Linking space–time variability of river runoff and rainfall fields: a dynamic approach

Statistical self-similarity in the spatial and temporal variability of rainfall, river networks, and runoff processes has been observed in many empirical studies. To theoretically investigate the relationships between the various time and space scales of variability in rainfall and runoff process we propose a simplified, yet physically based model of a catchment–rainfall interaction. The channel network is presented as a random binary tree, having topological and hydraulic geometry properties typically observed in real river networks. The continuous rainfall model consists of individual storms separated by dry periods. Each given storm is disaggregated in space and time using the random cascade model. The flow routing is modelled by the network of topologically connected nonlinear reservoirs, each representing a link in the channel network. Running the model for many years of synthetic rainfall time series and a continuous water balance model we generate an output, in the form of continuous time series of water discharge in all links in the channel network. The main subject of study is the annual peak flow as a function of catchment area and various characteristics of rainfall. The model enables us to identify different physical processes responsible for the empirically observed scaling properties of peak flows.     

Prediction and modelling of rainfall–runoff during typhoon events using a physically-based and artificial neural network hybrid model

Abstract

The accurate prediction of hourly runoff discharge in a watershed during heavy rainfall events is of critical importance for flood control and management. This study predicts n-h-ahead runoff discharge in the Sandimen basin in southern Taiwan using a novel hybrid approach which combines a physically-based model (HEC-HMS) with an artificial neural network (ANN) model. Hourly runoff discharge data (1200 datasets) from seven heavy rainfall events were collected for the model calibration (training) and validation. Six statistical indicators (i.e. mean absolute error, root mean square error, coefficient of correlation, error of time to peak discharge, error of peak discharge and coefficient of efficiency) were employed to evaluate the performance. In comparison with the HEC-HMS model, the single ANN model, and the time series forecasting (ARMAX) model, the developed hybrid HEC-HMS–ANN model demonstrates improved accuracy in recursive n-h-ahead runoff discharge prediction, especially for peak flow discharge and time.     

Long‐lead seasonal rainfall forecasting using time‐delay recurrent neural networks: a case study

A temporal artificial neural network‐based model is developed and applied for long‐lead rainfall forecasting. Tapped delay lines and recurrent connections are two different components that are used along with a static multilayer perceptron network to design a time‐delay recurrent neural network. The proposed model is, in fact, a combination of time‐delay and recurrent neural networks. The model is applied in three case studies of the Northwest, West, and Southwest basins of Iran. In addition, an autoregressive moving average with exogenous inputs (ARMAX) model is used as a baseline in order to be compared with the time‐delay recurrent neural networks developed in this study. Large‐scale climate signals, such as sea‐level pressure, that affect the rainfall of the study area are used as the predictors in the models, as well as the persistence between rainfall data. The results of winter‐spring rainfall forecasts are discussed thoroughly. It is demonstrated that in all cases the proposed neural network results in better forecasts in comparison with the statistical ARMAX model. Moreover, it is found that in two of three case studies the time‐delay recurrent neural networks perform better than either recurrent or time‐delay neural networks. The results demonstrate that the proposed method can significantly improve the long‐lead forecast by utilizing a non‐linear relationship between climatic predictors and rainfall in a region. Copyright © 2007 John Wiley & Sons, Ltd.     

A deterministic linearized recurrent neural network for recognizing the transition of rainfall–runoff processes

Characterizing the dynamic relationship between rainfall and runoff is a highly interesting modeling problem in hydrology. This study develops a deterministic linearized recurrent neural network (denoted as DLRNN) that deals with the system’s nonlinearity by recalibration at each time interval, and relates the weights of DLRNN to unit hydrographs in order to describe the transition of the rainfall–runoff processes. Case studies of 38 events, from 1966 to 1997, are implemented in the Wu-Tu watershed of Taiwan, where the runoff path-lines are short and steep. A comparison between the DLRNN and a feed-forward neural network demonstrates the advantage of DLRNN as a dynamic system model. It is concluded that DLRNN shows superiority in the performance of rainfall–runoff simulations and the ability to recognize transitions in hydrological processes.     

Rainfall‐runoff analysis based on competing linear impulse responses: decomposition of rainfall‐runoff processes

Many recent studies have successfully used neural networks for non‐linear rainfall‐runoff modelling. Due to fundamental limitation of linear structures, approaches employing linear models have been generally considered inferior to the neural network approaches in this area. However, the authors believe that with an appropriate extension, the concept of linear impulse responses can be a viable tool since it enables one to understand underlying dynamics of rainfall‐runoff processes. In this paper, the use of competing impulse responses for rainfall‐runoff analysis is proposed. The proposed method is based on the switch over of competing linear impulse‐responses, each of which satisfies the constraints of non‐negativity and uni‐modality. The computational analyses performed for the rainfall‐runoff data in the Seolma‐Chun experimental basin, Korea showed that the proposed method can yield promising results. Considering the basin characteristics as well as the results from this study, it may be concluded that three impulse responses are enough for rainfall‐runoff analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Real‐time recurrent learning neural network for stream‐flow forecasting

Various types of neural networks have been proposed in previous papers for applications in hydrological events. However, most of these applied neural networks are classified as static neural networks, which are based on batch processes that update action only after the whole training data set has been presented. The time variate characteristics in hydrological processes have not been modelled well. In this paper, we present an alternative approach using an artificial neural network, termed real‐time recurrent learning (RTRL) for stream‐flow forecasting. To define the properties of the RTRL algorithm, we first compare the predictive ability of RTRL with least‐square estimated autoregressive integrated moving average models on several synthetic time‐series. Our results demonstrate that the RTRL network has a learning capacity with high efficiency and is an adequate model for time‐series prediction. We also investigated the RTRL network by using the rainfall–runoff data of the Da‐Chia River in Taiwan. The results show that RTRL can be applied with high accuracy to the study of real‐time stream‐flow forecasting networks. Copyright © 2002 John Wiley & Sons, Ltd.     

New fuzzy neural network–Markov model and application in mid- to long-term runoff forecast

ABSTRACT

In this paper, a mid- to long-term runoff forecast model is developed using an ideal point fuzzy neural network–Markov (NFNN-MKV) hybrid algorithm to improve the forecasting precision. Combining the advantages of the new fuzzy neural network and the Markov prediction model, this model can solve the problem of stationary or volatile strong random processes. Defined error statistics algorithms are used to evaluate the performance of models. A runoff prediction for the Si Quan Reservoir is made by utilizing the modelling method and the historical runoff data, with a comprehensive consideration of various runoff-impacting factors such as rainfall. Compared with the traditional fuzzy neural networks and Markov prediction models, the results show that the NFNN-MKV hybrid algorithm has good performance in faster convergence, better forecasting accuracy and significant improvement of neural network generalization. The absolute percentage error of the NFNN-MKV hybrid algorithm is less than 7.0%, MSE is less than 3.9, and qualification rate reaches 100%. For further comparison of the proposed model, the NFNN-MKV model is employed to estimate (training and testing for 120-month-ahead prediction) and predict river discharge for 156 months at Weijiabao on the Weihe River in China. Comparisons among the results of the NFNN-MKV model, the WNN model and the SVR model indicate that the NFNN-MKV model is able to significantly increase prediction accuracy.

Editor D. Koutsoyiannis; Associate editor Y. Gyasi-Agyei     

Assessment of flooding in urbanized ungauged basins: a case study in the Upper Tiber area,Italy

The reliability of a procedure for investigation of flooding into an ungauged river reach close to an urban area is investigated. The approach is based on the application of a semi‐distributed rainfall–runoff model for a gauged basin, including the flood‐prone area, and that furnishes the inlet flow conditions for a two‐dimensional hydraulic model, whose computational domain is the urban area. The flood event, which occurred in October 1998 in the Upper Tiber river basin and caused significant damage in the town of Pieve S. Stefano, was used to test the approach. The built‐up area, often inundated, is included in the gauged basin of the Montedoglio dam (275 km²), for which the rainfall–runoff model was adapted and calibrated through three flood events without over‐bank flow. With the selected set of parameters, the hydrological model was found reasonably accurate in simulating the discharge hydrograph of the three events, whereas the flood event of October 1998 was simulated poorly, with an error in peak discharge and time to peak of −58% and 20%, respectively. This discrepancy was ascribed to the combined effect of the rainfall spatial variability and a partial obstruction of the bridge located in Pieve S. Stefano. In fact, taking account of the last hypothesis, the hydraulic model reproduced with a fair accuracy the observed flooded urban area. Moreover, incorporating into the hydrological model the flow resulting from a sudden cleaning of the obstruction, which was simulated by a ‘shock‐capturing’ one‐dimensional hydraulic model, the discharge hydrograph at the basin outlet was well represented if the rainfall was supposed to have occurred in the region near the main channel. This was simulated by reducing considerably the dynamic parameter, the lag time, of the instantaneous unit hydrograph for each homogeneous element into which the basin is divided. The error in peak discharge and time to peak decreased by a few percent. A sensitivity analysis of both the flooding volume involved in the shock wave and the lag time showed that this latter parameter requires a careful evaluation. Moreover, the analysis of the hydrograph peak prediction due to error in rainfall input showed that the error in peak discharge was lower than that of the same input error quantity. Therefore, the obtained results allowed us to support the hypothesis on the causes which triggered the complex event occurring in October 1998, and pointed out that the proposed procedure can be conveniently adopted for flood risk evaluation in ungauged river basins where a built‐up area is located. The need for a more detailed analysis regarding the processes of runoff generation and flood routing is also highlighted. Copyright © 2005 John Wiley & Sons, Ltd.     

Rainfall‐runoff modelling using artificial neural networks: comparison of network types

Growing interest in the use of artificial neural networks (ANNs) in rainfall‐runoff modelling has suggested certain issues that are still not addressed properly. One such concern is the use of network type, as theoretical studies on a multi‐layer perceptron (MLP) with a sigmoid transfer function enlightens certain limitations for its use. Alternatively, there is a strong belief in the general ANN user community that a radial basis function (RBF) network performs better than an MLP, as the former bases its nonlinearities on the training data set. This argument is not yet substantiated by applications in hydrology. This paper presents a comprehensive evaluation of the performance of MLP‐ and RBF‐type neural network models developed for rainfall‐runoff modelling of two Indian river basins. The performance of both the MLP and RBF network models were comprehensively evaluated in terms of their generalization properties, predicted hydrograph characteristics, and predictive uncertainty. The results of the study indicate that the choice of the network type certainly has an impact on the model prediction accuracy. The study suggests that both the networks have merits and limitations. For instance, the MLP requires a long trial‐and‐error procedure to fix the optimal number of hidden nodes, whereas for an RBF the structure of the network can be fixed using an appropriate training algorithm. However, a judgment on which is superior is not clearly possible from this study. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparison of static-feedforward and dynamic-feedback neural networks for rainfall–runoff modeling

A systematic comparison of two basic types of neural network, static and dynamic, is presented in this study. Two back-propagation (BP) learning optimization algorithms, the standard BP and conjugate gradient (CG) method, are used for the static network, and the real-time recurrent learning (RTRL) algorithm is used for the dynamic-feedback network. Twenty-three storm-events, about 1632 rainfall and runoff data sets, of the Lan-Yang River in Taiwan are used to demonstrate the efficiency and practicability of the neural networks for one hour ahead streamflow forecasting. In a comparison of searching algorithms for a static network, the results show that the CG method is superior to the standard BP method in terms of the efficiency and effectiveness of the constructed network's performance. For a comparison of the static neural network using the CG algorithm with the dynamic neural network using RTRL, the results show that (1) the static-feedforward neural network could produce satisfactory results only when there is a sufficient and adequate training data set, (2) the dynamic neural network generally could produce better and more stable flow forecasting than the static network, and (3) the RTRL algorithm helps to continually update the dynamic network for learning—this feature is especially important for the extraordinary time-varying characteristics of rainfall–runoff processes.     

Regionalizing mean annual flow and daily flow variability for basin‐scale sediment and nutrient modelling

River discharges vary strongly through time and space, and quantifying this variability is fundamental to understanding and modelling river processes. The river basin is increasingly being used as the unit for natural resource planning and management; to facilitate this, basin‐scale models of material supply and transport are being developed. For many basin‐scale planning activities, detailed rainfall‐runoff modelling is neither necessary nor tractable, and models that capture spatial patterns of material supply and transport averaged over decades are sufficient. Nevertheless, the data to describe the spatial variability of river discharge across large basins for use in such models are often limited, and hence models to predict river discharge at the basin scale are required. We describe models for predicting mean annual flow and a non‐dimensional measure of daily flow variability for every river reach within a drainage network. The models use sparse river gauging data, modelled grid surfaces of mean annual rainfall and mean annual potential evapotranspiration, and a network accumulation algorithm. We demonstrate the parameterization and application of the models using data for the Murrumbidgee basin, in southeast Australia, and describe the use of these predictions in modelling sediment transport through the river network. The regionalizations described contain less uncertainty, and are more sensitive to observed spatial variations in runoff, than regionalizations based on catchment area and rainfall alone. Copyright © 2006 John Wiley & Sons, Ltd.     

The physical framework of the dependence between channel flow hydrographs and drainage network morphometry

The physical basis of the linkage between magnitude and timing of channel flow hydrographs and drainage network morphometry is reviewed. Small Hortonian and structurally Hortonian networks are analysed using numerical runoff simulation. For Hortonian networks the variability of the geometry of individual channels and subcatchments within each Strahler order has generally little effect upon the overall character of the hydrograph in channels of higher order. If the network is also structurally Hortonian, the analysis of the simultaneous formation, travel, and concentration of the hydrographs in all channels of the network can be simplified to a sequence of one representative hydrograph per channel order. This approach is used in this study. Three major runoff processes control the flow hydrograph characteristics: the overland flow process which determines the water supply to the drainage network; the channel flow process which translates the hydrograph in space and time; and the drainage network process which concentrates and magnifies the flow at the junctions of the drainage network. Functional relations for the hydrograph peak, timing, and flow velocity are presented. For a given uniform rainfall and infiltration rate, the peak of the channel flow hydrograph is shown to increase geometrically with channel order, and its magnitude is directly related to the bifurcation ratio. The travel time of the peak also increases geometrically with channel order, and it is directly related to the channel length ratio over velocity ratio. The flow velocity of the peak changes in a downstream direction as a function of the bifurcation and slope ratio. It was also found that for negligible channel storage the channel flow and drainage network processes do not contribute significantly to the observed nonlinear response of a watershed to precipitation.     

Effectivity of dissolved SF6 tracer for clarification of rainfall–runoff processes in a forested headwater catchment

Understanding rainfall‐runoff processes is crucial for prevention and prediction of water‐related natural disasters. Sulfur hexafluoride (SF₆) is a potential tracer, but few researches have applied it for rainfall‐runoff process studies. We observed multiple tracers including SF₆ in spring water at 1‐ to 2‐hr intervals during rainstorm events to investigate the effectivity of SF₆ tracer in rainfall–runoff studies through the clarification of rainfall–runoff process. The target spring is a perennial spring in a forested headwater catchment with an area of 0.045 km² in Fukushima, Japan. The relationship between the SF₆ concentration in spring water and the spring discharge volume was negative trend; the SF₆ concentration in spring water becomes low as the spring discharge volume increases especially during rainstorms. The hydrograph separation using SF₆ and chloride ion tracers was applied for determining the contribution of principal sources on rainfall–runoff water. It suggested more than 60% contribution of bedrock groundwater at the rainfall peak and high percentage contribution continued even in the hydrograph recession phase. Based on observed low SF₆ concentration in groundwater after heavy rainfall, the replacement of groundwater near the spring with bedrock groundwater is indicated as a mechanism for water discharge with low SF₆ concentration during rainfall events. Consequently, rainstorm events play an important role as triggers in discharging water stored in the deeper subsurface area. In addition, SF₆ tracer is concluded as one of the strongest tracers for examining rainfall–runoff process studies. And, therefore, this study provided new insights into the dynamics of groundwater and its responses to rainfall in terms of SF₆ concentration variance in water in headwater regions.     

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.     

The effect of rainstorm movement on urban drainage network runoff hydrographs

This article explores the relations between network properties and the effect from moving rainstorms in terms of the peak response and time to centroid of hydrographs. A simple conceptual rectangular catchment is introduced with different configurations of drainage network simulated by the Gibbs stochastic model. The efficiency of the urban pipe networks varies widely compared with natural river networks; hence, the Gibbs model can be an appropriate approach to represent the network properties in urban drainage system. Simple cases of rainstorms moving with upstream and downstream directions and different speeds are considered to investigate the effect of rainstorm movement on urban drainage network runoff hydrographs. The results indicate that the effect of the direction and speed of the rainstorm movement varies significantly depending on the network properties. The relationship between storm speed and direction and the change in the peak runoff is dependent on the network configuration and network efficiency. In contrast to previous studies, this study indicates that the speed and direction of the rainfall movement that produces the maximum peak discharge changes depending on the network configuration. Copyright © 2011 John Wiley & Sons, Ltd.     

Effects of the catchment runoff coefficient on the performance of TOPMODEL in rainfall–runoff modelling

Effects of the catchment runoff coefficient on the performance of TOPMODEL in simulating catchment rainfall–runoff relationships are investigated in this paper, with an aim to improve TOPMODEL's simulation efficiency in catchments with a low runoff coefficient. Application of TOPMODEL in the semi‐arid Yihe catchment, with an area of 2623 km² in the Yellow River basin of China, produced a Nash–Sutcliffe model efficiency of about 80%. To investigate how the catchment runoff coefficient affects the performance of TOPMODEL, the whole observed discharge series of the Yihe catchment is multiplied with a larger‐than‐unity scale factor to obtain an amplified discharge series. Then TOPMODEL is used to simulate the amplified discharge series given the original rainfall and evaporation data. For a set of different scale factors, TOPMODEL efficiency is plotted against the corresponding catchment runoff coefficient and it is found that the efficiency of TOPMODEL increases with the increasing catchment runoff coefficient before reaching a peak (e.g. about 90%); after the peak, however, the efficiency of TOPMODEL decreases with the increasing catchment runoff coefficient. Based on this finding, an approach called the discharge amplification method is proposed to enhance the simulation efficiency of TOPMODEL in rainfall–runoff modelling in catchments with a low runoff coefficient. Copyright © 2004 John Wiley & Sons, Ltd.     

Coupling the Xinanjiang model to a kinematic flow model based on digital drainage networks for flood forecasting

The Xinanjiang model, which is a conceptual rainfall‐runoff model and has been successfully and widely applied in humid and semi‐humid regions in China, is coupled by the physically based kinematic wave method based on a digital drainage network. The kinematic wave Xinanjiang model (KWXAJ) uses topography and land use data to simulate runoff and overland flow routing. For the modelling, the catchment is subdivided into numerous hillslopes and consists of a raster grid of flow vectors that define the water flow directions. The Xinanjiang model simulates the runoff yield in each grid cell, and the kinematic wave approach is then applied to a ranked raster network. The grid‐based rainfall‐runoff model was applied to simulate basin‐scale water discharge from an 805‐km² catchment of the Huaihe River, China. Rainfall and discharge records were available for the years 1984, 1985, 1987, 1998 and 1999. Eight flood events were used to calibrate the model's parameters and three other flood events were used to validate the grid‐based rainfall‐runoff model. A Manning's roughness via a linear flood depth relationship was suggested in this paper for improving flood forecasting. The calibration and validation results show that this model works well. A sensitivity analysis was further performed to evaluate the variation of topography (hillslopes) and land use parameters on catchment discharge. Copyright © 2009 John Wiley & Sons, Ltd.     

Characteristics of surface runoff and throughflow in a purple soil of Southwestern China under various rainfall events

In order to harvest runoff to palliate water disaster as well as effectively manage irrigation and fertilizer application in the studied region, it is necessary to better understand the runoff processes. A newly designed runoff collection system for a plot scale was used to partition runoff under contrasting rainfall events into surface flow and subsurface flow to obtain characteristics of surface runoff and throughflow in a purple soil (Regosols in FAO taxonomy, Entisol in USDA taxonomy) of Sichuan, China. Under small rainfall (shower and drizzle), only surface runoff was observed. It is noted that, under shower, particularly with antecedent dry soil conditions, the highest peak surface runoff significantly lagged behind that of rainfall, because air‐locked soil pores of the top layer appeared temporally. Under rainstorm and downpour, surface runoff and throughflow both commenced and showed hysteresis. The hydrograph of surface runoff better resembled that of rainfall than throughflow did. The durations of throughflow discharge of post‐rainfall‐end were near the same (within 24 h) under various rainfalls and rather dependent upon the soil properties than the rainfall characteristics. Throughflow is about 60–90% of total runoff, and especially significant in a ploughed layer under downpour. The chloride concentration of throughflow was over twice that of surface runoff and rainfall, implying that throughflow contains more nutrients than surface runoff. Presumably, surface runoff was primarily governed by an infiltration‐excess or saturated excess‐infiltration mechanism under unsaturated or saturated soil conditions. Therefore, the management of water and fertilizer, and the harvesting of water flow in the ploughed soil layer, should be emphasized in this region. Copyright © 2005 John Wiley & Sons, Ltd.     

Use of the Connectivity of Runoff Model (CRUM) to investigate the influence of storm characteristics on runoff generation and connectivity in semi‐arid areas

Much attention has been given to the surface controls on the generation and transmission of runoff in semi‐arid areas. However, the surface controls form only one part of the system; hence, it is important to consider the effect that the characteristics of the storm event have on the generation of runoff and the transmission of flow across the slope. The impact of storm characteristics has been investigated using the Connectivity of Runoff Model (CRUM). This is a distributed, dynamic hydrology model that considers the hydrological processes relevant to semi‐arid environments at the temporal scale of a single storm event. The key storm characteristics that have been investigated are the storm duration, rainfall intensity, rainfall variability and temporal structure. This has been achieved through the use of a series of defined storm hydrographs and stochastic rainfall. Results show that the temporal fragmentation of high‐intensity rainfall is important for determining the travel distances of overland flow and, hence, the amount of runoff that leaves the slope as discharge. If the high‐intensity rainfall is fragmented, then the runoff infiltrates a short distance downslope. Longer periods of high‐intensity rainfall allow the runoff to travel further and, hence, become discharge. Therefore, storms with similar amounts of high‐intensity rainfall can produce very different amounts of discharge depending on the storm characteristics. The response of the hydrological system to changes in the rainfall characteristics can be explained using a four‐stage model of the runoff generation process. These stages are: (1) all water infiltrating, (2) the surface depression store filling or emptying without runoff occurring, (3) the generation and transmission of runoff and (4) the transmission of runoff without new runoff being generated. The storm event will move the system between the four stages and the nature of the rainfall required to move between the stages is determined by the surface characteristics. This research shows the importance of the variable‐intensity rainfall when modelling semi‐arid runoff generation. The amount of discharge may be greater or less than the amount that would have been produced if constant rainfall intensity is used in the model. Copyright © 2006 John Wiley & Sons, Ltd.     

Identification of physical processes inherent in artificial neural network rainfall runoff models

The emergence of artificial neural network (ANN) technology has provided many promising results in the field of hydrology and water resources simulation. However, one of the major criticisms of ANN hydrologic models is that they do not consider/explain the underlying physical processes in a watershed, resulting in them being labelled as black‐box models. This paper discusses a research study conducted in order to examine whether or not the physical processes in a watershed are inherent in a trained ANN rainfall‐runoff model. The investigation is based on analysing definite statistical measures of strength of relationship between the disintegrated hidden neuron responses of an ANN model and its input variables, as well as various deterministic components of a conceptual rainfall‐runoff model. The approach is illustrated by presenting a case study for the Kentucky River watershed. The results suggest that the distributed structure of the ANN is able to capture certain physical behaviour of the rainfall‐runoff process. The results demonstrate that the hidden neurons in the ANN rainfall‐runoff model approximate various components of the hydrologic system, such as infiltration, base flow, and delayed and quick surface flow, etc., and represent the rising limb and different portions of the falling limb of a flow hydrograph. Copyright © 2004 John Wiley & Sons, Ltd.