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1 47 9 Sherman(1932) Horton 9.1 A A Sherman
2 : Q Q(j) =Σ j i=0uh(j i)r e (i) (9.1) R e UH i, j t i j 10mm 9.1 (9.1) Σ L i=0uh(i) = 1 (9.2) UH(i) 1 (i =0, 1,,L 1,L) (9.3) L (L +1) t (9.1) R e (0) R e (1) R e (0) 0 0 R e (2) R e (1) R e (M) R e (M 1) R e (M) UH(0) UH(1).. UH(L) = Q(0) Q(1).. Q(N) (9.4)
3 9 49 M N N = M + L 0 UH(0),UH(1),,UH(L) UH(L),UH(L 1),,UH(0) Collins(1939) Collins R e UH = Q (9.5) R e (M +1,L+1) UH L +1 Q M + L +1 over-determined ( ) SE =(Q R e UH) T (Q R e UH) (9.6) UH UH =(R T e R e ) 1 R T e Q (9.7) T (9.7) (9.2) (9.3) 1 R e Q (9.2) (9.3) Bras (1990) S t S S S t S S(j t) = Σ j i=0uh(i) (9.8)
4 9 50 t 1 1 S t UH (0) = 1 S(0) t (9.9) UH (i) = 1 t (S(j t ) S((i 1) t )) (i =1, 2,,L ) (9.10) t 1 t t t A t R e Dirac delta IUH(t) S S(t) = t t UH(i) = 1 t (S(i t) S((i 1) t)) = i t 0 IUH(τ)dτ (9.11) (i 1) t IUH(τ)dτ (9.12) Synder(1938) U.S. Soil Conservation Service(Mockus,1957)
5 : U.S. Soil Conservation Service t/t p UH/Q p t/t p UH/Q p Synder t p [hr] Q p [m 3 /s] t[hr] t L [hr] t p = C t (LL c ) 0.3 (9.13) Q p = C pa t p (9.14) t = t p (9.15) 5.5 t L = 72+3t p (9.16) C t C p L [km] L c [km] A [km 2 ] t p 3 5 (Bras, 1990) U.S. Soil Conservation Service(Mockus,1957) t p [hr]
6 9 52 Q p [m 3 /s] t/t p vs UH/Q p t p Q p t p = D 2 + t l (9.17) Q p = 35.5 A t p (9.18) D t l Texas t l =2.55A 0.6 Ohio t l =0.96A 0.6 ( ) LLc b t l = a (9.19) (i 0 ) i Horton Horton Strahler (1952) Horton
7 9 53 ω d = max(ω u 1,ωu 2 + 1) (9.20) ω d ω u ω u 1 ω u 2 (1993) DEM ω d = max(ω u 1,ω u 2 +1, ω u m) (9.21) m ω u 1 ωu 2 ωu m (, 1975;, 1978; Scheidegger, 1965) Horton-Strahler Horton Strahler N i = R k i B (9.22) L i = R i 1 L L 1 (9.23) A i = R i 1 A A 1 (9.24) S i = R 1 i S S 1 (9.25) k i R B,R L,R A,R S N i,l i,a i,s i i Shreve (1966) Horton Strahler 246 Strahler Horton Shreve (1966) Rodriguez-Iturbe and Valdes (1979) i j IUH(t) =Σ n i=1 p i(0) dφ i(t) dt (9.26) n p i (0) i t i
8 9 54 Rodriguez-Iturbe and Valdes (1979) 3 φ 1 (t) = 1+ λ 3(λ 2 λ 1 q 13 ) (λ 2 λ 1 )(λ 1 λ 3 ) e λ 1t λ 3 λ 1 q 12 + e λ 2t (λ 2 λ 1 )(λ 3 λ 2 + λ 1λ 2 λ 1 λ 3 q 12 e λ 3t (λ 3 λ 1 )(λ 2 λ 3 φ 2 (t) = 1+ λ 3 e λ2t + λ 2 e λ 3t λ 2 λ 3 λ 3 λ 2 φ 3 (t) = 1 e λ 3t p 1 (0) = R2 B R 2 A (9.27) (9.28) (9.29) (9.30) p 2 (0) = R B R3 B +2R 2 B 2R B R A R 2 A(2R b 1) (9.31) p 3 (0) = 1 R B R b(r 2 B 3R2 B +2) R A R 2 A(2R b 1) (9.32) q 12 = R2 B +2R B 2 2R 2 B R B (9.33) q 13 = R2 B 3R B +2 2R 2 B R B (9.34) q 23 = 1 (9.35) q ij i j i j 0 λ i i Rodriguez-Iturbe and Valdes (1979) ( Agnese, et al. (1988); Karlinger and Troutman (1985); Karlinger, et al. (1987); Rodriguez-Iturbe, et al. (1979); Gupta, et al. (1980))
9 ds dt = R e Q (9.36) S = KQ (9.37) S (mm) Q [mm/hr] R e [mm/hr] K [hr] t [hr] K dq dt = R e Q (9.38) Q(t) =e t K Re t K /e K dt (9.39) Dirac-delta Q(t) h(t) { 1 h(t) = K e t K t 0 (9.40) 0 t<0 h(t) µ 1(h) µ 1(h) = (9.38) 0 t h(t)dt = K (9.41) K dq dt = Q (9.42) Q(t) =Q 0 e t K (9.43) Q 0
10 (9.36) K Q 2 Q 1 t = R e Q 2 + Q 1 2 (9.44) Q 2 Q 1 t Q 2 Q 2 = C 0 Q 1 + C 1 R e (9.45) C 0 = K t/2 (9.46) K + t/2 t C 1 = (9.47) K + t/ t (9.39) Q(t) =e t Re t K K /e K dt = Ce t K + Re (9.48) C t t =0 Q = Q 1 C = Q 1 R e (9.49) Q(t) =Q 1 e t K + Re (1 e t K ) (9.50) t t = t Q 2 Q 2 = Q 1 e t K + Re (1 e t K ) (9.51) 1 Q 1 2 c g = e t K Q 2 = Q 1 c g + R e (1 c g ) (9.52)
11 Q(t) =Q 1 e t K (9.53) Q 2 = Q 1 c g (9.54) K t c g K K = µ 1 (h) =µ 1 (Q) µ 1 (R e) (9.55) Nash Nash n A Nash (9.40) K (h(t)) h(s) =L(h) = 1 s 1 K (9.56) Nash (h n (t)) h n (s) =L(h n )= h n (t) = 1 (s 1 K )n (9.57) ( ) 1 t n 1 e t K (9.58) KΓ(n) K
12 : Γ(n) n Γ(n) =(n 1)! Nash n K µ 1(h n ) = nk (9.59) µ 2 (h n ) = nk 2 (9.60) nk = µ 1(O) µ 1(I) (9.61) nk 2 = µ 2 (O) µ 2 (I) (9.62) n K t 2 t n t (i 1) t i t A i 1, (i =1, 2,,n)
13 9 59 A(i) =A i /A (i =0, 1,n 1)) (9.63) A 0 t t = j (j 1) t j t A 0 R e (j) (j 2) t (j 1) t A 1 R e (j 1) Q(j) =Σ j i=1a(j i) R e (i) (9.64) (9.64) S S(j t) = Σ j i=0a(i) (9.65) IUH(t) = ds(t) (9.66) dt 9.3.2
14 9 60 S S(t) = IUH(t) = t t L t<t L (9.67) 1 t t L 1 t L t<t L (9.68) 1 t t L t L t L R e t t t L R e 0 <t t L 2t Q(t) = IUH(t τ)r e (τ)dτ = L t t 0 L R e t L t<2t L (9.69) 0 2t L <t Q p t L R e t L m 3 /s Q p = AR e (9.70) R e f Q p = far e (9.71) 40km 2 f t L Kraven ( (1985)) Clark 9.3.1
15 : 1945 Clark Clark Q (t) K 9.4 S ds dt = R e Q (9.72) 9.4.1
16 9 62 (9.72) S j+1 S j t = R e,j+1 + R e,j 2 Q j+1 Q j 2 (9.73) S j+1 t + Q j+1 2 = S j t + Q j 2 + R e,j+1 + R e,j 2 Q j (9.74) H S + Q H Q 9.3 t (1961) S = KQ p l (9.75) Q l (t) =Q(t + t l ) S t l t l K p t l ln S ln Q l (t), K p t l R e t l =0 dq dt = R e Q KpQ p 1 (9.76) Q (t) t l Runge-Kutta
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