Uncertainty estimates for surface runoff models

A lower bound for variance in surface runoff modelling estimates is advanced. The bound is derived using a linear unit hydrograph approach which utilizes a discretization of the catchment into an arbitrary number of subareas, a linear routing technique for channel flow effects, a variable effective rainfall distribution over the catchment, and calibration parameter distributions developed in correlating rainfall-runoff data by the model. The uncertainty bound reflects the dominating influence of the unknown rainfall distribution over the catchment and is expressed as a distribution function that can be reduced only by supplying additional rainfall-runoff data. It is recommended that this uncertainty distribution in modelling results be included in flood control design studies in order to incorporate a prescribed level of confidence in flood protection facilities.     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

GEOMORPHOLOGICAL TRANSFER FUNCTION CALCULATED FROM DIGITAL ELEVATION MODELS FOR DISTRIBUTED HYDROLOGICAL MODELLING

Recently, several attempts have been made to relate the hydrological response of a catchment to its morphological and topographical features using different hypotheses to model the effect of the drainage network. Several transfer functions were developed and some of these are based on the theory of a linear model, the geomorphological unit hydrograph. The aim of this paper is to present a methodology to automatically identify the transfer function, using digital elevation models for applications in distributed hydrological modelling. The transfer function proposed herein is based on the Hayami approximation solution of the diffusive wave equation especially adapted for the routing hydrograph through a channel network. The Gardon d’Anduze basin, southern France, was retained for applications. Digital elevation models were used to extract the channel network and divide the basin into subcatchments. Each subcatchment produces, at its own outlet, an impulse response which is routed to the outlet of the whole catchment using the diffusive wave model described by two parameters: celerity and diffusivity functions of geometrical characteristics of the channel network. Firstly, a geomorphological unit hydrograph obtained by routing a homogeneous effective rainfall was compared with the unit hydrograph identified by a lumped model scheme, then the distributed model was applied to take into account the spatial variability of effective rainfall in the catchment. Results show that this new method seems to be adapted for distributed hydrological modelling; it enables identification of a transfer function response for each hydrological unit, here subcatchments, and then simulation of the contribution of each unit to the hydrograph at the outlet. © 1997 by John Wiley & Sons, Ltd.     

Design hydrograph estimation in small and ungauged watersheds: continuous simulation method versus event‐based approach

The proper assessment of design hydrographs and their main properties (peak, volume and duration) in small and ungauged basins is a key point of many hydrological applications. In general, two types of methods can be used to evaluate the design hydrograph: one approach is based on the statistics of storm events, while the other relies on continuously simulating rainfall‐runoff time series. In the first class of methods, the design hydrograph is obtained by applying a rainfall‐runoff model to a design hyetograph that synthesises the storm event. In the second approach, the design hydrograph is quantified by analysing long synthetic runoff time series that are obtained by transforming synthetic rainfall sequences through a rainfall‐runoff model. These simulation‐based procedures overcome some of the unrealistic hypotheses which characterize the event‐based approaches. In this paper, a simulation experiment is carried out to examine the differences between the two types of methods in terms of the design hydrograph's peak, volume and duration. The results conclude that the continuous simulation methods are preferable because the event‐based approaches tend to underestimate the hydrograph's volume and duration. Copyright © 2011 John Wiley & Sons, Ltd.     

A spatially distributed Clark’s unit hydrograph based hybrid hydrologic model (Distributed-Clark)

ABSTRACT

A hybrid hydrologic model (Distributed-Clark), which is a lumped conceptual and distributed feature model, was developed based on the combined concept of Clark’s unit hydrograph and its spatial decomposition methods, incorporating refined spatially variable flow dynamics to implement hydrological simulation for spatially distributed rainfall–runoff flow. In Distributed-Clark, the Soil Conservation Service (SCS) curve number method is utilized to estimate spatially distributed runoff depth and a set of separated unit hydrographs is used for runoff routing to obtain a direct runoff flow hydrograph. Case studies (four watersheds in the central part of the USA) using spatially distributed (Thiessen polygon-based) rainfall data of storm events were used to evaluate the model performance. Results demonstrate relatively good fit to observed streamflow, with a Nash-Sutcliffe efficiency (E_NS) of 0.84 and coefficient of determination (R²) of 0.86, as well as a better fit in comparison with outputs of spatially averaged rainfall data simulations for two models including HEC-HMS.     

Storm‐event rainfall–runoff modelling approach for ungauged sites in Taiwan

This work develops a top‐down modelling approach for storm‐event rainfall–runoff model calibration at unmeasured sites in Taiwan. Twenty‐six storm events occurring in seven sub‐catchments in the Kao‐Ping River provided the analytical data set. Regional formulas for three important features of a streamflow hydrograph, i.e. time to peak, peak flow, and total runoff volume, were developed via the characteristics of storm event and catchment using multivariate regression analysis. Validation of the regional formulas demonstrates that they reasonably predict the three features of a streamflow hydrograph at ungauged sites. All of the sub‐catchments in the study area were then adopted as ungauged areas, and the three streamflow hydrograph features were calculated by the regional formulas and substituted into the fuzzy multi‐objective function for rainfall–runoff model calibration. Calibration results show that the proposed approach can effectively simulate the streamflow hydrographs at the ungauged sites. The simulated hydrographs more closely resemble observed hydrographs than hydrographs synthesized using the Soil Conservation Service (SCS) dimensionless unit hydrograph method, a conventional method for hydrograph estimation at ungauged sites in Taiwan. Copyright © 2008 John Wiley & Sons, Ltd.     

An investigation of lag times for rainfall–runoff–sediment yield events in small river basins / Une analyse des temps de réponse d'événements pluie–débit–transport solide au sein de petits bassins versants

Abstract

River basin lag time (LAG), defined as the elapsed time between the occurrence of the centroids of the effective rainfall intensity pattern and the storm runoff hydrograph, is an important factor in determining the time to peak and the peak value of the instantaneous unit hydrograph, IUH. In the procedure of predicting a sedimentgraph (suspended sediment load as a function of time), the equivalent parameter is the lag time for the sedimentgraph (LAG_s ), which is defined as the elapsed time between the occurrence of the centroids of sediment production during a storm event and the observed sedimentgraph at the gauging station. Results of analyses of rainfall, runoff and suspended sediment concentration event data collected from five small Carpathian basins in Poland and from a 2.31-ha agricultural basin, in central Illinois, USA have shown that LAG_s was, in the majority of cases, smaller than LAG, and that a significant linear relationship exists between LAG_s and LAG.     

On-line estimation of unit hydrographs using the wavelet-based LMS algorithm / Estimation en ligne des hydrogrammes unitaires grâce à l'algorithme des moindres carrés moyens à base d'ondelettes

Abstract

This study applies the discrete wavelet transform (DWT) to decompose the unit hydrograph, thereby generating parsimonious reparameterizations of the unit hydrograph. A model compression method is then employed to significantly compress the unit hydrograph requiring that fewer coefficients be estimated. Moreover, a wavelet-based linearly constrained least mean squares (WLCLMS) algorithm is also used to estimate on-line the wavelet coefficients of the unit hydrograph. The updated wavelet coefficients of the unit hydrograph, convoluted with effective rainfall input in the wavelet domain, allow for accurate prediction of one-step-ahead runoff in the time domain. The proposed approach allows the unit hydrographs to vary in time and accurately predicts runoff from a basin in Taiwan, thus making it highly promising for flood forecasting.     

Two‐dimensional continuous simulation of spatiotemporally varied hydrological processes using the time‐area method

Distributed, continuous hydrologic models promote better understanding of hydrology and enable integrated hydrologic analyses by providing a more detailed picture of water transport processes across the varying landscape. However, such models are not widely used in routine modelling practices, due in part to the extensive data input requirements, computational demands, and complexity of routing algorithms. We developed a two‐dimensional continuous hydrologic model, HYSTAR, using a time‐area method within a grid‐based spatial data model with the goal of providing an alternative way to simulate spatiotemporally varied watershed‐scale hydrologic processes. The model calculates the direct runoff hydrograph by coupling a time‐area routing scheme with a dynamic rainfall excess sub‐model implemented here using a modified curve number method with an hourly time step, explicitly considering downstream ‘reinfiltration’ of routed surface runoff. Soil moisture content is determined at each time interval based on a water balance equation, and overland and channel runoff is routed on time‐area maps, representing spatial variation in hydraulic characteristics for each time interval in a storm event. Simulating runoff hydrographs does not depend on unit hydrograph theory or on solution of the Saint Venant equation, yet retains the simplicity of a unit hydrograph approach and the capability of explicitly simulating two‐dimensional flow routing. The model provided acceptable performance in predicting daily and monthly runoff for a 6‐year period for a watershed in Virginia (USA) using readily available geographic information about the watershed landscape. Spatial and temporal variability in simulated effective runoff depth and time area maps dynamically show the areas of the watershed contributing to the direct runoff hydrograph at the outlet over time, consistent with the variable source area overland flow generation mechanism. The model offers a way to simulate watershed processes and runoff hydrographs using the time‐area method, providing a simple, efficient, and sound framework that explicitly represents mechanisms of spatially and temporally varied hydrologic processes. Copyright © 2015 John Wiley & Sons, Ltd.     

Does the incorporation of process conceptualization and tracer data improve the structure and performance of a simple rainfall‐runoff model in a Scottish mesoscale catchment?

A geomorphological instantaneous unit hydrograph (GIUH) rainfall‐runoff model was applied in a 31 km² montane catchment in Scotland. Modelling was based on flow path length distributions derived from a digital terrain model (DTM). The model was applied in two ways; a single landscape unit response based on the DTM alone, and a two‐landscape unit response, which incorporated the distribution of saturated areas derived from field‐validated geographic information system (GIS) analysis based on a DTM and soil maps. This was to test the hypothesis that incorporation of process‐information would enhance the model performance. The model was applied with limited multiple event calibration to produce parameter sets which could be applied to a spectrum of events with contrasting characteristics and antecedent conditions. Gran alkalinity was used as a tracer to provide an additional objective measure for assessing model performance. The models captured the hydrological response dynamics of the catchment reasonably well. In general, the single landscape unit approach produced the best individual model performance statistics, though the two‐landscape unit approach provided a range of models, which bracketed the storm hydrograph response more realistically. There was a tendency to over‐predict the rising limb of the hydrograph, underestimate large storm event peaks and anticipate the hydrograph recession too rapidly. Most of these limitations could be explained by the simplistic assumptions embedded within the GIUH approach. The modelling also gave feasible predictions of stream water chemistry, though these could not be used as a basis for model rejection. Nevertheless, the study suggested that the approach has potential for prediction of hydrological response in ungauged montane headwater basins. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Meixner functions for derivation of the unit hydrograph

The Meixner functions are utilized to relate the effective rainfall, the direct runoff and the unit hydrograph through linkage equations. The linkage equations are then employed to derive the unit hydrograph for given rainfall-runoff data on a small agricultural watershed. These functions are tested with regard to their ability to reproduce and predict the direct runoff hydrograph. The Meixner functions are found to be an effective analytical tool for hydrograph synthesis. Further, they compare well with the least squares and linear programming methods of the unit hydrograph derivation.     

UNIT HYDROGRAPH DERIVED FROM A SPATIALLY DISTRIBUTED VELOCITY FIELD

A unit hydrograph model is proposed in which the watershed is decomposed into subareas which are individual cells or zones of neighbouring cells. The unit hydrograph is found for each subarea and the response at the outlet to excess rainfall on each subarea is summed to produce the watershed runoff hydrograph. The cell to cell flow path to the watershed outlet is determined from a digital elevation model. A constant flow velocity is assigned to each cell and the time lag between subarea input and response at the watershed outlet is found by integrating the flow time along the path from the subarea to the outlet. The response function for a subarea is modelled as a lagged linear reservoir in which the flow time is equal to the sum of a time of translation and an average residence time in the reservoir. It is shown that the assumption of a spatially varying, but time-invariant, velocity field underlying this model produces a linear system model for all subareas whose outputs can be summed in the manner indicated. An example application is presented for the 8.70 km² Severn watershed at Plynlimon in Wales using a 50 m digital elevation model in which the cell velocity is calculated by modifying an average watershed velocity according to the terrain slope and the drainage area of each cell. The resulting model reasonably reproduces the observed unit hydrograph.     

A probabilistic framework to evaluate the uncertainty of design hydrograph: case study of Swannanoa River watershed

ABSTRACT

This study presents a probabilistic framework to evaluate the impact of uncertainty of design rainfall depth and temporal pattern as well as antecedent moisture condition (AMC) on design hydrograph attributes – peak, time to peak, duration and volume, as well as falling and rising limb slopes – using an event-based hydrological model in the Swannanoa River watershed in North Carolina, USA. Of the six hydrograph attributes, falling limb slope is the most sensitive to the aforementioned uncertainties, while duration is the least sensitive. In general, the uncertainty of hydrograph attributes decreases in higher recurrence intervals. Our multivariate analysis revealed that in most of the return periods, AMC is the most important driver for peak, duration and volume, while time to peak and falling limb slope are most influenced by rainfall pattern. In higher return periods, the importance of rainfall depth and pattern increases, while the importance of AMC decreases.     

An approach for evaluating the hydrological effects of urbanization and its application

This study focuses mainly on observing urban development in Taiwan's Wu‐Tu watershed from the perspective of urban hydrological theory. An approach is proposed for developing a method for incorporating available meteorological data to define the degree of change in a runoff hydrograph for urbanizing basins. The mean rainfall was estimated using the Kriging method. For calibration, two methods of calculating the effective rainfall (the Φ‐index method and the non‐linear‐programming (NLP) method) were used as model inputs, and the optimal global parameters of the linear reservoir model were then obtained from the shuffled complex evolution (SCE) algorithm. Twenty‐six (1966–1991) and eight (1994–1997) rainfall–runoff events were used for calibration and verification, respectively. The NLP method yielded better results than the Φ‐index method, especially for multipeak rainfall–runoff events. The regression equation determined the relationship between the parameters of the model and impervious areas. A comparison based on the results of the instantaneous unit hydrograph of the study area revealed that three decades of urbanization had increased the peak flow by 27%, and the time to peak was decreased by 4 h. The study simply describes the results of the impact of imperviousness on hydrological modelling. Copyright © 2002 John Wiley & Sons, Ltd.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

Exponential data fitting applied to infiltration, hydrograph separation, and variogram fitting

Most lumped rainfall-runoff models separate the interflow and groundwater components from the measured runoff hydrograph in an attempt to model these as hydrologic reservoir units. Similarly, rainfall losses due to infiltration as well as other abstractions are separated from the measured rainfall hyetograph, which are then used as inputs to the various hydrologic reservoir units. This data pre-processing is necessary in order to use the linear unit hydrograph theory, as well as for maintaining a hydrologic budget between the surface and subsurface flow processes. Since infiltration determines the shape of the runoff hydrograph, it must be estimated as accurately as possible. When measured infiltration data is available, Horton’s exponential infiltration model is preferable due to its simplicity. However, estimating the parameters from Horton’s model constitutes a nonlinear least squares fitting problem. Hence, an iterative procedure that requires initialization is subject to convergence. In a similar context, the separation of direct runoff, interflow, and baseflow from the total hydrograph is typically done in an ad hoc manner. However, many practitioners use exponential models in a rather “layer peeling” fashion to perform this separation. In essence, this also constitutes an exponential data fitting problem. Likewise, certain variogram functions can be fitted using exponential data fitting techniques. In this paper we show that fitting a Hortonian model to experimental data, as well as performing hydrograph separation, and total hydrograph and variogram fitting can all be formulated as a system identification problem using Hankel-based realization algorithms. The main advantage is that the parameters can be estimated in a noniterative fashion, using robust numerical linear algebra techniques. As such, the system identification algorithms overcome the problem of convergence inherent in iterative techniques. In addition, the algorithms are robust to noise in the data since they optimally separate the signal and noise subspaces from the observed noisy data. The algorithms are tested with real data from field experiments performed in Surinam, as well as with real hydrograph data from a watershed in Louisiana. The system identification techniques presented herein can also be used with any other type of exponential data such as exponential decays from nuclear experiments, tracer studies, and compartmental analysis studies.     

The relationship between runoff rate and lag time and the effects of surface treatments at the plot scale

Abstract

The manner in which both the seasonal and regional variations in storm duration, intensity and inter-storm period manifest in the runoff response of agricultural water supply catchments is investigated. High-resolution rainfall data were analysed for a network of 17 raingauges located across the semiarid (200–500 mm year^?1) agricultural districts of southwest Western Australia. Seasonal variations in mean storm duration, mean rainfall intensity and mean inter-storm period were modelled using simple periodic functions whose parameters were then also regressed with geographic and climatic indices to create spatial fields for each of these statistics. Based on these mean values, a continuous rainfall time series can be synthesized for any location within the region, with the rainfall depth within each storm being downscaled to 5-min time steps using a bounded random cascade model. Runoff from six different catchment surface treatments (“engineered” catchments) was simulated using a conceptual water-balance model, validated using rainfall—runoff data from an experimental field site. The expected yield of the various catchment types at any other location within the study region is then simulated using the above rainfall—runoff model and synthetic rainfall and potential evaporation time series under a range of climatic settings representative of regional climate variation. The resulting coupled model can be used to estimate the catchment area required to yield an acceptable volume of runoff for any location and dam capacity, at a specified reliability level, thus providing a tool for water resource managers to design engineered catchments for water supply. Although the model presented is specific for Western Australia's southwest region, the methodology itself is applicable to other locations.