Bivariate drought analysis using entropy theory

Transcription

1 Bvrte rought nlyss usng entropy theory Zengho Ho Deprtment of Bologl & Agrulturl Engneerng exs A & M Unversty, 3E Sotes Hll, 7 AMU, College Stton, exs Eml: hz7@tmu.eu Vjy P. Sngh Deprtment of Bologl & Agrulturl Engneerng exs A & M Unversty, 3 Sotes Hll, 7 AMU, College Stton, exs vsngh@tmu.eu Astrt Drought urton n severty re two propertes tht re usully neee for rought nlyss. o hrterze the orrelton etween the two rought propertes, vrte struton s neee. A new metho se on entropy theory s propose for onstrutng the vrte struton tht s ple of moelng rought urton n severty wth fferent mrgnl strutons. Prmeters of the jont struton re estmte wth Newton s metho. Monthly stremflow t from Brzos Rver t Wo, exs, re employe to llustrte the pplton of the propose metho to moel rought urton n severty for rought nlyss. Keywors: rought nlyss; jont struton; entropy theory; prnple of mxmum entropy Introuton Drought nlyss s mportnt for wter resoures plnnng n mngement. Yevjevh [967] use the run theory to efne rought s sequene of ntervls where wter supply remns elow wter emn. hs enles the hrterzton of rought n smple terms, suh s urton n severty, usng hyrologl vrles (e. g., stremflow). Drought urton n severty re two mn hrtersts tht hve often een use for rought nlyss. Drought urton n severty n e regre s rnom vrles n thus the prolty struton, ether seprte or jont, n e use for hrterzng rought. A trtonl wy to hrterze the rought urton or severty s se on fttng prolty ensty funton. Severl pprohes hve een propose for unvrte rought nlyss. he rought urton n e moele y geometr struton [Kenll n Drup, 99; Mther et l., 99] when t s trete s srete rnom vrle or y n exponentl struton when t s trete s ontnuous rnom vrle [Zelenhs n Slv, 987]. he gmm struton s generlly use to esre rought severty [Shu n Morres, 9]. However, the orrelton etween rought urton n severty nnot e hrterze y unvrte nlyss n lterntve multvrte pprohes hve therefore een use to moel the orrelton of rought vrles [González n Vlés, 3; Sls et l., ; Km et l., 6; Shu, 6; Nrjh, 7; Nrjh, 9]. Some vrte strutons hve een use for the jont struton of rought urton n severty, suh s the vrte Preto struton [Nrjh, 9]. he jont struton of rought urton n severty my e moele usng fferent strutons, n whh se the generl vrte struton oes not work. o ress ths ssue, the opul metho hs een pple to onstrut the struton tht s ple of lnkng two unvrte strutons to form vrte struton [Shu, 6; Shu et l., 7]. he jont struton n lso e onstrute from the ontonl struton n mrgnl struton [Shu n Shen, ]. Furthermore, nonprmetr methos hve lso een propose for vrte rought nlyss [Km et l., 3; Km et l., 6].

2 hs rtle proposes new metho for onstrutng the vrte struton of rought urton n severty wth fferent mrgnl strutons se on the prnple of mxmum entropy. he propose metho s pple for rought nlyss se on the monthly stremflow of Brzos Rver t Wo, exs. Metho. Prnple of mxmum entropy For ontnuous rnom vrle X wth prolty ensty funton (PDF) f( efne on the ntervl [, ], entropy s efne s mesure of unertnty expresse s [Shnnon, 948]: H ln x (.) For ontnuous rnom vrles X n Y wth PDF f(x, efne over the spe [, ] [, ], the Shnnon entropy n e efne s: H ln xy (.) he prnple of mxmum entropy ws propose y Jynes, [97] whh sttes tht the prolty ensty funton shoul e selete mong ll the strutons tht stsfy the onstrnts. Generlly, the onstrnts n e spefe s: x () where λ (=,,,m) re the Lgrnge multplers. Dfferenttng L wth respet to f n settng the ervtve to zero, the mxmum entropy struton n e otne s [Kpur, 989]: f ( m m x exp g ( g (... g ( ) () It hs een shown tht mny of the ommonly use strutons n e erve from entropy theory wth fferent onstrnts n the mxmum entropy struton n equton () norportes these strutons s spel ses [Sngh, 998]. For exmple, f the frst n seon moments re spefe s the onstrnts, the mxmum entropy struton s the norml struton. It n e seen tht the mxmum entropy struton s qute flexle.. Jont struton o erve the jont ensty funton f(x, of rought urton (X) n severty (Y), onstrnts for vrles X n Y nee to e spefe seprtely n jontly. It hs een shown tht from the prnple of mxmum entropy, the exponentl struton n e otne wth the onstrnts n the form of men, whle the gmm struton n e otne wth the onstrnts n the form of men n logrthm men [Sngh, 998]. he seprte onstrnts of vrle X n Y n e spefe orngly s: x x x (6) g ( x g =,, m (3) y y y (7) where m s the numer of onstrnts; g ( s the funton of x ng s the expetton of funton g (. he prolty ensty funton for the unvrte se n e erve orng to the prnple of mxmum entropy y mxmzng the entropy efne n equton (.) sujet to the onstrnts n equtons () n (3). he mxmzton n e heve usng the metho of Lgrnge multplers y ntroung the Lgrnge funton L: L ( log x (4) m ) ( ) ( g ( g ) ln y y ln y (8) he jont onstrnt n e spefe to moel the orrelton through the prout XY s: xy xy XY (9) Wth these onstrnts from equtons (6) n (9), the jont PDF n e otne y mxmzng the entropy n equton (.). Followng smlr steps n ervng the mxmum entropy struton n the unvrte se, the jont PDF n the vrte se n e otne s:

3 f ( 4 x, exp( x y 3 ln y x ().3 Mrgnl struton he mrgnl struton for rought urton X n e otne y ntegrtng the jont PDF f(x, gven y equton () over Y s: ( 3 4 f exp( x y ln y x y () Smlrly, the mrgnl struton for rought severty Y n e otne s: f exp( x y ln y x x () ( Prmeter estmton he entropy theory n e pple for prmeter estmton n Sngh [998] gve n ntrouton of the entropy se metho for estmtng the Lgrnge multplers for the ommonly use strutons. Lgrnge multplers n lso e estmte y mxmzng the funton [Me n Ppnolou, 984]: m g (3) Newton s metho n e pple for mxmzng the funton Г y uptng λ () wth some ntl vlue λ () equton elow: through the λ ( ) λ () H, =,,,m (4) where λ=[λ,,λ m ]; the grent s expresse s: g g( x, =,,,m () n H s the Hessn mtrx whose elements re expresse s: H, j g( g j ( x (6) g( x g j ( x,, j,..., m. Comments he steps ove emonstrte how to onstrut the vrte struton of rought urton n severty wth the onstrnts of men n logrthm men. In rel pplton, fferent onstrnts n e spefe for the urton n severty seprtely to form fferent jont strutons. Aorng to ertn mesures (e.g., the men squre errors of emprl n theoretl proltes), the most sutle onstrnts for eh vrle n e use to form the jont ensty funton of the urton n severty, whh n then e use for rought nlyss. he hrterst of the propose metho s tht rought urton n severty n e moele wth fferent mrgnl strutons (wth fferent onstrnts). 3 Results n susson Monthly stremflow t of Brzos Rver t Wo, X (USGS 896) for the pero from Jnury 94 to Deemer 9 ws use for rought nlyss. he men stremflow of eh month s use s the trunton level to efne the rought event. Sgnfnt orrelton exsts etween rought urton n severty n the vrte ensty funton n equton () ws use to moel them jontly. Hstogrms n the ftte mrgnl struton n equton () for rought urton n n equton () for rought severty re shown n Fgure. he ftte PDFs pture the generl pttern of the hstogrms. he emprl prolty estmte from the Grngorten s plottng poston formul n theoretl prolty re shown n Fgure. It n e seen tht generlly theoretl proltes ftte emprl proltes well. he Kolmogorov- Smrnov (K-S) gooness-of-ft test ws use to further test whether the oserve t n e moele wth the propose moel. he rtl vlues for the urton n severty t t % sgnfne level re.3 n.84. hs ntes the propose moel n e pple to moel the rought urton n severty t. he return pero for rought urton D greter or equl to ertn vlue n for rought severty S greter or equl to ertn vlue s n e efne s [Shu, 3; Shu, 6]: D E( L) P ( D ) (7.) D S E( L) P ( S s) (7.) where E(L) s the expete rought ntervl tme tht n e estmte from oserve roughts; D, S re the return peros efne for rought urton n rought severty, respetvely; S 3

4 P D (D ) n P S (S s) re exeene prolty of rought urton n rought severty tht n e estmte from equtons () n (), respetvely. he unvrte return peros of,,,, n yers efne y seprte rought urton n severty n then e estmte from equtons (7.) n (7.) n re summrze n le. For exmple, the rought urton for the yer return pero s roun 33.3 months n the rought severty for the yers return pero s roun.6 4 fs months. he jont return pero of the rought urton n severty n e efne y the rought urton n severty exeeng spef vlues. Speflly, the jont return pero DS of rought urton D n severty S n e efne s [Shu, 3; Shu, 6]: DS E( L) P( D, S s) (8) where P(D,S s) s the exeene prolty of rought urton n severty tht n e otne from the jont ensty funton n equton (). he jont return peros efne y equton (8) for fferent urton n severty vlues re shown n Fgure 3. he ontonl return peros re lso neee to ssess the rsk of wter resoures systems. he ontonl return pero D S s for rought urton gven rought severty exeeng ertn vlue n e efne s [Shu, 3; Shu, 6]: D S s (9) S P( D, S s) Smlrly, the ontonl return pero S D for rought severty gven rought urton exeeng ertn vlue n e efne s [Shu, 3; Shu, 6]: S D () D P( D, S s) he ontonl return peros re shown n Fgure 4. For exmple, gven the rought severty s> 4 fs month, the ontonl return pero of the rought urton exeeng month s roun yers. 4 Summry n Conluson A vrte struton se on entropy theory s propose for onstrutng the jont struton of rought urton n severty. he vntge of the propose metho s tht t s flexle to norporte fferent forms of mrgnl strutons of the rought urton n severty. Drought t efne y the monthly stremflow t Brzos Rver t Wo, exs re use to llustrte the pplton of the propose metho for rought nlyss. A goo greement s oserve etween the emprl n theoretl proltes of the rought urton n severty. he vrte struton s then pple to moel rought urton n severty jontly. Return peros of rought urton seprtely n jontly n the ontonl return peros re then estmte. he results show tht the propose metho s useful tool to erve the jont struton of rought urton n severty for rought nlyss. Referene González, J. n J. Vlés (3), Bvrte rought reurrene nlyss usng tree rng reonstrutons, Journl of Hyrolog Engneerng, 8(), Jynes, E. (97), Informton heory n Sttstl Mehns, Physl Revew, 6(4), Kpur, J. (989), Mxmum-entropy moels n sene n engneerng, John Wley & Sons. Kenll, D. n J. Drup (99), On the generton of rought events usng n lterntng renewl-rewr moel, Stohst Hyrology n Hyruls, 6(), -68. Km,., J. Vles n C. Yoo (6), Nonprmetr pproh for vrte rought hrterzton usng Plmer rought nex, Journl of Hyrolog Engneerng, (), Km,., J. Vlés n C. Yoo (3), Nonprmetr pproh for estmtng return peros of roughts n r regons, Journl of Hyrolog Engneerng, 8(), Mther, L., L. Perreult, B. Boée n F. Ashkr (99), he use of geometr n gmm-relte strutons for frequeny nlyss of wter eft, Stohst Hyrology n Hyruls, 6(4),

5 Drought Severty( 4 fs months) Prolty Prolty PDF PDF Me, L. n N. Ppnolou (984), Mxmum entropy n the prolem of moments, J. Mth. Phys, (8), Nrjh, S. (7), A vrte gmm moel for rought, Wter Resour. Res., 43, W8, o:.9/6wr64. Nrjh, S. (9), A vrte preto moel for rought, Stohst Envronmentl Reserh n Rsk Assessment, 3(6), Sls, J., C. Fu, A. Cnellere, D. Dustn, D. Boe, A. Pne n E. Vnent (), Chrterzng the severty n rsk of rought n the Poure Rver, Coloro, Journl of Wter Resoures Plnnng n Mngement, 3(), Shnnon, C. E. (948), A mthemtl theory of ommuntons, Bell Syst. eh. J., 7(7), Shu, J. (3), Return pero of vrte strute extreme hyrologl events, Stohst Envronmentl Reserh n Rsk Assessment, 7(), 4-7. Shu, J. (6), Fttng rought urton n severty wth twomensonl opuls, Wter resoures mngement, (), Shu, J., S. Feng n S. Nrjh (7), Assessment of hyrologl roughts for the Yellow Rver, Chn, usng opuls, Hyrologl Proesses, (6), Shu, J. n H. Shen (), Reurrene nlyss of hyrolog roughts of fferng severty, Journl of Wter Resoures Plnnng n Mngement, 7(), 3-4. Shu, J.. n R. Morres (9), Copul se rought severty urton frequeny nlyss n Irn, Meteorologl Appltons, 6(4), Sngh, V. P. (998), Entropy-Bse Prmeter Estmton n Hyrology, Kluwer Aem Pulshers. Yevjevh, V. (967), Ojetve pproh to efntons n nvestgtons of ontnentl hyrolog roughts, Hyrology Pper 3, Coloro Stte U, Fort Collns. Zelenhs, E. n A. Slv (987), A metho of stremflow rought nlyss, Wter Resoures Reserh, 3(), 6-68, o:.9/wr3p Drought urton (months) Drought severty ( 4 fs months) Fgure Comprson of the hstogrms n ftte PDFs Oserve heoretl 3 4 Drought urton (months) Oserve heoretl Drought severty ( 4 fs months) Fgure Comprson of the emprl n theoretl prolty Drought Durton (months) Fgure 3 Bvrte rought urton n severty return peros

6 Return pero (yers) Return pero (yers) s>. s> s>. 3 > > > Drought Durton (months)... 3 Drought Severty( 4 fs months) Fgure 4 Contonl return peros le Return pero efne y rought urton n severty seprtely Return Pero (Yer) Drought urton Drought severty (Month) ( 4 fs months)