A variable storage coefficient model for rainfall runoff computation / Modèle pluie débit basé sur un coefficient de stockage variable

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: A variable storage coefficient model for rainfall runoff computation / Modèle pluie débit basé sur un coefficient de stockage variable P. K. BHUNYA, P. K. SINGH, S. K. MISHRA & N. PANIGRAHY To cite this article: P. K. BHUNYA, P. K. SINGH, S. K. MISHRA & N. PANIGRAHY (008) A variable storage coefficient model for rainfall runoff computation / Modèle pluie débit basé sur un coefficient de stockage variable, Hydrological Sciences Journal, 5:, 8-5, DOI: 0.6/ hysj.5..8 To link to this article: Published online: 8 Jan 00. Submit your article to this journal Article views: 79 View related articles Citing articles: View citing articles Full Terms & Conditions of access and use can be found at

2 8 Hydrological Sciences Journal des Sciences Hydrologiques, 5() April 008 A variable storage coefficient model for rainfall runoff computation P. K. BHUNYA, P. K. SINGH, S. K. MISHRA & N. PANIGRAHY National Institute of Hydrology, Roorkee 47667, Uttaranchal, India Department of Water Resources Development and Management, Indian Institute of Technology Roorkee, Roorkee 47667, Uttaranchal, India pushpendras@yahoo.com Abstract The study explores the suitability of a two-reservoir variable storage coefficient (VSC) model with respect to the popular two-parameter gamma distribution (PGD) model (Nash, 957) for direct runoff hydrograph (DRH) derivation from a small test catchment. The proposed model is based on the concept of VSC that utilizes a varying storage coefficient K (by ratio r) instead of a uniform storage coefficient K as in the PGD model. Further, the proposed model uses unit-step rainfall instead of unit-impulse as input. The results show the PGD model to respond faster to a given amount of input (excess rainfall) than the VSC model. Based on the results it is found that the VSC model performs significantly better than the PGD model in deriving the shape of the direct runoff hydrograph. However, for a comparatively larger catchment the PGD model performs better than the VSC model. Similar to the PGD model, the VSC model has potential for synthetic unit hydrograph derivation. Key words variable storage coefficient; gamma distribution; direct runoff hydrograph; conceptual; rainfall runoff Modèle pluie débit basé sur un coefficient de stockage variable Résumé L étude explore la pertinence d un modèle à deux réservoirs avec coefficient de stockage variable (VSC) par rapport au modèle populaire de la distribution gamma à deux paramètres (PGD) (Nash, 957), pour la simulation de l hydrogramme de ruissellement sur un petit basin versant de test. Le modèle proposé est basé sur le concept de VSC qui utilise un coefficient de stockage K variable (selon le rapport r) au lieu du coefficient de stockage K uniforme du modèle PGD. De plus, le modèle proposé utilise en entrée une pluie unitaire par pas de temps à la place d une pluie unitaire impulsionnelle. Les résultats montrent que le modèle PGD répond plus vite que le modèle VSC à une quantité donnée en entrée (pluie nette). Il apparaît des résultats que le modèle VSC est significativement meilleur que le modèle VSC concernant la forme de l hydrogramme de ruissellement. Cependant, pour un bassin versant plus grand, le modèle PGD est meilleur que le modèle VSC. De même que le modèle PGD, le modèle VSC a du potentiel pour simuler un hydrogramme unitaire synthétique. Mots clefs coefficient de stockage variable; distribution gamma; hydrogramme de ruissellement; conceptuel; pluie débit INTRODUCTON The rainfall runoff relationship is one of the most important aspects of catchment hydrology, in which rainfall is considered as the input and runoff as the output of the catchment (system). Since the estimation of peak flow rate, time to peak, and temporal variation of flow volumes form the key objectives of a hydrologist, a number of rainfall runoff models based on various concepts and associated assumptions are proposed in the literature. In general, they may be broadly classified into three main categories (Singh, 988): (a) process-oriented or physical models, (b) conceptual models, and (c) empirical models. The linear time-invariant conceptual model of Nash (957), widely used for runoff estimation especially for determining peak runoff rates utilizes a spatially lumped form of the continuity equation and the linear storage discharge relationship. Using a cascade of n equal linear reservoirs of uniform storage coefficient K, Nash (957) derived the instantaneous unit hydrograph (IUH) model in the form of a two-parameter gamma distribution (PGD) that is the most popular and widely accepted approach to approximate the real system. The derivation of the classical Nash model is obtained under the assumption that watershed behaviour can be associated with a cascade of linear reservoirs where rain falls instantaneously on the nth reservoir (Fig. (b)). Holtan & Overton (964) proposed a double routing method for direct runoff hydrograph (DRH) derivation using two linear reservoirs in series with the same storage coefficient. Singh (964) derived an IUH model using two linear Open for discussion until October 008 Copyright 008 IAHS Press

3 A variable storage coefficient model for rainfall runoff computation 9 (a) u(t) Q (t) K = K K =K r Q(t) (b) δ(t) K Q (t) Q (t) K K K Q(t) Fig. Representation of catchment by (a) a two reservoir variable storage coefficient (VSC) model; and (b) a two-parameter gamma distribution (PGD) model. reservoirs with unequal storage coefficients. Later, Chow & Kulandaiswamy (97) considered the cascade of equal and unequal linear reservoirs for deriving a unit hydrograph from observed data. However, the determination of storage coefficient in such cases is difficult, even if some of the reservoirs are assumed equal, as one obtains repeated roots which complicate the estimation procedure. Koutsoyiannis & Xanthopoulos (989) used the parametric approach to identify the forms of IUH from eight known probability density functions. Jakeman et al. (996) developed a transfer function model (approach) to derive the IUH. The transfer function model is equivalent to a linear reservoir model with n branches all connected in parallel. The transfer function models are an ideal form for block diagram analysis and the interpretation of the model in serial, parallel and feedback connections of sub-systems that often have physical significance (Young & Garnier, 006). Wang & Chen (996) proposed an analytical solution of the linear, spatially-distributed model over a watershed using an ordinary differential equation to represent the relationship between the input, output and function of the sub-watershed based on the continuity equation and the linear storage discharge relationship. However, they assumed a uniform storage coefficient which was computed for the whole watershed using an optimization scheme. Jeng & Coon (00) critically examined various issues related with Nash s IUH, such as placing the instantaneous unit-effective rainfall at the farthest nth reservoir and representing the catchment by a series of identical linear reservoirs. They proposed a general form of IUH, in which the catchment area was divided into a number of unequal sub areas and the reservoirs were weighted with contributing area in terms of input. This method could perform poorer, if it is not adjusted well for the corresponding contributing area. Dooge & O Kane (00) provided a detailed review of models based on IUH theory including single reservoir, connected reservoirs models, and on a parallel or sequence basis. Jain & Sinha (00) applied the geomorphologic IUH approach (Rodriguez-Iturbe & Valdes, 979) to derive the flood hydrograph resulting from a known storm in a basin area. Nasri et al. (004) proposed a production function based model similar to the Nash model. More recently, Agirre et al. (005) and López et al. (005) developed rainfall runoff models in which the watershed is represented as a cascade of linear reservoirs. The model of López et al. (005) based on the geomorphic IUH approach, represents the watershed as a cascade of reservoirs in two ways: (i) from the sub-watershed network structured along the drainage network, and (ii) by areas limited Copyright 008 IAHS Press

4 40 P. K. Bhunya et al. by iso-distance curves to the outlet point. The model essentially includes the catchment geomorphology in its formulation and assumes a constant storage coefficient K similar to the Nash model to get a simplified formulation of IUH or in turn the DRH. Yang & Han (006) used a model derived by the unit pulse response of a given discrete transfer function. The model was numerically stable, physically realizable, parsimonious in parameters, and easy to implement in real time for its state and parameter updating. The major drawback of this model is the complications involved in estimation of its parameters. Therefore, in order to investigate the proposition that the storage coefficient (K) should vary in time and space (Singh, 00), the objectives of the present study are two-fold: (i) to develop a simple but hydrologically sound conceptual model based on the variable storage coefficient (VSC) approach (Wang & Wu, 98); and (ii) to test its validity on the field data and its performance with respect to the popular PGD model. TWO-RESERVOIR VARIABLE STORAGE COEFFICIENT (VSC) MODEL The proposed VSC model relies partially on the concept of Wang & Wu (98), which considers the following assumptions: (a) the storage coefficient for the series of two unequal linear reservoirs varies (increases or decreases) in geometric proportion; and (b) the input can be expressed by a unit-step function instead of a unit-impulse function. Here, it is worth emphasizing that a second order model is often adequate to simulate the system behaviour and, conceptually, it is equivalent to two linear reservoirs in series (Box & Jenkins, 976). Thus, the present study aims to develop a mathematical model based on the VSC approach considering two unequal linear reservoirs in series with storage coefficients varying in geometric proportion. The model formulation is described in the following section. Model formulation Considering the catchment as a system in which excess rainfall is the input that gets transformed into direct surface runoff at the catchment outlet through a transfer function, Ogata (970), Dooge (97), and Dooge and O Kane (00) suggested the following input output relationship using a linear differential equation as: n and Q t + a dq t n n n ()/ dt + an d Q() t / dt + L m m m m ()/ dt + a Q() t = b d I() t / dt + b d I() t / dt + L + b di() t / dt + b I() t 0 m m where a n, a n-,, a 0 and b m, b m-,, b 0 are constants; m and n are positive integers with n > m; and Q(t) and I(t) are the output and input at any time t. Using the Laplace transform, equation () can be rewritten as (Kreyszig, 99): A s Y s D s = B s X s D s () ()() () () () ( ) Q I where s is the Laplace transform variable and Y(s) and X(s) are the Laplace transforms of Q(t) and I(t), respectively, expressed as Y(s) = L{Q(t)}and X(s) = L{I(t)}, and: n n () = an ( s ) + an ( s ) + + a0 n n Q ( s) = anq( 0) ( s ) + [ anq ( 0) + an Q( 0) ]( s ) + + aq( 0) m m () s = bm ( s ) + bm ( s ) + + b0 m m () s = b I() ( s ) + [ b I () 0 + b I( 0) ]( s ) + b I( 0) A s D B D L (a) 0 () L (b) L (c) I m 0 m m + L (d) where I ( 0) and Q ( 0) are derivatives of input and output at time t = 0. Alternatively, equation () can be expressed as: Copyright 008 IAHS Press

5 () s H () s X () s + H () s Y 0 A variable storage coefficient model for rainfall runoff computation = (e) where H(s) = B(s)/A(s); and H 0 (s) = [D Q (s) D I (s)]/a(s). The terms H(s) and H 0 (s) are designated as the transfer functions of the system. Here H 0 (s) is dependent and H(s) is independent of the input and initial state of the system. For a zero initial state of the system, i.e. H 0 (s) = 0, equation () yields: s H s X s H s = Y s / X s (4) Y () = () () or () () ( ) Equation (4) forms the basis for the development of the proposed VSC model. Unit-step function and series representation Following Spiegel (97), the unit-step function can be used to represent various hydrological variables which change at discrete time intervals. If each of the m excess-rainfall blocks of magnitude ER, ER, ER,, ER m, occur in respective time interval Δt, then the input in terms of the unit-step function is expressible as: I(t) = ERu[] t + ( ER ER ) u[ t Δt] ( ER m ER m ) u[ t ( m ) Δt] ER mu[ t mδt] (5) where u(t) = for t 0, and u(t) = 0 for t < 0. The Laplace transformation of equation (5) yields: X () s = (ER / s) + [( ER ER )/ s] exp( Δts) + L (6) + ER ER / s exp ( m )Δts (ER / s)exp( mδts [( ) ] [ ] ) m m For the simplest case, if there is only one excess-rainfall block, then equation (6) reduces to: X ( s) = (ER / s) (ER / s)exp( Δts) (7) Figure (a) represents an example of catchment representation as a series of two unequal linear reservoirs with storage coefficients K and K (in units of hours), for the first and second reservoirs, respectively. If the storage coefficient varies (with ratio = r), then K = K and K = Kr. Then outflow from the first and second reservoirs can be computed as follows: First reservoir Application of a lumped form of the continuity equation and storage discharge relationship for the first reservoir yields: I () t Q () t S t) / dt = d ( (8) where I(t) is the inflow [L T - ] to the first reservoir, Q (t) [L T - ] is the outflow from the first reservoir, S [L] is the storage within the first reservoir, and ds (t)/dt is the rate of change of storage [L T - ]. The linear storage discharge relationship for the first reservoir is expressed as: S t) = KQ ( ) (9) ( t Coupling equations (8) and (9) yields: I t = Q t K Q t / d (0) () () () t + d Alternatively, equation (0) can be expressed as: Q () t [ + KD] I( t) = () where D is the differential operator. Equation () represents the outflow from the first reservoir and acts as input to the second reservoir. Operating Laplace transform on equation () one gets: [ + Ks] X ( ) Y ( s) = s () where Y (s) is the Laplace transform of Q (t), i.e. Y (s) = L{Q (t)} and the term [ + Ks] - represents the transfer function H(s) for the single linear reservoir model (Singh, 988). m 4 Copyright 008 IAHS Press

6 4 P. K. Bhunya et al. Second reservoir The linear storage discharge relationship for the second reservoir is expressed as: S (t) = Kr Q (t) () The continuity equation for the second reservoir with its input as equation () is expressible as: [ + KD] I() t Q ( t) = ds ( t) / dt Coupling equations () and (4) and simplification yields: Q () t = I() t / ( + Kr i- D) (5) i= Equation (5) represents the outflow Q (t) from the second reservoir due to unit step input I(t) at the inlet of the first reservoir. This will act as input to the third reservoir (if any), to get the resulting output from the system (Appendix B). The Laplace transform of equation (5) yields: Y i= = (6) i- () s ( + Kr s) X () s where Y (s) is the Laplace transform of Q (t), i.e. Y (s) = L{Q (t)}. From equation (6), the transfer function H(s) for the proposed VSC model having two linear reservoirs with varying storage coefficient can be expressed as: () s ( + Kr ) i- s H = (7) i= Equation (7) is the expression of IUH for the proposed VSC model, but expressed in the form of Laplace transform. Coupling equations (7) and (6) one gets: Y i= [ ] i- () s = ( + Kr s) ( ER /s) ( ER /s) exp( Δts) (8) Finally, the inverse Laplace transform of equation (8) yields the outflow in form of DRH as: Q () t = ER{ /( r) ( exp[ ( t Δt) / K] exp( t / K) ) r /( r) [ exp( t / Kr) exp( ( t Δt) / Kr) ]} + (4) for r (9) Equation (9) is the expression for the proposed VSC model. It may be noted here that Q (t) which is the output from the second reservoir also represents the system output, i.e. is equal to Q(t) for a system input equal to I(t). As this paper also presents a comparative study with the classical Nash model (PGD model), it would be appropriate to have a clear presentation of the IUH derived from the transfer function of the VSC model. Further, this type of analysis would be a promising step if the VSC model is to be applied for ungauged conditions. This is discussed in the following section. DERIVATION OF IUH FROM THE TRANSFER FUNCTION OF VSC MODEL As discussed earlier, equation (7) represents the Laplace transform of transfer function H(s) of the proposed VSC model. The inverse Laplace transform of equation (7) will yield the expression for IUH (let us denote its ordinate by q(t)) (Appendix A) as: q( t) = / K( r) exp( t / K) exp( t / Kr) for r (0) [ ] Copyright 008 IAHS Press

7 A variable storage coefficient model for rainfall runoff computation 4 where q(t) is the depth of runoff per unit time per unit excess rainfall [T - ]. The parameters K and r define the complete shape of the IUH for the VSC model. Now, applying the condition at time to peak t p, i.e. at t = t p, q(t) = q p (t); dq(t)/dt = 0. Equation (0) yields the expression for t p as: = Kr ln( / r) /( r) for r () t p Coupling equations (0) and () yields the expression for peak flow rate q p (t) as: qp t = / K( r) exp r ln(/ r) / r exp ln(/ r) /( r) for r () ( ) [ ( ) ( )] It can be seen that equations () and () can be used to develop a synthetic unit hydrograph (SUH) for ungauged catchments. The term synthetic denotes the unit hydrograph (UH) derived from watershed characteristics rather than from rainfall runoff data. For example, in an ungauged catchment, the salient points in the UH, i.e. peak flow rate q p and time to peak t p, can be estimated using a regional formula, for example the SCS (957) method. These values can be used in equations () and () to compute the VSC model parameters K and r, which can then be used in equations (9) and (0) to develop a DRH and SUH, respectively. However, the estimation of these parameters using equations (0) () is not easy, as it involves multiple roots. Therefore, simpler ways of calculating the parameters are discussed below. Parameter estimation of the VSC model The proposed VSC model (equation (9)) has two parameters, K and r. In the present study, the storage coefficient K is estimated using the Wilson (969) approach, and the geometric ratio r from the method of moments (Singh, 988). The former is a measure of temporary storage of the excess rainfall on the catchment before it can drain to the outlet, and can be derived from the relation (Wilson, 969): K = ΔV / ΔQ () where V is the volume under a narrow band of the recession of a hydrograph taken at its point of inflection on the recession limb, and Q is the rate of change of outflow rate over this band in time interval t. For geometric ratio r, the method of moment is used as follows. Taking log on both sides of equation (7), the following is obtained: + Ks + Krs (4) logh(s) = log [( )( )] Differentiating equation (4) and rearranging, one obtains: () s d H K = + H () s ds + Ks Kr Kr ( + s) A substitution of s = 0 in the above equation yields: d H(s) ds s=0 = [ K + Kr] Following Singh (988), the relationship between transfer function H(s) and moment M r can be expressed as: d r H ds () s r S = 0 = r ( ) M r Substituting the order of moment (r = ) and comparison of equations (6) and (7) yields: M = K( + r) (8) For estimating the first moment about the origin at t = 0, the following relationship (Nash, 957) can be used: (5) (6) (7) Copyright 008 IAHS Press

8 44 P. K. Bhunya et al. M = m Q m i (9) where m Q is the first moment of the direct runoff hydrograph (DRH) about the time origin divided by total direct runoff and is given by: m Q (incremental area of DRH moment arm) = (0) Total area of DRH and m i is the first moment of the excess-rainfall hyetograph (ERH) about the time origin divided by rainfall-excess and is given by: m i (incremental area of ERH moment arm) = () Total area of ERH Thus, the geometric ratio r can be estimated from equations (8) and (9) using the storm event data. The estimated values of parameters K and r for all the storm events considered in the analysis are given in Table. Table Estimated values of parameters of VSC and PGD models Event no. Date of storm event ER β VSC model: PGD model (mm) K (h) r K (h) n 4 / October 99 8 June 994 August September Two-parameter gamma distribution (PGD) model Based on the concept of n linear reservoirs of equal storage coefficient K, Nash (957) derived the instantaneous unit hydrograph (IUH) equation in the form of a gamma distribution function as: n ( t / K ) exp( t / ) q( t) PGD = / KΓ( n) K where n and K define the IUH shape and q(t) PGD is the depth of runoff per unit time per unit excess-rainfall (T - ). Here it is noted that, in development of equation (), the unit-impulse input is employed at the inlet of the first reservoir (Fig. (b)). This is one of the most popular models of the IUH for determining the direct runoff hydrograph (Singh, 99). For the condition at the time to peak, i.e. dq ( t) PGD / dt = 0 at t = t p, equation () results in the following: = t /( n ) n n p = e = ( ) K p and q tp ( n ) /Γ( n ) β where β is dimensionless. The empirical relationships developed for β and n are expressed as: (Bhunya et al., 00):.75 n = 5.5β +.04 for 0.0 < β < 0. 5 (4a).998 n = 6.9β +.57 for β 0. 5 (4b) For known β, the application of equation (4) gives accurate estimates of n and K, which has been demonstrated using field and simulated data by Bhunya et al. (00). () () COMPARISON OF IUHs OBTAINED USING VSC AND PGD MODELS Before application of the proposed VSC and PGD models to rainfall runoff data of the test catchment, it is appropriate to analyse the behaviour of IUHs obtained using both the models. Copyright 008 IAHS Press

9 A variable storage coefficient model for rainfall runoff computation 45 Figure shows the IUHs obtained with the VSC model (equation (0)) and three reservoir VSC (equation (B)) (for K = 5 h, r = 0.5) and the PGD model (for n = and, K = 5 h) (equation ()). It can be inferred that the PGD model IUH is more sensitive to any variation in number of reservoirs (n) than that of the VSC model. Further, the PGD model IUHs have a more elongated tail than those of the VSC model. The next section examines the workability of VSC and PGD models for DRH computation using rainfall runoff data of a small catchment VSC IUH for n = PGD IUH for n = 0.08 VSC IUH for n = PGD IUH for n = q(t) (h - ) Time (h) Fig. Instantaneous unit hydrographs due to VSC and PGD models for n =,. STUDY AREA The Kothuwatari catchment (7.9 km ) is a sub-catchment of the Tilaiya Dam catchment of upper Damodar Valley Corporation (DVC), Hazaribagh, India. The catchment is situated at the southeastern part of the Tilaiya Dam catchment between 4 7 and N and and E. In 99, the catchment was selected for Watershed Management under the Indo- German Bilateral Project (IGBP) for assessing the effects of soil conservation measures on runoff and wash load. Four storm events are considered in the analysis. The events considered in the application are -hour duration storms and the corresponding excess rainfall (ER ) amounts are given in Table. APPLICATION The performance of VSC and PGD models was evaluated using the two goodness-of-fit criteria, i.e. visual comparison and statistical measures. The visual approach is based on separate agreements between different segments such as rising limb, peak flow rate and recession segment of the observed and computed DRH, which is shown in Fig. for the storm event of 5 September 996. It is observed that the rising limb, peak flow rate, and the recession segment of VSC computed DRH are in much better agreement than PGD model. Similar results were obtained for the other three events (not shown here). Statistically, the goodness-of-fit is evaluated in terms of (i) special correlation coefficient (R s ), (ii) standard error (STDER), and (iii) relative error in peak flow rate (RE QP ). Copyright 008 IAHS Press

10 46 P. K. Bhunya et al. 5 0 Observed VSC Model PGD Model Discharge (m /s) Time (h) Fig. Observed and computed direct runoff hydrographs for the storm event of 5 September 996. Table Statistical measures for goodness-of-fit criteria of VSC and PGD models Criteria R s RE QP (%) STDER Event. no. VSC PGD VSC PGD VSC PGD The special correlation coefficient R s is expressed as (Eagleson & March, 965): N N N Rs = Qo () i Qc () i ( Qc () i ) / ( Qo ( i) ) (5) i= i= i= where Q o (i) and Q c (i) are the ith values of the observed and computed hydrograph ordinates, respectively, and N is the total number of hydrograph ordinates. A higher R s indicates a good fit and vice versa. The estimated values of R s for VSC and PGD models are given in Table. From the results it is observed that the values of R s vary between and and between and , for the VSC and PGD models, respectively; these values lie in the range of very good for the VSC model and poor to fair for the PGD model (Sarma et al., 97). Further, goodness-of-fit statistics are evaluated in terms of STDER, expressed mathematically as (US Army Corps of Engineers, 990): N = i= STDER ( Qo () i Qc () i ) wi N wi ( Qo ( i) + Qav )/ Qav / = (6a, b) where w i is the weighted value of the ith ordinate; Q o (i) and Q c (i) are the ith value of the observed and computed hydrograph ordinates, respectively; N is total number of ordinates; and Q av is the average of the observed hydrograph ordinates. Since the computed w i values (equation (6b)) are larger for larger Q values, the resulting high STDER value signifies larger non-matching areas on the upper portion of the hydrograph compared to non-matching areas in the lower portion below Copyright 008 IAHS Press

11 A variable storage coefficient model for rainfall runoff computation 47 Q av. A low STDER value represents a good fit, and vice versa; STDER equal to zero represents a perfect fit. The estimated STDER values for the VSC and PGD models are reported in Table. It can be inferred from Table that VSC model STDER values are less than those of the PGD model. This indicates that the VSC model performs better than the PGD model for a small catchment. Further, the relative error in peak flow rate (RE QP ) is expressed as: RE (%) Q Q / Q (7) ( p p ) 00 QP = (OBS) (COM) p (OBS) where Q p(obs) is the peak flow rate of the observed hydrograph (m /s), Q p(com) (Q p(vsc) or Q p(pgd) ) is the peak flow rate of the computed hydrograph (m /s). The RE QP values obtained by both the models for all the storm events are given in Table. The results show that the PGD model RE QP values are always positive and higher in magnitude than those of the VSC model, which indicates that the PGD model always underestimates the peak flow rates and performs less well than the VSC model. Comparison of the VSC vs the three-reservoir VSC model The VSC model considers only two unequal reservoirs in series for DRH computation; however, a good hydrological model should be stable, robust and accurate and, hence, it would be appropriate to check how the model responds if it has more than two (or three) reservoirs following the same storage properties. The expressions for the three-reservoir VSC model and its transfer function are derived in Appendix B. The three-reservoir VSC model is applied to the storm event data of 5 September 996 and the results are compared with those of the VSC model (Fig. 4). It is observed that the three-reservoir model underestimates the peak flow rate compared to the VSC model and matches well with the observed hydrograph; however, the difference is negligible (0.044%). Thus, it can be inferred that a model relying on the VSC concept with two reservoirs, i.e. the VSC model, is adequate to simulate the rainfall runoff process of a small natural catchment. 5 0 Observed Discharge (m /s) 5 0 VSC Model Three reservoir VSC Model Time (h) Fig. 4 Observed and computed direct runoff hydrographs by VSC and three reservoir VSC model for the storm event of 5 September 996. Copyright 008 IAHS Press

12 48 P. K. Bhunya et al. Comparison of VSC UH vs PGD UH The VSC IUH (equation (0)) and the PGD IUH (equation ()) models were applied to the published data of Gormel Ermenek Creek catchment (4 km ) located in Anatolia, Turkey (Haktanir & Sezen, 990). For comparison, the -hour UH is taken directly from Haktanir & Sezen (990). The estimated values of parameters of the VSC IUH and PGD IUH models are found to be K =.70 h, r =.46; and K =. h, n = 5.5, respectively. Figure 5 shows the comparison between the observed and computed UHs obtained using the VSC and PGD models. It is observed that the PGD model has a much better agreement than that of the VSC model, but both models underestimate the peak flow rates. However, the results indicate that the PGD model performs better than the VSC model for a medium-sized catchment. A similar trend was observed by Sarma et al. (97), who compared five rainfall runoff models, viz. the single linear reservoir, the Nash model, the double routing method, the linear channel linear reservoir model, and the IUH obtained by the Fourier transform method. They found that the Nash model performs best for larger catchments as compared to other models, and the single linear reservoir model performs best for smaller catchments. 6 5 Observed UH VSC UH PGD UH Discharge (m /s) Time (h) Fig. 5 Observed and computed UHs by VSC and PGD models for Gormel Ermenek Creek catchment (Haktanir & Sazen, 990). DISCUSSION The main aim of this study was to formulate a rainfall runoff model incorporating a variable storage coefficient for the series of two linear reservoirs instead of the uniform storage parameter that is generally used. The second goal was to derive relationships that relate peak parameters of the hydrograph, i.e. peak flow and time to peak, with the model parameters r and K, as well as to compare the workability of the proposed model with the PGD model using the real data of a small catchment. A brief discussion of the results is given below. The existing PGD model considers the basin to be a combination of n linear reservoirs having the same storage discharge relationship with respect to time and space. To signify the physical meaning properly, n should be an integer quantity; however, in practice, n is generally Copyright 008 IAHS Press

13 A variable storage coefficient model for rainfall runoff computation 49 obtained as a real quantity (Table ). The proposed VSC model considers two reservoirs (an integer number) which follow different storage discharge relationships (varying); hence, the proposed model is more realistic than the PGD model. This is evident from Table and Fig. : application of the PGD model to storm event data of the small test catchment results in hydrographs with more attenuation in peak, delayed time to peak and a more elongated tail than those of the VSC model. The study also reveals that the peak flow rates in the case of the PGD model are more sensitive to n than those of the VSC model (Fig. ) and, hence, a small error in n for the PGD model would affect the model performance more as compared to the VSC model. This makes the VSC model more suitable than the PGD model. Further application of the VSC and PGD models to the event data of a comparatively larger test catchment shows that the PGD model performs better than the VSC model. This is perhaps due to the fact that the VSC model suggests the estimation of K from a small strip of the recession segment of the observed hydrograph, whereas the recession segment of the hydrograph, in general, is a complex combination of the surface flow, interflow and the groundwater flow, each having different lag characteristics. Furthermore, for a small catchment, the surface flow mode will be more dominant than the interflow and groundwater flow and, hence, less error is encountered in K estimation. On the other hand, for a medium-sized catchment, the interflow and groundwater flow may play a more important role in K estimation so more error may be encountered in K estimation. In other words, for the VSC model, there is considerable variability associated with the selection of an appropriate value of the storage coefficient K. There is no unique value of K that can be obtained by considering any portion of the recession curve, as assumed by this model. A similar argument can also be made for the parameter r. This can be tackled efficiently if K and r can be related with β using a nonlinear relationship similar to that for the PGD model. This could be a further topic for research. CONCLUSIONS The following conclusions are drawn from the study:. The assumption of the constant storage coefficient in the popular PGD model is replaced by the varying storage coefficient in the VSC model. The workability of the VSC model is compared with that of the PGD model using the storm event data of a small catchment, and the results show that the VSC model yields a DRH closer to the observed DRH as compared to the popular PGD model.. The estimated lower values of STDER and RE and higher values of R s further support the workability and credibility of the VSC model for a smaller catchment compared to the PGD model.. The peak flow rates of a hydrograph derived using the PGD model are more sensitive to the number of reservoirs (n) compared to the VSC model. Therefore, any small error in n estimation would affect the performance of the PGD model, thus affecting the model results to a higher extent. This makes the VSC model more acceptable. 4. For a large catchment, the PGD model is found to perform better than the VSC model. 5. The ratio r can take any value greater than zero, i.e. r > The VSC model has potential for SUH derivation from small ungauged catchments, provided simple relationships can be developed for parameter estimation in terms of q p, t p, and β. These relationships will reduce the uncertainty in parameter estimation and enhance the model performance. REFERENCES Agirre, U., Goňi, M., López, J. J. & Gimena, F. N. (005) Application of a unit hydrograph based on sub watershed and comparison with Nash s instantaneous unit hydrograph. Catena 64,. Copyright 008 IAHS Press

14 50 P. K. Bhunya et al. Bhunya, P. K, Mishra, S. K. & Berndtsson, R. (00) Simplified two-parameter gamma distribution for derivation of synthetic unit hydrograph. J. Hydrol. Engng 8(4), 6 0. Box, G. E. P. & Jenkins, G. M. (976) Time Series Analysis Forecasting and Control. Holden-Day, San Francisco, California. Chow, V. T. & Kulandaiswamy, V. C. (97) General hydrologic system model. J. Hydraul. Div. ASCE 97(6), Dooge, J. C. I. (97) Linear theory of hydrologic systems. Agriculture Research service Tech. Bull. no. 468, US Dept Agric. Reprinted in 00: EGU Reprint Series,, European Geosciences Union, Kattenburg, Lindau. Dooge, J. C. I. & O Kane, J. P. (00) Deterministic Methods in Systems Hydrology. A.A. Balkema Publishers, Swets and Zeitlinger B.V., Lisse, The Netherlands. Eagleson, P. S. & March, F. (965) Approach to the linear synthesis of of urban runoff systems. Hydrodyn. Lab. Rep. 85. MIT Cambridge, MA, USA. Green, I. R. A. & Stephenson, D. (986) Criteria for comparison of single event models. Hydrol. Sci. J. (), Haktanir, T. & Sezen, N. (990) Suitability of two parameter gamma and three parameter beta distributions as synthetic unit hydrographs in Anatolia. Hydrol. Sci. J. 5(), Holtan, H. N. & Overton, D. E. (964) Storage flow hysteresis in hydrograph synthesis. J. Hydrol., 09. Jain, V. & Sinha, R. (00) Derivation of unit hydrograph from GIUH: analysis for a Himalayan river. Water Resour. Manage. 7, Jakeman, A. J., Littlewood, I. G. & Whitehead, P. G. (996) Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments. J. Hydrol. 7, Jeng, R. I. & Coon, G. C. (00) True form of instantaneous unit hydrograph of linear reservoirs. J. Irrig. Drain. Engng 9(), 7. Koutsoyiannis, D. & Xanthopoulos, T. (989) On the parametric approach of unit hydrograph identification. Water Resour. Manage., Kreyszig, E. (99) Advanced Engineering Mathematics. John Willey & Sons, Inc., Singapore. López, J. J., Gimena, F. N., Goňi, M. & Agirre, U. (005) Analysis of a unit hydrograph model based on watershed geomorphology represented as a cascade of reservoirs. Agric. Water Manage. 77, 8 4. Nash, J. E. (957) The form of the instantaneous unit hydrograph. In: General Assembly of Toronto. 4 September 957, vol. III, Surface Water, Prevision, Evaporation, 4. IAHS Publ. 45. IAHS Press, Wallinford, UK. Nasri, S., Cudennec, C., Albergel, J. & Berndtsson, R. (004) Use of geomorphological transfer function to model design floods in small hill side catchments in semi arid Tunisia. J. Hydrol. 87, 97. Ogata, K. (970) Modern Control Engineering. Prentice Hall, Englewood Cliffs, New Jersey, USA. Rodriguez-Iturbe, I. & Valdés, J. B. (979) The geomorphologic structure of the hydrologic response. Water Resour. Res. 5(6), Sarma, P. B. S., Delleur, J. W. & Rao, A. R. (97) Comparison of rainfall runoff models for urban areas. J. Hydrol. 8, Singh, K. P. (964) Nonlinear instantaneous unit-hydrograph theory. J. Hydraul. Div. Proc. ASCE 90(HY), 47. Singh, P. K. (00) A conceptual model based on unit step function approach with variable storage coefficient for estimation of direct runoff from a watershed of Tilaiya dam catchment in Upper Damodar Valley. Unpublished MTech Thesis (Soil and Water Conservation Engineering), G.B. Pant University of Agriculture and Technology, Pantnagar, India. Singh, V. P. (988) Hydrologic Systems, vol. I: Rainfall Runoff Modeling. Prentice Hall, Englewood Cliffs, New Jersey, USA. Singh, V. P. (99) Elementary Hydrology. Prentice Hall, Englewood Cliffs, New Jersey, USA. Spiegel, M. R. (97) Schaum s Outline of Theory and Problems of Advanced Mathematics for Engineers and Scientists. McGraw-Hill, New York, USA. USACE (US Army Corps of Engineers) (990) Flood hydraulics package. User s Manual for HEC-, CPD-A, Version 4.0, USACE, Washington, DC. Wang, G. T. & Wu, K. (98) The unit-step function response for several hydrological conceptual models. J. Hydrol. 6, 9 8. Wang, G. T. & Chen, S. (996) A linear spatially distributed model for a surface rainfall runoff system. J. Hydrol. 85, Wilson, E. M. (969) Engineering Hydrology. Macmillan, London, UK. Yang, Z. & Han, D. (006) Derivation of unit hydrograph using a transfer function approach. Water Resour. Res. 4, 006. Young, P. C. & Garnier, H. (006) Identification and estimation of continuous-time, data-based mechanistic (DBM) models for environmental systems. Environ. Model. Software, Copyright 008 IAHS Press

15 A variable storage coefficient model for rainfall runoff computation 5 APPENDIX A Derivation of IUH expression for VSC model Alternatively, equation (7) can be expressed as ( + Ks)( Krs) H ( s) = + (A) Using partial fraction expansion, equation (A) can be solved as: [ s + / K) ] / K( r) [ /( s / )] H ( s) = / K( r) /( + Kr (A) Taking inverse Laplace transform of equation (A) results: [ exp( t / K) exp( t / )] H ( t) = / K( r) Kr (A) Let H(t) = q(t), equation (A) reduces to: [ exp( t / K) exp( t / )] q( t) = / K( r) Kr (A4) which is the expression of the IUH for the proposed VSC model. APPENDIX B Third reservoir The storage discharge relationship for the third reservoir is expressed as: S ( t) = Kr Q ( t) (B) The continuity equation with its input as equation (5) is expressible as: I i= (B) i- () t ( + Kr D) Q () t = ds () t / dt Coupling equations (B) and (B) and taking Laplace transform, one obtains: i- ( + ) Y ( s) = I(s) Kr s (B) i= where Y (s) is the Laplace transform of Q (t), i.e. Y (s) = L{Q (t)}. Using equation (B), the transfer function for the three reservoir VSC model can be expressed as: ( + Kr ) i- s H ( s) = (B4) i= equation (B4) represents the IUH or transfer function for the three reservoir VSC model, but expressed in the form of Laplace transform. Substituting the expression for I(s) from equation (7), into equation (B) one obtains: i- ( + Kr s) [ ER/s (ER/s)exp( Δ )] (B5) i= Y(s) = ts Equation (B5) represents the Laplace transform of outflow from the third reservoir. Finally, operating inverse Laplace transform on equation (B5) yields: Q ( t) = ER { /( + r /( r) + r /( r r)( r )[( exp ( t Δt) )/( K exp ( t / K) )] [ exp ( t / Kr) exp ( t Δt) / Kr] )( r) [( exp ( t Δt) )/( Kr exp ( t / Kr ))]} Equation (B6) gives the output from the three reservoir VSC model. (B6) Copyright 008 IAHS Press

16 5 P. K. Bhunya et al. Derivation of IUH expression for three reservoir VSC model Alternatively, equation (B4) can be expressed as: ( + Ks)( + Krs) ( Kr s) H ( s) = + (B7) Using partial fraction expansion, equation (B7) can be solved as: [ + Ks) ] r /( r) [ /( + Krs) ] + r /( r)( r )[ /( Kr s) ] H ( s) = /( r)( r ) /( + (B8) Operating inverse Laplace transform on equation (B8) and let H(t) = q(t) one gets: q(t) = / K( r) [ exp( t / K) ( r ) exp( t / Kr) ( r) + exp( t / Kr ) ( r ) ] (B9) which is the expression of the IUH for the three reservoir VSC model. Received 4 December 006; accepted 5 November 007 Copyright 008 IAHS Press