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Testing probabilistic adaptive real‐time flood forecasting models

Testing probabilistic adaptive real‐time flood forecasting models Introduction The ability to issue flood warnings in a timely manner is an important part of mitigating the impact of flooding. In modern flood management practice, the issuing of warnings is guided by numerical models for forecasting future water levels or discharge. While one‐ and two‐dimensional hydrodynamic models of flood inundation are useful for management and planning purposes to issue warnings, it is often adequate to forecast only at specific points of interest where observations (either of water level or discharge) are available to evaluate the model forecasts. Extension to other river reaches where the risk of overtopping defences is significant is then generally related to different threshold warning levels at the forecast site. This paper outlines a method for producing forecasts at locations with observations based on time series models derived from available data. The models used, termed data‐based mechanistic (DBM) models (Young, ), are parsimonious time series models that can be interpreted in a mechanistic fashion (see for example Young and Beven, ; Young, , , ). These models are combined with an efficient data assimilation strategy to produce probabilistic forecasts of future observations. The implementation of these models in the UK National Flood Forecasting System (NFFS) is outlined using two case studies, the River Eden and River Severn. Before presenting the case studies, the following sections discuss where flood forecasting using DBM models may be applicable and outline the tools and techniques used for model building and data assimilation. DBM modelling for flood forecasting Applicability There are essentially two types of flood warning situation. The first type is where the natural response time of a catchment is insufficient to allow warnings with an adequate lead time to be based on rainfall measurements alone. In this case, both early warnings and event timescale warnings are necessarily dependent on rainfall forecasts, either extrapolated from radar rainfall fields (often called quantitative precipitation forecasts) or from numerical weather prediction models. In both cases, ensembles of rainfall predictions may be available. The second is where the natural response time of a catchment is sufficient to allow a short‐term warning with useful lead time to be issued during an event. In this case, the short‐term warning could depend only on rain gauge or radar rainfall data (quantitative precipitation estimates). It might also be possible to make use of measured river flow or level data for real‐time updating of the forecasts. This will generally only be the case for forecast points in larger catchments, but the use of data assimilation might then be highly valuable for accuracy in the prediction of new events. In this second type of situation, rainfall forecasts will then be more useful in early warning/flood alert decisions with longer lead times as provided, for example, by the UK National Flood Forecasting Centre (Price et al ., ). While DBM modelling can be used for the first type (Smith and Beven, ; Alfieri et al ., ; Smith et al ., ), it is ideally suited to the second of these types because as outlined in the following sections, the hydrologically interpretable model can be readily expressed in a state space form that allows robust data assimilation using linear filtering techniques. It is this situation that is considered in this paper. Representation of uncertainties in the flood forecast process Following the flooding in the summer of 2007, the UK government commissioned an independent review of flood risk management and response practices headed by Sir Michael Pitt. In its final report, the Pitt review notes that accurate warning at a specific location: … can probably only be aspirational in some cases due to the uncertainty and complexity of natural systems such as rainfall and water flow. (Penning‐Rowsell et al ., , Paragraph 20.30 ) And that There is some evidence that the public is more tolerant of uncertainty which has been honestly admitted than is often believed, and acknowledging uncertainty often carries fewer dangers. (Frewer et al ., ; Penning‐Rowsell et al ., , Paragraph 20.12) To acknowledge uncertainty in flood forecasts requires recognition of both the sources of the uncertainty and how they are handled in the forecasting process. Such information offers a starting point for the use of uncertainty representations within the frameworks for flood response. For example, the experience of the Environment Agency in Exercise Watermark, which set out to test the incident management response to severe widespread flooding in England and Wales, suggests that providing best, worst and most likely scenarios to response staff may be beneficial (Symonds, ). There are a number of different sources of uncertainty. In particular, we may identify: uncertainty in observed rainfall fields, only partially captured by interpolation of rain gauge data and calibration of radar data; uncertainty in antecedent conditions, only partially captured by running hydrological models continuously; uncertainty in model structures in representing run‐off generation and routing processes; uncertainty in calibration of model parameters, only partially captured by estimation of parameter distributions and their dependence on uncertainty in observations used in model calibration or data assimilation (which may be particularly high at flood levels). The DBM modelling framework addresses these sources of uncertainty in a number of ways. The first two sources are addressed through the use of noise terms in the data assimilation scheme that are designed to reflect the error in the evolution of the model states. This term also compensates for inadequacy in the model structure and uncertainty in the calibration of the model. The deductive method of model selection (see the following section) allows for the exploration of potential model structures, albeit that in operational forecasting, only one structure is utilised, while the fourth source is treated by the implementation of models within a stochastic parameter and error framework. In what follows, we present models that aim to predict water levels directly rather than estimates of discharge (see also Beven et al ., , 2011a ; Leedal et al ., ; Romanowicz et al ., ). This reduces the effects of rating curve uncertainties at gauged sites, but clearly, such models cannot be constrained by mass balance. This can actually be an advantage for flood forecasting, when we do not necessarily expect to have good knowledge of rainfall input data from either rain gauges or radar, and where rating curves may be extrapolated well beyond the range of available measurements (such as the major uncertainty in estimating the peak discharge for the Carlisle 2005 flood, see Horritt et al ., ). Data requirements Table outlines the minimum data requirements for a DBM model representing a rainfall–water level relationship. Similar data periods are required for a DBM model representing flood routing driven by an observed upstream water level. Higher temporal resolution in the data (e.g. 5 min) may be of use in some catchments. This, though, may require a continuous time formulation of the models (Young and Garnier, ; Garnier et al ., ; Young, ) rather than the discrete time version presented in this paper. Data requirements for the DBM methodology to be applied in a catchment (minimal data requirements in bold) Parameter Resolution Format Time‐range Unit Precipitation 15 min, 1 h ASCII tables 10 significant events 5+ years mm Discharge or water level 15 min, 1 h ASCII tables 10 significant events including baseflow periods 5+ years m 3 /s, mm ASCII, American Standard Code for Information Interchange; DBM, data‐based mechanistic. Modelling is often possible using only rainfall records as inputs. Temperature may be required if snowmelt is a significant run‐off generation mechanism (Young et al ., ; Smith et al ., ), but further observations such as global radiation or wind speed are not required. Unlike most hydrological models, there is no requirement for physiographical information (such as a digital terrain model or land surface characteristics) beyond readily available metadata such as catchment maps showing rain gauges and river connectivity. Care should be taken in selecting and processing the data before modelling. Data should be representative of the current catchment dynamics and, if possible, represent the variability within the catchment (e.g. rain gauges at a range of elevations). The observed input and output data should be checked for consistency. Periods of data that display some inconsistency (e.g. high discharge and no rainfall, single time steps with high precipitation values but no run‐off response) should be considered carefully because they might either represent important catchment dynamics or data of dubious quality that should not be included in the model calibration (Beven et al ., 2011b ). Changes in the water level during low flow periods that might be the result of changes in channel form during floods should also be noted because the rainfall–water level dynamics may not be well captured if they are not taken into account. Building DBM forecast models A number of papers (see Romanowicz et al ., , ; Beven et al ., 2011a ; Young and the references within) address the construction of DBM models of hydrological systems ‘offline’ using historical data. This section summarises this work focusing upon the estimation of a model of the hydrological response at a single gauge (be it water level or discharge) using a single known input series (often a catchment average precipitation estimate or an upstream water level/discharge). Multiple input series (e.g. from multiple rain gauges) may be explicitly incorporated into the DBM model by natural extensions of the methodology discussed later (e.g. Young, ). The use of single or multiple input series will be dependent upon the nature and size of the catchment. Before proceeding with details, it is important to note that the philosophy behind DBM modelling is not only that of fitting a model that is (in some predefined sense) statistically optimal and utilising this as a black box in forecasting. Instead, while fitted to optimise some criterion and maintain parsimony, a DBM model must also have a mechanistic interpretation as a representation of the physical system, for example, in defining pathways of response or feedback loops (see the later examples). Checking the validity of this interpretation against a perceptual model of the system gives some confidence that the predictions may be acceptable in situations not represented within the calibration data. The foundation of most DBM models is the linear transfer function. Eqn describes a discrete time linear transfer function between the observed output y = ( y 1 , … , y T ) and input u = ( u 1 , … , u T ), indexed by time, using the backward shift operator z −1 ( z −1 y t = y t −1 ) where v is a constant offset, ε t is a stochastic disturbance and t is the time step index. For the purposes of this paper, we consider v to be indicative of a baseflow output value so that the remainder of the model characterises the supposition of the event hydrographs. 1 y t − v = b 0 + b 1 z − 1 + … + b m z − m 1 + a 1 z − 1 + … + a n z − n u t − d + ε t The structure of the model is defined by three positive integers. Two of these, m and n , are the order of the transfer function numerator and denominator polynomials, respectively. The third d is number of time steps of pure, advective time delay. This limits the forecast lead time available from the observed input data. In some situations, a linear transfer function, coupled with data assimilation, may prove adequate for forecasting the future response of the system (see for example Lees et al ., ). More commonly, the nonlinear problems found in flood forecasting require the use of some additional model complexity. This usually takes the form of a nonlinear transformation of the input signal. This transformation of the input is often dependent on the current state of the system. A convenient means of representing this transformation (Young and Beven, ) is as a gain series f = ( f 1 , … , f T ), allowing the resulting model to be expressed as: 2 y t − v = b 0 + b 1 z − 1 + … + b m z − m 1 + a 1 z − 1 + … + a n z − n f t − d u t − d + η t where η t is a stochastic noise term. Models of this form using a number of different methods to calculate f have proved acceptable representations of a variety of catchments (see for example Young, ; Beven et al ., , ; Romanowicz et al ., , ; Ratto et al ., ; Leedal et al ., ; Smith et al ., , ). In applying the model outlined in Eqn , there are two challenges. The identification problem is that of selecting ( n , m , d ) and a suitable function F for computing f . The estimation problem is then that of estimating ( a 1 , … , a n , b 0 , … , b m , v ) and any parameters of the function F that are denoted as φ . The approaches taken to these problems are outlined in more detail in the following subsections with reference to the following commonly used sequence of steps: identify and estimate a linear transfer function using the observed input–output data; explore the properties of f using the nonparametric state dependent parameter (SDP) algorithm (Young et al ., ); if required, propose a function F to capture any state dependency; estimate φ along with the parameters of linear transfer function identified in Step 1; using the values of φ estimated in the previous step, compute an effective input series v = ( f 1 , u 1 , … , f T , u T ), using this as the input series repeat Step 1; if the order of the linear transfer function identified has altered, repeat the previous steps. Identification and estimation of a linear transfer function Consider the identification and estimation of the linear transfer function in Eqn as part of Step 1 of the previous sequence. Presuming v is given [often specified as min t = 1 , … , T ( y t ) ], a number of tools are available for the estimation of the parameters θ = ( a 1 , … , a n , b 0 , … , b m ) (see for example Ljung, ; Poskitt, ; Young, ). Within the DBM methodology, the most commonly used is the refined instrumental variable (RIV) method introduced in Young ( ; see Young for a more recent description). This provides a robust estimation methodology that produces reasonable parameter estimates when the error series has unknown, possibly nonstationary and heteroscedastic error properties. The RIV algorithm, as encoded in the Computer‐Aided Program for Time Series Analysis and Identification of Noisy Systems (CAPTAIN) toolbox (Taylor et al ., ), is very computationally efficient (typical evaluation times even on large data sets are less than 10 s). Identification of the model order ( n , m , d ) can therefore be performed by an exhaustive search over a range of plausible structures specified by the modeller. The identified model order is chosen by the modeller to provide a trade‐off between the fit of the model, the parsimony of the model structure, and the plausibility of the mechanistic interpretation of both the input non‐linearity and the transfer function in the DBM model (see for example Romanowicz et al ., ). The most common mechanistic interpretation of a linear transfer function seen in a flood forecasting context is that of parallel response paths. For example, a model with ( n , m ) = (2, 1) can be decomposed using partial fractions to give: 3 b 0 + b 1 z − 1 1 + a 1 z − 1 + a 2 z − 2 = β 1 1 + α 1 z − 1 + β 2 1 + α 2 z − 1 The first‐order terms represent two parallel tanks. These can be analysed in terms of their time constants (≈Δ t / log (− α i ) where Δ t is the model time step) and steady state gains ( β i /(1 + α i ), which should be greater than 0) to ensure that they are consistent with the modeller's concept of the system response. Note that higher order transfer functions can be decomposed in more than one way and that only one of the decompositions may have a mechanistic interpretation. Alternative decompositions can be found in, for example, Young et al . ( ). Identification and estimation of the input nonlinearity In keeping with the inductive approach prevalent in DBM modelling, the function F defining f is not presumed a priori. Instead, an empirical form of the function is derived by nonparametric analysis of the data. This allows the data to take precedent over the prior assumptions of the model builder by providing the modeller with a guide for specifying F (as well as some justification for pursuing this approach). The SDP tools that form a component of the CAPTAIN toolbox provide one method for proposing the empirical function form. The SDP algorithm has been described in detail in several papers (see for example Young et al ., ) and its use critiqued in Beven et al . ( 2011a ). In essence, the SDP algorithm iterates between fitting the linear transfer function and a nonparametric regression describing the dependency of one or more of the transfer function parameters on a known state. Because the SDP algorithm is only required to provide a guide for a subsequently optimised parameterisation scheme, it is not usually necessary to include a high‐order transfer function model structure within the algorithm set‐up. A standard method is to utilise a first‐order transfer function with a value of d taken from the identified linear model from Step 1. The state dependency of the single numerator parameter is then investigated. This arrangement often produces a good indication of the state‐dependent relationship linking observed output to the effective input series without the need to disentangle state dependencies among multiple SDPs. While the SDP algorithm provides a powerful tool for exploring the specification of F, two key challenges arise because of the nature of the system being studied. First, because of the rarity of extreme flood events and the difficulties in observing them, there are generally severe limitations in the data available for the crucial flood‐inducing levels. The identification of an input nonlinearity function at flood levels is therefore often reliant on a small number of data points, where both the state variable and the observation of the output may be subject to significant error. In forecasting, this can, at least to some extent, be compensated by the online data assimilation considered in the next section. The second challenge relates to the specification of the state variable. In several studies (for example Lees et al ., ; Romanowicz et al ., , ) where the time delay in the linear transfer function ( d ) has provided a suitable forecast lead time, the observed output has been used as the state variable in the belief that it provides a suitable surrogate for the catchment wetness. Alternative representations of catchment wetness, for example, an antecedent precipitation index (Smith and Beven, ) or the output of the optimal linear model from Step 1 (Laurain et al ., ), can also be considered. Potential parametric description of the input nonlinearity ranges from the comparatively simple, such as the power law: 4 f t = F ( y t ​ , φ ) = y t φ 0 < φ < 1 to more complex descriptions such as piecewise cubic Hermite interpolating polynomial splines or radial basis functions (e.g. Beven et al ., 2011a ). In some cases, such as when snowmelt influences the catchment run‐off, the nonparametric analysis may suggest that a simple physical model is appropriate (e.g. the degree day model used to predict snowmelt inputs in Smith et al ., ). Having proposed one or more potential formulation for the input nonlinearity, the parameters of the function require optimisation. This is typically performed using a numerical optimisation algorithm to select φ , which minimises a sum of squared errors criterion. For each trialled value of φ in the search, a corresponding θ is estimated using the instrumental variable methods discussed in the previous section. Following optimisation, the values of φ and θ should be checked to ensure that the model can be interpreted in a mechanistic fashion. Data assimilation within DBM A very important aspect of improving flood forecasts when knowledge of both inputs and appropriate models is subject to error (and uncertainty) is the use of data assimilation or adaptive forecasting. In what follows, we consider the assimilation of observations of the forecast water level variable to improve the forecasts of the observed hydrological response. The structure of the DBM model indicates that forecasts at lead times within the catchment lag d can be generated with observed input data. Forecasts at longer lead times require the use of forecast input data. If determinist forecasts, ideally revised every time step are available (e.g. a point summary of a probabilistic DBM model forecast), they can be substituted for the unknown inputs and similar techniques to those used below utilised (see Romanowicz et al ., ). This technique is used in the latter case studies for cascading the model forecasts. More complex situations where there are ensemble forecasts of the input are not considered here (but see Smith and Beven, ; Smith et al ., ). A DBM model can be cast within a general state space framework for nonlinear models (e.g. Liu and Gupta, ). The assimilation of observations of the output variable and generation of future forecasts consists of a filtering problem with components. The first is deriving the probability distribution of the states at time t + 1, given their distribution at time t and new pair of observations ( y t +1 , u t +1− d ). The second is to evaluate the predictive distribution of an observation n steps ahead by mapping of the stochastic noise terms given the initial distribution of the model states. Here, a Kalman filter approach is adopted in data assimilation. Extended forms of the Kalman filter have been developed for DBM models (Smith et al ., ). However, experience suggests that in this type of application, where the effective input can be computed from observed data, a state space formulation of the linear transfer function embedded in a simple predictor‐corrector implementation of the linear Kalman filter performs well. The mechanistic interpretation of the linear transfer function often allows the states of its spate space formulation to be defined in a meaningful way, for example, as a representation of the water following each of the parallel response paths. The variance of the system noise, which corrupts the evolution of the states in time, therefore gives an indication of the error about the different types of response. These variance terms, along with that of the observational noise, are often estimated by maximising the likelihood of the observed water levels based on forecasts at a given lead time (see Smith et al ., ). Considering one or more of the variance terms as time varying or dependent upon the magnitude of the forecast may be beneficial (Young, ; Smith et al ., ). Implementation in the E nvironment A gency of E ngland and W ales NFFS The DBM model scheme was implemented as an operational module within the UK NFFS, which is based on the Deltares DELFT‐FEWS forecasting framework (Werner et al ., ). The approach takes advantage of the open framework that allows DELFT‐FEWS/NFFS to be loosely coupled to any suitable forecasting model (Werner et al ., ). A list of models that have been coupled is provided in Weerts et al . ( ). Lancaster Environment Centre and Deltares have developed an implementation of the DBM methodology for online forecasting in the R open source statistical programming language (Ihaka and Gentleman, ) that can be coupled with the NFFS (Leedal et al ., ). This implementation was designed to be a modular, reusable framework that would easily scale for use at new sites and catchments in the future. It should facilitate advances in the use of DBM hydrological models. The framework for the DBM module prescribes a standard directory structure for holding the set of functions that process data, assimilate data, perform forecasting and write results to file. The structure is shown in Figure . Within the DELFT‐FEWS framework, the format of calls to a function is prescribed, i.e. each function must receive the expected format of variables. However, each model retains its own version of the main processing functions. This approach provides a compromise between code reuse, maintenance and flexibility. If a function is common to several models, it is straightforward to copy that function to each model. However, if a model requires a unique processing method, this is also possible. File structure for the data‐based mechanistic ( DBM ) module. is specific to a catchment, < ID > is specific to a forecast model within the catchment. For clarity, this structure shows a catchment with only a single model; in most cases, a catchment will contain several Folder< ID > directories. During real‐time operation: The required input data are retrieved from the NFFS database, are labelled with the station name and are stored in the Work directory of the DBM module. The past state of the model and data assimilation algorithm is loaded from the States directory. One cycle of forecast followed by assimilation is performed using the new data point that has been passed during the present time point. The new state of the model and data assimilation algorithm is saved in the State folder. The model iterates over the remaining input data to produce a segment of forecast. This is saved in the Work directory. The NFFS interface retrieves this section and incorporates it into the main NFFS database. Once the DBM module forecasts have been appended to the NFFS database, the usual tools are available to generate reports, statistics and visualisations (Werner et al ., ). R iver E den case study The DBM forecasting approach was applied to the Lower Eden catchment in Cumbria, UK. At the Sheepmount gauging station downstream of Carlisle, which is the lowest monitored point in the catchment, the total area is 2400 km 2 . The River Eden and its principle tributaries, the Rivers Eamont, Irthing, Petteril and Caldew, link a number of significant areas for run‐off generation (Figure ). These include the northern and eastern parts of the Lake District peaks (covering peaks such as Skiddaw and Helvellyn) and the areas of the Northern Pennines and Kielder Forest. These areas are typified by high annual rainfall totals and steep terrain. Flood warning lead times greater than 6 h may require the use of forecast precipitation. There are 31 river gauging stations and 16 telemetred rain gauges in the catchment. Map of the R iver E den around C arlisle (shaded area) showing river confluences and gauging sites. Inset map shows the location of the catchment in the UK . Using the DBM methods outlined, a simplified network of six submodels were identified and estimated using hourly data from October 2003 to March 2005. This includes the January 2005 Carlisle flood event. Figure illustrates the network of DBM submodels that represent either individual tributaries or reaches of the River Eden. Details of the structure of the DBM submodels are given in Table . The use of a semidistributed network of DBM models rather than a single input‐output model terminating at Sheepmount allows for the possibly complex spatial distribution of rainfall patterns within the catchment. The network approach also provides two related benefits: The network can provide forecasts at intermediate flood risk locations. The intermediate locations correspond to telemetry gauge sites, thus facilitating the assimilation of multiple streams of data during real‐time forecasting and providing some degree of robustness against station malfunction. In testing, the real‐time DBM model network with data assimilation performed well. Figure shows an example of the output generated from within the NFFS software, in this case for forecasts of up to 6 h lead time at Sheepmount. The handling of DBM models within NFFS has been configured to take advantage of the stochastic information available about the forecast with the best estimate forecast and 95% prediction confidence interval being shown. Data‐based mechanistic ( DBM ) model network of the lower E den catchment including the R ivers I rthing ( G reenholme), C aldew ( D enton H olme) and P etteril ( H arraby G reen). Locations mentioned in either river gauging stations or telemetered rain gauges (italic). Lower E den data‐based mechanistic forecasting model implemented within N ational F lood F orecasting S ystem. Results show 6‐h forecast ± 95% confidence intervals. A summary of the constructed models for the R iver E den Predicted location Observed inputs Nonlinearity f t Model order ( n m d ) Maximum lead time of forecast (hours) Great Corby Temple Sowerby, Udford PCHIP spline (2 2 4) 4 Greenholme High Row, Harescough, Crewe Fell, Aisgill PCHIP spline (2 2 6) 6 Linstock Great Corby, Greenholme PCHIP spline (1 1 1) 5 Denton Holme Carlisle, Geltsdale, Mosedale PCHIP spline (2 2 5) 5 Harraby Green Carlisle, Geltsdale, Mosedale PCHIP spline (1 2 5) 5 Sheepmount Great Corby, Greenholme PCHIP spline (1 1 1) 6 Locations in italics are rain gauges; the remainder are river level gauging stations. PCHIP, piecewise cubic Hermite interpolating polynomial. Validation A key component of any forecasting system is the continual assessment of operational models. To this end, it is informative to test the performance of the Eden DBM configuration using data extending to January 2010. The analysis was carried out in such a way as to simulate exactly the results that would have been generated had the set of models been run in operational mode. Figure shows results for the two largest events during the validation period for both 2 and 5 h forecast lead times. Lower E den data‐based mechanistic forecasting model for S heepmount showing the two largest events during the 2005–2010 validation period (left: 12 J anuary 2009, right: 20 N ovember 2009). Upper panels show simulated 5 h forecast; lower panels show 2 h forecast. The red line represents the observed water level, solid black line the expected value of the prediction, with the shaded area encompassing the 95% confidence interval for the prediction. For comparison with Figure , the dashed green and blue horizontal lines show the Environment Agency warning thresholds with codes RES FW NC3A and ACT OPS , respectively. The performance of the DBM model forecasts for Sheepmount can be summarised using the familiar probability of detection (POD) and false alarm rate (FAR) statistics. Here, the 99th percentile of the observed data (3.4 m) was used as the minimum flood event to include in the study. This allowed 27 events to be considered for POD/FAR during the validation period. The POD and FAR statistics for 2 and 5 h lead times are summarised in Table . POD and FAR are usually associated with deterministic model outputs. They can therefore only be evaluated for deterministic summaries of the probabilistic forecasts given by the DBM model. Results are presented both for the best estimate forecast, in this case the median value of the forecast distribution and the deterministic summaries resulting from taking the value at each time step, which according to the forecast has a 16% or 2.5% probability of being exceeded. These correspond to the mean value of the forecast plus one or two standard deviations, respectively. A possible application of these methods is to issue warnings at levels of confidence appropriate to the site risk or vulnerability. As can be seen in Table , the number of false alarms logically increases, as FAR is assessed at more cautious values (lower probabilities of exceedance at each time step). The high level of POD for the best estimate of the forecast value indicates that the model may be a little over‐cautious. This should be investigated in future refinements of the model scheme. Sheepmount POD and FAR statistics for 2 and 5 h (in parentheses) forecast lead times calculated using 2.9 m threshold as lower limit on peaks to include Statistic Mean Mean + 1 SD Mean + 2 SD POD 100% (100%) 100% (100%) 100% (100%) FAR 20% (36.8%) 29.4% (52%) 36.8% (58.6%) FAR, false alarm rate; POD, probability of detection; SD, standard deviation. The estimated error bounds should encompass the correct proportion of the data. This can be verified by investigating the ratio of observations inside and outside the forecast error bounds. Table presents these results for 2 and 5 h forecast lead times. Ideally, these values should be close to 32% and 5% for 1 and 2 standard deviations, respectively. These ratios tend to breakdown increasingly as more low flow data are removed from the sample. Because for flood forecasting, it is the high water levels we are interested in forecasting correctly (with appropriate error bounds), this highlights the importance of considering summary statistics calculated on appropriate subsamples of the entire observation record. Model efficiency and percentage of observations falling outside the estimated forecast uncertainty range for 2 and 5 h (in parentheses) forecast lead times at S heepmount Threshold (m) Number of observations Nash–Sutcliffe model efficiency for mean Observations outside 1 SD Observations outside 2 SD 0 35 007 0.98 (0.92) 17.1% (51.3%) 4.9% (17.9% 1 7 107 0.96 (0.88) 19.5% (34.3%) 7.1% (11.3%) 2 1 142 0.86 (0.59) 46.9% (50.0%) 21.0% (20.4%) 3 302 0.43 (−0.54) 73.2% (59.6%) 34.1% (24.8%) 4 32 0.62 (0.05) 68.7% (59.3%) 21.9% (12.5%) SD, standard deviation. It is also important to consider error in event timing as well as magnitude. While timing error is not available to the user in real time, it can be assessed postevent. Ideally, the point at which the observations cross a threshold level should fall within the period where the upper and lower bounds of a given prediction confidence interval intersect the threshold level. This is often a challenging performance measure, as the rising limb of a large flood event is difficult to forecast accurately. Table shows the results for 2 and 5 h forecast lead times at Sheepmount. Number of threshold crossings where the timing of the observed crossing lies within the two standard deviation ranges of the forecast Forecast lead time (hours) Number of crossings Number inside 2 SD of timing uncertainty 2 12 8 5 12 3 Data are for Sheepmount incorporating 11 RES FW NC3A and 1 ACT OPS threshold crossings (see Figure ). RES FW NC3A and ACT OPS are warning threshold codes. SD, standard deviation. R iver S evern case study A map of the upper River Severn, along with the gauges used in this study, is shown in Figure . The River Severn rises in the Cambrian mountains at a height of 741 m above ordnance datum (mAOD) and flows north‐eastwards through Abermule before meeting the Vyrnwy tributary upstream of Shrewsbury. Around the confluence of the Vyrnwy tributary, the valley is wide and flat with a considerable extent of floodplain. The river then flows through Montford to Shrewsbury and is joined at Montford Bridge by the River Perry that flows from the North. The lowermost point in the upper Severn catchment is defined as the gauge at Welsh Bridge in Shrewsbury. Average annual rainfall can exceed 2500 mm in the Cambrian mountains and the upper reaches of the Vyrnwy catchment. There are no significant flow control structures in the catchment except for Lake Vyrnwy in the upper reaches of Vyrnwy and the Clywedog Reservoir in the upper reaches of Severn. Map of the upper R iver S evern showing significant urban areas (shaded), river gauges (circles) and rain gauges (triangles). Inset map shows the location of the catchment in the UK . Flood events that have caused property flooding in the upper Severn catchment in recent years include Easter 1998, October 1998, October 2000, February 2002, February 2004 and January 2009, with the October 1998 event particularly severe in the upper Severn. DBM models for a simplified version of the upper Severn system were constructed in line with the methodology described earlier. A model of the system extending further downstream can be found in Romanowicz et al . . Three river level sites are modelled: Meifod on the Vyrnwy, Abermule on the Severn upstream of the Vyrnwy confluence and Welsh Bridge. These sites represent the outlets of the two main flow‐generating regions (Meifod and Abermule) and an observed location in the region of high flood risk (Welsh Bridge). Intermediate gauged sites between the Vyrnwy confluence and Welsh Bridge could be modelled at the cost of having to represent more complex floodplain interactions (e.g. Smith et al ., ). A schematic of the model is shown in Figure , with details of the models given in Table . The maximum lead time at Welsh Bridge can be extended by utilising the predictions at the two upstream locations as inputs to the model of this site. Data‐based mechanistic model network of the R iver S evern catchment including the R iver V yrnwy ( M eifod). Locations mentioned are either river gauging stations or telemetered rain gauages (italic). A summary of the constructed models for the R iver S evern Predicted location Observed inputs Non‐linearity f t Model order [ n m d ] Maximum lead time of forecast (hours) Abermule Cefn Coch, Dolydd, Pen‐y‐Coed y t 0.12 (2 2 5) 5 Meifod Llanfyllin, Pen‐y‐Coed y t 0.13 (2 2 2) 2 Welsh Bridge Meifod, Abermule None (2 2 16) 18 Locations in italics are rain gauges; the remainder are river level gauging stations. Calibration was performed using hourly data from 1 January 1999 to 31 December 1999. A single input series for each model was generated using a weighted sum of the observed inputs. These weights were optimised alongside the parameters of the nonlinearity. Figure shows a number of forecasts for the October 2000 flood event. The forecasts for Abermule and Meifod suggest that the nonlinearities identified in the calibration are suitable for modelling this event. As more data become available, recalibration of the nonlinearity and re‐identification of the linear transfer function structure may add additional benefits, particularly in improving the timing of the rising limb of the hydrograph. Validation forecasts for the O ctober 2000 flood on the R iver S evern for (a) A bermule with 5 h lead time; (b) M eifod with 3 h lead time; (c) W elsh B ridge with 6 h lead time; (d) W elsh B ridge with 16 h lead time. The red line represents the observed water level, solid black line the expected value of the prediction with the shaded area encompassing the 95% confidence interval for the prediction. The dashed horizontal lines show Environment Agency warning thresholds labelled in this work: Flood watch 103 (cyan), Flood warning (green) and Severe flood warning (blue). In the forecasts for Welsh Bridge, it is evident that as the lead time of the forecast decreases, the predictions increase in accuracy. The results at the longer lead time, where the effect on the forecast of the most recently assimilated observation is lower, suggest that the dynamics of this part of system may not be well represented by the model. These difficulties may be partially due to backwater effects or interaction with the flood plain that may be better represented by a more complex model structure (Smith et al ., ). Validation The performance of the River Severn DBM configuration is tested using data between January 2005 and August 2009. As with the Eden case study, the analysis was carried out in such a way as to simulate exactly the results that would have been generated had the set of models been run in operational mode. The POD and FAR statistics for 2 and 5 h lead times are summarised in Table . These were computed using the 46 events greater than 49 mAOD at Welsh Bridge during the validation period. As with the River Eden model, the high level of POD indicates that the model may be a little overcautious, and again, this should be investigated in future refinements of the model scheme. W elsh B ridge POD and FAR statistics for 2 and 5 h (in parentheses) forecast lead times calculated using 49 mAOD threshold as lower limit on peaks to include Statistic Mean Mean + 1 SD Mean + 2 SD POD 91.5% (100%) 100% (100%) 100% (100%) FAR 8.5% (50%) 15.1% (50.5%) 29.7% (63.3%) FAR, false alarm rate; mAOD, m above ordnance datum; POD, probability of detection; SD, standard deviation. Table presents results for the percentage of observations bracketed by the prediction intervals at 2 and 5 h forecast lead times. The desired percentages breakdown increasingly as more low flow data are removed from the sample. The results here suggest the distribution of forecast errors do not fit neatly within the Gaussian assumptions used within calibration of the variance terms within the data assimilation strategy. Alternative calibration strategies (e.g. Smith et al ., ) that are not reliant on the Gaussian assumptions have proved useful in such situations but do not currently form part of the UK NFFS deployment. Model efficiency and percentage of observations falling outside the estimated forecast uncertainty range for 2 h forecast lead time at W elsh B ridge Threshold (m) Number of observations Nash–Sutcliffe model efficiency for mean Observations outside 1 SD Observations outside 2 SD 47 40 152 1.00 (0.98) 9.7% (19.7%) 2.0% (5.4%) 48 13 656 0.99 (0.95) 16.7% (30.0%) 4.1% (10.8%) 49 3 918 0.97 (0.80) 22.8% (38.1%) 7.7% (15.7%) 50 661 0.95 (0.70) 26.9% (49.8%) 3% (15.1%) 51 135 0.59 (−1.45) 20.7% (51.1%) 0% (3.7%) SD, standard deviation. The performance of the model in forecasting the timing of rising limbs of the hydrograph is shown in Table . It shows that the model struggles to capture the timing of the rising limb, particularly at the highest water levels. This is keeping with the limitations of the representation of the system dynamics seen in Figure . Number of threshold crossings where the timing of the observed crossing lies within the two standard deviation ranges of the forecast Threshold Number of crossings Number inside 2 SD of timing uncertainty for 2 h lead time Number inside 2 SD of timing uncertainty for 5 h lead time Flood watch 54 41 28 Flood warning 42 22 18 Severe flood warning 6 2 0 Data are for 2 and 5 h forecast at Welsh Bridge. SD, standard deviation. Conclusions The outlined DBM modelling methodology for forecasting is effective in a flooding context as witnessed by the two example applications to UK rivers. Assimilation of observed river gauge data is important in improving forecasts of future water level or discharge. This is achieved in the DBM framework using an efficient computational scheme. The implementation of real‐time DBM forecasting models within the UK NFFS is demonstrated. Analysis of the forecasts produced for a validation period at two test sites shows that the Gaussian error assumptions used in the calibration of the data assimilation scheme are not valid. The forecast error bounds should therefore be considered as indicative because of the failure to characterise the underlying uncertainty in the forecasting process. However, the error bounds can be considered to contain information not present in a deterministic forecast. This is of use in helping operational forecasters issue flood warnings. The forecast results shown in this paper highlight why ongoing validation is important. This provides the basis for updating the forecasting models, something that can be readily achieved within the DBM framework outlined. It also allows the duty officer(s) responsible for issuing flood warnings opportunity to revise their incorporation of the additional information present in the probabilistic forecast in the forecast chain. Acknowledgements Application of the DBM methodology to flood forecasting was first carried out for the River Nith catchment in 1991 under contract to the Solway River Purification Board in 1991. More recent development work was supported by the Phase 1 of the Engineering and Physical Science Research Council‐funded Flood Risk Management Research Consortium and the European Commission Seventh Framework Programme project IMproving Preparedness and RIsk maNagemenT for flash floods and debriS flow events (IMPRINTS). This work and implementation within the NFFS were supported by the Environment Agency project SC080030 on probabilistic flood forecasting. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Flood Risk Management Wiley

Testing probabilistic adaptive real‐time flood forecasting models

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Wiley
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Copyright © 2014 The Chartered Institution of Water and Environmental Management (CIWEM) and John Wiley & Sons Ltd
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1753-318X
DOI
10.1111/jfr3.12055
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Abstract

Introduction The ability to issue flood warnings in a timely manner is an important part of mitigating the impact of flooding. In modern flood management practice, the issuing of warnings is guided by numerical models for forecasting future water levels or discharge. While one‐ and two‐dimensional hydrodynamic models of flood inundation are useful for management and planning purposes to issue warnings, it is often adequate to forecast only at specific points of interest where observations (either of water level or discharge) are available to evaluate the model forecasts. Extension to other river reaches where the risk of overtopping defences is significant is then generally related to different threshold warning levels at the forecast site. This paper outlines a method for producing forecasts at locations with observations based on time series models derived from available data. The models used, termed data‐based mechanistic (DBM) models (Young, ), are parsimonious time series models that can be interpreted in a mechanistic fashion (see for example Young and Beven, ; Young, , , ). These models are combined with an efficient data assimilation strategy to produce probabilistic forecasts of future observations. The implementation of these models in the UK National Flood Forecasting System (NFFS) is outlined using two case studies, the River Eden and River Severn. Before presenting the case studies, the following sections discuss where flood forecasting using DBM models may be applicable and outline the tools and techniques used for model building and data assimilation. DBM modelling for flood forecasting Applicability There are essentially two types of flood warning situation. The first type is where the natural response time of a catchment is insufficient to allow warnings with an adequate lead time to be based on rainfall measurements alone. In this case, both early warnings and event timescale warnings are necessarily dependent on rainfall forecasts, either extrapolated from radar rainfall fields (often called quantitative precipitation forecasts) or from numerical weather prediction models. In both cases, ensembles of rainfall predictions may be available. The second is where the natural response time of a catchment is sufficient to allow a short‐term warning with useful lead time to be issued during an event. In this case, the short‐term warning could depend only on rain gauge or radar rainfall data (quantitative precipitation estimates). It might also be possible to make use of measured river flow or level data for real‐time updating of the forecasts. This will generally only be the case for forecast points in larger catchments, but the use of data assimilation might then be highly valuable for accuracy in the prediction of new events. In this second type of situation, rainfall forecasts will then be more useful in early warning/flood alert decisions with longer lead times as provided, for example, by the UK National Flood Forecasting Centre (Price et al ., ). While DBM modelling can be used for the first type (Smith and Beven, ; Alfieri et al ., ; Smith et al ., ), it is ideally suited to the second of these types because as outlined in the following sections, the hydrologically interpretable model can be readily expressed in a state space form that allows robust data assimilation using linear filtering techniques. It is this situation that is considered in this paper. Representation of uncertainties in the flood forecast process Following the flooding in the summer of 2007, the UK government commissioned an independent review of flood risk management and response practices headed by Sir Michael Pitt. In its final report, the Pitt review notes that accurate warning at a specific location: … can probably only be aspirational in some cases due to the uncertainty and complexity of natural systems such as rainfall and water flow. (Penning‐Rowsell et al ., , Paragraph 20.30 ) And that There is some evidence that the public is more tolerant of uncertainty which has been honestly admitted than is often believed, and acknowledging uncertainty often carries fewer dangers. (Frewer et al ., ; Penning‐Rowsell et al ., , Paragraph 20.12) To acknowledge uncertainty in flood forecasts requires recognition of both the sources of the uncertainty and how they are handled in the forecasting process. Such information offers a starting point for the use of uncertainty representations within the frameworks for flood response. For example, the experience of the Environment Agency in Exercise Watermark, which set out to test the incident management response to severe widespread flooding in England and Wales, suggests that providing best, worst and most likely scenarios to response staff may be beneficial (Symonds, ). There are a number of different sources of uncertainty. In particular, we may identify: uncertainty in observed rainfall fields, only partially captured by interpolation of rain gauge data and calibration of radar data; uncertainty in antecedent conditions, only partially captured by running hydrological models continuously; uncertainty in model structures in representing run‐off generation and routing processes; uncertainty in calibration of model parameters, only partially captured by estimation of parameter distributions and their dependence on uncertainty in observations used in model calibration or data assimilation (which may be particularly high at flood levels). The DBM modelling framework addresses these sources of uncertainty in a number of ways. The first two sources are addressed through the use of noise terms in the data assimilation scheme that are designed to reflect the error in the evolution of the model states. This term also compensates for inadequacy in the model structure and uncertainty in the calibration of the model. The deductive method of model selection (see the following section) allows for the exploration of potential model structures, albeit that in operational forecasting, only one structure is utilised, while the fourth source is treated by the implementation of models within a stochastic parameter and error framework. In what follows, we present models that aim to predict water levels directly rather than estimates of discharge (see also Beven et al ., , 2011a ; Leedal et al ., ; Romanowicz et al ., ). This reduces the effects of rating curve uncertainties at gauged sites, but clearly, such models cannot be constrained by mass balance. This can actually be an advantage for flood forecasting, when we do not necessarily expect to have good knowledge of rainfall input data from either rain gauges or radar, and where rating curves may be extrapolated well beyond the range of available measurements (such as the major uncertainty in estimating the peak discharge for the Carlisle 2005 flood, see Horritt et al ., ). Data requirements Table outlines the minimum data requirements for a DBM model representing a rainfall–water level relationship. Similar data periods are required for a DBM model representing flood routing driven by an observed upstream water level. Higher temporal resolution in the data (e.g. 5 min) may be of use in some catchments. This, though, may require a continuous time formulation of the models (Young and Garnier, ; Garnier et al ., ; Young, ) rather than the discrete time version presented in this paper. Data requirements for the DBM methodology to be applied in a catchment (minimal data requirements in bold) Parameter Resolution Format Time‐range Unit Precipitation 15 min, 1 h ASCII tables 10 significant events 5+ years mm Discharge or water level 15 min, 1 h ASCII tables 10 significant events including baseflow periods 5+ years m 3 /s, mm ASCII, American Standard Code for Information Interchange; DBM, data‐based mechanistic. Modelling is often possible using only rainfall records as inputs. Temperature may be required if snowmelt is a significant run‐off generation mechanism (Young et al ., ; Smith et al ., ), but further observations such as global radiation or wind speed are not required. Unlike most hydrological models, there is no requirement for physiographical information (such as a digital terrain model or land surface characteristics) beyond readily available metadata such as catchment maps showing rain gauges and river connectivity. Care should be taken in selecting and processing the data before modelling. Data should be representative of the current catchment dynamics and, if possible, represent the variability within the catchment (e.g. rain gauges at a range of elevations). The observed input and output data should be checked for consistency. Periods of data that display some inconsistency (e.g. high discharge and no rainfall, single time steps with high precipitation values but no run‐off response) should be considered carefully because they might either represent important catchment dynamics or data of dubious quality that should not be included in the model calibration (Beven et al ., 2011b ). Changes in the water level during low flow periods that might be the result of changes in channel form during floods should also be noted because the rainfall–water level dynamics may not be well captured if they are not taken into account. Building DBM forecast models A number of papers (see Romanowicz et al ., , ; Beven et al ., 2011a ; Young and the references within) address the construction of DBM models of hydrological systems ‘offline’ using historical data. This section summarises this work focusing upon the estimation of a model of the hydrological response at a single gauge (be it water level or discharge) using a single known input series (often a catchment average precipitation estimate or an upstream water level/discharge). Multiple input series (e.g. from multiple rain gauges) may be explicitly incorporated into the DBM model by natural extensions of the methodology discussed later (e.g. Young, ). The use of single or multiple input series will be dependent upon the nature and size of the catchment. Before proceeding with details, it is important to note that the philosophy behind DBM modelling is not only that of fitting a model that is (in some predefined sense) statistically optimal and utilising this as a black box in forecasting. Instead, while fitted to optimise some criterion and maintain parsimony, a DBM model must also have a mechanistic interpretation as a representation of the physical system, for example, in defining pathways of response or feedback loops (see the later examples). Checking the validity of this interpretation against a perceptual model of the system gives some confidence that the predictions may be acceptable in situations not represented within the calibration data. The foundation of most DBM models is the linear transfer function. Eqn describes a discrete time linear transfer function between the observed output y = ( y 1 , … , y T ) and input u = ( u 1 , … , u T ), indexed by time, using the backward shift operator z −1 ( z −1 y t = y t −1 ) where v is a constant offset, ε t is a stochastic disturbance and t is the time step index. For the purposes of this paper, we consider v to be indicative of a baseflow output value so that the remainder of the model characterises the supposition of the event hydrographs. 1 y t − v = b 0 + b 1 z − 1 + … + b m z − m 1 + a 1 z − 1 + … + a n z − n u t − d + ε t The structure of the model is defined by three positive integers. Two of these, m and n , are the order of the transfer function numerator and denominator polynomials, respectively. The third d is number of time steps of pure, advective time delay. This limits the forecast lead time available from the observed input data. In some situations, a linear transfer function, coupled with data assimilation, may prove adequate for forecasting the future response of the system (see for example Lees et al ., ). More commonly, the nonlinear problems found in flood forecasting require the use of some additional model complexity. This usually takes the form of a nonlinear transformation of the input signal. This transformation of the input is often dependent on the current state of the system. A convenient means of representing this transformation (Young and Beven, ) is as a gain series f = ( f 1 , … , f T ), allowing the resulting model to be expressed as: 2 y t − v = b 0 + b 1 z − 1 + … + b m z − m 1 + a 1 z − 1 + … + a n z − n f t − d u t − d + η t where η t is a stochastic noise term. Models of this form using a number of different methods to calculate f have proved acceptable representations of a variety of catchments (see for example Young, ; Beven et al ., , ; Romanowicz et al ., , ; Ratto et al ., ; Leedal et al ., ; Smith et al ., , ). In applying the model outlined in Eqn , there are two challenges. The identification problem is that of selecting ( n , m , d ) and a suitable function F for computing f . The estimation problem is then that of estimating ( a 1 , … , a n , b 0 , … , b m , v ) and any parameters of the function F that are denoted as φ . The approaches taken to these problems are outlined in more detail in the following subsections with reference to the following commonly used sequence of steps: identify and estimate a linear transfer function using the observed input–output data; explore the properties of f using the nonparametric state dependent parameter (SDP) algorithm (Young et al ., ); if required, propose a function F to capture any state dependency; estimate φ along with the parameters of linear transfer function identified in Step 1; using the values of φ estimated in the previous step, compute an effective input series v = ( f 1 , u 1 , … , f T , u T ), using this as the input series repeat Step 1; if the order of the linear transfer function identified has altered, repeat the previous steps. Identification and estimation of a linear transfer function Consider the identification and estimation of the linear transfer function in Eqn as part of Step 1 of the previous sequence. Presuming v is given [often specified as min t = 1 , … , T ( y t ) ], a number of tools are available for the estimation of the parameters θ = ( a 1 , … , a n , b 0 , … , b m ) (see for example Ljung, ; Poskitt, ; Young, ). Within the DBM methodology, the most commonly used is the refined instrumental variable (RIV) method introduced in Young ( ; see Young for a more recent description). This provides a robust estimation methodology that produces reasonable parameter estimates when the error series has unknown, possibly nonstationary and heteroscedastic error properties. The RIV algorithm, as encoded in the Computer‐Aided Program for Time Series Analysis and Identification of Noisy Systems (CAPTAIN) toolbox (Taylor et al ., ), is very computationally efficient (typical evaluation times even on large data sets are less than 10 s). Identification of the model order ( n , m , d ) can therefore be performed by an exhaustive search over a range of plausible structures specified by the modeller. The identified model order is chosen by the modeller to provide a trade‐off between the fit of the model, the parsimony of the model structure, and the plausibility of the mechanistic interpretation of both the input non‐linearity and the transfer function in the DBM model (see for example Romanowicz et al ., ). The most common mechanistic interpretation of a linear transfer function seen in a flood forecasting context is that of parallel response paths. For example, a model with ( n , m ) = (2, 1) can be decomposed using partial fractions to give: 3 b 0 + b 1 z − 1 1 + a 1 z − 1 + a 2 z − 2 = β 1 1 + α 1 z − 1 + β 2 1 + α 2 z − 1 The first‐order terms represent two parallel tanks. These can be analysed in terms of their time constants (≈Δ t / log (− α i ) where Δ t is the model time step) and steady state gains ( β i /(1 + α i ), which should be greater than 0) to ensure that they are consistent with the modeller's concept of the system response. Note that higher order transfer functions can be decomposed in more than one way and that only one of the decompositions may have a mechanistic interpretation. Alternative decompositions can be found in, for example, Young et al . ( ). Identification and estimation of the input nonlinearity In keeping with the inductive approach prevalent in DBM modelling, the function F defining f is not presumed a priori. Instead, an empirical form of the function is derived by nonparametric analysis of the data. This allows the data to take precedent over the prior assumptions of the model builder by providing the modeller with a guide for specifying F (as well as some justification for pursuing this approach). The SDP tools that form a component of the CAPTAIN toolbox provide one method for proposing the empirical function form. The SDP algorithm has been described in detail in several papers (see for example Young et al ., ) and its use critiqued in Beven et al . ( 2011a ). In essence, the SDP algorithm iterates between fitting the linear transfer function and a nonparametric regression describing the dependency of one or more of the transfer function parameters on a known state. Because the SDP algorithm is only required to provide a guide for a subsequently optimised parameterisation scheme, it is not usually necessary to include a high‐order transfer function model structure within the algorithm set‐up. A standard method is to utilise a first‐order transfer function with a value of d taken from the identified linear model from Step 1. The state dependency of the single numerator parameter is then investigated. This arrangement often produces a good indication of the state‐dependent relationship linking observed output to the effective input series without the need to disentangle state dependencies among multiple SDPs. While the SDP algorithm provides a powerful tool for exploring the specification of F, two key challenges arise because of the nature of the system being studied. First, because of the rarity of extreme flood events and the difficulties in observing them, there are generally severe limitations in the data available for the crucial flood‐inducing levels. The identification of an input nonlinearity function at flood levels is therefore often reliant on a small number of data points, where both the state variable and the observation of the output may be subject to significant error. In forecasting, this can, at least to some extent, be compensated by the online data assimilation considered in the next section. The second challenge relates to the specification of the state variable. In several studies (for example Lees et al ., ; Romanowicz et al ., , ) where the time delay in the linear transfer function ( d ) has provided a suitable forecast lead time, the observed output has been used as the state variable in the belief that it provides a suitable surrogate for the catchment wetness. Alternative representations of catchment wetness, for example, an antecedent precipitation index (Smith and Beven, ) or the output of the optimal linear model from Step 1 (Laurain et al ., ), can also be considered. Potential parametric description of the input nonlinearity ranges from the comparatively simple, such as the power law: 4 f t = F ( y t ​ , φ ) = y t φ 0 < φ < 1 to more complex descriptions such as piecewise cubic Hermite interpolating polynomial splines or radial basis functions (e.g. Beven et al ., 2011a ). In some cases, such as when snowmelt influences the catchment run‐off, the nonparametric analysis may suggest that a simple physical model is appropriate (e.g. the degree day model used to predict snowmelt inputs in Smith et al ., ). Having proposed one or more potential formulation for the input nonlinearity, the parameters of the function require optimisation. This is typically performed using a numerical optimisation algorithm to select φ , which minimises a sum of squared errors criterion. For each trialled value of φ in the search, a corresponding θ is estimated using the instrumental variable methods discussed in the previous section. Following optimisation, the values of φ and θ should be checked to ensure that the model can be interpreted in a mechanistic fashion. Data assimilation within DBM A very important aspect of improving flood forecasts when knowledge of both inputs and appropriate models is subject to error (and uncertainty) is the use of data assimilation or adaptive forecasting. In what follows, we consider the assimilation of observations of the forecast water level variable to improve the forecasts of the observed hydrological response. The structure of the DBM model indicates that forecasts at lead times within the catchment lag d can be generated with observed input data. Forecasts at longer lead times require the use of forecast input data. If determinist forecasts, ideally revised every time step are available (e.g. a point summary of a probabilistic DBM model forecast), they can be substituted for the unknown inputs and similar techniques to those used below utilised (see Romanowicz et al ., ). This technique is used in the latter case studies for cascading the model forecasts. More complex situations where there are ensemble forecasts of the input are not considered here (but see Smith and Beven, ; Smith et al ., ). A DBM model can be cast within a general state space framework for nonlinear models (e.g. Liu and Gupta, ). The assimilation of observations of the output variable and generation of future forecasts consists of a filtering problem with components. The first is deriving the probability distribution of the states at time t + 1, given their distribution at time t and new pair of observations ( y t +1 , u t +1− d ). The second is to evaluate the predictive distribution of an observation n steps ahead by mapping of the stochastic noise terms given the initial distribution of the model states. Here, a Kalman filter approach is adopted in data assimilation. Extended forms of the Kalman filter have been developed for DBM models (Smith et al ., ). However, experience suggests that in this type of application, where the effective input can be computed from observed data, a state space formulation of the linear transfer function embedded in a simple predictor‐corrector implementation of the linear Kalman filter performs well. The mechanistic interpretation of the linear transfer function often allows the states of its spate space formulation to be defined in a meaningful way, for example, as a representation of the water following each of the parallel response paths. The variance of the system noise, which corrupts the evolution of the states in time, therefore gives an indication of the error about the different types of response. These variance terms, along with that of the observational noise, are often estimated by maximising the likelihood of the observed water levels based on forecasts at a given lead time (see Smith et al ., ). Considering one or more of the variance terms as time varying or dependent upon the magnitude of the forecast may be beneficial (Young, ; Smith et al ., ). Implementation in the E nvironment A gency of E ngland and W ales NFFS The DBM model scheme was implemented as an operational module within the UK NFFS, which is based on the Deltares DELFT‐FEWS forecasting framework (Werner et al ., ). The approach takes advantage of the open framework that allows DELFT‐FEWS/NFFS to be loosely coupled to any suitable forecasting model (Werner et al ., ). A list of models that have been coupled is provided in Weerts et al . ( ). Lancaster Environment Centre and Deltares have developed an implementation of the DBM methodology for online forecasting in the R open source statistical programming language (Ihaka and Gentleman, ) that can be coupled with the NFFS (Leedal et al ., ). This implementation was designed to be a modular, reusable framework that would easily scale for use at new sites and catchments in the future. It should facilitate advances in the use of DBM hydrological models. The framework for the DBM module prescribes a standard directory structure for holding the set of functions that process data, assimilate data, perform forecasting and write results to file. The structure is shown in Figure . Within the DELFT‐FEWS framework, the format of calls to a function is prescribed, i.e. each function must receive the expected format of variables. However, each model retains its own version of the main processing functions. This approach provides a compromise between code reuse, maintenance and flexibility. If a function is common to several models, it is straightforward to copy that function to each model. However, if a model requires a unique processing method, this is also possible. File structure for the data‐based mechanistic ( DBM ) module. is specific to a catchment, < ID > is specific to a forecast model within the catchment. For clarity, this structure shows a catchment with only a single model; in most cases, a catchment will contain several Folder< ID > directories. During real‐time operation: The required input data are retrieved from the NFFS database, are labelled with the station name and are stored in the Work directory of the DBM module. The past state of the model and data assimilation algorithm is loaded from the States directory. One cycle of forecast followed by assimilation is performed using the new data point that has been passed during the present time point. The new state of the model and data assimilation algorithm is saved in the State folder. The model iterates over the remaining input data to produce a segment of forecast. This is saved in the Work directory. The NFFS interface retrieves this section and incorporates it into the main NFFS database. Once the DBM module forecasts have been appended to the NFFS database, the usual tools are available to generate reports, statistics and visualisations (Werner et al ., ). R iver E den case study The DBM forecasting approach was applied to the Lower Eden catchment in Cumbria, UK. At the Sheepmount gauging station downstream of Carlisle, which is the lowest monitored point in the catchment, the total area is 2400 km 2 . The River Eden and its principle tributaries, the Rivers Eamont, Irthing, Petteril and Caldew, link a number of significant areas for run‐off generation (Figure ). These include the northern and eastern parts of the Lake District peaks (covering peaks such as Skiddaw and Helvellyn) and the areas of the Northern Pennines and Kielder Forest. These areas are typified by high annual rainfall totals and steep terrain. Flood warning lead times greater than 6 h may require the use of forecast precipitation. There are 31 river gauging stations and 16 telemetred rain gauges in the catchment. Map of the R iver E den around C arlisle (shaded area) showing river confluences and gauging sites. Inset map shows the location of the catchment in the UK . Using the DBM methods outlined, a simplified network of six submodels were identified and estimated using hourly data from October 2003 to March 2005. This includes the January 2005 Carlisle flood event. Figure illustrates the network of DBM submodels that represent either individual tributaries or reaches of the River Eden. Details of the structure of the DBM submodels are given in Table . The use of a semidistributed network of DBM models rather than a single input‐output model terminating at Sheepmount allows for the possibly complex spatial distribution of rainfall patterns within the catchment. The network approach also provides two related benefits: The network can provide forecasts at intermediate flood risk locations. The intermediate locations correspond to telemetry gauge sites, thus facilitating the assimilation of multiple streams of data during real‐time forecasting and providing some degree of robustness against station malfunction. In testing, the real‐time DBM model network with data assimilation performed well. Figure shows an example of the output generated from within the NFFS software, in this case for forecasts of up to 6 h lead time at Sheepmount. The handling of DBM models within NFFS has been configured to take advantage of the stochastic information available about the forecast with the best estimate forecast and 95% prediction confidence interval being shown. Data‐based mechanistic ( DBM ) model network of the lower E den catchment including the R ivers I rthing ( G reenholme), C aldew ( D enton H olme) and P etteril ( H arraby G reen). Locations mentioned in either river gauging stations or telemetered rain gauges (italic). Lower E den data‐based mechanistic forecasting model implemented within N ational F lood F orecasting S ystem. Results show 6‐h forecast ± 95% confidence intervals. A summary of the constructed models for the R iver E den Predicted location Observed inputs Nonlinearity f t Model order ( n m d ) Maximum lead time of forecast (hours) Great Corby Temple Sowerby, Udford PCHIP spline (2 2 4) 4 Greenholme High Row, Harescough, Crewe Fell, Aisgill PCHIP spline (2 2 6) 6 Linstock Great Corby, Greenholme PCHIP spline (1 1 1) 5 Denton Holme Carlisle, Geltsdale, Mosedale PCHIP spline (2 2 5) 5 Harraby Green Carlisle, Geltsdale, Mosedale PCHIP spline (1 2 5) 5 Sheepmount Great Corby, Greenholme PCHIP spline (1 1 1) 6 Locations in italics are rain gauges; the remainder are river level gauging stations. PCHIP, piecewise cubic Hermite interpolating polynomial. Validation A key component of any forecasting system is the continual assessment of operational models. To this end, it is informative to test the performance of the Eden DBM configuration using data extending to January 2010. The analysis was carried out in such a way as to simulate exactly the results that would have been generated had the set of models been run in operational mode. Figure shows results for the two largest events during the validation period for both 2 and 5 h forecast lead times. Lower E den data‐based mechanistic forecasting model for S heepmount showing the two largest events during the 2005–2010 validation period (left: 12 J anuary 2009, right: 20 N ovember 2009). Upper panels show simulated 5 h forecast; lower panels show 2 h forecast. The red line represents the observed water level, solid black line the expected value of the prediction, with the shaded area encompassing the 95% confidence interval for the prediction. For comparison with Figure , the dashed green and blue horizontal lines show the Environment Agency warning thresholds with codes RES FW NC3A and ACT OPS , respectively. The performance of the DBM model forecasts for Sheepmount can be summarised using the familiar probability of detection (POD) and false alarm rate (FAR) statistics. Here, the 99th percentile of the observed data (3.4 m) was used as the minimum flood event to include in the study. This allowed 27 events to be considered for POD/FAR during the validation period. The POD and FAR statistics for 2 and 5 h lead times are summarised in Table . POD and FAR are usually associated with deterministic model outputs. They can therefore only be evaluated for deterministic summaries of the probabilistic forecasts given by the DBM model. Results are presented both for the best estimate forecast, in this case the median value of the forecast distribution and the deterministic summaries resulting from taking the value at each time step, which according to the forecast has a 16% or 2.5% probability of being exceeded. These correspond to the mean value of the forecast plus one or two standard deviations, respectively. A possible application of these methods is to issue warnings at levels of confidence appropriate to the site risk or vulnerability. As can be seen in Table , the number of false alarms logically increases, as FAR is assessed at more cautious values (lower probabilities of exceedance at each time step). The high level of POD for the best estimate of the forecast value indicates that the model may be a little over‐cautious. This should be investigated in future refinements of the model scheme. Sheepmount POD and FAR statistics for 2 and 5 h (in parentheses) forecast lead times calculated using 2.9 m threshold as lower limit on peaks to include Statistic Mean Mean + 1 SD Mean + 2 SD POD 100% (100%) 100% (100%) 100% (100%) FAR 20% (36.8%) 29.4% (52%) 36.8% (58.6%) FAR, false alarm rate; POD, probability of detection; SD, standard deviation. The estimated error bounds should encompass the correct proportion of the data. This can be verified by investigating the ratio of observations inside and outside the forecast error bounds. Table presents these results for 2 and 5 h forecast lead times. Ideally, these values should be close to 32% and 5% for 1 and 2 standard deviations, respectively. These ratios tend to breakdown increasingly as more low flow data are removed from the sample. Because for flood forecasting, it is the high water levels we are interested in forecasting correctly (with appropriate error bounds), this highlights the importance of considering summary statistics calculated on appropriate subsamples of the entire observation record. Model efficiency and percentage of observations falling outside the estimated forecast uncertainty range for 2 and 5 h (in parentheses) forecast lead times at S heepmount Threshold (m) Number of observations Nash–Sutcliffe model efficiency for mean Observations outside 1 SD Observations outside 2 SD 0 35 007 0.98 (0.92) 17.1% (51.3%) 4.9% (17.9% 1 7 107 0.96 (0.88) 19.5% (34.3%) 7.1% (11.3%) 2 1 142 0.86 (0.59) 46.9% (50.0%) 21.0% (20.4%) 3 302 0.43 (−0.54) 73.2% (59.6%) 34.1% (24.8%) 4 32 0.62 (0.05) 68.7% (59.3%) 21.9% (12.5%) SD, standard deviation. It is also important to consider error in event timing as well as magnitude. While timing error is not available to the user in real time, it can be assessed postevent. Ideally, the point at which the observations cross a threshold level should fall within the period where the upper and lower bounds of a given prediction confidence interval intersect the threshold level. This is often a challenging performance measure, as the rising limb of a large flood event is difficult to forecast accurately. Table shows the results for 2 and 5 h forecast lead times at Sheepmount. Number of threshold crossings where the timing of the observed crossing lies within the two standard deviation ranges of the forecast Forecast lead time (hours) Number of crossings Number inside 2 SD of timing uncertainty 2 12 8 5 12 3 Data are for Sheepmount incorporating 11 RES FW NC3A and 1 ACT OPS threshold crossings (see Figure ). RES FW NC3A and ACT OPS are warning threshold codes. SD, standard deviation. R iver S evern case study A map of the upper River Severn, along with the gauges used in this study, is shown in Figure . The River Severn rises in the Cambrian mountains at a height of 741 m above ordnance datum (mAOD) and flows north‐eastwards through Abermule before meeting the Vyrnwy tributary upstream of Shrewsbury. Around the confluence of the Vyrnwy tributary, the valley is wide and flat with a considerable extent of floodplain. The river then flows through Montford to Shrewsbury and is joined at Montford Bridge by the River Perry that flows from the North. The lowermost point in the upper Severn catchment is defined as the gauge at Welsh Bridge in Shrewsbury. Average annual rainfall can exceed 2500 mm in the Cambrian mountains and the upper reaches of the Vyrnwy catchment. There are no significant flow control structures in the catchment except for Lake Vyrnwy in the upper reaches of Vyrnwy and the Clywedog Reservoir in the upper reaches of Severn. Map of the upper R iver S evern showing significant urban areas (shaded), river gauges (circles) and rain gauges (triangles). Inset map shows the location of the catchment in the UK . Flood events that have caused property flooding in the upper Severn catchment in recent years include Easter 1998, October 1998, October 2000, February 2002, February 2004 and January 2009, with the October 1998 event particularly severe in the upper Severn. DBM models for a simplified version of the upper Severn system were constructed in line with the methodology described earlier. A model of the system extending further downstream can be found in Romanowicz et al . . Three river level sites are modelled: Meifod on the Vyrnwy, Abermule on the Severn upstream of the Vyrnwy confluence and Welsh Bridge. These sites represent the outlets of the two main flow‐generating regions (Meifod and Abermule) and an observed location in the region of high flood risk (Welsh Bridge). Intermediate gauged sites between the Vyrnwy confluence and Welsh Bridge could be modelled at the cost of having to represent more complex floodplain interactions (e.g. Smith et al ., ). A schematic of the model is shown in Figure , with details of the models given in Table . The maximum lead time at Welsh Bridge can be extended by utilising the predictions at the two upstream locations as inputs to the model of this site. Data‐based mechanistic model network of the R iver S evern catchment including the R iver V yrnwy ( M eifod). Locations mentioned are either river gauging stations or telemetered rain gauages (italic). A summary of the constructed models for the R iver S evern Predicted location Observed inputs Non‐linearity f t Model order [ n m d ] Maximum lead time of forecast (hours) Abermule Cefn Coch, Dolydd, Pen‐y‐Coed y t 0.12 (2 2 5) 5 Meifod Llanfyllin, Pen‐y‐Coed y t 0.13 (2 2 2) 2 Welsh Bridge Meifod, Abermule None (2 2 16) 18 Locations in italics are rain gauges; the remainder are river level gauging stations. Calibration was performed using hourly data from 1 January 1999 to 31 December 1999. A single input series for each model was generated using a weighted sum of the observed inputs. These weights were optimised alongside the parameters of the nonlinearity. Figure shows a number of forecasts for the October 2000 flood event. The forecasts for Abermule and Meifod suggest that the nonlinearities identified in the calibration are suitable for modelling this event. As more data become available, recalibration of the nonlinearity and re‐identification of the linear transfer function structure may add additional benefits, particularly in improving the timing of the rising limb of the hydrograph. Validation forecasts for the O ctober 2000 flood on the R iver S evern for (a) A bermule with 5 h lead time; (b) M eifod with 3 h lead time; (c) W elsh B ridge with 6 h lead time; (d) W elsh B ridge with 16 h lead time. The red line represents the observed water level, solid black line the expected value of the prediction with the shaded area encompassing the 95% confidence interval for the prediction. The dashed horizontal lines show Environment Agency warning thresholds labelled in this work: Flood watch 103 (cyan), Flood warning (green) and Severe flood warning (blue). In the forecasts for Welsh Bridge, it is evident that as the lead time of the forecast decreases, the predictions increase in accuracy. The results at the longer lead time, where the effect on the forecast of the most recently assimilated observation is lower, suggest that the dynamics of this part of system may not be well represented by the model. These difficulties may be partially due to backwater effects or interaction with the flood plain that may be better represented by a more complex model structure (Smith et al ., ). Validation The performance of the River Severn DBM configuration is tested using data between January 2005 and August 2009. As with the Eden case study, the analysis was carried out in such a way as to simulate exactly the results that would have been generated had the set of models been run in operational mode. The POD and FAR statistics for 2 and 5 h lead times are summarised in Table . These were computed using the 46 events greater than 49 mAOD at Welsh Bridge during the validation period. As with the River Eden model, the high level of POD indicates that the model may be a little overcautious, and again, this should be investigated in future refinements of the model scheme. W elsh B ridge POD and FAR statistics for 2 and 5 h (in parentheses) forecast lead times calculated using 49 mAOD threshold as lower limit on peaks to include Statistic Mean Mean + 1 SD Mean + 2 SD POD 91.5% (100%) 100% (100%) 100% (100%) FAR 8.5% (50%) 15.1% (50.5%) 29.7% (63.3%) FAR, false alarm rate; mAOD, m above ordnance datum; POD, probability of detection; SD, standard deviation. Table presents results for the percentage of observations bracketed by the prediction intervals at 2 and 5 h forecast lead times. The desired percentages breakdown increasingly as more low flow data are removed from the sample. The results here suggest the distribution of forecast errors do not fit neatly within the Gaussian assumptions used within calibration of the variance terms within the data assimilation strategy. Alternative calibration strategies (e.g. Smith et al ., ) that are not reliant on the Gaussian assumptions have proved useful in such situations but do not currently form part of the UK NFFS deployment. Model efficiency and percentage of observations falling outside the estimated forecast uncertainty range for 2 h forecast lead time at W elsh B ridge Threshold (m) Number of observations Nash–Sutcliffe model efficiency for mean Observations outside 1 SD Observations outside 2 SD 47 40 152 1.00 (0.98) 9.7% (19.7%) 2.0% (5.4%) 48 13 656 0.99 (0.95) 16.7% (30.0%) 4.1% (10.8%) 49 3 918 0.97 (0.80) 22.8% (38.1%) 7.7% (15.7%) 50 661 0.95 (0.70) 26.9% (49.8%) 3% (15.1%) 51 135 0.59 (−1.45) 20.7% (51.1%) 0% (3.7%) SD, standard deviation. The performance of the model in forecasting the timing of rising limbs of the hydrograph is shown in Table . It shows that the model struggles to capture the timing of the rising limb, particularly at the highest water levels. This is keeping with the limitations of the representation of the system dynamics seen in Figure . Number of threshold crossings where the timing of the observed crossing lies within the two standard deviation ranges of the forecast Threshold Number of crossings Number inside 2 SD of timing uncertainty for 2 h lead time Number inside 2 SD of timing uncertainty for 5 h lead time Flood watch 54 41 28 Flood warning 42 22 18 Severe flood warning 6 2 0 Data are for 2 and 5 h forecast at Welsh Bridge. SD, standard deviation. Conclusions The outlined DBM modelling methodology for forecasting is effective in a flooding context as witnessed by the two example applications to UK rivers. Assimilation of observed river gauge data is important in improving forecasts of future water level or discharge. This is achieved in the DBM framework using an efficient computational scheme. The implementation of real‐time DBM forecasting models within the UK NFFS is demonstrated. Analysis of the forecasts produced for a validation period at two test sites shows that the Gaussian error assumptions used in the calibration of the data assimilation scheme are not valid. The forecast error bounds should therefore be considered as indicative because of the failure to characterise the underlying uncertainty in the forecasting process. However, the error bounds can be considered to contain information not present in a deterministic forecast. This is of use in helping operational forecasters issue flood warnings. The forecast results shown in this paper highlight why ongoing validation is important. This provides the basis for updating the forecasting models, something that can be readily achieved within the DBM framework outlined. It also allows the duty officer(s) responsible for issuing flood warnings opportunity to revise their incorporation of the additional information present in the probabilistic forecast in the forecast chain. Acknowledgements Application of the DBM methodology to flood forecasting was first carried out for the River Nith catchment in 1991 under contract to the Solway River Purification Board in 1991. More recent development work was supported by the Phase 1 of the Engineering and Physical Science Research Council‐funded Flood Risk Management Research Consortium and the European Commission Seventh Framework Programme project IMproving Preparedness and RIsk maNagemenT for flash floods and debriS flow events (IMPRINTS). This work and implementation within the NFFS were supported by the Environment Agency project SC080030 on probabilistic flood forecasting.

Journal

Journal of Flood Risk ManagementWiley

Published: Sep 1, 2014

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