CALIBRATION OF RIVER BED ROUGHNESS

CALIBRATION OF RIVER BED ROUGHNESS A. M. Wasantha La 1, M. ASCE Singuar vaue decomposition (SVD) is used to caibrate the Manning s roughness coefficients in a 1-D unsteady fow mode of the Upper Niagara River. The method is used to sove for the parameters after formuating the caibration probem as a generaized inear inverse probem. SVD is usefu in soving under-determined, overdetermined or even-determined probems, and can provide information to compute matrices describing parameter resoution, covariance and correation. This information is usefu in identifying the important parameter groups in the mode. Caibration is repeated with different numbers of parameter groups to determine the variation of the output error and uncertainty of the parameters with the parameter dimension. For purposes of comparison, the mode with a seected group of parameters is caibrated using Gauss-Newton method and minimax methods. The study shows the reationship of the parameters to the geometric ayout of the river and the gaging stations. INTRODUCTION Probems invoving mathematica modes are categorized as direct probems and inverse probems. In direct probems, everything about the mode is known, and the objective is to find the system output for a given input. Inverse probems are cassified as identification, detection and state reconstruction probems. Caibration is an inverse probem associated with identification, and is used to determine unknown constants or parameters in a mode. Direct or expicit parameter determination is not possibe in many noninear probems, incuding unsteady river fow networks. Indirect methods based on an output error criterion are commony used to determine bed roughness parameters iterativey. Agorithms used in caibration generay incude methods based on mathematica programming, optima contro or tria and error techniques. A summary of caibration methods used in groundwater probems is given in the text by Wiis and Yeh (1987). Athough there is theoreticay an infinite number of spatiay dependent 1 Senior Civi Engineer, South Forida Water Management District, 3301 Gun Cub Rd., West Pam Beach, FL 33416; formay Research Engineer, Dept. of Civi and Env. Engg., Carkson University, Potsdam, NY 13699. 1

bed roughness parameters in a river network, ony a finite number exists in a computer mode due to spatia discretization used with finite difference or finite eement methods. Out of this finite number, ony a imited number can be identified in most inverse probems (Beck, 1987). The current study investigates the use of Singuar Vaue Decomposition (SVD) to study parameters in a river network mode. The resuts are aso used to form parameter groups to simpify the probem and avoid getting a severey underdetermined systems. The study shows that some of the parameter groups can aso be associated with the geometric ayout of the river network and the pacement of observation stations. When identifiabiity is not a probem, popuar methods of caibration used in the past for open channe probems incude methods by Becker and Yeh (1972), Yeh and Becker (1973), and Fread, et a. (1978). Reated appications incude caibration of surface irrigation parameters by Katopodes et a. (1990). Error criteria commony used in the caibration are based on minimization of the sum of square of the errors in cacuations, the maximum absoute error or the bias. Fread (1978) used a method based on minimization of the bias, which required breaking down of the river into a number of singe channe reaches before caibrating each reach separatey. The Kaman fitering method used by Chiu, et a. (1978) is aso a vauabe too in caibration, adopted from optima contro theory. The SVD method has been previousy used by Wiggins (1972) and Uhrhammer (1980) to caibrate seismoogic parameters. Text by Meneke (1984) gives a description of inverse methods used in soving geophysica probems. The objectives of this study incudes determination of the structure, accuracy, and the optimum parameter dimension in the river network mode. Parameter covariance and correation matrices are aso determined in the study. The caibration is repeated using optimization methods based on the Gauss-Newton method and the minimax methods for comparison of resuts. 2

THEORETICAL CONSIDERATIONS The theoretica derivation of the equations required for caibration assumes that the numerica mode to be caibrated can be represented as a continuous function of the mode parameters ocay within the usefu range of the variabes. If y k j are the vaues of state variabes or water eves simuated by the mode at times k at ocations j, and if Y k j same variabes, the objective of the caibration is to find parameters x i, i are the physica observations of the 1 2 n that minimize the errors ε k j y k j Y k j, j 1 2 m k 1 2, using a specified criterion. n number of parameters; m number of observation stations; and number of time steps. The infuence coefficient method (eg., Becker and Yeh, 1972) uses parameter perturbations to define infuence or sensitivity coefficients. a k ji y k j x i y k j im x i 0 x i x i y k j x i for each i 1 2 n j 1 2 m k 1 2 (1) in which, a k ji sensitivity of the j th water eve with respect to i th parameter at time k; x i i th parameter; k time step number; number of time steps. Using matrix notation, (1) can aso be expressed as in which, the matrix A k m n time step. y k m 1 A k m n x k n 1 (2) is caed the infuence matrix. The superscript k is used to identify the Numerica computation of infuence coefficients is carried out by perturbing each of the parameters by a sma amount x i. For the current caibration of Manning s coefficients, x is 0.001. If n parameters are caibrated, the mode is run n 1 times to obtain a the m n eements of the sensitivity matrix. Generaized inear inverse probem The method used by Wiggins (1972), Ward et a. (1973) and Uhrhammer (1980) to sove inear inverse probems is used in the current study. Since there is a time series of errors at every gage, bias is used as the error indicator, and there are m error indicators for the probem. ε s j 1 y k j Yj k for j 1 2 m (3) k 1 3

in which, ε s j average error or bias at gage j, which is ε s in vector form. The corrections in parameters x needed to make ε s 0 can be determined by equating average y of (2) to -ε s and soving for et a. 1973). x. The resuting system of inear equations can be expressed as (Wiggins 1972, Ward 1 k 1 or, in matrix form, in which, a k jp x p 1 y k j Yj k for j 1 2 m p 1 2 n (4) k 1 A s 1 k 1 A s x ε s (5) A k ε s 1 The corrected set of parameters to be used in the next iteration are k 1 ε k (6) x r 1 x r ρ r x r (7) subjected to x r b x r 1 x r ub (8) in which r is the iteration number; ρ r parameter used to contro the step size; ub, b are specified upper and ower bound vaues. ρ r continued unti ε s 2 converges, and x 2 becomes sma 0 8 is used in the current study. The iterative procedure is x x δ in which δ machine precision of the computer or a arger vaue depending on the required parameter toerance. The m equations shown in (5) can be soved easiy using methods such as Gaussian eimination if m n and deta 0. When m n, the probem is overdetermined, and the east squares method eads to the soution x A T A 1 A T ε s as ong as deta T A 0. When m n, the probem is underdetermined, and the generaized or Penrose inverse soution is x A T AA T 1 ε s as ong as detaa T 0 (Nobe and Danie, 1975). None of these soution techniques can be used with rank deficient or i-conditioned matrices. The sensitivity matrix becomes rank deficient if there is at east one parameter that has no significant infuence on the any of the seected observations, or if there is at east one observation station that is not sufficienty affected by any of the parameters. Singuar Vaue Decomposition (SVD) is capabe of soving under, over, even or mixed determined probems as expained in detai by Meneke (1984). 4

SVD is based on the key theorem that a matrix A s can be decomposed into three matrices V, Λ and U such that A s UΛV T (9) in which U and V are m m and n n matrices of orthogona singuar vectors and Λ m n diagona matrix of singuar vaues of A s. Texts by Nobe and Danie (1975), Forsythe et a. (1977) and Press, et at., (1989) give detaied information about the method. With symmetric matrices, SVD gives eigenvaues and eigenvectors. The physica meaning of the terms and expressions of the SVD method are expained using (9) and (10). SVD essentiay diagonaizes the sensitivity matrix. After the decomposition, singuar vectors of ength n formed by the coumns of V give coefficients of the inear combinations of od parameters giving rise to new independent parameter groups. Singuar vectors of ength m formed by coumns of U give coefficients of the inear combinations of observations forming new observations groups. The new independent parameter groups are reated to groups of observations through the diagona eements in Λ. Both newy formed observation and parameter groups are independent of each other because of the orthogonaity of matrices V and U. Singuar vaues forming the diagona sensitivity matrix Λ reate these parameter groups to observations groups. The number of nonzero singuar vaues q in Λ is the rank of A s. It gives the maximum number of independent parameter groups that can be identified in the mode. SVD can be used to sove the system of equations shown in (5). The soution method repaces the previousy mentioned methods for m n, m n, m n or mixed determined probems. It works even if the matrices invoved are singuar (Meneke, 1984). The soution to the system of equations shown in (5) is obtained by decomposing A s into A s UΛV T. (5) then becomes Λ z d (10) in which, z V T x and d U T ε s (11) The soution to (10) can be written as x V 1 λ U T ε s (12) 5

in which λ are the diagona eements of Λ. If at east one λ i is zero or cose to zero such that the vaue is dominated by the roundoff error, the matrix is singuar. When this happens, 1 λ vaues corresponding to λ faing beow a sma cutoff vaue are a repaced by zeros (Forsythe, et a. 1975, Press, 1989). It can be shown that SVD determines the x for a probem that minimizes x 2 and ε s 2. The soution under these conditions is unique, and incudes the Gaussian, east square and Penrose cases mentioned before. The cutoff eve or the smaest singuar vaue λ min in (12) is usefu in controing the data errors in the soution. If it is sma, 1 λ becomes too arge, and errors in the soution wi be bown out of proportion. In a non singuar matrix, λ min is reated to the condition number, λ max λ min. A matrix is i-conditioned if it is too arge, or the reciproca, λ min λ max approaches foating point precision δ. For singe precision operations, δ 10 6 for most computers and the minimum cutoff possibe is δ λ max. Since data error can cause unnecessary error magnifications at sma cutoffs, a δ vaue of 10 3 was used in the current probem. Even arger vaues (10 2 ) were found to be sufficient during initia iterations. Methods of smoothing the cutoff have been discussed Wiggins (1972) and Uhrhammer (1980). The U and V matrices created by SVD can give additiona information about the parameter behavior. The foowing expression gives an idea about the parameter resoution in the computer mode (Wiggins, 1972): Parameter space resoution, R VV T (13) Reative size of the eements of R show reative resoution or independence of the corrections to the parameters. If the parameters are finey resoved, this matrix is expected to be cose to unity. If the parameters are not individuay resovabe but are resovabe as groups, the probem is characterized by compact resoution, showing square bocks or groups of dominant eements in the matrix. Covariance of the estimated parameters is defined as (Wiis and Yeh, 1987) ϒ cov ˆx E x ˆx x T ˆx (14) in which ϒis the parameter covariance matrix which can be computed as σ 2 A T s A s! determinant of A T s A s 1 when the 1 is not zero. σ 2 error variance at the output. When the matrix is iconditioned, this method fais. However, the foowing expression derived using the resuts of SVD 6

gives the covariance matrix for both singuar and non-singuar sensitivity matrices (Wiggins, 1972, Uhrhammer, 1980): Covariance ϒ 2 V σ2 Λ 2 VT (15) A vaue of 1 is used for σ 2 to obtain the reative vaues in the matrix eements. Uhrhammer (1980) referred to the covariance matrix as the uncertainty matrix, and used a damping parameter to damp out the effects of sma singuar vaues on 1 λ during a sharp cutoff. In the current study, a sharp eary cutoff λ min ( 10 3 ) as suggested by Wiggins (1972) is used for simpicity. The covariance matrix can be used to obtain the correation matrix (Uhrhammer, 1980) Correation among parameter corrections, ρ 2 i j ϒ 2 i j ϒ i iϒ j j (16) Correation among parameter corrections is usefu in determining the dependence of parameters and parameter groups. If there are parameters that are dependent as a group, and as a group if they are independent from other groups refecting a bock form in the matrix, caibration can be simpified by casting these groups as singe parameters. The correation matrix, resoution matrix, and V T are used in the current study to understand and isoate the groups of parameters. These groups can aso be reated to the ayout of the river network and gage positions. Gauss-Newton method The Gauss-Newton method as discussed by Wiis and Yeh (1987) considers the identification probem as an optimization probem aimed at minimizing output errors. This method cannot be used with underdetermined systems m n or systems with rank q n. It is used in the study as an aternative method to caibrate groups of parameters seected to satisfy these conditions. When using the method, the objective is to determine parameters x i i 1 2 n such that the summation of the squares of the errors is a minimum, subjected to parameter constraints given by (8). The sum of squares of errors is defined as m S j 1 k 1 w j k y k j Yj k 2 (17) over a the gages and a the time steps. The w j k are weighing factors, which are assumed to be 1 for the current probem. The parameter corrections that minimize S can be shown to be (Yoon and 7

Yeh 1976). C x D (18) in which eements i j of C and D are given by c i j n a k p i k 1 p ak p j ; 1 d i n k 1 p 1 a k p i y k p Yp k (19) The n n system of inear equations is soved to obtain the corrections x i, i 1 2 n. The parameters are updated using (7), in which ρ r can be assumed. Optima vaues of ρ r can be determined by the quadratic interpoation method suggested by Yoon and Yeh (1976). Iterations can be terminated when S converges and the parameter corrections are too sma to make physica sense, or when x x machine precision. Minimax method The objective of the minimax method used in the current study is to determine the parameters x i i 1 2 " n that minimize the sum of the absoute vaues of the maximum errors at each of the gages. The maximum errors e 1 e 2 # e m at gages 1 2 m at any time during the entire period are e i max The objective function seected is ε k i k 1 2 i 1 2 m (20) min e 1 e 2 e m (21) If the parameters x i i 1 2 " n are subjected to sma changes x i i 1 2 " n, then simuated vaues y k i i 1 2 " n k 1 2, change to k y$i y$i y k i i 1 2 " n k 1 2. Expressing y k i using (1), y k i can be expressed as y$i k y k i n a i j x j i 1 2 m k 1 2 " (22) j 1 The constraints used to determine maximum error can be set up by confining the new errors after a parameter correction to be e i for a stations. This condition is given by k y$i Y i e i for i 1 2 m and k 1 2 " k (23) 8

Substituting for y$i k using (22), (23) can be written as n a k i j x j e i Y k i y k i for i 1 2 " m and k 1 2 (24) j 1 n a k i j j j e i 1 Yi k y k i for i 1 2 m and k 1 2 " (25) Other conditions can aso be incuded to keep the soution within imits. Upper and ower bounds can be set to the parameters using % x i x i x i ub i 1 2 n (26) x i x i x i b i 1 2 n (27) in which, x i ub and x i b are the upper and ower bounds of parameters in (8). The objective function with 2 m 2 n constraints form a inear programming probem, which is soved for the corrections x i, i 1 2 n using the simpex agorithm. Parameters are updated using (7) unti the corrections are too sma to make physica sense or x x machine precision. APPLICATION TO THE UPPER NIAGARA RIVER The mode to be caibrated soves the Saint Venant equations using the four point impicit method. The numerica method is simiar to that used by Potok and Quinn (1979). The Saint Venant equations consist of the foowing continuity and momentum equations: Q t A t Q x Q 2 x A & ga h x 0 (28) p b τ b ρ 0 (29) in which, Q discharge; A fow area; h water surface eevation; x distance aong the river; t time; ρ density of water; p b wetted perimeter; τ b shear stress at channe bottom ρgq 2 n 2 b A 2 R' 1( 3) ; R hydrauic radius A p b ; p b wetted perimeter; n b Manning s coefficient. n b vaues for the discretized river sections are the unknown parameters in the mode to be caibrated. The map of the Upper Niagara River is shown in Fig. 1. In the map, the term NYPA stands for New York Power Authority hydropower intake, and SAB stands for the Sir Adams Beck power 9

pant intake on the Canadian side. The river network is discretized into 27 sections. The probem invoves caibration of the (n 27) unknown Mannings coefficients to minimize the errors at a the gaging stations. Houry water eves between Feb 13, 1989, and Feb 16, 1989, are used in the foowing sampe caibration to obtain Manning s coefficients for the range of water eves considered. A compete caibration requires data for a onger period of time, covering a wider range of water eves. The ist of 3 stations with known discharges and 9 stations with known water eves avaiabe to run the mode is shown in Fig. 1. Water eve gages are marked as G1... G9 and known discharges marked as Q10... Q12. The upstream end water eve at Fort Erie gage G1 is specified as the upstream boundary condition, and the downstream end discharges Q10, Q11 and Q12 at the contro dam and intakes are specified as the downstream boundary conditions. Water eve gage G2 is not used because its vaue is very cose (* 0.05 ft) to the vaue of G3, and both are represented by ony one node in the 1-D mode. The remaining 7 gages marked G3-G9 are used in the caibration. In the study, the observations and simuations at these gages are simpy referred to as Y i and y i i 1 m in which m 7. Caibration of individua mode parameters The generaized inear inverse formuation and singuar vaue decomposition are used to caibrate 27 bed roughness vaues n 1 n 27 of the 27 river sections in the 1-D hydrodynamic mode. SVD is needed in this case because there are ony 7 observation stations or 7 equations to sove (5) to determine 27 unknown parameters. A the Manning s coefficients are assumed as 0.025 in the first iteration. Figure 2 shows the sensitivity matrix A s indicating arge sensitivities of n 2 and n 3 to amost a the water eve gaging stations. In the figure, the circe diameters correspond to the sizes of the eements. n 2 and n 3 correspond to the rapids sections of the upper Niagara river where the drop in water eve is about 2 m and fow veocities reach approximatey 2.6 m/s. Figure 3 shows the convergence of a few seected stabe parameters n 2, n 6, n 11, n 13, and n 16, and the rapid reduction of the error variance in water eves with increasing number of iterations. The rapid convergence indicates that the method is capabe of minimizing the bias error as intended. Figure 3 shows that the method aso reduces the mean square error during caibration. Since the 10

probem is underdetermined, the soution shown in the second row of Tabe 1 is nonunique and has a arge random error. Parameter error variance is arge for reativey insensitive parameters as shown ater in Fig. 7. Even if the insensitive parameters have a arge variance, the mode gives a good error reduction as shown in Fig. 4 because the mode output is not affected by them. The set of 27 parameters with arge errors has a imited use. However, the by-products of the method such as the singuar vaues and singuar vectors can be used to understand the parameters of the mode in great detai as expained before. Three of the new ineary independent parameter corrections obtained using z V T x of (11) are shown beow. n+1 0 712 n 3 0 687 n 2 0 093 n 4 0 043 n 11 λ 1 81210 (30) n+2 0 594 n 11 0 364 n 16 0 317 n 21 0 253 n 10 λ 2 2269 (31) n+3 0 622 n 16 0 406 n 17 0 346 n 21 0 229 n 24 λ 3 1572 (32) in which, n+1 n+ 2 are the components of the vector z representing new independent parameter variations. These variations are made up of inear combinations of od parameter variations n 1 n 2. The coefficients in front of n 1 n 2, which form n+i, are obtained from the coumns of V. Variations of the new parameters, n+i are reated to variations in the mode output, h+i, through the reationship h+ i λ i n+i, which is simiar to (10) when the mode output variation is repaced by a negative output error. The singuar vaues λ i of the mode 81210, 2269, 1572, 827, 782, 476 and 22, (nearest integer) show the domination of the caibration by a sma number of parameter groups. The singuar vaues show that the rank of the (7 27) sensitivity matrix is 7 and that ony 7 independent parameter groups can be identified in the probem. The singuar vectors of V written as (30)...(32) show the three most important parameter groups. Singuar vaues λ i provide estimates of the sensitivities. The equations show that adjustment of the group of parameters, consisting of mainy n 3 and n 2 as indicated by the strong coefficients 0.712 and 0.687, can take care of about 93 % of the overa caibration. Parameters n 2 and n 3 represent the narrow rapids section of the river that has a controing effect on the overa water eves. Parameters n 11 n 16 form another independent group. The inear combinations of variations of the mode output are reated to n+1 n+ 2 through the equation h+ U T h. h+1 0 544 h 9 0 458 h 7 0 457 h 8 0 414 h 6 λ 1 81210 (33) 11

h+2 0 627 h 4 0 607 h 3 0 287 h 5 0 170 h 8 λ 2 2269 (34) h+2 0 930 h 5 0 265 h 4 0 210 h 3 0 110 h 9 λ 3 1572 (35) in which, h+1 h+2 are the eements of vector h+ made of inear combinations of variations of water eves h 1 h 2 h 7 which correspond to gages G3-G9. (33)-(35) show that the new parameter n+1 uses simuation errors in water eves h 9 h 7 h 8 h 6 for caibration, and parameter n+2 uses errors in h 4 h 3 for caibration. Approximate equaity of coefficients in (33) shows that a the gages have the same information required to caibrate n+1. Parameter n+ 2 however requires information mosty from h 4 and h 3 (G6 and G5) as shown by coefficients 0.627 and 0.607. Figure 5 shows the parameter resoution matrix obtained using (13). In the figure, parameter corrections in groups n 6 n 14 and n 16 n 24 are the east resoved. n 27 is the most resoved, because there are two gages at both ends of the reach. Parameters within a ong river reach without gaging stations in the midde are not sufficienty resoved, and behave as a group. Any further resoution of parameters within these zones is not possibe because observation points cannot distinguish between roughnesses of any two sections within a group if the group ies between two gages. It is aso unnecessary to resove the parameters any further uness the mode is used to estimate the water eves at the interior points. The accuracy of the parameter corrections is of the order 1 λ min as shown by (12). If a sma cutoff vaue λ min is seected, the parameters wi have arger errors. If a arge cutoff is seected, ony a few parameter groups wi be seected as expained before. With a very arge cutoff, the number of parameters may not be adequate to represent the physica system, and sufficienty minimize the errors. The next sections incudes determination of an optima parameter dimension based on the parameter uncertainty and the east square output error. The correation matrix in Fig. 6 shows the group behavior of the parameters. The figure shows the strong correation among parameter groups n 6 n 8, n 9 n 12, n 13 n 15, n 16 n 18, n 19 n 24, n 25 n 26. The map in Fig. 1 shows that these groups correspond to river reaches with water eve gages at both ends. Next sections describe the caibration using parameter groups. Caibration of parameter groups Caibration of the individua parameters in the previous section shows that they tend to group together as indicated by the correation, resoution and V matrices. In the SVD method expained 12

earier, singuar vectors in V coud impicity make use of these groups. However, 27+1 computer runs were needed for one iteration of the caibration. In this section, in order to reduce the number of runs and obtain more stabe parameters with ow error variances, 7 new groups of parameters are created from the origina parameters using the information avaiabe from the previous section. A singe parameter vaue is assigned to a the parameters in a group. The parameter groups created in this manner are shown in Tabe 2 and are marked as m 1 m 7. The number of groups is imited to 7 because it is the rank of A s. To study the effect of the number of parameter groups, or parameter dimension on the output error and parameter uncertainty, the 7 parameter groups are combined again with each other to form smaer numbers of parameter groups. Coumn 4 of Tabe 2 for exampe shows how parameters n 1 n 27 are combined to form 4 groups. Figure 7 shows the variation of the output error and parameter uncertainty with different numbers of parameter groups. The mean square bias error is used to measure the output error in which bias is the average difference between the observed and simuated water eves. An approximate estimate of the parameter error or uncertainty is computed using the expression σ 2 λ 2 min, which is approximatey equa to the infinite norm of the covariance matrix. Figure 7 shows that the parameter uncertainty increases and the output error decreases with increasing number of parameter groups. The figure aso shows that the output error variance does not reduce much when there are more than 3 parameters. With more than 3 parameters, the parameter uncertainty increases. The figure shows the existence of an optima parameter dimension of about 3 for the probem where both the output error and parameter uncertainty are ow. SVD with 7 parameter groups shown in Tabe 2 gave a mean square bias error of 7.23 10 after 7 iterations showing that the caibration is successfu even if these handpicked parameter groups are not exacty the singuar vectors in (30)-(32). The condition number of the 7 7 group sensitivity matrix is 150198/23.8, which shows that the matrix is nonsinguar, and it is possibe to use the Gauss-Newton method for comparison. The parameter resoution and correation matrices of the 7 parameter group caibration (not shown) indicate that the handpicked parameter groups are sufficienty resoved and independent. The third and fourth rows of Tabe 1 show the vaues of parameters obtained by caibrating the handpicked 7 and 3 groups of parameters using SVD. A these are even determined probems that can aso be soved using Gaussian eimination. The vaue of a singe parameter that fits the 13 4 m 2

entire river network is found to be 0.03022, with an output mean square bias error of 6.1 10 2 m 2. Caibration using Gauss-Newton and Minimax methods The Gauss-Newton and minimax methods are aso used to caibrate the same 7 handpicked parameter groups for comparison. The fina Mannings coefficients after 5 iterations of caibration using both methods are shown in Tabe 3 aong with the sum of error variances and mean square bias errors. The argest correction required on the Mannings coefficients after 5 iterations of the Gauss- Newton method is approximatey 0.00002, and the argest correction required after 8 iterations of the minimax method is approximatey 0.00005. The iterations were terminated at this point due to the smaness of the adjustment. During the caibration using the Gauss-Newton method, ρ was assumed as 0.75 during the first two iterations, and 1.0 afterwards. Larger vaues of ρ were avoided during first few iterations to avoid crashing the mode. The Gauss-Newton method required the east amount of computer time and the fastest rate of convergence. The minimax method is the most time consuming because of the inear programming (LP) agorithm. The computer time requirement increased with the number of time steps in the simuation runs because it increased the size of the inear programming probem. A Sparc 1 computer using the IMSL package needed 10-60 minutes of computer time for a singe iteration of the minimax, when compared with spit seconds needed by other methods. Tabe 3 shows that parameter vaues are approximatey the same for the most important parameter group m 1 consisting of n 1 n 6. Pots of the observed and simuated water eves using a three methods do not show any visuay detectabe difference. CONCLUSIONS SVD is usefu in identifying and understanding an arbitrary number of parameters when the equations invoved are overdetermined, even-determined, underdetermined or singuar. Caibration of the 27 parameters in the Niagara river mode using different numbers of parameter groups show that the mode has a maximum of 7 independent parameter groups, and the optimum parameter dimension is about 3. 14

The study shows that the underdetermined identification probem with 27 parameters can be reduced to an even-determined or an over-determined probem by creating a number of parameter groups equa to the rank of the sensitivity matrix (7) or ess. The groups can be created using the information from parameter space resoution matrices, correation matrices, singuar vectors or the geometrica ayout of the river system. Parameters corresponding to river reaches between gages appear as groups in the correation and resoution matrices, and therefore can be considered as groups. Once parameter groups are created such that the probem is neither underdetermined nor singuar, optimization methods such as the Gauss Newton and minimax methods can be used to carry out the same caibration. 15

ACKNOWLEDGEMENTS The study was partiay supported by the New York Power Authority through contract No. 029494-89 entited Upper Niagara River Ice Dynamics Simuation. Mr. Randy Crissman is the Contracting Officer s representative. The author wishes to thank Prof. H. T. Shen for having him in his research team. 16

APPENDIX I. REFERENCES Beck, M. B. (1987). Water Quaity Modeing: A Review of the Anaysis of Uncertainty, Water Resources Research, 23(8), 1393-1442. Becker, L., Yeh, W. W-G. (1972). Identification of parameters in unsteady open channe fows, Water Resources Research, 8(4), 956-965. Becker, L., Yeh, W. W-G. (1973). Identification of mutipe reach channe parameters, Water Resources Research, 9(2), 326-335. Chiu, Chao-Lin, (1978). Kaman Fiter in Open Channe Fow Estimation, J. of Hydr. Engg., ASCE, 104(8). Forsythe, E. George, Macom, A. Michae, and Moer, B. Ceve, (1977), Computer Methods for Mathematica Computations, Prentice Ha Inc. Forsythe, E. G., Macom A. M., and Moer, B. C. (1977). Computer methods for mathematica computations, Prentice Ha, Inc. Fread, D. L., and Smith, G. F. (1977). Caibration techniques for 1-D unsteady fow modes, J. of the Hydr. Div., ASCE, 104, Juy, 1027-1043. Katopodes, N. D., Jyh-Haw Tang, and Cemments, Abert J. (1990). Estimation of Surface Irrigation parameters, J. Hydr. Engg., ASCE, 116(5), 676-696. Meneke, Wiiam, (1984). Geophysica data anaysis: discrete inverse theory, Academic Press, Inc., New York. Nobe, B., and Danie, J. (1975). Appied Linear Agebra, Prentice Ha Pubishers, Engewood Ciff, NJ. Potok, A. J., and Quinn, F. H. (1979), A Hydrodynamic Mode for the Upper St. Lawrence River 17

for Water Resources Studies, Water Res. Bu., 15(6). Press, W. H., Fannery, B. P., Teukosky, S. A., and Verrering, W. T. (1989). Numerica Recipes, the Art of Scientific Computing, Cambridge University Press, Cambridge. Uhrhammer, R. A. (1980). Anaysis of Seismographic Station Networks, Bu. of the Seis. Soc. of America, 70(4), 1369-1379. Ward, S. H., Peepes, W. J., and Ryu, J. (1973). Anaysis of Geoeectromagnetic Data, Methods in Computationa Physics, 13, Academic Press, New York, 13, 226-238. Wiggins, R. A. (1972). The Genera Linear Inverse Probem: impications of surface waves and free osciations for surface structures, Rev. Geophysics and Space Physics, 10, 251-285. Wiis, Robert and Yeh, W-G. Wiiam (1987). Groundwater Systems Panning and Management, Prentice Ha, Inc. Yeh, W. W-G., and Becker, L. (1973). Linear programming and channe fow identification, Journa of the Hydrauics Division, ASCE, 99(11), 2013-2021. Yoon, Y. S., and Yeh, W. W-G. (1976). Parameter Identification in an Inhomogeneous Medium with the Finite-Eement method, Soc. Pet. Eng. J., 217-226. 18

APPENDIX II. NOTATION The foowing symbos are used in the paper A k A sensitivity matrix a k j i cross sectiona area of a channe A s k 1 Ak a k ji sensitivity of j th water eve with respect to i th parameter at time k. d intermediate variabe defined as U T ε s which give inear combinations of differences in water eves h i k m m j n n b n j n+j Q q r R R water eve at gage i time step number number of time steps number of observation stations group vaues of handpicked parameters number of parameters Manning s coefficient Manning s coefficient of river section j new parameter combination obtained using the SVD method discharge in a channe rank of the matrix, or number of singuar vaues used number of iterations hydrauic radius parameter resoution matrix U V m m and n n matrices of orthogona singuar vectors x i Yj k y k j x ε s ε s j a generic symbo for parameter i observed water eve at node j at time k (m) simuated water eve at node j at time k (m) vector of parameter corrections T n 1 n 2 vector consisting of summations of errors summation of errors at gage j over time 19

Λ diagona matrix of singuar vaues λ i i th singuar vaue ρ i j correation among parameter corrections τ b shear stress at channe bottom ϒ 2 covariance of parameter corrections Subscripts s summation over time r iteration number Superscripts k vaue at time step k 20

Figures Fig 1: Map of the upper Niagara River. Fig 2: Sensitivity matrix (Max diameter=31519); gages G3-G9 are marked as 1-7. Fig 3: Convergence of parameters and reduction of errors with iterations. Fig 4: Observed and simuated water eves after caibration. Fig 5: Parameter resoution matrix (Max. Diameter = 0.96). Fig 6: Correation matrix (Max diameter = 1.0). Fig 7: Variation of output error and parameter uncertainty with parameter dimension. 21

Tabe 1: Manning s Roughness Coefficients obtained using SVD Param. 1 2 3 4 5 6 7 8 9 Indiv..0273.0321.0352.0280.0226.0239.0237.0208.0248 7 groups.0329.0329.0329.0329.0329.0226.0226.0226.0229 3 groups.0327.0327.0327.0327.0327.0220.0220.0220.0238 Param. 10 11 12 13 14 15 16 17 18 Indiv..0267.0258.0211.0286.0269.0099.0166.0203.0200 7 groups.0229.0229.0229.0263.0263.0263.0194.0194.0194 3 groups.0238.0238.0238.0238.0238.0238.0220.0220.0220 Param. 19 20 21 22 23 24 25 26 27 Indiv..0239.0268.0284.0277.0133.0183.0346.0255.0223 7 groups.0261.0261.0261.0261.0261.0261.0261.0261.0007 3 groups.0238.0238.0238.0238.0238.0238.0238.0238.0238 KEY WORDS Bed roughness caibration, cana network, Open chane fow Hydrauics, Inverse probem, parameter identification, Singuar vaue decomposition, east square, Gauss-Newton, Linear Programming, 22

Tabe 2: Parameter groups formed by combining individua reach parameters; a groups are combinations of the 7 basic groups in coumn 2 Groups of Number of par. groups Origina and formation Parameters 7 5 4 3 2 1 n 1 n 5 m 1 m 1 m 1 m 1 m 1 m 1 n 6 n 8 m 2 m 2 m 2 m 2 m 2 m 1 n 9 n 12 m 3 m 3 m 3 m 3 m 2 m 1 n 13 n 15 m 4 m 4 m 3 m 3 m 2 m 1 n 16 n 18 m 5 m 5 m 4 m 2 m 2 m 1 n 19 n 26 m 6 m 5 m 1 m 3 m 2 m 1 n 27 m 7 m 5 m 3 m 3 m 2 m 1 Tabe 3: Resuts of the caibration of 7 handpicked parameter groups using different methods Group Min. Bias Least Sqr. Minimax 1 0.03289 0.03200 0.03296 2 0.02270 0.02539 0.02558 3 0.02282 0.02546 0.02504 4 0.02615 0.02504 0.02239 5 0.01948 0.02570 0.01816 6 0.02948 0.02541 0.02371 7 0.00701 0.02501 0.02408 Err. Variance (m 2 ) 0.00187 0.00959 0.05361 Mean Sq. Bias (m 2 0.00072 0.00848 0.05332 23

Parameter Individua 7 groups 3 groups 1 0.0273 0.0329 0.0327 2 0.0321 0.0329 0.0327 3 0.0352 0.0329 0.0327 4 0.0280 0.0329 0.0327 5 0.0226 0.0329 0.0327 6 0.0239 0.0226 0.0220 7 0.0237 0.0226 0.0220 8 0.0208 0.0226 0.0220 9 0.0248 0.0229 0.0238 10 0.0267 0.0229 0.0238 11 0.0258 0.0229 0.0238 12 0.0211 0.0229 0.0238 13 0.0286 0.0263 0.0238 14 0.0269 0.0263 0.0238 15 0.0099 0.0263 0.0238 16 0.0166 0.0194 0.0220 17 0.0203 0.0194 0.0220 18 0.0200 0.0194 0.0220 19 0.0239 0.0261 0.0238 20 0.0268 0.0261 0.0238 21 0.0284 0.0261 0.0238 22 0.0277 0.0261 0.0238 23 0.0133 0.0261 0.0238 24 0.0183 0.0261 0.0238 25 0.0346 0.0261 0.0238 26 0.0255 0.0261 0.0238 27 0.0223 0.0007 0.0238 24