bda6eofdei.ibiliti*lo nod.l difi.r.'n!hap6ofhalog@ shile oth.6 h.v.b..nrccomndded on lh. bais of dcductivc @dins,

'nE Eff€cr! ol Consrrucdon of Bd[atr lrlq ?€rc€trt.8€ Fr.qmncy DistibudoN .nd

Hydrolos'sls rc on n facd *ilh lsg. qudnaes of dat tequrirg ddlysi!, Sinc. n is

dimtu[ lo grap the toral d.ti DicluE lroh itbubdds such a TaIL 2.l a s.lul 6d n.p in

dcl! aalysG is ro plot $. d.r! a . lGqldE hklogmIn seLclina a p&rialu drtti.al fom fo. t li.qeftt c@e one nry b. r.npt d ro

sel6t a dislnburjon wilh a 1.4. ,unb.r ol plnnerea G@r.llt rhe noc pde.rc !dinribuaon h6, ihc b€|te. it will .d.pr ro d s.r oa d!r3. Holgq, for lh. spl. sia usually

.v.il.ble in hydrology, |he r.li$ility in 6tin trng norc rh& 2 or I pd@d.6 my b. quiL

bs Tnu a conp@ni* 06r b. m!d. b.rw6 flenbilily of the dislibulioh dd E|trbjlit of

Graphs 4 2.1 to 4 2.13 show Hinogrm3 dd Frcquncy Cwes for 13 dala sers. fi.gdpls indiqte that lho P..k noods of hi8h.r magnnud. htve ecur€d *nh smalls lrcqucncy

foi rhe dtu dRer @Grruction oa haftg.s

(^) Ar Sukkr (Th. .llet of Sltklr (B) od Cudd! (B) )Pcrcalscfrcqo.ftydisliburid3oall@dp6ts!lsutlu(B)b.toEdd .n.rrh.@n5iucton orsurbr (B) a wdl a G{ddu (B) rc d@ in Tlbh tlJ *nad 3how

All rhe dkrriburioG & Muldnod.l.

Th€ diskibutid b.foE sur*u (B) i! l,pb Kdic ln.r Sdd.ur (B) is Pl.ty

Kunic ed and Clddu (B) k noE Plrty Kunic (i a dDpe chegs)

(D

(i0

rrequen., oaloN noods decrees bui oly.ry tow ed v.ry hish floods

nc.€as, sDeiarly !ftq Glddu (B).

4J

(B) At Kor't Ohe.lfsr orSuk*ur(B)tnd orKot (a)rnd cuin|u (D rog.rh.r),Peioento3e fEquency distribuions of n@d pak d yori (B) heto@ dd oner ine

conslruclion olKoti (B) s w.ll6 Sukkq (B) !6sivon in Tnbb,r.2,2 which

(i) arr rhe dislriburioG lre Plaly Kuni. ed Multihodd (sn pe does not chege )(ii) Frequency of hodeBE noods d6cr.4q .nd of tow ed hiSh n@ds incr.es aner lhe

consltucrion bolh bsras.s (i.., Sutklr sd Kotri)(C) A. Kotri ( Th. €tf..k orsukklr (a) rd Korrt (E) ro!.rh.r r.d ot cuddu (B)).

Peee.hse frequ.ncy distributioB of flood pals alKotri (B) b.aorc ud rner |ne

consltucrion of Koln (B) s rc 6 Glddu (B) d. Sivd i! Trbr. a2J. whch sbos

(i) All lhe di$.ibuaoN se Pllry Kunic ed Mutribod.l (s. 6 lnd Sun( (B))

(ii) Frequd.y ot very low sd r.ry high noods i.cr.s6 Dd ot bw noods datlrean.r lhe consltucrion of$€ bar.g6, speidt aRd c!dd! (B) (simit& lo dE erT@s of

Co.fiici.nb of Skdn.lr rnd Kuraori3

43.r hportoc. of Cnc'ind qThe s D of o s;n6 is sd.6tty prcp.rlionrt b sahodr sizc .x@p( lhd tto.g a singte

ds 'l

sonefines decr.M wilh din&@ dow|r.e. In lhc majoriry of AM s.,is 6linacdvalue orq li€s betsea 0.3 md 0.8 bur 6ubid. thb ro8e 6 dd do ocd. UDusualty low, ssnall4 0.1 valu6 orC! occur in equ.rorjal r.gioB of nish r.infdl *hose fiood p@ducins

mshMisd ir lairt mirorn lrom r.s lo y.s. Low C, vaju.s al$ oeu fietativetympemeahle. hisn rai.fall cdlchmnts in lenpd.t! anes. Ldle vatues, C, > I, occDr in sodeAM sen6 wnich de well -behaved in h.Eoscneiry bur no3l ces of ldge C! 0re dle ro the

prese.€ of o.e or nore oudieA

In seneral, rh. vrlue of Ct, drinated ftoh AM sori.s ig tosiriv., roging fbm zqo lo 5 o ordoe, wirh rhe hsjorily ofldu6s tyins in tbo reg. o.4lo 0.6,c, &d c. tend b be posilively

coielat d Mce tle she fqrur.!..!s. high or lo* valu6 ofceh. Z= los e impli6 rhat C, inlogsDaceofAM is frcqldtly neSdivo.

Vdues oa Cr in AM seri6 vary fron 2,0 lo 3.0 lndividul vdu.s Ely vo.y wdely oulsrd.

lhese limils. Th. v.lre of C. obt in.d from such emples i! bound.d $ovq Kidy [35]

connned dcocri@lly dar torh C" and C. in rddon smpLs !E bound.d sd thd bounds olc !tuncrion or@plc sia N .bne i...,

o<c,<(x-Do5

'''("rt6.

Wh6. C, and Cs @ b.ed m u@r6r.d mhds.Hla [25a] foud $!r SkM6 of AM d.r. &nded !o a.ece eitl reod laglh dd

suss6i.d ..oreljon facror O + 8.5/N) byRnic!lo muftiply sdlspl.valu6oaC..aM sei6 a6 in fet raadon sd6 frcm d unlnoM dktdbuio.. Thd values ofcr

C, ud q oblaii.d fiod th.n &o bi6.d sstdat€s ol ihe populalion valu6. Tb6 bia h

imDorurl in m.ting inf@nes lron th. sMpl. momat, lor iBldc. by se ol mom.nr 6lio

diasms, rbout $. fom of 0r€ populdim di.tibu|16.

13.2 Rqul& sd Dit.!*ton

The valu.s of Ct and Cr for dillcrat periods d Sukk( (B), Kolri (B) dd aor iull

thra barug.5 rG 8rve. in Tabl6 4 3 1,4.3.2 ud 4 3.3, cpoctively. The oemddrsthn sudy @ obtaind froh smpL nomont a lolos:

-L

p, = c,, ,8, = ct

Fbm lhc bblcs, @ ger ln. follo ns informlrim lbour thc 5h!pe dd trDc ot rh.

dislribudoE for 6ch ol din€ d&,l! sn,:

lh. cune ir le$ shrply p.aked

,[t4p! , ,t=j.9,=

(i)

t,(D

(i0Pr= I 2l+ +ly Sli.sd, nolNormal,

tr=0.32r+ Ploiy Ku ic(<3 0), ie rhe cuiv. h6 ndoEr lul on ono sde

curv. wilh @orcr l!! on orc sid.

A. Sukr* B.foe Clddtr (B)

Pr=l.26F-> +ly S\.w.d, nor Nomd,

0:= I 396=> Pl.ly Kuni. (< 3 o), i e

,r, At Strkkur Aft.rGuddu (B)

{i) 0,{.533+ +lY Sk.w.d, nolNoml,

(lii) 0t: 0.652+ Platv Klnic (< 3 0), i .

5(D

(ii)

6.

(i)

(ii)

1.

(r)

(i0

(i)

th. cud. is l* siarDlv p.tr.d

ar Su*klr E lon Snkk!. (B)

S,= L72t +ly Skc*ad, not Nomd. not .longdcd to Rrl s

h= 5 034-> Llplo Kunic e 3 o) r e rhc cude is moF sndplv p'atcd dd h6 one

ar $kkur an.. Su|{iur (B)

0r{ ?99+ +ly Sk v€d, not Noft.! moE clonelled lo R ti S

9'=- o.Ol38=> Pldy Kurli€ (< 30) i..lne cury€ h4 afianerloD md share oftE

dislibulions cneSes ftoD lepto 10 platv

g,=o 83+ +lt Stdwd, nor Nom.l noc €longaLd lo Rtts

0,- o 669+ Ptaly Kurtc {< 3 O) i e. [tns top &d ffoffi rors

At Kotri B.fon Guddtr (B)

$,= 1.4?s=> +ly Sk wed. .ol Norml nor. elongaled lo R H s

8f 3 03> M€so Kunic , (}le cufl. looks like nomal cun' oo ono t00 omv

ar Korrl aft.lGoddu (B)

0,{ 666+ +t Stescd. nol Nomd, ndc .longd'rr b RH s

t:= -0.619 Phrv (uni. (< 3 o)' tha sh'p' or dicL'bunon chtls6 rom D6o lo Plrtv

0:: o.lo l=> Pl.tv Kunic (< 3 0) cudo h'vin8 on' 'drowdi

br'

(i)

(00t)

(i)

12.

C)

Ar Kotrr B.forc slkror (a)

h=o.2??+ +t st u!d, lpprox Nom.l.

!,= .0.412=> Plary K[ac (< 3.0). i.c 0.n . rop snn 'l.m*r t iL

U. At (orrt an.r suklof (B)

0) I'rr) 5r9+ +ly sre*ed. norNonol, mod elongded ro R.H.s.

Ol) nr-. O.3Ol+ Pldly Kudic(< 3.0)cup.havinsone narcwr |!il.

(t)

(iD

ai xotr &foc Kotri (f,)

D,= | ,M3+ +t Slt wed, nor Nod..l, ioE doeded lo RH.S.

h=-l.3t6+ Pldryr.lnic(< 3.0),l@klik mon l or m. sid. dt, on. riil wid..

ar Kotri Aft.f Ko6i (a)

tr=0.598+ +ty stewed, iol Nom.l, tuc 6long!1ed to RH S

tr-. 0.126+ Platy Kudic (< 3.0), hor. hDp6, dlrl€r lot, n{rowd tll on on side

a.

Ilbk of Co.ltrd..n fo. |n. lrrr. oaal r.sGddu (D) Surru G) Kolti (B)

s.D 51.08 259.6s r95.r2

038

c. o.5t- 0.s9

cr - 0.623,/ -0.65

suuu, (B) Korn (B)

SD 20s 12 t60.46

o3l 0,36

L 12 (Lar8o) 083

0.32

(iD

C,lies between0 3 md 0 8, vdisiibulions

C. G+vedd liesb.lw.en0 lnd 50

C"ind C. !e posirively coit lo&d, Vdistriburions

S D dd C" ar Sutku' dd Koki hlndS.s i.cr@ dd C. dec'G anet

Ci b€@mcs ncgahvu .ncr Glddu $niagc, v barascs.

1.4 Fidings

All de disrdbntions a€ Mulli mod.l .nd Dosililelv Ste*ed Gmilar lo lh.

r6ul6 on lh. b6is of meduB ot Ccnkal Tenddcr) which shos lhal li.quencl

ofloodsol hisher mosniludc d6ocis.s ud lheir masrnudg i.cre6.3ner

Magdude of CJ.o.-nom.lily oltll dislribltions, excepl Lr rhe ddaot

Kod (B) belore sulikur(a)

C" 'dd66 ed C. d.ctae fot tiE dda .| Sutkw (B) md Kofi {B) and th'

onnrudio. of 4ch of lhe bdcg.s, a\ept al Kolri (B) ,ad s!*tur (s) (r '

dde is eme eftd of oNtr*lion of be68q onCJ

c. al Gnddu (B) <c. al Sul"t( (B)< c! at Kotri (B) tot l3 v4C. ,r Koln (B) < C, at Sn*hr (A) ao! full d.lt

'I

The ragnnrde ol lossr oood p.ots d!cr.6 ud olhi3her fl@d pea[sincreses

and de€ is €huge in rhc !hop. ofsom. dist'iburioN rfler lhe coBtuclion of

baiiag6. sp€cialt.nd Cudd! (S),

Cr is + v€ for lnll dall at Suttu! (B) ed Koli (B) bul is negalive rne Clddu (B)

5 All lhe disrriburions ard Platy Klnic, cxc.pt for the dala at Sn*tur (B) bcrorc

sutkur (B) (beins Leplo Kunic) rnd tor lhc dala dl Kolri(B) betorecuddu(D)

(beinsM6oKun'd)

3. Ar Su*ku ad Korn b.ma6 lh. t.qudcy 6f v.!v lov od verv hign fl@ds

i.cr€&s dd ol tow no6& d.crc6.!, dpeciollv lnd conslruclron of cuddu

t. Ar Korri 6sas. s $o fEqu4y of modd:re doods d..|!g &d of ld* aC

llish noods ircr.q6 .nd d$ cdtnct16 of Sul*u ed vssi bd386.

All lht dstribudon eo Posiliv.ly st n€d, mulli modd sdpllry Kunic { qcDr 2 out of l3 sB ). Th. nasnitude of l4 0@d dccrca.! od orhEh

floods inc@s whilc lhc fcquacr of v.ry lo* trd v.ry hisi nood inq.es .i6 trl.

c@nMtd of bsns6. Cy ircr..56, Cr d*l!M dd Cr b.&m6 rc8!liv., dt.tcd8rtucld of Cdddu bdris! d inclt@ D tlE smplc aL

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Chapter 5

Prediction ofFloods by Weibull Ploning, lnd cunbel Analyticll Method:Th. Elfetr ol B.rng.j

51

A diskibution lor AM ooods c5lnot bc chos slcly on the basis or{priori th@reiical arSunents Th. chsacteristics of obsercd flood dala nust be

d€l..nined io 6 sitabl€ fdhion Md irk.n into rcmunt vhcn i dislribulion h

b€irg cnoF. Som. distributions 6 b€ .xclud.d ifit b knoM rha1 n doo

spl6 from rhcm do not hrrc cl@ctdistiq in @nmor with ob$ned n@d

Until 1970 lhc suitabihy of dy pfliol& diltribution fo. nood

frequency dalysis wa often jud8€d on th. bsis of physiql insp.crion of the

(bla or. prcbabilily plot On scl ! plot th. wpl. vslus of lh. htdrologi.al

r@rd apps.r r $ri.s of ploncd poi.c while dE cninAr.d disrdbu{on ofaprlialr fm, whos slihbility is b.ing dmined, muld app.d a a lin€ ot

curye The fom ofdistribution wh$€ lin€ or tuwe show€d best aere€n€.t with

rhe plott.d poinrs would then b€ chos6. Cwbel [21], in d.monslraling the usof EV disrribudonq n.de us of @nfrdcn@ i.tads a!o!t thc liic o. aRe of

thc tuIed distdbutioi otr rh. pobalility plor i or6.t \o help jndg. tdetuBnatu h. snitabiliry of th.y ditxibutiMJu 14M eri.t, ddta".

Il is now undcr$ood how€vd, bot[ fron th@rericrl s1.ris1i6 and from

lplit smple t*ts on nydroloajc.l drta, that &y sinele rMid of dlla fiom a

giv.n d'st.ibuton @ dGplay r plofi.d bch.vior which is quit. difmnt fron

rh. pm eT rchtioisn p. fils a $npL frcn a $6igh line popularion @uld

dhplay mrtcd d liurc on r prcba[ility plot ma.kably clos 10 a stRishr line

on a probabihy pbr Ther€for., rn r. is. disrinct polribility of@or when

choosing a disribution on the basis of i.sp{tion ofr probability plor. This is

nol the full ofih. ploi, p* se. bur std3 fioh the hjghly unenrin .itur. ofthc

A fdilid rcchniqu. B ro plor th. obsvld d.rr d ! prob.litiiy p.pauing r eitrblc plordng poritid aortuh |Id tla thc dd. n tu d by . nnighl lie.Thi! mthod b mr r@nftrd.d rcw ! d.y.. Hoy cv!. iri tfituriE felturc b ihar omc{r G how rdl rhc dn. fir th. $$i|ed dirribqrion.

Thc cfteB otfou plorriig -plofiig polirion fomts d €$iirurin8 rhc

plobllilny wi8lr.d nloictn! ofLNO), LP (3), GEv, rnd W.t by dhnbudonr wrcinv.srisrr.d by H.l'uiir ud BozdlM [2sbl.

In thjs ch.ptd w.ilull ptoning dd cu$.1 nlrlEd @ dies.d ed ts.d for

Findin8 lh..tr6ts of@rurrudion ofbr.6sd or pr.diqion ofnood. Co.lidcne

iitdd3 dc ctlql .d ..d e dm in srlph.$ Prub.biliry Pbq Pto(i.! Po{ ioru, Prob.bitt9 PrFr .nd W.ibul nolti.g

5.2.1 Pob.Dili9 p.pc. .!d Prcbrbiliry not

In 8.n.rd *hgl thc ondadE di!fiiburion frft{on Pr (x) is plorcd on

&nhnujc p.pcr vru rh. vrluc of X r i6ishr lirc dc mr eh. 10 sd. nddrlirc oi aithndic p.pq, P. (x) wolld h.vc to bc Sive! by thc qFdion P, (x) =u + b

ftus if Ihc orul.tirc di{riburion ol ! rcr of d.o plon n . nnisl tinc d dirhiEdcpiFr, th. d.ta follo*s I qifom diltdbotion. Prcholiliry p!p.r crD bc dcvdopcd sth$ uy c!ruhtivc disdburion d h. plottcd B r !r6itl|. li@. Cli.r.lty . *pslrctyp. ofprob.bility plpq is r.qlircd for c$r, oflhc ditrcrcar p.obrhility dturiblrios ro

plot a a llr|j8ht linc. TIE *rling of fi. prcblbility FFr m.y do h.v. ro ch.rg. sthc p.6@rd ofr pdiold dindhodon .n6gc.

Codru.iing pob.bility p.pa ir r proca ofrotr{oming tlE probalil'ry r.dcso th.t rhc relling @nulltirc odc b . nniaht lie. Thc rtu3lomrion tEhniqu.

u.b. illudnt d bytlsub lMFi'lt16lA prcblbility plot ir . plor of mSnirld.6s or probrlililt trcmiii.S rtr

prob.bahy ro eign ! d.h poi ir @dro y cLd.d ro u d.r.mining rh. ploui.s

position. lf orc it daling wirh . poplLlio4 ddmi.i.a th. plorri'rg posirioD ir @rcty

. nlt r of d.|miring th. ftrdion of rlE d.r. !du6 lcs! (gora) rhd or c{ud to rhc

v!lu. in quenio.. Thur ih. sdl6r (u861) topddion v.luc wqld ptol r 0.0 sd rhc

l$Ben Gnlll€0 popul.tion v!lu. sould plor !r .1.0. Asignins plonins loririo.s to

emplc data is not a sk.ighr foruard b@ue on, on ndo b. ture rhar a $mpl€

@nhins the smllqr a..l l.tAdt v:bq of th. untnoM popuhtion Thus plotting

positions of 0 ed I shNld b. rvoid.d for ernpl. d.rl! unls orc ha rddilion,l

infomation on tn popubtion limit3

Frequdct dalysis i md.ly a poc.dur. for Gtiding rh. ftqu€.cy ofoccuf.nc. or probability of occundc. of past dd / or tulur€ d€nr. Ths probabihy

plorting wnh or {irhout any dislriburion.l ssunpliotu is a nelhod of fiequency

aulyes,

Hydrologic frequenct analysb can bc mad. vilh ot wiihout maling my

disribltional assunptions. The pro@durc io b. followcd in €ith.r qs. is much the

sme. If no distriburional Gunptions de m.d€, the inv€rigalor n€rely plots the

obsfred data on any kjnd of papd ( noi o@swily prcbability p3 er ) dd ues rhis

he. besl judg@dt ro ddmin tbc rugritudc of pasr or tuture d€nts lbr vdious

r€tum p€riods. If an dalltic{l tehniqu. h uscd, ir is r@mmnded that tne data slill

b. plotl€d s rhat onc @ gcr cn ids of now vdl rhe dat! fii rhc asdcd &alytical

fom dd ro spor poEntial p.oblcms.

Flood pals do not o@r with lny lix.d p.ftd in imc or rDglDtude Timc

inltuls b.t*a 0od. vrry. L.rg. nood! n tuolly h.w la.gc rdum pdiod! od vi@

vere Tne d.ffnidor of th. Gluo Fiod my not involvc ey r.f€d@ to probabih,

Eo{a€r, a relario$hip belwn lh. prob.bility of o@tu@ of a {lmd tud its r.1un

p.riod betftd cd in jusifi€d A givd nood 9 with . r.lum pdiod T nay br *.eded

one in T yem H6ce th. probabilily of dc..dcn@ i! P (0r > 4) = l|I.The cunulative probability ofnoi-exe.d€ie, F (Qr) h givfl by .

F(O,\=P(O"<n\=t:fAbov. eqution is the basis for e*imating thc rognnudc of r floa, Qr, givo i13 t€tum

pdiod T subsriturins F (Q.) =r - l/T in . k.oM si.tisti@l di$dbunon tuicdon, oie

an $lvc ao. .n. nognitud. of Qt, Oi.n, th. drita d. plotl.d on prchability psp.r to

cnek wn€rner th.y follov a paniol& di31nbldo4 ro dder ms, ed b cndk fo.

5.2,2 Plo(i|!glolirions

Prolrbilitt plots rcqlir.6 inirilt .slimite ofrhe problbilily of mnqcc.dq@F= (F (Qr). rticb ir c.ttcd t ..plolling posirim. ptouing p6itioN rc .ts u$d to.siro1e p.rmEr.d by usins rh6 9rcb.liliry wisnl.d f,o@nr, (p\W0 nEdjod

Son. Connonly Uxd nortir! p6 io! Fotuur. ,E

Ploriing

T F= l- ln/V+l

'-:

/v + 0.12 i - o.44

,v i-05

N + 0.25 t 0.175

l/+ 025

N +O.2

M t-t

HoskinS i 0t5tv

N. No

nnk i. asdding ddq = N- n+rrot in d€@oding o.d.r - N- i +l

Plolring posnion fornula is civ€n by:

P(r>xq)=n (N+l) ''--------(5.1)wb@ n= ordr runber or nnk offi€ .vent,

N= Tolal nunrh.. of r@s,x=Q (c{0 @sc4)

(i) The data are amnged in decreasins oder ofnagnhude to IiDd m.

(ii) Tn€ Fobabilily P of.ach erenl being .qun.d to or *Fd.d (Ploltins positiod)

is calulaled by fornula (5.1) The W.ibull fomula is a @nprcn!. wnh morc

sllrislial ju$iliodon. Ifrh. N vrtua dc distnbu&d ujlomly bdvq 0 md

100 p€r@nl probability, th.. th€re must be N+ I intwals, Nlint rysls belwn

rhe data points .nd 2 int€rv0js a1 tbe cnds

(ii ) Th. Euiiene iii€rva|, T (the retun peiod or 6equ.ncr, h c.lculated s

r= (N+l!m

T=I/P

(5.2)

o3)On€ rhc d.r! ssies is id€ntili.d dd lanted, ed tbc w.ibull PlottiiS

points (Q,T) colculrled, a shph of masritud. (Q).vs p'obabilily' t(P( pxlP( x<x ) o. T )l cu b. plotted on probabilily pap$ wilh speoial s.tle or oiGumbel piobabilny p.r€. with rcduc€d 5.3l., i€ QVs.Yr to fit a disldbuion

Graphically) diesed i. stion 5.1.

Trbl6 5 I I ro 5.1.3 show cdolariotu fo. thc gnpns by W.ibull Plot for I tul d.t!

Ctrmb.b Atrrlttiql M.rhod

Tnc Evl dindhtiion, rl$ krcwn s Cumb.l disdbutior, is tne n6tfidely used di$dbutioo in n@d ft.qucncy rnalylis

CDL Fx(x)- axp I-€xp [-(x-O/dl, ..(J4),

wnere F(x) b rhe probdbilily of.on-dceddce for th€ valu. x,

t is th. l@tioi p!$*td,

P(X>rr=t-e-"M6r |rr-q,+ €q, €=0.57721

vdilnc: dl = 1--= = 1.645 4I

C@ffi.i6t ofSlcsrEss: C.= 1.1396C@troi.mofKonosb: G=5.4002

r,=q-oht-lnp]...............(5.6)

f r'lrr =d-a tn l-h 0-;) 1...o.?)L 'I

...(5.5)

(By iNqtbs CDF)

We hl@ u dt.rndiw fom:

rl-k+Kror. .....(5.8),vh@

r- = -91,*r'l-r.{ r -Ill l...rs.sr' 'L t ( r./tl

NoL: Kr ir th. rtrdudiz.<l OorlbL Vri.r. (r- o, o = l).

x n .l!o elLd Fr.q@icy F.dor.

5.3.2 Grntd" Eq! io!

X, = t + Kro-r b rh. choe'3 sc|@l cqu.lion of b/dolosicd fr.qlc'cy

X, = vrluc ofx with t .ct!m p6iod T,

7= nai ofx,

Kf fr.qudct f.ctor, wiich d.tod! upon T ed rI| fr.quaq dkdbudon.

.. (5.r0)

. o.t t)

,-=.l^^ l ]"** ,. Bar.drced\tiate., L T II ...(5.12),of

vi.

lo r"*,,0t T

valus of ," &d S" arc 8iv.n in labh ?.1 md T.ble ? 4, for ditrsern smpl.sia, io Eigiffiiog Hydrologt by K. Sub.d.y [ 36 ]

Anege the diehdge data in d.@ndin8 ordq &d no1€ tlrc spl€ size

N l{€re the amudl flood value is lh. vartuc X .

It, -+c'577 =Dulet's Conta\t\

tS" r1.282s,610,@ ,

53J PROCEDT'R.E

The alor equalioN e us.d und.r thc followiog prcc.dur. to dimr€ fte

flood nagnitude mtonding 10 a givo rcluo pdiod bas.d on & Amnal

Flood sdiet

C.lqlate T by WeibullPlo nsPolnioi Fomula -

D€rmire,- rnd S. lppropd.l. ro siv6 N.usins t!bl6 [ 36 ].

and o. L for rhe givd dau.

vii. Dct mirc th. rcquircd rr.

dii. Chdk wh.ilE rh. ote rd d!l. fl wd! wirh lhdrctical Gmbcl'sdr@. vrluc dirdbqrion.

Nor.: Cd@Ltiotu for $c Gr.thr by CunU Prcblbility Pipd & slDM

in Tdl.r 5l I to t.j.3.

t,1 VEruflCATION

{i) CNlculate vdu* of Xl for somc T< N, Ning Cunbelt iomulae

(ii) ?lot Xt.vs.T on Sdi Log or Gumb.l Probability Plot.

lfthe plor is . slr.idt lirc scn $d point (2 33, X ) li6 on it (wno N is 14.).thd Gamble's dktabltio. k . ,ro!.r fn.

By qtrspolsrioi, vdw of Xl for r>N @ b. d.t@nEd 6ily.

tIs GT'MDEL PROBADILITY IATXR

n is o .id for tlr gnphic.l !?tsa|ll!on ofcumbd ilisiribulior Bde

w r.qunc lbri$a sp€idly ftr*ed for vaioG vdue. orT.

(l) Forr $!b on 6si&.. w.6[l.ud eihm.ti! ldle of t'r vdtr*

( sdy lim -2 to 5)

(ii) lor sl.cr.d v.b4 orT ( Ssy 2. 10, 50, . .) Iind tnc vtlu6 of yr

(iii) For xr or Q vdud on ordi rq ue ather dith'rdic or log!'ithmi€

Ar Xr is a lincar tunction of yr, the poinls ( xn yr) fo' Gunbel

dislribution wil plol a! . ltoight lin. on clttbcl p.ob.bilily plot. This lioe an

be used for gr.phicd interpohtion snd ampol.tion. ThG Equaton of the

snanelt liffi, ctlLd "nod.l Jd W.lie tl@ b! GMb.l wthod," ht en d^a *tis obt ie! ud 3hoM inTabLNo.6.l.t 10TablcNo.6.l.l in Chaplq6.

Confid..c. Ljnir. Fo. Pr.dictior of noodr by Crnb.l A..tyljc.t M.rhod

SiM lhe v.lue of tlE vdi.l. f.r . siEn Etum p.riod, xn d.t€min d tyGuobcls nefiod m hav. .do.s du. to linir.d 6ple d.rl wed, rherefo.€,6nido@ limils &e deiralrl€.

Th6 @nfiden@ interyal indi€tc thc limit' lboul the calqld.d v.lue b€rwedshich lh. lrue value @ b€ eid to li. with epeiiic prcb.lity on sanpling €106 or y

For a confid€n@ probalilily C, thc 6nllden@ ioretu l of lh€ variare Xr isbounded by th€ !€lues X, af,d X1 givei by

x,a =Xr + f(c) sq . (5.D)

wnerc f(c) is. tunction ofth. @indcn@ problbiliryC Bddemined by us,ns rheLsbleoanomar 'dids(rcbdr$

C in% t0 68 80 90 95 99

{c) 0.674 1.00 1282 1.645 1.96 2.54

"** =bh . (5 14)

Kr= ft€quocf fadorgMaby.qu|lioi(s tl)

o".' = sr.ndrd ddiaioi of thc mpl.

Tabl€s 5 3.6 1 io 5.3.6 ll sboy th. @lculatcd valu* of C I's ar 8OpZ md 95%,

Gnphs 5 r I to 5. l. 13 show rh. f..quency cuN.! by W€ibull plofiing position,L S m.tbod and Confide.e inl.ryrls by Cunbel hcthod, L.S. nelhods in disNsed in

Tn. E|Tectl or Co.rtruction ol B.rn36 on Pi.dicrion ot pak Di!.rrrg. by

At S!Lt$ B.ngG

Mei dd q < xr , i..,.5 - yd flood, for .tl d.i! s.t3&<x0 for tne dara b.tor. rhc bs!8.t&>xro for rhe dau aftd bi@s6.

2

.].

% ditrcr6.€, for dif.rcnt T-y.es nood{ bdw the d.ti els, du' to thc

coisttuclior ofthe ba!8.r vui6 frcn

26./e to 43o/o for Gudtlu (B)2q/ob 45Yr for Sukku(B)

i . the ef€cts on Dcdidion of [@dt rr Suktur (B) ttuc to rnc @nrtrucrion ofGudd! (B)-and of Sukkw (B) e spprcndr.lv €qlal

l:1.27: 1.38lr l.lt:1.50

Bcf@ Glddu @)And Gddu @)

l: | 25:1.29l:1.32 145

ic. rhe etrects ofcrddu (B) > Thc ctrecls of Su*lu (B) and *b0>xs>xoii

4. Mdinum 0ood i! dpai.d to r.turn.n r

(i) 50 y@s

(v) 301ts

sins tul da13 of sukru @)ulinS d!l. bcrorc Grddu (B)Nirg d.rh .tq cflddu (B)using tu! beforc Sutlor (B)

uri.g datt .nq Suthr (B)

Hd@ th. nuitun tl@d it dpd€rt ro tctum .t Sukku (B) in lhc shorl'sl

D€riod (i.e. 15 y€s) uling th. d;tl ,nd Guddu (B) rnd in rhe longdt p'dod

i lo0 yd*) $ins th. d|t. b.forc Sukh. (B).

1a.2 Ar Koti B.mg.

M6 dd Qr < xr for dl &ta cn (a d Sutlotr (B))D, < xo for ihe d.t. b.rorc .ll b.rfl.sa (M .r sul&ur (B)Dr > x,o rn.r baras6, d@pt for rhc d.L .ild Su*lur (B)

2. % ditreioe, for ditr r.nt T-y.$ ll@ds, bet*€d th. do$ scG du. lo tbc

co.$rucrion of barlg€3 vdia fron

't94/,b 4e/"3lY"ro6e/'

41/oro22o/o

i.o. the ef€cts on prcdiclion of loodrlnd de Brotet due to sukrrr (B).

l: L26: I 16r:1.19:1.55

ior Sukku (B)

.t Koid(B) are the l€ast due ro Gudd!(B)

L

1ll9:lnl:1.15:1.50

11128:1.41I 1.39r I t6

x rm> x r>x m io €&h cs, a3 it SuEor (B)The efrects ofKoti (B) >The etr c!3 ofcuddu (B) > Th. .f@ts of sukkur (B)

Mdinum llood is dpat€d to rctud lnr(i) 100'ts usiDs tul dltr ofKotd @)(ii) l0oy4 uins dda bdore Kolri (B)(iii) 30y€s Bin8 dlll lnd Kot i (B)(iv) 20yaE using dar! b.foc sukru (B)(r) 50y4 using drlt ift6 Sukhr (B)(t) l00r!d using dal! b.forc Goddu (B)(ii) 20 yds urins d.tA rn* c!dd! @)

5.4.5

H.ncq w€ e dp€crins Minum llood to retum at Kotri (B) i. lle shon.dpcdod (i . 20y6) usins rh€ d a sct .nd Guddu (B) ed in fi. loi861 p.nod(100 ymhsing dra before Guddu(B) .d bdo€ Kotri (t r.

2

(ii)D' >rr for all rh. b.d.s6.

% difer€.@, for dif.rcnt T-y.es floods, vuia fton

b€tw4n Suklor (B) dd cuddu(B)

cuddu(B).

35.?70 , b.t{*nKorn6l oel. ; bdwn Koti

di€ad@ k h.tw*n ih.

xrm = I134:l4al: l3lr 142I I l2: 146

i e. lhe 6riG de l.Eali for Glddu (B) ald ladTh. ririos e sle itEd3in8 wi$ tift

4 Muinun 0@d it ciP.ct dto r.tnrid

(B) &d S!kku(B)(B) dd cuddu(B)

predictons at Kori (B) md

For Kotri (B)

for sukru(B).

(t(iD(iii)

usiis tult d.1s for clddu (B)$ins tuI dat. for for sukk$ (B)

usii8 tull dala for Kotri (B)

H€n@, we ee dparing m.nnun fl@d to rctun( i. 20 y.ds ) and at Kotd (B) i, tnc bngrst pcnod

C!dd! (B) in th. snon6i pldod

L Md ud q < x i, fd ![ th. drtr s.aD! < x o fd allth. dd! !.ls bcforc bu 86D; > x h ro, all iJE drla *tt ns b.E$e sd sle fo' ftll dar' er3

2 The .treds on pcdiclion of floodt ar Koti (B) @ rh€ 165l duc to Ouddu (B)

rid rr€atet due io Sukklt (B).itre'a;eas on prcoiaion i:ittoo<lt at Sutkur (B) doe to th' @hsltuction or

Guddu (B) md Suklur (B) ar lPProxitut.lv equal

3. x n > x $ > t 'o

for 3ll datt 3.t3 itd b!.Bg*' m"

"r..li "rctaa' fs) .r Kotd < Th. .trock of sul(ku (B) r Kord (B)

4. (t) MqinM [o.d to .aum .t Sul&i (B) .GquiB CDnd psiod ror rh'

d!!i .nd cuddu (B) .d lonscsr Fiod ror the dd! b'forc sul&u (B) (i '

(b) Maxinun fldd lo r.lur.

Crtldu (B) ed lon8.6l

at Kotri (B) r€qund shortdl P.dod ror rhc drt! !R.'

Pdiod for th. da1t befor€ rori (B) (i. fot 1wo

Corclusion

Tft..f6r3 of @mtrudion of bd.g6 o. a !h. dom itr€d dep6d upon th.

distane. | 00 - ye6 flood is largd rhe 50 - y.r4 flood (a dpdted) .lc

Mdinum 0ood is to rerun al cuddu (B) in thc ahonet p6iod and at Kord (B)

in the la€e p.dod.

5 Th. difcr€ncB belw.ei rhe prcdid.d R@ds !t Kord (B) dd at Guddu (B) >

b.two Kotri (B) and sukkur(B) > bctwcd sd&ur (B) dd Guddu (B)

6 Maximm fl@d is lo r€rum rr Guddu (B) i. tlE shond pqiod (20 rlt ud.t Kotd (B) in the lo.s6t rcriod ( l0o y4).

33

bl.5.l.l C.lcul.lior for Gtrnb.l Prob,bility P.p.r.nd w.ibull Plor Cudd! B.rrng.1962-1999

t.t6l

l1

t.la5 Lrl

t.t31

l. a

33 1.294

TrU.No:5.1.2 C.LrLlio! Aq GlDh|. Pnb.bilrr Prpd Ald W.lbo|r llol, SlltLlt (E)'' (l9lll - t910)

t.56)

tr l5l5

Lltt

0.1'

1.661 0.21t t.5570.211

5t 0.21

,1)

2t

23 t.21t ItD| 233 0.v2

t03l0.11 Lt26 t.t?l

33 0.?t3ltl

0.511

,.nt

n

td.ajj C.lol.ti.. ror a[. Cr4f, O. Cmb|. tu.liliv POs.!rl wdhdl tlo! K.ti aE tEo9i,r-lt )

1.533

n

Lll00lt7

ul

t.5x

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Chaptcr 6

Cotnpa rison of Prcdictiotr of noods By ]V.ibu ptotlire rd ModetsFor Ler3r Squlr.s snd cumbet Merhods

6.r Fbod PB{i.tio$ bt w.ibo ptod.g ! r-S .trd clnb.t n.rhod

6,1.1 lnrrud!.tion

ft.rc de rwo brlic dislincl ryp6 oflood pr.didion: (l) for rhc shon tun Md(2) for rh. long run This disliturion @dsponds to rhc lM pr&iiot problens ofoperations and of phnning The lwo ploblens call for ditr ror methodorosi6 forlh.n $biions. For op.r.rion l pnrpos.\ om w@ld cxrdd rhe pr*ft ofttirion inbth. tuturc b, d.r@inbdc +pliclrion of rhc nafurEl trs. For plming ,uQos.t one

roy bc salh6.d ro .xprcs the tutlrc io prcb{bilistic i.ms. This srudy is @nern€d

Plsnin8 of omd p.otcxon of a nv.r wil requ,rc s ir3 hos| b6ic spcl rhe

e{n.r.d dood pa*, *hich vdi6 wnh dm. &d sp.e. Flmd paks of ditredtretum pcnods in tidal innudc€d riv.r @n be .stier.d by using flood rouling a.dfrequ€ncy analysis r6hriqu*. By nej of rh4e proc.dur* rhe pr€dioioB of lh.chm8. in llood beh.vior in dq chmcls e all nadc posibtc.

6.r.2 Lmr Squrrc M.rhod (t_S M.rhod)

thc V&iabls X rrd y d. otd Ef...cd lo s indcpoddn Md d.p.nddrwiabld rcsp@lirct. Th. qirclion for ihe bn ft tinc, in th. mcthod of 16rsquar6, h that th. sun of att squ.o of ddistion of obsw.d poi s fro6 the

tirt.d tu.ciion h mininizd ro produe lcst sq!&s.Ler N a$n. rhrr rhc l6t squG tin! .pproinErirg rtr sr ofpoints(xr,yr,), (x,,y,),. .., (xN,yN) tu d. equaion

Th. €i(or i. rhc estihale of y ar {ch point (x,, y ) is €r $ lhat e ,=J, _ i,g .l( / "\ 4.L e, = L ly,- y, l= L tr,- a - bx,)

\ra,rl-c,u'd ci an Lc bdh tD$rtrc anrj n.grlr\c. ro ,vod rhc ncaauvc@nsd.rrh€sq,rEolrh..nou,., andtneirtumisddot.dbyzsu(hrh.l

z=u0,_ a _ b,,l'Thc critdi. 10 8.t ihe €quation ofthe b€sr 61 linc i! ro ninini& Z. ThDs

a dad=ataal>0,-d-bx,f

a't ab=atab(>A-a-b x, f )=0

-)-'

Ater cairying oui smmarionq

Ly,-NFbtJ ,=o

thc two .quatione b@me

,a Ly,=Na+b>,

=+ DtpD+btxi

a_il^'. 1^ tL

a-CrtCl)-Ot)E t|rNtt elf )

>',->y\o:=o

Solyiie the equlioFfi* ga

b=N Lr r -A,)ArJt tNL,1 -\r;,)1 )

Norc rh.t y=a+bx tine gcs throuah rhe poi.t ( r , t )Usina Minita! $ftwec rc ga .qurion ofrh. tine alo,B with rh. vdu.s ofr

adj F aod F. which m giv6. ii Tabtes 6 Ll,6.r 2 and 6.t 3 in Chapt.r 6.Th€ araphs oflitled line @ b€ uscd for inte,pot ion!s wc a GnroDotstion.

In lhi! chapr.r, rcd.k by L.S sd C!6bd r*rhod! e ddcloD.d sd ftprcdicrioG offloo& by Wcibul pioning. LS .td Cflmb.l Dlrho& & onDa.d.

6,2 Mod.k For Prcdl.tio. ofBood!

6,2. I Modch Dr L.S rnd CnDb.l Mcdlodr

Mod.ls for plldi.tion ordoods by L S. rnd c@b.t m€rhod .t Sur|(u(Er) Giv. ScB),Koid(B) ( s.6 Sd3) id rh@ b!r.s6 rogdnq @ shom h T.bt6 6.2.t.t,6 2 r.2 dd 6.2 l.l. BD6rivdv

Tdbh 6.2.1.t shws thalJd Mo.leh at&**tt (B):

0 lit.M$s for Gumb.l dcrhod < I'ttficeDt3 for L.S. nEdd for dt d.tl s€ts.Slops for cusb.lndh.d > Slops for L.S. trDd4 fd.! tu s.t3 (l.s% ro 4.d4).

(t Slopcs hlE in @!.d by 7tZ dd 69%, 6p6dFt, .Oa Slltor (B) dd .n.rOlddu (B), i... rh. .fiet! on the stopo ofnodcls !1 Sukrar (B) dE to Sukku (B)ir. lfs.r than d!. to both bamge€, i.c . Coddu (B) ed Suktu (B)

Gi) Th. iniercept ed dop. for tuI dat! li. bdwn rh€ir EluB for rh. d.la seE b.forcrnd aft6 t[. b.r.g6.

(o) L.S. nod.l i &..rt.d on rh. bsi! ofr-rsr, AdiR squ& ed F-l6! in acn ceTdbh 6.2. t.2 sh@s tltut ld Mod.tt at Kott (B):

(i) Iircr@pr! for cunbd method < Int@pB fo. L.S, for .I d.r. !.13 (a ar sukku. (B)).SloFs for Clnbl. ndnod > Slop6 for L.S, for !I dlrr !.lr Oy l.odl. ro 3.2y.), as

sukkur (B)).

Slop6 hav€ rrcE !.d by 940,6. | 58?o dd te4. Bp.diEly, .nq rhc @drucron ofKotn (B), Sul(k!. fB) lnd cudrlu (Br, ic. rhc.fcd! on fio 3lop. ofnodels !lKoki (B) are larg.st du. to sukkur (B) and soallsr duc ro .I rlu€ bdBg$,i . ,cuddu (B), suklor (B) and Korri (B).

The int€epr s.d 3lop. for tu! dar! lic halq dEn v.lG for rll. &t at b.foa.nddd rlE bong6 (!! !r Sur:rui (B))

L S nod€l is aecptlblc on tn. bsi! of t-r.sl, Adj-R !que! ud F-t.sl, for ![ dlra

('ii)

Tdbte 62-t.3 sl@s rtulu Moeb at alt Bu,as :

0 Ltd@pl3 for Cumble n rhod < Inr€@pts for L.S m6d.Slopc for Cumble m.thod > Slop.s for L.S mod.t

(ii) llrc lnrc'tcpr an! stopc lir' Sulkur (U) tic botw4tr rtnr latucs til (ilddu (U) aNlKord (B).

[i0 L.s mod.l is aept d(r-16r, AdjR,Frd), for![ data !.r3.

6,2.2 Coopr.ison of rtldidion oa Eoorh bt L.S. ind Cuobd Mdnod withWi.butt Plotdng

Valucs of Q, QB od Q6 (i c prcdicrion offlood) atof,g sith p€rqtagc ditrer€nc6ro' dillFot valu6 orT, obrain d by weibull ptonii& . L S. ed Clnbcl metho.jsarc prcgicd pair wis in T!bl6 6 2 2 r to6.2.2.11for@tof thc |] d!1!s.ts

Tabl. 6.2.2. I st'Ns thdt at Snthr (B) fu lntt ddtd (190t- r9e9).

(,) % di'IeGne berw€d predi.tiohs by L S n€thod lnd by cunbl€ n€rhod < 1.3%, fo,

(i) % dif.cne ber*q prcdicriois by L.s ud w.ibll ptoning < 9. l%, vT. ud

% dillrcn@ belwed pr.di.tions by Gubl. mcrhod dd Wcibull plo ing < 9 4%,

V T, qnd vri€s in si8n. Thc aig. ofditrercnc6 ir born €ses s smeThe largcr diferenc, in clch clsq i.dic{r. rhd prcdiclions by Wcib! plofiing e

(d) lQ Qol = 0.4e7', at r- 2.31.

As th. diffcien@ < 0.5%, Olmbet distriburio. i3 €xp.cr.d to bc . prop€r fit.

Tobh 6.2-2.2 sh@s that dt Stkhtt (B) beJde .o$ttr.tidt of Sukr'r (B) (t9ot -lgJ r):

O ./oditr r€nebdwenpredicrioisbyL.sndnododbyConbdo.ilbd< 1.4or,

V T (as for tu data).

(ii) (a) oZ dif€idcebdwen p.€dictionsbyL.S m.rhod a.d w.ibullplofiins< 6.7%, for dl T, a.d vai.s in s'9..

(b) % dif.rance belwen pr€didioi! by cnnbcl n.rh.d and $cibu ptonjis< 6.6%, V T, ad vui6 in siSn

Thc sbn of difr@nc in borh cls ir 3m.Thc diiter.n@s aE smdla thh rh6e for ftl dats, bur se lsa. .nougn b indiclre

rhtr tr.dicrioB !r. ror propd

I Q-Qcl =oe8%atr=233IIci@, Cunbel dhlrihution is noi expdrad to b€apropernlrahte 6.2.2.3 lhNt lttut at Sat,/. (D) a!t4 d,.e,Gti oJ s'khlt (B) (tq2- r gse):

% ditrffic. bcr**n pcdictions by L S ftrhod and by cuhbd n.rhod < 1.3%,

VT (as for bcfor. suLlur (B)).

(r) % ditr@.e bd*6 prcdidioB by L.s |!d W€ibult ptoning < 9.8% vT,

(b) % ditr rma bdwon rrcdicdo$ by cumbd mdhod ed w.iblll ploriing<9.47,,VT, rd wi6 itr si8n.

The sign ofdiftrcnc.r is sanc in boih cases.

Th. direoe have i.c@d .ftq Sotkur (B) &d hrsd % ditr r.nB indiqrc rhar

Pr.dicrjons &. rct proF.

0-U | = 0 7t% at T - 2.3i, hdce clmbel dinribution is no1 dpaned ro b€

r p.op€r fir Gimil.r ro b.foc Sll(kur (B)).

Tdble 6.2.2.,t shows itut at fuit,r (B) beJorc cuddr (B) ( 1962-1999):

% dift@ b.rwh pr.didioN by L.S .rd by Cflh!.t Mdhod < O.8yo, v T,(!, ror bcfo@ Guddu (B) ).

G) % ditr rcice beMc.n pr.didions by L s aid W.iboll plouing < 9 jol., v T

(b) % dif.ftre bc|t*h p.cdi.rion, by cflnb.t M.rM &d wcibul plo{ins<8.7%, vT, rid vdis itr sigi.

Thc sign of difcra@ is sle in born ce.L.ss % ditr€@cA indicat. lhlr Weibutt plorring ir @r a prcpq ,ir

lo- Q" l=ret.r=z.zt

As rhc diraeenc. > 0.5%, cumbct dkribution k not *pdcd to bd 2 Dop.r rir

fable 6.2 2 5. shMs thdt tt Sttht (B) afut Alt tdu (B) (t9O t - t96t:

[) % ditr rme bdwan prcdidiotu by L S &d bt Camb.t n.rhod <l.s% (stighdy

srcarcr rlRn rbr aulrddra).

(ii) (.) % difc.€n@ b€rw€d prcdictions by L S and wcibult ptori.g < I I 8%, V T,

O) % ditrer.ice belw*n prcdiclioro by Ounbel helnod and weibult plouing

<lloez,VT, dd varies in sign

(iii) rhc difr€renes hrv. insa$d aor rh. da(a of crddu (BXl'8.r rhd b.rorc

sokku. (B) )

Larg€r % diiterenccs indicdte ddta p.edicrions are nol prcper.

(N) lQ-8. = 012% alT = 23t

H.nce, (junbcl distnbuaon is.xpected robc I poper fit (d fortulldrrrbur conrqry

to dE esults uling dih tor befoc Guddu (B) ).

(iD

(iii)

tr)

Table 6.2.2.6 d1@s that 21 Kolti @)lu data (t901J999):

% difcren@ beiw@! predictions by L.S nod.l.nd by Cldbd m.lhod < 04%, VT(< d for Sutkur (B) tull).

(a) % dir.rcn@ b€iwed prcdidioro by L.S node! and Wciblllplouing<8 3olo.VT, and veies in sign

O) % dif.rcn€ bclw@ prcdictioB by Cumb.l m.rhod fd weibutt plonins

< 8 6%VT, and !&ies tu sigr

Th€ diferc.ces in both cases ar€ of sMe sicn.

Largd differ€nes indiqtc that pr.dicrioro e rcr prop€r.

l0-U 1=oaap/"JqT=233

As the diferdce < 0.5%, Cumbcl distribution is exp€cl€d to b. a p.op€r lll Gs folsukkur(B) tulldtla).

t'iD

(")

0

(,

(iD

lahle 6 2.2.7 shMs thar aI K.rn @) b.!M M'strcti@ o! S"ttv (B) (t901-1931):

% diilircn e bdwen prc<licliom by L,S model a6d by Glmbd ne$od <0.?%, VT,

G) % diltd.E b.tw@i Dtedicrions by L S nod.l.nd wabul plofiins <33%.

(b) % dif.r€nce bclwecn predictio$ by cunb€l mcihod and Weibult ptofiing

< 3 9.2, VT,and !di.s in si8n.

Smrlt.r ditrcr@ indic{e that L.S and W.ibull rbd.lc e a FoF ft .

1Q Q.1-o3e/.JnT =2t3& the dif.r.n@ <0.5ol0, Glhbel dislnbdior ir dp6r.d to b€ t prcpd frt, s tor

tull drta for Koni (B)

Tdble 62.2.8shrys that dt Kotti (B) altet tukrt (B) (1932-1c99):

% difrcme b€t@n prlilictioB by L.S nodd and by Clnb.t n thod < l. l./o , v T,

e for b.ro.. sukkur (B)).

(a) % dinde.e b€twcdr predictions by L.S modcl dd wejbullploling < 3.a%,

O) v. difft.e b€r'6 prcdictioN bycuobcl mrhod !d w€ibullptorrins

< 9 5%,VT, and v!n6 in !i8n.

The diF.r€hes in both ca* ar. of smc !i8n rnd r. lar8cr rhe rho$ for befor.

sukkur (B) - predictioro e nor pop.r.

\Q-Q a l=o801"/"at T =2.13.

As ih€ diffaie > 0 5%, Gumbel dishibulion is nor €xp.ct€d to be a tir

(@nt6ry to the dar! for bcfore Sul*ur (B)).

(iii)

Table 6,2,2.9 n@s hat at Kori (B) b.!@ Koni (8.) (t90t-1955):

(i) % diff€r..e beN€.n prcdictions by L.S ftodel ed by cumbd nerhod h v.ry smsll

(<0,4%). V T. u for tulldd!

(,ii)

(a) % dilf.rd@ betv@ Pr.dicrions by L S hod.l &d w.ibull plottng

< 4.6%, V T, and !!ri* in siSn

0) % din.rcnce bdw€en tr.diclions by Cumb.l Melhod and W.ibull tlollin8

< 4 2%.vT, and vdi* in sign.

The dilfqcnas are of em. sign, in bolh c.$, ud de sm.ll, i.. . Fedicrion se

lQ Q. =0534%JorT-23J

As lhe <titr r€ne > 0.5%, Crmbd disrribnlion is @t dp€cr.d to b. a frl (Co dry lo

Tabte 6.2.2. Io shM that at Kotn @) dJtet cdstructiat of Kotti (B) (t956t9ee)

(i) % ditrer.ie bdw€en pr€dictio.s bv L s ood.l and bv crdbcl o.lhod <l 2% v T,

e ror b€forc Koln (B) ).

(i) (a) % difeftncebetwe.i p.cdictio.sbvl S oodt'l dd wcibull plotting

< 9 7%,vT, and vari6 i. sien

(b) % dif.rcn@ b€tw€.n plcdicriom bv Gunbel rc6od ltd W'ibull plotting

< lO 3'l', VT, 0d vdi6 insign

(iii) IX. difrerch4s in bott (,g 8rc of sm. sig.

% diEeren@s have inqqs.d for lhe dars atd Kotd (B) (as at Sukkur (E), b'forc

ud aner $c .onstNdion of bdag6) dd ar. l&ge eio!8h to indic{te fiat th'

pr€dictaoN de no1 prop.r'

fu) lQ -Q.l=t t%J* r = 21j

As rhe di{f.rence > o 5%, Gunbel di$ribuiion is nol qpeded to be a propq ft (as

for b€fon Kot.i (B)).

(D

lit

Tabk 6.2.2.I ! sh@s thot at Kotti (B) beJe G'ler @) (1901-1962):

% ditrercnce bctw@. pediciions by L S nodcl .id bv Clmbd method <l 2%, V T

(a) % dillcrc@ berw€.n pr.dicriom by L S nod€l dd w.ibull plotting

< 4 9'2. V T, dd wri4 i. sign

(b) c/. dillqq€ b.r*ai prcdictiotrs by Gumb.l nethod !d w.iblll plofiing

< 4 2oZ, V T, dd vari.s io sg!'.

[,) The ditr€r€.@s ar. of sdc sign, sna €r thD atd Korri (B).

l0-Q. -O62r/"atT=231 - Olmb.l disrribodon i! tur. p@F. fit.(')

r.ht! 6.2.2. t 2 lJt*s 'ld

at kr' i (t) al.r Ohih (, (1962- 1919):

o)

% diUer.ne b€rwan prenictioG

< 9 3%, VT, and vlries in sign.

% ditrcra@ bcrwc.n pr.djdioB< 3 77o, V T, dd v.ries in sign

ditteren4 belw*.pr€dictionsby L.Snod€t .ndbyCumb.t method <1.7%,vT

by L.S nod.l ond W€abull plo{ing

by GuEbd nanod and Wcibult ptoiiiig

(ii)

0v)

As thc ditrd.nG ir both c.s6 e of $mc sisn .nd .re largd thh ro. b.forcGrddu (B) H.n@ 1h. prcdidion .r€ not prop.r.

% di,IeMes hlve insa$d .$d Cuddu @) (a for ltr sutd.ur (B) dd ,n.rKotri (B))

Q-Qa = |.44o/. dt T =2.J3.

A! lh. difse@ > 0.57a c!6b€l dturribltio, is ior | 6i (Sinilf ro bcf@ Kot i (BD

?ablc 6 2.2, 13 !h@s that at cdAt E) (1962J 999):

% ditrdcc bawen p'.dicrion by L. S rnd clnb€t < l8%, VT.G)v. di{r€racc b.rwen prcdidioo by r_ S and cumb.t< 8 6./0, lid vdi6 in sigi.(b)o/. ditrae@ b.rw@ pndicrion by L.S and Clmb.l < g.lcz,vT.nd v!ri6 in

Th. diffrd@s &. ldgd.nd8h to indi@te thlt p!.diclioE ec ru propq.

Q-Qe l=t.te JotT'=233As th. difercncc > o.syo, Cuhb.t dist,iburion is nor dpsred ro D. a plop4 fir

LS nnd Cunbd Mod.l3

Intccpt for Otrmbcl nodel < Intm.Pr for L S nEd.l for .ll dah s.ts !(t .ll

Slopcs for GDmbd oodel > Slope fo! L S nodel fo! all data lers lnd all banag€s

Th. .feis on slo[Es al Sdrlur (B) &€ grqrd for th. dara a8.. the onstruclion ofSukkur {B) tl6 lor rn.r rh. @Brrucljoi ofcuddu (D).

'l'ho.ll&ts on slopcs .r Korri (B) lrc lrrs*t d!. to sukkd (B) and snrtr.n du€ ro

crddu (B)

The inlercepl and slop. for tull d.t. li€ bet@n rhen vab* fo. b.fore snd .ncr rhe

bara86, in €ch cse. i e for Cumble a Ell a for L S nodel.

rcad Square mod.l iB .cceFed, for all dald ets and for all buraS.s

codparison ofpr.dictioE at Suklo. (B)

Percenlage dif.r.nes b€tEn prcdictions by I n thods ar. a! u.der;

tSmdCM G.M& G.Mlit Or Nor

sukkur(B)

<92

^nq sutklr (B) < l.l <98

Befo€ Guddu {B)

% ditr r..e bctwcn predictions by L.S by Gunbel Merhod is small, i.c < l 4%;

exccpt for an€. Glddu (B) ( < L8%)

L'gd diFffic berwd prcdictioB by L.S md Weibull or belwd Cunb€l adWcihull indiale fiat p'ldiclion by W€ibull m not propr

Fixing rolqance lev€l =0 5, c!6bel di5i.ibution is exp@led to be a propd fi lor tull

dail aid ror rhe d a aftd cuddu (B)

Th. difr.nes h.va inmsed .fi.r the @n$tuction of b@9.3

) Compairen of pr.diciioN ar Kori (B)

Perenlase difcroce! b.N€en pr€diction by 3 ftthods ar. as !nd.r:

c.M&

i Da'a <3.1

Fii

<t.2

SdoE cuddu (B) <t.2

I{ol.: r slandr for dif.r6l rdll

6.4 CoDclusion

Lrat Sqaur. model h r@ptcd for aU dara s.rs &d {o I bdag6. Crcst, Adj Rl

fhr lins for tull dar! e.is at Grch beraee atwry! li. bcrsed rh. tinc, for d,ta srbetor. ed,nd rh. bsrg6 by boLh rod.h, ic thftlre dldj of onsrtucron o

Tn! line! for cumbd model sr.rr at lowr valuc bur end up rr high€r valucs than toL S mod.l

Th€ difitEnla beteo predklbns of flood by LS Md Cflnbd models are smr foarr.durnbuton!. qcpr at Cflddu tB,, rner Clddu (B) at Kold (B) Md aRd clddt

PredGrions of nood by Wcibu ptorti.g e not M.prabte, due to the targe.differ.ne *irh predidiotr by cmb.t sd LS Modcl. ih6e diffqcica incr;rn.r th. onntudion ofb.na8sGonbcl distiibuliof, i3 nor €xp.cred ro be d prop.r ill in 9 dsta sd! oui of tj ser(,si"s l0 0_ l>oarrJ6)

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6

Chaptcr 7

Meihods of PrEnefer atrd Quarlil€ EstimrtioD of Pmbsbility Distributions

Ctdi@ of Probability distibuiiotu along with lheir desiptivc and pr€didionabilili€s. choic€ ofestimates a.d their propeni6like efficidcy, unbiased i.s, andusc ofl--nondts, Confidence and Robust.as ofD/E k disssd in rhis chapr€r

Probdiliry Dartriborion rur.rions

Lel the upp.r c.se lefier X denor. r randod vrnablq a.d a low.r case letter xd possible wlue of x. lbr ! raodod v.riable X, its cumulaive disliibuiion tunction,denoted by F . (X), is th. p@bability tbal the randod vdiable X is l.ss thin or equal

F (x) = Pr (x 5x)

F, (X) is (he ion-dcedane prcbability for the vdue xThe probabilily dcnsity tunction (pdo d.6cribd lhe r€lative lik€lihood ihal a

conrinuous mndon v.riabl€ X takes on diit€reii vllues, dd is the derivaliv. oflh.

| (t\= -lF tX, I

The rerum pqiod (some rim6 cilled $c ralr!@ inredal ) is onc.speifi.d rarher thm th. .xc..doe probab ny. For qdple rh€ antual nujnudnood flow dceded with e loz prcbabiliry in any year, or the chdce of I in 100, is

called 100-y€as flood , In geneml, Xr is 1fie T- year flood md fi€ r.latioNhip

b.twan TandPis Sivenby

r= IP

The relum pdjod, T, dpr€$6 that on arcrag. o.e flod grol.r rhan theT-y@r lood occuB in T-yar p€iod. The problbility ofT-y* flood being cxc@ded

J

.tCl'oic.of Preb.bilityDirtribtrliont

ln nany cou.tri* rhe choi@ of dist bution is argu.d in a 8€ndrl mannc., s

Th. (.hoe.) di$nbltion is

- wid.ly.. ... a@pt d,

- iimpl. @n@i.'rt lo $ply,- oNist6! odible o. rcbusr (low sibifi, ro odi6),- thcoGticdly wll b.s.d.

- doonqt€d i. the Glide wMO [5] lnd dwh@.No sD.cial m.tbod of pdm.tr $iimtion t pr€&d€d ed fic gophicdl mcthod

An ihponant probl€n in th. spplicltion ofEV distributions to tood is lhat of

choosing bdv€eo Typ6 1,2 6nd 3. Th.ory otr rs m n h in lhb cgrrd wh... lh.

e.d is 8rol6t. Thc djlEculry Md imlond@ of disti.euilhang b.rocd Evl ..dEV2 or bdw@ EV1 ad EVI k ju3l a grclt a thll ofdisdnglishi.g bcls@ Evl(or Evl or Evl) and LP3, for insrm, b@us. the difettB di3 bcrwn tlE

ndb6 of the fuily itscl4 in ih. nlnM in whict Q req wirn T. Th. EVI

ditlibudon pos*s6 a 6nire upp.r boud

Magnnud€ of the EVI vanal. il$ll incr@s linarly with l. T whil. $m.

olh.B inor.6e nonlinearly Vaial., wirh lrg. posilive Skdne$ incr46. mot.

npidly thd Evl vhile rho* with n.g!tiv. Skdn6s increae ldr r.pidly

Oath. ofi.r diskibutions, only th.log-.oml ha had ay lh.orctical supDon

Chow lrSl shrcd tbat irlbc AM [@d @u'd b€ .o.sidsed to be lh€ producl or I lar8.

numbcr oa andom eff@rs rficn ir wdld b. loA-nom.lly distributcd, b€lue tl'c

lo8!.nho ol $. vriit @uld b. @Ni&rcd to lhc uo of. Lrgc numb.r or Endom

.trets sd *ould th.rcfoE b. nom.lly dbr,ibui.d by th€ strd linil 1fi6@

Howw.r, to be valid 6 ! dcductirc th@ry, rh* cfr&ls elld h.vc ro b.

iddrifiable. F.iling rhis the distdbuion cu only be spponed by cnpiic.l dd!.

Thus !.gm.nrs 6noi, per sq ideiri& ! h.n ofdiddblrion tor ned! Empiric.jsuitability plays a duch taryd ole in disldbuio. choie rhs ! pdod r.asonins

slack er al {srl h.v. lhown that quantite .sinar$ wilh sma .xpededoPportln'ly design lGs (Wnicb is a ftnction of q@lil€ elioare bis rid m* ) eeobiain€d wa wh.n thc lssocd fod of nod.t disrribution is noi id.n qt ro lhepoDlrorion disrribuion prcvjd.d rhar rn mod.t dislribution is $ter.d ton ano.gdiinbutions which h.vc app.onndely rb. samc SkeM6s s rhc poputatioi

- lhedFtxbutioNnus havelheabitiq todqcnbeth.behav,o,ofd.t!ued

R&dom spl6 &on many of thc dbtrjblrioc rr.dirioMly used forfi.quency.mlysis do nor disptay the Me b€hdour of SkM6s s do obwed

Many ofth€ tladitionatty u*d thre parahet€r dktiburions !3, Lp3, CEV or EV2(w.ibnll ) * suficiendy fl.xibte to providc a mod.ratety good fil io ih. obseryeddltr Of the two widcty usd 1wc pdma.r dislnburioN Evl and LN2, thc lafier@ show a r€@Mbl. fir b a *id6 Eri.t, of ob$nd d.!. rhd @ $. fom* TheEVI sometins fir! obs.B€d dara w€ in hunid ctimat€s in which floods do nor vary

8,.atly from yw ro yor (bw q.)

Th. choie of . D/E pr@cdue ru$ l'L inro r@un! rh. pr.dicri!. lbi|ni*oa such prcedur6 along wirh oNidding dsdprirc abitiry tn vio of ln tict ofiblorulc k.owl€dgc of thc cor.ct fom of dislribltion of floods, rh. property ofrobnsrress dkcu$ed in *clion ?.rO is v€ry inponant io this conl€xr This dependsboth on the disdburion! chosen a.d frahod ofpar.mers €srjnalion.

Quesion of iir.Br is

sanplins stmd.rd .tror, inhcre,r

aeuaq (l&k of 6is0 dd .fiici.ncy (iiqe of

13.1

li se@d, distribuion ft.. turhod3 arc lN .6ci.nr rhan di$iburional

m.thod!. Thct arc by nature nor..ot!!r the dist.ibltioral nEthod! but rhis in itslfis insuflicidt lo l@mend th6. Th. si|tistiql mrm of flood *ri.s ( Sufrci.nrly

not well-known) dos 601p€rnil to n.k.. raso.lbl€, ifnor p.rfet, choice ofadistribution, whose us would yi.ld morc efii€imt qnanrilc eltinares lhat

dislribuiion-fr* n.thods. One D/E pr@.dur. is more €fficienl than snoth.r ifir has

small.r smpling vdia@ (=*2) thu th. othd

Panmdcr €sriMiion m.thods !rc u$ally n.ve b6 @mpecd uidd rh.

h@din8s of bi6, .Itcie.cy, $fficidcy lnd @isistaEy alrhough only th. 6Fl leoffi6. r@iv. much onsiddatioi by hydrclogrds.

ML c$imatd h.s oprin.l proFnie wh€. th. spls G) on *hich it is usd

adually are down from tn€ dttribution 6sun.d in lh€ preedur.. Ifrbc aamplc is

fror ! difercnl dhtdbution the optinrl piopenies are by no me3.s suarsnieed. Sin@

it is dttribltion sp€cifc it may not b. robulr Il5l.PWM i! a e!s, to lpply s odi.lry Dom€ntsr is usualb unbilsd dnd h

dmosl * efficicnt a ML. Ind..d in !n.ll spl6 PWM my b. d Gfficic s ML.

wirh. stubl. choi@ ofdilrribltion PWM.stimiion.l$ @ntrihltglo robunis.id is .(octive f.om that loi ofvicw [51.

PWM k superior 10 odio.ry momdr in hdpins to idori& dbidboriois.

dimar. thcir pdametss, dnd test lrypoih.5€ about valud of such pamcters.

L-moments which are dim.nrionless PWM Elios, ee also used to help

id enti& d hrributions l29l

Th. m€rhod adopted for dlimdion of Qr in ay silualion d.p.ndi on th€

amouni dd iy}. oriydrolqiel drl. N.ilabl. ri th€ sirc in qlcario. Propqti.s of.stimlo6 aE dioss.d in slion 7 7

7,5

The probabihy distibdions r.. tunction of a s.l ofpsam.r.ru as w.ll is rhe

r.ndom variables. when a probnbility disrribuion is el€ct€d for f.€qtrc.cy analysh

or rddon hyd.ologic vriablc auch u AM fled, e inporlst qucsrion is ro

cst'mte rh. p.rmd.rs oathar disribulion so rhil rhe r€quir€d qlaniitcs can b.calculal€d fron the lificd diskibution

A nunb* of n.thods can be used lor prrffi.rd sdmation Th.s. i.ctud€ (i)

MoM (iD MLM (iiD PwM (iv) Ls {v) Mdinum Enrrophy (ENT) (vi) Mixed

Mom.nrs (MIx) (vii) Gdcr.liz€d Mcrhod of Mon.nts (CMM) r. our $udy lh@ ofrhc nore @frnonly ureil m.thod ft @nside.cd, lM.ly (i) Melhod of Mon€.ts

(MoN{) (ii) Maximun Likely hood Method (I,!LM) (iii) Powd w.ightcd Momor

?.5.1 MOM

Estimale of rhc pa@ct* of a pdf e obt.i.ed in tle MOM by .qudins th.nooents of lhe slmplc with the mom.rs ofrhc Pdf.

For a dinribution which pahm.r.rs, d,4,, ., drwhich @ to bc 4iinated,

rhe fict k emplc moncnrs a'e set eilual to the @{6poidins popnlarioi mon.nrs

lhal .r. giv€n in t.rn5 of un*noM pdnd6 Th.e k equariotu lrc th.n solved

simultd€ously for lnc unknovn p'lmd.c

Ir h r€larively e3ry ndhod. MOM €slinltes ar. u ally iifdior lo MLM being h$.trcidt .sFcially for distibulions qilh lhrc. or norc nunb€r otpddel.A (Highly

biaFd in rclaiv€ly smu sdpl€t.

?.5.2 MLM

Estinalion by ML me$od involv6 thc cnoi@ of palmel€r 6timaLs th.l

produce a maxinun probabililyofoccurmce of lh€ob*ryadons

Fo. a dislribution with a pdf eivm by (x) ed p&m€leB at,a,, .,a, th.

litelihood tuhclion is d.fin d 6 the joint pdf of th. ob$dalrois co.ditioiil oi 8iv..value oflhe panm.t6 aL,a,,..,at in lh. fom:

Th. valu$ of a,,a:,. rdl rhat mximiz. th. likdinood tunction @

computed by parli.l diff...nriltion with r.spdt to a,,a:,...,dr rnd s.nins these

lrarrial dcrivrlivca cq(rl to z.ro. Thc rcs{lti'rs sd of cquarions nr. thcn $lvcd

simullaneouslyloobtainth.value!old,,z:,,dr,ie.

aa,

In @y ass it b diq

Patamet( 61iui6 vc obtrin d in thi! r.clhod, d in rn ce ofMotvi.

equating th. nom.trtr ofthc arnplc with the noMt! ofthc pdf

L- Mom.nts lre anoth.r way to $mm!riz. the slrlbtiel propeni€s ofrddonvdiables Tbb pronising nelhod he b4n d.v.lop.d by Horking I29l

=tl

ro muiniz. tn. nalnral lo$rithn of the likelih@d

ror r disrdbltioi with k pdMdsn 4, ,1,. .,r. , {hicn e€ to b€ dimted,

thc 6^r k smpl. mom..rr d. s.t .qul ro rh6 @Bponding popol.lion nommls

Th€ r€sulli.g .quadons d. rh€n lolv.d sinulte€orslt for tlE mkrcM pdd€t€E

d,C,, ,C1, D€tsil, of th. erin.tion proc.dur.s for eh of $e dinribuions

oNidded in lhis itudy by uling rhc abov. thr€eatimtion merhod are givo in

L-Mon rts @ be cxpr6s.d in tem ofliid combiMrioc orPWMs. Alrholsh

L-Mom.nts haE ben d.nn.d for a prcb.bility distributio\ bln in tdcrie d$lo0c. bc cnioaicd fron a fi.ite Mpl.. Th.s src almosl unbia&d Eslimnion ofsmph L-momenls is basd on a emple ofsi& n, ananged in amdng ordfI-clx ^<r,,< . <x,,,jbelh€ordercdsimplc

An unbiasd cstim.tor of Probability weiShtcd Mohd, B. ,E

4 ,'r ltl.l,., ( r.mom€nr orord.r,)

This may .hcmativ.ly be wnir€n as

,,'I,," (M4surc ofroc.tion )

, tll !,.;(, r)-.i9:+4AQ t)U -2

(r-t)(r-2) (t-,(,-')(,-2).(" .

))

Thc snplc L-6odc.ts are obtained by

Thc sampl. t-mome.t rrlios d delin.d by

t. /t. forr:l

tt = bo (Mq$r.of lodtion)

t, = 70\ 30b,+t24-bo

lt,l

| = t,ltt ( M.&E of s.de &d disp.cio., i..., C v)

Th€ vdus ofpa.am.tcs, bi's , r , ri 3 ed L nolMs fo. rh. dltr 3!G ofGuddu. Sukkur and Koui balg.r .re shown in Tabl* ?.3 4. I and 7 5.4,2.1 .5 4.j

The bound€d -n€s ofL nom.nt 6rior b o ad!d.96, bs.ue C! , Cr ,

4 ,4,.. qn trle ffbnddty trry. v.lu.s

ro z 3. I r.l <toFor x: o, r srisfie 0<t<lh6( l,l) is a na$rc of SkarB+ i... ra.tr tu r llffi orlGnGir, i.. Lcr.

Ttbl. ?,1,4,1Mtu Flos.n<t Mom.nr R.rio for Thf. Drra 5.6 oflndui tuv.r

Trhl. ?.5.1.2 E.tim 6 ofco.m.i.trt for PVyM" For A[ Pmb.bility Dittrtb tontN

cuddu (B) Full l3 701.658 421.454 303.104 245190Sukkur(B) full 99 622.879 166.591 266910 2t2 644(od (B) Full 99 451949 210.U8 197.108 l5?.082

T|bh t,5.4. 3 L - Moo..rr trd Rrrh. ror d. D.r. olTtG D.mg6 d h'|.u. R&r

N 5,D cv c, cr

Glddu (8) t3 701.653 25t.03t 0.158 0.612 2126622.419 205.708 oll0 I t20 r.68

|(oln(s) 159.645 0.t51 0830 1537

N

(Blt8 70 | 658 l4t.25l 2t.519 l6.t6l 0201 0.1tl 0 l16

B)622 419 0102 25156 19981 0 t1l 0 228 0.l8l

xmi (B) 99 451949 88.507 l4 424 ll.l95 0.r96 0.161 0149

An.r th. .srimation of p.r@rd of a di3rributi@. qudliL qrimr6 ( Xr )whi.n conBpond 10 difdar rcttm p.riod! my b. @npul.d. Th.,.rum p..iod is

'cl.tcd ro th. pobabihy or no.aacdde (F) by rh. ..hrion

T

..-. wh.rc F- F(Xr) is rh. probrb'l y of hlvins r ttood oa mlsirud. : xr Chow| | 4 prcpos 5 8@r bmuta t6r otot.rina x, a

Kr - fr.qucncy f.cror ehich is ! tundion of$. rctum Dsiod rd s6nd.6

rhc he$od adoflcd aor Bdmtion ot q in sy rituarion d€Fnds on rh..@nt rnd ryp. of hydrologidl d.r! rvJhbtc it ih.3it in q!.stion.

Th....r. rw typ6 of doB .!si|r.d vilh qudrit. din.t6. Th. tirsr ryD.rns fioh th. cumplions that rh. obFrvld dlta follow | ,r.riotd disdbutio.This .ror qn b. ch.ckcd by 8oodtr* of- Ur csr diss.d ir ctspr.r .ighr . Th.3@nd rylE as rh. .@r inhsot in p!m.r.B 6rin.rcd f.oh mdt spt6. Thi,crcr 6 b. rcduc.d by uliig ! d.rhod shich givc! minimuh vri'e pdM.l.rcslim s = > ninimum vlriNnc qledle. etNt4. Tti e.10. d b. cncck.d by

ihc ue of @nfd.@ int.N.t3 usirS rhc i.tomldon .Dour rh. !.npling ldise

Prep.ni6 olQu|niitg ErtihrtoB

MOM, ,lrh@gi ssy to .pply, d@ed is .or .. cfftci.nr

te .I lh. splc infomrion in mML 6rin[tjo4 dpcirtty in rhrc

MLn ihod is regard€<t s boig bcst b.@us ir is nod 'mci€it

That is the

snplinS varian* of lh. estimted ta6mele6 and hence of Qr is &slmptotiollv

smaller lhan by any othq 4tinarion melhod

ML eslimatc, are frequentlv bi&d bul @r6lioN for bi$ cd b€ found

ML dtintd may nor alwrvs b. fdible i' sall sFpl6 ftom lhta

MI-M i3 md efficienl, ie with smudt spli'g v'riancc of e$isal'd

pffametG and dinimum variMe for quantil6 atihar€s In $n€ cdes (Pason_r)

ns oflioahy is o.ly synptotic, i € nol good aor smal emPl's

M l f.cqenlly 8iv6 bi.!.d alrirol..' but bi6 qn b' bftal€d In ce o'

larse Nnber of pude16 *ith smll spl6 it is nol posiblc to g'r MLM

€s1imala. It rcquno highr @npllalional cforts.

Th€ opplication ofML estih.tion i3 no lo'ger unrtltlctive tom a nuneial

poinL of vicw bccause of lh€ widesPrctd u* ol prosr@m l€ cdcula|on and the

incrssd nufrb.r olmicro &d per$nil conputm tow avrilabl'

Tte PWM n€rhod tN goo<t sl.listicd ptopsli6 It wa origi'allv ns'd onlv

witb dknburions who* dttibldon tunctioA !(q), is dplicillv 'xpr6ibl€ in

MOM b relaiively 4v nethod MOM €sliDat's re usuallv infe or 10 MLM

being less c{icicnt €speiallv for distributioN tith thft ot not' nunbd of

padd.6 (Hi8!ly bas.d h r€larivclv sm.ll sPl€tPwM i3 ComPlrable to MLM compulitroc for PwM ar' sinple bur ror

distibutiotu, S E S de ditremt lt is Bdrd tha MLM for small spl*' i' $nc

M (i e. Explicil fomulae de po$ibl. for em' di*iburioit

The estih.ted !alu., 0, , nav diff.r trcm the true valu' Qr becNso or

(a) lnability of nodel choen (AM or PD) ro rcproduc' E populatio' Q'T

O) In@r@1 choi* ot distriburioi to de$rib€ the poPul'tion Yirhin the chose'

(d)

ln g€n.@1, ML nahods e knoM lo b. n6t efEciot i, ,syhproricdtylarg. enpl6 Whil. rhcre is m swer6 th.r ML b mon efficidl i. sr.lt sDl6it f.€q!.ft|y i! 30. Apcn fion €fnciercy, faibitity is horhd @nsidmrion. ForiBtdc., Matal.s Aid wallis I43l .epon.d rhar ML .srim.t6 were imeossibtc roobtain for som. LP3 samples using pt/ML €siimrrion meiho&.

ML stini€r hN optinal p.openis whcn the sampl6 G) o. which it is used

adually .rc d6wn fron th. disrribulion ssun.d in th. pr@.d!rc. Ifthc *hpl. i3

fron a dift rc.l distriblrion lhe optind prolEni.s rrc by no m.m guMtc.d. sineir is distribution, speilic it may .or b. ,olrler.

PWM is d 6y to apply s ordiMry momols, is !$aly mbiar.d .nd is

almo$ as €liici.nr I ML Inded in sda sa6pt6 pWM hay be d .mci.nt as MLWnh a suii.bl€ choice of disrribulion lWM.srimadon allo @nrnburca ro robusincsa.d is ollracrive fron rt$r poifl ofvicw.

PWM ar. slprrio. ro oidery mod.nts in helpinS bestimte lhen p!.rnet€G, and t.sr b/po1h4s lbout va|u6Whicb &c dinosionlds p\irM ratioq L mom.m. ec arso

,, {,nqsure olvarisbrlitt ofthe dimarcd vatucrrh.s.E of6rn&re, sr,qnrch 6 denne4 by (utuan., lt5las

Bid in lh. €stimaring prcc.duE (if rhh i! knoM to qi,! r @n6to tu b.

Smpling cror due ro rhe fad that pm,ftrd &e drinared from a tinir.

The .vaibble 'eord tsampl€ mry not b. a truty randon

r€quncd potulation) No €ontrol an b..xqcbed over rhis

can bc madc 'o

vdry rhe reemblcn.s of rheNumplon

It a@urts for th. cdor du. to irMll mols bll mr for the dor duc to th.

choi@ ofimppropnat€ dbtributioi. Thc S.E ofstimtc d.pcnd! in gd6d on th.

nclhod of p.metd 6tim ion. Coi!€quody, *h nEthod gi6 a difsst S.E ofqslimre Th. nort.6cic'n ndlbd b th which giB $. sEll6l S.E of$.

ln g.nqal, srad.(l @r of 6tim.t in.|t.g wilh T. population Cr .nd Cs

v.lu6, and is inve8cly p.oponiond ro 3.npL 3i&.

s I Qr ) = o € (T.C srlN"'

wlft o is popularion standad ddidion Md g ( ) h r tunclion which depqd

on the rom of rhe 9!dr disrriburion .nd on th. n.lhod of pdmet€r simtion. I ( )qould al$ depcnd on lh. forn ol thc etimari.g dbttjbllion from the pn€.t datr

h i. d@ th.r r poinr 6dtul. of r 6ui.4@ril. @r6pordiig ro . crum

Fiod my b. or@ 6l signif6@ unla tha. i! & itdic.rion of dic.@6.y oflhc sri@te Th@ has b€.n svsll srudie! rclot.d to pr.dicrion a@Ecy.

Risk an lysi. wa dl.nd.d 10 tim. d.p.nd. flmd nod€ls by Na€hnrb.l

r.d Kon€cny la6l Rdurn p..iod! @trdponding ro n6 .rrlrl flows wcdmiied by Rao 1531. Viclorov 164l dcmonrtr.tcd rhNr tb. €tr cr ofp.riod ofrc@rd

on food pr€didion on be very liong in the s.ns rhar [oo& aninared by using

diferent l0 y@ rdor& vaii.d !r much .t 595%

An approxhlie (l - a ) @.fido@ iir.dd for xris giva try

xt + t4 srt

wh@ t is th€ stanr!.rd Mml v.r.rt..

A roblst D/E prs.dure is rcltrivcly

dhinbudonal asumptio.s which n qesun.s rr..slmates ae bis, b, md roor h4n squlrc cror,

b = "\Q,_a,\r^,nft - lE\O- AlL-', l

Co.ndoce linits are rel{ted ro rh. na$r$ of dek. Bob€e and Morin lllluscd the dislributio. tuNriotu of the ordd staristic lor rhe lden (3) distnbuioi to

d.nv6 the @n6dae imervals dsciat.d wirh it.

7.10. Roburtnd ofD/E

insdnsilive to snDll changs in the

itu.. Th. cril€ria for ss$ing the*

A procedure lor 6titutin8 Qr is roblst if it yi.lds €slimares of qr which eSood (low bi6, nigh emciency) .v.i if lh. proc.dur. is ba$d o. M asmption*hich ie noltruc A Focedurc is .or roblst ifn yicl<b ter 6rin.i.s ofer $nq the

pro..durct assmprions d.pDn dcn sliddy &om wh.r is Io.. Sin@ ve do .orknow rhe distribnton of AM omds in iallr. ir b.hova us to st out dd find a

disribuiion and an 6ridaring troedurc ehich roA.rh.r ar. robusl whcn dating *nhdistribulions which siv€ nfldom enda whicl' hav. a 0ood,like bhaviour tt shoutd

bc pointed out lhat splir sanpl€ resll brsd on historicat AM Rood r€6rds ar€

inad.quate ro. t6ting the robush.ss of any distribuion and *timrion {D/E)

Broadly spaking an esridator i! robwi if I dtihdls er .,sosibly weU, de.if th. assunplions us.d in the csdmar4 ar. 5liShrly wong or ifthe dala are itl

behaved bfraue ofoutlie^ or m

A procedure is not robusr if it yi.lds p@r Atinats of Or whd the

proc.dur. s eftptions d.pan dd slighdy frcm whar is lrue Sine we do not

*now the distribudon of AM fl@ds in naorc it b€hov6 Ds ro Fk out ed nnd .distnbulion ed e 6iiming proedlre which tog.rhd re robun wh.n dqting with

whi.n sirc u&n $tnpl6 enidot rhlt i?lit sPh t6t! b!.dfo. t.$ing th. robultn6r of &y

hrE . Ilood-li|. bh!!io.. h should

on ltuoncd AM tlood l@rds d.dinriburion id drim.tion (D/E)

t5l

El

Chapt€r I

TestiDg The coodnes! Of Fit Ofprobabitfty Distributions

Introduction

Thc st.tistical dalysh of ftydrclogiot dar. dcp.nds lpon qr suc.* in

fitting a ih@i.lic,l probaliliry tu.dion to rh..mpidcat dishtutior offic obsd.dsplc. Blt the probl€n is b 16r how wclt the rh@tctial prcbalihy tun rion fitsrh€ crpiri*l disiriblrion. The purpose i! ro i.st rh. goodn s offil oa ihe rh@r.ri€sl

disttiburion lo lhc 6pnid disrribu on The siatisticat r€sls for rh€ purpo!. tred in

rhis sildy are Chi-square (ur), shimov- Kolnosorov (S-K), probabiliry plot

or€lalion co.€mcifl1 (PPcc) a.d Roor Men Squ... Enor (RMSE). w-r€sr and

A D l.st .r. uscd ro tesl the nomrtity s pcll s Soodnd of 6r of disrriblrion Thepoc.dur€s qn bc u..d lo rsl lhe disdblrioN $Odr.lt bur nor !s di$nninaloryl6ts ror choosing bdwen onc disriblion and an oIhs ( Cunw) Mo$ of $.n.lhods.vailabl. for sel6lion of dtriburions fon sm.I smpts ar. f,oi s.biirivcenouBh ro discriminate mo.g distriblrions

Ccftin goodness of fir stalGti6 may bc usd as the basis of. tesr of lhchypothois that a given sample ol d.b nly b! .eaa.d.d as hlvina been drawn

randonly from ! distributions of sp€cificd fom Exampt€s of tb6e are /, and S-K

sralnrica. Such l.sl can rejecl or by d.fdull accpt, the ru[ nyporh*is rbat a sanpl.of d.ta haw com. f.on a $ared dislribution.( Iffie ptuerqs ofrhe diribulionnam.d in th. tlypo1h6is ee 6lin.rcd fiom rh. Mde ir*!t $ is uMIy rhe q*,ihis is tak.n inio a@nr wh6 @unrinA dcgrds of aredo4 i.. i, <rermining rhc

distribltion of the r$ srarisrict Ths t4rs d. bc u*d ro tesr disribuions*paret.ly to dete.min€ wberhe. the data !r. in .@ord eith th€ disrnbudon or not

l. Btjt ia NERC (.31 plblkh.d r@lt5 of uing z' dd S-K tdts iid.tmining the $nalility ofdif.onr dindb{ti@ fo! AM nood 3diq. Th.$ r.5uhs

in g.n.ral nu$ be regrd€d e in@ncluliv. bdu$ of th{ lack of sl.tklicrl powrqhen cho$in8 bdw6 ali€m.lik!

Tolint t[. Goodtr63 of Fil

Thd. & rrc nEthods for. vi$rl judgnar ofgood6 of fit. Orc manod

h lo @mporc thc obswed rddiw t qu@cy clllG eith thc tholeiol rclliiw

fi.qu.ncy ore. Th. s@nd n rhod k to plot $e dr a on .gpodatc prcb.bilny

p!p.r .nd judg€ s to whe$er or not thc Bulti.g Plor is a s16ght line Th. stcp! for

conduoling .ach of lhe t6ts of G@dn s oflit u!.d in this dudy arc gircn 6 utdd

Lll Cni-Squrr.I6t [ 4 |

Oe of rlt nosr @n@nt u!.d tdt for goodN of 6t of @piric.l d.t. to

lpfificd thercricd fr.quaEy disrributid b thc Cni-squc t.$.This l*t mlt6 r@hDuLon h.rl6 the &tu 1 tumhd of obw|rioN &d rhc dp@tcd tumbcr of

oblwaliotu rhal f.ll in the cts i w.l'rh. dp6t€d nunbw t alcu!!Ld by

nuliplying rhc dp.cl€d r.lativ. fr.qudcy by $. tot.l nunbcr of obeNltioN.Tha

rn lbfttic is qlculated from fte r.htionship

x' = z@,-E,'l l4 (3 l)

wh.€ k isthctumbdofcLsintN d O, i! rh. otsltd.td q is lhe qpet.d

Nnbd ofo!*tutlioB in rhc hh da i d!d.h t bettd to defirc cls inicryd $ lhtr dp€ted .o. of obwdiN in o.h

int rysl ir lhc em. li h6 Chi tqun. dbitibunon silh k tl des.c of &ccdod,

s h.r o R th. iumbd ol Dr@.l$ cstrrh4.d flom lh. dda

th. lrypott4is tbar rhr dala dc f.on rhe lpeificd distribnrion is ,cjercd irlb6 cal.ulated vrtue of Chi{qle is guta, thu ir taht .n vahe at l- d t.v.l ofsignificanceand k-p-l d f, i.e,

r."r1.".*-",.

Ih. Chi-qure $disric dwty dcpdds oD ihc iumber ofctss intdalsdploy.d a! wcll as the cta5s limir In gd..rt, rh. nunh.r ofctas inr6rts, t,should b. gur€t or .quat to 5 od rh. dp€clcd v.tues of ab€ohre frequcn y for ochclass int.ryrl rhould al$ be greard thd or .qusl ro :t HAAN I23l

3,2.2 Th. Sniftov- Kotnogorov 16r

l| I a non,pdran€dc 1€si to b. usd ifno difinbution tunction ofa gjvcn rypet sp@ili.d and no qtinuiing or lcsring of iis pMmctcrs is pe.fom€d, ed rhc only@ndirion sp.cificd is rn.r lhe dGuiburion is @nriNous.

t r P\ (r) bc rhe @npt.rety !p6i6.d rh@raier @nutarirc dinributiontu.dion und.r rh. nutt ntpothais

Let Sr (x) b. thc sanpte anutative ddsity funcrion bed on n obseryarioosThis t.sl is.o.ducred usi.g rh. fo owing $.ps

Sl.p ll, Dctffii.. rh. ru'num d.vi.ron, D, dcfircd by

For aiy ob*ryed x, Sn (x)= k / n , wh.r€ k is th. nnmb€. orobseryaaons l€s than or equat to x.

F nu lF(rx) s. (x) | ............... (&2 )

Sl.p llL li fo. ih€ chos signincaicG ldcl, tlr. obw.d vdu. ofD b greair

th.n or .quol to th. diticd t$u|.l€d v.lu. of th. Kolr.oaorcv-

Sbiruv stalistic, fi. null hyTothdis rhd fi€ dpiric5l distribution

follos th. Nmed dhributio. t rcjded.

3.2.3 Prohbility Plot Corelrtion Co.mci.nt (PPCC)

Th. PPC .ocfficidn wd fiqt u.d a a c sioi for onpldig th€ eedres-oa-6t of alt.mriv. ditrihudoc, by Vog.l [65] 11 8iB a l|:.erc of thc od.larion

bd@n th. o.d..cd obsatiotu t'rnd rn 6r6ponding 6n.d quatniLc

ftese w,rEdd.rmincd by plorrrng posn'on3.lor e&h h, tud

( 8.1)

(r 0.r5 ) (8.4 )

vhre i is lh. rank i. ordsed obsd.rion, n is rlr tuple sire

Th. PPC @effici€nl p m€a!r6lh.linq ty ofth. problbilily plol providing

a quantna v. a$ds.nt offir and irs v!lu. nar 1.0 $89*t rh.t th. ob*tuaions

@uld haw ba ddM 6on th. 6!.d dirtributiont

the PPC @fici6l, p, oflhe 6tt d distdburioi .l ! bdr€e @ b. ohrin.d by :

(8 5), = i(, ,)("-")/{;r, o;(, ')'}"'

,= Ln,t, (8.6)

vo8.l 166l gjv6 6jncJ val!6 ofppcc for clEh.t &d Nom.l dtt.iburions.

3.2,1 Rmt Meu Sqtrrrc Error (RMSE)

Wh.n a distriburion is fitred to a spl€ of ob*d d.r. !t a sraton. rh.ddiaion oftte fifled dislribution from rh. obsd.d itat. dn b€ 6vrtuli.d by RMSEThe avcng€ RMSE dn b. used d a citdion for compdiig th. goodi€s-of-fir olalrernltivc dkrribulions In lh. ce ofregion.t &.tysis, ir is advanug.ous lo us rherc|anrc av.ng. RMSE rt @n b. qtculared ar a sislion tron,

RMSE =

(8 7)

. (8 3)l;i [+:]']"wh.rc, h, is th. obsrycd vltuc

"na /, i" rr," -.p,r"a "alu.

r.o, rr," nn"adi$nbdio.s Plofing posnion aomuta h rh6 sm. s for ppcic.

AhonS llrr€d di$ributions !h. one vith sma $r RMSE h6 lne h€sl ,i ro lhe data.

REMARI<S

Many hydbtostr diFolns. rhe usc oa Ch! squaE dd S-K l€sl for bringhydrologic frequdcy dhtribution!. Nenn6 lh. Chlsqwe resr nor lhc s-K lest ar€very pow€rtul sp€cially for 6n!ll sMpt6, be@u$ rh€ p.obabihy of aftepting th.nyporh.sis is vot higb whd ir i! h tact fatje. Th. nct pow€rtul re$ is ppCC.

3.3,1 W-T.rr ls.rlThc W-Sraiistics is obraincd by dividing thc squrc ofopp,opri{. lind

@nbi.a on oflhe smpte ordq srdisti6 by usual ,ynnetric otnats or vuide

rXb alio is both s..1. dd origin inldisr Dd hgre rlE irrini€ is .9ptupdarc for tr.d of lrypotheis ofmftrlity. Th. w-Slrlidic! ii d.6ncd by:

- =(2",,,)'fit,,_,r ... (8 9)

W-$lrhdca ie b@nded by ] O. This b th. rluimun vdue rhar th. W-st{risrics @likc and doot6 extrde nofr.lity ro @mp!t. the vdue ofw, giEn a @ftpl€te

redon suplc ofsia n, on. p.oed! a! follow!:

{ i ) order $. obefldioi to 8.t .n od.rcd Mpt.

(iii) ( r )

(b)

... (3.ro)

ffnis.ver r=,/ 2.6npuL, =Z "" ^0 "," y,)

rh€ vduc ofli rhv. bd trbullrcd by Shlpiro.rd WiI( I57l up

50

ifn ^

odd. l = (x - l )/ 2, rh.onpursrion isjlsr rhe eme sin(ii0(a).

(iv) compute t/ = D)/ S: .. (8ll)

Small valu6 ofw indicltc ron-norm.iiry

Wat tistiG ar difraot ignif@r ld.ls @ r&{t.i.d by

sh.pno .ad wilk l5?l ror n - !( l) 50.

A@.ding to Rot{on l3al, W-Sodnic provid6 llc b€sr rca ofmrnElity

EJ,2 A,D-Tst

This ldst t us.d ro check wn€thd a enpl€ of data m. fion a poputation

wnh specific disiributio, Ii is d modifi€ation of S-K rat &d 8iv* nor€ weiShr ro lhc

AD rcsl i3 nor distnbution fre as ir nak* use ofth. speific disr.ibulion

being tsted. Ir is nore sensitivc tud tabks ofciticat valu.s arc aho avajtable.

H,

Thc dda fotlows r sp€i6c disrdbution.

Th€ dara do€s nol fotlow tbc dinribution

T.st StarGtics: A: =-N-S

,k,€ s=il3:ll[z rrtr;,v ".,F s lh€ @mulative disribuiion

Y are lhe ordcrcd diskibltion.

. (3.12'

.. (3 t,

rh. hrtothBis tn.t the didiiburion is of .

^. 's sar€r the lh. crnical rrtu.

F (v,.,.,))l). L" 0

Criti€l Region: The t6t is on.-sided &dspaific fom ii rej€r.d iftn€ r.5r staristi6,

Not.: Minit.b rnd Dara ptol snvar€s de !*d ro ger the rsults of abov€ resl

Chapter 9

TRDDICTION OF FLOODS AT VARIOUS BARRAGES IY FITTING XTREME

VALUE ( EVI) AND GENETIALTZf,D EXTREME VALUE

(GEV) DTRIBUTIONS

9.t

Extrft vrlue (Ev I ) and Gmd.lizcd Ext'!re valu€ (cEV) prcb$ility

dist.ibu oi s are fittcd on lhe dala of P6al fl@d at Guddu, Sulkur and Koti

The lard.r.R ed lheir S.E s e erimGd by tltre m.rhods .d.ly,M€thod of Mod.nt! (MOM), Mdinum Uk.lihood M€rhod (MLM), and

Probabiliry W€ishen Monerr (PWM) The rduhs src hsred nsing xt -test, S-l(

lcsl, PPCC and RMSE The relatio.s betw Cs, Cx md b€twan ehple Lnoo. EtiG e usd ro nav. $mc ide ,bour th. p{.d dinrib{rion

Th€ dota of annual nood p€.Is at cuddu, dd sulklr beags lrom l90l !ore35, w0s anrltzcd by H B Bhuuo dnd N M Shaikh l8l and ltl, usi.g Fosq

typ.-1, I I t,H'a and Gunbel Extr.m. flmd fr€qucncy crd6.Th* dknbuftns wft 6r|.d by M. Id@ l.lal on foods drr! ofJ.hlnm

iivcr.t Rasul Wodd M.lroloSicd Oqaniatio. (WNIO) Itsl ale rc@m€.ds

lhcsc diskibutions Lrndwenr ud Walli! I39l conpar.d PwM! with sone

tr.djtion.l l6h.jqlcs in eslimlind Glnb€l psmd6 ed qumril.s ThB rh*dist.ibutions h.v. bdn s€l6r.d i. thir sfudy ro v.lid.te th. r@lrs ol preliNsludi€s The dsla, the freqrocy cufr6 dd HiltoSram of {ood Peaks are

The value of Sbistic s.h B Ma4 M.di!n, SD, C- C!, C- e&, e.pr.sdred in Tabl69. La 9.1.B and 9 Lc alonc *ilh rhos obrain.d in | 3 I md

l,, for 85 ya6 ( in rareith€sis ) Th. @nput€r sofrwarc SISS, Ex@|, Minnab

aod dat! plot werc uiql in thb study for Stltisti€l a$lysis.

M€rhodolog/

The permer€B of Gumbel dttdbution are €sdoal€d by MOM, MLM .nd

PWM, wh€.e s the paramd.c of CEV distribltion tre estimal.d o.ly by MOM

and PwM Quaitit.Gtidata (xr) aid lheir SEt . which @rcspond lo

difer€nl reru.n t€riods up to 100 ya6, ar€ aho alalai.d

9.2-r EV I dbt.ibtrtiotr

The Exrrem. value type I (Evl) distribution, whicn is lko knoM a Gombel

dislribDlion, is rhe most widely used distributio. in nmd frcquency amlysis {41

& l{01 ThepdfkSivflby.

/(,)=1"-of-f4ld l\t1 )wh€.€ a and ! ar. paEmctcrs

The lariablc X iakes the valu6 in rh€ nngc -@ < x @.

Thedishbution tunclion of x is givd by'

f r.-pr-lI l-;t I/q(x)=expl-e ILI

p, the lodtion pMmcter, i3 ih. mod. of th. dblribltion It dr

vaia.ce aid mean (af / Ai = 0 ror x =B)

d isa mqsd,e ofdrp€Bron rnd depends only on theveDnc.

, r'atu-btn,d and d'= ---.wheee=!

' ' '.(e 2J

. dbrribltion It d.p.nd. on lhc

{a}

(i)

PARAMETER DSTIMATION

MOM METEOD

p = .i - o.+soos ,tfl

-ta _: ln" = 0.?197 ,ln, . {el

(e4)

wber€ m' ed n: ar€ nom61! of lr .nd 2d o.d.r'

0i) MLMMETfiOD

- l,,t r- N \, fr),,",_i, st:'J _l r i,._ol;.r;r=q (e.r)::, \x;' )=,flE .quadon h a 6Nt b. etv.d ddytiqltv tt L elwd ndrivelt bv

NeMon's manod. An ioti.l v.lu. ofc B r.q!i.cd Dd 6 b. taka a 0F MOM

6rirate ofo TlEv.tucofa buPdd.d aa"-, = a" - F (a.)l F' (4.)......... tso

vhd€ P(a) i, thc ddivrtiE ol F (d) rid i! giHbv,. r -"\ ., /-t,\

F @ dF@ = Li,i et;J -i,t; I.-L i',' da o,i ;.i ai-t .(e7)

Th€ itmtion in €quation for o.n is rct .d uitil F(d) is sumciddv cloF to aro

Anq an *tinarc of A is obtiincd, w. us

i =a be (9 8)

"i+J

2"

ln ordd to obt.i. rhc Elu. of p@er.r d, t @mpor6 prcam i! C+ + brSuSe

h$ bd d.v.lopcd !.d is pr*med in App.ndix -A

(iio PvyM METEOD

a = 84 b")l bs(2)= /,/ bs(2)..............(e.e)

p =b. -0.571x151 d . (9.10)

wh.re b md b' arc plorting posirion .!rimr* ud 4 is th€ sdple L-hom.nr

(E} QU NTILEEIiTTMATES

Tl. disiriburion tunclion ofEvl (2) m b. obtdD.d io tne ire fom a

r=l d lo8 (-ro8 F) .. ... (9.1r)

Tn T-yed quetil.! ar. qldlared by eblriruting F=! (l/T), wh@ T is ctum

loel- tos(r -ril) ] (e 12)

( C ) STANDARD ERRORS OF QUANIILE ESlIMATDS

(r) MoM:

+=t( ir ) MLM:

( iii ) FrvM:

S,'=a

Irroer*o.sror*ooeorl

b*a

Ll5s94 +o r9r8? Y + r.l Y?l (9rt

r'l ......(e 14)

l.lt?8+0.45?41 +0.8046r11.... . ...(9.15)

t-

*r,.- r = S = - r"e [-roe (r -Vr) |

9.2,2 GEV Dirtriburion

The G@Blizld Bx1ronc v.luc (cEV) diddturion, inroduccd by Jciliso. [ ]31

@nbin6 into a sin8l. forn lhc thl.e po!s'bl. typ.! of limi!.s dktributio. for

extr€ne valu6. Th. di{ibution tu.ciion is:

.u,=* { -[,-,(?)]"'], -'.

f | (r-r)11 .= qp 1._dp | --llrrr0 .... .(q.16)t L d Jl

\v!lh x bqnd.d by u+ d/t totn.Dovc if r>0.'n ton bd@ irko. rl.r.i u adc @ locdid .rd $dc p.Dntrdlr cp..riv.ly , .nd dE 3hrF plr.rdd rd.[.miE wnii 6t!!d v.lu. diddbqtio. i! Fpr€qttcd. FrlE-Trpt .r Typcs

r, u ..d Ul @|Epond ro t - 0, k < O.nd t > 0,6p€tiv.t. Wha k = 0, th.GEv dikiburion r.duc ro tlE Cumblc dilrributi@,

Th€ itrwF disrributio. tundion ii

'{.)=/+atr-(-bsF}l/r, k>o

=/ -d los( - losF), t-0....(9.17)

Th€ nonaft of thc GEV dkrribution c€n b. o:pr.$.d in l.ms of thc salr@tu.nior\ r (.). For k> nR, fi. nkr\ vrrirtrcc md C. &.8ivo a.

/, =6 + (a/r){ r - f( '+

1) l (918)

(a)

(i)

di =(dhf tr( r+rt)- (r(r+*)I 1. ..... (e re)

- s4,(r){- r{r+ rt) r i r0 +rl(r + 2r)- 2 r' (r +r) },".^.' ir( | + 2,{)- r'(r +r)f

k,.,=*,,-F(tdlF (k,), .. .. ...(e2x)

MOM

App.orinalc rcLitionsnip b.rwd th. vtluc of t md c. it ohaif,.d

through Rcgr.$ioh AnalFis, N follows :

Fork>O(-2<Cs<l 14 ), _Kr

= l,k =o 2??648 - o.l22o16 cs + o.o6o2?8ca1 +o.ol6?59 Car

o oo58B cJ - o.oo244cJ{.oooo5cj.......... . (9.2t)

Tnc inirial valu€ lb is up dir.d by cqu.rion

F(l)=-c, 'Fl -&+3&8,-78iGT (921)

. (e 24)E u, dr t& c)+4

-4+4e+@,4 q4 !,l-s, +4 s,-d)@-rs,A)

{r(.:i)-r'(- \2.21)

(9.23)

s.=f(r+|}),and. .. ......1e25)

d,=f'(l

'rhc itffadon in.qudion (9.22) i!.cpcal.d untl

z.ro. on@ i kdtinncdssel A.d i 3

(iD PWM

The PWMs of ft. OEV didiibution ddtld

e.=(.'rI'[/,.

i =?.859C+29554Cr

I.'i'/

+ d)= r(r + d)Y (r +'t)...(e.26)

, ))l' '

lli{,-t'

/: (r) ir iunicicnrry cr6* ro

by lloliig .t cl [28], @

', -fl'-.('-; )l

+rI'r(r+r) (9 29)

(Apprenn.tc rdu.)...... (9 30)

^ 2h-b" lo8 2 ...............(9.31)rog 3

(bd br,b,ar.ft.lmplc.tdnttc.orfo.0r,F! )

(2 b, - b")iFG_;_;lii--F) . (9.J2)

(9r5)

Ial fi.

ed RMSE ee disussed in

D.la Plor Softwd. and ec

=a"-;{r(r-i) r}i

-i['

(9.1)

Tha disrribliion tunctio. ofx

( L.e /.j l (e r4)

(B)

Thc T-!@ owtle estinatd e obt.in.d as und€r

9.2.3 Coodner olnt r6tlTh. d.tils for @nducting

[a]!id in ch.pto 3 Tn

provided if, Tabl6-9 2 I I,

r2-ted , S-( tcst, PPCC

r6ulc ac obt.jncd using

rr'llrt I I

lor prr.m.tcr Atimtion.

rc tully dplain€d af,d d

++ hngurg. and Mi.iab

.l r

1l rl t.gf r.ll

0.t..1 RoL of SkdtrFs, (urtosi! .rd L EoF.trl RiticM.lfiodolo$r is giv€. in chapl.r 8

(r) c" dnd cr:

A Rama Chandra Pio l4 I has su8sc6t.d ih. relalion ship b€twn Cr and C, to.idenlification of par€.r p.oblbitity dbtiburiotu as followsl

Ck: t.8

2.0

9.0

Als for GEV D$dbution rh. rctdion ship i (.pproxi@rcly):

Cr=2 6951{.r35C!+I ?534qI{ M7C.j+O 037?C,.+O 0OI6C,J+o oo22c,6+ ..... ... ..... .,....(936)

whcre C, for cEV is @lold.d by .quarion ( 9 2O).

rr. ror rtpronnarina prcb$itiry dirtribrfio.s t! touows | 4

Hoskins [29] hB d.rdop.d rdllioNhip b.rwa r-mn .r nr,o3 u dd

(b)

t14

540 420.0

30

0.00

0121

03ll0. t6?

0t?015 0.16?

Also for CEV Dislribution ih€ Glstion ship i, (apprcxinutcly):

r1-0. 107+0. I roq tr+o 3484 h, { 667 t3 +O OO5? hi _ O.O42l rrr +. (9.3?)

Th.e relaioishiF gir rcN id.! rbour qndidare prc6.biliry dinnbu on ro b.

Norc: TIE fomuLc for rr, tj, r. md b, !ft si@ in I 4 I.

Disrihuron.Un,fom Expon.nrirl Oumbel

9.J Results And Discu33iotr

(ii) Piran.t.r.nd Qu.ntil. EstinnlB Grble 9.2.A, T.bl6 9.3.A )

9,3,4 GUDDU BARRAGE

(i) D6.riptiv. Sr.ti3ti6 ( Trbl.9.r.A)Craph I2 | shows $.r $per n@& of Ll nilion cu*6 or norc hav€

@@d€d in the y*B 1976, 1973, 1986 ud l9$. Graph 4.2.1 shows thal hieh

noods betwen lh€ rans.s oI (8?0-980), (980-10e0) a.d (1090n200) rholend

clsecs have oc@rcd 1.4 sd 4lim.s, rcsp.ctively, fton 1962lo 1999

Vrious Stdislics prcmr.n in T.blc9l A reval dal thse is a

subsrafial difq.na bcr@ Md ud M.dia4 *bjch i.cltlg with ir6eein mple siz nir b in @nfirnarion ro th. r€$lts giv€n by Niron, m Engtsn

expen in the hydrology [ 20 ]

Thc @.fici.nr ol Skemess is 0.612 forN = 38 and 0.523 for N:24vhich indi@16 tlur the dist.ibnrio. is +ly sk@€d, i.e. noo& ofhigh magnnrde

o@r norc rrquc.rly !s N inq6d

Table-9I A shows rhat Quanlil. Btimar$ using cumb€l by PWM yield

hisher fleds lh.n by t',U-rd dd MOM, in ord.r IWM estind.s yield stGhrly

hiener floods than by MOM Tbe di|Ideias b.tw4n rh€n v.ry fron o.l% ro

2%, l.ss for OEV than for Gomb€l

The estimated pahmer€G of Guobcl distribution by MOM, MLM and

PwM &d ofGEV distriburion hy MOM md PwM e p.*nt.d in Tabl€- 9 2 A

Quadil. .nimar6 for cEv > Qlutilc 6riml6 for clnbel, by @hd€thorl The difd.nc.s b€twen lh€n .r. vcry small.

| 50 nillion os6 nood i! d(p@Ld to psr d Guddu belgc, duri.g nexl 100

y€rs. by Gunbel, using MLM; &d l5t5 nillion cus fldd h qpeted by

GEv usina PWM, which b 0.4 million c!6a! (3 806) .bov. th. d€signed @p&ity

h should b. not d ihal Cumb.l's .xtr€m flood fr€qudcy cuN. usiig

data tbr 24 y€rsesridated L4 million cus4 floods tor t00yea6al Cuddu I 14 ]

(iii) s.Et orQurnril. [.!rlnnra b] Gunb.l (Trb|ts 9.J.A)

For T= 5,

s E (MOMyS E (MLM) - 1.08,

s E (PWMyS.E (MLM) - r.0?.

On th. bais of .stinlrt.d S.E'3

MOM 6 thc wor '. Thir b in

10, 20, 2t, 50, 75 &d l00lh.ratiosrre:

I 14,l 17,I 18, 1.20, l2l and I 22

LI0. L12, L12, Ll3, I 13 md Ll4

of Qoulil4r by Glmbcl, MLM is \h. b.sl a.d

@ntmlliotr to th. th@ry 4 rhe spl. siz is

{iv} Gooiltrd oafi( 16&

Cllculllcd Elu.s ofr:- t6t, S-K t€t, prcC .nd RMSE. fo. Cunbd and

GEv, are prentd in Tabl. 9.a,4

Cumb.l dbtribltion is eit.!I. of, rfi! bsir of t'?-tsi ar 5yo levd ddal$ on rne basis of PPCC (b.ing rpp'oxim.i.ly .qud to r.0) &d RMSE. It is

rejdted on the basi! of S-K r4t, bur s S-K 161 is no1 ! powdftl re$ [4], sGltbel disdbutio. saG to b. r propd dtutribuion, h is b.nr thM cEv by

RMSE.

Not (i) GEv disribuiion k nor rppropiiar. !s ir b r.je.red by )al - tcrt and

S-K t6t Thns we havc noi @lcuhr.d S E's ofQubril.! by cEV disrribltion.

Note(ii) M ld@es I44l had.eirhrused S.E ! nor I: -t.sr ud PPCC 16r (wbich

is md r€liable) MoE ovc. they ha!. !*d orly PWM hdhod for Gstinaiion and

S-k icst a Coodn* of fir r6r.H..@,th.k eggsrion for usins cEV disaiblrion

is not.@eptable, while usine lhe drt. ofl.d$ irc. cuddu.

TABLE 9,4,A. T6$ of Goodng of Fit 1 Guddo A.nt.) 1962-r99t

Not.: ' indi€r€s rcjectioh ofHypoihcsis

(r) Rol. ofc,, Cr lnd r-Mon.nt R.rioi

n$|!9.5.A. Obffi.d r6d Slgg6(.d v.lu6 olc, Cb rr rd ai (codd!) 1962-t999

F' indid$ rejdion of null Hypothsb (h. difcrcn@ bcrwen obs.prd a.d

suggened valu€s b€6m. gr4t.r th.n 20./o)

Abov. rabl€ sbows that Gunb€lis not s pop.r lit

CEv is beuer ft rhan Cumb.l on thc basis of k rnd

lh.baisofla, Cs md Cr

cr

9.3 B Slkkur Brrnge

(i) D$cnptiv.Srntirticr

Grath 3.?.2 shows $at

(T.bl.9.l.B)

s- PPCC RMSE

D

r33 10.32r 0.21 0,29' O.r8r 0,911 0,979 0,067 0.08t

suse6t.d Fo'Gumb.l[31

0 l5l 0 I7l t5l

o612 I t4 0 6t02 J26

mcured 6 times and Crlph 42.2 shows rhal hiSh floods above 750

rhousd @*c5 md berwen the ni86 of (750_s6o), (860_980), (980-1090)and(1090-1200)

rhousdd ee6 hav€ ocar.d 6, 3.4 rid 61in.s, speivdy, froD

Th. difd.n@ b.rw6n Md tnd M.die is sbdaniat ed iiq46wnh imr€e in rhc mptc sia fton 85 to 99 M.$ > M.dio > Modc, s lfic

(rabte9.I B)

Cs calol.red ftoi lnc dala tu r 12 whkh i! apprcximrety equal ro l. 14 sr.qun.d for G{obel (Tabt€ 9.5.8). H.ncc, cunbd\ is th€ 6Bi candidab

(D Ptnn.t r Dd QunriL tr(inrr€ (Trbtts 9,2.8 .nd T.bL 9.a,B)

The *rinars of puamrd !r. givd in Tail. 9.2.8 md Tabt. 9 j.B.shos !h.r MLM Q@ril. didte by C{nhd di! riburion yield tow6r noods nalr tclun pdiods The dif.cn@ bctl@ elurite diihates by MOM sdPwM &c vdy srutt (o I l% to O 33%). MOM *rirora yi.ld highdr ooodj

MOM for CEv yietdr stishily high.r eu.ndte sdn r* ln& prMV (!sfor Oumbel). The dif@nc.s incr.e f.om 0.22% ro O j4olo as rqum p.nodsincra* I 23 and l.t5 million €usq nood is.xp.cl€d, durins roo_ye4, byCunbd dislribudon ui.g MLM, and by OEV usiig rWM, r€spe.dv€ly.

11 should be nol.d thar conb€t's airm fl@ds ft.quencr qtu€, u!in8d a for 85 y@rs,Britui.d l.lghillioncu@0ood for nexr looyds [9 ]Qu&til. Blimlq by CEV aE higle $s $@ by c!nb.l. ft. diFscneiM.e for grets atum pcriods.

(iii) S.Et ofQtr..lit. f,ltio.r€ hy coEbd Dht.ib ion ( Trbt !9,3.8 )

T=5, t0, 20, 25,

s E(MOMyS E(MLM) =

S E(PWMYS E(MLM) =

r.11 r23 121

t.lt L14 I 16

128 tll l12I t6 t.t7 I t7

For Cumbel distribution MLM is lh. b.sr (most eflici€ ndthod) md MOM h

rhc *06r (le6t erlicicno

Not.: The S E s ofQuetile 6timllcs by GEv have oot b.d @ltularcd.

(ie)

TABLr e.4.8. T.n! orGoodn.$ olFil ( Sullur barr.C.) r90l-lgoa

s- PPCC RMSE

D

s.953 9,51' 0,21 0.t15 0.263r 0.989 0,986

Note i indialB rej.crion of Hypothgh

W. .ej.cl th€ hypolhesis tha! th6 dala is dislnbul€d leording ro cEv, 6ing Ilre$ aod S-K l€st bul not by RMSE This is th€ Eason aor nor c.lolaliDS S E s for cEVdadburion. Howev€., GDnbcl diskibution is acc.pt.d d

1r d =58< 1' o,r(4)=599, PICC =O.9S9 = LO

Dd =0.115<Doeri136

RMSE foi Cflhbel > RMSE for cEV

(r) Ihe Rol. ofc!, Cr ird t Mon.nt Rrliol

TablF9.5. B. Obercd md ErP..rcd vil!6 ol Cn Crt b Dd il(Suklor) l90ln999

Glmb.lI4l GEV

0.2.28 0l? 0 226

0.181 ot5 0168c, lt2 Ll I

Cr t.63 t40 5ll*i indicar€s reject6n of Nll tr?o1h6h (h. difisd@ bdwdn 6bko.n md

. $gg€sred v.tu6 b.ing gr@l.r lh.f, 2@/t.

CEv is nor laePred on lh. bdis ofCl(tumbel is ioi a@pt d on th. bsi3 of lr .nd Cr bul is .c!.p!.d on tlE bais ofcs and

CEv is b€nd llEn Gur{El b, t std Cr.

(r) D.$ripriv€ Strrtft! (T.bl. tl.C)

Gr.oh I 2 7 shos ln r SuDd floo& of982, ?91, 32t , ?99 , 3o4 and 764 (0o0

Gdph 4.2 ? show rn.l high lled! bliwcd 1he n.8.! of (5?0-6?5), (675_730),

i 080-335) and (88!990) (ooo oso hlw ocdtr d 6, 5, 5 and I tines, rcsp.ctivelv

i lablc- 91 c shows ihd the ditrercncc b.twcd Mdi and Median i! sub$etial, and

I rncroso wirl rhe sample size. The dda is +vly sk€wcd, for N= 85 ss well as fot N= I

I uhichindicaresthatth.aoodsof biShmagoitld.h.v€o@ur.dwthlssfrcquencv

dst h.ve occured in the r€m 1956 ,19?6 ,I994 ,1995,19t5 &d 1958, Bp@riv€lv

(ii) Pnnm.r.r rnd QnuriL f{lintr6 (T.bh qt. C r.d Trbl. 9.3,C)

Tabl.$gz C ahow rlE 6rinar6 of pddetd by rhE nelnods ofdinetion for cumbel ed cEv disrrib$otu. T.blq 9l.C shows rhe qualile 6ridal*ror borh rhc distriburions. For cumbel didnburiot\ MLM .etimat6 of lood de high6!MOM csrimata are lowdl ed lh€ difercnc* b.twn MOM lnd pWM .stimals krytom 0 5% ro I a% (incr4ing wnh rhe rerum pqiod). lor cEV, MOM stinates orfl@d ar. lower lhan PU'M Btinatesr rhc ditfercn@r ar. ema a.d incr€se with lhe

r.iurn period lrom 0.35% to 0 95%

cEv Quantile esdnai€s arc hisner than clnbele!6nrite €srinales, rhe

diferences b€ing very smau (i e < l%) l] 9?5 ni ion cus@s flood is exp€cl€d .t Kolri(B)durins loo-y€rs, using cunbd distnbution (MLM nethod), whioh iso rodilion.usq h'gher lhan irs d€sig€d apaciry

(iii) s.E,r of Qnantit. E inrt6 by cldb.l Diliriburior ( T.u6 9.3.c )

20, 25, 50,

121,t23

Ll0,I l0

S E {MOM t / S E (MrM) = r 04, LIl,l16,tt5.l13,

1.06, r 07, Lo8, 1 09,

Table94.C shows lhar cumbetdislribution h.c@prable on tbe bdis ofChi-squrre resl at 5% ldel. S-K !d, ppCC and RMSE Borh the distriburions ar€

rEepr.ble by PPcc md RMSE cEV is rcj.dcd on thc bsi! of chi-sque dd s_K

5 E (PWM)/S.E (MLM) =r 03,

Hence, MLM h the mosl eficienl mcrhod

MoM (as at Suklor and Cuddu baiiages)

Notc Th€ S E\ ofeu.ntitccsrimar.sbyc€V h.v€ nol been@tcutarcd

and PWM u nore c6d.nt rho

(i",

TABLE- 9.4.C T.sl! ot Goodr6s ol Fil ( rl Koin ) l9ol"l999

s- PPCC RMSE

D MLM PIVM

Clmbd GEV

t5r4 6.6t. 0.t36 0.lJl 0.!J. 0.t956 0.9954 0.052 0.055

Nor.: r indidr., Ejelion of Hypolh*is.

(v) Th. Rol. of C!, Cr.dd LMon.ni R.lid

Trblecl.C, vrlud olC, Cb t, rld.r {Kotd l90l.l09o

0.r60

(,3.5tr?

Above valu€ 8iv$ sode ida about frxing olth. distriburioN ,i... Gumb.l is noi a

prop€r fi1on ln€ basis ofcs a.d C r .

GEv is bdt( rho Guob.l on the bais ofc. lnd Cr..

t7t

9.4 Findings

(ii)

c.<l.B StrllurBrna.

(i)

{i) PwM quanril.4timat6 are high€r th.n MOM quudt€ .srinatcs, for borh

Quantil. .3lihal6 for cEV arc stighlly highq thd for Gunbel

(iii) MLM tuth.berrtrd MOM brhcworur onrh.bsisof SEs,cddt.rcdlor Cunbcl disrribuiion

(iv) GEV is rcj.cted by 12 - lest,S-K r.st, q and C( cuobet is betr.rltftnCEv bt t'1 test and RMSE bur nol on the basis ofcr Md ri ppCC

thu Ormbcl is re@hmended by lsing MLM

(y) 150 nillion us ttmd h dp6r.d dunng roo yas by Cumb.ldklribution. wina MLM mdhod of .srimion. as conplr.o ro trs

design€d capacny of t.t million cus@s l20 oillion cusG hld actu.llyposscd in 1976. Hene, rheie h a ned for redesigniog lh€ barlgc o!tdking suitsbl. deps ro $ve rhr srruclur. ofbs,Eg.

Qua.lilc esrihatB by MOM ar€ highcr

Th€ diffsenc. beired esdnates by

lbr Cuddu), for borb the dislribuions

than tho* by ?WM and MLM

MOM and PwM is very small Gs

(iD

(v)

Queril. csrinate by OEV > Quslilc erinllq by Clnbd

Th. ditre€ic.s i.c.qse fron I 5% to ?'/o as relurn p€riod incrces

For Olmbel, MLM b lh. hosl .6cian ddhod lid MOM is thc ledt

(iiD

Gunbel dd GEV both de @plcd on lhe bstu ofta lnd C, but &€ nor

a@.Pt.d on th. bdls ofcr. CEV t b.lls thd Gumb.l by tr pPCC and

Cr.

123 nillion d@ oood i! dpdtcd during l0O y.rfs, by Guobel

dkt.ibulion, using MLM ( rh. no$ etrcid n€rhod),which is 013

million cueB (i.e.lcFlo) I'ighd the the designed op&ity of 0.9 million

o* 12 $p.r n@dr high6 the its deigned ap.city hrd al@dy

p8!.d though Suklar bd.gq

CEV L rejd.d dd Glmb.l disriburio. is @ol.d otr fi. bsh of Chi-

squrre tcsr. S-Krd ud RMSE (s ror Guddu).

tl. r.sults obLin.d lor finin8 thc distdbnlions on p.* noods d Sul&ur

(B) 8rc Einih' io those obllincd for Glddu iB)

Thc data ofPak flood fton I90l lo 1999 rt Kotri b.u8. is + | y skaed

.nd ir not noml whi€h indicd.s th4 nood of hiSh m8nnud. hav€

(iv)

(vi)

(D

H.ncc, Cumb€l diedbution h r@nnended for turins th. usins MLM

(ii) td-M Quaniil. *timt6 ror bofi diskibutions ae rhe hrshe$.

MOM Quntil. .smates de highc. thm PWM eslihalB lhe diE rcnces

range f.on 0.4% to 1.4% (nore fo. Glnbel tllm tur GEV)

The diferdca bcr*ed Qudtil. €srihars bv both distibulioos

(corcsponding mclhods ot Brimarion) lre v.ry smtll largar for CEv rhm

for cunbel (sn. as fot Guddu (B) and Snkkur (B))

(iii) SE\ Qua.tilc etirurs by Gub.l n. miiiltrm for MLM and

maxinun fo. MOM. IyILM is lhc nod ef6.i6t for Cudbel di$tibulio'

(iv) Cunb€l is a plopd til (I:-res! S-K 161, Plcc, RMSE .nd cs)' €x@pl for

C, a.d cr CL GEv is ml appropiar. on [E bsis oflT-t.st lnd s-K t6t

bur n bertd ftan Gldb.l disrrib{ion b} q 8nd c(Both rhe distriburio$ arc rcceplablc by RMSE dd IPCC

(vi)

cunb.l disrnbution is b€rtr lh& GEV, by haionty oflh. cdtda,

MLM G rh. tusl cf6ci.t'r ndhod lot Gunh.l. PwM it b.lls ncrh.d rot

o 975 millio. d5.B llood is dpecl.d dunng 100-vart

dislnbudo. (using MLM) whi.h is 0.10 million o$6than thc d6ie!.d @pa.ny ol ba sG A $ps tood

cuses had ah.ady 9!sed tbrougl lh. banage i. 1956

llenoe, Gumb.l distribution is rcc.nmc.ded lor

ar.quency daia lsing MLM melhod d Kotn banag..

?

,5 Concluion:

Quntit c(imte. for Grnbd diltdhnion .r. loer ths for GEV

iD.crdly by MLM qtich it tlE md cfEi.nr rcthod ofcti tionforrh.dltlof

GEv i! not r.r9t d by z' $d S'K tcslt. lt b b.tter tl|rn Cl'n$cl

distibuliorbyl.rndqb mt !y PPCC rd RMSE, for dl {tc b|tn86

& fl@lt p..I. of nIgt[l. lc/i !o S hgls fim tlF dldgi.d 6p@itv

of b!r|g6 ll!rc.tthd pcsd 6 qD€ld lo pr.. thooEi tlEr\ e th.rc at I.G.d of cddisncd th. bd|g6.

Ho.c, Gmh.l dicliDulbo uriiS MLM t!.dF4 i. lg{Mtad.d for

1240(349.8)613.0 (6r10)651 5 (630)?01.7(665 0)l8 (24)

c.v

0.3? (0.26)

QD

1440 (114.0)

SD

25t.3\2J2.7)

c(

2.3?6(1916)

Cs

0612(0523)

I1570(lt2004ot 0G69 0325 0{?11.0

1200.00200)

SE

429 (J9.1)

0ll6

Nor€: ne v.!u6 if, thc p..6rlEsi. lrc for 24 ya6

. indidr* trlu6 in ( 000 Clsccs )

9r,r.t999

Notq Thevtu*in the oar.nthesis rc for 35 vd*

N

(39

522.87

(504.r)

alt5.0

(5E2.0)

516.0

(566.?)

306

i306)

1200

t200

s,D

205.1

r84,0

Q.D

102,5

(94.1)

0,305

(0.306)

s,E

20,8

09.9)

1.120

lt,26J

3_64

13.62)

Q,'

484.0

Qf689,0

(669)

407

(42r.0)

Df991,0

(810.0)

o,t7a 0,r81

. indiar6 v.lc in ooG.ll34

TiblF9.t.C. B ic arililtid (Kotri B.nrg.) l9ol-199

- l Meni" l ".o'"' l v"ae

99 (85) I lsr.cs {ar.o r I 420.0( rt9.0) I 356.10( 360.0

151.0 055.0 ) 912.0 (912.0) 0,130 ( 0.E2) a.537 ( 3.sr' )

s.D

t60.0 (t61.6)

Q.D

99.s0(94.50) 0.35 ( 0.35 )

s.E

16.21( r6.6)

Q,-

32?,0 ( 342.0)

Q"53s.50 ( 559.0 ) t86.0 (2t2,0)

Dr'

6lr.t0 ( 660.0)

0.t63 0.14,

Nct r Tdnnuc. tn il;pdorl*ise fd 85 )ds 'i.diord rds in 00cos'

TABI-E -9.2..4, Ptr.n.t B or Gnnbd ind GEV Dittriburior in 000 coxc!

Gnddtr Birr.r.) 1962-1999

MOM MLM

t9t.116 r95t5 xB.n2

588 608 585 ?55 58{.012

cEv

618 52 615 678

p 206 t6 208.016

0 028 0.027

ABLE-9.2,8

{sokkur B,dB.) l90lr99e

TABI,E -9.2.C. Prnm.t B of Gtrnb.J !trd GEV Dndbrrion i! OO0 od3(Ko(ri B!me.) lt0|-l99

MOM MLM

r60.202 150458 159.132

510371 5J2.440 5lt 025

cEv

529 0o 52608r

141.860 t4t.690

.009t -0.0888

MOM MLM

124l4l

J8o 149 J14.440 178.245

GEV

175.t l0 185 501

ll0 t20 128.907

0oll 00110

Tible9.3.A. Qu.ntil.4tin B 'nd

th'it S'E't (in Prrcntns) bv Gunb'lrnd

GEv dtuhbutio 't

Gudd! B'n'g" lt62-199

o"'nr . Ivhdilld6 li! 0U, cu!!!!l

T tGfv

MOM MLM MOM

.20 aa2.16

(62.49)

885.01

(5&32)

889.69

(62,30)

920.4 921,5

t0 rgtg,l1

(ra.9)

l03a.7a

11a.$rt

r0a!61

{a2-121

t0661 r069,8

20 .05 1t?0.10

{101-18)

?1.35

(91.51)

lllg.f,o

(lil2.26)

ltolt 1209.4

t241,6.01 I

121..80

l(11,1.63)

l72a,9r

(96,91t

185.43

o08.6?)

12$,r

5o .02 I

1352.5r

l(6t.t?)

136a,25 13?9.1?

(12r.66)

Ia?4.1 t186,o

!4,19

6a0.39)

l4tt3 t467,2t5 .013 1l(|2,4,5

oso.ao)

ra45.82

o23.?t

l0o .01 l419.2l

o59.Er)

tgt3.55

o30.82)

1521.4

o{8.?3)

1500.1 1515.0

r3l

T P

Qu lil. Mrrnnrd6 (in 000 .o.@)

GEV

MOM MLM TWM MOM

5 .20 110.66

(3rial

7$.r2

tn.t3\

169,71

(31,.3t)

?60.1 159.4

l0 ,10 890,88

(,|l.08)

871.02

(34,95)

aal.l3

(39.37)

33&9

.05

(54..1,1) (42.t6)

r025.4 1020.2

l5 l04a?a

(58.rrl

r0rJ.68

(4s.29) (52.5?)

1090,8

50 .02 ll55..l? llt9.52

(s3.r3)

ll5t.95

(62.21\

l2ll.l tt05.5

75 .013 1720.97

116.75\

ll8l.03

(57.821

t2t7.0r

t61.921

1291,6 t29t,2

100 .01 1261,32

t8r.02)

1224,57

(61.1r)

126it.06

(7r.95)

1361.0 lls3,9

T aLD9.i.B. Qu{ndl. Ertinrte and

GEv Dir(ribllioN ($.ir S.E'' (ln p.rcnthed) by Gumb.l'r .nd

Suklor B.rr.g. ) 190l-1999

TABL&9,3,C. Qurtrril.E tln.t6.nd th.tr S.Err (ir FFnrh6e) by Gunb.l rnd

GEV for Kotri B.rr[gc, 190r-1999.

T

O!.rail. M4nitud.! (in 000 curs)

GEV

MOM MLM MOM

5 .20 556.65

t2a.?51

512.99

121"19\

569.?l

(1a.3tl

575.r 57?.1

lo

(33.441 |l0.13)

66159

(3r.99)

20 .05

(42.25)

?63.68

(16.35)

?5?.51

(39.59)

?56.0 162.2

!5

(rs.l0)

?93.20

(3r.05)

?36.66

(12.391

144,6

50 -o2 8{5J2 ttr.53 t?6.4t 470., 811.8

?5 .0lJ 916.16

(59.rt)

937.56

(a9.tl)

92t 64

(54.10)

9i0,0 92&!

95413

(62,381

975.09

(52.691

965.63

(57.94)

951.8 963.7

TRf,DICTION OF trLOODS AT VARIOUS BARRAGf,S BY FITTINC

NORMAL AND LOC-NORMAL O) DISTRIBUTIONS

Chapter l0

I0.l Irlroduction

Nomal and Los-Normal disnibutions ar€ 6fi€d, @npar.d and r.sr.d

usin8 1116t, S-K Gst, PPCC, Ca , cr, RMSE and Mm. yr€tsonthcdluorflood p€rls al Glddu, Suklar and Kotn Baragd MOM, MLM md pWM are

uscd for.slimating the qulnlil.s lnd lhen Srandard E .ots for €ach distibution

Cof,fiden@ bdds ste als dhpn *hich auidc the relialitity of th. freqldcycur. Th. smplc L, moment ralios aE aho usd, s io chafl€r 9, for 8.(ingsomc idea abou t th. psrdr d islriburion

Ddpite much eforls by various hydrologists, St4risti@l16ts $8g6i lhar

lh.r. is no b.\l Probabrhty d'sinbunon lo' noods t her€ rs no,ason io 6peclthal a lngle distibution will apply lo all streams woddwid. H.nce, we ale rrying

ro ni sofre disrriburions on the d.ta of nood peaks al lh. banag6 on riva Indus

on the basis orene ofrhcn properlie w€ hld alredy 6(cd cumb.t a.d CEvdklibution on lh€ samedata in chall€r 9.

Hazen I25 r l h.d inkoduccd rhe Nordot probabitity t.pcr for.Mtysis oflrydrolosical dara Slek et al | 5, I ddonst4tcd lhat, ib thc !b*ne ofinaormarion about th. distnbulion of noods and Mnonic tos*s Nociar.d wirh

th. design of flood rcduction m.asur€s! tbe us ot lh€ Nomal di$ribution is b.lre,il'an orher dislribuiions $ch as Exlren€ Valuc, Lr8-Nomst or Weibull

The Nomt dislribuiion hN b@n std€d in liis stldy b@u$ ir his@nain usetur narh.matic.l plopenia. rt is a l*o p.lmei.! disl,ibulion, whicn is

bell shaped, conrinuous and synheric.t about the nqn; .oefllcient of skcwness

b.hg 2e ANOVA dd l*ting ofhy?orhais rely on eumplions ofnomatilyIt ir well L.own |hat oost hydrologjc variebts @1 b. iomalty disrnbured

I

I

bcqu$ 1te .dgc .€quired for sy @don vdiabl€ to b. .o.mally distributcd is

C@,6) How €v.., if the m6 ofa nndon vdiabl€ b about fou tim.s g@rer

tha its S.D, th. prcblbilily ofs mmal !.ndon v.ri!bl. b€ing n€aarive is wry

mall ed qn b. n.gler.d

Normal dGtibulion alno ilw.ys giv6 thc loscal values fo. hish now

analysis Afreid€r Argel I3 I had fiti.d Nomd disrriburion o. Mekong River

It phy, vqy inponmt role in slocnaldc hydrology in whici shlistid l@nniquB

h.rc b.6 stssi!.ly flployed. Thec iiclud. ftScssioi dalysis, statistiql

infc.G.es, dala g.ndrion rd timc Fris. ln rony ces. rlk Nomslity

dssunption is idposed upon son @mponcnt of1he model involved a.d in c.*ofun acceptanc. on. slarts to r€lu it or lo modify lh. nodels.

Lag-Nomal dkrriburion is s.l6t.d b@@ ir ha a long higory of u* iiw!l.r rsrcalt fl@d F.r dis.harg6 od r.ir6l i.tftnynmtion dab This

distribulion k @nv.nicnl 10 use bc@lsc of th. @e with whi.h its quarlil€s c..be der€min€d using normal rabl6. ftsk8h [ 49] r.pon€d thal LN (2) Ii$ oi $e

dar! of lllinois and B€rhlahmy [ ? I hed lin.d LN (2) ro ldoho Rrvd. Ir is usd by

Hen [ 25 a ], chow I 18 I ud nany orh.r Hrdrolosin' Chow h.d ddelopcd tgophical nerhod io d.rmine lh. fr.q!.ncy f.cior in LN ( 2 ) dkdbulion

LN (2) is oneofih€ rdr fr.qucntly Ned diliributionand is re@nmend.d

by wMO []sl It worh in the €se vh.re C" >01, r.g.rdl6s ofehpl. si&.Th. plrrhcr.G ofLN (2) m.y b. rhou8hl of a rh. nd and

v.riance, the n6 ud S.D, or the d6 Md C.v olthc lr8-NomBl vdar. ln

our study we h.vc @tuider€d mqn dnd S.D a, rh€ p@mdcc

th€re is l..k of asrEd upon cir.na for oomp.ring lifiing l€chniques for

sn.ll a.d modcdtc samplo sia. Th. s!ft dbtriburion My b. prmt€riz.d in

*vml *ay\ a rha m.lhod which b.sr drimat6 orc s.l of p.delc6 may not

bc thc b6l for eoth.r. MLM is prc(:r.d in (e ofn > 25 [ 4 ] Hd.q wc arc

inra.ftd in @npatiDg the relativ€ .fliciency of melhods ofgtinadon.

Historigran, Hi$osrah and frequ€ncy cuB€ alons wilh Mqn, Mcdkn,

Mode. S D, Cs. Ct, S E s. Sample Lnonents, Qr, Qr , .tc, re eme as shown in

chapFr 9 PSS, Excel, Minilab rnd D.la plot sonwar. d. u*d lor Stalisial

10.2 Methodology

10.2.1 Nomd Di ribuaion

The Probalility dmiry tunciion of a nomlly distdblred wialle X is given by

1, r')'

J t,l otlrwhet. tt and d are lhe paraneretr oflhcdisrribution. Th. variable X can uke any

vrlue in rhe ang. G6,b)Th. probability d.mny tunction orrh€ sbndrrd Norul vlri.te , k giv.n by

Ith)=:i* z, .. (102)

bo=2 5052367, bz=l 831204, 64=022647\A

66=0 1106469, br=4.0202490, bm{0019132

wh'ch can be nu m.riqlly a pp roxihaterl by a scrj6 of polynomiak such as,

.r (,,) = (t,j t h, t' t h, ri t. hx,' r h,,/")' f . (D . 00. :r )

In€qu.lion( l0 3 ), 0<, <6and

< (2.3)lo4 f(/) is d crcn tuictioi so rhat f (,)-(-u)diskiburion F(u) which is tne aru und.r thc prob.bility dcnsiry

Lni=l ::eadfi. ... (lo.4r:. J2E

Th. dinribltion turdim d h. NMiqIy .Poro'dtrt'rd I a I r'

F (i=t - J (,\b,q +b,q' + b,qr +,.a' +,,qJ)+€ ft) (1o'5)

o=--L 1to.o;, p {.2316419 rnd osr<o

b'=0.31918t530, br=-o.356563?E2, h-1.?314??93?

ba=-l 8212559t!, bt= r.330274429

Thc mr €(r ) in.qu.lion (lo. t ) i! <? txlo-r, edrctF!F(r )

(A) PARAMETERESTIMATTON

(i) METIIOD OF MOMENTS (MOM)

{i, =rri 6nd 4=J/4 . (r07)

(ii) MAXIIIUM UKLTHOOD METEOD (MLM) S'!' ! for MOM

( iii ) PWM METIIOD

A,=4 ard d,=G/)........ (loB)

QUANIILE ESTIMATES

For . siven r.tum p.riod T, th€ compo.ding trobabiliry oanon- dc.!dr@ is

F=I'UL Sloded noro.l veilre ", cor.lponding to a prob.biliry F oa non-

*cad.M. 6 b. olcuht.d 6 [4 ]:

I

f_

l:

G-2.51551?,

(i)

( 10.r0)

€(P)<4.5XlOr

+€(P) (ro e)

d,=1.432788, Cr=0.202853

C,-0.010323, dt=0.001r08

I

li

fi

w = J- z-t"e(P) , ro, p.o s

Whft P= l -F a.d th. prcbabilily of dc€dcncc €riot

For p >0.5, p is rcpl&.d by l-p z is compd.d with

Qlddc 6tin a e th.n cdularcd a

=4.r4 .. ( ro l| )

) STANDAFD ERRORS

(ii) MLM tu sn. f lor MOM

r" =l',41"'L 2l ( r0.12 )lLJ;

( rii ) !!!4s, = [r. o.srrr,']'"

NoL: (l) '? ir $m for rhe

dif.r.nt r€iu.n p.rio&. al

9.... . . (ro.n r

rhr.. daho& of drirltion bur i! difr.nr for

is difi.rdi for dif€ront par.nel.r Btimaiion

(2) s.E. s by PwM u€ 3lishdy L4a rls by MOM.

10.2.2 LN 2 D|IlTruEUnON

{r0rr)

Th. pdf of a logailhiic noo.lt distribu.d Eiabl. X with t*o parm.r.R is

,o"Jz" ... .. (10 14)

ph@ !, and 6r e $. nla.nd S.D. of lhc m!u.l logrithmr of x.Thc whbl. lo (x) ii tddddiud .s

Th. sl.nd&d @md w.iate u is obt.in d wirh pdfgiva by

JQ)= J+..lh. trc]6)

(A) &rrEg!!-&!!!E!$q

(i) [roM(- l

i,'tnni-d:12......ttot?). ;l=hl ;"',,,.t1 .. (ro r8). \I';I J

"-J

. (r0.20 )=]Ltv,,-t,i =".,

(ii) MLM

"'*I "o=

,--u l?)=

2 *J 'Ql ...ta\),i,=rnt,-5 (o zz)

. o0 24)

(r0.25),

.. (ro 26)

.. \t0 27 )

^'=-I; f {10 2E )

1"" - rl

j='''

,rt!11'''41. (r,'

,)'

n" l'-k'-"=((t

HcR erflrr) is *at@|.d as,lJ2,"hae, is rh.s|.rdad no,mal @are

obrdined fronqualront tO l5 r\oneapondmgtot -(l-trl2

(B) Qu'nril. E rinrrB

Thequmlile€stimates are obiained as

tnxr=tty+uot. (102J), tr = e

{c ) srrnd{rd Enoa

(i)

cv z

(ii)

sr

MLM

L u'IL .)

t{ I.

J; (10.29)

10.2.3 Goodner ofrt T.tl!

Tn ddlils fo.olducring f_i's! s-( r6t' PPCC tnd RMSE e dissd i' [4 ]

and t2 I.

10.2.4 RoL oa St rn4' Kurtosi! 'rd LMo|@t Rrti@

;h.o.dicat values ofc" q, rr and ! fo' v'rious distribliioN 'nd rormnke ror

4lcula'ing ( r a.d k for LN I I '

arc 8N'n i' la I Th* r'lil'ochiDs sive 5one

idd lbout ddidar€ Probdilirv diddhrio' $ b' ftt'd on av g|vd dtl'

I ruo Crrrnaa nrc I a I trs ses'stcd the Eldion shiP b'1wen Cr dnd C,

for iddifi€tion of p.rsn probsbililv dbtibudqts ae folloPs:

cr: l8 9.0l0

Al$ for LN(2) lhc tclttio' tbip is (lpPrcxltuielv):

cr=l+o 02565 C1+ l7205C,: +0041?t ci +004605C'.r _ 0 0048 ct) + __-_(1010)

wh.rc c"= I C" +c; '

{b) I-Mon.ni Ratios

Hosking has developed relationship be1w*n L-modent nlios b

apprcxidaring probability distribtrtions 6 follovs l4l000 000

0.l2l0331 017

0167 0 r5

Disdbulion:Uniform Exponendal Gumbel

Al!o lor LN (2J lh. relalion,h,p u (approxidarclvl:

t = o )22a24a 71 52 \t1 +o r22a bt -!.1364 t 6 *o I l l? t u + o l l4 \ N . 0o 3l)

rr n cabulal.d by using Log ofaclual obscPations Ov.. all low'rbond:

ft* ..htionships Sive sone ida aboli @dida|. probalilnv distribulion ro bc fillcd on

Nolq The aormulte fo. h. tr, ir and h ate sivd in I a I md I 15 l

Wrd a.d A Dr.si ar. used lo chdt lh. noftdnv of th. dlla

1l,e romul.cand procedur. is the sahe N given ii [ 54],I57l and [4]

l0-3 ltcsllts and Dis.usn'n

10.J. A Gtrddu Ban4e

D6c.iptir. Srrrhdcr ( T{ble9.l A)

Prnn.l.r ind Qurntil€ E3tintr€s

{i)

(ii)

Thc dri@t.d Prr.mc.t.B 16 pr.s.nt d in TabL l02A. and tlF Qu$tilc

cstimar€s .re giv.r io Tlbl€ ld.3.A.

{. ) Nom.l Dltrilutiotr

Q(Mr,M)=Q(MOM)Q ( Plutr ) rc lpprcrimatdy equ.l to Q ( MLM ), th. diflacE.s !r€ v.ry sm.ll,

(b) Log-Norn.l dlrtrib iotr

aw<aMw<QMTh. ditr€ltlB hly hoh 0 5% lo 1.5% i i@nns sior 'I..

(c) Noroil. v. r.g-Norml

Q by Nonsl dislribltid e sndld $o by rla-Nomd, tlE dif€tttB brtig

Q L% Ntrbt l0 Nqnol v&is ftom | 007to Ll52forMOM, fron l.02lo

Ll? for MLM .d fton r 07 to 1.26 fo Pwr4 ft. dif.rdre inclug for

T{bl}|0.4. A Rrtos ofQlrnlite Estinrr.j ( Log-Non{tvs NorDat)

cuddu 196!199

Tr t0 20 25 50 75

MOM t.007 r.05 r06 I lt Llll I 152

MLM 1.02 t.06 r0? Ll2 l.l5 1tl

t0? l13 r. t5 l2l | .21 t)6

143 and I 28 million cusc6 ilood i! expeol.d d Quddu, durins n.xt loo yaq using

MOM, for Lo8-Nomd ed Normal distriburioN, r6p.diveiy.liol.: I 50 nillion @ss ilood t exp@t.d lo pB.r cuddu betaa., durina nexl rOO

yff, by filling Glmbd, using MLM & in chaFs 9

( iii ) S, E 3 of Qnrnlir. Dsrinat6 ( Trbleto,3.A )

I!4!d S. E rv! : S E xd ( ditrftmd arc n.alipibt. ).glic

!!&!9r!]!! s E MoM < s. E ur.M = MoM k mor€.ltci€d rhan MLMThe.ario varies rrom 145 to I ?0, incrcasing wilh T,

T.b1610.5. A.Rrd6 of S.E,i of Qo.!tlt. IrtlD.r6 For Difr.Ltra R.rurn p.nodt

( rt C!dd! I r962a199

t0 20 25 50 t0o

tu/ S E ar forNom.l 1.001 0.997 0 993 096 0 993 0.996

u/S E Mod fo. Lo8-Nom.l l.a5 1.54 |57 164 1.67 l.?0

E {.9 NomdYS.E (Nodn l) by MOM l,0l |.06 1.10 r.t8 t22 1.25

F{ lne-NomdYs.E (Norb,l) by MIM t.46 t.61 l7t t.9l 2.M 2t2

Abov. rlbL indiarB ltllr S. E'. ofeurd. 6dmr6 by Log-Nomul d.hrye rn6 by Nmd dildribuiioa urin8 MOM rnd Mllvt TIE r.ri6 fo. MLM u.ldgd rhe for MOM i... I ,t6 lo 2.l2 $ conpqd to 1.Ol to I 25 ..d iM..!. yiu! T..ftleNom.l dirltributid untrg MOM ir b.nd $ot A- Nomd.

s- Ppcc RMSE

Normal LN (2)

D

Nom.l LN (2)

599 64J' ,2 8l 0.21 l.611 013 0 g?lt 0 9842 0 032 0 071

(iv ) Ta(s ofCoodness olritTABLE-r0.6.4 T.its ofGoodnss ofFit ( Coddu ) 1962n999

. i.dids rej@tion orHypothdh

Nomal Disrriburion is rcjEt€d onthebsh

LN (2) is ac@ptcd on rh. b4is otl':td &d

Log-Nomd is b.tler thu Nomal by PPCC

of I1Ls1ed S-K 16r ar t% l*cl

.nd &MsE.

TABLDTO.T.A Nomrlity T6l! ( cuddu ) 1962-t999

t Indicat* rejerion ofHyporh4is

The dara foi Cuddu b.iiig€ is nor normal .nd H. is rcjdt.d by both A D r*t

c, is 0360 vl ch de nor b€lon8 ro ( 0.2t.033 ) h indiat6 thtt ifi.

I lrlDlhcsis ol tic nonnalily on not b. aceplcd

R

r.062. 0.008 0.965. 0.0155

(ri ) C, Cr tnd I'Domdt R.tio3

Trblclo.&A vrhB of Cs , C! .od LMon.nr! ( Gtrddu ) 19621999

Both disdhutions are !ej.cl.d on the basit ofcs and cF

Normal Distnbdion is betq tho kg-Nomal by t and Ct but nol bv tl

Log- Nomal is rejdr.d t .

Nor.: t for bg-Nomal is cil@!.rcd by $ing lrg of rcrlal obwlr'ots

lU3.B Sukt!. D.mg.

(i)

(ii )

D6critriv. Sr.dttic, ( T.blF 9.1.B )

P.mn€r.r and Qu.nriLBrinitd ( Ttbl6 10.2.4 &TrbLl0.3.B)

Tablelo2.B stuws lh. v.lu$ oi paramer.rs by thre n€thods of estimaion

Ilble - lo.3.B shows the Qusnlila .stimtes for both $e distributions

Q (Mu\t) - Q @OM) sd Q (PwM) !rc lmall4 rha. Q ( MoM)bv t'/.to 4%

rag-Nomrl Dnitibtrtion

Q(No-M)esmlldtandQ(

Q ( MOM ) / Q ( MLM ) vui6 fonQ ( PwM )/Q ( MI-M ) vdi4 rrom

MOM ) ue lrs6t (i.e. dir.rot th.n ror

|.007 ro 1.029

L006 to 1.041

(b)

0.151 000 0l5l0.ll6 0.123

c. 06t2 000 I1266.

ck 100 5.87r

' indrcates rej.crion orH.

1e lhe difer€nces betwan qu.mil. estinarca by vsious nethods of esdnaiionarc ldss lhan 3%, dd incrase with T

Nomrl. V.lng-Nomrl

Q by Noftal disr.ibution e soralq rhan by Log_Nomal

QIbe Nontat lQ Nilnat van6 from l.0o5 to I t]5 forMOM, ron t02to! 15 for PWM ed fron LO2 !o l. r O for Mt_!t.The dif.crenas incr.e for targr rcrlrn p.riods lnd &e luaest for pWM.

Irblelo.4.B. R ioi ofQu.nrit. f,rtimir€ ( lag-Normnt .V! ,Nornrt ).t Srkkur

1.08 ud l.2l nilion cusa n@d is dt;iA;isE;,,i"8;aroojsq -by

distnbution usi.g PWM dd 6y t g-Nom.l usi.g MOi4 .eperivety

T,

t0 20 )5 50 75

MOM t04 r.06 lt0 I 125 I Ilt

MLM 099 t02 t0l 107 r.09 rt0

t02 I ll I tl5 I t5

Normrl S. E wM < S. E MLM by (t% to 3%) - pWM is thc n6t efEcient

mdhod This E difer€nt than rh. dp61.ions, b€sne q . for pWM is smlt.r lhe forMLM

Lo!nornrl S E Mov < S. EMLM > MOM is morc efficient than tvtl"M.

Th. mrio vari6 from I ll ro 1.48.

( jii ) S. E\ of Quartil. E.tinri.r ( Txbt. 10.3,D )

Irbl.l0,5, B. Ralio! ot S,t'r ot Qurnrit. Eltim{rs { Sutdiur , | 9Ol - r 9r9

T.

l0 20 25 50 75 100

S ENLN/S Ei{MlorNormal 1.05 106 107 108 108 103

S E/d/S E MoM lor LoA-Nmal l.rl l.l8 1.40 L45 147 t48

s E( tas-No'm.lYs E (Nomrl) by MoM |.00 t.05 t08 r.ll l.l7 ).21

s E (lns Norhal)/S E (Normal, by MLM I tl l5! 1.72 r80

Above &ble indi@t6 lhrl S

Nomd The diff.rmces u.E\ of Quiotil6 for LN (2) aclarge( for MLM rhrn for MOM,

liv) Tar ofGoodi.$TABLE I0,6.8. Tests of Goodn$i olfir (Sulkur) 190!1999

s- PPCC

Nomal LN(2)

RMSE

Nomrl LN(2)Norhri LNt2)

D

Nomal LN(2)

199 24101 5 84 0.It6 a 2641 01J5 0954 r 0938

r)

' Indicarq non acceptance of Ho

Above rabtc shoys lhat Nomal disribuio. ie rejtt.dlK rer ar 50o lev.l LN/2,irbeti.rrfuNoma.onrheban, ofPlCC and RMSE

6 rcjerqr by borh A D tsr) Md indicales non iomatiry of

TABLE IO,?.8. Nomi'ny T6.! { Suktur )

s rq{ron of null hyporhsis

The dara fo. Slkkur b.rmge is nor.omal dd Hqrrd w- les cv ts 0 305 which betoncs lo ( 025, o3l

c'sq P-Vrlue R

3,r57.0.01(M

("i)

(i)

(ii)

C., Cr rrd l-mon.nt Rrtlol

T.blFlO.8.E. Vrlu6 ol Cs. Cr tnd I'Mor.tr. turior I Sltklr ) 1901-1999

0 224 000 0 226

0.l8l o l2l o.l614

c.

cr 168 l0o

' rndrcarcs r.Jecnon of Ho

Nomal di$db{tion b @j4ted on $. bsis ofrr, t c' ddCr.

Ils-Nontl b reiar.d oily on lbe blsis of Cr LN (2) b b.ttd than th€ Normal

D6cripriv. Srititti6 ( T.ble9.r.C )

P.lu.l.. 8d Qurntil. L inttiont

The €sdnll.s ol pffi.lcB .r€ grv.n i. Ttbt. I0.2.C The Qrdile dimr6 arc

shown in T.blc l0 3 C and di$$!ed B und€.

Quantile .$ina6 by MrM are th. sane 6 by MOM Thcac are snallBl bv

P\tM Th. ditrercnc b.lween th. didal* bv PWM and MLM ar' very smdr.

varyiig Loh O l% ro 0.8% ( incisiiS witi ftrum p.nods ).

101

l^gNorhrl Ditdbuuon

Quantile 6tinlt s arc lttgest bv PWM 'nd

stnallest

Nomal disinbution) Th. dfreFe betw@ rhcn vary

Esiim;s ror Log_Noodl, bv ihe rhre€ nelhods are lars€t $an

(d for cuddu 8d Snkkut

In Qrmtil. &tiFrt ! of tng-Nord'l n r't

1901-1999 for difi.dt r.rurn P"dsTrblelo.a.C. P.rc4t.8' Incr'a'

Notud Dildburion ( Kot'i )

Tte difctcnces i. Qua.rile 'slimai's increase for larS€t

o 950 million d*ca flood n dpected ro pass throush

y@s, usins Log_Nomal di$nb$ion ( MOM ) which is

T , being larg.st bY PWM and

(orn Bafr8., dudng ndr r0o

?5 ttousand cuscs highcr lhan

Tlo 20 25 50 ?5 100

Nomal S E IMOM)S E (PWM) l.0ll l0l 1.009 1.008 | 008 r.007 I 007

Log-Nomal S E IMLMIs E(MOM) 0 963 L5? 1.59 1.67 l7l t7l

( a ) Nomal Disriibutioi: S. E3 of Qurntile Esriflrq by PwM are sm![q rhan

fiolc by MOM ( by abdt l% ) i..., P\\,M is norc.6ci.nt tttr MOM.This

@ll dc not l,lly wiih th. cmel! 8ircn by Ra t 5l I b@!e for rhis dch d

, ror PwM is sn.lld rh& for MOM

( b ) Log'Nomil Distribution S. Et by MLM ar. 50el0 to 7|pl. largq ihan by MOM

Th. difr€rhc6 iqq$ with rdun pqiodq i .. S E (MW) / S.E.(Motto .( | 50, I 73 ), i@3ing vith T thlr MOM i! mE .tnci.m thd MLM.

(c) Nohel. vs tas-Nomd:

S E (Nomal) / s E ( Log-Nomal) e (l !5, 2 I ).(iv) T6(l of Goodn6r ol llt

(iii) S. Ar, of Ondril. E ti!.r6.Trbltslo.s.C. Rrtid of $E ! of Qoruit. Etrin.r6 for Dilf.rot R.tum pcriods

T BLE-rO.6.C T4h ot Gtudnd or Fir r K6i,i \ r9o1-r999

' indioics hon eeptance ofNomal distribulion.

$ Krot PPCC RMSE

x

Norhd LN (2)

D

NorDrl LN (2) Nom'lLN(r)

5.9 t.74. 4.65 0.136 0.191. o.lt8 0,974. 0.991 0.ff,69 .067

Tabl. 10.6.C indidl€s Ihat No.mal disldburid is..jer.d on the b.sb oft,'l6lud S-K test al 5oZ l.v.l Log-Nom.li! beter than Nomal on th. basics of PPCC md

RMSE

i indicrl.s rqjdtion of null hypolh4is

On th€ basG of W-t6t i! is co.clud.d rhai rhe dara for thc Kotri BamSc is not

Nodnar Horcv.r it is Nomal on thc bsis of A D-r€r Mor@vcr. C.V is 0 36 which

dB nor bclongs lo ( 0 25, 0 33 ) tlcnce, the hwolhqis ofnon-.ornatirr is Mept.d

Tdbl6lo.?.C. Norm.lity T6t! (Kooi) l90l-199.

(Yi) Ct, Crand LMon.trt Ratid

Viluc olCs, Ck h.nd rr I Koln r l90t-1999.

No'g ri"diqr6 rhc Ej.(ion orrM iul I'yportrs. I 0rc dirlorcG bciu srcder dun ?0% ).

Above figurq i.di@G thal Nomul Disrribuiior is tcjdted on th. bsis ofe aftt C,

LN (2)is cjared hy Ct dd Ck but is bdl.r rha Nom.t by rr dd C,

R

0.001 0,0100

0 165

o l2lc, 0 8r0 000 Ll26'cr 1.517 100 5 87.

10.4. Findings

(^)

(i) D.ra b + ly !L.w.d &d mt Nomd ( A.D-lBi ed W-tqr )

M.a > M.dis > Mod. ( 3 ror Cuddu bt Gon'h.l ).

( ll ) For Nomd diribotio'r, Qundlc Fniml€ by MOI4 MLM lrd PwM

uc.lmst €qud. Fo. LN (2) QHd < Qru < Qm ( diratjc ar. v6y

0.I ).

( iii ) Fd Nomd di.tiborid, S.E'. ofQu! iL Erin|td by MOr4 PwM .nd

MLM @ lppuidtdy cqu.l. S.E Qw < s.E Qrd for rr8-Nomd.

Fd LN (2))rOM t tuc .6ci.nt thd MlJr.

( iv ) Q!6til6 6tiot6 ud rh.ir S.E! fd Nonnd Dnnbldon @ smdls

th$ for Lg-Nom.l.

( v ) Nomlt dttdbudd i! Ri.cr.d by xlr..q s-x tdt, Q , cN $d by a D

lg1 .id w-lcn . t!8-No4nl ii rcjcd.d br ca , c( md r..

( vi ) 1.443 million os nood b dpar.d .t Gudd!, dldng dl lO0 y@, by

turiig LN (2) @iig MOM, a @mp.t!d to Lt nillion os4 a n3

(B)

(i)

H6cc, LN (2) b l@Ilffid.d for noddins lhc d& .t Clddu bMlg.,

lsiia MOM.

Dxr ii +ly !t *!d .nd Md > Mcdie > Mod..

( ii) by PWM Nrc $ndlcr

MLM querilc cctin|ts

the orhes for Nom{l

@ lh€ srllet for t 8-

(iii) Quantil. dtimdes for Nomal di$ribdion 8re sdslls lhan for Log-

Normal, lp.cillly by PwM.

(iv) PwM i! no$.mci..t for Nomal distribution vhil. MOM is more

efncicnl thd MLM for Lg-Nom.l.

(v) Nomal diltriburion is Ri41dd on rh. bsir off , s-K-resr,b, q, q .nd Cr

whi!. I-.8-Nomdl is adept€d by all ( dccpt by CL ) md is b.trc. rhin

Normlbyt, C., PICC ud RMSE.

(vi) 125 million osr flood k cxp.d.d { Srtlo. bsaa. durina ndt I00

ycd by lifiin8 LN (2) using MOM ( rh. nod.6cidt hdnod )

Gubrcl dbiribnrio4 uinS MI-lr. h.d prcdi.led I 2l million @s.

( vii ) Th. rhl3 i! nor Nomrl, 4 r6r.d by A-D Md w-r6rs

(c)

(i)

(iD

Hdc., LN (2) is r€@nftdded for mod.lins the data. usins MoM.

IWM giv.s smallesl Qu.htile dimat.! and rheir S Es for Nonddi6rributioh MOM is nore efrcie thu Ml,L,l for Loq-Nornal

(iiD

(iv)

Q@tile 6tinat6 dd rncir S, E's ar. stullr for Nomal thm for LN {2)

IE djfferfta b€twan th. Qldtil. elrinArd by the rwo distributions

ar. trp ro 25% and b.r@n rh.ir s. E,3 &€ up ro I loez, by MLM.

Nomal didnbudon is r.j..r.d by tz-rd! S-( rd! W16! C. and b (nor

by cr, &d q ). LN (2) ir rcj.cr.d by c, and cr md is benr rhdNomal by t4 , PPCC md RMSE.

(v)

(vi)

Dala is not Nom4 a t$&d by A-D !.d W- reds

0.950 nillion tu&$ flood is cxpccr.d to pae.r (otri (B) during icxt

100 y€as, by i'nins LN (2) usin8 MO!4 ph.rc a 0.9?5 nillion @scs

fldd was dp€ct d by fiiiing Gumb.lby MLM. 0.981 oillion @sq had

ldually pded thloud th. hr.g. in 1956.

tt n@, ir t egadcd to lir LN(2) ui.g MOM on lh6lled dara of nv.r

10.5 Conclusion

Nomal dkrriburioi is cj61cd by Z' - ieri S - K le! A.D t6r md W r.st

for $€ d4ta of all rhre badagB. Alrhoue! Nomd dislriburion is norc €fi€imr

lhli LN (2) , y€t LN (2) t @omn nd.d for pr€dicrins fie loods, usioa MLM,

bec.use il is b€ll.r thd Normal dhlribution by 14 , C', , PPCC md RMSE for 6ll

MOM MLM

?0r.55E 70t.653 70t.658

251.194 251.194 250.025

6.,19! 6.49r 6.488

ol0.347 0J55 0.360

Pnnn.r€n of NorDal rd lag-Nomll

{ir 000'3 cNd) 196tr999

Dllrribn.ionr ( Glddu )

Trbl. -10,2.8. P.Em.r.E oa\omrl.nd lrg-Nom.l (2) Di.tnburio.r

in 000 @wr (Sukku E m8.) r90l-t99

MOM MLM

dl622.4D 622.479 622,419

20t55 205.55 r95.Sr5

rrB-Nodnl (2)

d.!80 6342 6,384

0.322 0J?

Trbl. -lo.AC P.riD.t B of liom.l.trd Loc-Ilod.l DirrdDulio..

( Korri ) l90l-r999

MOM MLM

4la!1.9,19 r|!i1.949 a5l.9a9

159.5!t 159.53t l5?.039

r.e-Nomd (2)

6.059 6.059 6.052

||.J€ 0.355 0J5r

T.bl.t03.A Qurnrit. E!lin.r6 rd rh.ir S.E r (in ptnnth6.r) by

Norml rrd Log-Norn.l dhrribotiom ( cuddu ) 196r-1999

T P

Qu.rtil. MrgDi.ld. (o0o fl!c,)

MOM MLM MOM MLM PWM

5 .20 913.la

(11.41

913.1r

(17.4)

912,16

{47.31)

334.60

152,99)

434.80

(59.s7)

349.43

t0 .10 1023,?3

(s5.oa)

1023,?3

(ss.0{)

t02t,2i

(s5.03)

1030.62

(55.57)

1039,1?

(80.78)

1042.12

20 .05 1115.04

(62.52'

1115.04

(62-5tl

ll1ttl

(62.63)

1169.16

(67.rU

Ita2.29

o0a.a7)

1183,49

llal.63

(6.r.E7)

lltl.63

(6r.87)

lt9.5a

(6s.0n)

t2tt,9L

(?1.a8) (rr2.an)

t23.t.66

50 .02 1217,11

(7r.8?)

1217.11

(ilr?)

I215.3?

(t2-03)

1341.44

(t4.63)

136t.03

o3r.35)

t37?.01

13 .013 1253.62

(?5,48)

1254.62

(? 8)

1256.03

(76.03)

11r5,66

(92.60)

t118,21

oss.l0)

1460.03

t00 .01 1236,25

{74.171

t23625

118,41)

1243.52

Q4.71)

l43t,l2

(98.36) (161,r0)

T.lrlcl0.3.B Qurntil. ddnit.! tud O.irS.E\( in P.nrth6i! ) by Norm.l rdrrg-Nordrl (2), 1901-1999 (Sutlor B.rng.)

T

Oo..i . M4rnud. (0oo c$d)

Log-Nornrl (2)

MOM MLM PIYM MOM MLM

5 .t0 195.93

(ra.rx)

?95.91

(24.0{)

1a7,61

(21.93)

113,63

(30.12)

768.52

(2r.s9)

l0 .10 136..13

(2?.E8)

8E6.,lil

(2?.EE)

8?3.60

126.66J

l9l.,16

(27.85)

479.74

(36.61)

8E9,3t

!o .05 96l.la

(31.19)

96l,la

(31.89)

91t4,6

(3{.rs)

1002-15

(l3.se)

tt3.600

(.6.56)

991.97

25 9t2,90

(r2.9)

9&t.90

(32.9)

965.f,t

(31.E9)

1036.q)

o5.56)

1016,09

(49.90)

r0!r.98

50 roa5.21

(!5.93)

l|N5.2l

{36.9!)

r02{65

(3413)

l1,|3.2l

(ar.8e)

llt5.13

(60.69)

r116.06

.015 107t.53

(349r)

l0?&d3

($.9r)

1056.45

(36.04) (as.7l)

r11t.27

(67.26)

1196.16

I t0t.24

(39.93)

l10l.2a

(5r.r8) (37.0s)

12la.09

(43.16)

ttt2.52

(72.0t)

t23t.5t

E dn.(d$d rh.ir S.Ett (in p.ntrrhdd) bv Nomd

Disiribudon ( (oki ) r90l-r99.

Qu.n(ih M4rrftld. (0oo cu$s)

T

MOM MLM MOM MLM

5 20 s46,26

(18.56)

586.26

08.66)

5U,t6

(18,42)

57t,72

Q4,11)

512.99

1t3.19)

590,41

l0 .10 656.50

(21.51)

6s6.50

(21.64)

653.3{l

(21.41)

&433

(21.87)

669,93

(32,t6)

699.32

20 .05 7t4,49

lt4.60t

114.49

(24.60)

710.33

12431)

752,9

{26.sO

167,20

(.l|.72)

802.1?

25 ?31.33

(2s.s2)

?31,33

l25.st)

127.t0

12s.29)

7E0.3?

(2&r0)

791.39

114.94)

8J,r,5?

50 .02

(23r3) (28.2r) (r&ot

865.3t

(33.24)

8E1.99

(s5.as)

9.t3.95

(29fl) (2tJ) (29.58)

915.54

(r6.35) (6r.94)

991.53

a7t.t3

{30.8?)

a23.23

(30.8r)

8l?.,11

(30.64)

950.?4

(33.6r) (66.?0)

1032.3?

Not : S E\ for PwM aE d ercurir.rr

Ch{ptcr l1

PRf,DICTION OF FLOODS AT VARIOUS BARRACES BY FITTINGPEARSON (3) AND LOGPf,ARSON (3) DISTRIBUTIONS

I l.l Intmduction

r..^on (3) a.d rls,P€en (3) disrribudoro rc 6it d, conpar.d and

r.sr.d nsins *rad, S-K r6q ppCC. Ca , cr ed RMSE on the dar6 of fl@d

Faks at Guddu, Sltdu .d Kotd Bmas6. MOM MLM ad pwM & us.d for

estimtirg rh€ queiiles ud rheir Sraidr.d Fms (S.Et) for och distriburion, as

ror cmb.l and cEv in cll.pld 9 ..d for Nonn l Jld rN (2) in ch!p16 loL-mon€ 6tios de als us.d for gctring sore idd abolr lhe Droi distribltioh

The .stinalee of noods of stecilicd frequdcy beyond rlrc r@rded nngcpl.y o. inpon.d rclc in rhe d6ien ofhydrtulic srructurG. The 6ddsr6 uerad. thrugn the u* of. paniotd poblbility dklritDdon tundion fited lo !

For.6l of noo& sn 9ft619 .cd@ th. dxge @i.d by tloo& ifprcp€r Prcblbilily dislriburio. ad .n .trci€'n idhod otsrimtion is us€d tor

for.@sting. Id.ntinc.rion of. probabiliiy disriburion of flood d.tt is @.net.dto Ih. lE*ny of cbdct.ri2irg ir . sF h.ric fom rhc d!r. @llet.d dd th.prcvj.ion ofrook for making sraiisri@l dalysis

OnG . distribudon is seleded for B., th. {.lysis b.@ms $ncdy

obj*riE and th. .xt arohtion ie &ro@tic.lh ndc Eom a drhddicjldelcmjnarion ofthe smplc Sratistic, i... n .q S.D ud q.Ditrqdi shlisli.iAns

shd bydrologisl have proposd ud u$d variour diltribulion dd dimlionmcthorls. for pr.diction of n@d3 of I'dus rivs r difaqt b.trlg6

Pr.dicrioi by Crmbcl , LN (2) Fosrsl[, pa@n Tylc | &d adullo@uBence of lloods suSge$ remodeting of the barag6. Nixon, H B Bhuro. N MShailh [ 8 ] md [9] had fiued veious Ska otua &d prohljlily disdbutions on ih.dda of Indus nvd at cuddu and Sullur bdag6 t r ditr€r€.r sple sjz6. n€n rsutrs

Snnnarr olrh. Rsurrt or pwiout ,rudi6

Above rabl. shows lh.r thd. is r n€.d for fifiina nill b.(.. disrnbudo4 ui.gson€ €Ilicimt nethod of .stinadon, to suSg€sl hr eiih€r .€no<teling thc bffrage orrak'ng sone suitable sr.ps ro mirimize rh. chee oalo*s

RcalizinS lh€ n€ed of*t6tion ofa particular probabihy disrribnlion and aefrci.nt nelhod of .sti@rio4 vadous invenigatos propG€d diferor distriburions to fira paniculr typ. ol hydrotogiql redon vdau.s rhc ditribulions cont.in pamm.t.sthal nlsr be esrinared fron sampl. <tala Math.m,ricaly, rhc more p.nmeters in arurclion,lhe morc lqible i is ih filling empiricil disfibulion Mo,e over, !* ofaAiwndhtriburion is subFcr lo rhc validily of €slinarion nerhods, ie , thc s.t.crion ofa

76

85

99

35

l8 cunbel(MLM)

150

6.75

9.50

975

950

3.53

t2 40

12.00

12.30

t2 50

12.10

141

15l

l4a

Fobabiliiy dist.ibution b dn opihizing problcm belw*n lh. flexibilfty .nd reliabiliry of

Hcne v€ arc usi.8 two norc distdbulioN, mely P (3) &d LP (3), tom'Ie ih.

compaircn beM.en dhtibutions 6id]-P(l) ir eld€d for fiftins th€ dara 6 ir is

dGiv.ly u*d ii USA ( 1 5 I and 001

h depods on C, as ils char.ci.rilti6, hcnce is inbnsiv.ly invdigar€d. Wh€n

CF 0. it be@m6 Lo8 iomE!. Bobe and Robiiall€te I I 01 iiv.stigaled dif€renl

ncthods of lirdng r (l) &d LP (3) dhnbudoN by Eins s4.d lons tm r@rds

Bousntoh 031 used LP(l) ih a study orflood.sii6.tes from shon rccords McMahon

ta4 dno.srrated rhlt LP (l) w6 sppliqblc to Aust.dian flood data Thcv inv€stisdt.d

fie efrels of s$pL d4 on pd.tr.r6 oflhe disltibution

Aashtar dd Eoob.e 16l ddelop€d conlid.ne I.twals (c.I s) for P(3) and

LP(l) Arrcra and Singn ttl compeed ditrerenl n€tho& of paramd€r estimation of lP

(l) by Moot. Cdlo sinuLtioi. On. of th. idPonel probl.B r€lar€d lo lhc u* ofLP

(l) is ih. v! ability in Cs.

Mardes dd Wallh t43l .onPsiis th€ qumtll. cstimale, bv P (l)' usine MOM

ad MLM, obe.ttd rhar MLM 6rimarq wc ls bi!!.d with smdld SE's Thc

diffe(€n@s belwen MOM and MLM se ldgd for slld sampl€s

PWM Ms apPlicd ro P (1) by Siddr and Dins [32] and tbcv found it a efrciehl

6 MOM. Shaligd a t lc t55l h.d ultz.d 0ood no*t iom 16srttotinIndiausinS

P (3)

ll.2 M.thodolo$/

ll.2.r P.aBon ( 3 ) Daliibltiotr

Thc pdfor Pwen ( 3 ) dhldbution is 8iv.n by

-_L-;l (l| r)

tti=f,6r' ",

r'(v\= --L7v''.''dv'' f\p) I'

wh@ o = Shap. pamel€r

I = sc.l. ptu.r.r1 - tocnion peuctd

(A) Prr.n.t r Ellimrlion

Thc dbdbuion tulrrim ofx rnd y e sih by

l1

. .(I2)

(ll.D

(|l.4)

(i) MOM

Plfuctd Gtihrtq e. obiained by rcplacin8 !r , ,& and 1' by th.i $mpl.

slimlB m L nh dd C. r! follow3;

.2.. ...... or.5)

01.6)

...... (11.4

(ii) MLM ( t.nria s.hriotr )ThcLik lyhoodlunotionof P(3 )isgivdby

.r-, +r r:r )r - l---1- | nl t:! l "?t ' )

Lar(r)l i.i\ d ./

tDg-likcly h@d tunction is Aiv.n by

..... (l r.8)

LDg L = -N Laa- N )-x(/-r )roe dt's r ( f )+( t - | )Ztogl', - |

-|it' -'tDi{fcr.nriating b.lh sid6 of€qution (119 )w

Np I I(,,-r) = o

r. t d, 0 ud 1 Md .quanns lo

(ll.e )

-Ivv(r)+Irosl4Jx-ta rtiI I ]=o'j:i\r /,

(l l.r0 )

ulrl)

( .12 )

a = |ir'

.L-.1,\ r ' /r\

whFv{r)=-+142=:-'\+ (ll.l3)op t \P)

Solviis.qutioB ( II lo ),01 ll)ud(ll 12) situltaMuiv' * gcr

-r) - ...... ..( 11.14 )

i = j;tr',',t (Il15)

For s gjvd inidal valuc of / *e od lwlud' ; a'd ]i fron cqu'lioit ( I l 14 )

.d( ll.ls ) Eqution( lll ll)onbeMr'n*

r - iroe(, /). I'losd-l/v(p) - 0 (rr16)

N.!bn nsation ndl&d k lppli.d to inPrcv' th' initial v!tu' of1 a

,,,,t - 1., - FIF 0r rD)

whcre ,' s rhe d.rivdivc of ]r si!.n by

26!7_iff-,*atffN>tl(\-rf

(ll.19,

Lft(',-t))Lt',-,1frr I- L.--!-- r, I - 1,,,,,,' 2t r2p, tmq, 252f. 24pt r'tzpo - "

......ltL22)

( .rE)

lr

I

l.]

n

|r

|.]

il

|l

ii

n

n

n

F

i!

n

|1

ll

|]

|l

in

0f

-t(l1.20 ,af

Y(r) r.s (/

ddv(t)= -r = (-0.5n2t5) (tuhr".@bq )

v'tr) = l-+-L*-L- t , | - I . lo! 2f' 6p' tof' 420' tof B2p"

@'dv'(t) = E, l6

(iiD I,IVM M.fiodItu PWMS girst by troltii8 1251 @ r! folos:

( r r.25

)

)

)(

r(P+tl2

p.2p)-3

whdc /, ( p, q ) is LEompleb B.t! tuncrior

A simplified agpropriale solulon ofequrion (lt.a follow:

For 4 > , l.r ," =t-rr th.n

I

( r 1.23

( l r.2a

23)beiv6byHdrins( )

036067 ,,, -05967 l. +o25t6t /",| - 21446\ t- + 2.5@96 t'-,0.77045 f. (lr 26)

(I| 27)

Fo r,<1, 6, ,.=3'1, rhq

; (l+02906,_)" L;'s8, ,t -oo44t flPutingvdue of ,in.qu.lions (ll.2l)a.d (1t.24). ve ger

d=.[s t"

Note Fo. n€adv. valu6 orb. d

.. ( .28)

.,.(l1.29)

UsinA lhc Wihon.Hil€ny M8fomation lor C. <

\=ai+i+K,la1i

,,2'c,

(l |.30)

I 0 ed iior C, as hieh a 2 0,

?(. c,>o ...(n lt)

(t MoM

?).'ftt2

1=a

'rt),,,v'..ff)"b.r. 0r 12)

)*r*,

ErcB olQu.trtil. E.ib.rs

*r,y,*l(t/,n*r

Kr is frdld.ncy arcror obtdn.d iron€qurion(l | 3l )

,1,[i

tr is s6ond @mn

aK. -rl I0/, c1l L

(iD MLM

lmonsi.id is Bdnai.d by atii. / ? \ \' I "l t^,?(. t).'f 'l iL'{?(" ?).'f {: *l

.... (11.33)

'r Iz;)'*t",,(fr ]'*,,{ #)'*,."[ #)1,;7)*,,",''fifl#]*,",t-,]fi)*"t0,,1 - 0,,a,

fl

r'1

.L

f-'

| | Y'(BlNd'Dl(p-2\

2

i#] orr5)

B2\pv (p

"a, (r )

cov (d, / )

D

Cr

9raraf

P-r,,ti!-td I

...0r.37)

.0r.38)

.. ( .39)

{aB

"r,o[i75 ":-r]rln;nLa;n-'P'(P)l-tl o I

Na'DL\R t) l

7,fi[,*tat-ffi) 0,.t

Panill ddivadq !c obrlircd fron .quldon ( I | .JO)

:]

9!,-ac,

rrr2 LP ( 3

rh€ rbok !.lud m .quttid (l I 11) we gd S.E! for MLM.

log x is a$uied to h.vc a

LP(3) dtl.ibulion. Tn

P( 3 ) distibltion tbd lh.

pdloltLP ( 3 ) dBltibut.d

drr( / )

For flood frcqudcy an tsis,

Thc disdbutio. tunction of

r{') = '-^ |dt\p) i,

bs(l'

)r:

(r

t

(1140)

tLI

I) i! 8iv

to8(t

.. (ll4!)

$bltiurcd in .q!.lio. (l | .42) gls

.-...(11.43)r(t)

C.

. . ol.4t

[('whcrc K= I tbr natuial log and K-(A) P.nn.l.r E$indtio{

{t MoM

(B) QlrniileEstin.tionThe T,year qu.ntile is obr.jned

Zr=logxt=pz+Xr

". = ",D

)",-;)"T),-?)(, gl'

01, - lq l'sI"l',) l

"{,-

-'('-

" -t'-

d, ...01.47)

.(l r 46)

T)043

Indidi Method of MondrsAft6 ukitrs rh. t4gaiihns ofrh. ob*Natioro, apply fomutae (t L5), (I I 6).nd

1rr Drosa /i, a una;

(iD MLM

By iteBtive ncrhod

To obl.in rh. Brimrer or a, /i ae, i lhe erc prc..durc ai rhat for paMit peS t used, wifi xirepl&ed by Log B, for rll i

Th€ dirccr applicariotr of PWM nethod ro Lp(3) dilrribudon ns .oi banrepo.tcd Howw€., PWM n.thod @ b. appticd 10 th. ftesfored obsdarioiz, = 14 (x , ), wherc z , is dttibuted s a P6on (l) wiabtc. The Droc€durc

ro. 6indlion ofpdm.t* is same 6 th|t for p(t)

c" =,/.[-iKr = Wikon,Hilfd9 .pproxin jon

.. . (lr 43)

. 0 r.49)

..0150)

. . 0l5l)

(c) S.E r orQurnril. E3tlnrte.

MOM

An approxi@r€ steddd dor i. rhis ce @ b. obllired iflogEithnio unns

e us.d in lhe .qu.lioro for S.E.'s of P (3) od may dFi b. @nvstcd lo

MLM

Filti.g ot LP (3) disfibutios ulirg ML mclhod k thc sm€ as friling of P(l)

distribuiioos ro th. loSdirhns of the dars. Thus rhc s.E's in th. loa*pace @n be

obtrincd usins thc .qlario8 for P (3)

11.2.3 Glodng ol F-rt T6trAs in discused if, Chapld 9.

11.2,4 Roleofsk rne.{ Kurrda rd l-nom.!r R!tN,As diiaissd in Cluptq 9.

ForPwson €): Ck= l+ |.5 c,:ror LP (3) * h$t rppli€d rotrrute for P (l) on $. log of!c1u.l obstlios.r.=o 1224+o 3Or t:1+0958h1-05?5136 +O l9413t

ll.3 RBula rnd Dilcussions

(i)

The rasuhs arc sam. asinChrpterg

(ii) Prnn.tcr rod QrrntiL cltlmdd ( T.blct 1.2.4 .nd 11.3.4 )

The €stinaled plr.ner.rs ol boln rh. ditibltio.s, hy thre m.thods, s.o@r.d i.Tablc ll2 A

The Qu& 1. 6rihris ar. prMt.d in Tabl. ll.3.A ed &e de$ed s

P..6on (a ) Dirrnburion

(b)

Th€ diferdB vs.y fron l% lo 3% I l&8.r fo.la€.r T,.

Tb. dilt.6e v!ry fron 5 % 10 6 %.

P(l).v3.LP(3)

J

T.bb-U.4.A.. Rrrios oa Qurtraih E rio.r.. ( cuddo ) 1962-199

10 20 50 ?5 100

P(3) l.0l l.(x l.0a t.0a r.08

LP(a) 1.07 r.08 t.08 t.ot 1.0, 1.09

P(3) r.ol r.02 1,01 1.05

LP(3) 1.05 1.05 r,o5 1,05 r.o5

MOM I-05 1.003 1.03 0.95

arurrQr.rr MLM t.(x)5

1.00 0.998 0,99a 0.993

Qrt / QEj vdis fiom t.01 to | .05 for MOt'i,

frofr I 005 lo 0 994 for MLM (apprcximal.v equll fo. 1, >20)

rnd fron L004 ro 0.999 for PWM flr ditrcrM dqce fo,ldgcr rctum

iqiods. I a0 ud I 52 sill'on as@s nood is qpd.d ro p!$ thrdgh Goddu

(B). in n.xt 100 y€aB, by P ( 3 ) usins MOM and by LP ( 3 ) using MLM.

{ni) s.Et ofQlrndL Ertin.r6 ( TrblFll3.A )

T.blel 1.5.A. Rrlior or S.[t. I Grdd! ) 19611999

l0 20 50 75 100

P (t) l.0r 1.02 1.02 1.02 1.01 L02

LP-' 0.91 0.E6 0.85 0,87 0.8r 0.?8

s.E (P-!) /S.E (LP-) MOM l.tl l.ll l ll

s.E (P.r) / s.E (LPr) MLM l.J) 13, tJ5 l.3t

P (3): S.E{w < S.F d ; th. difddc v.ry

LP (3): S.EMu > S EM ; th€ diF.rc.6 v:ry

P (3) .v!. LP (3) :

(a) Dy MOM : S.E nn > S.E !il) by lel., af,d smlld difcr.ncB i.e ror

lrg€r T'

(b) By MLM r sE't)> S.EL().

Th. diff€r.nes u.y f,om 22'lc to 39./", hrget for l.r8s T..

Hdce, LP(3) is .mcjcd rhd P (3), morc fo. MLM ths for MOM

liv) Good !d oaEil l6lt

TABlL ll.6.A. T6!r of G@dr4 ol Fit (Guddu) 1962_199

Abov. t!bl. indicars thd bodr the dbtnbutiotu do a@pt.d on the basi of 1,' teq S-K

tP(t) is bdtd th.n LP(l) otr thc bst ofPPCC ed RMSF

(v) C" Ci.nd l-dor. r|lid

T.blell.?-{. V.lod ofc" cb h .trd .. (Gddu) r96tl99

$ PPCC RMSD

D

P(3) LP(r) P(3) LP(3)P(l' LP(a) r(3) LPt3 )

3Ja 3.65 0.2r o.l9o 0.170 0.969t 0,9169 0.061s 0.0605

P(3)

0.t51 0l5l 0.t43

0lt6 0 127 0 t24

0 6t2 0.512 0 r00

cr 2 t2S 138 I 14.

Nore . rdiare non M.plue ofH.

Abov. rable indiqtd rhlt b{jrh lhe dinriburi@ r. M.ptlblc by h, q .ftt C..

Ll (3) h stsntt b.fid lhs p (3) oi tnc bsis ofq dd CrNoi€: rr and Cs for L P (3) e calcut.l.d using loA oflcrul obswdliong

ll.3.B, Sulkur B.me.

(i)

( ii ) Prrm.td ud Qurnlitc Estibi.!Tablell2.8 slb*s th. ulucs of p!@..6 fd P(3) ed rx (r), csr,hard by

thc thre @tho&. T$le lt.l.B. rhoM the quuritc enin.id tor b.lhrh. di8tribuiions, Thc qwrile estimars ar. disusd n under :

{r) PoBon(a)Dirriblriotr

MLM giv* rhe snall.li qunrilB dd PWM giB th€ hrg6t querih!

Qw / Qvu wis nom l006 ro 1.05 ed QwM / Qrd veie rrom l0l ro

1.04. Th. ntioe @ Srarr forluSaT..

(b) LP(3)

MLM giB lh. so.ll6t qudtile whiL MOM giv6 rh. larget quhnlqfi. difrtrdces d. sn.lld rhe for P ( 3 ).

AMU<ANM<AMOMOEM / QMu wid liom | 0l to | 01. grolc' for lsg.r T,

(c) P(3).v!.LP(3)Trbl.-lI.,t,B R riorofQurtrtir. &rim.t6( Srkkur, te0t.t999

MLM

Q €slioat.d by LP ( r ) arc smalld thd e by p ( 3 ) uring Mln,r dd pWM

( l% to 2% ), bur e laqcr usins MOM ( l% io s% ).r x.nd r.!r tritrion c1&no.d n a\Fr.d sr sutdur &rns. dudig d( loo js6 (un4 MtM) fd p (3) .id Lp (3),

(ill) 5.Es of Qu.ntit. Ejrin.td ( T.bbr r.3.B )

S.E s by MOM r€ laracr thd by WN (X% to|yo)MoM is 163 .fii.idr rh.. MLM

(b) LP(3)S E.'s by MOM.re lalgd the by MLM (4% to 8% ), i € \4LM is mre efficient

(c) P(3).vr.LP(3):

The dillcrcncB b€r*c.n the Brios dc up 10 l% for MOM md up ro 1elo for

MLM. A5 e S.E'3 by rx O) ar. 3mll6 lhe by P (l). LP (3) i! bcfid th& P

(l), s?6irlly for MLM

T.rk ofCoodnar ofFit

TAALE- rr-6.E. T6r! oa G..drd ol Fit (Suthd rq)1-1t9

Trblell.S. B. Rltid ol S.E'r ( Sutlur (B) ) l90l-1999.

l0 20 5o 75 l(x)

P-(jl 1.04 l.0t l,0l 1.0, t.o2 t.02

LP (]) t,04 1,06 1.07 1.08 1.05 1.05

s,E (LP-3t / S,E tPJ) MOM 0,99 0.99 0.91 0.998 0,99

s.E {LPn) /S.E tPJl MIn| 0,995 0.9? 0.94

* Pfcc RMSE

P(3) LP(3) r(3) LP(.')P(3) LP(!) Ptl) Lt(,tts93 5.81 0.1!6

rlndiarB non &eprre of null hyporhad.

Bolh th. dbtributio$ !r. Ej.crod on rh. bsi! of S-K 16r, bur &c a.@pr.d using t: ,

PPCC ahd RMSE. LP (3) b b€t!6 fi.' p (l) on ih. beis of ppcc &d RMSE.

(vii C!, Ck tnd r-nondt R.rioil

a

f

f

T

T.bblr.7.B. V.lu6 orc, Cr d r-Moodtn dor(Sottu) I9)l-199

P(3) Rd$ for LP(3)

0123 02r1 o.xt|0.131 0.148 0.1,1.1.

r.t20 l.ll0 1.t00

cr 3.680 4.850 4.815.

. indiclld Ej6ion ofHo.

AboE rable lnows th.r boih disrriburiotu arc a@Plsll|. oh ilr. b$t oatr .ndC' but noI by C( &d r. Morc owr, p (3) is b..is rha Lp (t) on de b!5b of

Notcr h ..d C. for LP (3) nc oiohtcd uling tog of&tual obhdiois.

I

(C) Korri Brns.

(i) D6criptiv.Shti.ti6As in Ch.pla 9.

(ii) P.dd.t r ud QundL E!liE.r6 (T.br. rt.2-c ud rrJ.c)

P!@erds dtinlt d by P(3) and LP(l), usinS th@ nElhod of srio.tion'

Prce dtl in Ttble Il2C Qu.nlil. E$inatd 8iv'n i' TabL ll'3C

(i) P (3):

Q{vo!y' Q Llo vdiB from 1 oo2 to l.oo8

a awr iQ rrwr veiB liom I 01 io I 04

Th. difrqcn6 betwc.n MOM ad MLM ue *.v s,nMLM give! smaller€stimd€sofQua iles than MOM

(b) LP{3) |

Q (Mdy' Q ftoMl ved fion I 02 ro I lo

Q (svy' Q(rc$ vset tm 1 007 to l 0l

Es$turd by PWM r. ndly eq!.1 to tho* bv PwM

MOM giv6 th. snalet qu!*ile 6lirlrr6Thus the eslih.rd ofquantil4 by LP(3) are ditrer€nl tt|n bv P(3)

2ll

Th. ..lio of QuMrild v{ia b.r1|@ ! OO2 dd LOO5, o.9o&d0.98, I Oo2 and

1.01, respediv.ly, by MOM, MLM rid rwM. Thus p (3) by Mr_M eiv6 rh. mallatquantils .srimat€s o 9 t and I oO hillio. @s fl@d is.xp€r€.t st Kofi {B).r6pcliwly, by P (3) and Lp 0), $ine MLM.

(.) P (.). vs. LP (3) :

Ttbl6rr.a.c. arrior of Qu.ntit. EsridrG ( Korri) l9|,r-r999

l0 20 50 75

Quou iQuw P(l) 1.002 1,004 1.005 1.006 1.0{E 1.008

LP (3) 1.02 1.05 r.0?

P (3) t.0l r.02 1,02 1.03 1.03 l.0a

LP(3) r.00? l.0l r.0l t.o2 1.02 1.03

MOM 1.00.' 1.005 r.002 1.002 1.00t

Qr tl, / QrP(r' MLM 0,98 0.95 0.93 0,91 0.90

1.001 l.ooa 1.010 t.oo5 l,0l

(iri) SE" (TrbFlrS,C)

r.b|cll,5.C. nxi6 of S.E t ( Kor.i ) l90l-r999

l0 20 50 ?5

P (3) 1.03 l.0t l.t| l.0l 1.05 |.O6

LP (l) 1.03 0.02 1.04 1.02 t.0l l.0t

s,E P (3) / S.E LP (3) MOM 0.9i 0.99 0.98 0.9 o.91 0.99

S.E P(3tS.EI}(3) MLM 0,991 o,9E 0.t96 0.91 0.96 0,94

ta) P(J): SE({o$/S E\vw,van.sberwm | 02lo | 06,

i.c., MLM tu norc etrrocnt rh.n MoM.

(b) LP (3): SE ftou sENLc) vais b.twco Lol ro |.04 ,

i e , MLM is mor. efficic tbd MOM (a it @G of P (3) ).

(.) P(a) .vr. LF(3)

(i) by MOM th6 ntio la.icr b€twe.0 9? bd 0.999.

Thus PO) i! b.ns ft rhan LP(3).

(ii) by Mlnl : Th. orio vancs bctw.0.94 sd 0.996

Thus P (r) k b.tt6 tu rno LP (3), a by MoM.

As P (l) grv6 $.llq S.E's thm LP (3), bolh bt MOM .td idn4dt 6tios esall6 by MLM rhd by MOM,So, P(3) is. b d fit thm LP(l) , sp€cially by MLM

$ PPCC RMSE

Dd

P(r) LP(3) P(r) LP(J)P(r ) LP(l ) P(3) LP(3)

4.45 424 0.tJ6 0.128 0.rr2 0.9946 0,9na 0,0!i66 0.069t

(iv) T6G otGoodtrd3 orrit

TABLf.l1,6,C. I6ti otGoodngs of Fit (Kotri (B) 1901-1999

T{bbrl.?.C, Vilu.s ofC,, Cb b atrd r. ftori) 190l-1999

Abovc t.bb snows thar both rh. di$nbudoB e rc.?r€d o. ih. basis ofalt r6rsMoreovcr, P (3 ) is b.ric fir lhan t,P (t) on rh. b6is ofppcc .nd RMSE

(v) C, Cr .nd LMomdt R.tio3

0 t6l 0.161 0 t65

0 Dl 0lll

q 0810 0 806 0 802

I517 3 965

Above l{bl. indid€ $rr borh rh. dtrrib'nioN ar. @prable on rhc basis ofh , q, Cr

hd cr . LP (3) b bciic! fir rhr. P (3) oi rne bsis of ta and Cr

(A) G{ddu(B)

Th. dara is +ly skew.d ed not Nonr.l

Qn.ntile 6tim.la by MOM e bKr for botn thc distijbsrions.

Th. S.Et by LP €) e sall.r rha by P(l), 6p€i.lt for lv{-r,

Bolh Oe dinribltioN rr. rcc.?rf! d thc brsi6 of X t- r*! S-K tcr,

PPCC, RMSE,C, md LnoNnr nlioi, *ce byCr

LP (3) is sliShlly bdt r th6 ?(3) by PICC and RMSE.

1.52 miuion cusds !loo<, in dFcld to p,$ through c!dd! (B), i. nql 100

yoB, by LP(3), usiig MLM, onpd.d to l.l hillion sds a! its dcsigncd

Heicc, LP(l) is rdommdd.d for dod.ling th. dau .r cuddu B.riage, !!ingMLM

(D

(iD

(E) suldor (B)

(iiD

Tn d,ia is +ly ikqcd, i.c. foo& ofhigh mlsnilud. nav. @ur'!d le

Quanlil. 6tim.l.s by MLM are smsll.st for both rhc disdburions, lid for LP (3)

ee sn ll.r th& for P (3).

n4-M k norc .mciar rhu MOM for r (3).

LP (3) h berd thd P (3), sp€cielly by MLM, urins s.E 3.

Borh dinributions rc r.j.cted by Cr md S-K rer blr e.p|ed on rhe bdis ot

rr le t, PPCC, RMsq Cr a'd q .

LP(l) is bet€r thm P(l) by PPCC and RMSE.

P (3) is slighdy berer rh& rx 0) by n &d Cr.

I 25 nillion cu!{s nood is qpored ar Slkkur b.rrage during n€xr I OO

yqrs\ by fitting LP (l), osirs MLM.

As th. .xiring qp&ity of SDkkur !m.9. i! 0.9 nittion cue6, hm@ th€r. kre€d for btjn8 sore sitlble ncdsa ro elc $c stocrurc

(iv)

(c) Korri (B)

(D

Men > Media. > Mod. ( 6 for cuddu ud Sukkur brrasc)

(ii) Quairil.s alio.r.s arc !@rtesr fo. Lp (3) by MLM snd for p (3) by MoM.P (l) by MLM Siva th. snEllat querit. sri@le.

(ii0 MLM i, norc .mciat thdn MOt\,t. sp{rdty fo, p (l).

Borh llE dislribtrrions a.. a@pt d on thc bsis of all rh. t.si, ofsiaiifi@* and

P (3) is bettd thai LP (3) on tbe bsb of PrcC and RMSE

LP (l) is b.(d tbm P (3) on th. bai! ort &d cr.0.91 miliom os Sood b qper.d !r Kotri bama., d$ng n6t ,00 ,ets, by

fitti.s P (3) !si.g MLM.

Hcncc, it is suSe6ted to talc slirrblc lt@um ro ew rhc @sily srructurc of

baiiasc f.om po$ible d.nases

ll.5 Conclusion

LP (3) is bener than I (3), rpccially usin8 MLM, for thc dat. oi Guddu

and Sukkur BMag6 oi rh€ basis offtajorily ofthe cin.d! lid t€srs usd ii rhis

P (l) is bcnd dM r" (l) ror Kori bddg. ont, on lh. blts oTRMSE

dd PPCC, but mr by S-K..d Z: - |.(!.A3 th.B is t ne€d for r.nod.ling ih. bd.g€, hac., LP (3) using MLM is

reconn.nd.d for mod€ling thc dat8 of n@d p€ak.

itr o0o. cutd ) l96t-1999

MOM MLM

I4369 lt9.ll

1068 1.071

-119.25 245 ll 15107

LP(t) 0102 0195 019

l23l.l5 334.10

-6.11 -0.02 .001

Tnbr. r.2.A. P.nD.r.6 of p (t).!d Lp (J) Dirriburion, cuddu

T.bL l lr.B. Prnm.r.6 of P (3) rnd Lp (i) Dittribrriotr! (Solhr)

in 000 r.uts ) l90l-1999

MOM MLM

PO) I15.t0 ll5.t4 r4219

319 .).085 z ll

255 8l 267 654 32100

LP (l) .0t7 151 158

2935 132 5l 3300

0ll

TrbL l t.2.C, Prnm.t d or P O) ,rd LP (3) Itlrtabotioni (Kotri)

itr 0(,0'. cud I 1901n999.

MOM MLM

P(3) 6621 62.24 E0.5{

5.81 408

/16.51 4155 t23.49

LP(r) .0t9 .019 .0194

f E2.95 38).51 ]]E.25

2.81 {.55

LP/3r /cnddn r r06r-raoo

T P

Qurrrnul. Mrsnltrd. tooorcuct

(3)MOM MLM MOM MLM

5 2090t 89

(56.09)

893 39

(s80)

491.1 839.71

(501)

E8?.64

(62 3)

8916

.101035 23

(69.89)

to49.17

(?0 3) (t.a) (58.2)20 .05

|53.67

(83 21)

lt95.5l

(84.6)

| !?8.9 1I]1.99

(7s 2)

|96.97

(64.8)

1180.8

04lt89 6l

(8?.46)

124121

(888)

)220.6 I153.57

(80 4)

t24t.84

(6E.5)

t22A 67

50 .02t296.33

(10015)

118004

(102 0)

ll46.l4 1240.70

(90 2)

t388.25

(78 5)

1349.1

75 0tlD55.83

o073?)

t459 57

(108.9)

l4l7.l3 1351.26

(93.r)

t46A.l2

(r9.9)

t420.4

100 ol1397.09

(ll2.40)

t51540

(1r4.5)

t461 | 40t.25

o05.5)

rs22.a2

(82.3)

r469 5

T.blell.l.A. Qu.nrit. Estinil. rndth.ir S.E\ (in p.nnrh6a) by

P

T.$FU.3.8. Qlr il. E id 6rrd rt irS.E r(hpln h6) bt

P (3) ud LP(ll ( Srllur l l90l-199.

T

Qmndtib M.striud. (0{0r.!@ )

MOM MLM MOM MLM

20714 708

(3 r.?l)

171749

{30 8)

768.95 771.9t5

(31.I) (29 D

162 6

t0 .10a97.249

(42.46\

492.169

(40.8)

989,55 906 049

(42.2)

479 n{40 6)

88? ?

20 05l0ll86t

(53.0?)

1006.t69

(51.5)

t022 65 tuo.647

(52.3)

988.11

(50 2)

l0ll0

25t04?.643

(56.44)

l04t.60

(55.D

t061,87 1084,655

(54.8)

'to22.41

(511)

t0460

50 oz1t56.285

/66.421

1149.x79

(65.r)

||82 i? t224.554

(66.?)

n2305

(6r.r)

1t65.0

15 .01J12t4.464

(?2 8D

12t0.958

(71.9)

t25t.15 1309. I l6

172.r\

118964

(63 6)

t224.O

1262.094

(17.041

t254 26t

(?5.8)

lt0o.87 D70.5lt

(?6.6)

1213.J9

(72 3)

12637

P 3l ind LPr3

T P

O!.n. iL i,r!!pl{d. (oootc!.d }

3)

MOM MLM MOM MLM

5 .20 515.47(r2.lr)

575.57(31.0D

574.402(32.0)

t73.40(12.9)

3114r(lr.2)

512(l |.4)

t0 .t0 664.83(42.45'

663.49(4r.13)

669.2t1(42.8)

662.49(42.4'

675.41(41.23)

667.2(4r.E)

20 05 ?46.05(52.r3)

t43.M(50.9D

756.405(52.8)

741.69(53.8)

714.65(51.8)

150.4(5r.2)

75 17r 02(55.16)

767.41(5r.?D

?at.442(55.9)

167.69(56.0)

806.47(53.9)

115(5.5r)

50 .oz 845.87(64.35)

8,10-41

162.E2'

865.0E5(63.8)

u7.13(65.6)

90J.?5(64.9)

861.1(65.r)

15 .0ll EE8.l,l8(66.19)

88.t52(64.r9)

9 .516(66.l)

886,t6o0.5)

964.45(68.2)

901.5(69.3)

t00 .01 9t7,57(m.2,

9t0.(66.0r)

944.126(6e.e)

9t5.7t(?0.8)

1006.49(69.9) (70.2)

TrD|}rr3.C. Q!..ril. Ertntt.. rnd li.ir S.E , (in p!n!lh66) ht

ChaDtcr 12

RANKING OT DFTRTEUTIONS AI{D COMPA.RISON O' METSODS OrDSIIMAlION

l2.l lntrcduction

L rhis chpr.r ri\ dirdbudoi!, fn d o,r thc &ta |r. migcd in

ord6 of mit.@ding io PACq RMS,E, ,t-rrd, S-K c!r, C. q, &d t . MoE(E rlr.c ndhod! of 6dmdo.{ $.d in rhb !tudy, |r! ds 6nr.d @.djng ioth.i. .trciclcy ( i..., uling S.E ! ).

l2,t Conp.rioD of Dtuidbuiion

12"2.1 RrtrLing tr ?PCC ud RMSE

(^) Rr .iry Dr PDCC

EV-r, r-N (2). CEV, LP (3), P (!).

TrhL l11l. A. n l|dnr otri. Dbt lbrri'd lr PIlcC

EVI GEV rN (2) P(3) LP (3)

PPCC0.9778

(3)0.vr88

(2',t0.9711

(5)0.9842o)

0.9698(6)

0.9769(a)

0.9891(|)

o,9662o)

0.9546(6)

0.989a(2)

o_98aa(5)

0.98a8(4)

0.9956o)

o,9954(2)

0,9787(6)

0,9951(3)

o 9946(4)

0,9928(5)

7 l1 6 t5 tlI IIl u

RMSEEVI GEV LN (2) P(r) LPO)

RMSE0.0671

(3)0.0824

(6)0.08t9

(5)00713

(4)0.06t5

(2)0.0605

(t)

(2) il) (6)0 0437

(5) (4) (3)Kofii 00521

(l)00552(2)

00673(5)

0.0668(4)

0 0566(3)

0.0691

{6)6 9 l6 tl 9 l0

u IV II III

(B) Rrnlirg bt RMSE

RT B Di

Noter valu6 ofPPCC dd RMSE h.E b6 6 ei.d td Ch.pt 69,lOdd ILos ill Grking is sis'*d ro . djnriburio on rtu bsL;t to|at hE, for @hcren4 speddy.

12,2J ConD.nr.n by Z:.nit 9K l6rt

ord.rof n.'ir EV.r. cEv. p ')LtotlIijt

T.bb 12,2,2 cc.pr.bittry of0. Di nblriGodrj|.i,onLrot,Et u€EV-l GEV LN (2) P (r) LP (l)

chi- I032. 2.43I

3.34ll

365llt

(4)

o.2l SKIII

0280. 1.61. t.t8. 019n

0ttI

(2)

5.99 chi- 5.953 9.53. u to. 584II

5.91IT

531I

(4)

0 116 s-K 0.1r5II

0.263. 0.264. 0.135I

0l?9r (2)

5.99 chi, 5.584

II4.35u I

(4)

o 116 s-K o t3lIv

0 33. 0191. 0,1l8II

0.128n

0.1l2I

(4)

2l t1

ichi- 3 3 l

s-K 2 2 2 2

t,,

Ncf.: (i) Rofu capir,l nnn5.R dsbt qisofpr!6tqg.Gt varu6 or t: ard S-K rd h!rc b€r @lLd.d {i6 dECt E 9, lO ald I l.(iiD ' dtur6 flibE ordE bt.

L cEv and Normal distributions arc not .@pr$I. by both lcnr.

Olh€r four disriibttioft &€ ac@piabl6 by Chisq tclr for dl ilc ba88pr

a.d by S-K 1€si only rr Kori2 Bebavior ofLP (3) b sihild lo blt better lhan thal oILN (2) by borh l6ls,

] LP (3), LN (2). P (3), Ev'L b th. ord.r ofmdil.

12,2.3 CoDp.rier by C' tr rnd Cr

TrbL 1t2.3 rhoritra f,b.in.r th. dlrtrtborlo! rn !.iirfrto.y .td dEir

ordd ol m.ril

DV-I GEV Nod.l LN(t) P (3) LP (3)

Cs

Kotri

NSSSSNSS

NSNSNS

NSs

NS

sss

sss

l53

NS

s

s

NS

s

sNsNSS

ss

s

s

s

s

s

s

3

5

NS

NS

NS

NS

NS

s

NS

NS

s

NS

NS

NS

NS

NS

s

NS

NS

s

1 29

3 6 2 2 25

(iDS for S isfdctory. NS for @l stisfldory.Enrrie in rhc abo* r!bl. haw bd @ll6r.d fiom chlptd 9. l0

2

l.

5

tx (l), P(3), cEv, LN (2) snd camb.l @ sdisfacrory in ordd of mdir

rr,3 Conprnsn of EstturoF

T.b|.12.3.A Quntil6 rnd Th.ir S.E t Ar cuddu (B)

Fron rhe above tahl. w€ not. tbai:

i Snrll€st S E i arc aorlP(3) usiig MLM.

n smallat Q!dtil.!.r. ror LN(2) u'i.r MoM

From rhe abovc rabl. ir is cl@ rhrt:

All th. dbtriburiotu m nor $lisfacrory by C*, for lh. dalt ofGuddu dnd Sukkur beages

! (3) is limila. to LP (l) md borh &e noE slisr&tory rhd

EVl, Nonul ud LN (2) disrriburioN @ mr srisf&rory aqlal

All lh. dtutriburiotu &. srisfactory by r!, for rlc d{a of Koti

Itc bchavior ofdara ar Kolri (B) by 14 lnd cr is difc.€nr ihm arsukku(B) rnd Guddu (B)

T g!ab!L-l!!Y1015 (75)

) LN (MOM)

l03r {55)

LP (3) (MLM)

1045 (58)l0

20 I t78 (9r) l169 (67) r97 (65)

25 tn4 (97) 1213 (7t) 1242 (68)

50 l]64 (lt4) 1347 (8t 1388 (78)

75 t4{! lr24' 1426 (9t) 1463 (80)

r50l (r3l) l43l (98) l52l (82)

R.rk! tr Itr rT lllI

T.bk 12I B Qu.!til6 tud Thdr S.E "

At SuU..tB,T Gmbd (MI.M) rN(MOM) Lr (3) (MrJ)10 87r (35) 891 (28) 880 (41)

20 979 \43\ 1002 (14) 988 (10)

iaa- t-;i . ^- ol)50 r1r9 (53) lt43 (42) ll28 (6D

1225 (6D 1248 (48) t2\1 (13)100

IIlllRbks I lI utrFom rne $ov€ LbL wc .ote lh :

i. S@116r S.E.'s e for LN(2) by MOM.

i'. snallBt Qldrild & for Cubbct by ML\r.

From ln abo!€ Lble wc ml€ lh{r:

i. Snrllesl S E.'s e for LN using MOM.

'i sd'rllesr Qurnlil* ar. for P(3) ulins MLM.

Nole AI thc vaiue pr.e.nred in T.bt6 t2.3.A, r2.J.B lnd | 2.3.C a.. 6ll.credlron chapt* 9.10 dd u.

T|bL l23.C Quuail6 |.d Ti.tr S.&! At Korri (B)

T canbcl (ll,I-lir) tN (MOM) P(3) ( ttllJt-{)

l0 670 (l l) 64 (p' 661 (41)

x 164 (31) 1s2 Q' 743 (5t)

2s ?91 (39) 780 (2E) 767 (54\

50 384 (46) 866 (3D &o (63)

915 (36) 331 (64)

r0o 9?t (5t) 9sl (39) 910 (66)

R&ks III II TI lIlr

12,4 coo.|nlo

Gtl$d t t .r-dar CEViULM i. tt !..r.ftd.d !!6odfor rl b6lt|g.r, t'oerftrr.N O) noodr of l0 io 30F!.d h.tgb

thln $. &ri8tld qpicity ofbur.go ir opcorn io 9$s {mogh thdnin rd 100 F|r. tl.t i, b€nat! Nomd(!g.d.l), d..9r by S.E.fb.dh.rr|aq MOM Lttc|dddd o.6od 6rtNO).

Ir P) !y Mltrt i' b.nr tu p O) b. rb dd. ofcudd Dds'*h,. hr|tr, P €) ty MI.I\{ n rd- |lI r.,p 6) tu rr. de otx..l

Fir'.Iy th. didthnionr m rr.ng6d in ord6 ofn|c ir u r,p (3).iouowd by Ouhb4 Ll,(a ald P(3).

l|

IIIItIItIIIIIIItT

tT

29

Chrpter l3

SUMMARY OF RESULTS AND RECOMMENDATIONS

lJ.l

{iiD

(D

{iD

ratirgof Drt. And D€criptiv. Sht!ri.!

The obj{tives frx.d fo. rhis study (as descibed in Chaler I )have ben rcbieved as follows

All rhe darr s.rs ft md.p.nd.nt. nol rcrddMd ilrr'on!ry, i..treid qGts (W-W t6t)Tbe dar. ar Kotri (B) b.for. and !li.r rh. @n tucrion ofoo.rbmas6 and .t Suklor (B) bdoc lad ana Sukhr (B) .nd Cflddu (B)

e homgtuus. i .., rh.y htc @m. fton id.tricd popltd'oni

Thc only outlid dd*r.d,! 'n

rh. d.u 3.1ofKod (B), i. , in

le56 (G,B rsr)

D.$ripriv. Srrri.iicr

Mqn > M.dian> Mod€ , V d!r! sct! lid !t.ll b.og*. At rne

dislribulions rc +1, stMd "I1E freq!@y of oood dd..ei wilh incrs& in rh@agnitude of

Avaag6 and Dispe.lions for lhc dsta lets rt SukLur (B) are smrllg

rhm for Guddu (B) bur I'atq rhan for Kor'i (B).

(B)

(iD

(iii)

!51

{B)

(iv)

Sk wr* rd Konork

All tb. dhdbutiotu e politivcl, skq.d, Muttihodat sd pl.ry- Kunic( exepr 2 our of l3 s.rr ).C, iicrcsA, q dEre.is ard Cr bdomd n.g!ri!q ai.. @ns|ociion ofclddubln gc or incq* in thc spl. 3i&.

w.ibull plo(itrg.trd nod.t by r.s d crEb.t

r€t Squarc mod.l h lcc€pi.d fo. ,I &ra s.rs .nd for dt b.d!8.s, wh . (tunb.lnod.l is nor a propd fir for 9 dll. sas (@t of t 3).

The lins for tull dAtt s.ts .r any b.rr8. atwly3 li. h.r*@ ti6 for lhc drl. s.lsbefoe Nd rncr thar barBgc.

Prcdicrioro offlood by weibull plofiing difld ftom rto$ by t-€sst Squ.rc modcls

Th. difrera|c€ ho.re and coNtuclion ofblrsaa.

(c)

(ii)

(iiD

133 Conp{rison of Disi.ibutio,!

(A) P8i.*lr.Comp.rn'on

(a) Oumb.t nsins Mr-l!,t b h.uq rhd cEv .(ii) LN {2r wrg MOM is batcr $s Nomj bydt@r.rh q@d S E,!_ (iii) LP rt) wins MLM is b.fid rhd p(l) in diorirv orrhc clg(B) Erpcr.{ n@d By B.u.r Dnlnbnlidr

(Million

(Million

Expel.d Fldd in tOO yd!cflmb.l LN(2) LP(li(r!tr_ro (MoM) (Mrrr)

l. to0 !20 (r 976) I t0 | 448 1.52| 2t lt914t098 (r956) 0.975 0.950 o.9l

I

r

(C) Rrnkiog ofSit Dilrriboriou

(i) By PPCC c!nh.l,

(ii) By RMSE:

(iii) By tr,r6t:

(iv) Byc.CkddL:LP(r),

0)

CEV &d Noma! di$dblton e rcjcci.d by zr - tosr dd s-K r*r

Over all Rdking LP(l), Guobd , LN(2), P(3) by lotding the s6

(D

(r)

LP(3),

(r)

LN(2), OEV,(2) (3)

cEv, P(3),(2.5) (2.5)

LN(2), r(3),

(2) (r)

PQ), GEV,

(2) (3)

rr(3), P (r)(4) (5)

LP (3). LN(2)(4) (5)

(4)

(4.5) (4.1

13..t ConparironoaE6tintoE

(i) 6ddu (B): MLM fo. LP (r) h mc .Sddr d'd MoM for LN (2)

R.lio of S.E's valy ftoD 6% to 15%.

(ii) Suld<ur @): MOM for LN (2) i! rcrc.frcidr thd MLM tur Gunbct

P..tio of S.E'r wy fron 2@/. to 25%

(iiD Kori (B) MOM for LN (2) ir F@.ricidl rne MLM fo. Globd.Ralio olS.E 3 viry fom 30./o ro 35%

Hei@, LP (3) nsins MI-LI !rd& firut, Gumbcl ulins MLM srMd! 2d dd LN (2) usiis MOM

,55

13,5 Conliriro! ol Rddb Abort Dktributionr ofthissrudywnb th6.of olh.r Ra.aruh.rs ,

Th. order of merir asigIed lo vrrious d6rriburioB is s unde

(D LP (3), G!nh.l, LN(2), P(3), cEv(MrM) G!trl'rxn OM)

( ) GEv, LN(2I Cunb€l LN{],( ii) P(3), u(l), crun$.I(i") Cumb.l, PPr, LP O)

(v) Glmbcl, LN(21, LP(l), P(l)(, GEV, LN(2). l(3)('iD Grmb.l, LN(2), LPO)(viiD ClDbcl, LP(3), LN(2)

Nor. LP (3) hae bd prerq€d by Mpo in B.!8tad6h, T. Ivajriat in

Ihriland bd is als used in USArrd Austntia.

P (3) is prcfdr.d by Buckd !d Oliwr, SinSh dd Ding Mai.ts .nd

Cumbcl is lsrd by BwDw of Brnglad6l\ NBh and Sind! Cn nS ed

LN (2) b u*d by consulmcr fim3 in B..gl.dBh.

GEV is prcf€n€d by FAP ofBbShd.sh, in LrK and by On€z.

tLnce, ii gd€ftI, rsulls ofth. pEsnr sludy ee in onnrnationto thc p.evious siudiest q@ th.r GEV is p..&r€d by J. Cnud€ry, and

FA'P in Bdglad6h dfld by Aifti in S@tlsd. Morc ovd, rhc.st$ ofthis (udy @n6fr rh. r@lts of Niroi sn dpcn who !tudied tlood palc!t Suklu an<i Koti Bm.gs.

13,6 R€.onnendatiotrs /Suggstions

Md h.s b@n sttuggling agdn r rh. c.rdroph. of noods dda@ rhc @rd.d li5rory of the wodd .W. @not fighr wirh oturq but @ onry

nd8alc aor protcrio./ pl!6rsd@. A! *. rc er e.c wh6 rhe spa [@<l

*ill l)@r h6e on rhc bas* of @rclurions obt.i.cd in rhis sludy edcr@mmcndarions /$gg6tioni ue gim s undq

I Co.sltucrion of ddvrcervoiB md dirp.Biotr structuB dl lieappropriate lo@tions would hclp onrrol flooding in the da dd ulitize

nin warer bmeicially.

2. Consttucrion of ef6live lood onrrol struclurcs lik€ spurs! dyks and

3 Conslruclion ofby-pss.hanftls ro p.ovid€.d€qur 0ood pror@tion

f&ilitiq for rh€ counrry.

4 Sysl€mrlic poc€due for *aluarinS p.otorion n rsr€3.5. Modifi@tion in th€ .pprc&h of pleni.a ..d forubdo. of nd

6 D.tail $udi6 and dttsis of nydrclolo,, llood .Lrrgei topo8nghy,

nv.r mr?hology, flood lftlr aisdng led ue, &a iftmd.lcd ddothcr relared mln€r b€ di.d dr.

1.

3.

|lt2.

Mod€l s:rudid ofvital 6nfd poirt! o. livca b. .o@@ged

Flood routing slndies be undd.lco ro d.rcrsin the ef€ct of {oodprol.clios dd bund conlttuction on dowmram flow!

9. Nllional nood prol.crion plan should b. r.vice.d sd updat€d w.ryy.rs, k@ping in vi€w the chang.d rivcr @nditioN .nd rcquichenls

t0. Ptop.r omuni@tion sysrem !o $ ro h.lp nnini4 d@gs due to

llood b. provided a. th. dd€lopMr snsE in l!@rc u@Prcvinci.l lts.eh 6tss b? orgdiz.d.

To apply lihe Scri6 Aiatts lior rBring rh. sr'tio@iry or dara and

REFDRf,NCES

L Annad, Ml, C.D. Sinclair, and A. w.duy 0988), - rngisric Fl@d FfqueicyAnalFis",.loudal of Hydnuli6, Vol,98,pp 205-224

2. Alcxrtrder, GN., A. Karoly dd A.B SBls ( I 969), ,,Eqlivalent Dhrriburions wjlhAppticrlioB to Rainfall s m Uppcr Bound to llood Dh;bulio.t.. ,ournat ofHyd.ology, Vol, pp 322-371

l. Arudor C. Arg.re (1981), " Srltisti.d tudysi! ofFlood ad Lw no$ oflh.MckonS Rivs ' A'nBis Snbnincd fo. rhc d€8ra of Ma.rs of Engin6i.8 jiInslirur. of Tdlnologl,, Bsgl@k. Thaih

4 (9rmKhddra Rao ( 2ooo ), rl@d Frequency A'alysis ,cRc pfts, London

5 Arora, K, and U P Singh (1989), ,. A ConpaBtive Evalualion ofEslihabrs ofih. Log Parsoh (LPl) Di$nbudon", Journll ofHydrlogy, Vol. I05, pp l9-t?

Ashkd, F., and B. Bob6 (1988), " ConfidcM Inlervds For Ftood Ee€.rs Und6a P@en I or l,g-Pee! I Disldburi@". Watd R6. Bul€ti4 Vol. 24. No j.

? Acthlanhh, N. (1977), Tlood ..alFis by Snm TEnsfomarion'., Joumat ofrh. Htdaulics Divisioq ASCE, vol. 103, No. 5, pp. 891,?8

3 Bhufio, llB.dd Shaikn NM (1987),.,ltmd Fr6que.cy AJEtysis otthc Indls&ivc. !t Cuddu B!tuse' , M€hr|n University Raarcb Joumat,Volume 6, No 2

9 Bhutro, H B and Shaikn N M. (1989), .. Flood Frequocy A$lysb of rh. Indus

Rivct .t Slklor B@ee' ,Mchr& UniwBity R66ch roumal,Volue 8, No. l.

l0 Bob.., B , lnd R. Robir.iarb (19?7), "Th. Ue of tl'e pemn Typ. 3 &d Log -P@6on Typ.l Dislribuliotu Rdisit.d i, w.r* Rmwe Res@ci_ vot B. No

Bob..,B and C, Mdn (t9?3) ,Dctermiialion dG t etual€s de Confiance dc laloi_ P.trsn UI pr les Sralhtiquc d , order ,,, tourul of Hyd,olo8y,

l5

t2.

21

l6

t7.

t3

19.

m.

21.

22.

25a

25 b.

26.

Boughtoa w.C. (1980), Frcqudcy Dslrihniotu for Amd nood", Wat.r

Reur*R.s.dt! VoL l6,No. 2,pp. 34?-360.

BNghlo'l,WC 0976) Tl@<l Elrimation Frcn Shorr R@rdr", toumd ofrnHydrtulicDivision, ASCE, pp,24!253

Bund Manual ( l97D@wi!cd), IniSltion & Powd D.9.dmmt con of Sifldh.

t? Cu.nme (1989), wo d m.ircloeic.l or8loiation, wMO-No: ?13, hydrotoAr

rcpon Nor 03, "Starbtiql Dttriburiois For Flood Frequdcy Aralysis , ca€va

chans, s'k and S.M Moor. (1983), "Flood frequency Artrlysis for smallWabsn & in Southcm lllinoir", VoL 19, No 2,pp 2?7-282.

chow , v 1 (1964).Edito. -ii4hicf , HNtrdbook of Appli.d Hydrology, Mc-

Gdw-Hill

Cho4 V.T. (1954), 'Thc Lo8-Prcbrbility trw dd its En8ii.6irs ApDliqtiom .Prcc€.din8s orrh. ASCE, 80 (5), pp l-25.

Danushkodi v. (1979), "Fl@d tloq Fr.q!.ncf by SCS-Tllo ConpNdProgor4" Jour.al ofrh. Hyddlic Ditision, AscE vol . pp. I | 23- | 135.

Flood Contrcl Pld fo. Si.dh ( 1978), Floorl Comisio4 lllmrbad.

Ctnble Ei. ( l94l) ' Thc rcrum Paiod offlood Flowr ' Am M{cl\ St.re. | 2

Guo, s.L., ed C.Cu.'u@. (19D, "Evdu.don of lhc Us€tulns of Hhodcaldd Pal.oSic.l llood in Qu. ilc Estidltion", t@rul ot Hydrolog,, Vol.129,9p.245-262.

HAAN, CT 09?7), Si.tbric.l M.tho& ii H/rolo$/, AnB, Iowa srare

H.B/ Bhulb'11932), Fluvial Chr[crdisli.r of rh. Indus Riv.l', Th€sis for tbeFar €e ofv E. Mehran Uruv.n'ly of Enginsing and Tehnology, Jmhor o

Hdei A. (1980), "Flood Flow!", trohn wilky & Bon! N€w York.

Haktonir.T. (1991), Stafttiol Mod.ling of mdimum now! in Tlrtkh rivdi',Hydrol. Sci.r. 36(4), 167J89.

HoskinS I R M. (1986 a), "Th. th@ry of Pobabilitt W.ighed Mommls", R*.R.p RC 1210, IBM Rcsrch Divhio., Yorkrown H€ighrs, N Y 10593

29

flilIItI

I

IIfIII

H

fi

hI

I

I

23

l7

l8

16.

lt.

14.

lt

l2

10.

27. H$kjn& J.RM. (1991 a). Approxietions for Ue ir CoBttucdng L-MomdlR.tio Diag'.fts", Res Rep. RC l?09?, IBM R.sch Division. yoftrovnHeighls, NY 10593.

Eoskin& ,. R M , watlis, J. R, and Wood, Eg€n.raliz.<l .x1r.me-value dislribdion by th.noftents , Tcchnom.rrie, 2?. 251 - 6t

Hosking, LnM (1990) t,Mon€.rs: An.tysis

usi.g Lin.ar Combinatios of ordd Slalisri*

Sociely B, Vol52, pp 105-124

JchiDddr Chewdhury M Abdut ksn. Find Epor( tgqr, " Scl4rion of

iTilr#1,.1'"r,- l@ron for Fr@d FEqucnN An rysis in Bmsradesh.

Jabiruddin Chowdhry & R.nnm (1986), "Ftood Action pld in Bangl.dcah,,

Iain, D , and V P Sinei r I o3b). . A Conpan&n oI Tr dstonation Merhods torlrood rr.qu.ncy An0tysis . waier RMUcs BuI.rin Vot 22. \O6.pp9Ol-912

JoldM4 A F (1955), -ft€ rr.qudcy Dbiriburion orrh. Amud Mlximun (orMnimur) vdu6 of M.rrctogcd EIod$., eur.rtv loumrl of rh. RoyalM€kolosiql Sdi€ry, 8?, t58

li.awat c.r.e(1988), Anatysis ofRi{um p.riods otMdi@o Ftood L.v€tsofChao Phny. Rjv€r in Banskdk ", A Th.sis subnix.d for rhe d.s..c ofM E,Alia Insritutc ofr.cnnobgy, Bangkok, ThsitMd

Ktby, W. (1969) On de Rlodoh o@!tr€ie ot M6jor Ftoods , WarcrR.source Rearch, vol.5, No 4 pp. 778-734

6>ubr'futi..t 1986), EnsinFng Hydrotogy., Td. M( cB*Hi Dlbhshing@mpMy hhii.d, Nd Dcht, si{h Rcprinr

tnwdy MD ed t.E. Nah 09?O) - A Coop.rison oaMciho& of Firlin8 fi.Doubl. E pon..tial Dndbution,, Jolmrt ofrlydruuticq Vot. ro,pp 259,2?5.

le6e, M.N. JM and N.C varltas (19?9)...F5imr,on ot paramcreB.ndQuo ihs of Wakebv Dinnbunon+ (qo dnkno*n r.e, Bodndt., JRR. vot rs,

F. (1985 b). Brinsrio. of(h.nahod or prcb.bihy-w.ighr.d

and Estimtion of Distributions

'', Jouhal of Royal Siaristiql

'o.

39 L.dw.h( J.M., N.C. Md.tt!, md r.R. W.[k. (19?9), probrbitity wishr.dMom.nt Conpeed yith 3om r.diriond reh.iq6 in E1liturins clmbdPdln|acB .nd Q@dlB , W.tcr Rcgclt 15, p9. lO55nO64.

rftgln.ir, D.P. lrd S. J. Butrd (19E2), " Gmb& Exlrsic vdu. IDltriblrion", A Nq toudd oftlE Hydtulic DMion ASCE, pp. 502-514.

Mrjun.drr, K.C. dd R.P. S.vhEy O965), ' enin[rd of Exrm V.lu6 byDif@nt DistributioD Fundiotu', w.tq 16@B l|elarch, Vot. I, No 3. pp

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Distribudon', W.t6 R6or6 R6..rc[ Vol. 9, No.2" pp. 281-289.

( M/Idr6(l9a), t .qu.ncy Pndicri@ tudyfi ofnood D.t| Ofrl|cl@ Rive

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3lNo.3"

/M.l Abd$ Srhr (1982, 'Pcsiod rbod F .quaE-y Arlt3b of Tl'lil.nd ", A

ft6b $bmnl.d for $. dryc. of M.E, Ari& Iillit'n ofTdhmlogy, Bsgkor,

Thlihd.

Nehh36d, H.P., rnd F. KdHy (1990), " RLI Ardy3i! ed nre D.p6dcn

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1l

4t.

'r7 N|tion.l R.qrch cou.ol comific (1988) "Eeituring Problbi|nie 0fExtm. nood., Malod! $d R@trlidd.d R6*cl|: N.rio..l Acld.my Per1

48!. NERC, (1975), nod Sludi6 R.ro ", Nlt EN.R6. C@mil, t ndon

48 b. Onoz, B. .nd M. B.'y&it 095), ' B.$-ft Dlrdbudm of r!96r AvrihblcFlood SmplB" , Iomd ofttydrelogt', Vol. l67.pp. 195-208.

49. Pdod\ A. (1981), "St.lirtrql Dacnin|lim ofD€i8r td now', routul ofHyd@loS', vol.51, pp l0+118.

54

50.

5t

52

tl

55

51.

58

t9

Pilgrim, D.H, (198?), Bddging rhe Cap Bclwcm Fl@d R6dar.h ed rietice .Warer Resourcs R€*rch, vol 22, No 9,pp. 1655-l?65.

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E{rcm€ volue Typ€ I Disiribution , .tournal ofHydr0ulics, vol 90, pp 251_268

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56.

,utl

)u'

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W!l.r Rsur6 R€grcn, Vol.2r,No 3, pp. 1657-1666.

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in Hydrclosy . A Tn sis Submjfi.d loi th. d.gte of ME tuid lnstitut. of

T€chnology Begkot, Thaled

us cmlosical sDwey (1e32) lir;/'t

t,.

65

63

Victorov, P (19?l), " Ef,.lt ofPcdod ofR@rd on Flood Pr.li.lior ",Iodal ofth.Hydraulic, ASCE.pp, 1853-1866.

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Yang Ch6s-T 098?), "Fl@d Prcdiction Synd in 3 Rivq Bain Tidal

I.fu6e", A Th*b Submiud fo. thc dc8t* of M.E A!i{n l6liiur. of

I@nmbs Buglok, Thr,hlld.

IIItIIItT

IIIIIIIIIIt

,-|

APPEI{DIX - A

Co|nlrarr Pfogr.[ For Tb P !..Cr Erttrdar OfG|rbd Dildtdr Ey Mffi Mftd tr C++ fr[.u!.

AdLrIb

llinclu&<iostree. h>finclude<conio. h>tinclude<Eth.b>{incLu&<io@oiP.h>

dotible Ex(doub]'e) ;double D* (double) ;

iiilit toqa,sle,t:3,660,613,5?8,5s1,683,34e'613'ioe . iogi,:z{ , rooz, 12oo ,5 41 , 1:-56 ,524 ,652

"t29 '446 '

iig liii, iii, irrs,3r:, rie:, grs, ses, eo 6,1086 '626 '113| 9BA , 5AA ,7 A'7 , 66',7 '

4L9 | ,

rnain O {I I--'---'-------'--/ I -----------------cout << setPlecision(5) <<setiosflags (ios; :fited) ,

textbackground(1);tertcolo. (15) ;cl'scrO;bool divideaYz€ro=f als€ ;

doubfe error=-l,doubl-e t, olata,cout << ltEnter iniial g!'ss:";

for (i-0;;i++) {

if (Dx{x):0.0) {divideBYZero=true;

)x=r-Fx (r) /Dx(x) ;tf (x!-0.0)erro!=(fabs (r-oldx) /a) i100;-----.i- Aiil << "Iteration' n<<

'( <<" Erlor: " <<

erEor << "t" << eudl;

n

n

i.f( (i+llt2o:0.0) {char c-getcho t

d.tch() ; !iii.ii"io.ooos l l (E*(t) <-0 0000001t!Fr(:)>--0.0000001) ) {

"""i-ii-;irii., " .. ' 44",srror:" <<'r!or << "t" <<

cn(U t

l)if (divideBYZero)i"rl. .l-;p".tru.ry divid€ bv zelo 'r!o!

trv 'ithdiff.lel G ut.a3. n;getch ( ) t

'atouble 9: (double r)(

fo. (i=0; i<39 t i++)A+-var (it {e:.I) (-var til /r} tfo! (i-O;i<38 ti++)B+'erp(-v.1til /at i.eturn A- (?01.554-:) iE,ldouble Dr (do!bl. t)(A-0. O rB'0.OrClO. O tillt tifo! (i-0 ti<38;i.++)A+=v!r ti.l r.:P(-w.1til/t) ;fo! (1-0; i<3Ati++)B+=€:p (-val tLl /r;fo! (i-0;i<38; i.++)c+-vrr til rval tj.l r.t P(-val tll /:)rcluln (1/ (tat) ) ac+B+(l/t) rA;l

n

n

l.i

fi

i''l

ir

lr

F

I

l'r

n

|1

n

ReulGOlConput r Prognm

fl

r|

II|

t|

n

tl

t|

t|

ll

I|

l|

I|

t|

I|

f,

1|

I|

n

I|

n

App.ddL - B

ltllpt. Of ltdi.€ t N O) Dtrdbrtbr To A.m.r patno.d D|.. Ar coddr &q. E, MLM ir.atod

,&

App€trdir -BExmpl€ ofFitting a LN (2) Distdbodon to As!.j Pqk Fl@d Dichc46 DaL

^n Exdde 6rfting a LN (2) dbrdbution ro r 5npl. of mnurt Fak flood d.ra

ar Guddu B.trosc ind.{imtion otd.sjgn lot fl@dr b|*d upon lh.lh6rydis@sscd in chlptd l0 is givcn bclow

r. Shlion Ds(ripriotr

Priod of r@rds 18 ys

p.. =2ln r,ln

b. Comp rriond P.ocdu{

St p l: Lisr th. d.iq tAk lo$rithtu (be'c' ), u stwn in Trble B-l

Sr.pt:Pe&n t r .$inaion by Mdimum li*clihood rdhod.Conptlc thc nd of Log.lralb.nEd d.la using thc cquato.

= Sln of@l (ii)/. =24.6728R8-6.491,*h@ n b rhe m. of d.[ poinl!.

^ {,/ ^\) Iol=ltlhr,-/,li,l ()t"'\ ') J

= s.16l(ili)/33-0.trJ73l33 -0.1257

<" = 0 35465Tn€ fitl.? LN (2) Diltriburion n obtrincd by

" T" ^lx, -expl t1 ,+uto yl ..(iiD

*h@ s is s sla.d&d Nomsl vidsc ed it3 valua Gn be obtuDed rromT.ble 8.2St.p 3r Conpuiition old.lign flood *tirrr*

For T-5, n _ 0.842, .I5 = 1030.62

For T= 50, , = 2.054, tt4= 7347.44

!o. T=100, r= 2126, xrw= l4at.rz

l. (x,)

t," -;,)

6 136

t9

Trbl.B.l. Ditr ol DiFh..C. ol Goddo B.r..g. r.d omprtiriont.

T.bl. D.2 values of th€ studard noml v&irre

1.1310.3,t2

Ap0.rdh- C

ltrre of lldt. GV Dbid- ft ld ft .rn .aDtdr!.DdArqa&l.|ltf.E I'WIb.a

nl

App.ndix . C

Er6pl. of nftirg . cEV Dirfibution to ^nn!.1

pat rrmd Dis.hi. D.t ,

^n qmpt. of tidng a cEV disrribution ro . sptc of d.ra ar Cnddu B!tr gc

lnd edmdns rne d*ign 0@d! b.r.d upon rh. rh@.y dios.d in cl,+|a 9. i3

.. Sr.dor Dd.riprionPJva: I.dusBange: cu.tdup.riod of ndrdr 38 ydgDai.: S.. Tabl. C I

b. Conpnrr{io.rt p.oediE

AnmBe th€daujr ed{ri.g ordcr Md drcr in @r. (i, otTlbt. C,2.fmm fie @mllhrbtu 4 3hoh in rh. T$te

Comput rior of firsr 3 spl. pWM,s ulinS.qudion

... (D

b o = sumof col. (ii)/n- zol.658

b, = Sun ofcol (iii) /o -azr 45a

h = $nof@l(iv)./n = to8.to?

:1

f

('1)',"t";')f.

Log 2

Sl.D 3:[i[3;" """-"* w t"t m.rho4 6ns.qudlo"" non

c= ]!rL

= 7.459n C + 2:9554 C,

0.021

log 3(iD

l-

(ii)

- r(r-i)= rtr-o.ozd

- rt.02?)= 0.988 8

'' C")

;= (zb,:b.ir I r . i l(r - z -')

\ -/'= 20E.O164

^ -f / "\ 'lp=bo+z_

lrlr+ eJ_rJ .(vi)

= 615.678

UsinS the dimtcd phrcrf{ lh. tutcd cEV is gs by rhe

" ;l r / ,',ilt, - , - 71r - j - ros l r - i ll | (q,,' ,' I

Bdn rioof Deig' Ftoodt

= u,, ur, * 39tg!!1f ,-{-,*f ,-l .)}-l

uu, L ( \ t t) |

... (v)

T- 5 y.aB, inc5 -ydnood i3

,5=921 456T = 50 ya4t th. 5G yd llood i,

rso= 1386 052T - 100 ycarq th. | 00 - y€ar flood h

t ro= l5l5 .5444

,l

I

l

I

T|t& C.r Ddr OfPdr Fbod Di*rrrg. (o00 Cuer)codduBd€G 1952 - t999

n

il

I|.]

:r

I

rrl

1:

Il:

r

Irr

a

|-

I

1962 441 t98l 129

1963 t982

t9u 733 t983 159

1965 @ t9a4 648

19{6 6tl t985 126

196? 678 t986 I t?3

1968 651 r987

1969 631 t988 I l6lt970 t989

t97l 6ll 1990 589

1972 l99t 606

t9?3 l0a4 t992 1086

t974 724 r9l 6

t975 t99a t7J

t976 t2o0 t995 938

t911 541 1996 588

t978 l156 t997 781

1979 524 1998 667

t980 652 1999 419

t11

T.Dl. C.2. Conpllrtio T.bL

S. NG

o (l)at-tj',.

{fii}a{*_3{8".

t212 t13 9.270

319 18.16'l,ot t3,oBl Lt!?

5 at9 15.2y1 1.114426 51slit 6.39643 ?1.83ta86 15.!2,1

5U |3.291 22.030t0 5al l!t.594 29.213

ll la,.l0t tt,021588 t?a.llo 48.55t

tl 5t9 t9t,02? t8.!696{16 2t7.9tt m.v,2

t5 613 2!t,tr5 8t.751613 2at.5t!

t7 210-602 lt2.Tr2t3 4A 7i1.7r9 tt23a

651 ll6.m2 la9.55a

N 33a-810

2l 660 1t6.155 l3l.28t61 t1t.567 2r0.15

lo3-rt5 2J5_152

633 4l..561 259.157

25 n9 !(D.10326 7lt27 159 533.3' t?0_3t228 n1 J6a.08l 'l(t.3912930 945 ?rO.6?5 5t6.081

988 80t.0 645.lltl(m a39.5t3l0l,l 93?.1t3 g,7.303

t(lB6 968.59| M.vt2I t56 lM2.2m 91t.111

16 !t63 lt00lt5 t039.016

lt?J ltal.ln I lo9,59l38 t2@ r20o r2o0

mt.6t7 t6015-10 t l7o€.103

12t.15 ,04.10?

n5

tl

n

n' It

t|

n

tl

t|

n

t|

fl

r|

n

r|

It

t|

rl

tl

r|

t|

n

n

ApF.dir - D

E!.qL Ol nrt|lS ? (3) Dldhortoo To TL An.ort p.rtnood DlchrrA. Ditr At c{ddu Arrns. By MOM

Appendix-D

E:rDDl.ofn ng P (3) Dl.trlhiloo Dy PwM to A.r!.t Pqt FroodDb.hi.g.r D.r.

An dMpl. of 6tti.s P (3) dinribrtion ro r MpL of$!|l|.l p.|t llood d.[ sl cuddub.ms. lnd c$imrtion ofd6ign p.rt floodt br.d upoi rh. Oco.y diss.d in chrpr.r

r Slttion Dsriprh!

Gudd!Pdiod of ndtrd: 38 Fs

56 Tr!1. B.I

0 + 0,2906 /-)(. ' 0.1882 r: + o.o.l42 ,:)

(r + 0.2906 +0 220?)(0.22M + o.lEtnlj.z2vr1 + 0.0{42(O2m1J'

f(, )d=r*t 1_--;+I lp + rtll

rtB r- r 14 6t4r = t3 642

Sl.p t:

SGp2:

b. coEp.htlon.l Pr! ..lrn

4= ?01.658, r,- l4l.25l q=0.612, C, = 0.358, t, = 0.153

ud r= 0.201(T.bLr 7.5.4.1, ?.5.r.2 ltd 7.5.a.3 )

P.mdd dimdd b PwM nalb<lSiE ,,- 0. t 53 it td dr lB

= a (2 | 7) (0. | 5tr = 02207

p

r(O +V2 )-r(5.u.)= 28.5s3

; = 6rr.t5nfE41). \2t5t3'

A - 1193!l

y-t'-af= m1.658- ( lrgjtr) (4.5rO - 151.066

TIE fded P O) Dblthoii@ b obtri'nd !y

t,-a h,i.-.,[r:E-lh. -\ r' 1r, -+ll+1,-!l,r | -r lc. 'o!,Lto\ o) ) l

rd rb Siva hr.hbB. Z Allddh B

s..ps: CsrFi-r- dddg! fo.d.didr.

For F5, r, - 0.7625, xt= 997.717

. Fd.rE5q tr, _ 2,5t4!, .Ilo = t:tat.lal

Fo. T= l0o, (, - 2.863, trd = t,t67.t4t

ll

IT

n

In

in

||

||

In

R

n

n

n

||

Ifl

tl

n

tl

11

n

il

|l

|l

ilnIJ

|l

Il

tl

n

li

tl

ll

|l

i]

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n

LN

. AmcrAt -P

rrr3 r" i;-rg, eri' m.rrl rx OL I O) .rd Ir (l|DIb

,19

App€ndir E.l

Mod.l! lor ctrnb.l rnd cEV Diuribudons

Distribution Banase Merhod Modeh

Curnbel

Guddu

MOM5=588.6+195.8log [-*['-ll

MLM r, = 585 8+199.5log *[,-r)]PWM

xr = 5E4.0+203 8 log *('rl]Sukkur

MOM r, = 530.4+ 160.2log l-* [ '- iJMLM

[-*( ;)l5 = 532.4+ 150.5 los

PWM= s:r.o * rse.r r"e [ .r[,,L)]

Kotri

MOM r, = 3E0.1+ 124 3 log , ill,*[

MLM \ - 3t8.4 + 129.7 loe [ *l' i)PWM \ - 378.2 + 127 -7 log *['-+Jl

GEV

Cuddu

MOM I l , ,,.-rl[ | ' rt J]

PWM

{ 't it'" 11" - 5r ,-a0s 0/0 mr,

I I

Sukkur

MOM

[ { I i]"'tlPWM ***,-,[, {,(,i)-"}]

,',r.r"n,"oral, { ,.f, f)."}]KoFi

MOM

PWM

Il1'**""'[ l'( ir'-]l

App€trdix 8.2

Mod.l lor NorD.t fid Log Nor.t Dsributioni

,;- ^'.t- ",

t.,

MLM

MLM

MOM& " (76.e r r0 63) +{,lle.2)+xr (?6.9)' t r063

MLM+ - o43.t r307)+(zt.l)+Kr

= (rr9 i.16r) +(l5t D+K, 01e.3),

., - 'rrr.r

. r D'. p$.e,*x, .,(rrs t)' " r rr

. (rr5 ra ' r0c)+ (26?.6J + Kr (lltl4trl09

- (14) 3 ' r.r r, +(]1r 0)+ K,

(66.2t 5 8t

a . 1or r ' e.n9 * 1nt r1 * r, \622r

r, - (30.51 4.o3) + (l2r 5) +(r (30 t:

'' - exD (0.010! lr3l2) + ({ 13) +Kr \23t.2

r, . cxp (0.0re! 13r.1 + 1- 0.0!) + Kr

! - ap(00let r{0.0) + c 0.0r) +Kr

- e\p l(00t7.2e.rt+ (-{ 7D + Kr \29 75

- ca (0. r57! 332 r) + 10.43) + Kr l(0.15?t , t42.5

- crp (o r1' reo.o) + { o.r) + Kr

- c\p (o ore" 32.e5) + (2.3 D + (r

r. -.4 (0.0lel! r32 r)+ c 0.51) +Kr

(0.r58t

(oor9)) :8295

(0or9l)1

=cxp l(0.01e.. ns 2t + t0 44J + Kr

Appendir E.3Mod.l of P (3) ntrd LP (3) Diltributio.t

t6l

APP.ndk - F

Cnpt! OlPltti!! DLltlbltior| Or Tlt lht OfPoknddt

Ar Gtrd.lu E.rfl!., S!*tur E|m3. Ald Koatl E n|!! DJ MOM

231

Ir r' ing r)i Dr{irbur 0tr. r,d tLitr Itrrr,,!r \,(,\l

.\ntcn,li\ l.) Fr'rtrcrrf DR"'ttr,'un..',11"' |l"rtr$ \t"\l

lr uirr.)f I'i{' iburions |aur,, Id, r,ts( Nlu\l

t:fli

llLl I l

'TTTT lI-" I ".1. Lt l-i"T''t-ri_il

if'irl-

rrl"t;

l;

:;]J