역산이론을이용한연안수질모형의매개변수추정 Parameter Estimation of Coastal Water Quality Model Using the Inverse Theory

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1 w w Áw œwz 17 «3 y, pp. 149~157, 역산이론을이용한연안수질모형의매개변수추정 Parameter Estimation of Coastal Water Quality Model Using the Inverse Theory y *Á **Á k** Hong Yeon Cho*, Bum Jun Cho** and Shin Taek Jeong** : x t» l w, n l w, y,, d š w d v w. sww wš, x w s d s w w y yw w x w w. w, ( w) w d w s w y» l w, n l w, y Á w y w,» j ùkû. wr, d w w, RMS d 5.0% w, e 0.95 w w q. w :, x, w, w, Abstarct : Typical water quality (WQ) parameters defined in the governing equation of the WQ model are the pollutant loads from atmosphere and watersheds, pollutant release rates from sediment, diffusion coefficient and reaction coefficient etc. The direct measurement of these parameters is very difficult as well as requires high cost. In this study, the pollutant budget equation including these parameters was used to construct the linear simultaneous equations. Based on these equations, the inverse problems were constructed and WQ parameter estimation method minimizing the sum of squared errors between the computed and observed amounts of the mass changes was suggested. WQ parameters, i.e., the atmospheric pollutant loads, sediment release rates, diffusion coefficients and reaction coefficient, were estimated using this method by utilizing the vertical concentration profile data which has been observed in Cheonsu Bay and Ulsan Port. Values of the estimated parameters show a large temporal variation. However, this technique is persuasive in that the RMS (root mean square) error was less than 5.0 % of the observed value ranges and the agreement index was greater than Keywords : inverse problem, model parameters, pollutant loads, pollutant release rates, pollutant budget equation 1. x y x w Áœ w s d w x. s yw dw š w» w ƒ yw w, w x (calibration & verification processes) v w. p, e x» w w GIGO (Garbage In, Garbage Out;» w»ƒ *w w Áw œw (Corresponding author, Coastal and Harbour Engineering Research Lab., Korea Ocean R&D Institute, Ansan PO Box 29, Seoul , korea. hycho@kordi.re.kr) ** Ÿ w m y œw (School of Civil, Environmental & Urban Engineering, Wonkwang Univ.) 149

2 150 y Á Á k ù ü, ƒ ù x w w l ƒ ù ) w yw x w. y» (»,, twát,,, ), w ( w [ ]) d ƒ w w. w w w ù, w w x w w (w w, 1995, 1996)., x x w» d š w w. d š w, x t» l w, n l w, y,. n l w n w x y dw w š ù, x w wš w (w w, 1997). w,» l w w dw ƒ w š, y ( s y ) x e w ù, x w ƒ., x w w y y dw ww» w x w w (, 2004). wš w x sww wš, w s s yw (inverse problem) w x w w. w, x w, ( w) w d w s w y» l w, n l, y Á w y (transformation rate) w w w Á mw, w» w w w s d s w. 2. (inverse problem) w w (Table 1 ). ƒ œ k ( ) w, d( œ) w x w. x d š w d ƒ w w š. Áw œw Hindcasting, Nowcasting, Forecasting œw w, (simulation, w ƒ w [ œ ] w d) w w. wr, ( ) (even-determined), (under-determined), (over-determined) xk w (Hensel, 1991). ƒ š, w(unique solution)ƒ. Table 1. Classification of the Engineering problems(lee, 2000) Problem Type Input Information (Model parameters) System (Model) Prediction Known Known Output Information Unknown (Computed) Identification Known Unknown Known Detection (Estimation) Unknown Known Known Remarks General problem type (simulation) Forward problem operation vs. design Model building (development) structure vs. parameter Inverse (Backward) problem controllable vs. uncontrollable

3 w x 151 ƒ w w. p (singular value) w w w ù, w p w wš š., ƒ w(multiple solution)ƒ., w w w(optimal solution) w sw, ƒ x š w œw w w. k y w š, w y y wš (Bennett & McIntosh, 1982; Legovic et al., 1990; Scales et al., 2001). w, w w», k x š (Gaspar, et al., 1990; Lee, 1994; Ancey et al., 2003). (2001) w (pipe network) k w w. ù, x w w,, w ww. w xk w, wš w x, w w. 3. x» 3.1 x p x wš w w p tx ù, w w š w (pollutant budget equation) w. w x w w ù,». w œ s s w, w w ù w w x w., w d wš, d(layer), š w w (Fig. 1). s w w w ƒ w w w w, s d w Fig. 1. Schematic Diagram of the Pollutant Budget Model. ww w (w w, 1999a, 1999b; Jorgensen and Gromiec, 1989). w (t 1, t 2 ; t -t = t) w w ƒƒ (d, 2 1 layer) w w. ø td( e»y i=0) ( Á táa S + ÁC 0 Á tá + ( )Á( C 0 - C 1 )/háa S Á t = M 0 (1) ø d( e»y i=nl) ( Á táa S + ÁC NL Á tá +( )Á ( ) /háa S Á t=m NL (2) C NL 1 C NL ø ( d i = 1, 2,... NL-1) C i ( C i 1 + ) ( ÁÁ tá +( )Á 2C i C i + 1 /háa S Á t = M i (3)»,,,, x, ƒƒ» l w (mg/m 2 -day), n l w (mg/m 2 -day), d y w w ( ) y (1/day), y (m 2 t 1 t 2 /day). C i = ( C i + C i ) 2, M i = t 1 t 2 ( C i C i )As h = i d (t 1, t 2 )» s³ s w w y, A S, =ƒƒ d t, d d (m 2 ), h= d Ì(m), t= s (second day). (1), (2), (3) w xk txw (4).

4 152 y Á Á k A s t C 0 t( C 0 C 1 ) h (1), (2), (3) tx 4 ( ) swwš, w ww d j x ( (4) )., wš w w d ww w w. w, w d y w w w w ( (5), (6), (7), (8) ). A= ( ) h 0 0 C 1 t C 0 2C 1 + C 2 0 C 2 t C 1 2C 2 + C 3 ( ) h 0 ( ) h 0 0 C NL 1 t C NL 2 2C NL 1 + C NL 0 C NL t( C NL 1 ) h A s t = M 0 M 1 M 2 M NL 1 M NL C NL (4) B = M 0 M 1 M 2 M NL 1 M NL (7) A x = B (8)», A = w (NL 4 Matrix), x = ( ) sww w (4 1 Matrix), B = y w (NL 1 Matrix), NL = w d wš w j (, NL 4)., (8) w w w(least square solutions) w w w (9) w w (Strang, 1980). x =(A T A) -1 A T B (9) 4.» 4.1 w s w w» w, w( ) w s( w s) x d mw y w ( d Fig. 2 ). w d w 5 d w ww, œ x w. A s t C 0 t( C 0 C 1 ) h 0 ( + ) h 0 0 C 1 t C 0 2C 1 C 2 ( ) h 0 ( + ) h 0 0 C 2 t C 1 2C 2 + C 3 0 C NL 1 t C NL 2 C NL 2C NL 1 0 C NL t( C NL 1 ) h A s t C NL (5) x = (6) Fig. 2. Monitoring Stations of the Cheonsu Bay and Ulsan Port.

5 w x 153 d w DO(, dissolved oxygen), COD(yw, chemical oxygen demand), SS (, suspended solids), TN(, total nitrogen), TP(, total phosphorus) w, w DO, COD, TN w. 2 5 s y w, ( w) 4 s y w. d d Table 2 w w. s³ (s³w MSL [mean Table 2. Measurement stations and water quality concentrations (a) Data in Cheonsu Bay (Pt. 1, North Jukdo: 'E, 36 33'N; ) Time & Date Data Set - T1 10:27(7/27) ph COD SS TN TP D D D D D Data Set - T2 17:12(7/27) D D D D D Data Set - T3 23:38(7/27) D D D D D Data Set - T4 06:23(7/28) D D D D D Data Set - T5 11:50(7/28) D D D D D (b) Data in Cheonsu Bay (Pt. 2: 'E, 36 33'N; ) Time & Date Data Set - T1 10:27(7/27) ph COD SS TN TP D D D D D Data Set - T2 17:12(7/27) D D D D D Data Set - T3 23:38(7/27) D D D D D Data Set - T4 06:23(7/28) D D D D D Data Set - T5 11:50(7/28) D D D D D sea level]» ) 15.0 m, 15.0 m. 4.2 (1), (2), (3) w yƒ w 2 yƒ w w w w w 1 w w w., w w w x w, d (Data Set - T1, T2, T3, T4, T5)

6 154 y Á Á k (c) Data in Ulsan Port (Pt. 0: ' 15E, 35 26'24N; ) Time & Date Data Set - T1 17:00(9/2) ph COD TN DO D D D D D Data Set - T2 01:00(9/3) D D D D D Data Set - T3 07:00(9/3) D D D D D Data Set - T4 13:00(9/3) D D D D D Data Set - T5 19:00(9/3) D D D D D d»( y) y w» w s³w s³w» s (Data Set - T12, T23, T34, T45) yw. w s³w» y w, w w, (Parameter [T1], [T2], [T3]) (Fig. 3). 4.3 w w x wš, w w, x d Fig. 3. Schematic Diagram of the Time-Series Data Set and Parameters. j y š q. 1» l w (mg/m 2 /day) COD, SS, TN, TP w w ƒƒ , , 0.518, , 2 ƒƒ , 9.060, 0.405, , w» l w (mg/m 2 /day) COD, TN, DO w w ƒƒ , 0.025, wr, 1 n l w (mg/m 2 /day) COD, SS, TN, TP w w ƒƒ 1.846, , 0.427, , 2 ƒƒ , 1.704, , , w n l w COD, TN, DO w w ƒƒ , 0.035, w d w w, w w w. RMS (root mean square) w y w w, COD, SS, TN, TP, DO w w ƒƒ , , , , , y w 5.0% w w eƒ de w. wr, Willmott (1981)ƒ w e (index of agreement) Ia ƒ

7 w x 155 Table 3. Estimated parameters using the inverse problems (1) Pt. 1 in Cheonsu Bay Time interval Conc. Parameters T1 T2 T3 Mean COD SS TN TP (2) Pt. 2 in Cheonsu Bay Time interval Conc. Parameters T1 T2 T3 Mean COD SS TN TP (3) Pt. 0 in Ulsan Port Conc. COD TN DO Parameters ƒ w w w. e (7) tx, 1.0 ƒ¾ eƒ d e ew w. N i = 1 ( ) 2 P i O i Ia = N [ P i O i + O i O i ] 2 i = 1 Time Interval T1 T2 T3», P i, O i»y ƒƒ w d y, d y ùkü, O i d y s³ w. COD, SS, TN, TP, DO w w e Ia ƒƒ 0.979, 0.987, 0.971, 0.969, d ƒ w e w q. 5. Mean s w x w w. w, w w x w w» l w, n l w, y, y w y w. d» j y š q ù, w (7)

8 156 y Á Á k Fig. 4. Comparison of the Measured and Estimated Mass Changes. w œw q., w w x w k w q.,» d sw d š w w wš yw w Á w ƒ v w. w x (inverse problem) w, w s d mw y w w x w w,»á» d v w w. w, w x w w w x w. ù, w w y w» w ƒ w d y mw y ƒ wš w w (Banks & Bihari, 2001). wr, w d z w z x y (indicator) q d š w y w, x j w w x w z mw. w» d yw w w, d t ww w (Hensel, 1991; Menke, 1989). w, w w d w k mw w ƒ ƒ w ƒ w w.

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