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METEOROLOGICAL APPLICATIONS Meteorol. Appl. 16: 35–40 (2009) Published online 21 January 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/met.120

On the propagation of uncertainty in weather radar estimates of rainfall through hydrological models Chris. G. Collier* Centre for Environmental Systems Research, School of Environment and Life Sciences, University of Salford, Greater Manchester, M5 4WT, UK

ABSTRACT: The generation of flow forecasts using rainfall inputs to hydrological models has been developed over many years. Unfortunately, errors in input data to models may vary considerably depending upon the different sources of data such as raingauges, radar and high resolution Numerical Weather Prediction (NWP) models. This has hampered the operational use of radar for quantitative flow forecasting. The manner with which radar rainfall input and model parametric uncertainty influence the character of the flow simulation uncertainty in hydrological models has been investigated by several authors. In this paper an approach to this problem based upon a stochastic hydrological model is considered. The errors in the input data, although they may be constrained, do propagate through the model to the flow predictions. Previous work on this error propagation through a fully distributed model is described, and a similar analysis for a stochastic hydrological model implemented in a mixed rural and urban area in north-west England carried out. Results are compared with those previously published for an American catchment. A possible approach to selecting flow forecast ensemble members is proposed. Copyright  2009 Royal Meteorological Society KEY WORDS

uncertainty; radar; hydrological model; stochastic; flow ensembles

Received 27 August 2008; Revised 19 November 2008; Accepted 3 December 2008

1.

Introduction

The generation of flow forecasts using rainfall inputs to hydrological models has been developed over many years and is now operational in many countries (see for example McEnery et al., 2005). The models used range from simple input-storage-output (ISO) models (Lambert, 1972; Collier and Knowles, 1986) and transfer function models (Young and Beven, 1994; Sempere Torres et al., 1992) to comprehensive distributed models such as the SHE model (Beven et al., 1980; Abbott et al., 1986); TOPMODEL (Beven et al., 1995; Beven, 1997); the TOPKAPI model described by Todini (1995); and the TCM/PDM model used by some regions of the UK Environment Agency (see for example Moore, 1985). However, the benefits of using complex models depends very much on the type of catchment, the input data and the application. Increasingly, input data to models may be derived from different sources such as raingauges, radar and high resolution Numerical Weather Prediction (NWP) models. Robbins and Collier (2005) compared the way errors in flow forecasts derived from an urban drainage model varied with increasing variability of rainfall. They found that raingauge inputs gave smaller errors than rainfall inputs from both radar and microwave links for uniform * Correspondence to: Chris. G. Collier, Centre for Environmental Systems Research, School of Environment and Life Sciences, University of Salford, Greater Manchester, M5 4WT, UK. E-mail: [email protected] Copyright  2009 Royal Meteorological Society

rain, but as the rain became more variable this ceased to be the case. In using data from different sources it is clearly necessary to combine these different data. One approach to this problem is to use a stochastic state-space model with a Kalman filter procedure allocating weights to each data form, based upon the respective uncertainty of each observation type and of the predictions (see Grum et al., 2002; Cornford, 2004). Recognizing that flow predictions made with such a model will nevertheless remain uncertain, a Bayesian post processor (Krzysztofowicz, 1999) may be used to analyse components of the output error associated with particular data input types as demonstrated by Collier and Robbins (2008) for an urban drainage modelling system. The manner with which radar rainfall input and model parametric uncertainty influence the character of the flow simulation uncertainty in a distributed hydrologic model has been investigated by Carpenter and Georgakakos (2004). Ensembles of flow simulations were investigated and it was found that radar rainfall uncertainties increased model flow uncertainties by steepening the inverse relationship between the ensemble dispersion flow diagram and the drainage area. Carpenter and Georgakakos (2006a) used a parsimonious model for spatially correlated radar-rainfall errors with radar-pixel error variance that depends on the magnitude of the observed rainfall. They confirmed that the ensemble flow range is dependent on subcatchment scale with a well-defined log – linear relationship. Previously, Georgakakos et al.

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(2004) had stressed the importance of including input rainfall uncertainty in procedures for generating flow using ensemble techniques. Collier and Robbins (2008) described another method of generating a representation of the likely error distribution using the output from a Bayesian post processor. The error distribution provides a probability forecast of the flow. This approach is simple to implement, and requires only knowledge of model performance as a function of flow and the error characteristics of the input data and the flow measurements which may of course change with time. In this paper this approach is reviewed in comparison with other methods of generating ensembles, and ways to present the uncertainty to users of discharge forecasts is considered. The errors in the input data, although constrained by the stochastic modelling approach, do propagate through the model to the flow predictions. Vivoni et al. (2007) describe this error propagation through a fully distributed model. A similar analysis for a stochastic model of the River Croal catchment (Figure 1), which contains within it an Urban Drainage System (UDS), is carried out and results compared with those using the distributed model.

2.

The River Croal catchment

The UDS of Bolton is located within the natural basin of the River Croal in north-west England. The River Croal catchment (143 km2 ) is a tributary of the River Irwell. The UDS feeds into the River Croal which has a short time-to-peak of around 3 h as shown in the sample hydrograph in Figure 2. Figure 3 shows the hydrometric network in and around this catchment. The squares show the 2 km radar grid generated from the Hameldon Hill C-band radar located just to the north of the catchment. Also shown are the raingauges, and microwave links used in the study reported by Robbins and Collier (2005), but not in the present work. The UDS area is 93 km2 , serving a population of 260 000. Rainfall runoff is transported together with domestic, commercial and industrial effluent along a network of about 1200 km of pipes having diameters from 150 to 1500 mm to an outlet treatment works at Ringley from where treated water is discharged into the River Croal. The UDS often has insufficient capacity to deal with this combined flow during only moderately heavy rainfall events, resulting in many occurrences of sewer

(a) River Irwell catchment, NW England

Higher Upper Irwell 105 km2

Irwell vale Lower Upper Irwell 51 km2

Croal 143 km2

Croal catchment

Roch 188 cm2

Bury Grounds Blackford Bridge

Bolton Farnworth

Lower Irwell 67 km2 Adelphi Weir

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measurement point

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L

0

4km

Main Outlet into the River Croal

Figure 1. (a) The River Croal subcatchment (143 km2 ) within the River Irwell catchment to the north of Manchester and (b) the Bolton Urban Drainage System (UDS) located within the Croal subcatchment: W – Water Street retention tank; S – Spa Road retention tank; L – Ladybridge retention tank. The main outlet of the UDS to the natural water course is shown. Also shown are the River Roch subcatchment (188 km2 ), the Lower Irwell subcatchment (67 km2 ), the Lower Upper Irwell subcatchment (51 km2 ) and the Higher Upper Irwell subcatchment (105 km2 ). Copyright  2009 Royal Meteorological Society

Meteorol. Appl. 16: 35–40 (2009) DOI: 10.1002/met

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Discharge (cumecs)

PROPAGATION OF UNCERTAINTY THROUGH HYDROLOGICAL MODELS 10 9 8 7 6 5 4 3 2 1 0 1

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Time (hours starting at 1300 UTC)

Figure 2. An example of the hydrograph of the River Croal on 2 July 2006 and the raingauge and radar estimates of the catchment rainfall. A fitted simple unit hydrograph model (dashed line) is shown on the actual hydrograph (solid line).

Figure 3. River Croal catchment in N.W. England showing the hydrometric network. The grid squares are 2 × 2 km. HH is the location of the Hameldon Hill C-band radar used in this study. The microwave links are now defunct, but were used in the study reported by Robbins and Collier (2005). Letters show the locations of raingauges.

overloading causing flooding and overflows into the natural water courses. Three large off-line tanks receive flows diverted from the main sewer in order to reduce the occurrence of flooding by holding back flood waters. The tanks have capacities of 2000 m3 (Ladybridge) and 10 000 m3 (Spa Road and Water Street). The complexity of the pipe system, the retention tanks and how they discharge into the natural water courses ensures that the behaviour of the River Croal discharge might be expected to demand a complex hydrological/hydraulic model. This is investigated in what follows.

upon assessing areas of similar hydrological behaviour such as that described by Wohling et al. (2006) may be adopted. However, the use of direct measurements of flow (or level) and catchment conditions using model state ensembles tends to damp the effect of outliers which may, or may not, represent real conditions. An alternative approach is to use a method which, as well as providing an updating procedure, may also combine different types of rainfall inputs, for example a state-space stochastic model. An example of a state-space stochastic model is Water Aspects (Grum et al., 2002) which has been applied to the River Croal. This model may be operated in both stochastic and deterministic modes. In the deterministic version of the model there are two equal sets of input (i.e. collections of signs) which always yield the same output sign if run through the model under identical conditions. No inner model conditions have a stochastic behaviour. The stochastic version of the model has components having random character which are not explicit in the model input, but only implicit or ‘hidden’. Therefore, identical inputs will generally result in different outputs if run through the model under, externally seen, identical conditions (Refsgaard, 1996). In this paper, as in Collier and Robbins (2008), constant errors are assumed as given in Table I, throughout each of the cases studied. The model comprises a rainfall plane and the hydrological system. All measured data are considered to be different ways of looking at the same thing. Given the presence of any observations, the state of the rain plane and the hydrological system is updated by weighting between what the model has predicted and the available observations. The weighting is done using a Kalman filter based on the uncertainties of the predictions and observations respectively. Further details are given in Collier and Robbins (2008).

4.

Model performance

Rainfall measurements from the hydrometric network are assimilated into the stochastic state-space model. Estimates of the error variances associated with each measurement technique, and with the model, have been pre-determined. Flow predictions are provided at regular intervals (in this case 50 time step intervals) from the start of the event, and these are then compared to the observed flow at the catchment outlet. Table I. Errors used in the stochastic models (Stoch 1: manually defined errors; Stoch 2: errors estimated by optimization over period October–December, 2004). The radar errors are the variance of the error rather than the mean bias. Parameter

3. A stochastic state-space model of the River Croal, NW England To allow model discharge predictions to be updated using actual measurements of flow, procedures based Copyright  2009 Royal Meteorological Society

Flow gauge Raingauge Radar Reservoir Water

Stoch 1

Stoch 2

Units

4.0 2.0 3.0 100

3.9 2.1 0.5 83

m2 s−1 mm h−1 mm h−1 m3

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The mean errors used in two versions of the stochastic model for the rainfall and the river flow are given in Table I. The radar errors represent the error variance not the mean bias of the estimates. The stochastic model realization reduces the variance in the peak flow simulations using rainfall inputs from raingauges (an average 12% improvement), radar (up to a 28% improvement) and a combination of both (up to an 11% improvement) (see Collier and Robbins, 2008). Interestingly the use of raingauge inputs produces the smallest errors. This is probably due to the availability of a quite dense raingauge network in the catchment (Figure 3), and limitations in the radar quality control procedures. Ground clutter is removed, but no other quality control procedures are applied. Collier and Robbins (2008) showed that the stochastic model using radar input approaches the level of performance using raingauge input as the error associated with the radar data reduces. The smallest error (0.5 mm h−1 ; Stoch 2) objectively derived for the radar data is achieved using a model parameter optimization procedure carried out over a three month period (October–December 2004). Lumping all the statistics together the model performance as a function of the input error in the rainfall was examined. The stochastic model produces smaller timing errors than the deterministic model for all input errors. However, errors in peak flows are only smaller for the stochastic model when the input error is less than around 3 mm h−1 for this particular catchment. This 3 mm h−1 error threshold corresponds to about a 15% error in peak flow. Provided the error in the rainfall is constrained to be less than 3 mm h−1 the stochastic model outperforms the deterministic version of the model.

in the UDS, and released after peak flows have occurred (Section 2). This too may result in increased discharge biases. When the Mean Absolute Error (MAE) defined as the sum of the differences between the actual and the modelled flows divided by the number of events is used (Figure 5), the results between the two studies are consistent. As the rainfall MAE increases, the discharge MAE also increases for all lead times. For the present study the MAE gives comparable values of MAE to the January 0–1 h ahead frontal values of Vivoni et al.’s study. The River Croal catchment includes a large urban drainage system and is paved over a large part of its area. This might have been expected to result in a difference between the studies. This does seem to be the case for the MAE analysis for the October 0–1 h ahead squall line case reported by Vivoni et al., and the Croal analysis. It is also the case for the bias errors as noted previously. Discharge bias errors are larger for the Croal catchment than for the Baron Fork catchment as the rainfall bias increases. This may also be due to the different model formulations used. Further work is needed to investigate this behaviour. 6. Generating flow forecast ensembles and probability forecasts The improvement possible using a stochastic modelling approach depends upon the errors associated with the 1.6 1.5

Error propagation through the stochastic model

A number of studies have examined the propagation of radar estimation errors (see for example Borga, 2002; Sharif et al., 2002, 2004), and more recently Vivoni et al. (2007) have examined the propagation of errors arising from forecasts based upon radar data input to a fully distributed model which differ in magnitude with forecast lead time. The experiments of Vivoni et al. (2007) were carried out for two flood events, a winter frontal event and an autumn squall line. A similar error analysis has been carried out using the River Croal stochastic model with the discharge and radar rainfall biases for six events being plotted in Figure 4. The six events occurred during August and September, these being a mixture of frontal and convective events. Also shown in the figure are the results reported by Vivoni et al. (2007). Interestingly the differences between this study and the present study are not great for rainfall biases less than 1.1. Thereafter the rainfall bias leads to an increased discharge bias. This may be due to the failure of the stochastic model to cope with input errors which approach the model assumed errors (Table I). Also for large rainfall water is diverted into the retention tanks Copyright  2009 Royal Meteorological Society

1.4 Discharge bias

5.

1.3 1.2 1.1 1 0.9 0.8 1

1.1 1.2 Rainfall bias

1.3

Figure 4. Propagation of radar rainfall bias errors to river discharge forecast bias errors in the River Croal catchment using a stochastic model (diamonds – August–September), and in the Baron Fork, USA catchment (Vivoni et al., 2007) using a distributed model (squares – January; triangles – October). Key: diamond Croal August–September; square January frontal; triangle October squall line. Meteorol. Appl. 16: 35–40 (2009) DOI: 10.1002/met

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Discharge MAE

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Figure 5. Illustrating the discharge Mean Absolute Error (MAE) as a function of the rainfall MAE multiplied by the catchment area for various rainfall forecast lead times in hours. Note that the River Croal data for zero lead time are consistent with the points for the Baron Fork catchment (Vivoni et al., 2007) having lead times for the January cases of 0–1 h. Key: diamond Croal 0 h; square January 0–1 h; triangle January 1–2 h; diagonal cross October 1–2 h; diagonal cross with vertical bar October 2–3 h; dot January 2–3 h; roman cross October 0–1 h.

rainfall and flow inputs to the model. If the errors exceed the mean input errors used to formulate the stochastic model then no advantage is gained. By modifying the likelihood parameters in a Bayesian post processor Collier and Robbins (2008) were able to construct the error range for hydrographs derived using the deterministic model, and two versions of the stochastic model. Together these hydrographs comprise an ensemble which may provide a probabilistic density function for the forecast flows. More usual methods of generating ensembles of flow forecasts involve varying hydrological model parameters as in the Generalized Uncertainty Estimation (GLUE) procedure (Beven and Binley, 1992), or using as input to a model an ensemble of rainfall forecasts generated by adding a noise component to the deterministic nowcast, which has the spatial and temporal characteristics of the rainfall field (see for example Germann et al., 2008). Whatever procedure is used it is straightforward to extract from the discharge ensemble a forecast of the flows exceeding specific values, or using a rating curve, specific levels. The spread of the ensemble represents the uncertainty of the forecasts. These forecasts can then be used with cost-loss analyses (see for example Lee and Lee, 2007) taking account of socio-economic factors (see for example Haynes et al., 2008) to aid civil defence decision makers. However, problems arise when the actual flow is represented by an outlier of the ensemble. Statistical procedures may be used to minimize the impact of, or reject, outliers, and therefore successful forecasts of extreme flows are unlikely. Quality control procedures have been developed to assess the viability of model hydrographs using knowledge of actual or modelled peak flows or a statistical analysis of a substantial part of the measured hydrograph (Collier, 2009). These procedures take no account of how errors are likely to propagate through Copyright  2009 Royal Meteorological Society

models as a consequence of model structures. An additional approach may be to constrain the model forecast hydrograph ensemble (reduce the ensemble dispersion) by identifying those model peak flows which exceed the modulus of the expected values given the known amplification factor as discussed previously. For the River Croal it is known that the radar rainfall bias shown in Figure 4 is about 1.1 giving a discharge bias of around 1.2. This is consistent with the observed and modelled hydrograph shown in Figure 2 for which the discharge bias is about 1.04. For the ensemble case discussed by Collier and Robbins (2008), however, the deterministic hydrographs indicate discharge biases much larger than this, and therefore they are suspect. The stochastic hydrographs given in the example reported in this paper have the modulus values of the required peak flow amplification factors of less than 1.2. A challenge is to use the model performance analysis to assess whether a discharge ensemble member, which meets the bias criteria but is an outlier of the ensemble, is a valid forecast. If it is not valid, then it can be removed from the procedure producing the probability forecast. If it is valid, then further tests are needed to assess the probability associated with it. A possible approach is as follows. From the ensemble derived for the River Croal by Collier and Robbins (2008), Figure 6 shows the constructed cumulative Probability Density Function (PDF) that an ensemble member is a valid realization for this case as a function of the peak discharge MAE. This has been constructed for the Stoch. 1 model (Table I) having a rainfall error of 3 mm h−1 . For each ensemble member a value of peak discharge MAE may be evaluated, and the appropriate ‘probability of validity’ may be obtained from the graph. It is clear that the peak discharge MAE must be determined reliably if this approach is to be adopted for real-time use. A decision may then be taken as to whether to retain each Probability that ensemble member is a valid realisation (%)

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Figure 6. Cumulative probabilities that an ensemble member will be a valid representation of the hydrograph using a standard Gaussian distribution for the River Croal on 10–11 September 2004. The rainfall error specified for the stochastic model version 1 (3 mm h−1 ) has been used. From this information, ‘reliability’ bounds may be specified from which valid members of the ensemble constructed by Collier and Robbins (2008) may be selected (see text). Meteorol. Appl. 16: 35–40 (2009) DOI: 10.1002/met

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member in turn in subsequent statistical analyses. During real-time operation the peak flow forecast ‘error’ must be evaluated against recently observed maximum peak flows for similar rainfall inputs. However, the success of this approach remains to be tested. In Figures 4 and 5 we have compared a lumped (stochastic) model with a distributed hydrological model. Carpenter and Georgakakos (2006b) showed that a lumped model and a distributed model produce statistically different flow ensembles. Hence, the uncertainties in discharges from these model types are likely to be different, and we should bear this in mind in drawing conclusions from these figures. It is necessary to use bias and MAE graphs constructed for a particular type of model to constrain uncertainty in ensembles generated only using that type of model and no other.

7.

Concluding remarks

The use of hydrograph ensemble outputs from models, however they are produced, are increasingly being used to quantify the uncertainty in flow forecasts. However, ways of constraining the ensembles to provide more reliable uncertainty measures are needed. This paper has outlined one simple approach based upon knowledge of how rainfall bias errors and Mean Absolute Error (MAE) are propagated through hydrological models. Such a technique is important for using radar data which are likely to have bias errors which will vary with time and in space, and which propagate through hydrological models in different ways depending upon the model structure. References Abbott MB, Bathhurst JC, Cunge JA, O’Connell PE, Rasmussen J. 1986. An introduction to the European Hydrological System – System Hydrologique European, ‘SHE’ 1: History and Philosophy of a physically-based Distributed Modelling System, 2: Structure of a Physically-based Distributed Modelling System. Journal of Hydrology 87: 45–77. Beven KJ. 1997. TOPMODEL: a critique. Hydrological Processes 11: 1069–1086. Beven KJ, Binley AM. 1992. The future of distributed models – model calibration and uncertainty prediction. Hydrological Processes 6: 279–298. Beven KJ, Lamb R, Quinn P, Romanowicz R, Freer J. 1995. Topmodel. In Computer Models of Watershed Hydrological Water Resource Publications, Singh VP (ed.). Highlands Ranch: CO, USA; 627–668. Beven KJ, Warren R, Zaoui J. 1980. SHE: Towards a Methodology for Physically-based Distributed Forecasting in Hydrology, Vol. 129. IAHS Publication: Wallingford; 133–137. Borga M. 2002. Accuracy of radar rainfall estimates of streamflow simulations. Journal of Hydrology 267: 26–39. Carpenter TM, Georgakakos KP. 2004. Impacts of parametric and radar rainfall uncertainty on the ensemble streamflow simulations of a distributed hydrologic model. Journal of Hydrology 298: 202–221. Carpenter TM, Georgakakos KP. 2006a. Discretization scale dependencies of the ensemble flow range versus catchment area relationship in distributed hydrologic modelling. Journal of Hydrology 328: 242–257. Copyright  2009 Royal Meteorological Society

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Meteorol. Appl. 16: 35–40 (2009) DOI: 10.1002/met