Statistical Analysis of Flood Peak data of North Brahmaputra Region of India based on the methods of TL-moment

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1 Sascal Aalyss of Flood Pea daa of Noh Bahmapa Rego of Ida based o he mehods of TL-mome Sobh Dea & Mda Boah Absac: TL-mome mehod has bee sed a aalyss o deeme he bes fg dsbo o seam flow gagg ses of he Noh Bahmapa ego of Ida. Thee exeme vale dsbos vz. geealzed exeme vale dsbo geealzed logsc dsbo geealzed Paeo dsbo ae fed fo hs ppose sg he mehod of TL-mome. The pefomaces of he dsbos ae valaed sg hee goodess of f ess amely elave oo mea sqae eo elave mea absole eo ad pobably plo coelao coeffce. Fhe TL-mome ao dagam s also sed o cofm he goodess of f fo he above hee dsbos. Fally goodess of f es esls ae compaed ad geealzed exeme vale dsbo s empcally poved o be he mos appopae dsbo fo descbg he aal flood pea sees fo he majoy of he saos Noh Bahmapa ego of Ida whe he paamees ae esmaed by sg TL-mome mehod. Key Wods: TL-momes Exeme vale dsbo Qale fco Noh Bahmapa ego.. INTRODUCTION Floodg he plas ad valleys dg he ay seaso s a commo hazad Noh- Eas Ida. I cases mmese desco of cops popey ad eve of lfe he ego. Ifomao o flood magde ad he feqeces ae eeded fo desg of Hydalc sces sch as dams spllways oad ad alway bdges clves ba daage sysems flood pla zog ecoomc evalao of flood poeco pojecs ec. Thee ae hdeds of dffee mehods ha have bee sed fo esmag floods. Hosg J.R.M. Walls J.R. ad Wood E.F. 985 sed he mehod of pobably- weghed momes mehod fo he esmao of paamees of exeme- vale dsbo. Hosg 99 Hosg ad Walls 997 odce he cocep of L-momes as paamee esmao mehod fo vaos pobably dsbos flood feqecy aalyss. Sce he hs pocede has bee sed fo flood feqecy aalyss by vaos eseaches. L-momes based egoal flood feqecy aalyss was caed o by Paada e al. 998 Kma e al. 999 Aem ad Hamacogl 6 Modaes 7 Saf 8 ad Hssa ad Pasha 8 o develope flood feqecy elaoshp fo boh gaged ad gaged cachmes of dffee egos. Kma ad Chaejee 5 sed L- momes o develop egoal flood feqecy elaoshp fo boh gaged ad gaged cachmes of Noh Bahmapa ego of Ida. Lae Bhya e al. sed LH-momes fo egoal flood feqecy aalyss of he same ego ad a compaave sdy has bee made bewee L-momes ad LH-momes. I he pese sdy a aemp has bee made o aalyze he sascal modelg of flood pea daa sg Tmmed L-momeTL-mome. Elam ad Sehel developed he TL-momes as a geealzao of he L-momes ad wh moe advaages ove L-momes ad coveoal momes. TL-momes assg zeo wegh o exeme obsevaos hey ae easy o compe ad moe obs ha L-momes whe sed o esmae fom a sample coag oles. A few sdes have bee made egadg he TLmomes mehod; see Hosg 7 Asqh 7 Moeem 7 Moem ad Selm 9 Noa e al.. So fa o goos wo bag he wo by Shab e al. psed coeco o he applcao of TL-mome flood feqecy aalyss a effo has bee made o fd he bes fg pobably dsbo o descbe flood pea daa of Noh- Bahmapa ego sg TL-mome. Depame of Mahemacal Sceces Tezp Uvesy Napaam Tezp-788Assam Ida e-mal: sobh@ez.ee.

2 Fo hs ppose hee -paamee exeme vale dsbos vz. Geealzed Exeme Vale dsbo GEV Geealzed Logsc dsbo GLD Geealzed Paeo dsbo GPD ae cosdeed. The esmao of he paamees fo each dsbo has bee doe sg he mehods of TL-mome.. DATA Sees of aal maxmm pea flood daa of e seam flow gagg ses lyg he Noh Ba ego of he ve Bahmapa Fge. s cosdeed hs sdy. The Bahmapa ve bas exeds ove a aea of 58 m ad les Tbe Bha Ida ad Bagladesh. The daage aea of he bas lyg Ida s 9 m whch foms ealy 5.9% of he oal geogaphcal aea of he coy. The mea aal afall ove he bas excldg Tbe ad Bha s abo mm. The mea aal pea floods of hese ses vay fom 99.6 o m /s ad he cachmes aeas age fom 8 o m. The ame of he gagg ses ad legh of he ecods fo each ses ae gve Table. Table.Name of he ses alog wh sample ses Seal o. Name of ses Sample sze Moas Noa Boola Dhas Pacho Bels Jabhaal Sbas Be Sash Fge. Idex map of Bahmapa ve sysem

3 . METHODOLOGY I ode o descbe he behavo of exeme flood a a pacla aea s ecessay o defy he dsbos whch bes f he daa ad he pefomace of a pacla dsbo depeds o he mehod of he esmao of he paamees. The good esmao of he paamees may be obaed by selecg he pope mehod of esmao. I hs sdy he paamees fo each of he afoesad dsbos ae esmaed sg he mehod of TL-Mome. METHOD OF TL-MOMENT The fdameal seps of TL-momes ae esseally he same as L-momes. Le... be a sample fom a coos dsbo fco F. wh qale fco QF ad le : :... : deoes he ode sascs. The he h L-mome s gve by E :... I TL-momes defed by Elam e al. he expecaos em E Eq. : wll be eplaced by E :.Tha s fo each he cocepal sample sze wll be ceased fom o ad wo oly wh he expecaos of he ode sascs by mmg he smalles ad lages fom he cocepal sample. Ths Y :... Y : he h TL-mome s defed as E... : Fo TL-momes yelds he ogal L-momes ad whe he he h TL mome s defed as E :... I o sdy we have cosdeed TL-momes fo o esmae he paamees of each of he afoesad dsbos. Whe he fs fo TL-momes ca be expessed as E ] 6β 6 [ : β ` E [ : : ] 6 β β β E [ :5 :5 :5 ] 5β β 6β β 5 E [ 5:6 :6 :6 :6 ] β 5 5β β β The aleave expessos fo he fs fo TL-momes whe ae β

4 d Q d Q d Q d Q The TL- coeffce of vaao τ TL-coeffce of sewess τ ad TL-coeffce of oss τ ae defed as τ τ τ. The h TL-mome ca be esmaed fom he sample by eplacg : E wh s based esmao E : : ˆ whch ca be obaed fom he esls esablshed by Dowo 966 as l l l E : : ˆ Ths he h sample TL-mome l s defed as... ˆ : E l whch o smple e-aageme gves he aleave fom l : Now we ae a poso o dscss he TL-mome fo each exeme vale dsbo. TL-momes fo Geealzed Exeme Vale GEV Dsbo: The pobably desy fco fo GEV s gve by x x x f exp ξ ξ whee x ξ < fo > ad < x ξ fo <. Qale fco of GEV: 6 Q Q ξ whee

5 Q [ log ]/. The combg he dees -5 wh 6 we ge he fs fo TL-momes fo GEV as a sysem of eqaos volvg he paamees ξ ad [see Eq. 7- ] ξ Γ 6Γ Γ 5 5 5Γ I he evalao of he of he paamees he sample TL-momes l may be sed decly. So he shape paamee ca be esmaed by mecally solvg he hghly o lea eqao gve by 5 l 5 ˆ τ l 9 I ode o solve Eq. fo mecally we fs geeae dffee vales fo he eval [ ] fo sable sep sze ad hose vales ae sed o calclae he gh had sde of he Eq.. If we deoe he appoxmae gh-had sde by he symbol τ fo a pacla vale of he s chose sch a way ha τ τ s mmm. The esmaes of he ohe wo paamees ad ξ ae ˆ ˆ ˆ l ˆ ˆ 6Γ ˆ ˆ ˆ ξ l ˆ ˆ ˆ ˆ ˆ Γ ˆ. TL-momes fo Geealzed Paeo Dsbo GPD: The pobably desy fco fo GPD s gve by x ξ f x whee ξ < x ξ fo > ad ξ x < fo. Qale fco of GPD: Q ξ Q whee 5

6 Q [ ]/ The fs fo TL-momes of GPD ca be obaed as. 5 ξ The esmaes of he paamees of GPD ae he gve by ˆ 5 ˆ τ 9 9 ˆ τ l ˆ 6 ˆ ˆ ˆ ˆ ˆ ˆ 5 ξ l. ˆ ˆ TL-momes fo Geealzed Logsc Dsbo GLD: The pobably desy fco fo GLD s gve by x ξ f x x ξ whee < x ξ fo > ad ξ x < fo <. Qale fco of GLD: Q ξ Q whee Q [ { / } ]/. 6

7 The fs fo TL-momes of GLD ca be obaed as π ξ s π π s π 5 π 8s π π 7. 8s π The esmaes of he paamees of GLD ae he gve by ˆ 9 ˆ 5 τ l s πˆ ˆ ˆ ˆ π ˆ ˆ π ˆ ξ l. s πˆ ˆ GOODNESS OF FIT GOF The ex sep o aalyss s o evalae he pefomace of he dsbos. The ess appled fo jdgg he goodess of f fo he fed dsbos fo aal flood pea daa ae elave oo mea sqaed eo RRMSE elave mea absole eo RMAE ad pobably plo coelao coeffce PPCC. Whle he fs wo ess volve he assessme o he dffeece bewee he obseved vales ad expeced vales of he assmed dsbos he las oe meases he coelao bewee he odeed vales ad he coespodg expeced vales. The fomlae fo he ess ae ˆ : RRMSE Q F : : Qˆ F RMAE : : { Qˆ F Q F} PPCC b { Qˆ F Q F} : whee : s he obseved vales of he h ode sascs of a adom sample of sze Q ˆ F. s he esmaed qale vales assocaed wh he h Ggoe plog poso F. ad Q F Q ˆ F. The smalles vales of RRMSE ad RMAE coespod o he bes fg dsbo whee as he case of PPCC he dsbo wh he comped PPCC closes o dcaes he 7

8 bes. We addoally appled TL-mome ao dagam whch ca be daw by plog TLoss τ as odae ad TL-sewess τ as abscssa. The smple explc expessos fo τ ems of τ fo he assmed dsbos ca be we as τ 8 A τ whee he coeffces A ae gve he Table. Alhogh hs s a cde mehod ca povde some sghs o he seleco of he bes fg dsbo. τ A GPD GEV GLD A A.6.9 A A A.68.7 Table. Polyomal appoxmaos of as a fco ofτ. ` The obseved sample TL-sewess fo all he e saos ae sbsed place of τ Eq. o ge he esmaed TL-oss τ fo he assmed dsbos. These comped vales τ τ fo each dsbos alog wh he obseved ae ploed o he TL mome ao dagam. Fo a pacla sao he dsaces bewee τ τ ad fo all dsbos ae compaed ad evalaed. The dsbo coespodg o he smalles dsace s cosdeed o be he bes. 8

9 . Dhas.5. Be Noa TL-oss Boola Sbas Pacho Jabhaal Moas Bels Sash Obs GPD GLD GEV TL-sewess Fge. TL-Mome Rao Dagam fo Aal Flood Pea Daa of Noh Bahmapa Rego. RESULTS AND DISCUSSION The fs sep o aalyss volves he esmao of paamees fo each of he afoesad dsbo sg he mehod of TL-mome. The paamees fo each of GEV GPD ad GLD dsbo ae esmaed fo each of he e seam flow gagg ses sg he mehodology as meoed above ad he esmaed vales ae gve Table. The compaos ae caed o sg he sofwae Malab 6. The ex sep o aalyss volves he seleco of he bes fg dsbo o of he hee caddae dsbos. The pefomaces of he dsbos ae assessed wh he help of hee goodess of f ess whch ae meoed he seco.. The esls of all GOF ess ae gve he Table. As he esls Table s see ha he mmm RRMSE ad RASE appeas a GEV dsbo all he ses. 9

10 Table:. Esmaes of he paamees fo each dsbo sg TLMOM. Saos GPD GLD GEV TLMOM ξ Moas Noa Boola Dhas Pacho Bels Jabhaal Sbas Be Sash TLMOM ξ TLMOM ξ B he PPCC es he esls vaes fom se o se. I hs ess he vale of PPCC closes o oe appeas a GEV dsbo fo ses Moas ad Dhas a GPD dsbo fo he ses Boola Bels Jabhaal ad Be. Table :Thee goodess of f es esl fo each sao cosdeed hs sdy. RRMSE RASE PPCC STATIONS GPD GLD GEV GPD GLD GEV GPD GLD GEV Moas Noa Boola Dhas Pacho Bels Jabhaal Sbas Be Sash I emag ses hs vale appeas a GLD dsbo. Nex we hogh he esls obaed fom TL-mome ao dagam whch ca be see he Fge. As llsaed by Fge fo he ses Be ad Noa he bes fg dsbo s GLD fo Boola he bes fg dsbo s GEV. Fo he ohe ses excep Dhas ad Sash he bes fg dsbo s GPD. I he Fge s also see ha he obseved sample vales fo

11 Dhas ad Sash ae vey fa fom all he hee dsbos. As a esl s ahe dffcl o selec oe pacla dsbo fo hese ses wh he help of hs dagam. Table 5. Bes fg dsbos accodg o GOF ess based ad TL-mome ao dagam. Saos LMOM RRMSE RASE PPCC Moas GEV GEV GEV GPD Noa GEV GEV GLD GLD Boola GEV GEV GPD GEV Dhas GEV GEV GEV Pacho GEV GEV GLD GPD Bels GEV GEV GPD GPD Jabhaal GEV GEV GPD GPD Sbas GEV GEV GLD GPD Be GEV GEV GPD GLD Sash GEV GEV GLD TL-MOMENT RATIO DIAGRAM We smmaze he esls based o he hee goodess of f ess ad he TL-mome ao dagam o decde he bes fg dsbo fo a pacla se Table 5. Fom he Table 5 s obseved ha GEV s fod o be he bes amog ohe fg dsbos de RRMSE ad RMAE ess b pefom pooly PPCC ad TL-mome ao dagam. Whle GLD ad GPD dsbos shae he same a he pefomace de PPCC es GPD obseved o be bes TL-mome ao dagam. Fom he above dscsso we ca coclde ha GEV s he bes fg dsbo followed by GPD ad GLD o descbe he aal flood pea daa ove he saos cosdeed fo hs sdy. Acowledgemes: The fs aho wold le o has Depame of Scece ad Techology Gov.of Ida fo povdg facal sppo de Wome Sces Scheme A. REFERENCE Asqh W.H. 7. L-Momes ad TL-Momes of he Geealzed Lambda Dsbo. Compaoal Sascs & Daa Aalyss Aem I.A. Hamacogl N.B. 6 Assessme of egoal floods sg L-momes appoach: he case of he Rve Nle. Wae Reso Maag Bhya A. Boah M. ad Kma R. Regoal Flood Feqecy Aalyss of Noh-Ba of he Rve Bahmapa by Usg LH-Momes Wae Reso Maag Elam E.A. Sehel A.H. Tmmed L-momes. Comp Sa Daa Aal 99. Hosg J.R.M. Walls J.R.997 Regoal Feqecy Aalyss: A appoach based o L-momes Cambdge Uvesy Pess Cambdge UK. Hosg J.R.M. Walls J.R. ad Wood E.F. 985 Esmao of he geealzed exeme- vale dsbo by he mehod of pobably- weghed momes Techomecs Hosg J.R.M. 99 L-momes: aalyss ad esmao of dsbos sg lea combaos of ode sascs. J R Sa Soc Se B 5 5. Hosg J.R.M. 7 Some heoy ad paccal ses of mmed L-momes Joal of Sascal Plag ad Ifeece 7 9. Hssa Z. Pasha G.R. 8 Regoal flood feqecy aalyss of he seve ses of Pjab Pasa sg L-momes. Wae Reso Maag Paada B.P. Kachoo R.K. Shesha D.B. 998 Regoal flood feqecy aalyss of Mah-Sabama Bas Sbzoe -a sg dex flood pocede wh L-momes.Wae Reso Maag. Kma R. Chaejee C. 5 Regoal flood feqecy aalyss sg L-momes fo Noh Bahmapa Rego of Ida. J Hydol Eg. Kma R. Sgh R.D. Seh S.M. 999 Regoal flood fomlas fo seve sbzoes of zoe of Ida. J Hydol Eg.

12 Kma R. Chaejee C. Kma S. Loha A.K. Developme of egoal flood feqecy elaoshp sg L-momes fo Mddle Gaga Plas Sbzoe f of Ida. Wae Reso Maag Modaes R. 7 Regoal feqecy dsbo ype of low flow Noh of Ia by L-momes. Wae Reso Maag 8 8. Moem I.A. 7 L-Momes ad TL-Momes Esmao fo he Expoeal Dsbo. Fa Eas J. Theo.Sa Moem I.A. ad Selm Y.M. 9 TL-momes ad L-momes Esmao fo he Geealzed Paeo Dsbo. Appled Mahemacal Sceces. -5. Noa A.T. Ab El-Magd TL-momes of he expoeaed geealzed exeme vale dsbo Joal of Advaced Reseach Saf B. 8 Regoal flood feqecy aalyss sg L-momes fo he Wes Medeaea egoof Tey. Wae Reso Maag Shab A.B.Dad Z.M. ad Aff N.M. Regoal aalyss of aal maxmm afall sg TLmomes mehod Theo Appl Clmaol