Consistency between hydrological models and field observations: linking processes at the hillslope scale to hydrological responses at the watershed scale

The purpose of this paper is to identify simple connections between observations of hydrological processes at the hillslope scale and observations of the response of watersheds following rainfall, with a view to building a parsimonious model of catchment processes. The focus is on the well‐studied Panola Mountain Research Watershed (PMRW), Georgia, USA. Recession analysis of discharge Q shows that while the relationship between dQ/dt and Q is approximately consistent with a linear reservoir for the hillslope, there is a deviation from linearity that becomes progressively larger with increasing spatial scale. To account for these scale differences conceptual models of streamflow recession are defined at both the hillslope scale and the watershed scale, and an assessment made as to whether models at the hillslope scale can be aggregated to be consistent with models at the watershed scale. Results from this study show that a model with parallel linear reservoirs provides the most plausible explanation (of those tested) for both the linear hillslope response to rainfall and non‐linear recession behaviour observed at the watershed outlet. In this model each linear reservoir is associated with a landscape type. The parallel reservoir model is consistent with both geochemical analyses of hydrological flow paths and water balance estimates of bedrock recharge. Overall, this study demonstrates that standard approaches of using recession analysis to identify the functional form of storage–discharge relationships identify model structures that are inconsistent with field evidence, and that recession analysis at multiple spatial scales can provide useful insights into catchment behaviour. Copyright © 2008 John Wiley & Sons, Ltd.     

How well can the subsurface storage–discharge relation be interpreted and predicted using the geometric factors in headwater areas?

Headwater storage–discharge (S–Q) remains one of the least understood processes, and there is renewed interest in the S–Q relation. How well can the S–Q relation be interpreted mechanistically using geometric factors? In this paper, the hillslope storage Boussinesq and hillslope storage kinematic wave equation were adopted to guide the theoretical derivations. Analytical solutions were derived based on the hsKW equation for nine idealized hillslope aquifers, which were subdivided into two groups, i.e. hillslope aquifers with exponential hillslope width function (C1) and hillslope aquifers with Gaussian hillslope width function (C2). We found that analytical expressions of the S–Q relation can be derived for C1 hillslope aquifers. For more compound hillslope aquifers, i.e. C2, no explicit S–Q relation can be obtained. The whole subsurface recession after a rainstorm is simulated by applying the initial saturation condition. We found that the simulated S–Q processes can be characterized by a two‐phase recession, i.e. quick and slow recession. The time (t_b) at the dividing point of the quick and slow recessions depends on the geometric factors, such as the plan and profile curvature. In the quick recession for C1, many of the S–Q curves can be described as linear or quasi‐linear functions, which indicate that linear reservoir models can be applied approximately for recession simulations. However, during the slow recession phase of C1 and during the whole recession of C2, the S–Q relations are highly non‐linear. Finally, we compared the hillslope storage kinematic wave and hillslope storage Boussinesq models for simulating subsurface water recession after a rainstorm event in a real‐world headwater catchment (G5) in China. Through comparison of the recession slope curves, we found that the simulated results of the models employing the Gaussian hillslope width function match the observed hydrograph. The results indicated that appropriate organization of the hillslope geometric factors enhances our ability to make S–Q predictions. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic Forecasting Models of the Monthly Streamflow for the Blue Nile at Eldiem Station

Egypt is almost totally dependent on the River Nile for satisfying about 95% of its water requirements. The River Nile has three main tributaries: White Nile, Blue Nile, and River Atbara. The Blue Nile contributes about 60% of total annual flow reached the River Nile at Aswan High Dam. The goal of this research is to develop a reliable stochastic model for the monthly streamflow of the Blue Nile at Eldiem station, where the Grand Ethiopian Renaissance Dam (GERD) is currently under construction with a storage capacity of about 74 billion m³. The developed model may help to carry out a reliable study on the filling scenarios of GERD reservoir and to minimize its expected negative side effects on Sudan and Egypt. The linear models: Deseasonalized AutoRegressive Moving Average (DARMA) model, Periodic AutoRegressive Moving Average (PARMA) model and Seasonal AutoRegressive Integrated Moving Average (SARIMA) model; and the nonlinear Artificial Neural Network (ANN) model are selected for modeling monthly streamflow at Eldiem station. The performance of various models during calibration and validation were evaluated using the statistical indices: Mean Absolute Error, Root Mean Square Error and coefficient of determination (R2) which indicate the strength of fitting between observed and forecasted values. The results show that the performance of the nonlinear model (ANN) was much better than all investigated linear models (DARMA, PARMA and SARIMA) in forecasting the monthly flow discharges at Eldiem station.     

Trends in rainfall and runoff in the Blue Nile Basin: 1964–2003

Most of the water from the Nile originates in Ethiopia but there is no agreement on how land degradation or climate change affects the future flow in downstream countries. The objective of this paper is to improve the understanding of future conditions by analysing historical trends. During the period 1964–2003, the average monthly basin‐wide precipitation and monthly discharge data were collected and analysed statistically for two stations in the upper 30% of the Blue Nile Basin and monthly and 10‐day discharge data of one station at the Sudan–Ethiopia border. A rainfall–runoff model examined the causes for observed trends. The results show that, while there was no significant trend in the seasonal and annual basin‐wide average rainfall, significant increases in discharge during the long rainy season (June to September) were observed at all three stations. In the upper Blue Nile, the short rainy season flow (March to May) increased, while the dry season flow (October to February) stayed the same. At the Sudan border, the dry season flow decreased significantly with no change in the short rainy season flow. The difference in response was likely due to the construction of weir in the 1990s at the Lake Tana outlet that affected the upper Blue Nile discharge significantly but affected less than 10% of the discharge at the Sudan border. The rainfall–runoff model reproduced the observed trends, assuming that an additional 10% of the hillsides were eroded in the 40‐year time span and generated overland flow instead of interflow and base flow. Models concerning future trends in the Nile cannot assume that the landscape runoff processes will remain static. Copyright © 2010 John Wiley & Sons, Ltd.     

Linearized seismic waveform inversion using a multiple re-weighted least-squares method with QR preconditioning

Linearized inversion methods such as Gauss‐Newton and multiple re‐weighted least‐squares are iterative processes in which an update in the current model is computed as a function of data misfit and the gradient of data with respect to model parameters. The main advantage of those methods is their ability to refine the model parameters although they have a high computational cost for seismic inversion. In the Gauss‐Newton method a system of equations, corresponding to the sensitivity matrix, is solved in the least‐squares sense at each iteration, while in the multiple re‐weighted least‐squares method many systems are solved using the same sensitivity matrix. The sensitivity matrix arising from these methods is usually not sparse, thus limiting the use of standard preconditioners in the solution of the linearized systems. For reduction of the computational cost of the linearized inversion methods, we propose the use of preconditioners based on a partial orthogonalization of the columns of the sensitivity matrix. The new approach collapses a band of co‐diagonals of the normal equations matrix into the main diagonal, being equivalent to computing the least‐squares solution starting from a partial solution of the linear system. The preconditioning is driven by a bandwidth L which can be interpreted as the distance for which the correlation between model parameters is relevant. To illustrate the benefit of the proposed approach to the reduction of the computational cost of the inversion we apply the multiple re‐weighted least‐squares method to the 2D acoustic seismic waveform inversion problem. We verify the reduction in the number of iterations in the conjugate'gradient algorithm as the bandwidth of the preconditioners increases. This effect reduces the total computational cost of inversion as well.     

The effects of controlled drainage on subsurface outflow from level agricultural fields

Daily outflow frequencies and recession curves were used to identify differences in storage–outflow relationships between two different drainage systems, conventional and controlled drainage. A three‐year (1996–1999) field drainage experiment was carried out on a loamy sand soil in southern Sweden. Plots with an area of 0·2 hectares were drained by conventional subsurface drainage (CD) or by controlled drainage (CWT1 and CWT2). The controlled drainage system allowed the groundwater level in the soil to be varied during the year. It was kept at least 70 cm below the soil surface during the growing season but allowed to rise to a maximum of 20 cm below the soil surface during the rest of the year. Measurements were performed to record precipitation, drain outflow and groundwater levels. Daily values of outflow were divided into 10 categories, based on the size of outflow. Recession curves of hourly measurement of outflow were selected. They behaved like single reservoirs and a linear storage–outflow model was applied. Least squares estimates of the parameters initial outflow, initial storage volume and retention constant were calculated. Controlled drainage had a significant effect on total drain outflow and outflow pattern during the three years of measurement. The total drain outflow was 70% to 90% smaller in CWT than in CD. The analysis revealed that the initial outflows were higher, the retention constant and the temporary storage lower in CWT. The hydrological impacts of the reduction in temporary storage were higher peak flow, shorter lag time and shorter recession time and these effects increased with an increased groundwater level. Copyright © 2003 John Wiley & Sons, Ltd.     

A new saturated/unsaturated model for stormwater infiltration systems

Infiltration systems are widely used as an effective urban stormwater control measure. Most design methods and models roughly approximate the complex physical flow processes in these systems using empirical equations and fixed infiltration rates to calculate emptying times from full. Sophisticated variably saturated flow models are available, but rarely applied owing to their complexity. This paper describes the development and testing of an integrated one‐dimensional model of flow through the porous storage of a typical infiltration system and surrounding soils. The model accounts for the depth in the storage, surrounding soil moisture conditions and the interaction between the storage and surrounding soil. It is a front‐tracking model that innovatively combines a soil‐moisture‐based solution of Richard's equation for unsaturated flow with piston flow through a saturated zone as well as a reservoir equation for flow through a porous storage. This allows the use of a simple non‐iterative numerical solution that can handle ponded infiltration into dry soils. The model is more rigorous than approximate stormwater infiltration system models and could therefore be valuable in everyday practice. A range of test cases commonly used to test soil water flow models for infiltration in unsaturated conditions, drainage from saturation and infiltration under ponded conditions were used to test the model along with an experiment with variable depth in a porous storage over saturated conditions. Results show that the model produces a good fit to the observed data, analytical solutions and Hydrus. Copyright © 2008 John Wiley & Sons, Ltd.     

Investigation of flow-rainfall co-variation for catchments selected based on the two main sources of River Nile

The co-variation of rainfall and flow was assessed in four selected catchments of the River Nile which has two main sources including the White Nile (in the Equatorial region) and the Blue Nile (from the Ethiopian highlands). The selected catchments included Kyoga and Kagera (from the Equatorial region), as well as Blue Nile and Atbara (in Sudan and Ethiopia). In each catchment, the flow-rainfall co-variation was investigated at both seasonal and annual time scales. To explain aggregated variation at larger temporal scale while investigating the possible change in catchment behavior, which may interfere with the flow-rainfall relationship, rainfall-runoff modeling was done at daily time scale using data (falling within the period 1949–2003) from Kagera and Blue Nile i.e. the major catchment of each region where the River Nile emanates. Correlation analysis was conducted to assess how well the variation of flow and that of catchment-wide rainfall resonate. The co-occurrence of the changes in observed and simulated overland flow was examined using the quantile perturbation method (QPM). Trends in the model residuals were detected using the Mann–Kendal (MK) and cumulative rank difference (CRD) tests. The null hypothesis H ₀ (no correlation between rainfall and flow) was rejected at the significance level α of 5% for all the selected catchments. The temporal changes in terms of the QPM anomalies for both the observed and simulated flow were in a close agreement. The evidence to reject the H ₀ (no trend in the model residuals) was generally statistically insufficient at α = 5% for all the models and selected catchments considering both the MK and CRD tests. These results indicate that change in catchment behavior due to anthropogenic influence in the Nile basin over the selected time period was minimal. Thus, the overall rainfall-runoff generation processes of the catchments did not change in a significant way over the selected data period. The temporal flow variation could be attributed mainly to the rainfall variation.     

Conceptual rainfall–runoff model with a two‐parameter,infinite characteristic time transfer function

A two‐parameter transfer function with an infinite characteristic time is proposed for conceptual rainfall–runoff models. The large time behaviour of the unit response is an inverse power function of time. The infinite characteristic time allows long‐term memory effects to be accounted for. Such effects are observed in mountainous and karst catchments. The governing equation of the model is a fractional differential equation in the limit of long times. Although linear, the proposed transfer function yields discharge signals that can usually be obtained only using non‐linear models. The model is applied successfully to two catchments, the Dud Koshi mountainous catchment in the Himalayas and the Durzon karst catchment in France. It compares favourably to the linear, non‐linear single reservoir models and to the GR4J model. With a single reservoir and a single transfer function, the model is capable of reproducing hysteretic behaviours identified as typical of long‐term memory effects. Computational efficiency is enhanced by approximating the infinite characteristic time transfer function with a sum of simpler, exponential transfer functions. This amounts to partitioning the reservoir into several linear sub‐reservoirs, the output discharges of which are easy to compute. An efficient partitioning strategy is presented to facilitate the practical implementation of the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of a data‐based mechanistic modelling (DBM) approach for predicting runoff generation in semi‐arid regions

This paper addresses the application of a data‐based mechanistic (DBM) modelling approach using transfer function models (TFMs) with non‐linear rainfall filtering to predict runoff generation from a semi‐arid catchment (795 km²) in Tanzania. With DBM modelling, time series of rainfall and streamflow were allowed to suggest an appropriate model structure compatible with the data available. The model structures were evaluated by looking at how well the model fitted the data, and how well the parameters of the model were estimated. The results indicated that a parallel model structure is appropriate with a proportion of the runoff being routed through a fast flow pathway and the remainder through a slow flow pathway. Finally, the study employed a Generalized Likelihood Uncertainty Estimation (GLUE) methodology to evaluate the parameter sensitivity and predictive uncertainty based on the feasible parameter ranges chosen from the initial analysis of recession curves and calibration of the TFM. Results showed that parameters that control the slow flow pathway are relatively more sensitive than those that control the fast flow pathway of the hydrograph. Within the GLUE framework, it was found that multiple acceptable parameter sets give a range of predictions. This was found to be an advantage, since it allows the possibility of assessing the uncertainty in predictions as conditioned on the calibration data and then using that uncertainty as part of the decision‐making process arising from any rainfall‐runoff modelling project. Copyright © 2001 John Wiley & Sons, Ltd.     

Implementing a nonlinear groundwater module in the soil and water assessment tool (SWAT)

The Soil and Water Assessment Tool (SWAT) is widely used in modeling water quantity and quality. In the original SWAT, groundwater flow is calculated using a linear‐reservoir model, with outflow proportional to storage. However, observations show that this assumption is not always applicable; for example, macropores in Karst formations would seriously affect the groundwater behavior. A nonlinear groundwater algorithm was introduced in a new version of the SWAT model, called ISWAT. The Shenandoah Valley area in the Eastern U.S., which includes a number of geologic formations including Karst, was selected to test the modified ISWAT model. Parameter ESTimation (PEST) was coupled with ISWAT to auto‐calibrate the nonlinear parameter values. Ten years of record at 15 stream gauges were used to calibrate the model. The nonlinear ISWAT, statistically and visually, performed better in stream discharge estimation especially during baseflow recession and low‐flow periods. This indicated that the nonlinear algorithm can better represent groundwater behavior. The coupled ISWAT‐PEST approach can be used in future stream discharge simulation. Copyright © 2013 John Wiley & Sons, Ltd.     

Recession curve estimation for storm event separations

An accurate recession curve model is important for separating individual flow events, which is especially difficult over catchments in regions with a maritime climate where frequent rainfall events cause the flows to rise before they reach the baseflow level. The traditional recession curve equations are based on static linear and nonlinear reservoir models. These models work quite well for ground water dominated recession curves, but not so well when the direct runoff is significant in the recession part. In this study, a new modelling methodology is explored based on self-adaptive parameters in the linear and nonlinear reservoir models. It has been found that the adaptive forms performed better than the static ones, especially when a window for the adaptive parameter estimation is properly selected. While the nonlinear adaptive model had better accuracy over the linear one, it could become unstable if its window is too narrow, indicating that more research work is needed to find an useful pattern for the window size. A comparison between the recession curve models and PDM model (a rainfall-runoff model) has shown that they agreed quite well in most winter events, but less so in the summer.     

Sediment balances in the Blue Nile River Basin

Rapid population growth in the upper Blue Nile basin has led to fast land-use changes from natural forest to agricultural land.This resulted in speeding up the soil erosion process in the highlands and increasing sedimentation further downstream in reservoirs and irrigation canals.At present,several dams are planned across the Blue Nile River in Ethiopia and the Grand Ethiopian Renaissance Dam is currently under construction near the border with Sudan.This will be the largest hydroelectric power plant in Africa.The objective of this paper is to quantify the river flows and sediment loads along the Blue Nile River network.The Soil and Water Assessment Tool was used to estimate the water flows from un-gauged sub-basins.To assess model performance,the estimated sediment loads were compared to the measured ones at selected locations.For the gauged sub-basins,water flows and sediment loads were derived from the available flow and sediment data.To fill in knowledge gaps,this study included a field survey in which new data on suspended solids and flow discharge were collected along the Blue Nile and on a number of tributaries.The comparison between the results of this study and previous estimates of the sediment load of the Blue Nile River at El Deim,near the Ethiopian Sudanese border,show that the sediment budgets have the right order of magnitude,although some uncertainties remain.This gives confidence in the results of this study providing the first sediment balance of the entire Blue Nile catchment at the sub-basin scale.     

Hysteresis‐based flood‐wave analysis using the concept of strain

Hysteresis represents a loop in a rating curve and is a phenomenon which closely resembles that occurring in stress–strain curves used for studying the elastic properties of solid substances in engineering mechanics. Earlier hysteresis‐based studies used for defining floodwave propagation in open channels have qualitatively shown that hysteresis is an index of energy loss during floodwave propagation. Using the concept of elasticity, this paper introduces a new term called flow strain (defined as the ratio of change in discharge to the initial discharge) for investigating hysteresis. The usefulness of this new term is evaluated with use of four dam‐break studies. The study reveals that:

• 1 flow strain is a function of three wave speeds, Seddon speed, Lagrange speed, and elastic speed;
• 2 a single linear reservoir concept frequently used in flood routing is a specific variant of the Seddon speed formula;
• 3 the non‐linear storage–discharge relationship, widely used in overland flow modelling, is a variant of the kinematic wave representation;
• 4 the discharge ordinates on the recession part of a hydrograph follow a simple first‐order autoregressive form;
• 5 the hysteresis, phase difference and logarithmic decrement all define attenuation and are indices of energy loss during floodwave propagation.

Copyright © 2001 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.     

Genetic programming approach for flood routing in natural channels

In recognition of the non‐linear relationship between storage and discharge existing in most river systems, non‐linear forms of the Muskingum model have been proposed, together with methods to calibrate the model parameters. However, most studies have focused only on routing a typical hypothetical flood hydrograph characterized by a single peak. In this study, we demonstrate that the storage–discharge relationship adopted for the non‐linear Muskingum model is not adequate for routing flood hydrographs in natural channels, which are often characterized by multiple peaks. As an alternative, an evolutionary algorithm‐based modelling approach, i.e. genetic programming (GP), is proposed, which is found to route complex flood hydrographs accurately. The proposed method is applied for constructing a routing model for a channel reach along the Walla Walla River, USA. The GP model performs extremely well with a root‐mean‐square error (RMSE) of 0·73 m³ s^?1 as against an RMSE of 3·26 m³ s^?1 for routing the multi‐peaked hydrograph. The advantage of GP lies in the fact that, unlike other models, it establishes the routing relationship in an easy and simple mathematical form. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of urbanization on streamflow in the Atlanta area (Georgia,USA): a comparative hydrological approach

For the period from 1958 to 1996, streamflow characteristics of a highly urbanized watershed were compared with less‐urbanized and non‐urbanized watersheds within a 20 000 km² region in the vicinity of Atlanta, Georgia: in the Piedmont and Blue Ridge physiographic provinces of the southeastern USA. Water levels in several wells completed in surficial and crystalline‐rock aquifers were also evaluated. Data were analysed for seven US Geological Survey (USGS) stream gauges, 17 National Weather Service rain gauges, and five USGS monitoring wells. Annual runoff coefficients (RCs; runoff as a fractional percentage of precipitation) for the urban stream (Peachtree Creek) were not significantly greater than for the less‐urbanized watersheds. The RCs for some streams were similar to others and the similar streams were grouped according to location. The RCs decreased from the higher elevation and higher relief watersheds to the lower elevation and lower relief watersheds: values were 0·54 for the two Blue Ridge streams, 0·37 for the four middle Piedmont streams (near Atlanta), and 0·28 for a southern Piedmont stream. For the 25 largest stormflows, the peak flows for Peachtree Creek were 30% to 100% greater than peak flows for the other streams. The storm recession period for the urban stream was 1–2 days less than that for the other streams and the recession was characterized by a 2‐day storm recession constant that was, on average, 40 to 100% greater, i.e. streamflow decreased more rapidly than for the other streams. Baseflow recession constants ranged from 35 to 40% lower for Peachtree Creek than for the other streams; this is attributed to lower evapotranspiration losses, which result in a smaller change in groundwater storage than in the less‐urbanized watersheds. Low flow of Peachtree Creek ranged from 25 to 35% less than the other streams, possibly the result of decreased infiltration caused by the more efficient routing of stormwater and the paving of groundwater recharge areas. The timing of daily or monthly groundwater‐level fluctuations was similar annually in each well, reflecting the seasonal recharge. Although water‐level monitoring only began in the 1980s for the two urban wells, water levels displayed a notable decline compared with non‐urban wells since then; this is attributed to decreased groundwater recharge in the urban watersheds due to increased imperviousness and related rapid storm runoff. Copyright © 2001 John Wiley & Sons, Ltd.     

Testing a conceptual hillslope recession model based on the storage–discharge relationship with the Richards equation

The conceptual recession model based on the storage–discharge relationship was proposed to account for the unsaturated–saturated water storage interaction. The recession model was formulated by combining the constitutive storage–discharge relationship with the integral balance equation for unsaturated and saturated water storage. The functional form of the constitutive storage–discharge relationship was determined from the spatial integration of the Richards equation. The performance of the recession model was tested by comparing with the solution of the Richards equation for different simulation geometric shapes and soil types. The conceptual recession model incorporating the unsaturated–saturated water storage interaction was in good agreement with the recession response of the Richards equation. However, the recession model that neglected the unsaturated–saturated water storage interaction was comparable to the Richards equation only for soils with the weak interaction between unsaturated water storage and saturated water storage. This result suggests the important role of the unsaturated–saturated water storage interaction in the formulation of the recession process when the derivative of the functional relationship between the unsaturated water storage and saturated water storage becomes significant. Copyright © 2007 John Wiley & Sons, Ltd.