1 Introduction

Regionalisation of conceptual rainfall-runoff models is a popular approach to estimate flows in ungauged catchments (Post et al. 1998; Sefton and Howarth 1998; Kokkonen et al. 2003). Numerous regionalisation methods have been proposed in the literature for predicting catchment model parameters (Bloschi 2005). Among the most widely used techniques are linear regression analysis between the model parameters and physiographic catchments atributes. Typically, linear multiple regression are used where each model parameter is estimated indepedently from the others (e.g. Post and Jackman 1996, 1999; Sefton and Howarth 1998). Often in regionalization studies, the predictive focus has been on a certain flow regime, in perticular, estimation of flood indices for ungauged catchments has received a great deal of attention (e.g., Farquharson et al. 1992; Mimikou and Gordios 1989, Zirnji and Burn 1994). Nathan and McMahon (1990) considered low flow characteristics, which may be of importance to ecological health of a river system.

The first major step in in the developemnt of relationship between rainfall and runoff was the theory of unit hydrograph proposed by Sherman (1932). Clark (1945) developed instantaneous unit hydrograph (IUH) model by assuming that the outflow hydrograph for any storm is characterized by the translation and storage effect of separable sub-areas of basin. Pure translation of the direct runoff to the outlet viz. the drainage network is described using the channel travel time, giving thereby an outflow hydrograph which ignores water storage effects. Nash (1957) proposed a conceptual model based on a cascade of equal linear reservoir for derivation of IUH for a natural watershed. Nash (1957) and Dooge (1959) suggested a two-parameter gama type model in which respons of instantaneous unit rainfall was represented by gama function of n numbers of identical linear reservoirs. Considering the importance of rainfall-runoff modelling for ungauged or partially gauged watersheds, Rodriguez-Iturbe and Valdes (1979) introduced geomorphological instantaneous unit hydrograph (GIUH) that used geomorhological parameters of the watershed for development of IUH which was further elaborated by Gupta et al. (1980). Rodriquez-Iturbe et al. (1982a and b) proposed geomorphoclimatic instantaneous hydrograph (GcIUH) as a link between climate, gemorphologic structures and hydrologic response of a basin. Koutsoyiannis and Xanthopoulos (1989) emphasized the advantages of parametric approaches for derivation of unit hydrograph in order to establish a relationship between the unit hydrograph (UH) and catchments characteristics.

Azward and Muzik (2000) developed spatially varied time area based GIUH that employed a cell structure and routes the spatially distributed excess rainfall from one cells to other following the maximum down slope deviation to the watershed outlet. Merz and Bloschi (2004) examined the performance of varoius methods of regionalising the parameters of a conceptual model in 308 Austrian catchments. Parajka et al. (2005) invetisgated the performance of a range of methods for transposing catchments model parameters to ungauged catchments using data of 320 Austrian catchments and found Kriging approach and similarity approaches are the best. Heuvelmans et al. (2006) investigated the use of neural nets for regionalisation. Bardosy (2007) discussed and analysed a different approach for transfer of entire parameter sets from one basin to another if the model performance (as defined by Nash-Sutuclife efficiency) on the doner catchment is acceptable. Ahmad et al. (2009) has proposed that time of concentration (T c ) and storage coefficient (R) of Clark’s IUH can be determined using optimization based on downhill simplex optimization technique. The model resulted satisfactory efficiency of more than 95 % during validation and root mean square error less than 6 % during validation and sensitivity analysis indicated that surface runoff hydrograph is more sensitive to R compared to T c . Jaiswal et al. (2010) developed regional relationships using geomorphological and fluvial characteristics of watersheds for determination of parameters of GIUH based Clark model.

Choi et al. (2011) proposed a new methodology to estimate Nash model parameters based on concept of geomorphologic dispersion stemming from spatial heterogeneity of flow path within a catchment. The characteristic velocity in the model was estimated using digital elevation model and statistical features of historical events. Ghumman et al. (2011) applied a downhill simplex optimization technique to optimize regional Nash model parameters (n & k) for computation of direct surface runoff. The performance of model was adjudged by model efficiency and concluded that direct runoff predicted with regional Nash model parameters in 57 events in six catchments has given model efficiency of 67 %. Sarkar and Rai (2011) used Soil Conservation Services-Curve Number (SCS-CN) method for computation of rainfall excess and Nakagami-m distribution for GIUH and then UH for a basin in Ganges river system. The generated UH has been routed with the help of kinematic wave approach at a gauged point on river Bhagirathi and ultimately a flood hydrograph was developed by adopting 100 year return period 1-h rainfall.

2 Nash Model

The Nash model is one of the most widely used models in applied hydrology. Nash (1957,1958, 1959, 1960) proposed a cascade of n number of identical linear reservoirs as a model on which to base the derivation of IUH’s for natural watersheds. The linear reservoir assumed in Nash model are fictious reservoirs in which the storage is directly proportional to the outflow from it. Using the convolution equation and the impulse response function for linear reservoirs, the IUH corresponding to the Nash Model can be easily obtained as follows:

$$ u(t)=\left(\frac{1}{ k\varGamma (n)}\right){\left(\frac{t}{k}\right)}^{n-1}{e}^{-t/k} $$
(1)

where u(t) is the ordinate of IUH at time t, n and k are the shape and scale parameters respectively. When the Eq. 1 is differentiated eith respect to time (t) and condition of du(t)/dt is applied at t = t p , the time to peak (t p) can be obtained as:

$$ {t}_p=\left(n-1\right)k $$
(2)

By putting the value of k from Eq. (2) to Eq. (1), u(t) can be expressed as:

$$ u(t)=\left(\frac{{\left(n-1\right)}^n}{t_p\varGamma (n)}\right)\left(\frac{t}{t_p}\right){e}^{-t/{t}_p} $$
(3)

Rodríguez-Iturbe and Valdes (1979) suggested the following equation for computation of shape parameter (n) of Nash model with the help of geomorphologic parameters.

$$ \frac{\left(n-1\right)}{n}{e}^{-\left(n-1\right)}{\left(n-1\right)}^{n-1}=0.58{\left(\frac{R_B}{R_A}\right)}^{0.55}{R_L}^{0.05} $$
(4)

Where R B , R L and R A are the bifurcation ratio, length ratio and area ratio respectively. Rosso (1984) suggested finally the following equations for computation of n and k using the iterative computing scheme proposed by Croley (1977) :

$$ n=3.29{\left(\frac{R_B}{R_A}\right)}^{0.78}{R_L}^{0.07} $$
(5)
$$ k=\frac{0.44}{V\left(n-1\right)}{\left(\frac{R_B}{R_A}\right)}^{0.55}{R_L}^{-0.38} $$
(6)

For estimation of dynamic parameter velocity (V), Rodriguez-Iturbe et al. (1979) made an assumption of equilibrium state of basin. According to this assumption, the flow velocity and discharge at any moment during the storm can be considered as constant throughout the basin. This characteristic velocity for the basin as a whole changes throughout as the storm progresses and may be termed as equivalent velocity (V e ). Similarly, the discharge was considered as equilibrium discharge (Q e ). Using this principle, the peak velocity (V p ) which may be the equivalent velocity at the time of highest rainfall intensity (i p ) is estimated by developing a relationship between equilibrium velocity (V e ) and rainfall intensity (i) in the following form:

$$ {V}_e=a*{i}^b $$
(7)

where, Ve is the equivalent velocity in m/sec, i is the intensity of rainfall excess in mm/hr and a and b are the watershed specific coefficient and exponent respectively.

In case of gauged catchments, where velocities and corresponding discharges passing through a gauging section are known from observations, assuming the equilibrium state of basin, the equilibrium discharge (Q e ) may be considered as the multiplication of rainfall intensity and contributing basin area (A). The intensity of rainfall excess (i) in mm/hr for an equilibrium discharge (Q e ) in m3/sec can be computed with the help of the following equation

$$ i\kern0.5em =\frac{Q_e}{0.2778A} $$
(8)

Using different pairs of V e and i, the coefficient a and exponent b can be computed using least square method. For an ungauged basin, bed slope, geometric properties of gauging section and Manning’s roughness coefficient (N) are used to determine different pairs of V e using Manning’s equation and Q e at different depths of gauging section. Graphs may be plotted between depth v/s area of cross-section and depth v/s discharge. The Q e for a known value of rainfall excess is estimated using Eq. 8. The corresponding depth and area of cross-section may be obtained using graphs between depth v/s area of cross-section and depth v/s discharge. Knowing cross-sectional areas, the V e for different values of i can be computed and a relation in the form of V e  = a * i b may be developed.

3 Study Area

The study area is the central part of India, administratively known as Madhya Pradesh (M.P.) state. This region is very rich in natural resources and many important tributaries of river Ganges and peninsular rivers originate from this part. The Narmada, Chambal, Parvati, Sindh, Son, Ken, Betwa, Bina, Dhasan, Bearma etc. are some of the important rivers that originate from this part of country. This region can be characterized by undulating topography, deciduas type of forest, small valley lands with productive soil and semi arid climate. The average rainfall in this region is about 1,100 mm, which is near to the national average. The base map showing the location of watersheds has been presented in Fig. 1. The geomorphological parameters of these watersheds (WS-1 to WS-41) have been depicted in Table 1. The catchment areas of watersheds vary from 0.77 sq. km to 518.67 sq. km. Similarly, bifurcation ratio and length ratio range from 2.00 to 5.48 and 1.11 to 5.98 respectively.

Fig. 1
figure 1

Location map showing watersheds selected for the study

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Table 1 Geomorphological characteristics of the watersheds selected for study
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4 Methodology for Regional Approach

In the present paper an attempt has been made to develop the regional relationships for computation of peak velocity (V p ) at the time of peak rainfall intrensity (i p ) and scale parameter (k) of Nash model. The at-site, at-site regional and regional only approaches have been used for computation of ordinates of IUH and subsiquently the UH and direct surface runoff (DSRO) has been determined.

4.1 Estimation of Velocity

The development of V e & i relationship is considered a difficult task for field engineers and water resoueces managers. In this study, regional relationships have been developed for computation of the coefficients a & b using basin characteristics of nine gauged and six ungauged watersheds. Separate regional relationships have been developed to compute coefficient a and exponent b of V e -i relationship using catchment area (A) and average slope (S) of watersheds.

4.2 Estimation of Model Parameters

The shape parameter (n) of Nash model in can be computed easily with the help of geomorphologic parameters. For estimation of scale parameter (k), a regional relationship have been developed between the k and the geomorphologic and fluvial characteristics, 37 watersheds (WS-3 to WS-9 and WS-12 to WS-41) have been selected to derive the relationship during calibration, while remaining four watersheds (WS-1, WS-2, WS-10 and WS-11) have been chosen for validation. In the calibration, the scale parameter and ordinates of arbitrary GIUHs for all 37 watersheds have been determined for different rainfall excess ranging from 1 mm/hr to 40 mm/hr. Various combination of morphological and fluvial characteristics have been tried to develop regional relationship with scale parameter and in turn, a relation between a morpho-fluvial factor L Ω /(R 0.43 L  * V p ) and k has been derived to regionalize the scale parameter (k) of Nash model. The factor, L Ω /(R 0.43 L  * V p ), represents the combined effect of geomorphologic and fluvial characteristics of a watershed.

Employing the above method, at-site, at-site regional and regional only IUH and UH are determined for few known storms. In the site specific approach, the geomorphologic parameters and site specific relationship between V e & i of the respective site are extended to derive the DSRO; while in the at-site regional approach, site specific relationship between V e & i has been used to compute peak velocity (V p ) and the regional relationship for computation of shape (n) and scale parameter (k) have been used to derive the IUH, UH and subsequently the DSRO. In case of regional only analysis, the regional relationships of V e , n and k were used to compute the IUH, UH and DSRO. The relationships derived from analysis of watersheds used in calibration were extended to WS-1, WS-2, WS-10 and WS-11 for validation. The flow chart showing various steps in regionalization of parameters for development of model has been given in Fig. 2.

Fig. 2
figure 2

Flow chart for development of regional Nash model

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The performance of at-site, at-site regional and regional only approaches are evaluated in comparison to the observed runoff data using spatial correlation coefficient (SC), integral square error (ISE), relative mean absolute error (RMAE), root mean square error (RMSE), relative error in peak (REP), Nash & Sutcliffe efficiency (Nash and Sutcliffe 1970) and coefficient of residual mass (CRM). The SC gives the measure of the degree to which two variables are linearly related and varies between −1 and 1. The high value of SC indicates strong correlation. The ISE is a measure of system performance formed by integrating the square of the system error over a fixed interval of time; smaller the ISE value closer is the match. The RMAE is a measure indicating how close forecasts or predictions are to the eventual outcomes and the RMSE is the square root of the mean-squared-error. The RMAE and RMSE ranges from 0 to infinity, with 0 corresponding to the ideal. The REP is the measure of deviation in two peaks. Nash-Sutcliffe efficiency is widely used statistics in hydrology and reaching toward 100 % indicative of closer match in most of the observations. The CRM is a measure of the tendency of the model to overestimate or underestimate the measurements (Bhadra et al. 2008). Positive values for CRM indicate that the model underestimates the measurements while negative values for CRM indicate a tendency to overestimate. For a perfect fit between observed and simulated data, the values of CRM should be equal to 0.0. Formulae for various goodness of fitness parameters used in the study are given below.

  1. a)

    Spatial correlation Coefficient (SC)

$$ SC\kern1em =\kern1em \frac{2\kern0.5em {\displaystyle \sum_{t=1}^n\kern0.5em {Q}_o(t)\kern0.5em {Q}_c(t)\kern1em -{\displaystyle \sum_{t=1}^n\kern0.5em {\left[{Q}_c(t)\right]}^{\kern0.5em 2}}}}{{\displaystyle \sum_{t=1}^n\kern0.5em {Q}_o(t)}} $$
(9)
  1. b)

    Integral Square Error (ISE)

$$ ISE\kern1em =\kern1em \frac{\kern0.5em {\left[\kern0.5em {\displaystyle {\sum}_{t=1}^n\kern0.5em {\left\{\kern0.5em {Q}_o(t)\kern0.5em -\kern0.5em {Q}_c(t)\right\}}^{\kern0.5em 2}}\right]}^{\kern0.5em 0.5}}{{\displaystyle \sum_{t=1}^n\kern0.5em {Q}_o(t)}} $$
(10)
  1. c)

    Relative Mean Absolute Error (RMAE)

$$ RMA\kern1em =\kern1em \frac{\frac{1}{n}\kern0.5em {\displaystyle \sum_{t=1}^n\kern0.5em \left[\kern0.5em {Q}_o(t)\kern1em -{Q}_c(t)\kern0.5em \right]}}{Q_{op}} $$
(11)
  1. d)

    Relative Mean Square Error (RMSE)

$$ RMS\kern1em =\kern1em \frac{{\left[\kern0.5em {\displaystyle \sum_{t=1}^n\kern0.5em {\left\{\kern0.5em {Q}_o(t)\kern1em -\kern1em {Q}_c(t)\right\}}^2}\right]}^{\kern0.5em 0.5}}{n} $$
(12)
  1. e)

    Relative Error in Peak (REP)

$$ REP=\frac{Q_{op}-{Q}_{cp}}{Q_{op}} $$
(13)
  1. f)

    Nash-Sutcliffe Efficiency (η)

$$ \eta =\frac{ IV- RV}{ IV}*100 $$
(14)

where Q o (t) = observed discharge at time t; Q c (t) = computed discharge at time t; Q op = observed peak discharge; Q cp = computed peak discharge; n = no. of observation, IV = initial variance, RV = remaining variance. If \( {\overline{Q}}_o \) is the mean observed discharge, the IV and RV can be expressed as:

$$ IV={\displaystyle \sum_{t=1}^n{\left[{Q}_o(t)-{\overline{Q}}_o\right]}^2} $$
(15)
$$ RV={\displaystyle \sum_{t=1}^n{\left[{Q}_o(t)-{Q}_C(t)\right]}^2} $$
(16)
  1. g)

    Coefficient of residual mass (CRM)

$$ CRM=\frac{{\displaystyle \sum_{i=1}^n\left[{Q}_o(t)-{Q}_c(t)\right]}}{n*{\overline{Q}}_o} $$
(17)

5 Results and Discussion

Using the methodology explained above, an attempt has been made to develop a regional relationship for determination of scale parameter (k) of Nash model in data scarce central India region using geomorphologic and fluvial characteristics of the basins. In the analysis, regional relationships for computation of coefficients (a & b) of V e and i relationship have been developed. Using at-site, at-site regional and regional only approaches, the IUH, UH and the DSRO for different events have been computed in these watersheds. The results obtained from the analysis have been compared with the observed results using different performance evaluation parameters.

5.1 Regional Relationship for Estimation of Equilibrium Velocity (Ve)

The estimation of peak velocity requires the gauge discharge data or information of section and Manning coefficient (N) of the watershed. In this study, an attempt has been made to develop a regional relationship between fluvial characteristics with basin characteristics. From the analysis, it has been observed that slope and catachment area play an important role in deciding the coefficients a & b of V e -i relationship. The following equations have been developed for estimation of a & b.

If \( A\sqrt{S} \) is less than 3.5

$$ a=0.239A\sqrt{S}+0.351 $$
(18)
$$ b=0.127A\sqrt{S}+0.195 $$
(19)

If \( A\sqrt{S} \) is equal to or greater then 3.5

$$ a=0.279{\left(\frac{L_{\varOmega }}{\sqrt{S}}\right)}^{0.249} $$
(20)
$$ b=0.114{\left(\frac{L_{\varOmega }}{\sqrt{S}}\right)}^{0.333} $$
(21)

The graphical representation for computation of a & b are given in Fig. 3.

Fig. 3
figure 3

Regional relationship for computation of a & b of equilibrium velocity (Ve)

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5.2 Development of Regional Parameters of Nash Model

For development of regional relationship, the geomorphological parameters of 37 watersheds (WS-3 to WS-9 and Ws-12 to Ws-41) and Ve-i relationship of gauged watersheds have been used as inputs. In case of ungauged watersheds, peak velocity (V p ) has been computed using regional Eqs. (7) and (18) to (21). The scale parameter k and ordinates of IUH’s for different arbitrary excess rainfall intensity ranging from 1 mm/hr to 40 mm/hr for all the watersheds have been computed. Various combination of morphologic and fluvial characteristics have been tried to define relationship with k for the region. Finally, the following mathematical relationship between a morpho-fluvial factor [L Ω /(R 0.43 L  * V p )] and k has been found the most appropriate for estimation of scale parameter for ungauged watersheds.

$$ k=0.2143\frac{L_{\varOmega }}{R_L^{0.43}\kern1em {V}_p} $$
(22)

The graphical representation of regional relationship between L Ω /(R 0.43 L  * V p ) and k has been presented in Fig. 4.

Fig. 4
figure 4

Regional relationship betwen L Ω /(R 0.43 L  ∗ V p ) and scale parameter (k)

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5.2.1 Calibration of Regional Model

The regional relationship for computation of k has been developed using geomorphologic parameters of 37 watersheds. For calibration, few storms of seven watersheds (WS-3 to WS-9) which were used in developing the regional relationship have been analyzed. These watersheds have been selected for performance evaluation due to availability of observed discharge data. The ordinates of excess rainfall for selected storms have been computed using φ − index method. In this method, a uniform value of loss rate (φ − index) has been computed by trial and error method to make the volume of excess rainfall equal to the volume of (DSRO). The at-site, at-site regional and regional only IUH, UH and corresponding DSRO for these storms have been computed. From the observed point observations, smoothened flood hydrograph has been prepared and straight line base flow separation technique was used to compute DSRO from the flood hydrograph. The at-site, at-site regional and regional only DSRO for few known storms are presented in Fig. 5. It can be observed from Fig. 5 that site specific, at-site regional and the regional only approach exhibit a close resemblance with the observed data. The model results for at-site, at-site regional and regional only approach including i p , V p , n,,k, peak runoff (Q p ) and time to peak (T p ) for few storm events during calibration were given in Table 2.

Fig. 5
figure 5

Comparison of site specific and regional model during calibration

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Table 2 Parameters of Nash model during calibration
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The statistical correlations between the observed and the computed values of the DSRO’s representing, SC, ISE, RMAE, RMSE, REP and CRM are given in Table 3. Bhadra et al. 2008 has used coefficient of residual mass (CRM) for evaluating the performance of GIUH models and in the present study, CRM varies between −0.12 and 0.25 for the site specific approach, −0.12 to 0.31 for at-site regional approach and -0.11 and 0.24 for regional only approach. It has been observed that model efficiency varied from 59.7 % to 92.2 % for the site specific approach, 40.6 % to 94.4 % for at-site regional and 68.6 % to 98.7 % for the regional only approach; while RMAE varied between 0.06 and 0.39 for the site specific approach, 0.04 to 0.40 for at-site regional and 0.03 to 0.42 for the regional only approach.

Table 3 Evalution of performance of at-site, at-site regional and regional Nash model during calibration
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5.2.2 Validation of the Regional Model

The developed model has been validated with WS-1, WS-2, WS-10 and WS-11 which were not included in the development of the regional relationship during calibration. The at-site, at-site regional and regional only DSRO for few storms have been computed (Fig. 6). In the at-site approach, parameters of Nash model (n, k) and peak velocity (V p ) have been computed using site specific fluvial and morphologic information. The at-site regional approach used site specific relationship for computation of k, while V p was computed with the help of regional relationship developed during calibration. In regional only approach, all parameters of Nash model and V p were derived from regional relationships developed for the study area. The model parameters, peak discharge and time to peak for all three approaches during validation are given in Table 4.

Fig. 6
figure 6

Comparison of site specific and regional model during validation

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Table 4 Parameters of Nash model during validation
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The parameters showing the performance of regional model during validation are presented in Table 5. It can be seen from Table 5 that the REP varied from −0.21 to 0.51 for site specific, −0.18 to 0.55 for at-site regional and −0.21 to 0.48 for regional only approach. Similarly, RMSE varied from 0.05 to 8.96 for site specific, 0.06 to 8.60 for at-site regional and 0.05 to 9.89 for regional only approach, which imply a close match. And thus, it validates the approach for application in the region of Central India. As the model results during calibration where changes have been made to develop relationships and validation with independent data indicate satisfactory results in terms of model evaluation, the relationship developed for computation of peak velocity and model parameters of Nash model can be used in other watersheds of Central India region where water resources planning is difficult due to non-availability of gauge discharge data.

Table 5 Evalution of performance of at-site, at-site regional and regional Nash model during validation
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6 Conclusions

The Nash model is the most widely used rainfall-runoff model in the field of hydrology and proposed regional approach for computation of model parameter and peak velocity have been found successful to derive the flood hydrograph for ungauged watersheds. The regional relationships developed for computation of peak velocity suggested that the basin characteristics may be very useful for computation of peak velocity for geomorphology based rainfall-runoff models. The scale parameter can reasonably be estimated using morpho-flluvial characteristics of watershed linking to equilibrium peak velocity, rainfall intensity, bifurcation ratio, length ratio and area ratio. The regional relationships developed using geomorphic and fluvial characteristics of 37 watershed and verified on four watersheds (not used in calibration) indicated that relative error in peak (REP) varies from −0.58 to 0.38 for site specific approach, −0.15 to 0.54 for at-site regional and −0.18 to 0.55 for regional only approach which is fairly accurate in regional approach in comparison to site specific approach.

The results of the regional approach have been found in close match with the observed data. The regional model developed based on the analysis of watersheds in the Central India region of India can successfully be used in other ungauged small and medium watersheds in the region for rainfall-runoff modeling and designe flood estimation knowing geomorphologic characteristics of the watersheds. The proposed analysis could provide a wide range of application in the field of rainfall-runoff modeling, derivation of flood hydrograph and design flood estimation particularly in ungauged catchments and catchments with limited data.