バイアス補正を考慮するカルマンフィルタを導入した実時間流出予測

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1 京都大学防災研究所年報第 5 号 B 平成 19 年 4 月 Auals of Disas. Prev. Res. Ist., Kyoto Uiv., No. 5 B, 27 * ** * ** (, 25) 6 (, 26) (1974) ( 1984; 1992; ( ), 26; Wood ad Szollosi- Nagy, 1978; Kitadis ad Bras, 198; O Coell ad Clarke, 1981; Refsgaard, 1997) (198) (1982a, 1982b, 1983) - 1 -

2 (Evase, 1994) ( Moradkhai et al., 25; Weerts et al., 26; Romaowicz et al. 26; Vrugt et al. 27) Kim et al. (27) HBV Wohlig et al. (26) SAC-SMA Seo et al. (23) 6 Fread ad Mig (1993) FLDWAV (21) Shiiba et al. (2) Friedlad (1969) Dee ad Da Silva (1998) Friedlad (1969) (833 km 2 ) ( 25) 1/25, 3km 3km - 2 -

3 25 m ( 21) m ( 24) (, 25) km p p 3.4 j j 1 x j [m3 /s] x j 1 x j x 1 j d j ( 1 ) (, 26) x j = C 1 x j 1 + C 2 x 1 j 1 + C 3x 1 j + C 4 d j (1) C1 C4 cδt/δl 2X C 1 = 2(1 X)+cΔt/Δl cδt/δl +2X C 2 = 2(1 X)+cΔt/Δl 2(1 X) cδt/δl C 3 = 2(1 X)+cΔt/Δl 2cΔt C 4 = 2(1 X)+cΔt/Δl (2) (3) (4) (5) c [m/s] Δt -1 t j-1 c < Δl /Δt c Δt Δl j l Δt -1 t j-1 c > Δl / Δt Δl j Δl / c Fig. 1 Multi-step, Multi-reach stabilizatio method for Muskigum Cuge model ( c = dx I da = m N ) 1 m ( xo B ) m 1 m l (6) m : (=5 /3) i : [rad] N : Maig [m 1/3 s] B : [m] Δt : [s] Δl : [m] x o x o = x 1 j 1 (2) (5) X X = 1 ( ) 1 xo (7) 2 BcIΔl 3.2 Multi-step, Multi-reach (1) Δl Δt C i (i =1,...,4) X (1999) Multi-step, Multi-reach Multi-step, Multi-reach Δl/Δt c Δl Δt C i (i =1,...,4) X Δl Δt ( 3km) 1 c Δl/Δt cδt c Δl/Δt Δl/c Fig. 1 (c <Δl/Δt) Multi-step, Multi-reach - 3 -

4 (1) ((j 1)+cΔt) ((j 1) + 2cΔt) x j (j 1) j x 1 j 1 x 1 j Fig. 1 (c >Δl/Δt) 3.3 Multi-step, Multi-reach Multi-step, Multi-reach Fig. 1 (c <Δl/Δt) Multi-step, Multi-reach c/δt x j (1) (1) x j (8) x j = A 1 x j 1 + A 2 x 1 j 1 + A 3x 1 j + A 4 d j (8) (8) A i (i =1,...,4) A 1 = C l 1 (9) ( ) A 2 = {A 2,l A 2,m = C 1 A 2,m 1 + C 2 1 m 1 l ( ) +C 3 1 m } ; m =1,..., l, A 2, = (1) l { m 1 A 3 = A 3,l A 3,m = C 1 A 3,m 1 + C 2 } l m +C 3 ; m = l, A 3, = (11) l A 4 = C 4 l (12) l (= Δl/cΔt) Multi-step, Multireach t (= cδt/δl) { m 1 A 1 = A 1,t A 1,m = C 3 A 1,m 1 + C 2 } t m +C 1 ; m =1,... t, A 1, =) (13) t d 1 x 1 d 2 x 2 d 3 Dowstream x 3 x 4 Upstream d 4 x 5 d 5 Fig. 2 Example of river chael etwork composed of five segmets A 2 = ( {A 2,t A 2,m = C 3 A 2,m 1 + C 2 1 m 1 ) t +C 1 ( 1 m t ) ; m =1,..., t, A 2, = } (14) A 3,m = C t 3 (15) A 4 = C 4 t (16) Multi-step Multi-reach (8) 3.4 (8) Fig. 2 Figure 2 5 (8) 3 x 3 4 x 4 2 x 2 x 2 = A 2 ( 1 x 3 + x ) ( 4 + A 2 2 x x 1 ) 4 +A 2 3x A 2 4d 2 (17) A A 1 1 x 1 1 A 2 1 A 2 1 x 2 1 x 3 1 A 4 1 x 4 1 x 5-4 -

5 A 1 3 A 1 2 A 2 3 A 2 2 A 2 2 = A 3 3 A 4 3 A 4 2 A 5 3 A 1 4 A A 3 4 A 4 4 A 5 4 x1 1 x2 1 x3 1 x4 1 x 1 d 5 5 d 1 d 2 d 3 (18) d 4 (18) x j x = A x 1 + D d (19) x (p 1 ) d (p 1) x = A x 1 + D d + (2) A D (p p) E{ } = (21) E{ T m} = Q δ m (22) E{ } δ m Q (p p) y (r 1 ) H (r p) y = H x + (23) E( ) = (24) E( T m )=Rδm (25) R (r r) (2) (23) Kalma(196) (2) D d D d ( ) (2) (23) (p 1 ) (26) (27) x = A x 1 + D d + B + (26) y = Hx + C + (27) B (p p) C (C : r p ) 4.2 x z =[x T T ]T (2p 1 ) z (Jazwiski, 197) ẑ =Φ ẑ 1 + D d (28) P z () =Φ P z( 1)Φ T + Q (29) [ ] 1(3) K z() =P z ()L T L P z ()L T + R ẑ = ẑ + K [ z() y L ẑ ] (31) P z() =[I K z()l ] P z () (32) ẑ (2p 1) z K z() (2p r) P z () z z P ( 1) (2p 2p) L (r 2p) D d (2p 1) Q (2p 2p) [ ] A B [ ] = L = H C (33) I - 5 -

6 D d = [ D d 4.3 ] [ ] Q Q = (34) (26) (27) (p 1) p (2p 2p) Friedlad (1969) (2p 2p) x (p p) Friedlad (1969) ( Hsieh ad Che, 1999 ; Igagi, 2) Friedlad (1969) Friedlad (1969) Igagi (1981) (1) (26) ( ) x = A x + D d + (35) y = H x + (36) x = A x 1 + D d (37) P x () =A P x ()A T + Q x (38) [ ] K x() = P x ()H T 1(39) H P x ()H T + R x = x + K x() [ y H x ] 1 (4) P x() = [ I K x()h ] P x () (41) x P x() x ˆX ˆX ˆX = A ˆX 1 + D d + B (42) [ ] ˆX = ˆX + K x() y H ˆX C (43) (2) U, V, S x x ˆX ˆX ˆX = x + U (44) ˆX = x + V (45) U V (p p ) X x (37) (42) (44) (45) ˆX x = U = A ( ˆX 1 x 1 ) + B =(A V 1 + B ) (46) U V 1 U = A V 1 + B (47) ˆX x (4) (43) (44) (45) ˆX x = V = ˆX x + K x() [ ] H ( ˆX x ) C = [ U K x()(h U + C ) ] (48) S (r p ) S = H U + C (49) V U V = U K x()s (5) (47) (49) (5) V 1 U, S, V (44) (45) U V (3) β β ˆ = ˆ 1 (51) r (r 1) r = y H x [ ] = y H ˆX U (52) (52) 1 2 C C ( ) r = y H ˆX C (53) +(H U + C ) (54) - 6 -

7 = S + v (55) (55) v (y H ˆX C ) p v E{v v T } = H P x ()H T + R (56) P β () =P β( 1) (57) K β () =P β ()ST [ ] 1 H P x ()H T + S P β ()ST + R (58) [ ] ˆ = ˆ + K β () r S ˆ (59) P β () = [ I K β ()S ] P β (6) P x () r S (47) (49) (5) x (42) (43) ˆx ˆx (4) ˆx = x + U ˆ (61) ˆx = x + V ˆ (62) ˆP x ˆP xβ (Igagi (1981) ˆP x () =E{(ˆx x )(ˆx x ) T } = P x ()+U P β ()U T (63) ˆP xβ() =E{(ˆx x )(ˆ ) T } = U P β () (64) ˆP x() =E{( xˆ x )( xˆ x )} = P x()+v P β ()V T (65) ˆP xβ () =E{(ˆx x )(ˆ ) T } = V P β () (66) (37) (41) (51) (6) (61) (66) Friedlad (1969) 4.4 P x P β Dee ad Da Silva (1998) P β P β P x (r r) (p ) b E[ x x ] (67) b x x (61) (67) b b E[ˆx x ] E[U ˆ 1 ] (68) = U (69) ˆb = U ˆ U T (7) ˆb = U ˆ U T (71) ˆP b () = U ˆP β () (72) ˆP b () = U ˆP β () (73) (49) (5) V V = U K x()[h U + C ] (74) ( C =) V =[I K x()h ]U (75) - 7 -

8 V (62) ˆx ˆx = x + V ˆ = x +[I K x()h ]U ˆ = x [I K x()h ]ˆb = x + K x()[y Hx ] [I K x()h ]ˆb = x ˆb + K x()[y H ( x ˆb )] (76) x x (37) x 1 x x = A ˆx 1 + D d (77) ˆx 1 x (Dee ad Silva (1999)) b K b () (58) U K b () U K β () K b () =P b ()HT [H P x ()H + H P b ()H + R] 1 (78) (59) U ˆb ˆb = ˆb K b ()(y H x + H b ) (79) ˆx ˆb (63) (79) ˆP x () = P x ()+P b () (8) P x P b ˆP x () P b P b P x (81) P b P b () =γ ˆP x () P x () =(1 γ) ˆP x () (81) γ P b P x 1 γ 1 γ γ x = A ˆx 1 + D d (82) ˆb = ˆb 1 (83) ˆx = x ˆb (84) P x () =A P ()A T + Q x () (85) ˆP x () = 1 1 γ P x () (86) P b () = γ 1 γ P x () (87) K b = P b ()HT [H P b HT +H P x H T + R ] 1 (88) ˆb = ˆb K b [y H ( x ˆb )] (89) K x = P x ()H T [H P x ()H T + R ] 1 (9) x =( x ˆb ) +K x[y H ( x ˆb )] (91) P x() =[I K x()h ] P x (92) - 8 -

5. 1 Katsura River Basi (833 km 2 ) Lake Biwa Step 1 1 Step 2 d Step 3 x Step 4-1 ˆb 1 (= ˆb

+1Step 6 Step 9 Step 7 +2 6. 6.1 ( : 833 km 2 ) 1992 1993 Fig.

Kameoka Yodo River Basi (7,281 km 2 ) 1 km Teryuji Katsura Fig.

9 5. 1 Katsura River Basi (833 km 2 ) Lake Biwa Step 1 1 Step 2 d Step 3 x Step 4-1 ˆb 1 (= ˆb ) (ˆx = x ˆb ) Step 5 P x () ˆP x () P b () Step 6 y Step 7 (88) (92) Step 8 Step 2 Step 7 +1Step 6 Step 9 Step ( : 833 km 2 ) Fig ( ) (1992 ) Gaugig Sta. Kameoka Yodo River Basi (7,281 km 2 ) 1 km Teryuji Katsura Fig. 3 Locatios of gaugig statios i the Katsura River Basi (1993 ) ( : 1 ) ( : 1 ) 6.2 (2) 1 [m 6 /s 2 ] 1 [m 6 /s 2 ] γ.5 ( 25) 6.2 (1) Figure , 1, 2, 3 () - 9 -

10 '92, Katsura, -hr '92, Katsura, 1-hr Kalma filter Bias '92, Katsura, 2-hr Kalma filter '92, Katsura, 3-hr Kalma filter Kalma filter Fig. 4 Aalysis (-hr) ad predictios with 1, 2, 3-hr lead time for 1992 evet at Katsura '93, Katsura, -hr '93, Katsura, 1-hr Kalma filter Bias '93, Katsura, 2-hr Kalma filter '93, Katsura, 3-hr Kalma filter Kalma filter Fig. 5 Aalysis (-hr) ad predictios with 1, 2, 3-hr lead time for 1993 evet at Katsura (Kalma filter) () (Bias) - 1 -

11 '92, Teryuji, 3-hr Kalma filter (two statios) Kalma filter (two statios) Simulate '93, Katsura, 3-hr (g =.25) 45 (g =.5) (g =.75) 4 Bias (g =.25) Bias (g =.5) 35 Bias (g =.75) Fig. 6 Predictio results with 3-hr lead time. Compariso of the results filtered with observed discharge oly at Katsura ad both at Katsura ad Kameoka (two statios). Fig. 7 Effect of the parameter γ o predictio result with 3-hr lead time ad bias estimatio 4 7 m 3 /s 1 m 3 /s Fig (2) Figure 6 3 5km 1 km 4 (3) γ γ

12 '93, Katsura, Augumeted Filter hr 1-hr 2-hr 3-hr Fig. 8 Predictio results for 1993 evet at Katsura with Adaptive Kalma Filter.5 γ Figure γ γ P b γ (Bias) γ γ =.5 (4) 4. Figure Fig. 4 (Kalma filter) 3 ( )

13 (B) ( : ) ( ) (26) :,. (21) :,, No. 691 / II - 57, pp ( ) (26) : [ ]. (25) :,, No. 83 / II - 73, pp (198) : - kiematic wave -,, 23, B-2, pp (1982a) :,, 25, B-2, pp (1982b) :, 25, B-2, pp (1983) :,, 26, B-2, pp (1984) : 28, pp (26) :,, 49, B, pp (24) :,, 48, pp (1992) :,, 36, pp (1974) :,, 221, pp (21) :,, 45, pp (1999) : Multi-step, Multireach Muskigum-Cuge, Vol. 12, No. 5, pp Dee, D.P. ad Da Silva, A.M. (1998) : Data assimilatio i the precese of forecast bias, Q.J.R. Meteorol. Soc., 124, pp Evase, G. (1994) : Sequetial data assimilatio with a oliear quasi-geostrophic model usig Mote Carlo methods to forecast error statistics, J. Geophys. Res., 99, pp. 1,143 1,162. Freed, D. L. ad Mig, J. (1993) : A Kalma filter ehaced real-time dyamic flood routie model. I Proceedigs of XXV Cogress of Iteratioal Associatio for Hydraulic Research, Tokyo. Friedlad, B. (1969) : Treatmet of bias i recursive filterig, IEEE Tras. o Automatic Cotrol, Vol. 14, No. 4, pp Hsieh, C. ad Che F. (1999) : Optimal solutio of the two-stage Kalma estimator, IEEE Tras. o Automatic Cotrol, Vol. 44, No. 1, pp Igagi, M. (1981) : A alterate derivatio ad extesio of Friedlad s two-stage Kalma estimator, Vol. 26, No. 3, pp Igagi, M. (2) : Optimal ad suboptimal separate-bias Kalma estimators for a stochastic bias, Vol. 45, No. 3, pp Jazwiski, A. H. (197) : Stochastic processes ad filterig theory, Academic Press, New York. Kalma, R. E. (196) : A ew approach to liear fil

14 terig ad predictio problems, J. of. Basic Egieerig, 82, pp Kim, S., Tachikawa, Y. ad Takara, K. (27) : Applyig a recursive update algorithm to a distributed hydrologic model, Joural of hydrologic egieerig, ASCE, Vol. 12, No. 3, pp Kitadis, P. K. ad Bras, R. L. (198) : Real-time forecastig with a coceptual hydrologic model 1. aalysis of ucertaity, Water Resour. Res., Vol. 16, No. 6, pp Moradkhai, H. M., Hsu, K. L., Gupta, H. ad Sorooshia, S. (25) : Ucertaity assessmet of hydrologic model states ad parameters: sequetial data assimilatio usig the particle filter, Water Resour. Res., Vol. 41, W512. O Coell, P. E. ad Clarke, R. T. (1981) : Adaptive hydrological forecastig - a review, Hydrol. Sci. Bull., Vol. 26, No. 2, pp Refsgaard, J. C. (1997) : Validatio ad itercompariso of differet updatig procedures for real-time forecastig, Nord. Hydro., Vol. 28, pp Romaowicz, R. J., Youg, P. C. ad Beve, K. J. (26) : Data assimilatio ad adaptive forecastig of water levels i the river Sever catchmet, Uited Kigdom, Water Resourc. Res., Vol. 42, W647. Seo, D. J., Kore V. ad Cajia, N. (23) : Real-time assimilatio of radar-based precipitatio data ad streamflow observatios ito a distributed hydrological model, Proc. of symposium HS3 held durig IUGG23 at Sapporo, July 23, IAHS Publ., o. 282, pp Shiiba, M., Laureso, X. ad Tachikawa Y. (2) : Real-time stage ad discharge estimatio by a stochastic-dyamic flood routig model, Hydrol. Process., 14, pp Weerts, A. H. ad El Serafy, G. Y. H. (26) : Particle filterig ad esemble Kalma filterig for state updatig with hydrological coceptual raifall-ruoff models, Water Resourc. Res., Vol. 42, W943. Wohlig, T. ad Leartz, F. ad Zappa M. (26) : Techical ote: updatig procedure for flood forecastig with coceptual HBV-type models, Hydrol. Earth Syst. Sci., Vol. 1, pp Wood, E. F. ad Szollosi-Nagy, A. (1978) : A adaptive algorithm for aalyzig short-term structual ad parameter chages i hydrologic predictio models, Water Resourc. Res., Vol. 14, No. 4, pp Vrugt, J. A. ad Robiso B. A. (27) : Treatmet of ucertaity usig esemble methods: compariso of sequetial data assimilatio ad Bayesia model averagig, Water Resourc. Res., Vol. 43, W1411. Real-time Flood Forecastig Icorporatig Kalma Filter with Bias Correctio Takahiro SAYAMA, Yasuto TACHIKAWA*, Tomoyuki HIRATA** ad Kaoru TAKARA *Graduate School of Urba ad Eviromet Egieerig, Kyoto Uiversity **Nomura Research Istitute Syopsis As a data assimilatio method of a distributed raifall-ruoff flood predictio system ad river discharge observatio data, this study proposes a filterig method of Muskigum Cuge river routig models. Applicatio of the covetioal Kalma filter to river routig models is ot effective because hillslope models have sigificat impact o the flood predictios. I order to overcome this problem, the proposed method estimates biases iduced by raifall-ruoff models as well as state variables i the filterig algorithm, so that the filterig has effect o the predictios with lead time of few hours. Demostrated flood predictios at the Katsura river basi show that the bias correctio improves the accuracy. Keywords:distributed raifall-ruoff model, Muskigum Cuge, flood predictio, data assimilatio

京都大学防災研究所年報第56号(平成24年度)

京都大学防災研究所年報第56号(平成24年度) 京都大学防災研究所年報第 56 号 B 平成 25 年 6 月 Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 56 B, 203 Upgrading the Integrated Hydrological Model for the Kiso River System (II) Yoshinobu SATO, Motohiro HONMA