P. Johnson and M. Clements

Transcription

1 Determination of instantaneous unit hydrographs by least squares polynomials P. Johnson and M. Clements Abstract. After reviewing the dichotomy of approach used in deriving unit hydrographs, the theory of least squares polynomials and their application to the convolution integral is detailed. A method for deriving the instantaneous unit hydrograph using a least squares polynomial technique is outlined and results of applying the method to three real flood events are discussed. Resume. Après avoir examiné la dichotomie d approche utilisé en dérivant des hydrographes uni&, la théorie des moindres carrés polynomials et son application a l intégrale de circonvolution a été détaillé. Une méthode pour deriver l hydrographe unité instantané en utilisant la technique des moindres carrés polynomials et esquissée et les résultats d application de la méthode à trois cas réels de crues sont examinés. INTRODUCTION During the last 15 years several methods have been developed for the derivation of unit hydrographs of natural catchments using recorded rainfall and stream discharge data. The methods have been derived mainly to overcome the basic problem of numerical instability experienced when the technique of successive ordinate evaluation by forward substitution is used. As previously described by Eagleson et al. (1966), these methods may be considered as belonging to one of two approaches, namely black box analysis, and parametric system synthesis, respectively. In black box analysis, little consideration is given initially to the conceptual characteristics of the unit hydrograph. Instead, well-known mathematical and statistical techniques are employed to determine appropriate numerical values of unit hydrograph ordinates, or coefficients of a functional series representing the unit hydrograph. Examples of such methods include, least squares regression analysis (Snyder, 1955), harmonic analysis (O Donnell, 1960), Laguerre series analysis (Dooge, 1965), Wiener- Hopf optimization with constraints (Eagleson et az., 1966) and spectral analysis with Fourier series (Bayazit, 1966). A novel approach, in which rainfall data are not required, was recently added to this list, Delaine (1970). In this, ordinates of the unit hydrograph are determined through identification of equivalent factorial values of two polynomial expressions. Both expressions represent successive discharge values measured at discrete time intervals for two distinct storms. The method is demonstrated with synthetic data only. In contrast to black box analysis, parametric system synthesis seeks firstly to represent the unit hydrograph by a conceptually meaningful analytical expression, and then secondly to determine suitable parametric values to enable the expression to best represent a particular catchment response. The first phase of this process, the conceptual representation, has been much discussed. Two notable early contributions were made by Clark (1 945) and O Kelly (1955). Since these, several significant studies have been reported and include Nash (1957), Dooge (1959), Sin& (1962), and Kulandaiswamy (1964). The second phase, parametric evaluation, is not SO well investigated; approximate graphical methods are often adopted. A well-known objective method however

2 484 P. Johnson and M. Clements was proposed and applied to a two-parameter model (Nash, 1959). In this method, a solution is achieved through a special linkage equation which enables moment transforms of the instantaneous unit hydrograph to be calculated from previously computed moment transforms of the hyetograph and storm hydrograph. Parameter values are related to the moment transforms. An analogous and equally objective method may also be adopted using Laplace transforms (Johnson, 1970). The potential of applying integral transforms to solve different parametric models has yet to be realized. Of the two approaches, black box analysis would seem to yield the more accurate results. This impression was recently reinforced by Laurenson and O Donnell(l969) and is believed to be due primarily to there being greater elasticity in unit hydrograph shape using black box analysis than there is usually associated with conceptual expressions. For this reason therefore unit hydrograph derivation is perhaps best carried out using a more reliable black box technique. Such a situation is of course undesirable for it is surely the wish of most hydrologists not only to be able to derive a unit hydrograph accurately, but also to understand the reasons for its shape. It may be argued therefore that a general method is needed which makes good use of the merits of black box analysis and yet is capable of being interpreted in a conceptual manner. The Laguerre series method as proposed by Dooge purports to do this to a limited extent and the coefficients of other series solutions might also enabke parametric evaluation. A method which may also fulfil both requirements is developed in this paper. The method depends on two theoretical aspects. Firstly, it is assumed that a function may be represented by a polynomial and it is shown that the coefficients of the polynomial are simply determined for best fit by least squares analysis. Secondly, coefficients of a polynomial representing the storm hydrograph are identified with equivalent coefficients of the instantaneous unit hydrograph (I.U.H.) and Mellin transforms of the hyetograph, through the convolution integral. POLYNOMIALS AND LEAST SQUARES ANALYSIS (Milne, 1949; McCracken and Dorn, 1964) It is assumed that any functionf(x) (continuously or discreetly defined), existing in the interval e Q x Q b, may be approximated by the polynomial For best fit it is also assumed that the sum of squares of differences between the function and its polynomial approximation wil be a minimum. The sum of squares of differences over the interval e < x,< b is represented by the integral Ja Since the value of the integral wil vary, depending upon coefficient values in the polynomial and since both the integral and its second derivative are always positive, it follows that a minimum wil exist. This can be determined through the general relationship ar -- -o a pk fork=o, 1,..., n

3 i.e. or ai b - = 2Ja [P(x)-f(x)]xkdx = o a pk b f(x).2& =s, P(x).xkdx Determination of instantaneous unit hydrographs 485 for k=o, 1,..., n Substituting for P(x) from (I) f (x). 2 dx = Jub (Po + p1x + p2x p" x") xk dx or where n mfk = 2 Pj.Akj j=o mfk = fork = O, 1,..., n b f(x) xk dx (a kth order Mellin transform) and bj+k+i -,j+k+i Akj = j+k+l The relationship represents n + i linearly independent equations: Aoo~o +Aoi~i +Ao2~ AonPn=mfo Aiop0 +AllPl fa12p2 +..' +Al"Pn = mfl A o ~ + o Anipi fan-2~ Ann& = mfn and sincef(x) is known, and its limits a and b are defined, the y1 + 1 unknown coefficients, po, pi..., pn, can be determined uniquely having first calculated the values Of Akj and mfk for all k and i. By representing the set of equations in matrix form, L4l(P) = (mf) the solution may be represented formally as (P) = L41-'(mf> (24 (2b)

4 486 P. Johnson and M. Clements POLYNOMIALS AND CONVOLUTION The relationship between effective rainfall intensity i(7), the I.U.H. u(0, t), and the storm hydrograph 4(t), in a stationary-lumped-linear-system, is given by the convolution integral q(t) = lof i(~).u(o, t - T) d7 (3) It is assumed that both the I.U.H. and the storm hydrograph can be represented by polynomials of degree n. n or u(o,t-7) = u0+u1(t-7)+u2(t-7) U,(t-7)n The general term (t- T ) in ~ (4b) is expanded to give ti-'( (- 7)j (t - 7)j = ti + jt+'(-- 7) + j r+) i 4 k=o where (i) is the binomial coefficient. Equation (5) is substituted into (4b) to give Equations (4a)and (6) are substituted into (3) to give where mik is the kth order Mellin transform of the hyetograph. Replacing j - k by p

5 the right-hand side becomes, after rearranging, Determination of instantaneous unit hydrographs 487 Writing j for p and k for i, In this equation, coefficients of like powers oft must be equal, and thus which is an expression representing y1 + 1 linear equations, 40 = uomio-ulmil + u2miz - u3mi (-1)" un min ql = ulmil- 2uzmi2 + 3u3rni (-1)"-'Un min-i 42 = u2 mi2-3u3mi3 +,.. + (-I)n-2un min-2 etc. Written in matrix notation these equations become (4) = [MiI(u) for k < j The formal solution for uo, ul,..., un, is therefore METHOD (u) = [MiI-'(q) (7) To solve equation (7) and hence to define an I.U.H. in the form of a polynomial, Mellin transforms of the hyetograph and polynomial coefficients of the storm hydrograph must first be calculated. Since numerical evaluation of Mellin transforms and determination of polynomial coefficients through equation (2) are theoretically possible, determination of the I.U.H. is feasible. A procedure which makes use of the theoretical aspects is summarized below. (1) The polynomial degree, 'n', is chosen. (2) Mellin transforms of the storm hydrograph, of order ranging from zero up to and including IZ, are computed. (3) Polynomial coefficients for the storm hydrograph, qj, are determined by solving equation (2a) or (2b) (qj E pi and mfi E mqj). In forming the matrix [A], the lower limit of integration, 'a', is zero and the upper limit of integration, 'b', is equal to the duration of the storm hydrograph. (4) Mellin transforms of the hyetograph (this is usually represented as a simple bar diagram), of order ranging from zero up to and including IZ, are computed. The matrix [Mi] is then formed.

6 488 P. Johnson and M. Clements (5) Polynomial coefficients of the I.U.H., u,, are determined by solving equation (7). APPLICATION AND RESULTS The ability of the method to determine a reasonably accurate and realistic I.U.H. was studied initidly with synthetic data. Results of these tests were sufficiently successful to warrant the method's application to real flood data. Three flood events were analysed. The first was recorded on the Ashbrook catchment in southeast England and was first analysed by Nash (1957). The other two events occurred on the River Dovey catchment in North Wales. Results of the analyses are summarized in Figs. 1, 2 and 3 which illustrate the derived I.U.H.s, and the actual storm hydrographs compared with those hydrographs computed by convolution. In obtaining the computed curves polynomials of degree 16 were used. Convolved hydrographs in all three cases are noted to be good approximations to the actual storm hydrographs. Differences between the two hydrographs are further I U H -POLYNOMIAL DEGREE 16 I U -. L P Y (3 a d I U u> - o [ fi 300 ACTUAL STORM HYDROGRAPH /CONVOLVED STORM HYDROGRAPH, FIGURE 1. TIME ihrsl I.U.H. and storm hydrographs for the River Dovey - flood 1.

7 Determination of instantaneous unit hydrographs 489 c IUH- POLYNOMIAL DEGREE 16,ACTUAL STORM HYDROGRAPH,CONVOLVED STORM HYDROGRAPH FIGURE 2. O TIME ihrsl I.U.H. and storm hydrographs for the River Dovey - flood 2. illustrated for Ashbrook in Fig. 4 in which it is seen that large relative errors occur only at both ends of the hydrograph. These features are generally common to all three examples and are not unexpected in view of two unrealistic features of the 1.U.H.s. It wil be noted that the derived 1.U.H.s have non-zero values at the origins and tails which exhibit waviness. Apart from these two defects, the shapes of the 1.U.H.s are generally realistic. During the analyses, no attempt was made to force the L.S.P. solution into better agreement with observed data by imposing constraints. Imposition of rational restraints (e.g. making the constant coefficient in the I.U.H. polynomial zero) is of course a possible way by which better results may be obtained, but before these are adopted it is contended that several numerical aspects to improve results should first be investigated (e.g. the effect of number of data on the rising limb of the storm hydrograph). Work on both these aspects is now being carried out. In this connection the results of one investigation are of interest. The affect of polynomial degree on residual variance of fit using Ashbrook data is shown in Table 1; results for River Dovey flood data show a similar pattern. The last two columns of data represent the residual variance of difference between the actual storm hydrograph and respectively,

8 490 P. Johnson and M. Clements I l O00 CTUPL STORM HYDROGRAPH,CONVOLVED STORM HYDROGRAPH FIGURE 3. TABLE 1. TIME lhrrl I.U.H. and storm hydrographs for Ashbrook. Residual variance between computed and actual storm hydrographs Residual variance for normalized data (X Polynomial degree Initial L.S.P. fit to n storm hydrograph Convolved hydrograph Il

9 Determination of instantaneous unit hydrographs 49 1 POLYNOMIAL DEGREE 16 ~ ~~ O IS FIGURE 4. TIME Ihro) Relative error of convolved hydrograph far Ashbrook. and (1) the L.S.P. representing the storm hydrograph, calculated from equation (2), (2) the finally convolved hydrograph. Two important features are indicated in these results. Firstly, residual variances in both columns are similar in magnitude for the same degree of polynomial. This implies that errors in the derived I.U.H. occur mostly as the result of initial lack of fit of an L.S.P. to the actual storm hydrograph. Hence, if this particular aspect can be improved, the potential accuracy of the method wil be much greater. Secondly, high values of residual variance are noted to occur for polynomials with degrees 4, 9, 13, 18 and 22-23, but especially for degree 9. This undesirable feature suggests a cycle of potential inaccuracy related to polynomial degree and its reason as yet is not understood. CONCLUSIONS In as much as the method has only been used to analyse three real flood events it is not yet possible to comment on general accuracy to be expected. Analyses have proved however that the L.S.P. technique outlined in this paper is capable of yielding realistic results with an acceptable degree of fit. In its basic form, the L.S.P. method seems to promote unrealistic end conditions

10 492 P. Johnson and M. Clements of the I.U.H. These features are the main cause for local error occurring at both ends of the convolved hydrograph. It is believed that if a better initiai polynomial fit to the storm hydrograph can be obtained by adopting either more precise numerical evaluation or by using rational constraints, the method will produce results with a high degree of accuracy. The problems of producing more accurate results, and the possibility of using polynomial representation to determine parametric values of conceptual models, are currently undergoing investigation. Acknowledgements. The authors are grateful to Professor P. Novak for the interest shown during the development of the method and to Mrs D. Moran who patiently typed the script. They acknowledge also the assistance given by Mr D. Harrison, Engineer to the Gwynedd River Authority in supplying hydrological data. Ashbrook data were used with the permission of Professor J. E. Nash, University of Galway. Advice from DI J. Firth is also acknowledged. IIEFERENCES Bayzit, M. (1966) Instantaneous unit hydrograph derivation by spectral analysis and its numerical application. CENTO Symposium on Hydrology and Water Resource Development: Ankara. Clark, C. O. (1945) Storage and the unit hydrograph. Trans Amer. Soc. civ. Engrs 110, Delaine, R. J. (1970) Deriving the unitgraph without using rainfall data. J. Hydrol. 10, Dooge, J. C. I. (1959) A general theory of the unit hydrograph. J. geophys. Res. 64 (21, Dooge, J. C. I. (1965) Analysis of linear systems by means of Laguerre functions. J. SIAM Control, ser. A, 2 (3), Eagleson, D. S., Mejia, R. and March, F. (1966) The computation of optimum realisable unit.hydrographs. Wat. Resour. Res 2 (4), (Discussion 4 (1) (1968), ) Johnson, P. (1970) Calculation of the instantaneous unit hydrograph using Laplace transforms. J, Hydrol. 9 (2), Kulandaiswamy, V. C. (1964) A basic study of the rainfall excess-surface runoff relationship in a basin system. Ph.D Thesis, University of Illinois, Urbana. Laurenson, E. M. and O'Donnell, T. (1969) Data error effects in unit hydrograph derivation. Proc. Amer. Soc. ciu. Engrs, Hyd. Div. 95, no. HY6. McCracken, D. D. and Dom, W. S. (1964) Numericalhfethods and Fortran Programming: Wiley. Milne, W. E. (1949) Numerical Calculus, p. 242: Princeton Univ. Press.. Nash, J. E. (1 957) The Form of the Instantaneous Unit Hydrograph: Assoc. Int. Hydrol. I.U.G.G., Toronto. Nash, J. E. (1959) Systematical determinations of unit hydrograph parameters. J. geophys. Res. 64 (i), O'Donnell, T. (1960) Instantaneous unit hydrograph derivation by harmonic analysis. Eaux de Surface, pp : IAHS Publ. no. 51. OKelly, J. J. (1955) The employment of unit hydrographs to determine the flows of Irish arterial drainage channels. J. Instn civ. Engrs 4, Singh, K. P. (1962) A non-linear approach to the instantaneous unit hydrograph. Ph.D. Thesis, University of Illinois, Urbana. Snyder, W. M. (1955) Hydrograph analysis by the method of least squares. Proc. Amer. Soc. ciu. Engrs, Hyd. Div.,81, no. 793, 25 pp.