Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Book Review

Abstract

The instantaneous unit hydrograph (IUH) of a watershed is the result of one instantaneous unit of rainfall excess distributed uniformly over the watershed. Although the geomorphological characteristics of the basin remain relatively constant, the variable characteristics of storms cause variations in the shape of the resulting hydrographs. It is, therefore, inadequate to use one typical IUH to represent the hydrological response generated from any specific storm. In this study, a variable IUH was derived that directly reflects the time-varying rainfall intensity during storms. The rainfall intensity used to generate the variable IUH at time t is the mean rainfall intensity occurring from the time t—T _c to t in which T _c is the watershed time of concentration. Hydrological records from three watersheds in Taiwan were used to demonstrate the applicability of the proposed model. The results show that better simulations can be obtained by using the proposed model than by using the conventional unit hydrograph method, especially for concentrated rainstorm cases.     

A self‐organization algorithm for real‐time flood forecast

The group method of data handling (GMDH) algorithm presented by A. C. Ivakhnenko and colleagues is an heuristic self‐organization method. It establishes the input–output relationship of a complex system using a multilayered perception‐type structure that is similar to a feed‐forward multilayer neural network. This study provides a step towards understanding and evaluating a role for GMDH in the investigation of the complex rainfall–runoff processes in a heterogeneous watershed in Taiwan. Two versions of the revised GMDH model are implemented: a stepwise regression procedure and a recursive formula. Eleven typhoon events in the Shen‐cei Creek watershed, Taiwan, are used to build the model and verify its usefulness. The prediction results of the revised GMDH models and the instantaneous unit hydrograph (IUH) model are compared. Based on the criteria of forecasting precision and the rate and time of peak error, a much better performance is obtained with the revised GMDH models. Copyright © 1999 John Wiley & Sons, Ltd.     

Runoff estimation for an ungauged catchment using geomorphological instantaneous unit hydrograph (GIUH) models

A geomorphological instantaneous unit hydrograph (GIUH) is derived from the geomorphological characteristics of a catchment and it is related to the parameters of the Clark instantaneous unit hydrograph (IUH) model as well as the Nash IUH model for deriving its complete shape. The developed GIUH based Clark and Nash models are applied for simulation of the direct surface run‐off (DSRO) hydrographs for ten rainfall‐runoff events of the Ajay catchment up to the Sarath gauging site of eastern India. The geomorphological characteristics of the Ajay catchment are evaluated using the GIS package, Integrated Land and Water Information System (ILWIS). The performances of the GIUH based Clark and Nash models in simulating the DSRO hydrographs are compared with the Clark IUH model option of HEC‐1 package and the Nash IUH model, using some commonly used objective functions. The DSRO hydrographs are computed with reasonable accuracy by the GIUH based Clark and Nash models, which simulate the DSRO hydrographs of the catchment considering it to be ungauged. Inter comparison of the performances of the GIUH based Clark and Nash models shows that the DSRO hydrographs are estimated with comparable accuracy by both the models. Copyright © 2007 John Wiley & Sons, Ltd.     

A practical estimation of Clark IUH parameters using root selection and linear programming

The values of the parameters of the Clark instantaneous unit hydrograph (IUH) are often relying on the subjective decision of the researcher, which leads to large variations of their values. Therefore, an objective method minimizing the subjective judgement in the IUH modelling procedure while providing a reduced range of acceptable values is proposed. The proposed method uses a basin average IUH to mitigate the robustness problem of the Clark IUH parameters. Using linear system theory, the z‐transform is applied to the average IUH and then the IUH polynomial is factored into the recession and time‐area curve (TAC) components based on a convolution relation between the Clark IUH parameters. During this calculation, the root selection method was adopted to verify the storage coefficient R from the recession component and a linear programming technique was applied for determining the TAC for the basin of interest. The Wi River basin was used to test the applicability of the proposed method. The results showed that the components of a single reservoir and the TAC for Clark IUH were separated effectively, and acceptable values for the parameters were obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of a regionalized Clark IUH model with limited hydrologic data availability

To aid prediction of the flow hydrograph in a basin with limited data, a practical approach to determining a regionalized Clark instantaneous unit hydrograph (IUH) model is presented. The proposed model is described in terms of the synthetic time–area concentration curve, the concentration time, and a special regional similarity value that is valid in the whole basin. The latter was estimated from a Monte Carlo testing procedure based on the normal probability distribution of transformed regional similarity values composed of the time of concentration and the storage coefficient in gauged basins. The time–area concentration curve and the concentration time were calculated from a rational equation as in conventional methods. The method of transformation adopted was the Box–Cox power transformation, which is known to make non‐normal values resemble normal data. By introducing the regional similarity value into a Clark IUH, a statistically best estimate of IUH for given data conditions and its quantified degree of uncertainty were realized. The Wi River basin in Korea was used to test the applicability of the regionalized Clark IUH. The performance of the suggested methodology was evaluated by assuming an ungauged sub‐basin at the site. The results showed that the IUH model developed in this work was an effective tool, predicting a reliable hydrograph within the study area even though only limited data were available. Copyright © 2008 John Wiley & Sons, Ltd.     

Multi‐site calibration using a grid‐based event rainfall–runoff model: a case study of the upstream areas of the Nakdong River basin in Korea

Most runoff analyses using a grid‐based distributed model use one parameter group calibrated at the outlet of a watershed, instead of dividing the watershed into subwatersheds. Significant differences between the observed value and the simulation result of the subwatersheds can occur if just one parameter group is used in all subwatersheds that have different hydrological characteristics from each other. Therefore, to improve the simulation results of the subwatersheds within a watershed, a model calibrated at every subwatershed needs to be used to reflect the characteristics of each subwatershed. In this study, different parameter groups were set up for one or two sites using a distributed model, the GRM (Grid based Rainfall‐runoff Model), and the evaluations were based on the results of rainfall–runoff analysis, which uses a multi‐site calibration (MSC) technique to calibrate the model at the outlet of each site. The Hyangseok watershed in Naeseong River, which is a tributary of Nakdong River in Korea, was chosen as the study area. The watershed was divided into five subwatersheds each with a subwatershed outlet that was applied to the calibration sites . The MSC was applied for five cases. When a site was added for calibration in a watershed, the runoff simulation showed better results than the calibration of only one site at the most downstream area of the watershed. The MSC approach could improve the simulation results on the calibrated sites and even on the non‐calibrated sites, and the effect of MSC was improved when the calibrated site was closer to the runoff site. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of geomorphologic and hydrological characteristics in watershed saturated areas using topographic‐index threshold and geomorphology‐based runoff model

The objective of this paper is to investigate the variation of geomorphology and runoff characteristics in saturated areas under different partial contributing area (PCA) conditions. Geomorphologic information and hydrologic records from two mid‐size watersheds in northern Taiwan were selected for analysis. The PCA ratio in the watershed during a storm was assumed equal to the ratio of the surface‐flow volume to the direct runoff volume from measured hydrologic data. The extents of PCA regions were then determined by using a topographic‐index threshold. Consequently, the geomorphologic factors in saturated and unsaturated areas could be calculated using a digital elevation model, and these factors could then be linked to a geomorphology‐based IUH model for runoff simulation, which can consider both the surface‐ and subsurface‐flow processes in saturated and unsaturated areas, respectively. The results show that geomorphologic characteristics in the saturated areas vary significantly with different PCA ratios especially for higher order streams. A large PCA ratio results in a sharp hydrograph because the quick surface flow dominates the runoff process, whereas the hydrologic response in a low PCA case is dominated by the delayed subsurface flow. Copyright © 2007 John Wiley & Sons, Ltd.     

APPLICATION OF UNIT HYDROGRAPH TECHNIQUES TO SOLUTE TRANSPORT IN CATCHMENTS

Current models of solute movement in catchments are based on rainfall–runoff models and are consequently biased towards processes which determine the magnitude and timing of water flux. It is shown here that the instantaneous unit hydrograph (IUH), or runoff response function, obtained from a hydrograph is fundamentally different from the residence time distribution which governs the response to solutes/tracers. Using hydrometric and tracer data obtained from a small (25 ha) catchment in the humid tropics a modification of the IUH technique is demonstrated which also allows approximate modelling of the tracer data. New features of the modified conceptual model are identified with known hillslope processes.     

Performance of a conceptual and physically based model in simulating the response of a semi‐urbanized watershed in San Antonio,Texas

The need for accurate hydrologic analysis and rainfall–runoff modelling tools has been rapidly increasing because of the growing complexity of operational hydrologic and hydraulic problems associated with population growth, rapid urbanization and expansion of agricultural activities. Given the recent advances in remote sensing of physiographic features and the availability of near real‐time precipitation products, rainfall–runoff models are expected to predict runoff more accurately. In this study, we compare the performance and implementation requirements of two rainfall–runoff models for a semi‐urbanized watershed. One is a semi‐distributed conceptual model, the Hydrologic Engineering Center‐Hydrologic Modelling System (HEC‐HMS). The other is a physically based, distributed‐parameter hydrologic model, the Gridded Surface Subsurface Hydrologic Analysis (GSSHA). Four flood events that took place on the Leon Creek watershed, a sub‐watershed of the San Antonio River basin in Texas, were used in this study. The two models were driven by the Multisensor Precipitation Estimator radar products. One event (in 2007) was used for HEC‐HMS and GSSHA calibrations. Two events (in 2004 and 2007) were used for further calibration of HEC‐HMS. Three events (in 2002, 2004 and 2010) were used for model validation. In general, the physically based, distributed‐parameter model performed better than the conceptual model and required less calibration. The two models were prepared with the same minimum required input data, and the effort required to build the two models did not differ substantially. Copyright © 2012 John Wiley & Sons, Ltd.     

Implementation of an automatic calibration procedure for HYDROTEL based on prior OAT sensitivity and complementary identifiability analysis

Efficiency of hydrological models mostly depends on the quality of the calibration performed prior to use. In this paper, an automatic calibration framework for the distributed hydrological model HYDROTEL is proposed. The calibration procedure was performed for three watersheds characterized with different hydroclimatological conditions: the Sassandra located in Ivory Coast, Africa, and the Montmorency and Beaurivage watersheds located in Quebec (Canada). Results of one‐a‐time (OAT) sensitivity analysis showed that the order of the most sensitive parameters differs for each watershed. Thus, the sensitivity depends on the hydroclimatic and physiographic characteristics of the watersheds. Co‐linearity indices showed that all model parameters were identifiable, that is, none of the studied parameters could be explained by a combination of the other parameters. Following these findings, an automatic calibration was run. Results indicated there was good agreement between simulated and measured streamflows at the outlet of each watershed; Nash–Sutcliffe efficiency (NSE) ranging between 0.77 and 0.92 and R² ranging from 0.87 to 0.97. When comparing NSE and R² values obtained using a process‐oriented, multiple‐objective, manual calibration strategy, a slight increase in model efficiency was reached with the automatic calibration procedure (4.15% for NSE and 2.95% for R²) improving predictions of peak flows for the Montmorency and Beaurivage watersheds (temperate climate conditions) and flows beyond the rainfall season in the Sassandra watershed. The proposed automatic calibration procedure introduced in this paper may be applied to other distributed hydrological model. Copyright © 2013 John Wiley & Sons, Ltd.     

Development and application of a physically based landscape water balance in the SWAT model

Watershed scale hydrological and biogeochemical models rely on the correct spatial‐temporal prediction of processes governing water and contaminant movement. The Soil and Water Assessment Tool (SWAT) model, one of the most commonly used watershed scale models, uses the popular curve number (CN) method to determine the respective amounts of infiltration and surface runoff. Although appropriate for flood forecasting in temperate climates, the CN method has been shown to be less than ideal in many situations (e.g. monsoonal climates and areas dominated by variable source area hydrology). The CN model is based on the assumption that there is a unique relationship between the average moisture content and the CN for all hydrologic response units (HRUs), and that the moisture content distribution is similar for each runoff event, which is not the case in many regions. Presented here is a physically based water balance that was coded in the SWAT model to replace the CN method of runoff generation. To compare this new water balance SWAT (SWAT‐WB) to the original CN‐based SWAT (SWAT‐CN), two watersheds were initialized; one in the headwaters of the Blue Nile in Ethiopia and one in the Catskill Mountains of New York. In the Ethiopian watershed, streamflow predictions were better using SWAT‐WB than SWAT‐CN [Nash–Sutcliffe efficiencies (NSE) of 0·79 and 0·67, respectively]. In the temperate Catskills, SWAT‐WB and SWAT‐CN predictions were approximately equivalent (NSE > 0·70). The spatial distribution of runoff‐generating areas differed greatly between the two models, with SWAT‐WB reflecting the topographical controls imposed on the model. Results show that a water balance provides results equal to or better than the CN, but with a more physically based approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Modelling the impact of agroforestry on hydrology of Mara River Basin in East Africa

Land‐use change is one of the main drivers of watershed hydrology change. The effect of forestry related land‐use changes (e.g. afforestation, deforestation, agroforestry) on water fluxes depends on climate, watershed characteristics and spatial scale. The Soil and Water Assessment Tool (SWAT) model was calibrated, validated and used to simulate the impact of agroforestry on the water balance in the Mara River Basin (MRB) in East Africa. Model performance was assessed by Nash–Sutcliffe Efficiency (NSE) and Kling–Gupta Efficiency (KGE). The NSE (and KGE) values for calibration and validation were: 0.77 (0.88) and 0.74 (0.85) for the Nyangores sub‐watershed, and 0.78 (0.89) and 0.79 (0.63) for the entire MRB. It was found that agroforestry in the watershed would generally reduce surface runoff, mainly because of enhanced infiltration. However, it would also increase evapotranspiration and consequently reduce baseflow and overall water yield, which was attributed to increased water use by trees. Spatial scale was found to have a significant effect on water balance; the impact of agroforestry was higher at the smaller headwater catchment (Nyangores) than for the larger watershed (entire MRB). However, the rate of change in water yield with an increase in area under agroforestry was different for the two and could be attributed to the spatial variability of climate within the MRB. Our results suggest that direct extrapolation of the findings from a small sub‐catchment to a larger watershed may not always be accurate. These findings could guide watershed managers on the level of trade‐offs that might occur between reduced water yields and other benefits (e.g. soil erosion control, improved soil productivity) offered by agroforestry. Copyright © 2016 John Wiley & Sons, Ltd.     

The geomorphological unit hydrograph from a historical‐critical perspective

In this paper we present a brief overview of geomorphological instantaneous unit hydrograph (GIUH) theories and analyze their successful path without hiding their limitations. The history of the GIUH is subdivided into three major sections. The first is based on the milestone works of Rodríguez‐Iturbe and Valdés (Water Resources Research 1979; 15 (6): 1409–1420) and Gupta et al. (Water Resources Research 1980; 16 (5): 855–862), which recognized that a treatment of water discharges with ‘travel times’ could provide a rich interpretation of the theory of the instantaneous unit hydrograph (IUH). We show how this was possible, what assumptions were made, which of these assumptions can be relaxed, and which have become obsolete and been discarded. The second section focuses on the width‐function‐based IUH (WFIUH) approach and its achievements in assessing the interplay of the topology and geometry of the network with water dynamics. The limitations of the WFIUH approach are described, and a way to work around them is suggested. Finally, a new formal approach to estimating the water budget by ‘travel times’, which derives from a suitable use of the water budget equation and some hypotheses, has been introduced and disentangled. Copyright © 2015 John Wiley & Sons, Ltd.     

Phosphorus export dynamics and hydrobiogeochemical controls across gradients of scale,topography and human impact

Concentration‐discharge (c‐Q) plots are routinely used as an integrated signal of watershed response to infer solute sources and travel pathways. However, the interpretation of c‐Q data can be difficult unless these data are fitted using statistical models. Such models are frequently applied for geogenic solutes, but it is unclear to what extent they might aid in the investigation of nutrient export patterns, particularly for total dissolved phosphorus (TDP) which is a critical driver of downstream eutrophication problems. The goal of the present study was therefore to statistically model c‐Q relations (where c is TDP concentrations) in a set of contrasting watersheds in the Northern Great Plains—ranging in size from 0.2 to 1000+ km²—to assess the controls of landscape properties on TDP transport dynamics. Six statistical models were fitted to c‐Q data, notably (a) one linear model, (b) one model assuming that c‐Q relations are driven by the mixing of end‐member waters from different landscape locations (i.e., hydrograph separation), (c) one model relying on a biogeochemical stationarity hypothesis (i.e., power law), (d) one model hypothesizing that c‐Q relations change as a function of the solute subsurface contact time (i.e., hyperbolic model), and (e) two models assuming that solute fluxes are mostly dependent on reaction rates (i.e., chemical models). Model performance ranged from mediocre (R² < 0.2) to very good (R² > 0.9), but the hydrograph separation model seemed most universal. No watershed was found to exhibit chemostatic behaviour, but many showed signs of dilution or enrichment behaviour. A tendency toward a multi‐model fit and better model performance was observed for watersheds with moderate slope and higher effective drainage area. The relatively poor model performance obtained outside these conditions illustrates the likely importance of controls on TDP concentrations in the region that are independent of flow dynamics.     

Effects of spatial aggregation of soil spatial information on watershed hydrological modelling

Many researchers have examined the impact of detailed soil spatial information on hydrological modelling due to the fact that such information serves as important input to hydrological modelling, yet is difficult and expensive to obtain. Most research has focused on the effects at single scales; however, the effects in the context of spatial aggregation across different scales are largely missing. This paper examines such effects by comparing the simulated runoffs across scales from watershed models based on two different levels of soil spatial information: the 10‐m‐resolution soil data derived from the Soil‐Land Inference Model (SoLIM) and the 1:24000 scale Soil Survey Geographic (SSURGO) database in the United States. The study was conducted at three different spatial scales: two at different watershed size levels (referred to as full watershed and sub‐basin, respectively) and one at the model minimum simulation unit level. A fully distributed hydrologic model (WetSpa) and a semi‐distributed model (SWAT) were used to assess the effects. The results show that at the minimum simulation unit level the differences in simulated runoff are large, but the differences gradually decrease as the spatial scale of the simulation units increases. For sub‐basins larger than 10 km² in the study area, stream flows simulated by spatially detailed SoLIM soil data do not significantly vary from those by SSURGO. The effects of spatial scale are shown to correlate with aggregation effect of the watershed routing process. The unique findings of this paper provide an important and unified perspective on the different views reported in the literature concerning how spatial detail of soil data affects watershed modelling. Different views result from different scales at which those studies were conducted. In addition, the findings offer a potentially useful basis for selecting details of soil spatial information appropriate for watershed modelling at a given scale. Copyright © 2011 John Wiley & Sons, Ltd.     

Integrated modeling as a decision‐aiding tool for groundwater management in a Mediterranean agricultural watershed

A decision‐aiding methodology for agricultural groundwater management is presented; it is based on the combination of a watershed model, a groundwater flow model, and an optimization model. This methodology was applied to an agricultural watershed in northeastern Greece. The watershed model used was the Soil and Water Assessment Tool (SWAT), which provided recharge rates for the aquifers. These recharge rates were imported in the well‐known MODFLOW groundwater flow model. Both models were calibrated and verified using field data. Then, the nonlinear optimization problem was solved by a piecewise linearization process, in which the Simplex algorithm was applied sequentially. Apart from several pumping and climate change sensitivity scenarios, a land use change scenario and a climate change scenario, combining the three models, were tested, showing the ability of this methodology to be used in the decision‐making process. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparative assessment of two distributed watershed models with application to a small watershed

Distributed watershed models are beneficial tools for the assessment of management practices on runoff and water‐induced erosion. This paper evaluates, by application to an experimental watershed, two promising distributed watershed‐scale sediment models in detail: the Kinematic Runoff and Erosion (KINEROS‐2) model and the Gridded Surface Subsurface Hydrologic Analysis (GSSHA) model. The physics behind each model are to some extent similar, though they have different watershed conceptualizations. KINEROS‐2 was calibrated using three rainfall events and validated over four separate rainfall events. Parameters estimated by this calibration process were adapted to GSSHA. With these parameters, GSSHA generated larger and retarded flow hydrographs. A 30% reduction in both plane and channel roughness in GSSHA along with the assumption of Green‐Ampt conductivity K_G‐A = K_s, where K_s is the saturated conductivity, resulted in almost identical hydrographs. Sediment parameters not common in both models were calibrated independently of KINEROS‐2. A comparative discussion of simulation results is presented. Even though GSSHA's flow component slightly overperformed KINEROS‐2, the latter outperformed GSSHA in simulations for sediment transport. In spite of the fact that KINEROS‐2 is not geared toward continuous‐time simulations, simulations performed with both models over a 1 month period generated comparable results. Copyright © 2006 John Wiley & Sons, Ltd.     

Use of a watershed hydrologic model to estimate interbasin groundwater flow in a Costa Rican rainforest

The watershed hydrologic model TOPMODEL was used to estimate interbasin groundwater flow (IGF) into a small lowland rainforest watershed in Costa Rica. IGF is a common hydrological process but often difficult to quantify. Four‐year simulations (2006–2009) using three different model approaches gave estimates of IGF that were very similar to each other (10.1, 10.2, and 9.8 m/year) and to an earlier estimate (10.0 m/year) based on 1998–2002 data from a budget study that did not use a hydrologic simulation model, providing confidence in the new estimates and suggesting each of the three model approaches is viable. Results show no significant temporal variation in IGF during 2006–2009 (or between this period and the earlier study from 1998–2002). Simulations of the 16 consecutive 3‐month periods in 2006–2009 gave 16 values of IGF rate with a mean (10.1 m/year, standard deviation = 0.6 m/year) very similar to the estimates above from the 4‐year simulations. This suggests the modified version of TOPMODEL can be used to model stream discharge and estimate IGF for sub‐annual time periods during which change in water storage is not necessarily equal to zero. Thus, simple watershed models may be used to estimate IGF based on even relatively short calibration periods, making such models useful tools in the study of this widespread hydrological process that affects water and chemical fluxes and budgets but is often difficult and costly to quantify. Copyright © 2013 John Wiley & Sons, Ltd.