Application of an improved adaptive importance sampling method in extreme hydrological analysis

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1 Š ñ { d ñ ` ` U Û M Vol.32, No Sstems Engineering Theor & Practice Oct., 2012 : (2012) : TV122.5 : A "!#$%&'(*)+*,-*.*/02143#*5 687:9, ;:<>= (@ACBEDFCBGEHIJHKLFMNCOQP, RS ) T U VXWZYZ[]\X^]_X`XacbZdXe]fZgZhZiX, k]lzmznzozpzqzrzs, t]u L- v, wzx hz zz{z Z}Z~ pzqzz `] X, ƒz ZnX ]Z `]ˆX ]ŠZ. k]œz ZŽZ Z Z Z 24h {Z Z rzs, š œ Ÿšœ š p 213mm `œš { 200mm. ª «š ^œ_š`ša b dšeœf g h iš œ ± ² ³š ÂœÃ eœf g hš`œä Å Æ Ç VšÈ nš œ, µ š œ¹ g h `œº» ¼ ½ ¾ ~šàœ `. MC,. É Ê, Ë ^œ_š`šaìb dšeœf g h iš]íš ]Î lzï ÐZÑ p qšòœ. Ó Ô ÕÖ^œ_ aìb dšeœf g hš ; l m n o p q ³š ; nš œ ; L- v Application of an improved adaptive importance sampling method in extreme hdrological analsis TONG Yang-bin, XU Yue-ping (Institute of Hdrolog and Water Resources Engineering, Zheiang Universit, Hangzhou , China) Abstract This paper proposes an improved adaptive importance sampling method that can be used for variables following a GEV distribution. The L-moment method is emploed to build the relationship between sample statistics and distribution parameters. In the case stud, the 24-hour design storm of Yungang Basin in Zheiang Province is considered, and the probabilit of extreme precipitation is calculated for two rainfall gauge stations, Jinzhuling and Xianrentan. The results show that the proposed method performs well in simulating extreme rainfall, and the iterative number decreases as the number of samples increases. Compared with traditional MC simulation, the improved adaptive importance sampling method has better efficienc. Besides, this method has huge potential to be used for other distributions. Kewords improved adaptive importance sampling; GEV distribution; extreme hdrological events; L- moment method 1 ØÚÙ Û Ü Ý ß à á â ã ä å æ Ý Û, è é ê ëíìïî ðíìïñ òíìïó ôíìïõ öíìï ø ù ú û ü ý, þ ÿ Ý Ý, 98 ñ ì 2008 ì ó ô ù Û Ü Ý á â ã ä ß., ñ ò! "#$ % & ' ( ) * +, Û. -., / , ó : ; Ý á â < ä = C Û Ü Ý á â ã ä [1]. ñ D ð@e F@G@H I@H J@K@L M@N OZ F P Q. R S ð T U. V J K W X Y Z [ \ ] ^@_ ` á âzã ä å az Ý d. b / c O e f, Y g h i & k [2 5] ìml n o p M N [6] ì q r [7] ù s ú t, T u U v w x l n z ã ä. / ñ D { ü ý, R S u U / GLUE J } ~ é V M N [8 9], O t! R S Û Ü / U, ƒ ß v 2 Š@ Œ@ Ž@ [10]. ñ D F Û ÝZ w x ú. T T U. Ž Û ; M N, O M Š } Pearson- Cì h Pearson- Cì e _ Û, ~ é -. U ú 4 h Kappa M Šíì 5 h Wakeb M Š ù ; ` š œ L 2 c 9 Ÿ : : ª «( ); ± ² ³ ± µ (2010DFA24320); ¹ ² º» ¼ ½ ¾ ( ) À Â Ò : Ã Ä Å (1984 ), Æ, Ç È É Ê Ë, Ì», Í Î Ï Ð : Ñ Ñ Ó Ô ; Õ Ö µ : Ø Ù (1975 ), Ú, Ç È Û Ü Ë, Ý, Í Î Ï Ð : Ñ Ñ Ó Ô, uepingxu@zu.edu.cn.

2 «é _ / d ß ß ß 2346 ßáàáâäãáåáæ Û Ü M Š W ê U R S J K ñ D Û M Š ò, ó ô î R S / ñ D 2 õ öùøùúùûùüäýáþáÿ U 2.1 Û Ü Ý Ý Û Ü Òááè 32 w x. ë D ì í î é ï Ž ð ñ d R S, L ~ e d M N U. á â ã ä Ý ( ) w x, F O Monte Carlo (MC), ð E I H R S Ž Û Ü Ý á â ã ä æ J K. v, U MC J K W S Z ë X l, Û î F G. ) û!, Y g R S Ü " íì R S Eì M # R S ù ù. R S ò U p 2 $ (change of measure) Û Ü Ý á â ã ä, ò U % M Š Ž R S, & ' ( ( ) z F. Œ I H 2 x ] ) ã ä * ß p + h f(x),, w(x) = f(x)/g(x), w(x) - + h, ÿ. /102+ h, g(x) ÿ R S (importance sampling, 3 4 IS) * p : ; < % = > * @2A1B2C D E F (G H I J K L M E F ) N O P Q, R S T U V W X E F 8 Y Z [, \ ] ^ _ ` T D > [10]. ` T 8 c d e f g h : i s(x i ) r k l m H I J K, n = > p = P {s(x i ) r} o p. q W r s t 5 I {s(xi)}, < u v w I {s(xi)} = g(x) z { f(x), } ~ M N ƒ 8 T U, n l 8 ˆ k { 1, x s(x i ) r 0, x s(x i ) < r p = I(x)f(x) dx = I(x)w(x)g(x) dx (2) ˆp IS = 1 N N I{s(X i ) r}w(x i ) (3) d Š 8 Œ ; g Ž 8 = > t 5 g(x), ^ _ ` T D > š œ 8 IS e f1@ i, IS ƒ Ÿ m 5 g(x θ), 51B2 θ < e p. e f k ª, ˆ. Bucher [11] Š Y P ƒ 1@2^ ± ² m ³ µ ` T (adaptive importance sampling, AIS) e f. Stadler Ro [12] ¹ ³ºµ ` T» k 5 e f ¼ 5 ef ½¾, À 5 e f ƒ k ½ : KÂ (Ãe ) f KÄ5 f, ÅÆÇÈ8 ÉÊÌËÍ½ f ˆ8 5, G KÂ KÄ [13] 5, Î Z Ï Ð ` T t Ñ Ò Ó Ô Õ K Ä 5 ˆ f Ö ØÇ, Ù X f Ú ~ M mûnyüýtu, ÛTUÚ ˆßà8_áâ t58 5, Ö ãzäå, u ÉÊÌËÍ e f p = > J K ; o æ D è é ê ë b í î K Â f 8 ì g h : 1) Ù l ß à Â µ init Ú { ï ð ` T (ñ ò ó ô ƒ ) 8 Â µ 0, Å Æ ~ M m Û (n ) T U x; ± V W H I E F 8 T U xe, u 5 ne Ê s ; õ ns = 1 Ê s ö 1 Å Æ, ˆ Â µ is (1)øùe σ is (1) = > P is (1). ú û ü ý [12] 8 þ ÿ, K Â øùe = > 8 g h : ˆµ = ˆσ 2,f = 1 p w(n i ) 1 N f 1 N f (1) p n i w(n i ) (4) (x f,i ˆµf )2 (5) ˆp f = 1 N N g(x i ) (6)

3 i p h 8 N N N + a b p 8 10, :! ) õ " # ns = ns + 1, Â e $ k % 8 ` T t Ú Ö ö ns Å Æ, Å Æ & K Â a µ is (ns)øùe σ is (ns)øù = > P is (ns). " # í î b + 3) ª, ' ( ) * è, - ë K Ä 5 f ; T U 8 Ä 5 ˆ z { ² Â, Ä 5 Â e 8 ˆ $ k m Å Æ. ` T 8 5, /. Ö h m Å Æ, Ã K Â f 4. K Ä 5 f 8 ð 5 ; : ó ô ƒ 1@76 æ 8 m 8, 9 : Z < t 5 ;1@2 H I J K æ Y Z L M 8 E F, \ ] < Ï Ð Â. 3 =>@BADCEGFIHIJGKILIM N O P Q i 8 ; m R [ 8 ³ºµ ` T ef, u N w S T Š } U V \ óô ƒ, ] d c1@7w X U a [13 15] [16 17] ; N V \ ó ô ƒ 8, Y u ; Z [ 9 \ ] 8 ^ ü U, g _ ` ø N ˆ b c Ô Ô. Š H I d c J K 8 e > ƒ 1@, f g H ƒ (generalized extreme value, GEV) ; 7 8 m, Æ ) D É h N [18 20]. f g H ƒ 8 = > t 5 Ã ƒ t 5 g h : u1@, = f(x) = α 1 e (1 k) e (7) F (x) = e e (8) { k 1 log{1 k(x ξ)/α}, k 0 (x ξ)/α, k = 0 GEV ƒ ƒ k k l 5 køm 5 α n o 5 ξ. k < 0, ; GEV- qºâ ƒ, r s k Frechet ƒ p ; k > 0, ; GEV- t2â ƒ, r s k Weibull ƒ p ; k = 0, ; GEV- u â, h s k Gumbel ƒ. O v Õ f g H ƒ, ^ ± ² w Ö 8 ³ºµ ` Tf. U k ó ô ƒ, ˆ T U 8 Â e í z ˆ., x & ² % t 5 8 5, /. Ö h m ; { Š H ƒ 1@, ˆ T U 8 Â e., N Z & t 5 8 5, } ~ m % 8 ˆ e f, U ü s S k K 5 f., H ƒ 8 Ä 5 N Ï Ð ƒ 8 Â ( m Ã ó ô ƒ N ), } h N ƒ K Ä 5 f. b í î K 5 f 8 ì g h : 1) ` T. Ù ß à 8 f g H ƒ Å Æ ~ M m Û (n ) T U x; ± V W H I E F 8 (ne ) T U + xe; /. 6 2 H ƒ 5 Ã T U ˆ 8 \ ƒ 5 kø αø ξ. N O ˆ + K Â f K Ä 5 f Î ; Šf ƒ 5 +, T ƒ 5 q W ² L- Š f [21]. \ L- Š f 8 U = Œ ø g 4 Ž L- Š ˆ 8 ˆ Y ü ý [21 22]. & 4 Ž L- Š ˆ., xy : TU8 U 5 (L-CV)ø ô 5 (L-skewness) ` 5 (L-kurtois), u ˆ g h : τ = l 2 /l 1 τ 3 = l 3 /l 2 τ 4 = l 4 /l 3 ] L- Š ø L-CVø L-Skewness L-Kurtois Ã H ƒ 5 8 \ g h [21] : ˆl 1 = ξ + α[1 Γ (1 + k)]/k ˆl 2 = α(1 2 k )Γ (1 + k)/k ˆl 3 = 2(1 3 k )/(1 2 k ) 3 ˆl 4 = [5(1 4 k ) 10(1 3 k ) + 6(1 2 k )]/(1 2 k ) (11)H ƒ 8 5, $ k Å Æ 8 ß à 5. 2) õ ns = 1 Ê s ö 1 Å Æ, í î ß à 5 $ k ` T = > ƒ 8 5, M n T U, ± u1@28 H I T U, (10) ø (11)ˆ H ƒ 8 5, ƒ k k is (1)ø α is (1) ξ is (1); ] ö 1 Å Æ. H I J K 8 = > p is (1) (6)ˆ. 3) ns = ns+1, í î ˆ & ± 8 5 $ k % 8 ` T t Ú Ö ö ns Å Æ. k d Å Æ É 8 [, ƒ m Å Æ 8 É, À ö ns Å Æ ˆ 8 5 = > k k i (ns)ø α i (ns)ø (9) (10) (11)

4 F a " b ³ Ĩ IšG IœI IIŸI ξ i (ns) p i (ns). ] ö ns Å Æ. ` T t s k k is a (ns)ø α is (ns)ø ξ is (ns) p is (ns). í î b + 4) ª 3), ' ( ) *, Ï Ð O û Y ó 2 8 z K Â f. 32 [13], 5 = > Ê ª «- ± ² ³ F T µ ¹ º» ¼, k % ¹ ½, ³ O ¾ ³ F 251.8km 2, ì Â ³ F Ÿ Ã Ä Å Æ Ç â È É, c Ê Ë, d ^ ;1@, ; ¹ º b c1@2a S m. U ü \ ] 8 } U k 24 p [22] d c x, ] x > a ; m H I d c J K, a k m r d c. W Ì Í Ô ± ² ³ L- Š f ˆ F 8 ˆ b c, & ² N d Î 8 ˆ b c, Ï Ð u É Ê1Ë 24 p ˆ b c V \ f g H ƒ. U ü Ñ ] i ³ 24 p ˆ b c 8 5 Ò ƒ t Ó k f g H ƒ, Ú ƒ w Ö ³ºµ ` T f h G K Ó ± ² ³ f Å Æ H I J K 8 Y [. Ê 1 9 s k F Ô Õ Ö Ø Ù ½ c Ú 8 ß à ƒ Ó ø a 8 Û x > a m J K Š 5 Ò 8 = >. ` T T U Ó n ƒ Û 200ø 500ø 1000ø 2000; Å Æ 8 Ü W Ý Û ß àáâ ã ä å æ è é ê ë ì í î ï ð ñ ò ó ô õ ò ó ö ò ó a ú û ü ý þ ÿ œ ø ù E E-3 62 km, À Â d 1.45%. 4.2 Õ ½ Ú ø N ` T Ó Î Ö ² ½ Å Æ. Ê 2 Ê 3 ; K Ó f Å Æ 8 ˆ É, ± ² \ ] 8 Ó, f Ï Ð H ƒ 8 Ó, : H I J K 8 = > Ô. 2 ß àáâ ã ä å æ è é ó ó ð ñ ò ó ô õ ò ó ö ò ó ø ù (%) E E E E E E E E ß àáâ ã ä å æ è é ó ó ð ñ ò ó ô õ ò ó ö ò ó ø ù (%) E E E E E E E E Ê1@2ö 2 ; f 9 ~ 8 z Ó, Y :1Ë ± : ` T Ó, 9 ~ 8 z Ó. Õ H ƒ 8 Ó, k l Ó Š Å Æ 8 U"!, ] m Ó n o Ó 4 Õ"#. $ 8 Ý ; r Å Æ 8 = > Ã Ê Ò = > 8 4 Õ Ý p, f %, 4 Õ Ý Î p 0.5%, Ê1Ë2Å Æ = > ¼ 7 Ü Q 5 Ò = >, h G w Ö ³ºµ ` T f Z o & Å Æ H I J K 8 L M. 6 2('2 h Y ),

5 «í X c b " ƒ X ª V X & 10, :! 2349 ` T Ó, Å Æ N, g n k ½ É 4 o p ; ` T Ó k 2000, Å Æ É. 4.3 * +1 2, - k ² Ö m È. w Ö 8 ³ºµ ` T e f 8 D É ˆ, 8 Monte Carlo (MC) e f, G } ` T e f Ú Å Æ X H I J K, Ï Õ ½ e f 8 Å Æ É Ö ('2. Å Æ T U Ó \ 1000 / à, /. ƒ Û 1 0 ø 10 0 ø 50 0 ø ø Å Æ É g Ê 4 9 s. \ Ê : ±, Å Æ = > 8 4 Õ b 5 Ý T U Ó8 R S] 1p, { 1 p u 3 4. g É Ý Š 1% : h 6 ` T 7 É, N O < MC f 8 ~ : 6 9 :, ; ; _ Ó f 9 ~ 6 9 : Ó.= F : ü(>6 24 p 2 d A B"6 U, MC f ` Û C 0 9 : D E"6 Ï"G H, I ª < U V W X (>6 U, J K L M N 6 O P. Q R S T N N 1 Y Z [ 9 \, ] ^ _ ` a [ 9 7 b. 4 defghi MC klmnopq rstu vwx z{ } ~ } ~ (%) (%) 5 ƒ X U V W X W œ Z ˆ Š Œ Ž [ T š, S Ÿ ª «b, ] ± ² ³ µ \ ¹ º» ¼, ½ ¾ Ž S À Z a, Â Ã K Š Œ Ž [ Ä Å X MC À(ÆÇ. È É Ê(Ë, Ì ˆ Š Œ Ž [ _ M Í Î Ÿ Ï Ð Ð Ø, ] ^ N N Ñ Ò Z [ Ó b, Ž Ô Õ Ÿ (Ö S. Ù"Ú U"V"W"X «" "U"V ", T" "Û"Ü"Ý" ""ß"à"á Å"š "â"ã, ä"m"å" "Ž" ", æ""_ ª ««ï V «ö S è Ö, é ê S è ë ì í î. ð Ã ñ ò(öé K ó ô K ë S õ ë ì Ö. ø ù ú û [1] üý, þÿ, Gemmer M. [J]., 2008, 19(5): Jiang T, Su B D, Gemmer M. Trends in precipitation extremes over the Yangtze River Basin[J]. Advances in Water Science, 2008, 19(5): [2] Lu D, Yao K. Improved importance sampling technique for efficient simulation of digital communication sstems[j]. IEEE Journal on Selected Areas in Communications, 1988, 6(1): [3] Sadowsk J S, Bucklew J A. On large deviations theor and asmptoticall efficient Monte Carlo estimation[j]. IEEE Transactions on Information Theor, 1990, 36(3): [4] Chen J C, Lu D Q, Sadowsk J S, et al. On importance sampling in digital communications Part I: Fundamentals[J]. IEEE Journal on Selected Areas in Communications, 1993, 11(3): [5] Sadowsk J S. On the optimalit and stabilit of exponential twisting in Monte Carlo estimation[j]. IEEE Transactions on Information Theor, 1993, 39(1): [6] De A, Mahadevan S. Ductile structural sstem reliabilit analsis using adaptive importance sampling[j]. Structural Safet, 1998, 20(2): [7],,. r! " # $ % & ' [J]. ( ) $ % *, 2007, 19(18): Zhou H, Qiu Y, Wu X J. Rare event simulation method based on importance sampling technolog[j]. Journal of Sstem Simulation, 2007, 19(18): [8] Kuczera G, Parent E. Monte Carlo assessment of parameter uncertaint in conceptual catchment models: The Metropolis algorithm[j]. Journal of Hdrolog, 1998, 211(1/4):

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LA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce > ƒ? @ Z [ \ _ ' µ `. l 1 2 3 z Æ Ñ 6 = Ð l sl (~131 1606) rn % & +, l r s s, r 7 nr ss r r s s s, r s, r! " # $ s s ( ) r * s, / 0 s, r 4 r r 9;: < 10 r mnz, rz, r ns, 1 s ; j;k ns, q r s { } ~ l r mnz,