INTRODUCTION
The process of rainfallrunoff in any watershed is a very complex process that
depends on many properties of watershed such as area, overland slope, land use,
etc. For a true determination of this process, plenty of hydrological models
have been developed from the past until now and Nourani et al. (2007)
have made a list of these models that shown in Fig. 1.
Due to the complexity of rainfallrunoff process and absence of data to describe in detail the character of heterogeneous and of spatially distributed inputs, simulation of the rainfallrunoff process is generally based on the conceptual models. The linear reservoir model presented by Zoch in 1934 is the oldest and simplest model with high level of application in relation to simulation of rainfallrunoff process which is the base of most other conceptual models (Chow et al., 1988). Certainly the cascade of linear reservoirs model with equal storage coefficients is the first conceptual model which uses the real meaning of linear reservoir with a mathematical base to present an explicit mathematical formulation for Instantaneous Unit Hydrograph (IUH) of the watershed (Nash, 1957). Then, Dooge (1959) developed a more complete model for calculation of IUH considering the effect of flow transition and adding the meaning of linear channel to the Nash’s model. But since the equation of IUH was not easily solvable for complex applicable problems, different simplified models out of the abovementioned model were propounded.
For instance, Wang and Chen (1996) and Jeng and Coon (2003) have presented
approaches on the basis of cascade of linear reservoirs model. Also some computer
models on the basis of linear reservoirs concept have been introduced in recent
years which have been briefly explained by Singh and Woolhiser (2002).

Fig. 1: 
Hydrological models (Nourani et al., 2007) 
But these models have a great number of parameters for estimation and are usually able to calculate the hydrograph only at the watershed outlet, so hydrologists have decided to create and develop semidistributed models (Nourani and Mano, 2007). Such models have been introduced on the basis of geomorphological routing concept and representing Geomorphological Unit Hydrograph (GUH).
It was at the end of 70th decade that utilizing this type of routing started for those watersheds with lack of complete observation data and it was tried to determine most of parameters according to the watershed morphology. Therefore reserving routing model is presented by Boyd (1978) and Boyd et al. (1979) on the basis of incorporating geomorphological properties of the watershed. RodriguezIturbe and Valdes (1979) and Gupta et al. (1980) presented a geomorphological instantaneous unit hydrographs based on the theory of exponential distribution of required time that a drop of water measures in a specific way in the watershed. Rosso (1984) proposed a model which expresses the Nash's IUH parameters as functions of the Horton's indexes. Karnieli et al. (1994) and Hsieh and Wang (1999) presented different models for geomorphological routing by a similar method to the Boyd (1978). López et al. (2005) and Agirre et al. (2005) developed geomorphological instantaneous unit hydrographs based on cascade of reservoirs that have one uncertain parameter. Nourani and Mano (2007) used TOPMODEL and kinematics wave approaches to present a model that all parameters of this model were linked to geomorphologic properties except one uncertain parameter.
Furthermore, by developing GIS tools in hydrological science and rainfallrunoff modeling, it is possible to determine all hydrological and morphological parameters of the watershed precisely and easily by DEM (Digital Elevation Model) maps (Jenson and Domingue, 1988; Maidment et al., 1996; Olivera and Maidment, 1999; Maidment, 2002).
In this study, first, theories of the Nash and SCS models as two classic and most applicable models are considered and then an application of a new model is shown. This is based on the linear reservoirs cascade concept that represents the structure of the watershed by its subwatersheds, which are defined as the terrain portion draining to a channel of the drainage network and kinematics wave. At the end the result of this model for Ammameh watershed, a small watershed in central Iran, would be compared with results of Nash and SCS models.
All of the model geomorphological parameters are determined by GIS tools and only one uncertain parameter would be determined and calibrated using rainfall  runoff data sets.
THE NASH MODEL
The formulation of Nash’s IUH was obtained under the assumption that,
watershed behavior can be associated with a cascade of n equal linear reservoirs
each having lag time of k, where unit rainfall instantaneously is imposed on
the upper reservoir. With the above assumption, a Gamma distribution with parameters
n and k is derived for IUH (Nash, 1957):
where, h(t) is IUH of Nash’s model, Γ(n) is a Gamma function and
n, k could be determined by moments method as (Singh, 1988):
In which M_{1} and M_{2} are the first and second moments of the functions and I, Q are inflow and outflow hydrographs, respectively.
Nash used his model in 1962 for some British catchments. He has established experimental relations between watershed properties and n, k parameters and presented Nash’s synthetic model (Singh, 1988).
THE SCS MODEL
SCS method is one of the methods that can be used for computing UH in watersheds with insufficient data. First, lag time, t_{l}, could be determined with regard to rainfall continuity then watershed physical properties such as area, main river length, average slope, CN (Curve Number) are used in order to make synthetic unit hydrograph for the watershed (Chow et al., 1988).
THE SLRC MODEL
SLRC (Semidistributed version of Linear Reservoirs Cascade) model, proposed
in this research, is a semidistributed version of linear reservoirs cascade.
In this model, the watershed is divided into subdivisions with regard to topography,
then each subwatershed is substituted by one linear reservoirs cascade, thus
the watershed is represented by reservoirs that distributed according to the
watershed morphology. Finally a nonlinear equation (i.e., kinematics wave) is
used for flow routing through the watershed main channel. Meantime, main precipitation
is divided proportionally between subwatersheds on the basis of areas.

Fig. 2: 
Operation of SLRC model 
n this model, outflow of each subwatershed is determined with regard to geomorphology
properties and using Nash’s synthetic model equations (Singh, 1988):
where, S_{0} is average land slope, A is area (km^{2}), L is the longest flow path in the drainage network (m) for ith subwatershed.
Then the determined runoff applies to the main river momentarily as lateral
flow. Thereafter flow is routed in the main channel using the kinematics wave
equation. It can be mentioned that, kinematics wave equation may be considered
as a reliable routing model for such watersheds with high land slope. The circumstance
of the model operation is illustrated by Fig. 2.
Dispensing the term of pressure and acceleration in SaintVenant equation,
in shallow water and using the Manning’s formula, kinematics wave equation
can be obtained as (Chow et al., 1988):
with coefficients as follow:
where, q is lateral flow, S'_{0} is channel slope, n_{m} is Manning coefficient and B is channel width. By solving this nonlinear partial differential equation, flow discharge in any time (t) and distance (x) can be obtained through the watershed main channel.
However, in SLRC model Eq. 4 is used as the adjusted form
of:
where, ,
is a correction coefficient due to the assumptions and condensing applied to
the model and is the only determinable parameter of the model.
In SLRC, Digital Elevation Model (DEM) might be used for determining channel
slope. Manning coefficient could be obtained with regard to the land use and
plant coverage of the watershed. Whereas, using large scale maps for extracting
some hydrological properties such as channel width is not suitable and also
using small scale maps for hydrological modeling is not economic, so an equation
was proposed by Bandaragoda et al. (2004) in order to estimate channel
cross section width as:
where, A_{upstream} is upstream drainage area (km^{2}), a =
0.0011, b = 0.518 and B is channel width (m). Correctness of Eq.
7 has been reported by other researches (e.g., Nourani and Mano, 2007) but
unlike b, a may change from a watershed to the other greatly. A_{upstream}
is calculated at each location with distributing area linearly along the main
river. ,
a watershed parameter with no dimension, is the only parameter which should
be calibrated using rainfallrunoff data. As a matter of fact,
is used for correcting due to the applied assumptions and existence of uncertainty
in the estimated parameters such as n_{m} and a.
Implicit finite difference method (Chow et al., 1988) is used in SLRC
model for solving Eq. 6 and all used geomorphological parameters
are extracted by GIS tools.
In order to determine the uncertain parameter of the model (),
direct search method (Yue and Hashino, 2000) may be used among other optimization
schemes.
EFFICIENCY CRITERIA
For a more complete analysis of the suitability of the models, Nash and Sutcliffe
index (E) (1970), correlation coefficient between observed and calculated data
(R) and ratio of absolute error of peak flow (RAE_{P} (%)) are used
in the current study. These indicators are defined as follows:
where, Q_{i, obs} is observed discharge at t = i, Q_{i, sim} is simulated discharge at t = i, No is the number of observed data and Q_{P obs}, Q_{P sim} are observed and simulated peak discharges, respectively.
WATERSHED DESCRIPTION
The Ammameh Watershed, one of the subwatersheds of Jajrood in upstream of
Latian Dam, is located in south area of Central Alborz, near Tehran (Capital
of Iran) with an area of 37.2 km^{2} between the heights of 1900 and
3868 m. From topology point of view, it is a mountain area. Figure
3a shows aerial photograph of Ammameh watershed.
Figure 3b shows DEM of Ammameh watershed which illustrates
elevation condition of watershed and also DEM is a tool for eliciting geomorphological
properties of the watershed. This map obtained from topography map of the watershed
(scale: 1/25000) using GIS.
Regarding vegetation coverage, about 200 hectares (5% of watershed area) include
gardens and grass and the remained has vegetable coverage of bushes. Vegetation
coverage of watershed extracted from aerial photographs is shown in Fig.
3c. Some of the events in this watershed have been registered from 1990
with time intervals of 30 min. In order to consider the rainfallrunoff by the
models, watershed is divided into 5 subwatersheds using GIS tools (Fig.
3d).
RESULTS AND DISCUSSION
Due to the lake of registered data, 8 events were used for comparing simulated and observed direct hydrographs; 6 events for calibrating and 2 events for verifying. In order to find the observed direct hydrograph of any event, first, base flow specified by fixed gradient method and then observed direct hydrograph calculated. Next, penetration of each event calculated by the use of continuous theorem and fixed penetration rate method, finally this penetration deducted from observed hyetograph in order to find the excess hyetograph of the event.
Specifications of each event are shown in Table 1. In the
Table 1 first column is the number of event, the second is
the date of event, third is the height of precipitation, fourth is the equivalent
height of direct runoff, fifth is the loss rate and the last is the transform
rate of rainfall into runoff in the watershed.Geomorphological parameters extracted
by GIS and necessary parameters for calculating determinable parameter of SLRC
model are shown in Table 2.

Fig. 3: 
(a) Situation map, (b) DEM, (c) Vegetation coverage and (d)
Subwatersheds 
Table 1: 
Rainfallrunoff events data 

Table 2: 
Geomorphological parameters of SLRC model 

Table 3: 
Calibration results of Nash, SCS and SLRC models 

Table 4: 
Verification results of Nash, SCS and SLRC models 

We have respectively the number, area, the length of the greatest drainage
of subwatershed, average slope of subwatershed, length of the river, river
slope of the subwatershed, n (number of reservoirs) and k (lag time or storage
coefficient) as the introduced amounts in Eq. 3. Last column
is channel width at the subwatershed outlet, determined by Eq.
7. Average Manning coefficients are n_{m} = 0.024 for subwatersheds
1, 2, 4, 5 and n_{m} = 0.03 for subwatershed 3, which obtained considering
vegetation coverage extracted from aerial photography using ERDAS IMAGINE software.
The watershed considered as a single piece with no subwatershed for modeling
by SCS method. Curve Number (CN) was chosen 79 according to the vegetation coverage.
Lag time parameter was chosen as the SCS variable parameter and for calculating
optimum value of the parameter, direct search method (Yue and Hashino, 2000)
was used and results are given in Table 3.
Calibration results of Nash’s model with determinable parameters calculated
by moments method (Eq. 2) and SCS and SLRC models with parameters
calculated by direct search method are presented by Table 3.
The graphs of calibration step and observed direct hydrographs are shown in
Fig. 4.
Then the models were verified by two events data sets that results are presented
by Table 4. Average values of parameters, obtained from calibration
process, were used in verification step. Verification graphs and observed direct
hydrographs are shown in Fig. 5.
Considering Table 3 and 4, average values
of efficiency criteria in SLRC model are greater than average values of criteria
in Nash and SCS models. With a view to the fact that Nash model has two degrees
of freedom, calibration result must be better because a twoparameter model
may have better fitness on the observed data. Also having two parameters, can
increase more dependency on the accuracy of the determined values of the parameters
and existing error in determined parameters in calibration step can decrease
accuracy of the model in the verification step.

Fig. 4: 
Calibration process graphs of Nash, SCS and SLRC models 

Fig. 5: 
Verification process graphs of Nash, SCS and SLRC models 
Considering the result of SCS model shows that, use of bad and inadequate
modeling of physical parameters in SCS model causes untrustworthy result.
According to the results presented in Fig. 4 and 5,
in spite of considering just one calibrated parameter in the SLRC model, it
can properly detect the rising limbs of the hydrographs which are usually depended
on the storm properties as well as recession limbs which are more related to
the watershed geomorphological factors. The existence of both geomorphological
and calibrated parameters in the model formulation may lead to this capability.
In high discharges, a watershed usually shows linear behavior (Pilgrim, 1976)
and nonlinear models may have weak performance in such situation. However,
taking advantage from Nash equation as a linear routing model may help to the
SLRC model in order to catch the peak discharges more appropriately, as indicated
by RAE_{p} index in Table 3 and 4.
CONCLUDING REMARKS
On the basis of obtained results, distinguished properties of SLRC model are:
• 
SLRC model is a geomorphological model which is on the basis
of watershed physical properties, covers area with different properties
and also considers them in modeling. 
• 
Formulation only depending on a single parameter. 
• 
SLRC model is a semidistributed model and against the lumped models such
as Nash’s model, also can consider hydrological condition of the interior
parts of the watershed, therefore this is realized that capability of SLRC
model is more than the other classic model such as Nash and SCS models and
also it has ability to simulate rainfallrunoff properly. 
• 
Accompaniment of a nonlinear distributed routing model (i.e., kinematics
wave) by a linear lumped rainfallrunoff model (i.e., Nash’s model)
gives a suitable ability to the SLRC model in order to be a reliable runoff
routing model. Particularly for watersheds which have variant geomorphological
properties such as Ammameh watershed which the variation of the land elevation
is more perceptible in it (from 1900 to 3868 m). 
Proposed model in this paper was considered in a watershed which was divided on the basis of same interval lines, but this model could be used for watersheds which were divided into subwatersheds on the basis of joint point in the drainage network.
It is also suggested to contemplate the seasonality and temporal effects accompanied by physical and geomorphological factors in the formulation of the presented model. This accompaniment and using more storms data may extend the model abilities and efficiencies; this proposal can be considered as a new research plan for the future work.