Unit hydrograph estimation with multiple events and prior ... - CiteSeerX

Ilydrological Sciences -Journal- des Sciences IIydrologiquest37,5,

10/1992

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Unit hydrograph estimation with multiple events and prior information: I. Theory and a computer program M. BRUEN & J. C. I. DOOGE Centre for Water Resources Research, University College Dublin, Dublin 2, Ireland

Abstract The method of regularization for estimating unit hydrographs is expanded to allow the inclusion of prior information about the unit hydrograph shape. This may give smooth estimates without any loss in volume. The method is illustrated with prior information from a regression on catchment characteristics and with catchment lag determined from the data. A computer program to implement the method is given together with a sample calculation.

Estimation de l'hydrogramme unitaire a partir d'événements multiples et de l'information antérieure: I. Théorie et un programme d'ordinateur Résumé On développe ici la méthode de régularisation pour estimer les éléments des hydrogrammes unitaires en se servant de renseignements supplémentaires. Par exemple on se sert des caractéristique physiques du bassin ou du temps de réponse. On explique une méthode pour incorporer ces renseignements dans le calcul. On donne un programme d'ordinateur et un calcul en exemple.

INTRODUCTION Unit hydrograph theory assumes that a catchment acts on an input of effective precipitation in a linear and time-invariant manner to produce an output of direct storm runoff (Dooge, 1959). For continuous input and output series that relationship is expressed mathematically by an equation in which the direct storm runoff is the convolution of the effective precipitation with the instantaneous unit hydrograph (i.e. the impulse response) of the catchment. When the precipitation is expressed in terms of successive volumes in equal intervals of time and the storm runoff is expressed in terms of ordinates sampled at the same interval, the convolution equation reduces to a linear matrix equation form characteristic of linear stationary systems (Dooge, 1973) involving ordinates at the same interval of a finite period unit hydrograph. The problem of estimating such a unit hydrograph thus requires an estimate of a vector, h, from an overdetermined system of linear equations formed from error-prone measurements: Xh = y + e Open for discussion until I April 1993

(1)

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M. Bruen & J. C. I. Dooge

where: X is an n by m observation matrix, formed from the effective precipitation series; y is an n element right hand side vector, formed from the storm runoff series; h is an m element vector of unknown unit hydrograph ordinates; and e is an n element noise vector. Essentially the solution for h is a multiple linear regression problem.

ORDINARY LEAST-SQUARES The method used most often to solve equation (1) is least squares (Lawson & Hanson, 1974). In its basic form the estimate h of the vector, h, is the solution of the "normal equations": XTXh = XTy

(2)

In the case of unit hydrograph derivation, errors in the effective precipitation and direct runoff data tend to be magnified in the unit hydrograph estimated from equation (2). A useful upper bound for that magnification of relative error is provided by the condition number of XTX. This is a unique real-valued functional of the matrix equal to the ratio of the square roots of its maximum and minimum eigenvalues (Dooge & Bruen, 1989). Bruen & Dooge (1984) pointed out that, for pulse response estimation for linear systems with isolated data, the XTX matrix has a symmetric Toeplitz structure. This means that the matrix is made up of only m independent values and the same value occurs for every element of the matrix which has the same magnitude of difference between its row and column indices. This property allows an efficient algorithm, due to Zohar (1974), and programmed by Farden (1976), to be used to solve equation (2). For this method, Dooge & Bruen (1989) showed that the condition number of XTX, and thus the amplification of error, remained finite for all inputs for the simple case of two unit hydrograph ordinates. One of the features of unit hydrographs derived by the ordinary least squares method from error prone data is that their shape may not correspond with that anticipated by an experienced hydrologist, who would expect a unit hydrograph to have the following characteristics: (a) zero ordinates for all times prior to the start of precipitation excess (causality); (b) a volume of unity in cases where the input and output series have equal volume (conservation); (c) a smooth shape with positive skew, no negative ordinates and no high frequency oscillations (heavily damped system and nature of input and output); and (d) a unimodal shape typical of the vast majority of derived unit hydrographs (ovoid shape of catchment). Results which deviate from those expectations may be due to:

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(i)

model error, when the catchment response is not approximately linear or time-invariant; and/or (ii) errors in the data and their possible amplification during the estimation. This paper is concerned with the generalization of the least squares method to cope with the latter problem. For example, Fig. 1 shows the precipitation excess and direct runoff for a flood event at Almondell on the River Almond. Figure 2 shows the unit hydrograph derived from those data by ordinary least squares. It satisfies expectations (a) and (b) above but not the others. It has negative ordinates, is not smooth and has a number of peaks. The first step in the generalization of the least squares method is aimed at estimating a smooth unit hydrograph. 140

ê LU

10

15

20 25 Time (h)

30

35

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Fig. 1 Data for sample calculation: storm, event on Almond at Almondell.

R E G U L A R I S A T I O N AND R I D G E R E G R E S S I O N The problem of estimating a pulse response of a system from a record of its input and output is incorrectly posed in the sense that small perturbations in the data can lead to large deviations in the resulting estimate. Tikhonov (1963, 1965) developed a method, which he named regularization, for computing stable solutions for such incorrectly posed problems. A variant of the method was applied to pulse response estimation by Kuchment (1967) and subsequently by Anderssen (1971) and Bruen & Dooge (1984). The method was also applied to difficult multiple regression problems by Hoerl & Kennard (1970b), and was subsequently applied to a number of areas including regression of the mean annual flood on catchment characteristics (Acreman, 1985). A simple form of the method was called ridge regression by Hoerl &

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M. Bruen & J. C, 1. Dooge 0.12i

Almond at Almondell

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20 Time (h)

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Fig. 2 Unit hydrograph (OLS: ordinary least squares). Kennard (1970a) and this name is widely used in the operations research field. In ridge regression, the estimate of h is taken as the solution, h, of the modified normal equations: (XTX + kI)h = XTy

(3)

where k is a scalar parameter and / is an m by m identity matrix. This gives a biassed estimate of h and the bias is proportional to k. If the matrix XTX is symmetric Toeplitz then XTX + kl is also symmetric Toeplitz and Zohar's (1974) efficient algorithm can be used for estimating pulse responses by ridge regression. One of the effects of using XTX + kl with a positive k instead of XTX is to increase both the maximum and minimum eigenvalues by adding k to both and consequently to reduce the condition number and thus the magnification of any data errors. Figure 3 shows two unit hydrographs estimated by regularization from the storm shown in Fig. 1. For k = 0.05 the recession was smoothed somewhat, but still was irregular. For k = 1.0, the unit hydrograph was smooth overall. However, the volume of the latter unit hydrograph was less than for the ordinary least squares estimate. In practice, any negative ordinates would be set to zero and the unit hydrograph normalized to have unit volume. Regularization or ridge regression can be interpreted in a number of different ways each of which provides a different perspective on the reason for its success. These are: (a) for positive values of k, the condition number of the coefficient matrix of equation (3) is less than that of equation (2) and this reduces numerical instability in the solution; and (b) regularization is numerically equivalent to providing additional information about the solution in the form of m extra equations. The right hand

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20 Time (h)

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Fig. 3 Unit hydrograph (SLS: smoothed least squares). sides of equations (2) and (3) are the same but an additional term has been added to the left hand side in equation (3). In the next section, it is shown that replacing equation (2) by equation (3) is equivalent to combining the information in equation (1) with the additional equation: h = 0+e

(4)

where e is an error vector. The parameter, k, in equation (3) represents the user's uncertainty about the quality of the data. It controls the degree of constraint or smoothness imposed on the final estimate and so the method was called smoothed least squares (SLS) by Bru en & Dooge (1984) who gave a computer program to calculate the estimate and demonstrated its use in determining unithydrographs.

USE O F ADDITIONAL INFORMATION In the absence of any data on effective precipitation and direct runoff, the hydrologist may use a synthetic, or conceptual, unit hydrograph (Nash, 1958; Dooge, 1973). A standard shape is used and its parameters are often estimated from physical characteristics of the catchment. The method described here allows such a synthetic unit hydrograph to be used in conjunction with measured data. Consider the two possible extreme cases. If full reliance is placed on only the measured data then the ordinary least squares method, equation (2), can be used and any additional information about the unit hydrograph is ignored. If, in contrast, full reliance is placed on a synthetic shape, for example one fitted by a regional regression on catchment characteristics, then the rainfall and runoff data are ignored and the unit hydrograph is given by:

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h = hp + e

(5)

where hp is the synthetic shape and e is an associated error vector. Both sources of information can be combined in an overall least squares estimate of the unit hydrograph by using a weighted combination of equation (1) and equation (5). If a weighting matrix 0 is applied to the equations for hp then the full set of equations can be written as: X h = 0

y Qhp

+

e

where Q is an m by m weighting matrix. It could be a diagonal matrix with different weights w,- for each unit hydrograph ordinate, i.e. diag(wl, a2, ..., wm) or have the same weight for all terms, i.e. wl, where / is an m by m identity matrix. The augmented coefficient matrix on the left hand side has n + m rows and m columns and the augmented right hand side vector has n + m elements. The least squares solution to equation (6) is given by: (XTX + QTQ)h = XTy + 0 r Q hp

(7)

If the same weight is given to the additional information for each ordinate, i.e. 0 = ul, then the solution becomes: (XTX + kI)h = XTy + khp

(8)

where k is equal to or. Comparison of equation (8) with the standard ridge regression (i.e. smoothed least squares) solution given by equation (3), shows that the latter implicitly assumes additional information of the form h = 0. Although this does smooth the solution it reduces the values of the estimated ordinates, violating continuity, and some volume correction is required (Bruen & Dooge, 1984). The coefficient matrix in equation (8), and thus the condition number, is the same as in equation (3) and the estimate can be obtained with Zohar's (1974) efficient method. The value oik can be chosen subjectively to reflect the weight to be given to the additional information provided by equation (5). If the synthetic shape, h , is obtained completely independently of the measured input and output data and if equations (1) and (5) are interpreted in a probabilistic manner and e and e are assumed to be normally distributed with zero means and variance matrices of a2I and ozl respectively, then equation (8) approximates the posterior maximum likelihood estimate of the parameters, using both the prior information and the input and output data, when k is s2/aj7, where sz is the sum of squares of the residuals of the OLS solution of equation (1) alone (Bruen, 1985). Here k clearly is a weighting of the relative worth of each source of information.

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E X A M P L E O F USE O F ADDITIONAL INFORMATION The use of smoothed least squares with prior information on the basis of equation (8) is illustrated for the River Almond at Almondell. An a priori shape for the unit hydrograph and a priori values for any parameters involved were required. In this example, the simple unit hydrograph shape recommended by the UK Flood Studies Report (NERC, 1975) was used. Its rising limb is a straight line and its recession is formed by two straight lines (Fig. 4). It is completely determined by specifying its time to peak, T . In the absence of any other information, the time to peak may be estimated from a regression on catchment characteristics as given in the Flood Studies Report (NERC, 1975): -0.38;UA4(l•0.14, + URBAN) -1.99 T. = 46.6RSMD-UAS1085~Ui*MSL

(9)

where: Tp is time to peak of the response (h); MSL is main stream length (km); 51085 is the stream slope between 10% and 85% of its length (m km"1); URBAN is the fraction of the catchment urbanized; and RSMD is the daily rainfall volume of 5-year return period less the effective mean soil moisture deficit (mm). For the Almondell catchment, the above variables had the values: RSMD = 39.9 mm; S1085 = 5.49 m km"1; MSL = 30.24 km; and URBAN = 0.099. The time to peak, given by equation (9) was thus 7.46 h and this completely specified the a priori, unit hydrograph. Using this shape as prior information, with the regularization parameter k equal to 0.05 and 1.0, gave the estimated unit hydrographs shown in Fig. 5. The degree of smoothing was similar to that in Fig. 3, but with no reduction in volume. There is a kink in the curve however between 18 and 19 h, i.e. approximately the duration of the FSR unit hydrograph. The unit hydrograph ordinates for times greater than this are estimated only from the data. The kink

Tp

ordi

c

—••J **1

Qp

sz

"5.

sz ' 0.5 Qp c

/, ^^

BSa^. ! ^ S .

1,2 Tp

2Tp

Tp 2.52 Tp

\ .

Time

Fig. 4 Unit hydrograph shape (Flood Studies Report).

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does not occur if the a priori unit hydrograph is smooth for the full duration of the estimated unit hydrograph, for example a conceptual model consisting of two linear reservoirs in series, Fig. 6.

1 £ o

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20 Time (h)

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Fig. 5 Unit hydrographs (SLS-CC: smoothed least squares with prior information).

! !

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Fig. 6 Unit hydrographs (SLS-CC: smoothed least squares with prior information). The regression equation (9) applies to catchments in the British Isles. For other catchments, and even when the catchment characteristics are not known, the time to peak may be approximated by the catchment lag. This may be estimated from the data using the method of moments. The first moment of the input series is subtracted from the first moment of the output series to give the

Unit hydrograph estimation: I. Tlieoty and computer program

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first moment or lag, Tlag, of the response. If several events are available, the lag is estimated as the average of the individual lags calculated for each event. The more events used in the average the more reliable should be the estimate. For the Flood Studies Report unit hydrograph, the time to peak is 0.86 times the lag. This is not independent prior information and so the resulting estimate does not necessarily approximate maximum likelihood. However, any simple conceptual model gives the shape expected of a unit hydrograph and is better able to bias the estimate towards that shape than towards the zero vector as in the ordinary regularization method.

CONCLUSIONS The main purpose of estimated unit hydrographs is their use to predict the direct runoff due to any excess precipitation event which has not been used in the estimation of the unit hydrograph. The only reliable method of assessing its usefulness in this respect is a split sample analysis (KlemeS, 1986). The method described here is compared in this way with ordinary least squares and smoothed least squares (regularization) in the following paper (Bruen & Dooge, 1992). Using prior information in the way described represents a trade-off. It can produce more acceptable and stable unit hydrographs, but at the expense of an optimal linear model fitting of the data. The method described here does not involve a significantly more complex computer program. Bruen & Dooge (1984) pointed out that ridge regression for unit hydrograph derivation required only a single extra arithmetic addition more than the least squares method. This is less than the computation required to derive a unit hydrograph by the least squares method and then smooth it. Including the prior estimate of the unit hydrograph does not involve an undue amount of extra programming. A computer program which can be used for all three methods is given in Appendix A.

REFERENCES Acreman, M. C. (1985) Predicting the mean annual flood from basin characteristics. Hydrol. Sci. J. 30, 37-49. Anderssen, A. S. (1971) Studies in system identification. PhD thesis, University of Queensland, Brisbane, Australia. Bruen, M. (1985) Black-box methods of systems analysis applied to modelling of catchment behaviour. PhD thesis, National University of Ireland. Bruen, M. & Dooge, J. C. 1. (1984) An efficient and robust method for estimating unit hydrograph ordinates.7. Hydrol. 70, 1-24. Bruen, M. & Dooge, J. C. I. (1992) Unit hydrograph estimation with multiple events and prior information: II. Evaluation of the method. Hydrol. Sci. ./. 37(5), 445-462. Dooge, J. C. I. (1959) A general theory of the unit hydrograph. J. Geophys. Res. 64(2), 241-256. Dooge, J. C. I. (1973) Linear theory of hvdrologicsvstems. Tech. Bull. no. 1468, USDA, Washington, DC, USA. Dooge, J. C. I. & Bruen, M. (1989) Linear algebra and unit hydrograph stability. J. Hydrol. I l l , 377-390. Farden, D. C. (1976) Inversion of Toeplitz equations: Computer program. Trans. IEEE AP-24, 906-907.

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Hoerl, A. E. & Kennard, R. W. (1970a) Ridge regression. Biassed estimation for nonorthogonal problems. Technometrics 12, 55-67. Hoerl, A. E. & Kennard, R. W. (1970b) Ridge regression. Applications to nonorthogonal problems. Technometrics 12, 69-82. Klemes, V. (1986) Operational testing of hydrological simulation models. Hydrol. Sci. ./. 3 1 , 13-24 Kuchment, L. S. (1967) Solution of inverse problems for linear flow models. Sov. Hydrol. Selected Pap. 2, 194-199. Lawson, C. L. & Hanson, R. J. (1974) Solving Least-Squares Problems. Prentice-Hall, New Jersey, USA. NERC (1975) UK Flood Studies Report. Natural Environment Research Council, UK. Tikhonov, A. N. (1963) Solution of incorrectly formulated problems and the regularization method, (in Russian). Dokl. Akad. Nauk SSSR 151, 501-504. Tikhonov, A. N. (1965) Improperly posed problems of linear algebra and a stable method for their solution. English translation in Sov. Math. 6, 988-991. Zohar, S. (1974) The solution of a Toeplitz set of linear equations. J. Assoc. Comput. Mach. 2 1 , 272-276.

APPENDIX A : COMPUTER P R O G R A M

KER6TMB

Purpose This subroutine estimates a pulse response (unit hydrograph) for a discrete, linear, time-invariant, single input-single output system from data for one or more isolated events. It allows prior information about the catchment lag to be used to bias the estimate. This gives a more stable estimate. The pulse response suggested in the UK Flood Studies Report is used as the shape of the bias function. This program is an extension of subroutine KER6TM given by Bruen & Dooge (1984).

Method The method is an extension of the method of regularization (Kuchment, 1967), otherwise called ridge regression (Hoerl & Kennard, 1970a) or smoothed least squares (Bruen & Dooge, 1984). The normal equations (equation (6)) are solved using an efficient algorithm for Toeplitz equations (Zohar, 1974) programmed by Fard en (1976).

Language and availability The subroutine was written in FORTRAN by M. Bruen and a listing is included in Appendix B.

Calling arguments The subroutine is called in the usual way: CALLKER6TMB(XS,YS,Z,NXS,NYS,NEV,NZ,R,TP,IER) where:

Unit hydrograph estimation: I. Theory and computer program

XS

YS

Z NXS

NYS

NEV

NZ

R

TP

1ER

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is a REAL vector containing, in sequence, the values for the input series for each event. This must be supplied by the calling routine and is not altered by KER6TMB; is a REAL vector containing, in sequence, the values for the corresponding output series for each event. This must be supplied by the calling routine and is not altered by KER6TMB; is a REAL vector which, if KER6TMB is successful, is the estimate of the pulse response; is an INTEGER vector containing the number of input series values in each event. These must be set by the calling routine and are not altered by KER6TMB; is an INTEGER vector containing the number of output series values in each event. These must be set by the calling routine and are not altered by KER6TMB; is an INTEGER variable containing the number of events available for the estimate. It must be set by the calling routine and is not altered by KER6TMB; is an INTEGER variable containing the number of values required in the estimated pulse response. It must be specified by the calling routine and is not changed by KER6TMB; is a REAL variable containing the smoothing constant, k. It must be set by the calling routine and is not altered by KER6TMB. If R is set to zero then the ordinary least squares estimate is calculated. is a REAL variable containing an estimate of the time to peak of the pulse response of the catchment expressed as the number of data measurement intervals. If no information about it is available, TP should be set to 0.0 It must be set before calling KER6TMB and is unchanged by it; and is an INTEGER variable which will be set to 0 if KER6TMB did not detect any problem, and set to 999 otherwise. Its value should always be checked by the calling routine after KER6TMB finishes.

Restrictions (a)

(b) (c)

The dimension allocated to internal arrays restricts the maximum number of pulse response values to not more than 150. This restriction can easily be removed by redimensioning the arrays AC, CC and WORK to the required number of pulse response terms, and changing the value assigned to MAXDIM accordingly. To solve the set of Toeplitz equations, this subroutine calls Farden's (1976) subroutine TPSLV, a listing of which is given in Appendix D. The input data should be in comparable units and the catchment lag, if specified, should be in terms of the number of data measurement intervals.

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Sample calculation The following simple sample problem can be used to check the program. Use the following data from two independent events, measured at hourly intervals: 1st event Input values :1.0, 2,0, 3.0 Output values : 0.045, 0.355, 0.845, 1.305, 0.995, 0.73, 0.505, 0.39, 0.285, 0.21, 0.15, 0.115, 0.05, 0.02 2nd event Input values :3.0, 2.0, 1.0 Output values : 0.2, 0.8, 1.2, 1.05, 0.85, 0.525, 0.48, 0.265, 0.28, 0.125, 0.175, 0.12, 0.075 and assume the catchment characteristics are: 51080 = 30 m km"1; RSMD = 80 mm; MSL = 2 km; and URBAN = 0.02. The above data give an estimated catchment lag of 2.35 h. The values of the calling arguments for KER6TMB are thus: XS = 1.0, 2.0, 3.0, 3.0, 2.0, 1.0 YS = 0.045, 0.355, 0.845, 1.305, 0.995, 0.73, 0.505, 0.39, 0.285, 0.21, 0.15, 0.115, 0.05, 0.02, 0.2, 0.8, 1.2, 1.05, 0.85, 0.525, 0.48, 0.265, 0.28, 0.125, 0.175, 0.12, 0.075 NXS = 3, 3 NYS = 14, 13 NEV = 2 NZ = 12 R = 0.5 TP = 2.35 The results given by the program are: 1ER = 0, i.e. no problems were detected Z = 0.0849,0.2396,0.2274,0.1490,0.1033,0.0500,0.0545,0.0319, 0.0287, 0.0165, 0.0187, 0.0092 Note the values of Z are rounded to four places of decimals and have not been normalized to have unit volume.

APPENDIX B: LISTING OF SUBROUTINE KER6TMB SUBROUTINE KER6TMB(XS,YS,Z,NXS,NYS,NEV,NZ,R,TP,1ER) CCCCCCCCCCC-

Estimates linear pulse response. Using Smoothed Least Squares (Régularisation) with or without additional "prior" information about time to peak of catchment. Pulse response from Flood Studies Report with the given time to peak is used as the "prior information" M.Bruen, Department of Engineering hydrology, UCG. July 1991.

DIMENSION^S(l) ,YS(1) ,Z(NZ) , NYS (NEV) , NXS (NEV) , AC (150) , * CC(150),WORK(150) DOUBLE PRECISION DTEMP C- Arrays are dimensioned for maximum C- of 150 response values.

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MAXDIM=150 C- Check problem dimensions 1ER = 999 IF((NZ.GT.MAXDIM).OR.(NZ.LE.O))GOTO 900 C- Number of pulse response terms is within required limits. IER=998 IF(NEV.LE.0)GOTO 900 C- There is at least one data event to use. IER=997 IF(R.LT.0.0)GOTO 900 C- The régularisation parameter is not negative C- All checks o.k. so far CC- Initialise arrays for auto- and cross- products. DO 10 1=1,NZ AC(I)=0.0 10 CC(I) = 0.0 IDX = 0 IDY = 0 C- Add in contributions from each event to autoC- and cross- products DO 60 1=1,NEV C- For each input/output event ... C- Get number of input values in the I th. event. NX = NXS(I) C- Check that its o.k. C- (no upper limit imposed by this program) 1ER = 995 IF(NX.LE.0)GOTO 900 C- Its o.k., now get number of values in output series. NY=NYS(I) C- Check that there are sufficient values. 1ER = 994 IF (NY.LT.NX)GOTO 900 1ER = 0 C- There are sufficient values, C- calculate auto and cross products. DO 20 J=1,NX LIMUP=NX-J+1 DTEMP=0.0D0 ITEMP=IDX+J-1 DO 30 K=1,LIMUP C- Note : summation is in Double Precision 30 DTEMP=DTEMP+XS(IDX+K)*XS(ITEMP+K) 20 AC(J)=AC(J)+DTEMP ITEMP1=MIN0(NY,NZ) DO 40 J=1,ITEMP1 LIMUP=MIN0(NX,NY-J+1) DTEMP=0.0D0 ITEMP=IDY+J-1 DO 50 K=1,LIMUP C- Note summation is in Double Precision 50 DTEMP=DTEMP+XS(IDX+K)*YS(ITEMP+K) 40 CC(J)=CC{J)+DTEMP IDX=IDX+NX 60 IDY=IDY+NY CC- Scale the auto and cross product series. T1=AC(1) AC(1)=1.0 CC(1)=CC(1)/T1 DO 70 1=2,NZ AC(I)=AC(I)/T1 70 CC(I)=CC(I)/T1 CC- Add in prior information if R.GT.0.0 CIF (R .GT. 0.0) THEN CRégularisation is required AC(1) AC(1) THEN + R about time to peak is given. CIF(TP.GT.O.O) Prior =information

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M. Brueii & J. C. I. Dooge CALL IOHUH(Z,NZ,1.0,TP,IER) IF (IER.EQ.O) THEN DO 80 I = 1, NZ CC(I) = CC(I) + R * Z(I) ENDIF

80 ENDIF ENDIF

C- SOLVE SYMMETRIC TOEPLITZ EQUATIONS. CALL TPSLV(NZ,AC,CC,Z,WORK) 900 RETURN END

APPENDIX C: LISTING OF SUBROUTINE IOHUH This routine calculates the DT unit hydrograph shape specified in the Flood Studies Report (NERC, 1975) for given values of time step, DT, and time to peak, TP. Note only the shape is used and scaled according to TP and DT. For time steps other than 1 h this is not the same as the DT unit hydrograph corresponding to the 1 h Flood Studies unit hydrograph. cccCcccccccccccc-

SUBROUTINE IOHUH(UH,NZ,DT,TP,1ER) Generates ordinates of a DT-unit pulse response sampled at intervals of DT. Uses the shape of the Standard Unit Hydrograph in the Flood Studies Report for the specified time to peak, TP. M.Bruen, Department of Engineering Hydrology, UCG, 1991 Calling arguments are .... UH DT-unit Pulse response (unit hydrograph) values (OUTPUT) NZ Number of values required in pulse response (INPUT) DT Time step (hours) between values (INPUT) TP Required time to peak (hours) (INPUT) 1ER Error flag, set to 0 if everything o.k. (OUTPUT)

DIMENSION UH(NZ) c- Check value of lag. IF (TP .GT. 0.0) THEN TB=2.52*TP TW=1.7*TP QP=0.817*DT/TP QP05=QP*0.5 TEMPI = QP05/(TW-TP) TEMP2 = QP05/(TB-TW) T=0.0 DO 10 1=1,NZ T=T+DT IF (T.LT.TP) THEN F=QP*T/TP ELSEIF (T.LT.TW) THEN F=QP-TEMP1*(T-TP) ELSEIF (T.LE.TB) THEN F=TEMP2*(TB-T) ELSE F=0.0 ENDIF 10 UH(I)=F 1ER = 0 ELSE cnegative or zero lag not allowed. 1ER = 999 ENDIF c-

Unit hydrograph

estimation: I. Tlieory and computer

program

RETURN END

APPENDIX D: LISTING OF SUBROUTINE TPSLV This routine, published by Farden (1976), is given here for completeness. SUBROUTINE TPSLV(N,R,PS,W,E) FARDEN,D.C., TRANS IEEE, AP-24, PPS.906-907,1976 DIMENSION R(N),PS(N),W(N),E(N) NM1=N-1 CINV=1.0/R(1) DO 101 1=1,NM1 R(I)=CINV*R(I+1) 101 PS(I)=CINV*PS{I) PS(N)=CINV*PS(N) W(1)=PS(1) E(l)=-R(l) AMB=1.0-R{1)*R(1) DO 150 1=1,NM1 11=1+1 THET=PS(I1) ET=-R(I1) DO 105 L=1,I LI=I-L+1 THET=THET-W(L)*R(LI) 105 ET=ET-R(L)*E(LI) C1=THET/AMB C2=ET/AMB 11=1+1 IBAR=Il/2 DO 110 L=1,I LI=I1-L 110 W(L)=W(L)+C1*E(LI) DO 115 L=1,IBAR LI=I1-L C3=E(L) E(L)=C3+C2*E(LI) 115 Received 5E(LI)=E(LI)+C2*C3 September 1990; accepted 9 March 1992 IF(IBAR-I/2)12 5,12 5,12 0 120 E(IBAR)=C3*(1.0+C2) 125 W(I1)=C1 E(I1)=C2 150 AMB=AMB-C2*ET RETURN END C-