GIUH Based Transfer Function for Gomti River Basin of India

Journal of Spatial HydrologyVol.9, No.2 Fall 2009 GI UH Based Tr ansf er Funct i on f or Gomt i Ri v er Basi n of I ndi a R. K. Rai 1 , Alka Upadhyay 1 , S. Sarkar 2 , A. M. Upadhyay 3 , and V. P. Singh 4 Abst r act Geomorphologic instantaneous unit hydrograph (GIUH) can be used as a transfer function for modeling the transformation of excess rainfall into surface runoff, in which excess rainfall is an excitation (i.e. production function) to the hydrologic system. These models can be used to predict / forecast the temporal variation of the surface runoff at the outlet of ungauged basin, which is useful in the hydrologic / environmental engineering applications. The present study deals with the geomorphometric investigation and provides an efficient solution approach to derive the GIUH based transfer function and thus geomorphologic unit hydrograph (GUH) for the basin. Since, Gomti river basin is ungauged, therefore, to test the effectiveness of the approach two cases were considered. Firstly, the approach was tested on the catchment for which published UH data was available; and secondly, the approach was applied for the Gomti river basin for the derivation of GUH. To verify the derived GUH of the Gomti basin, a comparison was performed with the synthetic unit hydrograph (SUH) obtained from the Central Water Commission (CWC) procedure. Based on the comparison of the result, it may be revealed that the GUH with dynamic flow velocity of 0.68 m/s was close to the SUH. Key words: Basin, CWC, Direct runoff; Geomorphology; Gomti river; Geomorphologic instantaneous unit hydrograph; Transfer function; Geomorphologic unit hydrograph, SRTM, Synthetic unit hydrograph, Ungauged basin I nt r oduct i on Water resources planning, development and operation of various schemes requires accurate estimation of hydrologic response of the basin. In surface hydrology, rainfall-runoff and soil erosion process are very important and needs good understanding. These hydrological processes are nonlinear and involve various climatic, topographic, soils, land use information. To develop a good understanding and hydrological model for these processes, a reliable and wide variety geophysical and hydro-meteorological data are required. These models may be broadly classified as empirical, conceptual, lumped, and physically based distributed models. Empirical models have their own limitations as they are site specific, whereas, conceptual models are flexible and based on the simplification/approximation of physical concepts of the processes. Therefore, conceptual model has a room for theoretical simulation of individual components of the process. For example, Nash based instantaneous unit hydrograph (IUH) 1 DHI (India) Water, Environment & Health, B-220, L. Ground Floor, C. R. Park (Outer Ring Road), New Delhi-110019, India ([email protected]; [email protected] ) 2 Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee-247 667, Uttarakhand, India. 3 Alaknanda Centre for Basin Management (Water & Environment), 8/38-B, Sector-8, Indira Nagar, Lucknow-226016, Uttar Pradesh, India 4 Department of Biological and Agricultural Engineering, Texas A & M University, 2117 TAMU, Scoates Hall, College Station, TX 77843-2117, USA (e-mail: [email protected] ) Rai et.al./ JOSH Fall 2009 (9)24-50 model (Nash, 1957) is based on the concept of ‘a cascade of linear reservoirs’ in the watershed. The physically based distributed models on the other hand, need rigorous computational skill for solving the governing equations and therefore, require a significant quantum of watershed information along with event based rainfall and runoff data for the calibration of model parameters (Singh and Frevert, 2002 a, b). It is the fact that most of the Indian as well the watersheds of developing countries do not have sufficient historical hydrological records and detailed watershed information needed for physically based distributed models. Therefore, a simplified representation of the rainfall-runoff process through the input-output linkage without a detailed physical description of the process can be used and this is the basis of the linear theory of hydrologic systems. For mathematical description of linear hydrologic system, the unit hydrograph (UH) or unit pulse response function can be employed for system identification in hydrologic analysis (Sherman, 1932; Dooge, 1959, 1973; Chow et al., 1988; Singh, 1988a). Therefore, geomorphology based approach becomes a one of the most popular modeling tools for the computation of runoff hydrographs under these circumstances (Rodriguez-Iturbe and Valdes, 1979; Rodriguez-Iturbe et al., 1982; Yen and Lee, 1997; etc.). Geomorphology of a river basin describes the status of topographic features of the surfaces and streams, and its relationship with hydrology provides the geomorphological control on basin hydrology (Jain and Sinha, 2003). Geomorphology reflects the topographic and geometric properties of the watershed and its drainage channel network. It controls the hydrologic processes from rainfall to runoff, and the subsequent flow routing through the drainage network. The role of basin geomorphology in controlling the hydrological response of a river basin is known for a long time. Moreover, for any infrastructural development, it is very useful tool for first hand overview of the basin. It is advantageous in case of laying out the urban drainage and irrigation canal system, aqueducts, study the physiographic impacts on environment, and selection of silt disposal site, hydropower site (Sarkar and Gundekar, 2007), recharge zone, percolation tank, retention tank, dam site, etc. This drainage network of the river basin can provide a significant contribution towards flood management and water logging program (Jain and Sinha, 2003). Earlier works have provided an understanding of basin geomorphology-hydrology relationship through empirical relationships (Snyder, 1938; Horton, 1945; Taylor and Schwartz, 1952; Singh, 1988b; Singh, 1989; Jain and Sinha, 2003; etc.). Snyder (1938) proposed synthetic unit hydrograph approach (SUH) for ungauged basin as a function of catchment area, basin shape, topography, channel slope, stream density and channel storage; and derived the basin coefficient by averaging out other parameters. Furthermore, the hydrologic response is a function of climatic parameters, landuse, soil parameters and topography and therefore, for any physical based models requires time to time change in their parameters due to variation encountered with respect to the gradual climatic changes and landuse of the watersheds (Rodriguez-Iturbe et al., 1982). However, the geomorphological parameters are mostly time-invariant in nature and therefore, geomorphology based approach could be the most suitable technique for modeling the rainfall-runoff process for ungauged catchments. The rainfall pattern, in general is undergoing a change due to global changes in atmospheric conditions. Further, because of different activities in the watershed, its land use is also having a gradual changes and this has an impact on the characteristics of the runoff produced from the watersheds. Thus, the geomorphological instantaneous unit hydrograph (GIUH) (Rodriguez-Iturbe and Valdes, 1979) and further simplified by Gupta et al. (1980) is a hydrological model that relates the geomorphological features of a basin to its response to rainfall. They can be applied to ungauged basins having scarce hydrologic data (Al-Wagdany et al., 1998). The derivation of Journal of Spatial Hydrology 30 Rai et.al./ JOSH Fall 2009 (9)24-50 the GIUH uses the assumption that a stream of a certain order has a known linear response function of the familiar or complex probability distributions (Rodriguez-Iturbe and Valdes, 1979; Kirshen and Bras, 1983; Rinaldo et al., 1991; Jin, 1992; Fleurant et al., 2006; etc.). The effect of linear channels in the hydrologic response was introduced by Kirshen and Bras (1983). Thus, the GIUH based transfer function approach is applicable in such a situation where rainfall data is available but runoff data are not, and it is a more powerful technique for the flood estimation than the commonly used parametric Clark model (Clark, 1945) and Nash’s cascade technique (Nash, 1957) (Yen and Lee, 1997; Bhaskar et al., 1997; Jain et al., 2000; Lohani et al., 2001; Kumar et al., 2002; Sarangi et al., 2007; Bhadra et al., 2008; etc.). Another advantage of GIUH technique is its potential for deriving the unit hydrograph (UH) using the geomorphologic characteristics obtainable from topographic map / remote sensing, possibly linked with geographic information system (GIS) and digital elevation model (DEM) (Rodriguez-Iturbe and Valdes, 1979; Rosso, 1984; Sahoo et al., 2006; Kumar et al., 2007; etc.). However, the GIUH technique is applicable for the estimation of the direct runoff component of the stream flow and hence, can be used to generate the direct runoff hydrograph (DRH). Once the DRH is computed, the flood hydrograph can be simply obtained by adding the base flow component. Therefore, based on the foregoing discussions, and the interrelationship between the geomorphology and hydrology, the present study was carried out with the objectives: (i) to present an efficient solution approach for generating geomorphologic transfer function i.e. GIUH; (ii) to test the applicability of proposed approach on the published data of Bhunya et al. (2008) (iii) to investigate the geomorphology of the Gomti river basin and estimate the dimensionless Horton’s ratios; and (iv) to derive the geomorphologic unit hydrograph (i.e. GUH) for Gomti river basin using the Horton’s ratios. Geomor phol ogi c Par amet er s Certain characteristics of the drainage basins reflect hydrologic behavior and are therefore, useful, when quantified, in evaluating the hydrologic response of the basins. These characteristics relate to the physical characteristics of the drainage basin as well as of the drainage network. Physical characteristics of the drainage basin include drainage area, basin shape, ground slope, and centroid (i.e. centre of gravity of the basin). Channel characteristics include channel order, channel length, channel slope, channel profile, and drainage density. Basi n or der and channel or der Drainage areas may be characterized in terms of the hierarchy of stream ordering. The order of the basin is the order of its highest-order channel. It can be stated that the uppermost channel receives water from overland surface and direct towards the outlet through the drainage network. This upper most channel joins another channel and form the higher order channel, and so on. The first order channel is defined as those channels that receive water entirely from overland surface and does not have tributaries. The junction of two first order streams forms the second order stream. It means the higher order drain carries more water than the lower order drains. A second order channel receives flow from the two first order channels and from overland flow from surface and it might receive the flow from another first order stream. When the two second order streams joins together forms the third order stream, and so on. This scheme of stream ordering is referred to as the Horton-Strahler ordering scheme (Horton, 1945; Strahler, 1957). A watershed is described as first-, second-, third-, or higher order, depending upon the stream order at the outlet. Journal of Spatial Hydrology 31 Rai et.al./ JOSH Fall 2009 (9)24-50 Basi n ar ea Basin area is defined as the area contained within the vertical projection of the drainage divide on a horizontal plane. Some areas in the drainage basin do not contribute to the runoff and are termed as closed drainage. These area may be lakes, swamps etc. Basi n shape The basin shape may influence the hydrograph shape, especially for small watersheds. For example, if the watershed is long and narrow, then it will take longer time for water to travel from the most extreme point to the outlet and the resulting hydrograph shape is flatter. For more compacted watershed, the runoff hydrograph is expected to be sharper with a greater peak and shorter duration. Numerous symmetrical and irregular forms of drainage areas are encountered in practice. To define the basin shape, a multitude of dimensionless parameters were used to quantify and these are: form factor, shape factor, elongation ratio, circulatory ratio, and compactness coefficient (Table 1). These factors involve watershed length, area and perimeter. The watershed length can be defined as the length of the main stream from its source (projected to the perimeter) to the outlet. Clearly, the form factor is less than unity and its reciprocal, the shape factor, is greater than unity. A square drainage basin has the shape factor equals to unity, whereas the long narrow drainage basin would have a shape factor greater than unity. The elongation ratio, circulatory ratio, and compactness coefficient approach to unity as the watershed shape approached to circle. Table 1 Watershed shape parameters (Singh, 1988) Parameter (Authors) Definition Formula Value Form factor 2 Watershedarea (Watershedlength) 2 / A L = < 1 Shape factor, s B 2 (Watershedlength) Watershedarea 2 / L A = > 1 Elongation ratio Diameter of circle of watershedarea Watershedlength 0.5 1.128 A L = 1 s Circulatory ratio Watershedarea Area of circle of watershed area 2 12.57 r A P = 1 s Compactness coefficient Watershed perimeter Perimeter of circle of watershed area 0.5 0.2821 r P A = 1 > A = watershed area; L = watershed length; P r = watershed perimeter Basi n sl ope Basin slope has a pronounced effect on the velocity of overland flow, watershed erosion potential, and local wind systems. Average basin slope is defined as (Singh, 1989) / S h L = (1) where S is the average basin slope (m/m), h is the fall (m) (i.e. difference in maximum and minimum elevations), and L is the horizontal distance (m) over which the fall occurs. Journal of Spatial Hydrology 32 Rai et.al./ JOSH Fall 2009 (9)24-50 Dr ai nage basi n si mi l ar i t y Strahler (1957) hypothesized the drainage basin similarity as follows: 2 / b A L C = (2) where, A is the area of the drainage basin [L 2 ], L b is the length of the basin [L] and C is the constant for basin similarity. The value of C will be nearly equal for geomorphologically similar basins. Dr ai nage densi t y Drainage density is defined as the length of drainage per unit area. This term was first introduced by Horton (1932) and is expressed as / d D L A = (3) where, D d is the drainage density, L is the total length of the drains in the basin and A is the area of the drainage basin. Eq. (3) can be further expanded as follows: , 1 1 i N i j i j d L D A O = = = ¯¯ (4) where, is the highest order of the stream, is the number of O i N i ÷ th order drain and is the length of , i j L j ÷ th stream of th order drain. Above relationships can be simplified as follows: i ÷ 1 1 1 1 b d L R r D A r O÷ O ÷ = · ÷ (5) where, 1 L is the average length of the first order drain ( ) and ratio r is defined as follows: 1 1, 1 1 / N j j L N = = ¯ / l r R R = b (6) where l R and b R are the stream-length, and bifurcation ratio, respectively. Det er mi nat i on of Hor t on’ s r at i o Three of Horton’s ratios namely bifurcation ratio (R b ), stream-length ratio (R l ) and stream-area ratio (R a ) are unique representative parameters for a given watershed and are fixed values for a given watershed system. Bi f ur cat i on r at i o b R The number of channels of a given order in a drainage basin is a function of the nature of the surface of that drainage basin. In general, the greater the infiltration of the soil material covering the basin, the fewer will be the number of channels required to carry the remaining runoff water. Moreover, larger the number of channels of a given order, the smaller is the area drained by each channel order. A dimensionless parameter based on the number of channels with respect to their order is termed as bifurcation ratio and is useful in defining the watershed response. The bifurcation ratio is given as follows. 1 / b i i R N N + = (7) Journal of Spatial Hydrology 33 Rai et.al./ JOSH Fall 2009 (9)24-50 where R b is the bifurcation ratio, and i N 1 i N + are the number of streams in order i and i+1 respectively, and O is the highest stream order of the watershed. The value of 1, 2,......., i = O b R for watersheds varies between 3 to 5. This law is an expression of topological phenomenon, and is a measure of drainage efficiency. St r eam- l engt h r at i o l R This refers to length of channels of each order. The average length of channels of each higher order increase as a geometric sequence, which can be further explained as: the first order channels are the shortest of all the channels and the length increase geometrically as the order increases. This relation is called Horton’s law of channel length and can be formulated as follows. 1 1 i i L L R ÷ = l (8) where i L is the average length of channel of order i, l R is the stream-length ratio. The l R is mathematically expressed as follows. 1 / l i i R L L + = (9) where, , 1 1 i N i j i L N = = ¯ i j L (10) The lengths of channels of a given order are determined largely by the type of soil covering the drainage basin. Generally, more pervious the soil, longer will be the channel length of a given order. Also, higher is the l R more will be the imperviousness. Generally it varies between 1.5 to 3.5. St r eam- ar ea r at i o a R The Channel area of order i, i A is the area of the watershed that contributes to the channel segment of order i and all lower order channels. It can be quantified as: 1 1 i i A A R ÷ = a (11) where i A is the average area of order i and a R is the stream area ratio, and are given as follows: 1 / a i i R A A + = (12) , 1 1 i N i j i A N = = ¯ i j A (13) where is the total area that drains into the stream of order i. , i j A th j Der i vat i on of Geomor phol ogi c Uni t Hydr ogr aph ( GUH) This section describes the potential of GIUH for deriving the SUH, and solution procedure adopted to solve the Nash based GIUH and GUH. Back gr ound of GI UH based appr oach The GIUH theory was introduced by Rodriguez-Iturbe and Valdes (1979) by relating the peak discharge and time to peak discharge with the geomorphologic characteristics of the catchment and a dynamic Journal of Spatial Hydrology 34 Rai et.al./ JOSH Fall 2009 (9)24-50 velocity parameter. This pioneering work of Rodriguez-Iturbe and Valdes (1979), which explicitly integrate the geomorphologic details and the climatologic characteristics of the basin, in a framework of travel time distribution, is a boon for stream flow synthesis in the basin having no or scanty information of flow data. This formulation of GIUH is based on the probability density function (pdf) of the time history of a randomly chosen drop of effective rainfall arrived to the trapping state of a hypothetical basin, treated as a continuous Markovian process, where the state is the order of the stream in which the drop is located at any time. The value at the mode of this pdf produces the main characteristics of GIUH. Rodriguez-Iturbe and Valdes (1979) derived the peak and time to peak characteristics of the IUH as function of Horton’s order ratios (Horton, 1945), and are expressed as follows (Kumar et al., 2007). 0.43 1.31 / p l q R V O = L l · (14) 0.55 0.38 0.44( / ) ( / ) p b a t L V R R R ÷ O = (15) where q p is the peak flow (h -1 ), t p is the time to peak (h), L O is the length of the highest order stream (km), V is the dynamic velocity parameter (m s -1 ). Multiplying eqs. (14) and (15) a non-dimensional term is derived at. * p q t p p (16) 0.55 0.05 * 0.5764[ / ] p p b a l q t R R R = The term is not dependent on the velocity and thereby on the storm characteristics and hence, it is a function of only the geomorphologic characteristics of the basin. The dynamic velocity parameter in the formulation of GIUH incorporates the effect of climatic variation. Rodriguez-Iturbe et al. (1979) showed the dynamic velocity parameter of the GIUH can be taken as the velocity at the peak discharge time for a given rainfall-runoff event in the catchment. Valdes et al. (1979) compared the GIUHs for some real world basins with the IUHs derived from the discharge hydrograph produced by a physically based rainfall- runoff model of the same basins and found them to be remarkably similar. For few of Indian catchments this GIUH based approach was successfully demonstrated (Kumar et al., 2002; 2007). In the present study, the Nash based GIUH model was developed for the Gomti river basin. * p q t The Nash model The Nash model (Nash, 1957) is based on the concept of routing of the instantaneous inflow through a cascade of linear reservoirs with equal storage coefficient. The Nash model can be expressed as follows: 1 1 ( ) ( / ) exp( / ) ( ) n u t t k t k k n ÷ = I ÷ (17) where u(t) is the ordinates of IUH (hour -1 ), t is the sampling time interval (hour), n and k are the parameters of the Nash model, in which is the number of linear reservoirs, and k is the storage coefficient (hour). n A unit hydrograph (UH) of desired duration (D) may be derived by using the following expression: 1 ( , ) [ ( , / ) ( , ( ) / )] U D t I n t k I n t D k D = ÷ ÷ (18) where U(D, t) denotes the ordinates of D hour UH (hour -1 ), t is the sampling time interval (hour), ( , / ) I n t k is the incomplete gamma function of order n at , and D is the duration of UH (hour). ( / ) t k Geomor phol ogy based par amet er est i mat i on of Nash’ s GI UH Journal of Spatial Hydrology 35 Rai et.al./ JOSH Fall 2009 (9)24-50 The complete shape of the GIUH can be obtained by linking the and of the GIUH with scale (k) and shape (n) parameters of the Nash model. By equating the first derivative with respect to t of eq (17) to zero, t becomes the time to peak discharge, . Thus, taking the natural logarithm of both sides of eq. (17), differentiating with respect to t and by simplification eq. (19) is derived. p q p t p t 1 ( 1) ln[ ( )] n u t t k c ÷ = ÷ + c t ) (19) Equating eq. (19) to zero results in by replacing t with t p , ( 1 p t t k n = = ÷ (20) Simplifying the value of t p from eq. (20) in eq. (17) and simplifying yields 1 1 exp[ ( 1)] ( 1) ( ) n p q n k n ÷ = ÷ ÷ · ÷ I n (21) Combining eqs. (20) and (21) results: 1 ( 1) exp[ ( 1)] ( 1) ( ) n p p n q t n n n ÷ ÷ · = ÷ ÷ · ÷ I (22) Equating eq. (22) with eq. (16) results in: 1 0. ( 1) exp[ ( 1)] ( 1) 0.5764[ / ] ( ) n b a l n n n R R R n ÷ ÷ · ÷ ÷ · ÷ = × I 55 0.05 (23) The Nash parameter n, can be obtained by solving eq. (23) using the Newton-Rapson method. Procedural step of estimation of n using Newton-Rapson method is given in appendix-I. The Nash’s parameter k for the given velocity V is obtained using eqs. (15) and (20) and the known value of the parameter n as follows: 0.55 0.38 0.44 1 ( 1) b l a R L k R V R n ÷ O = · · · ÷ (24) The derived values of n and k are used to determine the complete shape of the Nash based GIUH using eq. (17). Subsequently, the D-hour UH can be derived from eq. (18). The direct runoff hydrograph (DRH) is estimated by convoluting the excess rainfall hyetograph with the UH. Appl i cat i on of t he Model and Resul t s Gomti basin is ungauged basin; therefore, testing the accuracy of derived GUH for this basin using the aforesaid approach is quite difficult. Thus, for systematic application of the proposed approach, two cases were considered: (i) Case I: Application of the model on published data of Burhner catchment taken from Bhunya et al. (2008); and (ii) Case II: Application of the model for Gomti river basin. Case I : Appl i cat i on of t he model on publ i shed dat a of Bur hner cat chment India is divided into seven hydro-meteorological zones and is further divided into twenty six hydro- meteorological sub-zones [i.e. sub-zones 1(a) to 1(g), 2(a) to 2(c), 3(a) to 3(i), 4(a) to 4(c), 5(a), 5(b), 6 and 7] (Jain et al., 2007). The Burhner catchment was picked up from hydro-meteorological subzone 3(c) which comprises of Upper Narmada and Tapi basins. It lies between 80º36' and 81º23' E longitudes and 22º00' and 22º56' N latitude. It has a continental type of climate with hot summers and cold winters and receives most rainfall from the south-west (SW) monsoon during the months of June to October. Mean annual rainfall varies approximately from 800 to 1600 mm. The main soil group is black soil, comprising Journal of Spatial Hydrology 36 Rai et.al./ JOSH Fall 2009 (9)24-50 different types, namely, deep black, medium black and shallow black soil. It is sixth order catchment comprised of about 4103 km 2 area. Length of maximum order stream is 138 km, whereas R a , R b , R l are 3.96, 3.523, 1.787 respectively. Peak discharge, Q p of hydrograph is found to be 69.02 m 3 /s, whereas, time to peak t p is of the order of 11 hour and velocity is 4.15 m/s. Using the aforementioned geomorphologic information, the GUH was derived for the Burhner catchment and compared with the observed UH reported in the study by Bhunya et al. (2008). The comparison of the GUH and observed UH is depicted in Fig. 1. The comparative result of UH for the catchment indicates the suitability of the technique for the derivation of UH and thus DRH. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 5 Time (hour) Q p ( m 3 / s ) 0 Observed UH Derived GUH Fig. 1 Comparison of derived GUH and observed UH of Burhner catchment (Case I) Case I I : Appl i cat i on of t he model f or Gomt i r i ver basi n The river Gomti is one of the major tributary of river Ganga originated from the Pilibhit district of the Uttar Pradesh and joins the river Ganges at Varanasi after traveling approximately 662.5 km. It drains approximately 30407.2 km 2 area of the central-east part of the Uttar Pradesh. Gomti river basin extended between the latitude of 25°23'12.62'' to 28°46'58.75'' N and longitude of 79°57' 33.76'' to 83°11'13.25'' E and belongs to hydro-meteorological subzone 1(f). The digital elevation model (DEM) and the extracted drainage network of the basin using the ArcGIS are shown in Fig. 2(a) and 2(b), respectively. The shape of the basin is nearly oblong in nature. Topography is nearly undulating with high terrains at upstream end of the basin. The maximum at upstream end and minimum elevation at downstream end of the basin are 227 m and 58 m above msl, respectively. The climate of the basin is ranging from semi arid to sub humid tropical with average annual rainfall at different locations is 850-1100 mm. Approximately, 75 percent of total rainfall is due to the occurrence of North-West Monsoon (Mid June to Mid October). The mean minimum and maximum temperature over the basin is 4.5° to 46.0° C with daily mean sunshine of 8 hours. The relative humidity varies between 10-90 percent. The potential evapotranspiration experienced in the basin is nearly 1300 mm. Major soil group experienced over the basin is sandy loam to clay loam, however, large variation can be observed throughout the basin. The land use in the basin can be broadly categorized as: agriculture, non-agriculture, pasture fellow, forest, etc. Major crops cultivated in the basin are paddy, wheat, sugarcane, potato, and other vegetable crops depending on the irrigation facilities. Journal of Spatial Hydrology 37 Rai et.al./ JOSH Fall 2009 (9)24-50 Fig. 2 Digital Elevation Model (DEM) and extracted drainage network of the Gomti river basin To extract the geomorphologic features of the basin, SRTM (Shuttle Radar Topography Misson) data of 90 x 90 m resolution was used. Based on the study by Haase and Frotscher (2005) it can be stated that the SRTM data sets are of major relevance for providing terrain information in large and transboundary river basins for handling the regional environmental problems, and could be applied in meso / macro- scale river network and terrain analyses. Besides these, the application of the SRTM data is cost-efficient and informative and is web-based world wide available and appropriately implemented into a geographic information system (GIS)-framework. The DEM of Gomti river basin (Fig. 2a) from SRTM data set was processed in the ArcGIS 9.1 platform to delineate the basin boundaries and drainage networks (Fig. 2b). The stream ordering was carried out using the DEM of the basin using Strahler’s scheme followed by establishing the flow direction, flow accumulation etc. The extracted stream network of different orders is shown in Fig. 3(a), and the highest order of the basin was recorded as 6. Maximum length of the river is found to be 662.5 km. As stated earlier, the aerial contribution to the runoff by different order is also important to understand the runoff and erosion process, and therefore, it needs to be extracted and shown in Fig. 3(b). The number of streams of different orders, length, and corresponding area are presented in Table 2, whereas, the extracted geomorphologic parameters such as drainage area, perimeter of the basin, length of the basin, maximum and minimum elevation, watershed relief, relief ratio, elongation ratio, mean slope, drainage density, stream frequency, circulatory ratio, farm factor, Horton’s bifurcation ratio (Fig. 4), length ratio (Fig. 5), stream-area ratio (Fig. 6), etc. are summarized in Table 3. Journal of Spatial Hydrology 38 Rai et.al./ JOSH Fall 2009 (9)24-50 Fig. 3 Streams of different orders and corresponding aerial coverage for Gomti river basin y = -1.4418x + 8.5298 R 2 = 0.9971 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 Order L n ( N ) (R b = 4.283) 3 (a) Fig. 4 Estimation of bifurcation ratio (R b ) of the basin Journal of Spatial Hydrology 39 3 (b) Rai et.al./ JOSH Fall 2009 (9)24-50 y = 0.7173x + 0.7604 R 2 = 0.8347 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Order L n ( L ) (R l = 2.218) Fig. 5 Estimation of stream length ratio (R l ) of the basin y = 1.5342x + 1.329 R 2 = 0.9943 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 Order L n ( A ) (R a = 4.772) Fig. 6 Estimation of stream area ratio (R a ) of the basin Table 2 Number, length, and area for streams of various orders of Gomti basin Order Number Length (km) Area (km 2 ) 1 1316 4422.90 18577.16 2 291 2243.21 23670.74 3 65 1222.95 28525.32 4 12 797.90 29549.77 5 4 645.34 30252.09 6 1 63.82 30407.20 Journal of Spatial Hydrology 40 Rai et.al./ JOSH Fall 2009 (9)24-50 Table 3 Extracted geomorphological parameters of the Gomti river basin Parameters Value Area (km 2 ) 30407.2 Perimeter (km) 1639.4 Length of Basin (km) 481.0 Maximum elevation (m) 227.0 Minimum elevation (m) 58.0 Relief (km) 0.169 Mean slope (km/km) 0.00027 Stream Characteristics Numbers Mean Length (km) Mean Area (km 2 ) 1 st order stream 1316 3.36 14.12 2 nd order stream 291 7.71 81.34 3 rd order stream 65 18.81 438.85 4 th order stream 12 66.49 2462.48 5 th order stream 4 161.34 7563.02 6 th order stream 1 63.82 30407.20 Ratios Bifurcation ratio (R b ) 4.283 Length ratio (R l ) 2.218 Stream-area ratio (R a ) 4.772 Horton ratio (r) 0.518 Drainage density (D d ) 0.304 Drainage frequency (F s ) 0.055 Elongation ratio (Re) 0.144 Circulatory ratio (Rc) 0.145 Farm factor (Ff) 0.121 Shape factor (Fs) 8.24 Compactness factor (Fc) 2.63 Using the aforesaid approach of Nash based GIUH and estimated Horton’s ratios (i.e. R b , R l , and R a ) the parameter n of the Nash model was optimized using the Newton Rapson method (i.e. eqs. 25 through 31). The optimized value of n was found to be 3.17. The parameter k of the Nash model was estimated for the derived Horton’s dimensionless parameters (i.e. R b , R l , and R a ) at different dynamic velocity of flows using eq. (24). The dynamic velocity of flow can be computed using the approach suggested by Kumar et al. (2002). The estimated values of n and k for various flow velocities are given in Table 4. Using the estimated values of the n and k at different velocity of flows V, the ordinates of instantaneous unit hydrograph (IUH) was computed using eq. (17), and finally the 1-hour UHs were derived for different velocity of flow using the relationship given by eq. (18). The 1 h-UH at different velocities are depicted in Fig. 7. Journal of Spatial Hydrology 41 Rai et.al./ JOSH Fall 2009 (9)24-50 0 1000 2000 3000 4000 5000 6000 7000 0 25 50 75 100 125 Time (hour) U H ( m 3 / s ) GUH (V = 0.5 m/s) GUH (V = 0.68 m/s) GUH (V = 0.75 m/s) GUH (V = 1.0 m/s) GUH (V = 1.5 m/s) GUH (V = 2.0 m/s) GUH (V = 2.5 m/s) UH: Computed by CWC Fig. 7 GUH at different velocities and observed UH (CWC procedure) for the Gomti river basin Table 4 Estimated values of Nash parameters n and k at different velocity, V V (m/s) n k (hour) V (m/s) n k (hour) 0.50 3.1665 18.0463 3.00 3.1665 3.0077 1.00 3.1665 9.0232 3.50 3.1665 2.5780 1.50 3.1665 6.0154 4.00 3.1665 2.2558 2.00 3.1665 4.5116 4.50 3.1665 2.0051 2.50 3.1665 3.6093 5.00 3.1665 1.8046 To validate the derived GUH for the Gomti river basin, a comparison was presented with the UH computed from the procedure adopted by CWC. The CWC method is well accepted by field engineers / hydrologist for the derivation of UH on the ungauged catchment in India for the estimation of design flood to install the waterway of bridges/cross drainage structures across small and medium streams. In this method a set of equations for UH parameters (viz., peak flow rate, time to peak flow rate and time base of the hydrograph) as a function of slope, length of the longest stream and drainage area were developed for the regions based on the regional analysis of the observed data of 42 representative sub-watersheds (40 sub-watersheds data were collected by Northern and North-eastern Railways, India and 2 sub- watersheds data were collected by Ministry of Transport, India). Computational step to derive the SUH using CWC technique is given in Appendix-II. The computed 1- hour SUH using the CWC procedure is shown in Fig. 7. It can be revealed from Fig. 7 that the derived GUH with flow velocity of 0.68 m/s (approx) is close to the SUH computed by CWC procedure. Since, CWC technique is independent of the most important variable of the basin i.e. climatic parameter (i.e. dynamic flow velocity) and does not account for the geomorphologic parameters such as characteristics of the drainage network, contributing area of the different order drains and their lengths. Thus, the error associated in the computation of flood hydrograph may be more. Besides this, CWC technique does not give the actual shape of the hydrograph. Hence, the GUH based approach becomes more realistic which provide the complete shape of the UH at different Journal of Spatial Hydrology 42 Rai et.al./ JOSH Fall 2009 (9)24-50 values of dynamic velocity of flow. Once the dynamic flow velocity is estimated from field visit, the actual UH of the basin can be easily obtained. The average dynamic flow velocity in the Gomti river varies between 0.65 to 0.75 m/s. The DRH and thus the flood hydrographs for the river Gomti can be easily obtained by convoluting the rainfall excess (mm) with ordinates of GUH. Using the results obtained from the analysis, a relationship between the peak flow rate, dynamic flow velocity, and time to peak flow have been sought for Gomti river basin and are expressed as follows. Q p = 0.0289 x V 0.9982 (R 2 = 0.999) (25) t p = 19.834 x V -1.043 (R 2 = 0.993) (26) t p = 0.488 x Q p -1.0452 (R 2 = 0.993) (27) where, Q p is the peak flow rate (m/s/mm), V is the dynamic velocity of flows (m/s), and t p is the time to peak (hours). Based on the velocities, the peak flow and time to peak can be directly obtained. Equations (24) through (27) can be used as a ready reference to the field engineers for design of hydraulic structures. Concl usi ons This study comprised of geomorphological investigation and development of GUH for Gomti river basin of India. Based on the geomorphological analysis, Gomti river basin is of sixth order and almost flat terrain except at the upstream end of the basin. A detailed feature useful to engineering and environmental studies / application for the basin is summarized in tabular form (Tables 2 and 3). Also, the extracted drainage network of the basin has wide range of application such as installation of the artificial drainage system and laying down the canal system, site identification for rain water harvesting, groundwater recharge and waste disposal, etc. However, the main theme of the paper is to develop a GUH for the Gomti river basin for the estimation of flood hydrograph. To test the accuracy of the GIUH technique, Case-I was considered to derive the GUH for the Burhner catchment of India. Based on the comparison of the derived GUH and observed UH, it may be concluded that the proposed approach is suitable and efficient for the derivation of UH. On the other hand, Case II presents the application of GIUH technique for the derivation of GUH for Gomti river basin. Three of Horton’s ratios of the Gomti river basin were coupled with the parameters of the Nash’s IUH model and the GUHs were derived for different values of dynamic velocity of flow (Fig. 7). The obtained GUH with dynamic flow velocity of 0.68 m/s (approx.) shows closer agreement with SUH derived from CWC approach. Since, CWC approach is independent of climatic parameter (i.e. dynamic flow velocity) and geomorphologic characteristics other than the slope, drainage area and length of main stream; therefore, the resulting UH may have higher computational error. Hence, the GUH based approach will be superior to the CWC approach. Based on the analysis of the results, three equations (i.e eqs 24 through 27) were also developed for the estimation of peak ordinates of direct surface runoff hydrographs for Gomti basin provided the excess rainfall depth is available. Concluding, it may be remarked that the proposed technique, which was not yet developed for the Gomti river basin of India will, helps the engineers for accurate estimation of the flood hydrograph as well as for the modeling of pollutants transport. Besides this, the described technique is cost-effective and quite accurate for determining the GUH and flood hydrograph for any catchment / basin (gauged or ungauged) as it requires DEM of the catchment that can be freely obtained from SRTM source. Journal of Spatial Hydrology 43 Rai et.al./ JOSH Fall 2009 (9)24-50 Appendi x - I : The New t on- Rapson met hod t o est i mat e n From eq. (23), the overall function can be defined as follows: 0.55 1 ( 1) exp[ ( 1)] ( 1) 0.5764 ( ) n b l a R n f n n n R ÷ ÷ = ÷ ÷ · ÷ ÷ × I 0.05 R (28) The first derivative of f with respect to n can be expressed as follows: 1 1 1 1 1 1 exp( 1) ( 1) exp( 1) ( 1) ( ) ( ) ( ) 1 1 exp( 1) ( 1) exp( 1) ( 1) {ln( 1) 1} ( ) ( ) n n n n n f n n n n n n n n n n n n n n n n v ÷ ÷ ÷ ÷ ÷ ' = ÷ + · ÷ ÷ ÷ + · ÷ · I I ÷ ÷ ÷ ÷ + · ÷ + ÷ + · ÷ × ÷ I I + (29) where ( ) n v is a Psi function also known as digamma function and is expressed as follows: ( ) ( ) [ln{ ( )}] ( ) d z z z dz z v ' I = I = I (30) The Psi function for any real number was evaluated using the following set of equation (eqs. 31a through 31d). (1) 0.57722 v ¸ = ÷ = ÷ (31a) (1/ 2) 2ln(2) 1.96351 v ¸ = ÷ ÷ = ÷ (31b) 1 1 1 ( ) ( 2) n k n k n v ¸ ÷ ÷ = = ÷ + > ¯ (31c) 1 1 1 2ln(2) 2 ( 1) 2 2 1 n k n k n v ¸ = | | | | + = ÷ ÷ + > | | ÷ \ . \ . ¯ (31d) The Gamma function of a dummy variable z, ( ) z I can be defined as follows: 1 0 ( ) exp( ) z z t · ÷ I = · ÷ í t dt (32) The optimization algorithm for Newton-Rapson method is expressed as follows: ( _ old) _ new _ old ( _ old) f n n n f n = ÷ ' (33) The optimization terminated when the following condition is achieved. ( _ new _ old) 0.00001 n n ÷ s (34) Appendi x - I I : CWC pr ocedur e of UH comput at i on f or Gomt i r i ver basi n Gomti basin falls under sub zone 1(f) of the hydro-meteorological homogeneity. Considering the sub zone 1(f), the relations established between physiographic and unit hydrograph (UH) parameters developed by Central Water Commission, India (CWC, 1985) are applied for derivation of 2- hour SUH for an ungauged catchment. The following steps are involved for derivation of 2 hour unit hydrograph. The Journal of Spatial Hydrology 44 Rai et.al./ JOSH Fall 2009 (9)24-50 parameters used for developing SUH as proposed by CWC are given in the Fig. 8. Using the suggested procedure a 2 hour SUH has been developed for the Gomti river. Fig. 8 Components of SUH for CWC procedure Step 1: Analysis of ‘catchment physiography’ Physiographic parameters viz. the catchment area A, length of the longest stream L and equivalent stream slope are determined from the catchment area plan for the estimation of S / L S , where the equivalent stream slope can be calculated as follows. S ( ) 1 2 i i i L D D S L ÷ ÷ = ¯ (35) where is the length of the i th segment in km, i L 1 i D ÷ , i D are the depths of the river at the point of intersection of ( ) and th contours respectively from the base line (datum) drawn at the level of the point of study in meters and 1 i ÷ i L is the length of the longest stream. For Gomti river basin equivalent slope is found to be 5.2 m /km Step 2: Determination of ‘peak value for 2-hour SUH’ The following equations is used for the peak discharge per unit of catchment area (m 3 /s/sq km) for the 2-hour SUH of Gomti river basin. p q 0.649 2.030/( / ) p q L S = (36) The peak discharge (m 3 /s) for the 2 hour SUH is obtained as follows p Q p p Q q A = × (37) Step 3: Determination of ‘time from center of effective rainfall duration to the peak value for 2-hour SUH’ Journal of Spatial Hydrology 45 Rai et.al./ JOSH Fall 2009 (9)24-50 The time to peak (hr) for 2-hour SUH is estimated using the following relationship. p t 1.308 1.858/( ) p t q = p ws. (38) Step 4: Estimation of the ‘ time to peak ( ) for 1-hour SUH’ m T The time to peak for 2-hour SUH (Figure 8) is computed using the following relationship. m T / 2 m p r T t t = + (39) Step 5: Estimation of ‘ time base for 2-hour SUH’ Time base (hour) is computed as follows. B T ( ) 0.779 7.744 B p T t = (40) Step 6: Estimation of ‘time widths for different % values of peak discharge of 2-hour SUH’ The time width parameters (in hr) , , and (Figure 8) can be calculated using the following set of equations. 50 W 75 W 50 WR 75 WR 0.990 50 2.217 /( ) p W q = (41) 0.876 75 1.447 /( ) p W q = (42) 0.907 50 0.812 /( ) p WR q = (43) 0.791 75 0.606 /( ) p WR q = (44) (Note: the values in the parenthesis refer to Figure 8) Step 7: Computation of ‘unit depth’ Parameters viz. p p 0 5 5 are used and a 2-hour SUH is generated. The runoff depth ( d ) under the SUH is computed as foll Q , m T , t , B T , 50 W , 5 WR , 7 W , 7 WR o 0.36 i r Q t d A × × = cm (45) In case, the depth of runoff ( ) for the SUH is not equal to 1.0 cm, suitable modifications may be made by smoothening the graph so as to contain a unit volume ( i,e, 1 cm d × catchment area). Step 8: Cross checking of Time Base of SUH by ‘triangular unit hydrograph (TUH) method’ Conventionally the time base of a TUH is adopted as follows. B T B T =2.67× (46) m T This value comes out to be same as CWC method and therefore, the SUH derived from CWC method is accepted for the Gomti river basin. Unit depth calculation for the derived SUH is done. B T Step 9: Conversion of 2 hr UH to 1 hr UH by S-Hydrograph method To derive 1-hour SUH from 2-hour SUH, S-hydrograph technique was used. Journal of Spatial Hydrology 46 Rai et.al./ JOSH Fall 2009 (9)24-50 Ref er ences dany, A. S., and Rao, A. R. (1998). “Correlation of the velocity parameter of Al-Wag three geomorphological instantaneous unit hydrograph models.” Hydrol. Process., 12, 651-659. A. Pnigrahi, N., Singh, R., Raghuwanshi, N. S., Mal, B. C., and Tripathi, M. P. (2008). “Development of geomorphological instantaneous unit hydrograph mo Bhadra, Bhunya Central Kumar, del for scantly gauged watersheds.” Environmental Modelling & Software, 23, 1013-1025. r, N. R., parida, B. P., and Nayak, A. K. (1997). “Flood estimatio Bhaska n for ungauged catchments using the GIUH.” J. Water Resour. Plng. and Mgmt., 12 (4), 228-238. , P. K., Berndtsson, R., Singh, P. K. and Hubert , P. (2008) Comparison between Weibull and gamma distributions to derive synthetic unit hydrograph using Horton ratios, water Resources Research , Vol. 44, W04421, doi:10.1029/2007WR006031 Water Commission Research Designs (CWC) (Directorate of Hydrology [Regional Studies]) & India Meteorological Dept. & Ministry of Surface Transport (1994). Flood Estimation Report for Western Himalayas - Zone 7, Design Office Report No. WH/22/1994, CWC, New Delhi. .T., Maidme Chow, V nt, D.R., and Mays, L.W. (1988). Applied Hydrology, McGraw-Hill Book Company, New York. Clark, C. O. (1945). “Storage and the unit hydrograph.” Trans. Am. Soc. Civ. Engr., 110, 1945-46. Dooge, J.C. (1959). “A general theory of the unit hydrograph.” J. Geophys. Res., 64(2), 241-256. J.C. (1973). Linear theory of hydrologic Systems, Technical Bu Dooge, lletin No. 1468, U.S. Department of Agriculture, Agricultural Research Service, Washington, D.C. t, C., Kartiwa, B., and Roland, B. (2006). “Analytical Fleuran model for a geomorphological instantaneous unit hydrograph.” Hydrol. Process., 20, 3879-3895. D., and Frotscher, K. (2005). “Topography data harmonization and uncertainties applyin Haase, g SRTM, laser scanner and cartographic elevation models.” Advances in Geosciences, 5, 65-73. R. E. (1945). “Erosional development of streams and their drainage b Horton, asins: Hydrophysical approach to quantitative morphology.” Bull. Geol. Soc. Amer. 56, 275–370. K., Singh, R. D., and Seth, S. M. (2000). “Design Jain S. flood estimation using GIS supported GIUH approach.” Water Resour. Manage., 14, 369-376. , and Sinha, R. (2003). “Derivation of u Jain, V. nit hydrograph from GIUH analysis for a Himalayan river.” Water Resour. Manage., 17, 355-375. . (1992). “A deterministic gamma-type geomor Jin, C.-X phologic instantaneous unit hydrograph based on path types.” Water Resour. Res., 28, 479-486. , D. M., and Bras, R. L Kirshen . (1983). “The linear channel and its effect on the geomorphologic IUH.” J. Hydrol., 65, 175-208. R., Chatterjee, C., Lohani, A. K., Kumar, S., and Singh, R. D. (2002). “Sensitivity a Kumar, nalysis of the GIUH based Clark model for a catchment.” Water Resour. Manage., 16, 263-278. R., Chatterjee, C., Singh, R. D., Lohani, A. K., and Kumar, S. (2007). “Runoff estimation for an ungauged catchment using geomorphological instantaneous unit hydrograph (GIUH) model.” Hydrol. Process., 21, 1829-1840. A. K., Singh, R. D., and Nema, R. K. (2001). ‘‘Comparison of geomorpholog Lohani, ical based rainfall- runoff models.’’ CS(AR) Technical Rep., National Inst. of Hydrology, Roorkee, . E. (1957) Nash, J . “The forms of instantaneous unit hydrograph” Int. Ass. Sci. and Hydrol., Pub. 1, 45 (3), 114-121. , A., Marani, A., a Rinaldo nd Rigon, R. (1991). “Geomorphological dispersion.” Water Resour. Res., 27, 513-525. Journal of Spatial Hydrology 47 Rai et.al./ JOSH Fall 2009 (9)24-50 Rodriguez-Iturbe, I., and Valdes, J. B. (1979). “The geomorphologic structure of hydrologic response.” Water Resour. Res., 15 (6), 1409-1420. Rodriguez-Iturbe, I., Gonzalas S. M., and Bras, R. C. (1982). “The geomorpho-climatic theory of the Rosso, 0, 914-920. Sarangi, A., Madramootoo, C. A., Enright, P., and Prasher, S. O. (2007). “Evaluation of three unit Sarkar, ekar, H. G. (2007). “Geomorphological parameters: are they indicators for y, ka, 225. e-Hall, New Jersey Singh, V 988b). Elementary Hydrology, Prentice-Hall, New Jersey 07632, USA ations. Water Resources Publications, Highlands Ranch, CO. eting, Part 2, p. Strahler “Quantitative slope analysis.” Bull. Geol. Soc. Amer. 67, 571–596. ys. Union 33, 235–246. en, B. C., and Lee, K. T. (1997). “Unit hydrograph derivation for ungauged watersheds by stream-order laws.” J. Hydrologic Engrg., 2 (1), 1-9. instantaneous unit hydrograph.” Water Resour. Res., 18 (4), 877-886. R. (1984). “Nash model relation to Horton order ratios.” Water Resour. Res., 2 Sahoo, B., Chatterjee, C., Raghuwanshi, N.S., Singh, R., and Kumar, R. (2006). “Flood estimation by GIUH-based Clark and Nash models.” J. Hydrologic Engrg., 11 (6), 515-525. hydrograph models to predict the surface runoff from a Canadian watershed” Water Resour. Manage., 21, 1127-1143. S., and Gund installation of a hydropower site?” International Conference on Small Hydropower, Kand Srilan Sherman, L.K. (1932). “Stream flow from rainfall by the unit-graph method.” Eng. News Rec., 108, 501- 505. Singh, V. P. (1988a). Hydrologic Systems: Vol I, Rainfall-Runoff Modeling, Prentic 07632, USA. . P. (1 Singh, V. P. (1989). Hydrologic Systems: Vol II, Watershed modeling, Prentice-Hall, New Jersey 07632, USA. Singh, V.P., and Frevert, D.K. (Eds.) (2002a). Mathematical Models of Large Watershed Hydrology. Water Resources Publications, Highlands Ranch, CO. Singh, V.P., and Frevert, D.K. (Eds.) (2002b). Mathematical Models of Small Watershed Hydrology and Applic Snyder, F. F. (1938). “Synthetic Unitgraphs.” Trans. Am. Geophys. Union, 19 th Annual Me 447. , A. N. (1956). Strahler, A. N. (1957). “Quantitative analysis of watershed geomorphology.” Trans. Am. Geophys. Uni., 38, 913-920. Taylor, A. B., and Schwartz, H. E. (1952). “Unit-hydrograph lag and peak flow related to basin characteristics.” Transact. Amer. Geoph Y Journal of Spatial Hydrology 48 Rai et.al./ JOSH Fall 2009 (9)24-50 Not at i ons & Abbr evi at i ons b R = bifurcation ratio i N = number of i-th order streams O = highest stream order of the basin l R = stream length ratio i L = average length of i-th order stream i L = length of i-th order stream a R = stream area ratio i A = average area of i-th order stream rains into the stream of order i , i j A = total area that d th j d D = drainage density p q = peak flow rate L O = length of highest order stream ( , ) / I n t k = incomplete gamma function of order n at ( / ) t k I = Gamma function ( ) n v = Psi function of n ( ) ' I z = derivative of gamma function A = drainage area of the basin C = constant for drainage basin similarity n unit hydrograph place u be reservoir (Nash parameter) atershed perimeter CWC = Central Water Commissio D = duration of UH DEM = digital elevation model DRH = direct runoff hydrograph exp IS = exponential function mation system G = geographic infor GIUH H = geomorphologic instantaneous GU = geomorphologic unit hydrograph h = fall in the basin hydrograph IUH = instantaneous unit k = storage coefficient (Nash parameter) er which fall (i.e. h) take L = horizontal distance ov sin L b = length of the ba m f linear n P = n r o = w r Q p = peak flow rate r = / l b R R S = average slope of the ba Topograp sin TM hic Mission SR =Shuttle Radar SUH = synthetic unit hydrograph t = sampling time interval t p = time to peak Journal of Spatial Hydrology 49 Rai et.al./ JOSH Fall 2009 (9)24-50 Journal of Spatial Hydrology 50 drograph , t) u(t) = ordinate of IUH UH = unit hydrograph V = dynamic velocity of flow TUH = triangular unit hy U(D = ordinates of D hour UH Rai et.al./ JOSH Fall 2009 (9)24-50 model (Nash, 1957) is based on the concept of ‘a cascade of linear reservoirs’ in the watershed. The physically based distributed models on the other hand, need rigorous computational skill for solving the governing equations and therefore, require a significant quantum of watershed information along with event based rainfall and runoff data for the calibration of model parameters (Singh and Frevert, 2002 a, b). It is the fact that most of the Indian as well the watersheds of developing countries do not have sufficient historical hydrological records and detailed watershed information needed for physically based distributed models. Therefore, a simplified representation of the rainfall-runoff process through the input-output linkage without a detailed physical description of the process can be used and this is the basis of the linear theory of hydrologic systems. For mathematical description of linear hydrologic system, the unit hydrograph (UH) or unit pulse response function can be employed for system identification in hydrologic analysis (Sherman, 1932; Dooge, 1959, 1973; Chow et al., 1988; Singh, 1988a). Therefore, geomorphology based approach becomes a one of the most popular modeling tools for the computation of runoff hydrographs under these circumstances (Rodriguez-Iturbe and Valdes, 1979; Rodriguez-Iturbe et al., 1982; Yen and Lee, 1997; etc.). Geomorphology of a river basin describes the status of topographic features of the surfaces and streams, and its relationship with hydrology provides the geomorphological control on basin hydrology (Jain and Sinha, 2003). Geomorphology reflects the topographic and geometric properties of the watershed and its drainage channel network. It controls the hydrologic processes from rainfall to runoff, and the subsequent flow routing through the drainage network. The role of basin geomorphology in controlling the hydrological response of a river basin is known for a long time. Moreover, for any infrastructural development, it is very useful tool for first hand overview of the basin. It is advantageous in case of laying out the urban drainage and irrigation canal system, aqueducts, study the physiographic impacts on environment, and selection of silt disposal site, hydropower site (Sarkar and Gundekar, 2007), recharge zone, percolation tank, retention tank, dam site, etc. This drainage network of the river basin can provide a significant contribution towards flood management and water logging program (Jain and Sinha, 2003). Earlier works have provided an understanding of basin geomorphology-hydrology relationship through empirical relationships (Snyder, 1938; Horton, 1945; Taylor and Schwartz, 1952; Singh, 1988b; Singh, 1989; Jain and Sinha, 2003; etc.). Snyder (1938) proposed synthetic unit hydrograph approach (SUH) for ungauged basin as a function of catchment area, basin shape, topography, channel slope, stream density and channel storage; and derived the basin coefficient by averaging out other parameters. Furthermore, the hydrologic response is a function of climatic parameters, landuse, soil parameters and topography and therefore, for any physical based models requires time to time change in their parameters due to variation encountered with respect to the gradual climatic changes and landuse of the watersheds (Rodriguez-Iturbe et al., 1982). However, the geomorphological parameters are mostly time-invariant in nature and therefore, geomorphology based approach could be the most suitable technique for modeling the rainfall-runoff process for ungauged catchments. The rainfall pattern, in general is undergoing a change due to global changes in atmospheric conditions. Further, because of different activities in the watershed, its land use is also having a gradual changes and this has an impact on the characteristics of the runoff produced from the watersheds. Thus, the geomorphological instantaneous unit hydrograph (GIUH) (Rodriguez-Iturbe and Valdes, 1979) and further simplified by Gupta et al. (1980) is a hydrological model that relates the geomorphological features of a basin to its response to rainfall. They can be applied to ungauged basins having scarce hydrologic data (Al-Wagdany et al., 1998). The derivation of Journal of Spatial Hydrology 30 Basin order and channel order Drainage areas may be characterized in terms of the hierarchy of stream ordering. useful. This upper most channel joins another channel and form the higher order channel. Physical characteristics of the drainage basin include drainage area.).. GIUH./ JOSH Fall 2009 (9)24-50 the GIUH uses the assumption that a stream of a certain order has a known linear response function of the familiar or complex probability distributions (Rodriguez-Iturbe and Valdes. 2000. Jin. Kumar et al. 1945. in evaluating the hydrologic response of the basins. 1957).. Once the DRH is computed.). Channel characteristics include channel order. Bhadra et al. channel length.al. 2006. Jain et al. Rosso. the flood hydrograph can be simply obtained by adding the base flow component. 1945) and Nash’s cascade technique (Nash. The first order channel is defined as those channels that receive water entirely from overland surface and does not have tributaries. Kirshen and Bras. (ii) to test the applicability of proposed approach on the published data of Bhunya et al. Another advantage of GIUH technique is its potential for deriving the unit hydrograph (UH) using the geomorphologic characteristics obtainable from topographic map / remote sensing. and the interrelationship between the geomorphology and hydrology. The effect of linear channels in the hydrologic response was introduced by Kirshen and Bras (1983). 2008. Journal of Spatial Hydrology 31 . and so on. 2006. A watershed is described as first-. possibly linked with geographic information system (GIS) and digital elevation model (DEM) (Rodriguez-Iturbe and Valdes. This scheme of stream ordering is referred to as the Horton-Strahler ordering scheme (Horton. 2001. Rinaldo et al. (2008) (iii) to investigate the geomorphology of the Gomti river basin and estimate the dimensionless Horton’s ratios. second-. These characteristics relate to the physical characteristics of the drainage basin as well as of the drainage network. The junction of two first order streams forms the second order stream. basin shape. The order of the basin is the order of its highest-order channel. or higher order. 1997. 1979. 1984. 1991. 1997. It can be stated that the uppermost channel receives water from overland surface and direct towards the outlet through the drainage network. depending upon the stream order at the outlet. Sahoo et al. and drainage density. and so on. ground slope.e.. GUH) for Gomti river basin using the Horton’s ratios. when quantified. the present study was carried out with the objectives: (i) to present an efficient solution approach for generating geomorphologic transfer function i. Sarangi et al.. Therefore.. 1979.. It means the higher order drain carries more water than the lower order drains. the GIUH technique is applicable for the estimation of the direct runoff component of the stream flow and hence. and it is a more powerful technique for the flood estimation than the commonly used parametric Clark model (Clark.. etc.. can be used to generate the direct runoff hydrograph (DRH). Strahler. channel profile. 1992. 1983.e. and (iv) to derive the geomorphologic unit hydrograph (i.. Geomorphologic Parameters Certain characteristics of the drainage basins reflect hydrologic behavior and are therefore. Bhaskar et al. Kumar et al. and centroid (i. 2007. Fleurant et al. third-. based on the foregoing discussions.. When the two second order streams joins together forms the third order stream. Thus. the GIUH based transfer function approach is applicable in such a situation where rainfall data is available but runoff data are not. Lohani et al. channel slope. A second order channel receives flow from the two first order channels and from overland flow from surface and it might receive the flow from another first order stream.Rai et. 1957) (Yen and Lee. 2002.e. 2007. etc. centre of gravity of the basin). However.). etc. swamps etc. For more compacted watershed. Basin shape The basin shape may influence the hydrograph shape. For example.57 A  Pr 2 0. then it will take longer time for water to travel from the most extreme point to the outlet and the resulting hydrograph shape is flatter. shape factor. the form factor is less than unity and its reciprocal. difference in maximum and minimum elevations). the shape factor. The watershed length can be defined as the length of the main stream from its source (projected to the perimeter) to the outlet. if the watershed is long and narrow. Numerous symmetrical and irregular forms of drainage areas are encountered in practice. whereas the long narrow drainage basin would have a shape factor greater than unity.al. elongation ratio. 1988) Parameter (Authors) Form factor Shape factor. Average basin slope is defined as (Singh. and L is the horizontal distance (m) over which the fall occurs. A square drainage basin has the shape factor equals to unity. h is the fall (m) (i. the runoff hydrograph is expected to be sharper with a greater peak and shorter duration.2821 Pr  A0. is greater than unity. and compactness coefficient (Table 1). circulatory ratio.128 A0. Clearly. area and perimeter.Rai et. Pr = watershed perimeter Basin slope Basin slope has a pronounced effect on the velocity of overland flow.e. Journal of Spatial Hydrology 32 .5 1 1 1 A = watershed area. especially for small watersheds. watershed erosion potential. Some areas in the drainage basin do not contribute to the runoff and are termed as closed drainage. These area may be lakes. a multitude of dimensionless parameters were used to quantify and these are: form factor. Table 1 Watershed shape parameters (Singh. and compactness coefficient approach to unity as the watershed shape approached to circle. and local wind systems. These factors involve watershed length. Definition Formula Value <1 >1 Watershed area (Watershed length) 2  A / L2 Bs Elongation ratio Circulatory ratio Compactness coefficient (Watershed length) 2 Watershed area Diameter of circle of watershed area Watershed length Watershed area Area of circle of watershed area Watershed perimeter Perimeter of circle of watershed area  L2 / A 1. The elongation ratio. To define the basin shape. 1989) S  h/ L (1) where S is the average basin slope (m/m). circulatory ratio. L = watershed length./ JOSH Fall 2009 (9)24-50 Basin area Basin area is defined as the area contained within the vertical projection of the drainage divide on a horizontal plane.5  L 12. L1 is the average length of the first order drain (   L1. and bifurcation ratio. L is the total length of the drains in the basin and A is the area of the drainage basin. respectively. Eq. Bifurcation ratio Rb The number of channels of a given order in a drainage basin is a function of the nature of the surface of that drainage basin. This term was first introduced by Horton (1932) and is expressed as Dd  L / A (3) where.al. stream-length ratio (Rl) and stream-area ratio (Ra) are unique representative parameters for a given watershed and are fixed values for a given watershed system./ JOSH Fall 2009 (9)24-50 Drainage basin similarity Strahler (1957) hypothesized the drainage basin similarity as follows: A / Lb 2  C 2 (2) where.Rai et. Moreover. A is the area of the drainage basin [L ]. j A (4) where. A dimensionless parameter based on the number of channels with respect to their order is termed as bifurcation ratio and is useful in defining the watershed response. Rb  Ni / Ni 1 (7) Journal of Spatial Hydrology 33 . (3) can be further expanded as follows: Dd  of  L i 1 j 1  Ni i. j is the length j  th stream of i  th order drain. the smaller is the area drained by each channel order.  is the highest order of the stream. Determination of Horton’s ratio Three of Horton’s ratios namely bifurcation ratio (Rb). Dd is the drainage density. j / N1 ) and ratio r is defined as follows: (6) r  Rl / Rb where Rl and Rb are the stream-length. the fewer will be the number of channels required to carry the remaining runoff water. larger the number of channels of a given order. The value of C will be nearly equal for geomorphologically similar basins. Lb is the length of the basin [L] and C is the constant for basin similarity. In general. the greater the infiltration of the soil material covering the basin. N i is the number of i  th order drain and Li . Above relationships can be simplified as follows: L1 Rb 1 r   1  A r 1 N1 j 1 Dd  (5) where. The bifurcation ratio is given as follows. Drainage density Drainage density is defined as the length of drainage per unit area. Generally.al. Rl is the stream-length ratio.Rai et./ JOSH Fall 2009 (9)24-50 where Rb is the bifurcation ratio.. more pervious the soil. j is the total area that drains into the j th stream of order i. j Ra  Ai 1 / Ai Ai  1 Ni A j 1 Ni (13) where Ai . higher is the Rl more will be the imperviousness..5 to 3. Background of GIUH based approach The GIUH theory was introduced by Rodriguez-Iturbe and Valdes (1979) by relating the peak discharge and time to peak discharge with the geomorphologic characteristics of the catchment and a dynamic Journal of Spatial Hydrology 34 . Generally it varies between 1.. longer will be the channel length of a given order. Stream-area ratio Ra Ai is the area of the watershed that contributes to the channel segment of (11) The Channel area of order i. 2. Li  1 Ni L j 1 Ni i. The average length of channels of each higher order increase as a geometric sequence. The value of Rb for watersheds varies between 3 to 5. It can be quantified as: Ai  A1 Ra i 1 where Ai is the average area of order i and Ra is the stream area ratio. Also. and solution procedure adopted to solve the Nash based GIUH and GUH. j (10) The lengths of channels of a given order are determined largely by the type of soil covering the drainage basin. This law is an expression of topological phenomenon.. Ni and Ni 1 are the number of streams in order i and i+1 respectively. Stream-length ratio Rl This refers to length of channels of each order. (8) Li is the average length of channel of order i. and are given as follows: (12) i. order i and all lower order channels.  and  is the highest stream order of the watershed.. i  1.. which can be further explained as: the first order channels are the shortest of all the channels and the length increase geometrically as the order increases. Derivation of Geomorphologic Unit Hydrograph (GUH) This section describes the potential of GIUH for deriving the SUH. The Rl is (9) Rl  Li 1 / Li where. This relation is called Horton’s law of channel length and can be formulated as follows.5. Li  L1 Rl i 1 where mathematically expressed as follows. and is a measure of drainage efficiency... t / k ) is the incomplete gamma function of order n at (t / k ) . I (n. Rodriguez-Iturbe and Valdes (1979) derived the peak and time to peak characteristics of the IUH as function of Horton’s order ratios (Horton. t / k )  I (n. treated as a continuous Markovian process.05 The term (16) q p * t p is not dependent on the velocity and thereby on the storm characteristics and hence./ JOSH Fall 2009 (9)24-50 velocity parameter. Geomorphology based parameter estimation of Nash’s GIUH Journal of Spatial Hydrology 35 . V is the dynamic velocity parameter (m s ). in a framework of travel time distribution. the Nash based GIUH model was developed for the Gomti river basin.al.. (t  D) / k )] D (18) where U(D. n and k are the parameters of the Nash model.55  Rl 0. Multiplying eqs. (1979) compared the GIUHs for some real world basins with the IUHs derived from the discharge hydrograph produced by a physically based rainfallrunoff model of the same basins and found them to be remarkably similar.. (14) q p  1. and k is the storage coefficient (hour). The dynamic velocity parameter in the formulation of GIUH incorporates the effect of climatic variation. tp is the time to peak (h). Valdes et al. t is the sampling time interval (hour). -1 (15) L is the length of the highest order stream (km). t) denotes the ordinates of D hour UH (hour-1). This pioneering work of Rodriguez-Iturbe and Valdes (1979).38 where qp is the peak flow (h-1). The Nash model The Nash model (Nash.44 ( L / V ) ( Rb / Ra )0. and D is the duration of UH (hour). which explicitly integrate the geomorphologic details and the climatologic characteristics of the basin. is a boon for stream flow synthesis in the basin having no or scanty information of flow data. where the state is the order of the stream in which the drop is located at any time. The value at the mode of this pdf produces the main characteristics of GIUH. (1979) showed the dynamic velocity parameter of the GIUH can be taken as the velocity at the peak discharge time for a given rainfall-runoff event in the catchment. t )  1 [ I (n. In the present study. t is the sampling time interval (hour). This formulation of GIUH is based on the probability density function (pdf) of the time history of a randomly chosen drop of effective rainfall arrived to the trapping state of a hypothetical basin.43 V / L t p  0.5764[ Rb / Ra ]0.55 Rl 0. it is a function of only the geomorphologic characteristics of the basin. The Nash model can be expressed as follows: u (t )  1 (t / k ) n 1 exp(t / k ) k  ( n) (17) where u(t) is the ordinates of IUH (hour-1). (14) and (15) a non-dimensional term q p * t p is derived at. 2002. A unit hydrograph (UH) of desired duration (D) may be derived by using the following expression: U ( D. 1957) is based on the concept of routing of the instantaneous inflow through a cascade of linear reservoirs with equal storage coefficient. For few of Indian catchments this GIUH based approach was successfully demonstrated (Kumar et al. 1945).31 Rl 0. 2007). in which n is the number of linear reservoirs. q p * t p  0.Rai et. Rodriguez-Iturbe et al. 2007). and are expressed as follows (Kumar et al. . The Nash’s parameter k for the given velocity V is obtained using eqs.44 L k V R   b   Ra  0. Procedural step of estimation of n using Newton-Rapson method is given in appendix-I. (15) and (20) and the known value of the parameter n as follows: 0. It has a continental type of climate with hot summers and cold winters and receives most rainfall from the south-west (SW) monsoon during the months of June to October. 2007). (23) using the Newton-Rapson method. (20) and (21) results: qp  tp  (22) Equating eq. t p ./ JOSH Fall 2009 (9)24-50 The complete shape of the GIUH can be obtained by linking the q p and t p of the GIUH with scale (k) and shape (n) parameters of the Nash model. 5(b). 4(a) to 4(c). Case I: Application of the model on published data of Burhner catchment India is divided into seven hydro-meteorological zones and is further divided into twenty six hydrometeorological sub-zones [i. the D-hour UH can be derived from eq. can be obtained by solving eq. 2(a) to 2(c). differentiating with respect to t and by simplification eq.5764[ Rb / Ra ]0. (20) in eq. (18). two cases were considered: (i) Case I: Application of the model on published data of Burhner catchment taken from Bhunya et al. By equating the first derivative with respect to t of eq (17) to zero.55  Rl 0.al. (19) is derived. (19) t  t p  k (n  1) qp  (20) Simplifying the value of tp from eq.Rai et. (2008). The Burhner catchment was picked up from hydro-meteorological subzone 3(c) which comprises of Upper Narmada and Tapi basins. (22) with eq. (16) results in: (n  1)  exp[(n  1)]  (n  1)n 1  0. 5(a). sub-zones 1(a) to 1(g). (17) and simplifying yields 1 exp[(n  1)]  (n  1) n 1 k  ( n) (n  1) exp[(n  1)]  (n  1) n 1  ( n) (21) Combining eqs. 6 and 7] (Jain et al. (17). (19) to zero results in by replacing t with tp. t becomes the time to peak discharge.55  Rl 0. It lies between 80º36' and 81º23' E longitudes and 22º00' and 22º56' N latitude. The direct runoff hydrograph (DRH) is estimated by convoluting the excess rainfall hyetograph with the UH. Application of the Model and Results Gomti basin is ungauged basin. Subsequently. comprising Journal of Spatial Hydrology 36 . Thus. 3(a) to 3(i).05  ( n) (23) The Nash parameter n.   1 ( n  1)  ln[u (t )]      t t   k Equating eq. and (ii) Case II: Application of the model for Gomti river basin. taking the natural logarithm of both sides of eq. testing the accuracy of derived GUH for this basin using the aforesaid approach is quite difficult. The main soil group is black soil. for systematic application of the proposed approach. Mean annual rainfall varies approximately from 800 to 1600 mm. Thus.e. therefore.38  1 (n  1) (24) The derived values of n and k are used to determine the complete shape of the Nash based GIUH using eq. (17). Length of maximum order stream is 138 km. 80 70 60 Qp (m3/s) 50 40 30 20 10 0 0 10 20 30 Time (hour) 40 50 Observed UH Derived GUH Fig.5° to 46. wheat. The shape of the basin is nearly oblong in nature. non-agriculture. namely. The comparative result of UH for the catchment indicates the suitability of the technique for the derivation of UH and thus DRH.787 respectively. 1 Comparison of derived GUH and observed UH of Burhner catchment (Case I) Case II: Application of the model for Gomti river basin The river Gomti is one of the major tributary of river Ganga originated from the Pilibhit district of the Uttar Pradesh and joins the river Ganges at Varanasi after traveling approximately 662.62'' to 28°46'58. (2008). 75 percent of total rainfall is due to the occurrence of North-West Monsoon (Mid June to Mid October). large variation can be observed throughout the basin. The relative humidity varies between 10-90 percent.al. The climate of the basin is ranging from semi arid to sub humid tropical with average annual rainfall at different locations is 850-1100 mm. respectively.2 km2 area of the central-east part of the Uttar Pradesh. Peak discharge.15 m/s. 1.02 m3/s. It drains approximately 30407. Journal of Spatial Hydrology 37 . time to peak tp is of the order of 11 hour and velocity is 4. Approximately. whereas. 3.76'' to 83°11'13. medium black and shallow black soil. Topography is nearly undulating with high terrains at upstream end of the basin.0° C with daily mean sunshine of 8 hours. deep black. Rl are 3. the GUH was derived for the Burhner catchment and compared with the observed UH reported in the study by Bhunya et al.Rai et. etc.523. respectively. The digital elevation model (DEM) and the extracted drainage network of the basin using the ArcGIS are shown in Fig. Major crops cultivated in the basin are paddy. 2(a) and 2(b). The potential evapotranspiration experienced in the basin is nearly 1300 mm. Qp of hydrograph is found to be 69.96. whereas Ra. The land use in the basin can be broadly categorized as: agriculture. 1. Gomti river basin extended between the latitude of 25°23'12. Using the aforementioned geomorphologic information. Rb. forest.75'' N and longitude of 79°57' 33. It is sixth order catchment comprised of about 4103 km2 area. however. and other vegetable crops depending on the irrigation facilities. sugarcane. The mean minimum and maximum temperature over the basin is 4. potato.5 km. The comparison of the GUH and observed UH is depicted in Fig. Major soil group experienced over the basin is sandy loam to clay loam.25'' E and belongs to hydro-meteorological subzone 1(f)./ JOSH Fall 2009 (9)24-50 different types. The maximum at upstream end and minimum elevation at downstream end of the basin are 227 m and 58 m above msl. pasture fellow. 2 Digital Elevation Model (DEM) and extracted drainage network of the Gomti river basin To extract the geomorphologic features of the basin. The DEM of Gomti river basin (Fig. length.1 platform to delineate the basin boundaries and drainage networks (Fig. mean slope. 3(a). and the highest order of the basin was recorded as 6. and therefore. the aerial contribution to the runoff by different order is also important to understand the runoff and erosion process. the application of the SRTM data is cost-efficient and informative and is web-based world wide available and appropriately implemented into a geographic information system (GIS)-framework. relief ratio. etc. Horton’s bifurcation ratio (Fig. Besides these. flow accumulation etc. and could be applied in meso / macroscale river network and terrain analyses. stream frequency. farm factor. 2b). 2a) from SRTM data set was processed in the ArcGIS 9. The number of streams of different orders. The stream ordering was carried out using the DEM of the basin using Strahler’s scheme followed by establishing the flow direction.Rai et. elongation ratio. perimeter of the basin. 6). The extracted stream network of different orders is shown in Fig. it needs to be extracted and shown in Fig.al. maximum and minimum elevation./ JOSH Fall 2009 (9)24-50 Fig. and corresponding area are presented in Table 2.5 km. watershed relief. Journal of Spatial Hydrology 38 . Maximum length of the river is found to be 662. As stated earlier. stream-area ratio (Fig. the extracted geomorphologic parameters such as drainage area. 3(b). Based on the study by Haase and Frotscher (2005) it can be stated that the SRTM data sets are of major relevance for providing terrain information in large and transboundary river basins for handling the regional environmental problems. 5). length ratio (Fig. drainage density. whereas. SRTM (Shuttle Radar Topography Misson) data of 90 x 90 m resolution was used. length of the basin. 4). are summarized in Table 3. circulatory ratio. al.Rai et./ JOSH Fall 2009 (9)24-50 Fig.283) 3 (a) 5 6 7 Fig. 3 Streams of different orders and corresponding aerial coverage for Gomti river basin 8 7 6 5 Ln (N) 4 3 2 1 0 0 1 2 3 Order 4 y = -1.9971 (Rb = 4. 4 Estimation of bifurcation ratio (Rb) of the basin Journal of Spatial Hydrology 3 (b) 39 .5298 R2 = 0.4418x + 8. 218) 1 0 0 1 2 3 Order 4 5 6 7 Fig.7604 R2 = 0.90 2243.74 28525.7173x + 0.5342x + 1.772) Fig.95 797. and area for streams of various orders of Gomti basin Order 1 2 3 4 5 6 Number 1316 291 65 12 4 1 Length (km) 4422.21 1222.20 Journal of Spatial Hydrology 40 .al.8347 (Rl = 2.90 645.Rai et.16 23670.09 30407.82 Area (km2) 18577./ JOSH Fall 2009 (9)24-50 6 5 4 Ln (L) 3 2 y = 0.34 63.329 R2 = 0. length.77 30252.32 29549. 5 Estimation of stream length ratio (Rl) of the basin 12 10 8 Ln (A) 6 4 2 0 0 1 2 3 Order 4 5 6 7 y = 1.9943 (Ra = 4. 6 Estimation of stream area ratio (Ra) of the basin Table 2 Number. 71 18.36 7. and finally the 1-hour UHs were derived for different velocity of flow using the relationship given by eq. The optimized value of n was found to be 3. Rb. The 1 h-UH at different velocities are depicted in Fig.283 2. The parameter k of the Nash model was estimated for the derived Horton’s dimensionless parameters (i. 7.63 Value 30407.144 0. Rl.02 30407.00027 Numbers 1316 291 65 12 4 1 Mean Length (km) 3.e.218 4.34 438.772 0.121 8. and Ra) the parameter n of the Nash model was optimized using the Newton Rapson method (i.12 81. Rl.17. eqs.0 58. The dynamic velocity of flow can be computed using the approach suggested by Kumar et al.al.145 0./ JOSH Fall 2009 (9)24-50 Table 3 Extracted geomorphological parameters of the Gomti river basin Parameters Area (km2) Perimeter (km) Length of Basin (km) Maximum elevation (m) Minimum elevation (m) Relief (km) Mean slope (km/km) Stream Characteristics 1st order stream 2nd order stream 3rd order stream 4th order stream 5th order stream 6th order stream Ratios Bifurcation ratio (Rb) Length ratio (Rl) Stream-area ratio (Ra) Horton ratio (r) Drainage density (Dd) Drainage frequency (Fs) Elongation ratio (Re) Circulatory ratio (Rc) Farm factor (Ff) Shape factor (Fs) Compactness factor (Fc) 4. and Ra) at different dynamic velocity of flows using eq. Journal of Spatial Hydrology 41 . Using the estimated values of the n and k at different velocity of flows V.24 2.49 161. (24).82 Mean Area (km2) 14.34 63.20 Using the aforesaid approach of Nash based GIUH and estimated Horton’s ratios (i.48 7563.4 481.81 66.304 0.518 0. the ordinates of instantaneous unit hydrograph (IUH) was computed using eq.Rai et.0 0.0 227.055 0.e. Rb. (2002).169 0. (18). (17).85 2462. 25 through 31).e. The estimated values of n and k for various flow velocities are given in Table 4.2 1639. 50 5. the error associated in the computation of flood hydrograph may be more.0463 9.00 4.8046 To validate the derived GUH for the Gomti river basin. It can be revealed from Fig.5 m/s) GUH (V = 2.5 m/s) GUH (V = 0.0 m/s) GUH (V = 2. CWC technique is independent of the most important variable of the basin i.1665 3.hour SUH using the CWC procedure is shown in Fig.0232 6.6093 V (m/s) 3.1665 3. The CWC method is well accepted by field engineers / hydrologist for the derivation of UH on the ungauged catchment in India for the estimation of design flood to install the waterway of bridges/cross drainage structures across small and medium streams. Besides this. V V (m/s) 0.68 m/s) GUH (V = 0.1665 k (hour) 18.00 2. length of the longest stream and drainage area were developed for the regions based on the regional analysis of the observed data of 42 representative sub-watersheds (40 sub-watersheds data were collected by Northern and North-eastern Railways./ JOSH Fall 2009 (9)24-50 GUH (V = 0.0077 2.0154 4.50 2. contributing area of the different order drains and their lengths.e. time to peak flow rate and time base of the hydrograph) as a function of slope.75 m/s) GUH (V = 1. the GUH based approach becomes more realistic which provide the complete shape of the UH at different Journal of Spatial Hydrology 42 .1665 3. The computed 1. Since.0051 1. dynamic flow velocity) and does not account for the geomorphologic parameters such as characteristics of the drainage network. a comparison was presented with the UH computed from the procedure adopted by CWC.0 m/s) GUH (V = 1. In this method a set of equations for UH parameters (viz. 7 GUH at different velocities and observed UH (CWC procedure) for the Gomti river basin Table 4 Estimated values of Nash parameters n and k at different velocity.1665 3.50 4..00 3.5 m/s) UH: Computed by CWC 7000 6000 UH (m3/s) 5000 4000 3000 2000 1000 0 0 25 50 75 Time (hour) 100 125 Fig.00 n 3. 7 that the derived GUH with flow velocity of 0.1665 3. Hence. 7. CWC technique does not give the actual shape of the hydrograph.68 m/s (approx) is close to the SUH computed by CWC procedure.1665 k (hour) 3. India and 2 subwatersheds data were collected by Ministry of Transport.1665 3.50 n 3.50 1.2558 2.5116 3.al. climatic parameter (i. peak flow rate.1665 3.5780 2.00 1.e.Rai et. India).1665 3. Thus. Computational step to derive the SUH using CWC technique is given in Appendix-II. the described technique is cost-effective and quite accurate for determining the GUH and flood hydrograph for any catchment / basin (gauged or ungauged) as it requires DEM of the catchment that can be freely obtained from SRTM source. A detailed feature useful to engineering and environmental studies / application for the basin is summarized in tabular form (Tables 2 and 3). The DRH and thus the flood hydrographs for the river Gomti can be easily obtained by convoluting the rainfall excess (mm) with ordinates of GUH. Gomti river basin is of sixth order and almost flat terrain except at the upstream end of the basin. site identification for rain water harvesting. and tp is the time to peak (hours). dynamic flow velocity) and geomorphologic characteristics other than the slope. which was not yet developed for the Gomti river basin of India will. the resulting UH may have higher computational error./ JOSH Fall 2009 (9)24-50 values of dynamic velocity of flow. helps the engineers for accurate estimation of the flood hydrograph as well as for the modeling of pollutants transport. Three of Horton’s ratios of the Gomti river basin were coupled with the parameters of the Nash’s IUH model and the GUHs were derived for different values of dynamic velocity of flow (Fig.043 tp = 19.0289 x V0. On the other hand. Based on the velocities. Once the dynamic flow velocity is estimated from field visit. Qp = 0. The obtained GUH with dynamic flow velocity of 0. Case II presents the application of GIUH technique for the derivation of GUH for Gomti river basin. Also. drainage area and length of main stream.993) (26) -1.999) (25) -1.65 to 0.e eqs 24 through 27) were also developed for the estimation of peak ordinates of direct surface runoff hydrographs for Gomti basin provided the excess rainfall depth is available. Based on the comparison of the derived GUH and observed UH. it may be concluded that the proposed approach is suitable and efficient for the derivation of UH.al. groundwater recharge and waste disposal.68 m/s (approx. Using the results obtained from the analysis. the peak flow and time to peak can be directly obtained. To test the accuracy of the GIUH technique. Based on the geomorphological analysis. the extracted drainage network of the basin has wide range of application such as installation of the artificial drainage system and laying down the canal system. the GUH based approach will be superior to the CWC approach. Equations (24) through (27) can be used as a ready reference to the field engineers for design of hydraulic structures. therefore. Qp is the peak flow rate (m/s/mm).) shows closer agreement with SUH derived from CWC approach. Conclusions This study comprised of geomorphological investigation and development of GUH for Gomti river basin of India. V is the dynamic velocity of flows (m/s). Besides this. it may be remarked that the proposed technique.993) (27) where. etc.0452 2 tp = 0. dynamic flow velocity. the main theme of the paper is to develop a GUH for the Gomti river basin for the estimation of flood hydrograph.Rai et. However. Based on the analysis of the results. and time to peak flow have been sought for Gomti river basin and are expressed as follows.9982 (R2 = 0. Case-I was considered to derive the GUH for the Burhner catchment of India. Concluding. the actual UH of the basin can be easily obtained. Since. Journal of Spatial Hydrology 43 . The average dynamic flow velocity in the Gomti river varies between 0. 7). Hence.e.488 x Qp (R = 0. CWC approach is independent of climatic parameter (i. three equations (i. a relationship between the peak flow rate.75 m/s.834 x V (R2 = 0. 5764  b   Rl 0. The Journal of Spatial Hydrology 44 . 31a through 31d). the overall function can be defined as follows: R  (n  1) (28) exp[(n  1)]  (n  1)n 1  0. Considering the sub zone 1(f).hour SUH for an ungauged catchment./ JOSH Fall 2009 (9)24-50 Appendix-I: The Newton-Rapson method to estimate n From eq. (23). the relations established between physiographic and unit hydrograph (UH) parameters developed by Central Water Commission.55 (29) The Psi function for any real number was evaluated using the following set of equation (eqs. 1985) are applied for derivation of 2.  (1)    0.  ( z ) can be defined as follows: ( z )  t 0   z 1  exp ( t ) dt (32) The optimization algorithm for Newton-Rapson method is expressed as follows: n _ new  n _ old  f (n _ old) f (n _ old) (34) (33) The optimization terminated when the following condition is achieved.57722 (31a)  (1/ 2)    2 ln(2)  1.al. India (CWC.00001 Appendix-II: CWC procedure of UH computation for Gomti river basin Gomti basin falls under sub zone 1(f) of the hydro-meteorological homogeneity. (n _ new  n _ old)  0.05 f  Ra   ( n)  The first derivative of f with respect to n can be expressed as follows: 1 n 1 f exp( n  1)  (n  1) n 1  exp( n  1)  (n  1) n 1  (n)  ( n)  ( n) n 1 n 1  exp(  n  1)  ( n  1) n 1  exp(  n  1)  (n  1) n 1  {ln( n  1)  1} (n)  ( n) where  ( n) is a Psi function also known as digamma function and is expressed as follows: d ( z )  ( z )  [ln{( z )}]  (30) dz ( z ) 0.96351  (n)     k 1 k 1 n 1 (31b) (31c) ( n  2) n   n      2 ln(2)  2    2 2 k 1  k 1   1   1   (n  1) (31d) The Gamma function of a dummy variable z. The following steps are involved for derivation of 2 hour unit hydrograph.Rai et. Using the suggested procedure a 2 hour SUH has been developed for the Gomti river.Rai et. Fig. 8.649 The peak discharge Q p (m /s) for the 2 hour SUH is obtained as follows 3 (36) Qp  q p  A (37) Step 3: Determination of ‘time from center of effective rainfall duration to the peak value for 2-hour SUH’ Journal of Spatial Hydrology 45 . Di 1 .al. 8 Components of SUH for CWC procedure Step 1: Analysis of ‘catchment physiography’ Physiographic parameters viz. the catchment area A . q p  2.2 m /km Step 2: Determination of ‘peak value for 2-hour SUH’ The following equations is used for the peak discharge per unit of catchment area q p (m3/s/sq km) for the 2-hour SUH of Gomti river basin./ JOSH Fall 2009 (9)24-50 parameters used for developing SUH as proposed by CWC are given in the Fig. length of the longest stream L and equivalent stream slope S are determined from the catchment area plan for the estimation of L / S . where the equivalent stream slope S can be calculated as follows.  Li  Di 1  Di  (35) S L2 where Li is the length of the i th segment in km. Di are the depths of the river at the point of intersection of ( i  1 ) and i th contours respectively from the base line (datum) drawn at the level of the point of study in meters and L is the length of the longest stream. For Gomti river basin equivalent slope is found to be 5.030 /( L / S )0. 36  Qi  tr cm A (45) In case. Step 9: Conversion of 2 hr UH to 1 hr UH by S-Hydrograph method To derive 1-hour SUH from 2-hour SUH.791 (41) (42) (43) (44) (Note: the values in the parenthesis refer to Figure 8) Step 7: Computation of ‘unit depth’ Parameters viz. the depth of runoff ( d ) for the SUH is not equal to 1. The runoff depth ( d ) under the SUH is computed as follows.67  Tm This is accepted for the Gomti river basin. WR50 and WR75 (Figure 8) can be calculated using the following set of equations.907 WR75  0. TB =2.al.744  t p  0.0 cm.812 /( q p ) 0. Unit depth calculation for the derived SUH is done. Step 8: Cross checking of Time Base of SUH by ‘triangular unit hydrograph (TUH) method’ Conventionally the time base TB of a TUH is adopted as follows. TB  7. the SUH derived from CWC method Journal of Spatial Hydrology 46 .308 Step 4: Estimation of the ‘ time to peak ( Tm ) for 1-hour SUH’ The time to peak (38) Tm for 2-hour SUH (Figure 8) is computed using the following relationship. (39) Tm  t p  t r / 2 Step 5: Estimation of ‘ time base for 2-hour SUH’ Time base TB (hour) is computed as follows. W75 .447 /( q p ) 0. d 0.858 /( q p )1.779 (40) Step 6: Estimation of ‘time widths for different % values of peak discharge of 2-hour SUH’ The time width parameters (in hr) W50 . S-hydrograph technique was used./ JOSH Fall 2009 (9)24-50 The time to peak t p (hr) for 2-hour SUH is estimated using the following relationship. (46) TB value comes out to be same as CWC method and therefore. suitable modifications may be made by smoothening the graph so as to contain a unit volume ( i. W50 . TB . W75 . 1 cm  catchment area). t p  1. W50  2.876 WR50  0. WR75 are used and a 2-hour SUH is generated. Tm .Rai et. WR50 . t p . Q p .217 /(q p ) 0.e.990 W75  1.606 /( q p ) 0. ” Water Resour. (2000). R... Roorkee. Singh. D. “The forms of instantaneous unit hydrograph” Int. 64(2). D.. Pnigrahi.” Water Resour.” Water Resour. Manage.. R. 65.” Trans. R. K. (2001).C. Clark. 1468. Marani. P.S.” Advances in Geosciences. 12 (4). National Inst. R. C. (1991). Lohani.al. CWC.. L. (1945). Civ.’’ CS(AR) Technical Rep..C. “A general theory of the unit hydrograph. E. Journal of Spatial Hydrology 47 . and Sinha.. Geophys. (2006). K. 12. “Correlation of the velocity parameter of three geomorphological instantaneous unit hydrograph models. & Ministry of Surface Transport (1994). D. E. and Tripathi. B. 1829-1840.. water Resources Research .. 513-525. A.. 28. “The linear channel and its effect on the geomorphologic IUH.. Process.. 27. N. J. D.... 14. WH/22/1994... and Hubert . Manage. N. and Frotscher.. K.. and Singh. Kartiwa.. 21. A. S. Hydrol. P. Geol. 65-73. 175-208. 479-486. Rinaldo./ JOSH Fall 2009 (9)24-50 References Al-Wagdany. R.” J.. Water Resour. C. McGraw-Hill Book Company. and Mgmt. A. Res. Chatterjee. “Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative morphology. (2005). L. Soc. Lohani. M.” J.. 355-375. S. Flood Estimation Report for Western Himalayas .” Hydrol. U. Jin. (1998). (1983). Process. Bhunya.T. D. New York. 3879-3895... Haase.. (2008) Comparison between Weibull and gamma distributions to derive synthetic unit hydrograph using Horton ratios. “Geomorphological dispersion. 1.. S. Amer. 23.. 110.” Hydrol. and Nema. R. (2007). (1945).. “Development of geomorphological instantaneous unit hydrograph model for scantly gauged watersheds. A. Vol. R. R. “Design flood estimation using GIS supported GIUH approach. Kirshen.” Water Resour. Washington. B. Technical Bulletin No. Soc. P. K. 20. Kumar. 275–370. 1013-1025. Bhadra. Res. A. Singh.. O. B. Applied Hydrology. J.. Design Office Report No. 241-256. Lohani. N. D. (2008). K. Engr. and Mays.” Water Resour. K. and Hydrol. doi:10. R. Department of Agriculture. Manage. and Bras. Chatterjee. Linear theory of hydrologic Systems. (1997). 228-238.. Kumar. S. Pub. A. Plng. Res. 5.Zone 7.. and Rigon. 45 (3). K. Dooge.R. “Flood estimation for ungauged catchments using the GIUH. (1992). V. Mal. R. (1959). M. Kumar. A.. of Hydrology.. Bhaskar. P.. Singh. “Topography data harmonization and uncertainties applying SRTM. A. 16. 651-659. Agricultural Research Service. J. (1973).” J. Nash. 44. parida. Chow. B.C. Raghuwanshi. C.. New Delhi.. “Runoff estimation for an ungauged catchment using geomorphological instantaneous unit hydrograph (GIUH) model.. 369-376. D.. C. Am. 1945-46. 56. 114-121. Maidment. and Rao. P. R. “Derivation of unit hydrograph from GIUH analysis for a Himalayan river. D. “Sensitivity analysis of the GIUH based Clark model for a catchment. Horton.W.” Hydrol.. and Nayak. Jain S. V. and Roland. K. Singh.. Berndtsson. Sci.. and Kumar. A. (2002).” Bull. Ass. “Storage and the unit hydrograph. (2003). Fleurant. M.” Environmental Modelling & Software. ‘‘Comparison of geomorphological based rainfallrunoff models. R.-X. (1957). and Seth. Dooge. Singh. R. K. 263-278. W04421. “Analytical model for a geomorphological instantaneous unit hydrograph.Rai et. “A deterministic gamma-type geomorphologic instantaneous unit hydrograph based on path types.. Jain. C. laser scanner and cartographic elevation models. S.1029/2007WR006031 Central Water Commission Research Designs (CWC) (Directorate of Hydrology [Regional Studies]) & India Meteorological Dept. C. R. (1988). Process. R. 17.. Hydrologic Engrg.P. Singh.. A. Sherman. “The geomorphologic structure of hydrologic response. S. 38. (1956). Madramootoo. C. T..” Trans. B. Strahler. Prentice-Hall. “Flood estimation by GIUH-based Clark and Nash models. B.. Part 2. News Rec. and Lee. “Quantitative analysis of watershed geomorphology. and Gundekar. Union 33. V. N. (2007). New Jersey 07632.. O. (1984). Manage. I. and Prasher.. A.” Water Resour. Mathematical Models of Large Watershed Hydrology. Sarkar.... C. D. H. and Bras. Elementary Hydrology. S. Singh.” Bull. Geophys. B. 67. 1-9. 501505. Prentice-Hall. R. 15 (6). F. and Kumar. 2 (1). G. “Quantitative slope analysis. and Valdes. CO.. M. (1988a).K. (1979). Kandy. USA Singh... “Geomorphological parameters: are they indicators for installation of a hydropower site?” International Conference on Small Hydropower.K. (1989). New Jersey 07632. USA.. Enright. Res. F. p. Sarangi. V. Rosso. 914-920. “Stream flow from rainfall by the unit-graph method. Watershed modeling. 447. and Schwartz. 235–246. “Nash model relation to Horton order ratios. Hydrologic Systems: Vol II. Highlands Ranch. Journal of Spatial Hydrology 48 . “Unit-hydrograph lag and peak flow related to basin characteristics. and Frevert.” J. Uni. R. 18 (4). “The geomorpho-climatic theory of the instantaneous unit hydrograph.K. (Eds... Hydrologic Systems: Vol I. Amer. C. Sahoo. Singh. Geol. Prentice-Hall. 571–596. Am. (Eds. Union. A.. D. Highlands Ranch.” Water Resour. I. C. (1932). 19th Annual Meeting. New Jersey 07632.S. Soc. P. N.. K. A. USA. Srilanka.” Water Resour.” J. Yen. A. Res. (1982). P. (1957). P. 515-525. Water Resources Publications.” Trans.. (2007). Am. (2006).P. R... (1938). Gonzalas S. B. Amer. Res. Mathematical Models of Small Watershed Hydrology and Applications.” Eng. Strahler. Singh. (1997). 913-920. 11 (6). “Evaluation of three unit hydrograph models to predict the surface runoff from a Canadian watershed” Water Resour. (1952). Snyder. 877-886. H. N. 21.. V. R. 1127-1143. Hydrologic Engrg. (1988b). Singh. 1409-1420.. “Unit hydrograph derivation for ungauged watersheds by stream-order laws. V. J. 20. Geophys. 225. Geophys.. Taylor. and Frevert. 108. L.” Transact.. Raghuwanshi.) (2002a). P. Chatterjee./ JOSH Fall 2009 (9)24-50 Rodriguez-Iturbe. V. CO. E. Rainfall-Runoff Modeling. Water Resources Publications.) (2002b).al. Rodriguez-Iturbe.Rai et. “Synthetic Unitgraphs. / JOSH Fall 2009 (9)24-50 Notations & Abbreviations Rb = bifurcation ratio = number of i-th order streams = highest stream order of the basin = stream length ratio = average length of i-th order stream = length of i-th order stream = stream area ratio = average area of i-th order stream = total area that drains into the = drainage density = peak flow rate = length of highest order stream = incomplete gamma function of order n at (t / k ) = Gamma function = Psi function of n = derivative of gamma function = drainage area of the basin = constant for drainage basin similarity = Central Water Commission = duration of UH = digital elevation model = direct runoff hydrograph = exponential function = geographic information system = geomorphologic instantaneous unit hydrograph = geomorphologic unit hydrograph = fall in the basin = instantaneous unit hydrograph = storage coefficient (Nash parameter) = horizontal distance over which fall (i.e.Rai et. j j th stream of order i Dd qp L I (n.al. h) take place = length of the basin = number of linear reservoir (Nash parameter) = watershed perimeter = peak flow rate = Rl / Rb = average slope of the basin =Shuttle Radar Topographic Mission = synthetic unit hydrograph = sampling time interval = time to peak Ni  Rl Li Li Ra Ai Ai . t / k )   ( n) ( z ) A C CWC D DEM DRH exp GIS GIUH GUH h IUH k L Lb n Pr Qp r S SRTM SUH t tp Journal of Spatial Hydrology 49 . Rai et. t) u(t) UH V = triangular unit hydrograph = ordinates of D hour UH = ordinate of IUH = unit hydrograph = dynamic velocity of flow Journal of Spatial Hydrology 50 .al./ JOSH Fall 2009 (9)24-50 TUH U(D.