The importance of rainfall estimation to flood forecasting is highlighted by results obtained by Bell and Moore [6] as part of the HYREX (HYdrological Research EXperiment) research programme. Raingauge data from a special network of 49 tipping-bucket raingauges in the 132 km² Brue catchment in southwest England was used with the PDM (Probability Distributed Model) rainfall–runoff model to explore the sensitivity of flow simulations to rainfall. A catchment of this size in the UK would typically only have one raingauge available for use in flood forecasting. Fig. 2 shows the spread of rainfalls measured by the raingauge network obtained over two-day periods for a stratiform and a convective storm. The cumulative hyetographs highlight the significant spatial variability of rainfall within the catchment during convective rainfall; a degree of variability is still apparent for the stratiform storm. Using the PDM rainfall–runoff model, calibrated using the catchment average rainfall, with each raingauge in turn produces the ensemble of simulated flow hydrographs shown in Fig. 3. The spread of hydrographs, particularly for the convective case, serve to highlight the importance of rainfall estimation when flood forecasts are to be based on use of a rainfall–runoff model. It is results of this kind that continue to stimulate research on improved spatial estimation of rainfall, including the use of weather radar.

Whilst a raingauge provides a measurement of rainfall at a point, weather radar provides a volume-integrated estimate of rainfall aloft through its measurement of radar reflectivity and its relation to rain drops. The size and height of the sampled volume increases with range from the radar, reducing spatial resolution and making it a potentially poorer estimate of rainfall at ground level. Some of the problems associated with radar rainfall measurement are summarized in Fig. 4. Use of radar scans at different beam elevations allows the vertical profile of radar reflectivity to be characterized and used in inferring rain rate at the ground as a function of range. Corrections used in the UK network radars are reviewed by Harrison et al. [15] and Lynch et al. [21], whilst Borga et al. [8] provide a useful insight into the relative magnitude of different forms of correction for one radar in the UK network.

Whilst physics-based corrections to radar data normally take precedence, it is generally acknowledged that combination with point raingauge measurements can have real benefits to rainfall estimation accuracy. In some cases, this may take the form of a periodic adjustment to the radar bias with reference to a raingauge network. For operational purposes, the Hyrad system [25,35] employs a multiquadric surface fitting scheme to spatially interpolate radar adjustment factors (formed as a ratio of gauge to raingauge estimates) in merging rainfall estimates from radar and raingauge networks. The surface used for dynamic adjustment is recalculated every 15 min. Moore [26] reports on how the density of the raingauge network used for merging impacts on rainfall estimation accuracy. Wood et al. [40] have distinguished between a climatological, long-term adjustment for bias and a dynamic adjustment. The latter aims to make allowance for more short-term variations in the relation between reflectivity and rain-rate, for example corresponding to changes in drop size distributions during the passage of a storm. A combination of climatological and dynamic adjustment is proposed, where the efficacy of the latter depends on the raingauge density in relation to the spatial variability of the storm. This is being considered in a future upgrade to Hyrad.

The greatest uncertainty in rainfall relates to forecast estimates. The state-of-the-art indicates that simple advection forecasts based on inferring storm movement from time-displaced radar images work best, for lead times out to, say, 6 h. The forecasts are normally smoothed with increasing lead-time, towards the current field-average value in the case of Hyrad [25,34] and towards a numerical weather prediction (NWP) forecast on a 15-km grid in the case of the Nimrod UK product [14]. A probabilistic extension of Nimrod, called STEPS, is under development and capable of providing ensembles of rainfall forecasts [37]. Following workers in the USA [12,20] and France [1], Bell and Moore [5] have pursued a more conceptual water balance approach to storm rainfall forecasting. The basic premise is to use a simple representation of storm dynamics combined with frequent state correction of the water volume in an atmospheric column, as inferred from multi-scan radar measurements. Inferring storm development in this way has yet to be demonstrated to have convincing benefits; however, improvements to multi-scan strategies in the UK have yet to be explored in this context. The longer-term goal relating to rainfall forecasting in the UK is to pursue higher resolution (moving from circa 12 km to 4 and 1 km) NWP formulations capable of representing the initiation and development of convection. At present, NWP rainfalls out to 1.5 days on a 15 km grid are used in support of flood warning in England and Wales, via the Hyrad system, serving primarily as a first-alert flood indicator.

The basic sources of information for rainfall estimation are raingauge network data and weather radar products. Without further reception, processing, and support tools, these sources cannot easily be used to provide an initial warning of impending flooding or as direct input to flood forecasting models. Hyrad has transformed the use of weather radar for hydrological use in the UK by providing a software system that is easy to use for the visualization of a variety of radar, and weather-related, products and that interfaces to flood forecasting systems.

Fig. 5 shows the main components of Hyrad: data reception, spatial image database (SIDB), hydrology kernel (anomaly removal, radar-raingauge merging, rainfall forecasting, catchment average rainfall estimation), client visualization, and system monitoring/scheduling/control. Mosaics of windows can be configured to visualize rainfall fields at different scales (European, regional and catchment) and in different ways (animated images highlighting motion, cumulative fields revealing anomalies, catchment average time-series portraying storm profiles): an example display is shown in Fig. 6.

The hydrology kernel removes static and transient anomalies in fields of different resolution and combines them to form a composite image. Rainfall fields can be estimated by radar-raingauge merging and raingauge-only interpolation and rainfall forecasts derived from these fields. These local Hyrad products, along with any external source products, can be processed to derive catchment average estimates for feeding forward to flood forecasting and modelling systems. The activities within the kernel are summarized in Fig. 7.

The generic, configurable form of Hyrad allows other spatial products to be received, visualized, and processed: examples are NWP rainfall and temperature and MOSES (Met Office Surface Exchange Scheme) soil moisture, evaporation, and runoff products.

In England and Wales, rainfall forecasts are used as a form of first-alert for flood warning and as a basic ingredient of a ‘Flood Watch’ service for areas not served by flood forecasting systems. The Daily Weather Forecast provides 6-h-interval forecasts for areas out to 1.5 days (and less frequent ones to 5 days), as typical amounts and most likely maximum along with a 3-category measure of confidence. Heavy rainfall warnings are triggered to be sent when certain conditions apply: these are presented in probability form for some regions. Monitoring of the performance of this service is done routinely and feeds into the post-event assessment of flood warning provision.

Jones et al. [17] report on a detailed consideration of methodologies for assessment aligned to operational requirements for monitoring. A set of performance statistics are identified for assessing different forms of forecast, which includes a form of continuous Brier Score for use with probabilistic forecasts. A PC tool has been developed to help assess the Heavy Rainfall Forecast [18]. The study included a methodology to compare the relative performance of different forecast sources. This indicated there is skill in the forecasts, in relation to naïve forecasts such as persistence, but this is weak reflecting the difficulty of rainfall prediction. Whilst not conclusive, due to the limited extent of the comparison dataset, using radar rainfall out to 6 hours and NWP rainfall out to two days could prove as good on average as the ‘added-value’ daily weather forecast product. NWP rainfall forecasts are now routinely made available to the flood warning process via Hyrad.

Flood modelling and forecasting involves far more than the traditional crafts of rainfall–runoff modelling, channel flow routing and forecast updating. Commenting on progress in these traditional crafts will be deferred to subsequent sections. The theme of systems will be continued here to highlight the connectivity of the rainfall and flood forecasting system chain. Fig. 8 presents a schematic design of an advanced flood forecasting and modelling system based on the river flow forecasting system or RFFS [25,28]. A shell-kernel architecture is presented, in which the shell is receptive to incoming data from hydrometeorogical systems such as Hyrad and from hydrometric telemetry networks. These external data sources feed a modelling database, which is also receptive to locally generated forecasts produced by the modelling kernel. Observation and forecast data can be visualised on user PCs and flood warning reports generated and disseminated via linked applications.

At the heart of the modelling kernel is the information control algorithm or RFFS-ICA [28], which serves as a generic, configurable forecast construction environment. This allows forecasts to be constructed using any set of models in any river network configuration, using data files to define the arrangement independent of the coding of the models (which are plug-in-and-play) and construction of the forecasts. The concept of a two-layer configuration of models is used in which a model component layer allows a model algorithm layer to be embedded within it. This provides a level of modularity that can be exploited to considerably simplify commonly occurring assemblages of model elements to form larger building blocks with internal structures that frequently repeat. For example, a headwater model component structure may typically combine precipitation merging and snowmelt codes with rainfall–runoff models and updating procedures and possibly local stage-discharge relations; the elements are configured as model algorithms.

A forecast requirement layer allows forecasts at the same location to be constructed using different models, as defined in the model component layer. It also defines the inputs and connectivity of the model network. Fig. 9 presents an example of a complex Model Network schematic, in this case as used for flood forecasting throughout northeastern England. The ICA can be scheduled to run in harmony with telemetry polling. The use of a model state set ensures that the results can be equivalent to a seamless continuous running of the system. The state set may be improved at times of late receipt of telemetry data by manually starting a run from a previously saved state set.

An example flood forecast from a Flood Forecasting and Modelling System is shown in Fig. 10. This illustrates a forecast obtained from the PDM rainfall–runoff model for a river basin in northeastern England. State correction is used to initialise the model to produce a flow at the forecast time-origin close to the observation of flow. Two forecasts are presented, one based on perfect foreknowledge of rainfall and the other based on a negligible rainfall forecast. The importance of a good rainfall forecast becomes clearly evident for higher forecast lead-times. The modelling and forecast updating methods used to produce this forecast are discussed in the next section.

To some researchers, flood forecasting is synonymous with rainfall–runoff modelling and a major preoccupation. The previous section has highlighted that forecasting systems can involve complex networks of interconnected models of different kinds, representing processes as diverse as snowmelt and the hydrodynamics of rivers under the influence of tide and barrier control. This diversity is added to by the need to develop procedures to assimilate up-to-the-minute observations of river flow or level so as to improve model forecast performance, via so-called updating methods. Selected issues concerning these modelling and updating requirements will be considered here.

8.1 Rainfall–runoff modelling toolkits

With the recognition that rainfall–runoff models share a limited set of common elements, there is a trend towards building ‘model toolkits’ that encompass these basic elements and can be configured to represent different forms of catchment behaviour. This was one motivation behind building the PDM (Probability Distributed Model) as one of the more popular rainfall–runoff models used in practise in the UK [23–25]. The PDM is presented in Fig. 11 in the structural form most commonly used in practise. Both fast and slow response routing components, here more loosely referred to as surface and subsurface storages, can be represented by the Horton–Izzard equation [10]. This equation results from continuity and the simple nonlinear storage form of momentum equation, q=kSn, where q is flow rate per unit area, S is water storage per unit area and k−1 (the storage time constant) and n are parameters of the relation. Solutions for different values of n are provided that cater for behaviours characteristic of channel and groundwater translation systems (see Appendix A of [31]). As an alternative to the Horton–Izzard formulation, either routing storage can be represented by a cascade of two linear reservoirs. Solution of this system takes the form of a transfer function (TF) model with dependence on two past outputs and the present and past input [9,36]. Whilst the TF model has four parameters, this is linked directly to the two physical time constants of its equivalent reservoir storage model and its continuity constraint; imposing equality of time-constants reduces the parameters to one.

8.2 Channel flow routing models

A unique feature of real-time flood forecasting is the ability to use observations of the state of the basin up to the current time to improve model performance. Most commonly, the observations relate to river level or flow but other relevant observations on the state of the basin, such as groundwater well levels and soil moisture content might be used. The most natural method of forecast updating is state correction in which a direct or related measurement of a model state is used to adjust its value so as to improve forecast performance. River flow, as the most commonly used observation for updating, can be used to calculate a model error which, when factored and added to the state value to be corrected, provides a basis for updating the state. If the model states chosen for correction are the fast and slow response flows from the PDM model and the factors are chosen to be the proportion of each flow to the total model flow, then the corrected flows will sum to the observed flow. Introducing relaxation coefficients into the factors allows more conservative adjustments to be made; these coefficients may be arrived at through optimisation using historical records. Note that by adjusting the two model flows, the water contents of the stores that they derive from are also state corrected indirectly. The equations for correction are set down in [25].

The above outline of ‘empirical state correction’ has its origins in the more formal theory of Kalman filtering [13,16], which employs the same basic form of adjustment, but defines the factors as gains which are estimated according to the relative uncertainties associated with the model and observation estimate of each state to be corrected. The theory is optimal for linear models and observation systems and if the uncertainties are correctly characterised. Approximate solutions exist relevant to the nonlinear systems and forms of error encountered in hydrology. Computations based on such solutions can be highly complex and there is no guarantee that they will provide better forecasts than ones based on the simple empirical scheme described above. This is the basis for recommending the latter for practical application.

The same underlying form of correction can be used to adjust model parameters in real-time, treating these essentially as extended state variables. This approach is still used when models take very simple forms, such as those based on pure TF model concepts, where the inadequate model dynamics are compensated for by shifting the parameter values to keep the forecasts on track. The approach is not generally recommended as the need for time-varying parameters is a diagnostic of an imperfect model structure and highlights the need to account for the variation within the model to predict future variations. It might for example diagnose the need for an explicit moisture accounting model component to remove the need to track a time-varying gain within a TF model.

Errors in the model inputs can be treated with similar state correction schemes. If flow observations are used as the basis of correction, then the inputs are related only very indirectly and with time delays, making robust adjustment difficult. For this reason, input correction of this form is generally not practised. Note that a major reason for employing state correction of the normal form is that errors in the rainfall input propagate through a rainfall–runoff model as errors in the store water contents, which are more reliably corrected at this stage.

A very different approach to updating focuses on the predictability of the model errors themselves. It is a common feature of models that transform one or more inputs to an output that errors in the model output series – the so-called simulation-mode errors – exhibit runs of over-estimation and under-estimation. This serial dependence can be used as the basis of predicting future errors which, when added to the modelled output for the same time, provides the updated forecast required. A simple structure for an error model is obtained as a weighted linear combination of a certain number of past simulation-mode errors. This structure is referred to as autoregressive, or AR, and the number of past errors included referred to as the model order p indicating that there is this number of weights or AR parameters. An extension of this idea is to also include a weighted combination of q past one-step ahead forecast errors in the predictive equation for the model error. This is referred to as the moving average or MA component and, when used in combination with the AR part, is referred to as an ARMA error predictor. It can be the case that a high order AR error model may be better represented by a low order ARMA error model. If the ARMA model structure is well identified and appropriate, the one-step-ahead prediction errors should approximate white noise with an absence of serial correlation. Unlike state correction, the methodology is applied externally to the ‘process’ model and can be applied to any model including rainfall–runoff, channel flow, and hydrodynamic forms. The level of success achieved is a function of the strength of the serial dependence in the simulation-mode errors. Unfortunately this can be weakest in the vicinity of the rising limb and the peak of the flood hydrograph and strongest on the recession limb where correction is least important. It is normally used in preference to state correction when time-delay effects are judged to make such correction unreliable: for example, for multi-reach channel flow routing. Variants can be developed based on the use of proportional, rather than the normal additive, errors [25].

10.1 Lessons from model intercomparisons

10.2 Flood warning for ungauged locations and area-wide forecasting

Developing a new generation of flood model on a distributed grid has great appeal, not only as a solution to forecasting for the ungauged location. Such area-wide models can also provide the basis of a first-alert flood warning system when fed by hydro-meteorological information from weather radar and numerical weather prediction model forecasts out to one or two days ahead. They can be configured to provide national, European and even global coverage. Offering an indicative level of warning, they serve to complement more detailed flood forecasting systems operating at a regional level and with higher levels of data assimilation. Such systems offer the potential to map flow dynamically down a river network and to delineate areas of inundation, at least at an indicative level, enhancing the present flood watch service.

Arguably the greatest challenge confronting flood forecasting and warning is to develop methodologies capable of embracing forecast uncertainty into the decision-making process. This will demand close cooperation between meteorological and hydrological communities to make real scientific progress. Providing probability estimates of rainfall fields for past and future times that are credible is a difficult challenge, and may need to encompass consideration of varying measurement and modelling methods at different times. Propagating probability estimates of rainfall through networks of hydrological models – with errors in their structure, parameters, and states – to obtain probabilistic flow forecasts that are plausible is not straightforward.

Pierce et al. [37] present a first-step approach to probabilistic forecasting by generating an ensemble of radar-rainfall forecasts from a stochastic advection-based scheme and using these in the PDM rainfall–runoff model; it is assumed that errors in rainfall forecasting dominate those associated with the hydrological model. The difficulties of a decomposition approach to characterising uncertainty may lead to more pragmatic methods being explored, such as empirical analysis and characterisation of the forecast errors themselves.

Given a plausible probability forecast of river flow and a loss function for the location to be warned, then a probability forecast of damage costs can be derived based on standard decision theory. This probabilistic damage cost forecast can be used as the scientific basis for deciding if, and how early, to warn. Developing this methodology into a practical decision-support system for flood warning across a region has the potential to be of great benefit to flood defence operations.