,zt 5ll SHAH ABDUL LATIF UNIVERSITY KHAIRPUR STATISTTCAL STUDY OF FLOOD FRf,QUENCY ANALYSIS 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 TII 310.5491t MEM a -_'-\ Decemb€r 20(M.
309
Embed
Statistical Study of Flood Frequency Analysis of Indus River at Various Barrages in Sindh byProfessor Abdul Ghani Memon
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
,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.
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$d high flows sihultan€ouslr, rhe amularivdeffecl of w h ich Senerated sp€r R ood conditions rr varioue controt poims The sir ual ion
resull.d in qide sprad <lama8! to thc piivat. propeny and dmaee ro lh. rrigaion,D.ainage and Inlrartuclur€ inctudins Road6, Schools md H€trh feitities The irritisl
estimates ndieted thar 1he damlg* in th. order of Rs 60 billion equivdt€nl to US
Rnyinq Yelow, .rc., in y€r t998 wrious Arn ricd aM Europtu count es and Chimfa@d sedous floods probt€m ofih€ c€.tury
!loo& in th. Indus bdin ee gd.rally qusd by the oau,,l evcnk lik. healyrainfall in lhe !pp.r otchEcnts ofrir€G o. ,.jnfall sp.rieposed on !now,m.tl. H€r7rainf,ll in m@nsoon s.en is @s.d by lropic.t d.prcions Aqerslty fomcd in rh.BayofB€ngrl, shcofwhich nnve innialty in. Norlh _w*tcrty ed lhcn in a
Northely dii.crion to ihe f@r hills oflhe Weslrn Himalayas wiih consequent
orograpnicenhancem.ntof raiofath Flows.xcadiigbankfilcapeitisof Ueivscluonels r€sults in inuDdation ofldge rers, fajlurc ofwucr @nv€yMcc sy$fl, and
!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.$.d thousn
rhe b@s., s&ly in 1942, r9B, €tc. Tn€ bs!8. .t Korri, 6npt.r.d in | 955, hB ihe
desiged dpacity ofo 875 dillioi .uses It crDc und€. ndihum pr.ssuE in 19j6, topas ihe flood dilchrrg. of 0.982 million cuscs whicn ws als !bov. ns d€si8n€d
I. ord.r to rclG !h. Foblm ofsddy, n ir [email protected] lo cstiolt wirh
praision thc hiShdt po$ibl.llood lil@ly to rrivc ar Clddn, Suktur lnd Kotri, at
lest foi rhc n€ l00y.s. Thair mlld b.po$ibl.lo sgSBr to th!dtbonti6bnenEd for rdodcling the bm.ga @rdingly !d .tso to .doF olhd [email protected]: construction of snall dac for lrooge ofwlr.r ro stop rh. ttow of hieh p€ak
Two s.parat. asp€cts ofsch choi.e de inpondt Th6e de rh. descriptive
and pred,crivc propeniB ofrhe choFn m.thod The de$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
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$E ror hl8b.r pal dircn.rgdli N* blnds ![ich pre spsd ofooodr wd6 oG rlE lcnd rcslrinS in rh.
incras. of pdt .iva tlow.
ii Morc ..pid Motrin thc upps c.rchneft ofrhc dd sFi.n.iii [email protected] 6ic in th. erchlM$ ofthe ldbubry nEd.
Thus qelhe( propheh predicr lik.lihood of. psk oa l5 0 lac and I I 0 lm
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
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 $n.bitiri of ditr rcnr din.ibution lo nod.t noodarequcncics L-nomeni r io di.Brams v€.. lsd by rhefr for s.lerion of
(5) lrr(J and woo 1681 Btimared dnud oood probabitirica usins louri.r s.ries.(6) Smnh 16ll r.pr6dl.d bsin sl. ii fl6d pol disrriburio4 uinS a
Lo8- Nomrl modet
(?) Bou8hlon Jl2l deletoped and thcn a thr.cpemtd distrib{tion to th. mualflood dara 6om ?8 etctftnts in 6t.m Ausrrati. Ee dddred nontinsr
. rehlionsnip betw*. the tiequacy telol sd a double logrithfric tu.cion ofthc r@@@ inr.ryal.
. (3)l Anhed d at J r I cohparcd rhe Log-Logislic disrnburio n toOEV,LN(t),andp
(3) disrribudons by usina dara from [email protected] ir M foud lo perfom b.rre,rhln orh€r distjbutions lnd hene ws r@nhetuled for tudbc analy,is
(ll) Majumdar md saehiey {4rl conducted e a$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
(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
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
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
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..
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 $|tiowity . Thus
Iidu! ric d!r. d ach ba!g. @ @ldrdcd to b. irdcFdort Dd !|[email protected]... rrcd .xint
l.J.t Tdl! for [.orog.!.lty A.d Strtb.rtilyMal{N - WnITNEY (M-W) U TIST it !r.d to t !t for honbg.nchy dds.tiowily ofr dlll !a l58l
W.stn !.? null Hyporiatu s
(i) fio:TlE l{o sd6 or!i4 nr sd ,: (r, <,,r ) .onE tin id6tic.l
Er: Tha tN e6d6 e mr ila icrl.
(ii) Thc aignifi@oc lcvcl h !.r !t o - O05
(iii) To ..rry out th. t4!i *! .mgc .ll n, +n I ob€qv.rioln of tn @mbir.d
splcs in thc odd of ifuering tugninrd. .rd Bi8. thc ruk,|,2,....,n'}nr to thq!. In ce oftiB w aign tlE !wrg. of .d nnkjRr= S@ofmt!lssigrEd to obr. |lb in
'lnpLR1= sufr of nnl3 i. em9L 2.
u, =laa+ala+rll2l-R'
we cho$ 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:
(ii) All th. mcer.s of @mr.t r.nddcy !.d disFRion ar Suktlr ircr.e [email protected]. 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|[email protected] (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
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 $rp6 of Q- T relarions is of Cdl Dractrcd
sitnrfid@ it o{ €Ilediv. ramn.nddioN e to b. madc by Itood hyd@logsr5 The
histosu alone, bed uurlt on 50 or L*r ob*o.lio.s, is ulbl. lo oid lhe t6t oa
behoriour of 0oods neds ro b. srldicd. Thi! b.ha our is r€fl4ted in rh. higher dineBionl.$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.
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 $. d.r! a . lGqldE hklogmIn seLclina a p&rialu drtti.al fom fo. t li.qeftt c@e one nry b. r.npt d ro
sel6t a dislnburjon wilh a 1.4. ,unb.r ol plnnerea [email protected] rhe noc pde.rc !dinribuaon h6, ihc b€|te. it will .d.pr ro d s.r oa d!r3. Holgq, for lh. spl. sia usually
.v.il.ble in hydrology, |he r.li$ility in 6tin trng norc rh& 2 or I [email protected] 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
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
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 [email protected] 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
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
t
8d
5
g
e e
IeE
e
$ Ia
e
;.E:4"9i{bsE; =rt
6€6
Is o 9
inE
€-g
EEg
i EtF;6ae!c aE€ R R
^! !
!!iEE,jg!!:s.E:die
c € €
I c
U5i5a1A i,{ Ei rbEfl'drEis a a
3
r49,
EFdf
8
€x Eii88
I
NB
::8 8
3
8 8 a8
R
R
8E
E
a'
c
6
s
!3
,E
ta
t
8s
3
aR& F&
EEA6
;^
alb-9
ts6EEE
E
e5
E
3C^9 E
?iie: i eEE3€6<:e{
e*d
3..
h-€eE ! e9fixEi
RIg
i:cglt5s€6;iEi€eg
o9
a
69
{!ts;,i
33*a
9-o
Fu n+s p
s8 a
5^aa Rtg aEF 6
,t II
I 9 0a
^d5 i:
in E'
ra
68
33
n
tgE_8t 3
eg
E
=z
k
I
I
I
.sE
g e
6
pP
f
E f
a
a
b
E
5
Ix
6
€i Er= c!: :,iSqi!!gF; a8:
€ G tsg
ii
€d !qIa 8
€ qF
; -:EESFE
:6 =e
6
sG
€
:a
df
B;-B8E+irr8
RI
3gi-
E3
6 I
d6
:E
t5
85
2,
tEa 8T
3d E
q8
8g x
8
88
E€
a
!ai€E5d€
33.EE
ES
!Er*na+i -e
3!"9 e&;gFAEE;e
:gr oL
a
'g peE€€
sc
aell!
o8$€*d6_ e :a
EE 6Z
6 3o*d.
I
I
!i
e i9EEEA v=
6Z
;z8
g
#3
aa
ta
g^
EA\5
5Ftt
:.g
n
tl
n
il
n
|'l
li
tl
'tln
c[l,I|
n
l.l
|'l
l.I
n
|'|
n
|.l
l1
a
€
€
E
E
I a
a
EP
lr ti
Eg3R
2z
eaI'tiE
I 2a
ea5E e
zz
€I €b !d5E
t4
F3I
:
T
E
r{rbrl
I
E
tl
I|
I|
tl
||
tIT
I
na
F
I
!3TE
g!d
tT.
(,
IIT
IIIn
ttIn
t
!8a
I.
g
s
II5
IF
I|
b
ilPI
3
d2
t
I
t
;:c{!
-ia
t-
dtt:I
e!
t
t
a:a
s
iltc
b
e
b
t!
9::9
0
IIIIIIn
Il
rIIIllIIIIII
'-aI
gET
I
aP
t
i
Pesge.o.n;
lsliIEIEIEI'IItslr!t\
I't
G3s
i
s
t|
I|
t|
I|
l|
lIT
'ltIIIT
IIT
Il|
ttl
Ec
a
3lI
gr
?
II
19l5Ittilili]IIEllIElElbtlr-IElallIBlEli
l-I
I
I
\
?
'l
ta
E
!
E
.t
ll
I
E
Ia
I
!
I
n
n
:n|l
n
l.l
|l
n
|.l
|l
q
5
i:
F!J
t;
5
3$B
3
t
2
I
Iro.nb!l
rfI|
f|
I|
R
n
tItn
IIIt
,lItt
'|In
E
P
5
5liIt
i6
g
=
r
!I€3
xII!itI'
6
!
.l
3
p
tg2
R5t
I
E
I
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 [email protected] 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
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,
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
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)
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 [email protected]€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! $ftwec rc ga .qurion ofrh. tine alo,B with rh. vdu.s ofr
adj F aod F. which m giv6. ii Tabtes 6 Ll,6.r 2 and 6.t 3 in Chapt.r 6.Th€ araphs oflitled line @ b€ uscd for inte,pot ion!s wc a GnroDotstion.
In lhi! chapr.r, rcd.k by L.S sd C!6bd r*rhod! e ddcloD.d sd ftprcdicrioG offloo& by Wcibul pioning. LS .td Cflmb.l Dlrho& & onDa.d.
6,2 Mod.k For Prcdl.tio. ofBood!
6,2. I Modch Dr L.S rnd CnDb.l Mcdlodr
Mod.ls for plldi.tion ordoods by L S. rnd [email protected] m€rhod .t Sur|(u(Er) Giv. ScB),Koid(B) ( s.6 Sd3) id rh@ b!r.s6 rogdnq @ shom h T.bt6 6.2.t.t,6 2 r.2 dd 6.2 l.l. BD6rivdv
Tdbh 6.2.1.t shws thalJd Mo.leh at&**tt (B):
0 lit.M$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%,
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
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)
(v,)
I
E
E
l&&
9
frIttdla
t:5
li
E?
EI
IE
ln
t5
jI
d
T
6
*99+6R9Ir-tdl&
bn
e
6
;
g
b?
g
I,e
EI€l
;l11;
I
ts
E
3
E
5
,i
I
'a
J
z
:s
E!
I-: d s
EE!1i
E
6
i
&
t
iid
f
0d
i
6
5
II F
x
d
a^ !g
d
g
L
5E
3::*
d
eE
c€
g ed
E
5
x
?
!r.! 3.
!:ii
EE:d
R
6
I
T
& 5
T Ificx
ti
+E-_o
t
- .!l
da€
t eI
d:6
T
a I s
F a xEI
a
a 8
!
t!a"l
ItE:,
EEa€g 69I :- .l-a 6E-: r€E ir: EX.€ :gE ,EO
t{5'!
l.i
t€
E
E
s
e e
: q $
3
d I
B a a F3
ae
a a
E
E
€
s
d
A
q a B
a t
{ G
*
3
€ I 5 a
ae
5
3B
E5
8
6
a
: €
a
3 p
ai
a
I
,
d
F
€
a
d a B
F ;
eEF
a
3
r3
i5
'
J
t
E
d
I
3
E X a
d6 p
eI
a
3
t
f
E
q
3 * 8
2
'
I€EE
:
c
: f
3
q a
a a aI
a6g
3
I a
a
:
d
d
!
a
a
d
J E
R E
R a
t
'I
e
I
3
I3
3
€
E
,
d
5E
dd
E
1
o
'r
s R
s e
F a
5
s d
3 3
e
€ 3 3
?
8
a
s
x3
d
a H
T a {
-
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 [email protected] 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.
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 $ter.d ton ano.gdiinbutions which h.vc app.onndely rb. samc SkeM6s s rhc poputatioi
R&dom spl6 &on many of thc dbtrjblrioc rr.dirioMly used forfi.quency.mlysis do nor disptay the Me b€hdour of SkM6s s do obwed
Many ofth€ tladitionatty u*d thre parahet€r dktiburions !3, Lp3, CEV or EV2(w.ibnll ) * suficiendy fl.xibte to providc a mod.ratety good fil io ih. obseryeddltr Of the two widcty usd 1wc pdma.r dislnburioN Evl and LN2, thc lafier@ show a r€@Mbl. fir b a *id6 Eri.t, of ob$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 [email protected]. is more €fficienl than snoth.r ifir has
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:
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.
rhc he$od adoflcd aor Bdmtion ot q in sy rituarion d€Fnds on rh..@nt rnd ryp. of hydrologidl d.r! rvJhbtc it ih.3it in q!.stion.
Th....r. rw typ6 of doB .!si|r.d vilh qudrit. din.t6. Th. tirsr ryD.rns fioh th. cumplions that rh. obFrvld dlta follow | ,r.riotd disdbutio.This .ror qn b. ch.ckcd by 8oodtr* of- Ur csr diss.d ir ctspr.r .ighr . Th.3@nd rylE as rh. .@r inhsot in p!m.r.B 6rin.rcd f.oh mdt spt6. Thi,crcr 6 b. rcduc.d by uliig ! d.rhod shich givc! minimuh vri'e pdM.l.rcslim s = > ninimum vlriNnc qledle. etNt4. Tti e.10. d b. cncck.d by
MOM, ,lrh@gi ssy to .pply, d@ed is .or .. cfftci.nr
te .I lh. splc infomrion in mML 6rin[tjo4 dpcirtty in rhrc
MLn ihod is regard€<t s boig bcst b.@us ir is nod 'mci€it
That is the
snplinS varian* of lh. estimted ta6mele6 and hence of Qr is &slmptotiollv
smaller lhan by any othq 4tinarion melhod
ML eslimatc, are frequentlv bi&d bul @r6lioN for bi$ 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$rib€ the poPul'tion Yirhin the chose'
(d)
ln g€n.@1, ML nahods e knoM lo b. n6t efEciot i, ,syhproricdtylarg. enpl6 Whil. rhcre is m swer6 th.r ML b mon efficidl i. sr.lt sDl6it f.€q!.ft|y i! 30. Apcn fion €fnciercy, faibitity is horhd @nsidmrion. ForiBtdc., Matal.s Aid wallis I43l .epon.d rhar ML .srim.t6 were imeossibtc roobtain for som. LP3 samples using pt/ML €siimrrion meiho&.
ML stini€r hN optinal p.openis whcn the sampl6 G) o. which it is used
fron a dift rc.l distriblrion lhe optind prolEni.s rrc by no m.m guMtc.d. sineir is distribution, speilic it may .or b. ,olrler.
PWM is d 6y to apply s ordiMry momols, is !$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
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 [email protected] 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
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
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 [email protected],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 [email protected]€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,
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 [email protected] 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
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).
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 [email protected] 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 [email protected]\ r (.). For k> nR, fi. nkr\ vrrirtrcc md C. &.8ivo a.
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 [email protected] 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 $@ by c!nb.l. ft. diFscneiM.e for grets atum pcriods.
(iii) S.Et ofQtr..lit. f,ltio.r€ hy coEbd Dht.ib ion ( Trbt !9,3.8 )
T=5, t0, 20, 25,
s E(MOMyS E(MLM) =
S E(PWMYS E(MLM) =
r.11 r23 121
t.lt L14 I 16
128 tll l12I t6 t.t7 I t7
For Cumbel distribution MLM is lh. b.sr (most eflici€ ndthod) md MOM h
rhc *06r (le6t erlicicno
Not.: The S E s ofQuetile 6timllcs by GEv have oot b.d @ltularcd.
W. .ej.cl th€ hypolhesis tha! th6 dala is dislnbul€d leording ro cEv, 6ing Ilre$ 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
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
(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 $ve rhr srruclur. ofbs,Eg.
Qua.lilc esrihatB by MOM ar€ highcr
Th€ diffsenc. beired esdnates by
lbr Cuddu), for borb the dislribuions
than tho* by ?WM and MLM
MOM and PwM is very small Gs
(iD
(v)
Queril. csrinate by OEV > Quslilc erinllq by Clnbd
Th. ditre€ic.s i.c.qse fron I 5% to ?'/o as relurn p€riod incrces
For Olmbel, MLM b lh. hosl .6cian ddhod lid MOM is thc ledt
(iiD
Gunbel dd GEV both de @plcd on lhe bstu ofta lnd C, but &€ nor
[email protected] on th. bdls ofcr. CEV t b.lls thd Gumb.l by tr pPCC and
Cr.
123 nillion d@ oood i! dpdtcd during l0O y.rfs, by Guobel
dkt.ibulion, using MLM ( rh. no$ 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
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
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 $e
dar! of lllinois and B€rhlahmy [ ? I hed lin.d LN (2) ro ldoho Rrvd. Ir is usd by
Hen [ 25 a ], chow I 18 I ud nany orh.r Hrdrolosin' Chow h.d ddelopcd tgophical nerhod io d.rmine lh. fr.q!.ncy f.cior in LN ( 2 ) dkdbulion
LN (2) is oneofih€ rdr fr.qucntly Ned diliributionand is [email protected]
by wMO []sl It worh in the €se vh.re C" >01, r.g.rdl6s ofehpl. si&.Th. plrrhcr.G ofLN (2) m.y b. rhou8hl of a rh. nd and
v.riance, the n6 ud S.D, or the d6 Md C.v olthc lr8-NomBl vdar. ln
our study we h.vc @tuider€d mqn dnd S.D a, rh€ p@mdcc
th€re is l..k of asrEd upon cir.na for oomp.ring lifiing l€chniques for
sn.ll a.d modcdtc samplo sia. Th. s!ft dbtriburion My b. prmt€riz.d in
*vml *ay\ a rha m.lhod which b.sr drimat6 orc s.l of p.delc6 may not
bc thc b6l for eoth.r. MLM is prc(:r.d in (e ofn > 25 [ 4 ] Hd.q wc arc
inra.ftd in @npatiDg the relativ€ .fliciency of melhods ofgtinadon.
Historigran, Hi$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'
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.
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.
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
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 $ncdy
obj*riE and th. .xt arohtion ie &[email protected] ndc Eom a drhddicjldelcmjnarion ofthe smplc Sratistic, i... n .q S.D ud q.Ditrqdi shlisli.iAns
shd bydrologisl have proposd ud u$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$pL d4 on pd.tr.r6 oflhe disltibution
Aashtar dd Eoob.e 16l ddelop€d conlid.ne I.twals (c.I s) for P(3) and
LP(l) Arrcra and Singn ttl compeed ditrerenl n€tho& of paramd€r estimation of lP
(l) by Moot. Cdlo sinuLtioi. On. of th. idPonel probl.B r€lar€d lo lhc u* ofLP
Th. difrqcn6 betwc.n MOM ad MLM ue *.v s,nMLM give! smaller€stimd€sofQua iles than MOM
(b) LP{3) |
Q (Mdy' Q ftoMl ved fion I 02 ro I lo
Q (svy' Q(rc$ 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), $ine MLM.
(.) P (.). vs. LP (3) :
Ttbl6rr.a.c. arrior of Qu.ntit. EsridrG ( Korri) l9|,r-r999
Nor. LP (3) hae bd prerq€d by Mpo in B.!8tad6h, T. Ivajriat in
Ihriland bd is als used in USArrd Austntia.
P (3) is prcfdr.d by Buckd !d Oliwr, SinSh dd Ding Mai.ts .nd
Cumbcl is lsrd by BwDw of Brnglad6l\ NBh and Sind! Cn nS ed
LN (2) b u*d by consulmcr fim3 in B..gl.dBh.
GEV is prcf€n€d by FAP ofBbShd.sh, in LrK and by On€z.
tLnce, ii gd€ftI, rsulls ofth. pEsnr sludy ee in onnrnationto thc p.evious siudiest q@ th.r GEV is p..&r€d by J. Cnud€ry, and
FA'P in Bdglad6h dfld by Aifti in S@tlsd. Morc ovd, rhc.st$ 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
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 [email protected] 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$nbudon", Journll ofHydrlogy, Vol. I05, pp l9-t?
Ashkd, F., and B. Bob6 (1988), " ConfidcM Inlervds For Ftood Ee€.rs Und6a P@en I or l,g-Pee! I Disldburi@". Watd R6. Bul€ti4 Vol. 24. No j.
? Acthlanhh, N. (1977), Tlood ..alFis by Snm TEnsfomarion'., Joumat ofrh. Htdaulics Divisioq ASCE, vol. 103, No. 5, pp. 891,?8
Haktonir.T. (1991), Stafttiol Mod.ling of mdimum now! in Tlrtkh rivdi',Hydrol. Sci.r. 36(4), 167J89.
HoskinS I R M. (1986 a), "Th. th@ry of Pobabilitt W.ighed Mommls", R*.R.p RC 1210, IBM Rcsrch Divhio., Yorkrown H€ighrs, N Y 10593
29
flilIItI
I
IIfIII
H
fi
hI
I
I
23
l7
l8
16.
lt.
14.
lt
l2
10.
27. H$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