Time-varying cascade model for flow forecasting

Hydrological forecasting - Prévisions hydrologiques (Proceedings of the Oxford Symposium, April 1980; Actes du Colloque d'oxford, avril 1980): IAHS-AISH Publ. no. 129. Time-varying cascade model for flow forecasting S. K. SPOLIA Indian Institute of Technology, New Delhi, India Abstract. Considering the lag time in a cascade model of equal linear reservoirs to be proportional to the number of reservoirs, a stochastic cascade model for the determination of the unit impulse response function is proposed. This model can be used for sampling a number of impulse response functions, with lag time of the cascade belonging to the Erlang density function, for use in planning and design. The cascade model is shown to be deterministic if the average lag time is used as a parameter instead of lag time sampled from the Erlang density function. Two versions of the deterministic cascade model are then formulated for use in flow forecasting. These versions are termed time-invariant and time-varying depending upon whether the average time lag parameter is considered to be constant or time varying. Modèle en cascade à temps variable pour la prévision des débits Résumé. En admettant que le temps de réponse dans un modèle en cascade de réservoirs linéaires est proportionnel au nombre de réservoirs, on propose un modèle stochastique en cascade pour déterminer un certain nombre de fonctions de réponse aux impulsions unitaires. Ce modèle peut être utilisé pour l'échantillonnage du nombre d'impulsions de la fonction de réponse, le temps de réponse de la 'cascade' correspondant à la fonction de densité d'erlang, en vue de la planification et de la mise au point des projets. Le modèle en cascade s'avère déterministe si le temps de réponse moyen est utilisé comme paramètre au lieu de tirer au hasard ce temps de réponse de la fonction de densité d'erlang. Deux versions du modèle en cascade sont présentées pour utilisation dans la prévision des débits. Ces versions sont appelées modèle à temps constant ou modèle à temps variable suivant que le paramètre temps de réponse moyen est considéré comme constant ou variable avec le temps. INTRODUCTION Most of the linear conceptual basin models for the transformation of rainfall excess into surface runoff are time invariant and are formed from simpler components, either the linear reservoir and/or a channel or both. All hydrological systems are, strictly speaking, time varying if not nonlinear. Chiu and Bittler (1979) used a time-varying storage time delay constant K(t) to develop the time-varying impulse response function of a linear reservoir to describe the time variability of the relationship between rainfall excess and surface runoff. Mandeville and O'Donnell (1973) introduced the concept of time variance for the impulse response function of paired combinations of a linear reservoir and/ or a linear channel and other linear models such as time-area methods and multiple reservoir cascades. The common features of these studies and those of Dooge (1959) and Nash (1960) (who consider the system to be time-variant) are a deterministic approach, lumped system and determination of the impulse response function of the system. In this study a cascade model of equal linear reservoirs for the lumped stochastic system for the identification of the impulse response function is proposed. A further deviation from the earlier approach of representing the total lag time (T) of a system in terms of the storage delay time of each time-invariant and time-varying linear reservoir of the cascade (K and K(t) respectively) is made by considering the storage delay time of each reservoir as a function of the total lag time. Cascade models for a lumped deterministic hydrological system, time invariant and time varying, are then formulated for use in flow forecasting. 319

320 S. K. Spolia TIME-INVARIANT STOCHASTIC CASCADE MODEL Nash (1960) put forward a deterministic cascade model of equal reservoirs to represent basin behaviour in the transformation of rainfall excess into surface runoff. The storage discharge relationship of each reservoir is given by S i (t) = KQ i (t) (1) where S,-(0 is the storage in the /th reservoir at time t; Qi(t) is the discharge from the z'th reservoir at time t; and K is lag time in the reservoir. One important property of this model is that it conserves flows, that is, all flows that enter the cascade at the input point (first reservoir of the cascade) either leave the output point (last reservoir of the cascade) or remain stored inside the system. Thus the cascade system is defined by the following n first-order differential equations (/ = 1, 2,..., ri): 7<2/(0 = 7[G/-i (0-<2/(0] (2) at K where n is the number of reservoirs in the cascade. If /(?) is the input into the cascade, Q(t) is the output from the cascade and S(t) is the storage in the cascade, then: and ( I(t) for i = 0 0/(0= \ nm.. (3) 1(2(0 for i = n S(0=S(0)+p[/(T)-fi(T)]dT (4) Jo Alternatively it can also be shown that storage in the cascade is S(t)=Ky" &(0 + S(0) (5) If it is required to compute storage in a cascade, then equation (5), often computes S(t) with less numerical error than does equation (4) (Manetsch, 1976). Let Tbe the lag time through the cascade with probability density function f(t). Let T be introduced in equation (2) so that the cascade system be defined by the following n first-order differential equations (i = 1, 2,..., «): 7 0/(0 = 7 [Ô/-1 (0-0/(0] (6) at T then equations (3) and (5) are still applicable to the cascade model representing the stochastic process with K replaced by T/K; K now represents some fraction of T in the stochastic model. The system of first-order differential equations given by equation (6) can be combined into a single equation: V a i D"- i Q(t)=I(t) (7) The unit impulse response h(t, r) of the linear stochastic hydrological system of equation (7) is given by hit ' T) K ftkx"^1 = Trn\T) ex^tkit) (8)

Time-varying cascade model 321 which is obtained from the solution given by Nash (1960) by replacing K by T/K. The impulse response function is independent of the time of occurrence r of a unit impulse. Equation (8) yields a family of response functions for various values of T sampled from the density function/(t). If K = n, i.e. when delay time is distributed in the cascade in proportion to the number of reservoirs then the impulse response function and the probability density function describing the delay times of flow population passing through the cascade are both respectively, given by the Erlang density functions (Manetsch, 1966): h{ur)= i±\. W»-i e xp(-/ir/r) (9) in \ TI f(t )= ±ljl\ (r)»-i. ex p(-nr/mr) (10) Tn \UT' where the mean, variance and the mode of/(2") are given respectively by Mr > MÎï,jn and n(n 1 )//x r ; as n ->, f(t) -* N(jx Ti 0 ). Substitution of the mean delay time H T for T in equation (9) gives the deterministic impulse response function. TIME-VARIANT DETERMINISTIC MODEL In flood forecasting problems a deterministic model is invariably used; thus, the impulse response function obtained from equation (9) with T replaced by Mr can be used to obtain the unit hydrograph. The unit hydrograph when convoluted with the input will give the forecasted flow rates. The convolution process can be avoided if flow rates are directly expressed in terms of the input rates. This is accomplished as follows: The best representation of the solution of the first-order equations describing a deterministic cascade is obtained by replacing 7 with fx T in equation (6). If these equations are then integrated over an interval (t, t + At), one obtains: ' ( + A( dqiir) r t+ At K dr= [ft-1 (0-0,(01 dr (11) dr Jt \x T Using the Euler integration approximation t + At f(u)du~-< Atf(u) (12) this set of equations can be written as: KAt Q t (t + A0 = ft(0 + [ft _ i (0 - ft(01 for i = 1, 2,..., n (13) Recognizing that Q 0 (t) = I(t) and ft, (0 = Q(t), the outflow from the cascade can be found by induction as: 0 for i < n Q(i) = {(/AtK\[^i-n [i - 1 -/' Ê:)E:r(';;-;'K"-'H) *""*"

322 S. K. Spolia When K = n belongs to an Erlang density function so that [o for i < n 0(0 = Am 1-/ n -) I (J) for z > n Mr * <] = l \ n -1 where (. I is the binomial coefficient i!//'! (/ - /')! ; and F is given by / Am \ \ Mr / (14) (15) TIME-VARYING DETERMINISTIC MODEL In modelling hydrological systems, one often encounters processes in which the delay time parameter is time-varying. A case is now considered where /u T is a function of time n T (t). The storage in the z'th reservoir with Mr(0 distributed in proportion to the number of reservoirs is given by MO S i (t) = J^Q i (t) The conservation of flow requires that ds,(0 Q t -x{t)-qi(t) àt which on insertion in equation (16) gives the set of n first-order differential equations (z'= 1,2,...,«): dqi(t) Qi-i(t)-Qi(t) 1 + At n T (t) If dii T (t)/dt can be approximately expressed as d/i r (0 ii T (t) - vat - &t) (18) At At the set of equations (17) can be integrated to obtain flow Qj(t) over the interval (t, t + At) which on using Eulef's integration approximation, equation (12), gives the following set of equations (for i = 1,2,...,n): Q,(t + At) = Q i (t) H T(t - At) - nat nat MO MO e*-i(o (20) Knowing that Q 0 (t) =I(t), the outflow Qi(t) from the first reservoir can be computed by setting / = 1 in equation (20) for a fixed interval At. This outflow forms the inflow into the second reservoir. Again, equation (20) can be used for i = 2 to obtain the outflow from the second reservoir and so on. Thus the outflow from the last reservoir of the cascade is: 0 for i < n 0(0 : /(/) 1 x àt n' l = K! MO F{m)ii T (rn) I i - 1 m = k, m j (16) (17) for z > «(21)

where Time-varying cascade model 323 *X0 = 7X MrO) mi =(i n K)+ I M-l F(m)n 2 T (m) m=k,m y is the sum of the products of FQri) and Mr( m ) taken m, at a time for m varying between A^ and i 1 ; and 2i-i F(m)n T (m)= 1 m = ic, 0 fi' ' Mr(0=l for/>i-l DISCUSSION AND CONCLUSIONS The majority of linear cascade models proposed to explain the transformation of excess rainfall into surface runoff are deterministic and time invariant. After the formulation of a stochastic model for the unit impulse function, expressions are developed for the time-invariant and time-varying deterministic cascade model. The stochastic model can be used to obtain a family of response functions with lag time sampled from an Erlang density function. The method of moments can be used for the computation of its parameters. The proposed time-invariant and time-varying cascade models can be used for flow forecasting. The time-invariant model can be identified from the observed data on inflow and outflow following the straightforward method of moments. However, this task is not simple in the case of the time-varying cascade model. Although, Mandeville and O'Donnell (1973) and Mandeville (1970) have considered the theoretical aspects of identification of such models, further experience is necessary in the identification of such a model. One of the possibilities is to assume a polynomial function of t for Mr(0 an d evolve a best function through a large number of iterations. Alternatively, it may be possible to fit a suitable empirical time function from observed values of fi T of various durations for a given basin. Another resource that can be followed is to formulate the model in an equivalent state space formulation and use the extended Kalman filter approach. All these possibilities need to be thoroughly examined before the model can be used in practice. REFERENCES Chiu, C. and Bittler, R. P. (1969) Linear time varying model of rainfall-runoff relation. Wat. Resour. Res. 5, no. 2, 426-437. Dooge, J. C. I. (1959) A general theory of the unit hydrograph. /. Geophys. Res. 64, no. 2, 241 256. Nash, J. E. (1960) A unit hydrograph study, with particular reference to British catchments. Proc. Instn. Civ. Engrs. 17,249-282. Mandeville, A. N. (1970) Time variant linear catchment system. MSc Thesis, Imperial College, University of London, London, UK. Mandeville, A. N. and O'Donnell, T. (1973) Introduction of time variance to linear conceptual catchment models. Wat. Resour. Res. 9, no. 2, 298-310.

324 S. K. Spolia Manetsch, T, J. (1966) Transfer function representation of the aggregate behaviour of a class of economic processes. Inst. Elect. Electron. Engrs. Trans. Automat. Control AC 11, 693 698. Manetsch, T. J. (1976) Time varying distributed delays and their use in aggregative models of large systems. Inst. Elect. Electron. Engrs. Trans. Sys., Man. and Cybn. SMC-6 no. 8, 547-553.