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Dynamic Model of Daily Rainfall, runoff and sediment yield for a Himalayan Watershed
J. agric. Engng Res. (2000) 75, 189}193 doi:10.1006/jaer 1999.0498, available online at http://www.idealibrary.com on
Dynamic Model of Daily Rainfall, runo! and sediment yield for a Himalayan Watershed Akhilesh Kumar; Ghanshyam Das Department of Soil & Water Conservation Engineering, College of Technology, G.B. Pant University of Agriculture and Technology, Pantnagar, US Nagar 263 145, (UP) India; e-mail of corresponding author: [email protected] (Received 15 December 1998; accepted in revised form 15 October 1999)
A dynamic model of the daily rainfall, runo! and sediment yield process was developed and applied on the upper Himalayan catchment of the Ramganga river, comprising an area of 1010 km2. The estimated and predicted values of daily sediment yield from the developed model compared very well with the corresponding measured values of daily sediment yield. ( 2000 Silsoe Research Institute
1. Introduction Information on suspended sediment yield (washload) from a catchment is very often required for planning, designing and evaluation of soil conservation projects, design and operation of reservoirs, environmental and water pollution control measures, and drought and #ood control programs. Development of washload models is still in its stages of infancy. The available soil loss models can be categorized into two groups, one based on stormwise analysis and the other on a yearly basis. Stormwise models are either sediment graph models (RendonHerrero, 1974; Williams, 1972; Das & Agarwal, 1990) or total sediment yield models (Williams, 1978; Das & Chauhan, 1990), and the yearly models are for average soil erosion per annum (Wischmeier & Smith, 1965; Elwell, 1978). These models are empirical in nature and consider the watershed as a non-deterministic system for simpli"cation in calculations, which is not the true representation of the natural process. Natural systems and their relationships are dynamic, where the output represents the memory of the system. The rainfall, runo! and sediment yield process is also a dynamic relationship, based on the memory content of the watershed system. It has been observed in Canada that a "rst-order dynamic model on monthly basis and a second order dynamic model on daily basis adequately describe the sediment yield process with runo! as the input (Sharma et al., 1979). It has also been observed by them that a secondorder discrete dynamic model with seven parameters 0021-8634/00/020189#05 $35.00/0
adequately represents the daily sediment yield of the Thames river in Southern Ontario, Canada (Sharma & Dickinson, 1979). They also found that the watershed #uvial system exhibits weak memory on daily basis. A dynamic model, using identi"cation techniques, was developed to predict water table #uctuations (Tabrizi et al., 1990). They were of the view that the same technique can also be adopted to model sediment yield process. The rainfall-runo! models on a daily basis were developed by assessing the dynamic response characteristics of river #ood (Jackman et al., 1993). Dynamic models were developed, by considering the e!ects of the watershed memory, to estimate daily sediment yield (Kumar, 1993) and daily runo! volume (Kumar & Das, 1998) for a Himalayan watershed in India. In the development of sediment yield models, generally, the runo! has been considered as an important input variable. However, it has been observed that the sediment yield at any time depends on the magnitude of the sediment generation and subsequent transportation of the eroded soil to the outlet of the watershed. The sediment generation process is governed by rainfall, runo! and catchment characteristics while the transportation process is mainly in#uenced by runo! parameters and watershed characteristics including the con"guration of the watershed #uvial system. It is, therefore, felt that the inclusion of the rainfall in the sediment yield modelling process may help in a more realistic assessment.
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Fig. 1. Naula watershed: altitude in m; d, sediment and runow measurement post; m, raingauge station
middle of June to the end of September, with a mean annual rainfall of 1015 mm. Hydrologic data of the watershed is collected by the Divisional Forest O$ce (Soil Conservation) Ranikhet. Automatic rain gauges are installed in the watershed at two stations including the Naula station, and nonrecording rain gauges at eight other stations, spread up throughout the watershed. To measure the runo!, stage recorders are installed at four stations including the Naula station, and sediment observation posts are also installed at these stations. Automatic #oat-type recorders are installed to record stage hydrographs of the #ood water. The average concentration of sediment on daily basis, i.e. the amount of sediment present in one unit volume of runo!, is determined by analysing the small representative samples collected by using a bottle sampling method. The daily sediment yield on any day from the watershed is then determined by multiplying the average sediment concentration with the total volume of runo! on that day. The daily runo! volume is determined from the daily hydrographs on the basis of the stagedischarge rating curves prepared for the site. 3. Model hypothesis
In the present study, an attempt has been made to develop dynamic sediment yield model, considering both rainfall and runo! as the input variables, to estimate the sediment yield from a catchment on a per day basis. The model was applied on a Himalayan sub-catchment of the Ramganga river called in this study as Naula watershed (Fig. 1), to test its applicability and capacity to estimate and generate daily sediment yield for the catchment.
2. The study area The watershed (Fig. 1), considered for this study, called &Naula watershed', is the catchment upstream to the Naula gauging station of the Ramganga river. It consists of two gauged sub-catchments and a small area drained by small rivers and streams. The Naula watershed selected for this study comprises an area of 1010 km2, where the maximum and minimum elevations range from 3112 to 792 m above mean sea level with undulating irregular slopes. The watershed is located between the latitudes of 29344@N and 3036@20@@N, and between the longitudes of 7936@15@@E and 79331@15@@E in the Ranikhet forest subdivision of Ramganga river catchment (Uttar Pradesh, India). The watershed is located in a Himalayan sub-tropical area and has a climate with a mean annual temperature of 303C and a mean minimum temperature of 183C. The precipitation in the watershed mainly occurs in the form of rainfall from the
Sediment out#ow from a catchment is a function of rainfall, runo! and watershed characteristics. Invariably, for the conceptualization of system models for sediment yield processes, the watershed is considered to be a lumped system or a black box, where no consideration is to be given to the physical processes operating within the system. The e!ects of rainfall and runo! on sediment yield is a dynamic process, in the watershed system. The sediment yield at any time is dependent not only on the present values of the rainfall and runo!, but also on the previous values of rainfall, runo! and sediment yield, i.e. the data which are in the memory of the system. Such situations are particularly true in cases of large catchments, where the eroded soil takes a considerable time to travel to the outlet. Previous rainfalls govern the antecedent moisture conditions of the watershed and its ability to produce runo!, and the runo! which is still in the watershed's #uvial system may either gain in suspended sediment due to land slides or channel scour, or lose it due to deposition on the channel bed, depending on the nature of #ow parameters. The sediment yield from a catchment at any time, therefore, is a combined e!ect of the present rainfall and runo! values and the previous values of the rainfall, runo! and sediment yield. Thus, logically it appears that a dynamic model is likely to be a better representation of the rainfall-runo! process. Mathematically, such a concept can be expressed as >"f [R, Q, (K¸SCP)]
(1)
DY N AM I C M O D EL O F D A I LY R AI N FA LL R U N O FF A N D S ED I M EN T
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dynamic model is described as i~6 i~6 i~5 > "A # + A R #+ B Q #+ C > t 0 i t`1~i i t`1~i i t~i i/1 i/1 i/1 (4) Fig. 2. Schematic representation of rainfall, runow and sediment yield process in a watershed: K, soil erodibility factor; L, slope length factor; S, slope gradient factor; C, crop cover factor; P, supporting conservation practice factor
and the system analogy of it is expressed as in Fig 2, where: R, Q and > are daily rainfall, runo! and sediment yield, respectively; K is the soil erodibility factor; ¸ is the slope length factor; S is the slope gradient factor; C is the crop cover factor; and P is the supporting conservation practice factor. The combined e!ect of the terms K, ¸, S, C and P of the Universal Soil Loss Equation of Wischmeier and Smith (1965) represents the watershed system characteristics. A functional form of the above process can also be described as >"f (R , R , 2 , R , Q , t t~1 t~n t Q , 2 , Q , > , 2 ,> ) (2) t~1 t~n t~1 t~n where subscripts t, t!1, etc. denote the time at respective lags in days. The above equation falls in the multipleinput, single-output (MISO) category since it has a single output in response to a number of input variables.
4. Model development At "rst, a preliminary study of the model [Eqn (2)] was conducted on the study watershed, where it was observed that the sediment yield on any particular day is the combined e!ect of rainfall and runo! on that day, and rainfall, runo! and sediment yield of the "ve previous days. Accordingly, for this watershed, a time lag of "ve days was considered for the development of functional relationships between input and output variables of the model. This gave the number of input variables in the model as 17, having the following form: > "f (R , R , 2 , R , Q , t t t~1 t~5 t Q , 2, Q , > , 2, > ) t~1 t~5 t~1 t~5
(3)
4.1. ¸inear dynamic model To develop the model, at "rst the model structure was considered to be linear. The basic structure of a linear
The values of model parameters (A , A , B and C ) of 0 i i i the equation are to be determined through a multiple regression analysis. To obtain these values for the watershed model, at "rst monthwise models were tried, but the models gave a very poor correlation. In the next attempt, a generalized model with 17 variables for the daily sediment yield data of the entire monsoon season was tried, and the model parameters were computed by considering the daily data series of a three year period (1980}1982). It was found that the generalized model gave a much better correlation as compared to monthwise models. To reduce the number of variables of the model, and to make the model structure more compact and better representative of the rainfall, runo! and sediment yield process, the input variables found signi"cant at the 5% level of signi"cance were retained, and the remaining variables not signi"cant at this level were dropped, by applying the technique of stepwise regression. At "rst, the least signi"cant variable was removed and the process was repeated until all the remaining variables were found to be signi"cant at the 5% level. A linear dynamic model applicable at the 5% level of signi"cance was then obtained, which was of the following form: > "0)00221Q !0)000613Q !0)00103Q (5) t t t~2 t~3 The model in Eqn (5), however, yielded a poor value of coe$cient of multiple determination R2 equal to 53)3%. At the same time, in this model, the rainfall terms were not found to be signi"cant at the 5% level of signi"cance.
4.2. Non-linear dynamic models To improve upon the performance of the linear dynamic model, non-linear forms were then tried. A dynamic non-linear sediment yield model is described as i~6 i~6 ln > "ln A # + A ln R # + B ln Q t 0 i t`1~i i t`1~i i/1 i/1 i~5 # + C ln > (6) i t~i i/1 The values of model coe$cients of the equation were similarly determined as in case of the linear model. After eliminating the non-signi"cant terms, the following
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non-linear model consisting of variables signi"cant at the 5% level of signi"cance was obtained: ln > "!1)17#0)423 ln R #0)28 ln R t t t~2 #0)273 ln Q #0)47 ln > #e . (7) t~2 t~1 where e is the error component. The coe$cient of multiple determination R2 for the model in Eqn (7) was obtained as 82)3%, which is fairly acceptable in the case of sediment yield models, as such models represent a very complex hydrologic process.
5. Error component The composition of the developed model can be divided into two components, namely, the deterministic and the stochastic. The deterministic component consists of terms obtained through a multiple regression analysis, and the stochastic component represents the error values e which may have accrued due to faults during sampling, t measurement and recording of the data. A dynamic model is an approximation of the true physical process; therefore, its "nal error component is equivalent to the &white noise' in the system, which is supposed to corrupt the system and contaminate the output values. Attempts were made to model the error values e of the developed t non-linear model, but all these models gave a very poor (less than 16%) correlation. It showed that the values for e or residuals of the model had no correlation with the t present or past values of the measured rainfall and runo! data series, and these are independent, randomly distributed and probably cannot be modelled. The magnitude of independence and randomness of values for e was tested by computing the auto-correlat tion functions (ACF) and the auto-correlograms. The
Fig. 3. Autocorrelation of residual error series e for sediment t
series is considered to be a randomly distributed and independent white-noise series when the ACF are not signi"cantly di!erent from zero, or are within the limits of twice the standard error (SE) of the series (Kottegoda, 1980). The residuals, i.e. the di!erence between measured and predicted values of daily runo!, and their ACFs at di!erent lags were calculated. The standard error (SE) of the residual series was found to be equal to 0)0975 at the 5% level of signi"cance. From the autocorrelogram (Fig. 3), it was also observed that at the 5% level of signi"cance, most of the ACF values were within twice the standard error (SE) limits. This meant that the series for e is independent, randomly distributed and cannot be t modelled. Therefore, the error component was dropped from the main model, and the "nal model which represents the rainfall, runo! and sediment yield process of the study watershed was expressed as ln > "!1)17#0)423 ln R #0)28 ln R t t t~2 #0)273 ln Q #0)447 ln > . t~2 t~1
(8)
6. Model testing and veri5cation After the development of model, it was tested on the daily data series of the years 1980}1982, to check its performance. The estimated values of daily sediment yield from the model were compared with the corresponding values of measured daily sediment yield for the monsoon months of these years. It was observed that, at the 5% level of signi"cance, the estimated values from the model and the measured values of daily sediment yield were in fair agreement. The applicability of the model was checked by applying it on the data series of the years 1979 and 1983, respectively, which were not considered for the development of the model. These data were collected by the Ranikhet Forest Subdivision (Soil Conservation). It was then found that, for all the four monsoon months (June}September), the values of daily sediment yield predicted by using the model were in good agreement with the corresponding values of measured sediment yield. The model was also used to synthetically generate a daily sediment yield data series for future conditions by using the past measured daily rainfall and runo! data and the model predicted values of daily sediment yield as the input information, through a step regression technique. Graphical comparison of measured, predicted and synthetically generated values of daily sediment yield for the month of August and September is shown in Figs 4 and 5, respectively, for the year 1983. It was observed that the model predicted and generated values of daily sediment yield compared fairly well with the corresponding values of daily sediment yield. The coe$cient of multiple determination of the
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estimated and synthetically generated values of daily sediment yield were found to be in fair agreement with the corresponding measured values. It was also observed that the generation of daily sediment yield values on a monthwise basis provides a better prediction than the generation of daily sediment yield values, considering the entire monsoon season at a time. The error component present in the model has been found to be independent and randomly distributed &white noise'.
References
Fig. 4. Measured, estimated and generated daily sediment yield for August, 83: , measured; , estimated; , generated
model was found to be 82)3%, which indicated that the model is able to explain more than 80% of the variability. Thus, the model was considered to be a good representation of the rainfall, runo! and sediment yield process of the Naula watershed.
7. Conclusion An input}output time-invariant dynamic model has been developed to represent the daily rainfall, runo! and sediment yield process in the Naula watershed of Ramganga river catchment in the Himalayan region of Uttar Pradesh (India). The coe$cient of multiple determination for the model was found to be 82)3%. The model
Fig. 5. Measured, estimated and generated daily sediment yield for September, 83: , measured; , estimated; *---*, generated
Das G; Agarwal A (1990). Development of conceptual sediment graph model. Transactions of American Society of Agricultural Engineers, 33(1), 102}104 Das G; Chauhan H S (1990). Sediment routing model for mountainous Himalayan regions. Transactions of American Society of Agricultural Engineers, 33(1), 95}99 Elwell H A (1978). Modelling soil losses in Southern Africa. Journal of Agricultural Engineering Research, 23(2), 117}127 Jackman A J; Littlewood I G; Whitehead P G (1993). An assessment of the dynamic response characteristics of stream #ow in the Balquhidder catchments. Journal of Hydrology, 145(3/4), 337}355 Kottegoda N T (1980). Stochastic Water Resources Technology. The Macmillan Press Ltd, London Kumar A (1993). Dynamic models of daily runo! and sediment yield for a sub-catchment of Ramganga river. PhD Thesis, Department of Soil and Water Conservation Engg, College of Technology, Pantnagar, India (unpublished) Kumar A; Das G (1998). A dynamic runo! model for a Himalayan watershed. International Journal of Agricultural Engineering, Asian Association of Agricultural Engineers, 79(3 and 4), 201}210 Rendon-Herraro O (1974). Estimation of washload produced on certain small watersheds. Journal of Hydraulic Division Proceedings, American Society of Civil Engineers, 98(5), 835}848 Sharma T C; Dickinson W T (1979). Unit step and frequency response functions applied to the watershed #uvial systems. Journal of Hydrology, 40(1/2), 323}335 Sharma T C; Hines W G S; Dickinson W T (1979). Input}output model for runo!-sediment yield processes. Journal of Hydrology, 40(1/2), 299}322 Tabrizi M H N; Jameluddin H; Billings S A; Skaggs R W (1990) Use of identi"cation techniques to develop a water table prediction model. Journal of American Society of Agricultural Engineers, 33(6), 487}494 Williams J R (1972). Sediment yield prediction with Universal equation using runo! energy factor. Present and Prospective Technology for Predicting Sediment Yield and Sources Proceedings, In: Sediment Yield Workshop, 20}30 November, Oxford, USDA Sedimentation Laboratory Williams J R (1978). A sediment graph model based on instantaneous unit sediment graph. Journal of Water Resources Research, 14(4), 659}664 Wischmeier W H; Smith D D (1965). Predicting rainfall erosion from crop land east of rocking mountain. US Department of Agriculture, Agriculture Hand book, No 282, 48 pp
Dynamic Model of Daily Rainfall, runo! and sediment yield for a Himalayan Watershed Akhilesh Kumar; Ghanshyam Das Department of Soil & Water Conservation Engineering, College of Technology, G.B. Pant University of Agriculture and Technology, Pantnagar, US Nagar 263 145, (UP) India; e-mail of corresponding author: [email protected] (Received 15 December 1998; accepted in revised form 15 October 1999)
A dynamic model of the daily rainfall, runo! and sediment yield process was developed and applied on the upper Himalayan catchment of the Ramganga river, comprising an area of 1010 km2. The estimated and predicted values of daily sediment yield from the developed model compared very well with the corresponding measured values of daily sediment yield. ( 2000 Silsoe Research Institute
1. Introduction Information on suspended sediment yield (washload) from a catchment is very often required for planning, designing and evaluation of soil conservation projects, design and operation of reservoirs, environmental and water pollution control measures, and drought and #ood control programs. Development of washload models is still in its stages of infancy. The available soil loss models can be categorized into two groups, one based on stormwise analysis and the other on a yearly basis. Stormwise models are either sediment graph models (RendonHerrero, 1974; Williams, 1972; Das & Agarwal, 1990) or total sediment yield models (Williams, 1978; Das & Chauhan, 1990), and the yearly models are for average soil erosion per annum (Wischmeier & Smith, 1965; Elwell, 1978). These models are empirical in nature and consider the watershed as a non-deterministic system for simpli"cation in calculations, which is not the true representation of the natural process. Natural systems and their relationships are dynamic, where the output represents the memory of the system. The rainfall, runo! and sediment yield process is also a dynamic relationship, based on the memory content of the watershed system. It has been observed in Canada that a "rst-order dynamic model on monthly basis and a second order dynamic model on daily basis adequately describe the sediment yield process with runo! as the input (Sharma et al., 1979). It has also been observed by them that a secondorder discrete dynamic model with seven parameters 0021-8634/00/020189#05 $35.00/0
adequately represents the daily sediment yield of the Thames river in Southern Ontario, Canada (Sharma & Dickinson, 1979). They also found that the watershed #uvial system exhibits weak memory on daily basis. A dynamic model, using identi"cation techniques, was developed to predict water table #uctuations (Tabrizi et al., 1990). They were of the view that the same technique can also be adopted to model sediment yield process. The rainfall-runo! models on a daily basis were developed by assessing the dynamic response characteristics of river #ood (Jackman et al., 1993). Dynamic models were developed, by considering the e!ects of the watershed memory, to estimate daily sediment yield (Kumar, 1993) and daily runo! volume (Kumar & Das, 1998) for a Himalayan watershed in India. In the development of sediment yield models, generally, the runo! has been considered as an important input variable. However, it has been observed that the sediment yield at any time depends on the magnitude of the sediment generation and subsequent transportation of the eroded soil to the outlet of the watershed. The sediment generation process is governed by rainfall, runo! and catchment characteristics while the transportation process is mainly in#uenced by runo! parameters and watershed characteristics including the con"guration of the watershed #uvial system. It is, therefore, felt that the inclusion of the rainfall in the sediment yield modelling process may help in a more realistic assessment.
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Fig. 1. Naula watershed: altitude in m; d, sediment and runow measurement post; m, raingauge station
middle of June to the end of September, with a mean annual rainfall of 1015 mm. Hydrologic data of the watershed is collected by the Divisional Forest O$ce (Soil Conservation) Ranikhet. Automatic rain gauges are installed in the watershed at two stations including the Naula station, and nonrecording rain gauges at eight other stations, spread up throughout the watershed. To measure the runo!, stage recorders are installed at four stations including the Naula station, and sediment observation posts are also installed at these stations. Automatic #oat-type recorders are installed to record stage hydrographs of the #ood water. The average concentration of sediment on daily basis, i.e. the amount of sediment present in one unit volume of runo!, is determined by analysing the small representative samples collected by using a bottle sampling method. The daily sediment yield on any day from the watershed is then determined by multiplying the average sediment concentration with the total volume of runo! on that day. The daily runo! volume is determined from the daily hydrographs on the basis of the stagedischarge rating curves prepared for the site. 3. Model hypothesis
In the present study, an attempt has been made to develop dynamic sediment yield model, considering both rainfall and runo! as the input variables, to estimate the sediment yield from a catchment on a per day basis. The model was applied on a Himalayan sub-catchment of the Ramganga river called in this study as Naula watershed (Fig. 1), to test its applicability and capacity to estimate and generate daily sediment yield for the catchment.
2. The study area The watershed (Fig. 1), considered for this study, called &Naula watershed', is the catchment upstream to the Naula gauging station of the Ramganga river. It consists of two gauged sub-catchments and a small area drained by small rivers and streams. The Naula watershed selected for this study comprises an area of 1010 km2, where the maximum and minimum elevations range from 3112 to 792 m above mean sea level with undulating irregular slopes. The watershed is located between the latitudes of 29344@N and 3036@20@@N, and between the longitudes of 7936@15@@E and 79331@15@@E in the Ranikhet forest subdivision of Ramganga river catchment (Uttar Pradesh, India). The watershed is located in a Himalayan sub-tropical area and has a climate with a mean annual temperature of 303C and a mean minimum temperature of 183C. The precipitation in the watershed mainly occurs in the form of rainfall from the
Sediment out#ow from a catchment is a function of rainfall, runo! and watershed characteristics. Invariably, for the conceptualization of system models for sediment yield processes, the watershed is considered to be a lumped system or a black box, where no consideration is to be given to the physical processes operating within the system. The e!ects of rainfall and runo! on sediment yield is a dynamic process, in the watershed system. The sediment yield at any time is dependent not only on the present values of the rainfall and runo!, but also on the previous values of rainfall, runo! and sediment yield, i.e. the data which are in the memory of the system. Such situations are particularly true in cases of large catchments, where the eroded soil takes a considerable time to travel to the outlet. Previous rainfalls govern the antecedent moisture conditions of the watershed and its ability to produce runo!, and the runo! which is still in the watershed's #uvial system may either gain in suspended sediment due to land slides or channel scour, or lose it due to deposition on the channel bed, depending on the nature of #ow parameters. The sediment yield from a catchment at any time, therefore, is a combined e!ect of the present rainfall and runo! values and the previous values of the rainfall, runo! and sediment yield. Thus, logically it appears that a dynamic model is likely to be a better representation of the rainfall-runo! process. Mathematically, such a concept can be expressed as >"f [R, Q, (K¸SCP)]
(1)
DY N AM I C M O D EL O F D A I LY R AI N FA LL R U N O FF A N D S ED I M EN T
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dynamic model is described as i~6 i~6 i~5 > "A # + A R #+ B Q #+ C > t 0 i t`1~i i t`1~i i t~i i/1 i/1 i/1 (4) Fig. 2. Schematic representation of rainfall, runow and sediment yield process in a watershed: K, soil erodibility factor; L, slope length factor; S, slope gradient factor; C, crop cover factor; P, supporting conservation practice factor
and the system analogy of it is expressed as in Fig 2, where: R, Q and > are daily rainfall, runo! and sediment yield, respectively; K is the soil erodibility factor; ¸ is the slope length factor; S is the slope gradient factor; C is the crop cover factor; and P is the supporting conservation practice factor. The combined e!ect of the terms K, ¸, S, C and P of the Universal Soil Loss Equation of Wischmeier and Smith (1965) represents the watershed system characteristics. A functional form of the above process can also be described as >"f (R , R , 2 , R , Q , t t~1 t~n t Q , 2 , Q , > , 2 ,> ) (2) t~1 t~n t~1 t~n where subscripts t, t!1, etc. denote the time at respective lags in days. The above equation falls in the multipleinput, single-output (MISO) category since it has a single output in response to a number of input variables.
4. Model development At "rst, a preliminary study of the model [Eqn (2)] was conducted on the study watershed, where it was observed that the sediment yield on any particular day is the combined e!ect of rainfall and runo! on that day, and rainfall, runo! and sediment yield of the "ve previous days. Accordingly, for this watershed, a time lag of "ve days was considered for the development of functional relationships between input and output variables of the model. This gave the number of input variables in the model as 17, having the following form: > "f (R , R , 2 , R , Q , t t t~1 t~5 t Q , 2, Q , > , 2, > ) t~1 t~5 t~1 t~5
(3)
4.1. ¸inear dynamic model To develop the model, at "rst the model structure was considered to be linear. The basic structure of a linear
The values of model parameters (A , A , B and C ) of 0 i i i the equation are to be determined through a multiple regression analysis. To obtain these values for the watershed model, at "rst monthwise models were tried, but the models gave a very poor correlation. In the next attempt, a generalized model with 17 variables for the daily sediment yield data of the entire monsoon season was tried, and the model parameters were computed by considering the daily data series of a three year period (1980}1982). It was found that the generalized model gave a much better correlation as compared to monthwise models. To reduce the number of variables of the model, and to make the model structure more compact and better representative of the rainfall, runo! and sediment yield process, the input variables found signi"cant at the 5% level of signi"cance were retained, and the remaining variables not signi"cant at this level were dropped, by applying the technique of stepwise regression. At "rst, the least signi"cant variable was removed and the process was repeated until all the remaining variables were found to be signi"cant at the 5% level. A linear dynamic model applicable at the 5% level of signi"cance was then obtained, which was of the following form: > "0)00221Q !0)000613Q !0)00103Q (5) t t t~2 t~3 The model in Eqn (5), however, yielded a poor value of coe$cient of multiple determination R2 equal to 53)3%. At the same time, in this model, the rainfall terms were not found to be signi"cant at the 5% level of signi"cance.
4.2. Non-linear dynamic models To improve upon the performance of the linear dynamic model, non-linear forms were then tried. A dynamic non-linear sediment yield model is described as i~6 i~6 ln > "ln A # + A ln R # + B ln Q t 0 i t`1~i i t`1~i i/1 i/1 i~5 # + C ln > (6) i t~i i/1 The values of model coe$cients of the equation were similarly determined as in case of the linear model. After eliminating the non-signi"cant terms, the following
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non-linear model consisting of variables signi"cant at the 5% level of signi"cance was obtained: ln > "!1)17#0)423 ln R #0)28 ln R t t t~2 #0)273 ln Q #0)47 ln > #e . (7) t~2 t~1 where e is the error component. The coe$cient of multiple determination R2 for the model in Eqn (7) was obtained as 82)3%, which is fairly acceptable in the case of sediment yield models, as such models represent a very complex hydrologic process.
5. Error component The composition of the developed model can be divided into two components, namely, the deterministic and the stochastic. The deterministic component consists of terms obtained through a multiple regression analysis, and the stochastic component represents the error values e which may have accrued due to faults during sampling, t measurement and recording of the data. A dynamic model is an approximation of the true physical process; therefore, its "nal error component is equivalent to the &white noise' in the system, which is supposed to corrupt the system and contaminate the output values. Attempts were made to model the error values e of the developed t non-linear model, but all these models gave a very poor (less than 16%) correlation. It showed that the values for e or residuals of the model had no correlation with the t present or past values of the measured rainfall and runo! data series, and these are independent, randomly distributed and probably cannot be modelled. The magnitude of independence and randomness of values for e was tested by computing the auto-correlat tion functions (ACF) and the auto-correlograms. The
Fig. 3. Autocorrelation of residual error series e for sediment t
series is considered to be a randomly distributed and independent white-noise series when the ACF are not signi"cantly di!erent from zero, or are within the limits of twice the standard error (SE) of the series (Kottegoda, 1980). The residuals, i.e. the di!erence between measured and predicted values of daily runo!, and their ACFs at di!erent lags were calculated. The standard error (SE) of the residual series was found to be equal to 0)0975 at the 5% level of signi"cance. From the autocorrelogram (Fig. 3), it was also observed that at the 5% level of signi"cance, most of the ACF values were within twice the standard error (SE) limits. This meant that the series for e is independent, randomly distributed and cannot be t modelled. Therefore, the error component was dropped from the main model, and the "nal model which represents the rainfall, runo! and sediment yield process of the study watershed was expressed as ln > "!1)17#0)423 ln R #0)28 ln R t t t~2 #0)273 ln Q #0)447 ln > . t~2 t~1
(8)
6. Model testing and veri5cation After the development of model, it was tested on the daily data series of the years 1980}1982, to check its performance. The estimated values of daily sediment yield from the model were compared with the corresponding values of measured daily sediment yield for the monsoon months of these years. It was observed that, at the 5% level of signi"cance, the estimated values from the model and the measured values of daily sediment yield were in fair agreement. The applicability of the model was checked by applying it on the data series of the years 1979 and 1983, respectively, which were not considered for the development of the model. These data were collected by the Ranikhet Forest Subdivision (Soil Conservation). It was then found that, for all the four monsoon months (June}September), the values of daily sediment yield predicted by using the model were in good agreement with the corresponding values of measured sediment yield. The model was also used to synthetically generate a daily sediment yield data series for future conditions by using the past measured daily rainfall and runo! data and the model predicted values of daily sediment yield as the input information, through a step regression technique. Graphical comparison of measured, predicted and synthetically generated values of daily sediment yield for the month of August and September is shown in Figs 4 and 5, respectively, for the year 1983. It was observed that the model predicted and generated values of daily sediment yield compared fairly well with the corresponding values of daily sediment yield. The coe$cient of multiple determination of the
DY N AM I C M O D EL O F D A I LY R AI N FA LL R U N O FF A N D S ED I M EN T
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estimated and synthetically generated values of daily sediment yield were found to be in fair agreement with the corresponding measured values. It was also observed that the generation of daily sediment yield values on a monthwise basis provides a better prediction than the generation of daily sediment yield values, considering the entire monsoon season at a time. The error component present in the model has been found to be independent and randomly distributed &white noise'.
References
Fig. 4. Measured, estimated and generated daily sediment yield for August, 83: , measured; , estimated; , generated
model was found to be 82)3%, which indicated that the model is able to explain more than 80% of the variability. Thus, the model was considered to be a good representation of the rainfall, runo! and sediment yield process of the Naula watershed.
7. Conclusion An input}output time-invariant dynamic model has been developed to represent the daily rainfall, runo! and sediment yield process in the Naula watershed of Ramganga river catchment in the Himalayan region of Uttar Pradesh (India). The coe$cient of multiple determination for the model was found to be 82)3%. The model
Fig. 5. Measured, estimated and generated daily sediment yield for September, 83: , measured; , estimated; *---*, generated
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