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Discharge Interval method for uncertain flood forecasts using a flood model chain: City of Kulmbach

Article in Journal of Hydroinformatics · July 2019 DOI: 10.2166/hydro.2019.131

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Discharge Interval method for uncertain flood forecasts using a flood model chain: city of Kulmbach Md Nazmul Azim Beg, Jorge Leandro, Punit Bhola, Iris Konnerth, Winfried Willems, Rita F. Carvalho and Markus Disse

ABSTRACT

Real-time flood forecasting can help authorities in providing reliable warnings to the public. Ensemble Md Nazmul Azim Beg (corresponding author) Rita F. Carvalho prediction systems (EPS) have been progressively used for operational flood forecasting by European Marine and Environmental Sciences Centre, Department of Civil Engineering, hydrometeorological agencies in recent years. This process, however, is non-deterministic such that University of Coimbra, – uncertainty sources need to be considered before issuing forecasts. In this study, a new Rua Luís Reis Santos Pólo II, 3030-788 Coimbra, Portugal methodology for flood forecasting named Discharge Interval method is proposed. This method uses E-mail: [email protected] at least one historical event hindcast data, run in several ensembles and selects a pair of best Jorge Leandro Punit Bhola ensemble discharge results for every certain discharge level. Later, the method uses the same Iris Konnerth parameter settings of the chosen ensemble discharge pair to forecast any certain flood discharge Markus Disse Chair of Hydrology and River Basin Management, level. The methodology was implemented within the FloodEvac tool. The tool can handle calibration/ Technical University of Munich, Arcisstrasse 21, 80333 München, validation of the hydrological model (LARSIM) and produces real-time flood forecasts with the Germany fl associated uncertainty of the ood discharges. The proposed methodology is computationally Winfried Willems fi efficient and suitable for real-time forecasts with uncertainty. The results using the Discharge Interval IAWG, Of ce for Applied Hydrology, Water Resources Management and method were found comparable to the 90th percentile forecasted discharge range obtained with the Geoinformatics, Alte Landstr. 12-14, 85521 Ottobrunn, Ensemble method. Germany Key words | calibration, forecasting, hydrological modelling, uncertainty, validation

INTRODUCTION

The economic loss within the European Union due to flood atmospheric physics and also because of the limited resol- issue exceeded 60 billion euros from 1998 to 2009 with ution of simulated atmospheric dynamics (Lorenz ; 1,126 fatalities (EEA ; Kauffeldt et al. ) making Buizza et al. ; Kauffeldt et al. ). Uncertainty may flood resilience one of the prominent issues for many also arise from the inherent issues with the structures of a cities. The loss increased in the past decade as a result of cli- hydrological model (Renard et al. ). Combining these mate change, increasing city population and increase in per reasons, hydrological forecast models contain uncertainty capita wealth (EEA , ). Improved disaster risk man- to a great extent (Beven & Binley ; Boyle et al. ; agement through early warning information is one of the Refsgaard et al. ; Wani et al. ). Assessing uncertainty critical procedures to reduce flood losses. Significant suc- in the model results is an integrated part for hydrologic mod- cess in flood forecasts lies in the accuracy to predict the elling and considered necessary in research and practice, state of future atmospheric conditions. Yet, numerical especially when models are used for water management weather forecasts are still inaccurate due to the limitation issues (Beven ; Refsgaard et al. , ; Vanden- in mathematically representing the non-linear and complex berghe et al. ; Todini ; Barbetta et al. ). doi: 10.2166/hydro.2019.131 926 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Different methods have been developed for uncertainty With the advancement of radar technology, it is possible quantification in flood forecasting. Some methods focus on to obtain spatial rainfall data (Codo & Rico-Ramirez ). the model input uncertainty such as meteorological input Recent improvements of radar rainfall data accuracy and forcing and model initial states as, e.g., van Andel et al. resolution have shown possibilities (Pedersen et al. ) () and Li et al. (), respectively, and others focus to replace point rain gauge data by spatially variable rainfall on the hydrological model parameters such as Benke et al. forecasts in the near future. However, rain gauge data are () as well as the hydrological model itself (Coccia & still considered better regarding measurement accuracy Todini ; Deletic et al. ). (Muthusamy et al. ). One of the significant challenges Among different uncertainty estimation approaches, of using rain gauge data is to convert the point measure- Bayesian inference methods are very popular as they can uti- ments to spatially distributed data as most of the popular lise multiple parameter values within model structure hydrological models are lumped catchment area models limitations. One example is the Generalized Likelihood and for which the point measurements must be upscaled Uncertainty Estimation (GLUE) methodology, described for the whole catchment area. The uncertainty contribution by Beven & Binley (). A large number of Monte Carlo of the spatial distribution error becomes important when simulations are required in GLUE, yet accepting parameter point rainfall data are needed to be interpolated to an set criteria is subjective and based on a user-defined area. Geospatial simulations such as Kriging is also a threshold (Dotto et al. ). The results can be sensitive to better option for spatial rainfall interpolations as this the choice of the threshold value (Freni et al. ). The method considers spatial dependence structures of the Bayesian Model Averaging (BMA) method is considered a data (Mair & Fares ; Ly et al. ). Kriging with external better approach as it optimises the model posterior density drifts showed effective and reliable ways to improve the by estimating different weights (Raftery et al. ; Vrugt quality of spatial rainfall distribution (Berndt et al. ; & Robinson ). BMA applies predictive probability den- Cecinati et al. ). sity function considering weights of discrete bias corrected In the FloodEvac project, a real-time flood forecasting forecasts. This method reflects the relative contributions to tool was developed which can forecast flood discharges the predictive skill of the model by discouraging ensembles and flood extents with the inclusion of uncertainties (Disse with lower weights, which can be useful in reducing compu- et al. ). The tool can be utilised using Ensemble-based tation costs of running large numbers of ensembles (Raftery prediction to consider both model input and model par- et al. ). Hydrological Ensemble Prediction System ameter uncertainties with flood discharge forecast. The (widely known as HEPS or only EPS: Ensemble Prediction current study explains the hydrological forecast efficiencies System) is one of the most practised methodologies to pre- and proposes a new alternate methodology to reduce fore- dict river flow, which is mainly based on the BMA cast computational time by optimising the ensembles of method. In this process, the system generates an ensemble the forecast. Reporting past performance of the forecast sys- of river flow forecasts for the same forecast period consider- tems is given the highest priority by the hydrologists, ing a range of probabilistic assessments of future river flow researchers and end users to evaluate the forecast perform- instead of only one projection. Several case studies are pub- ance of a hydrological model (Wetterhall et al. ). For lished in the literature indicating its decent performance in this reason, the proposed methodology is validated by hind- forecasting flood, particularly in the case of issuing flood casting a few historic flood events. alerts with more confidence (Roulin ; Regimbeau et al. ; Cloke & Pappenberger ; Laurent et al. ; Brown ). The advantage of EPS forecasts is that FLOODEVAC TOOL AND LARSIM MODEL it can be instantly used for flood forecasting. The main input of a rainfall–runoff model is the rainfall, The FloodEvac tool, developed under the FloodEvac pro- which however does not have any easy procedure to be ject, allows the simulation of the rainfall–runoff process recorded accurately at both temporal and spatial scales. and includes uncertainty from different sources (Leandro 927 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

et al. ; Disse et al. ). The tool can be run in simu- sub-catchment has its own properties including elevation, lation or forecast modus. The former is suitable for length of watercourses and schematic river cross sections reproducing specific flood events or for the simulation of with roughness coefficients. The channel flow routing is long time series (e.g., yearly) while the latter is suitable for done using translation-retention method. It simulates the real-time flood forecasting. The model chain includes a rain- hydrologic processes for one sub-catchment for a defined fall uncertainty module, an uncertainty and calibration period. The resulting output hydrograph is the input infor- module for the hydrological model, and a link to several mation for the next element according to the general hydraulic models. model structure rules. The model structure can be both In the rainfall module, rainfall data can be introduced in grids or hydrological sub-catchments based. It considers a three different ways: using (1) hourly observed/forecast rain- soil module with storage capacities in calculating the flood fall from German Meteorological Services (DWD), routing consisting of three storages at upper, middle and generated/collected at each rain gauge location; (2) gener- lower soil. All three may contribute to the discharge com- ated rainfall based on historical data; or (3) generated ponents, modelled as a linear storage system (Figure 1(a)). rainfall based on synthetic data. Uncertainty can then be The hydrological model includes 34 parameters that allow added to catchment rainfall based on the sequential con- modelling of different processes such as direct discharge, ditional simulation (Seo et al. ). interflow and groundwater flow (please see Haag et al. LARSIM (Large Area Runoff Simulation Model) is a () and Ludwig & Bremicker () for a complete conceptual hydrological model used in the tool. The catch- description of the parameters). ment parameters can be subdivided either in gridded or LARSIM is already in use at the Upper Main catchment irregular sub-catchments. The hydrological processes are area for operational flood forecasting, operated continuously calculated in a series of sub-catchments connected by by the Flood Forecast Centre of Bavarian Water Authority. flood routing elements in a predetermined sequence. Each The model has been calibrated for historical flood events

Figure 1 | (a) LARSIM hydrological model structure, adapted from Disse et al. (2017) and (b) flowchart of FloodEvac tool with Discharge Interval method, adapted from Leandro et al. (2017). 928 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

(Laurent et al. ) where each sub-catchment has its own set are accessible at each rain gauge points from 2005 to 2015 of 34 calibrated parameters. The current work uses the same at a temporal resolution of 1 hour. There are also 55 dis- initial setup as the operational forecast model which is charge measuring stations in this area. The discharge data referred to as ‘base model’ for the rest of the work. are also available at hourly temporal resolution, and col- In forecast modus, the FloodEvac tool can generate lected by Bavarian Hydrological Services. The Kulmbach ensembles by sampling LARSIM parameters, using a beta city contains 92.8 km2 with a population of nearly 26,000 or a normal probability distribution function. The beta func- (Bhola et al. ). Part of the White Main river passes tion can produce skewed distribution and hence considers through the city accumulating river flow from Schorgast the asymmetric uncertainty parameter intervals around the and upper White Main. The Red Main river also connects calibrated parameter set when generating the ensembles. Kulmbach at the downstream side of the city at the west. Flood waves coming from the first two rivers are more sig- nificant than the third as they are connected at the CATCHMENT AREA AND DATASETS upstream part of the city and both flows sum up together at the city entrance. There are three discharge measuring The current study focuses on the city of Kulmbach located in gauges at these three rivers, namely, Kauerndorf, Ködnitz the catchment area of the Upper Main in Bavaria. The total and Unterzettlitz, respectively. All three are located just out- catchment area is 4,244 km2 (Figure 2). A total of 77 rain side the city perimeter. Flood discharge forecast of these gauges are available in the catchment area. Historical data three gauges is considered in the current study.

Figure 2 | Case study map of Upper Main catchment (at the top left corner) and zoomed view of the location of the discharge gauges and the city of Kulmbach. 929 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Between 2009 and 2014, four different flood events were actual DWD measured hourly rainfall at rain gauge locations seen in Kulmbach (Figure 3). These flood events occurred was available to this study and is therefore used here as forecast during January 2011 (referred to as Jan 2011 event), January input data. The current work hindcasts the mentioned four 2012 (referred to as Jan 2012 event), December 2012 flood events using multi-ensemble-based ‘Ensemble method’ (referred to as Dec 2012) and May 2013. Within these four and a proposed new method named ‘Discharge Interval events, Jan 2011 was the highest and had a hundred-year method’. The latter method needs a selection of best ensemble return period (Bhola et al. ). The recorded discharge member pairs which requires one flood event result utilising was 92 m3/s and 74 m3/s at Kauerndorf and Ködnitz, multi-ensemble-based hindcast along with a well-calibrated respectively. However, at Unterzettlitz, the flood discharge base model. The flood event of Dec 2012 is used to exemplify was recorded as 252 m3/s. The May 2013 event was theforecastingtool(inFigure 3). This event was considered second highest, where the recorded highest flows were for the selection of pairs of the Discharge Interval method, as around 50 m3/s at Kauerndorf, 100 m3/s at Ködnitz and it is advised not to use a very high flood event for calibration 185 m3/s at Unterzettlitz. During the Dec 2012 flood, Köd- purposes of a hydrological model (Laurent et al. ). nitz gauge recorded close to 65 m3/s flood discharge which was higher than that of Kauerndorf flow, close to 40 m3/s. The last event of Jan 2012 was slightly less signifi- METHODS cant in terms of observed highest flood discharge. A hydrological model result may have contributions from Generation of rainfall and rainfall uncertainty different uncertainty sources. However, in this work, the uncer- tainty estimation in the forecast was done at two stages: (a) The rainfall uncertainty module checks observed or forecast rainfall (input) uncertainty and (b) parameter uncertainty. The rainfall data at the available stations and distributes the data

Figure 3 | Historical observed discharge data at the three river gauges near the city of Kulmbach, namely, Kauerndorf, Ködnitz and Unterzettlitz with the four selected most recent flood events. The solid and the dashed lines represent flood discharge from the base model simulation and observed data, respectively. Dec 2012 event was chosen for the selection of best pair ensembles and analysis of Discharge Interval method, and the remaining three events were chosen for validation. 930 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

within the whole catchment considering sequential con- parameters were identified as most sensitive including ditional geospatial simulation. Previously, Wilks () EQD, beta, BSF, EQB, EQI, EQD2, A2 and Dmax (Härder extended a Markov-chain based approach for precipitation ). In this work, only these eight parameters were con- occurrence combined with exponential distribution for non- sidered for uncertainty analysis out of the 34 available zero amounts to multiple sites, successfully reproducing parameters in the LARSIM model. Table 1 shows the means, variances and interstation correlations. The depen- description and value range of these considered parameters. dence structures may be captured in a more complex Later, the hydrological model was set for auto calibration, manner by copula-based approaches such as Bárdossy & applying historical observed flow data of five years (2009– Pegram (). Studies in order to simulate precipitation 2013) at Ködnitz and Kauerndorf gauges. This step assigned in complex spatial terrains are based on multifractals a set of 34 model parameters at each of the 81 sub-catch- (Deidda ), multiplicative cascades (Gupta & Waymire ments. In a next step, flood discharge from the original ; Nykanen & Harris ) or Gaussian random fields operational model (Laurent et al. ; Haag et al. ), cur- (Shah et al. ). rently in use by the Flood Forecast Centre at the Bavarian In this work, rainfall is not considered as normally dis- Water Authorities (base model) was compared with the tributed and continuous; on the contrary, the intermittency one obtained using the automatic calibrated model. Similar (alternation of rainfall and no rainfall) and the positive results were obtained between the two models, which indi- skewness of the distribution of nonzero rainfall are con- cated that the hydrological response of the study area sidered within the geospatial simulation, which is a similar remained unchanged from 2009 to 2013 considering the approach to Wilks (). Therefore, a discrete–continuous base model calibration. Hence, the base model parameters mixed distribution is considered, using two variants. The dis- were kept unchanged and considered as the calibrated par- crete part of the distribution is empirically recorded via the ameters in this work. proportion of zeros in the total sample, and the continuous In the end, the ensemble was generated randomly from part is mapped on the three-parametric gamma distribution a given parameter range, as described by Härder (). as well as by a nonparametric kernel density. The whole Different normal or beta probability distribution curves geostatistical simulation is implemented using two different were prepared for each sub-catchment such that the curves R-packages, namely, gstat (Gomez-Hernandez & Journal consider the base model parameter as the median value ; Pebesma ) and RandomFields (Schalather et al. and empirical parameter range as the extreme values. ). Later, one set of parameters was chosen randomly for The LARSIM model takes input from distributed rainfall each of the sub-catchments considering the eight parameters data for the catchment at a spatial resolution of 1 km × 1 km. mentioned in Table 1. The remaining 26 parameters were Preliminary tests showed that the variation in the spatial kept unchanged as the base model. interpolation of the rainfall distribution has less effect on discharge forecasts when compared to the effects of vari- Forecast of flood discharge using FloodEvac Ensemble ation in hydrological parameters. Therefore, and for this method work, only 10 rainfall simulation sets were considered to estimate the uncertainty of the spatial rainfall distribution Before the hindcasting process can start, a warm-up period (Figure 4). is required. The model is simulated using observed precipi- tation and temperature data for one year of the warm-up Generation of parameter uncertainty period until 49 hours before the forecast initialisation time. At this stage, the model uses the base model parameters. An analysis of parametric uncertainty was performed in The model results are stored in an ‘initial state file’ at the LARSIM base model to derive the most sensitive parameters end of the one-year warm-up simulation time. As such, we of the model regarding flood discharge at the upstream ensure that the internal model state condition of the gauges of Kulmbach, i.e., Ködnitz and Kauerndorf. Eight basins is as close as possible to the real conditions. Later, 931 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Figure 4 | Uncertainty simulation of rainfall distribution map. Data show 1 hour accumulated rainfall distribution in mm. this initialisation state is used to simulate each forecast run the flood forecasting for at least 49 hours ahead of the initial of the ensemble. However, in the case of starting the simu- time (Ludwig & Bremicker ), to allow the model suffi- lation from an ‘initial state file’, it is recommended to start cient additional warm-up period before producing reliable 932 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Table 1 | Parameters considered for uncertainty analysis of the hydrological model

Parameter Recommended empirical name Unit Description value range (Härder 2017)

EQB [–] Gauging size for the retaining constant of the basis discharge storage 10,000–50,000 EQI [–] Calibration variable for the retaining constant of the interflow storage 1,000–5,000 EQD [–] Calibration variable for the retaining constant of the slow runoff storage 500–1,000 EQD2 [–] Calibration variable for the retaining constant of the fast runoff storage 10–500 A2 [mm/h] Threshold value, if reached the surface runoff will be assigned to the fast runoff storage 0.5–4.0 BSF [–] Exponent of the soil moisture saturation area function for adjustment of the share of 0.01–0.5 runoff as a function of the soil storage load beta [1/d] Drainage index for the deep seepage (to the basis discharge storage) 0.000002–0.1 Dmax [–] Index for lateral drainage to the interflow storage in the area of large grain sizes 0–10

results. Therefore, each forecast simulation was run for 61 pattern and the model uses the same parameter set that hours. In this process, the tool collects 49 hours of observed was generated at the first stage. hourly rainfall followed by 12 hours of forecast rainfall data. These 61 hours of rainfall data are passed through the rain- Forecast of flood discharge using the Discharge Interval fall distribution uncertainty module, and 10 different rainfall method uncertainty datasets are prepared. Later, 50 different par- ameter sets are produced using the parameter uncertainty In the case study, the weather forecasting frequency is 1 hour. module. These 50 parameter sets are combined with the In the case of flood forecast mode, the new results would 10 rainfall uncertainty cases, repeating one rainfall scenario become outdated at the end of an hour as updated weather with every five parameter sets in sequential order, thus, forecasts become available. However, the Ensemble making 50 sets of hydrological models for the Upper Main method takes a significant fraction of an hour in flood dis- catchment. These 50 models are simulated, and the results charge forecasting. For real-time forecasts, it is intended to of discharge ensembles are stored. In this way, the run- reduce the forecast time requirement. In this section, we pro- time required for producing a forecast for the next 12 pose an alternative option to produce similar uncertainty hours is around 25 minutes on a desktop with 3 GHz Intel bands of forecast within a much shorter period. This method- processors and 16 GB of RAM, running in parallel mode ology is termed as ‘Discharge Interval method’ hereinafter. using three cores. Later, the first 49 hours of simulation The Discharge Interval method is proposed to optimise results are deleted, and the last 12 hours of forecast data the Ensemble method in order to improve the compu- are stored. tational time required for the forecast model. We chose New weather forecast/updated rain gauge data become different parameters for different flood intervals in order available at every hour in this case. The tool can receive real- to properly forecast all the flood levels with the intention time observed rainfall and discharge data at every hour. For to forecast the possible maximum and minimum flood dis- this reason, the forecast process repeats at every hour. In the charges instead of forecasting all the ensemble members. next forecast process, the model simulates a new ensemble However, this method requires prior knowledge of the of 50 simulation runs of 61 hours simulation each, of hydrological response in the catchment area before the which the first 48 hours are a repetition of the previous actual forecast, which can be obtained by hindcasting a simulation; the 49th-hour data use the latest available real- flood event using the Ensemble method. This event is con- time observed data and the last 12 hours data use the sidered as the best pair selection event. newly available weather forecasts. At this stage, the tool In the proposed method, the possible observed dis- regenerates 10 sets of rainfall simulations using the same charge is divided into predefined discharge intervals (ΔQ), 933 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

based on the selection event. A pair of flood discharge of all the ensemble simulations for the flood discharge fore- ensembles is selected for each interval in such a way so cast at gauge Ködnitz. that the ensemble pair can bracket the maximum numbers As in the hindcast, the rainfall data are available at every of observed discharge data within them. At a later stage, hour; the Ensemble method forecast was done using 50 only the selected best pair ensemble parameter sets are ensemble members to limit the computation time required. used to forecast the possible maximum and minimum dis- The forecast quality might have been improved if more charge for any other events within the same catchment. uncertainty runs were used. However, it can be seen from The pairs are expected to give a maximum and a minimum Figure 5 that the ensemble of these 50 members could effec- limit of flood discharge forecast. Larger intervals will lead to tively bracket the observed discharge data at both very few best pairs. Very small intervals will lead to many Kauerndorf and Ködnitz. best pairs, which loses the benefits against using the ensem- The quality of the forecast data is assessed from each ble in the first place. Figure 1(b) shows a flowchart of the hourly lead of the forecast. Figure 6 shows the scatter plot whole process. of ensemble forecasts and corresponding observed dis- charge data. The inconsistency between forecast and observed data increases with increasing forecast lead time, UNCERTAINTY ANALYSIS OF THE ENSEMBLE as indicated by their increasing mean error (BIAS) and METHOD root mean square error (RMSE) values. It is apparent that the model is excellent for forecasting flood up to 4 hours The uncertainty in the Ensemble method forecast is assessed in advance as it shows a narrow spread at the scatter plot 3 in this section. The flood forecast results of Dec 2012 event with small BIAS (around 0.25 m /s) and the RMSE is 3 at gauge Ködnitz is used as an example. The event had two below 3 m /s. The uncertainty in forecast within 5 or 6 peaks, on 24 and 28 December, respectively. The peak dis- hours lead time is slightly worse as these values increase. charges were recorded as 64.2 m3/s and 58.4 m3/s on 24 However, the scatter plots at a lead time of 7 to 12 hours and 28 December, respectively. Figure 5 shows the results are even wider with higher RMSE, which is an indication

Figure 5 | All forecast uncertainty results from the Dec 2012 event at Kauerndorf and Ködnitz gauges. Each solid line indicates one forecast ensemble member of 12 hours long and dashed line indicates observed discharge over the whole time. 934 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Figure 6 | Scatter plots of observed vs forecast data for each hourly forecast lead for Dec 2012 event. Each scatter point represents one temporal data from simulated ensemble result, and corresponding observed data (adapted from Beg et al. (2018)).

of higher uncertainty. The shape of the scatter spread as well cause could be the lack of high temporal resolutions or qual- as the BIAS and the RMSE values are almost consistent at ity of input data (rainfall measurements). 10 to 12 hours lead time, which indicates that at a lead In assessing the uncertainty of the basin response to time of more than 9 hours, the forecast uncertainty becomes uncertain flood discharge forecast, confidence intervals are stable and stops increasing. calculated. According to the procedure described, for each It can be seen from different forecast leads, especially at time point of the forecast horizon, the discharges were fore- 7 to 12 hours, that LARSIM underestimated the high flood casted 12 times using a 50 member ensemble of uncertainty peaks. According to Ludwig & Bremicker (), the runs. In this way, each temporal result is predicted 600 935 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

times, which is used for the statistical assessments for each the flood discharge is found within the 98% confidence time step. interval of the simulated discharges. A confidence interval chart is shown for all the simulated forecast data (Figure 7). We calculate the confi- dence intervals of the whole forecast data in two parts: IMPROVING THE FORECAST COMPUTATIONAL considering a forecast lead time up to 6 hours and 12 EFFICIENCY: DISCHARGE INTERVAL METHOD hours. In calculating the confidence intervals at a specific date and time, all the forecasted data at that time were con- Defining the best pairs sidered for calculating the corresponding percentiles. Comparing with the observed flood discharge of the area, The Discharge Interval method requires one ensemble- it can be seen that the model can forecast the rising limb based flood hindcast result to optimise the forecast hydrolo- of the flood peak reasonably well. Both rising limbs of gical model parameters. As mentioned earlier, data at the observed flood discharge lie within the 90th percentiles Ködnitz gauge point from Dec 2012 event were chosen for of the simulated results at both 6 hours and 12 hours lead this purpose. forecasts. The peak discharge predicted in the model simu- In Ködnitz river gauge, the highest warning level (warn- lation is slightly earlier than the actual flood peak time. ing level 4) is equal to 69.1 m3/s (Bavarian State Office for However, the uncertainty interval is found higher at the the Environment ). Considering this issue, the possible flood peak as the range of 90th percentile band at 12 observed discharge is divided into seven segments with an hour lead is 42 and 78 m3/s, respectively. The similar interval of 10 m3/s starting from 0 discharge. Each possible uncertainty band is found considerably lower at 6 hours combination of ensemble member pair is investigated con- lead, varied between 48 and 65 m3/s. The falling limb of sidering the number of observations. For this part, all the

Figure 7 | Confidence intervals in flood forecasting at Ködnitz for Dec 2012 event (adapted from Beg et al. (2018)). 936 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

ensemble results, each having 50 members of 12 hours of different at each of the 81 sub-catchments. Out of these temporal forecast data were checked, and a list of all the 34, the first eight parameters, as mentioned in Table 1, are possible member pairs were listed. In this case, the current randomly selected and the remaining 26 have the same data had 50 members in each ensemble which creates a value as those in the base model. Ensemble member 2 also total of (50!/(2! × 48!)¼) 1,225 possible pair combinations. has 34 different parameters at each of the sub-catchments. In the next step, the tool checks which pair of ensemble However, eight of these parameters at each sub-catchment members can bracket the maximum numbers of observation are different from those assigned at ensemble member 1 discharge of a certain interval within them. A pair of mem- and the remaining 26 are the same. In this way, the whole bers containing the maximum numbers of observation is combination of parameters is directly related to better agree- selected and termed as ‘best pair’ for that discharge interval. ment of discharge observation at a certain interval. Later, the best pairs were applied to the other three events, e.g. whenever the latest observed discharge is within that Validation of the Discharge Interval method certain interval, the forecast model would use only the best pair parameter sets to forecast the next 12 hours of To check the effectiveness of the proposed Discharge flood discharge. The best pairs found for the current Interval method in the above section, we applied the model setup are listed in Table 2. methodology to hindcasting, using three other flood event For example, on 23 December 2012 at 19:00 (Figure 8), scenarios. The chosen three additional flood scenarios before starting the forecast for the next 12 hours, the fore- were (Figure 3): January 2011, January 2012 and May 2013. cast tool checks the latest available observed flood The methodology was also applied to the gauge locations discharge at Ködnitz gauge and finds that the observed of Kauerndorf, Ködnitz and Unterzettlitz, as these were flow at that time was 45.2 m3/s. Then the tool forecasts located just at the upstream of the city of Kulmbach. the next 12 hours of discharge using parameters of ensemble For these cases, the hindcast was run using both Ensem- member number 10 and 16. After 1 hour, on 23 December ble method and Discharge Interval method, utilising the 2012 at 20:00, the forecast tool repeats its forecast. At that same combination as described earlier. The seven ensemble time, the latest available observed forecast was 52.1 m3/s, member pairs chosen from the Dec 2012 event were used in and the forecast tool chooses a new pair of ensemble mem- the Discharge Interval method to check if they can still be bers to forecast the next 12 hours of flood, which are able to forecast the discharge appropriately at these three member number 32 and 45. However, it should be noted gauges. The results are shown in Figure 9. that the Upper Main river catchment area contains 81 sub- During the hindcast of January 2011, the observed dis- catchments and each of the ensemble member set contains charge showed two peaks with a maximum discharge different parameters at each sub-catchment level. As such, close to 80 m3/s at both Kauerndorf and Ködnitz and ensemble member 1 contains 34 parameters which are 180 m3/s at Unterzettlitz. The flood forecast predicted a similar flood peak at Kauerndorf and Unterzettlitz, however, much higher peak discharge at Ködnitz (160 m3/s). The Table 2 | Observed best pair sequence in the work seven pairs of selected ensemble members showed qualitat- ively similar flood prediction to the 90th percentile of the Discharge interval (m3/s) Ensemble member best pair Ensemble method. During the first flood peak between 6 0 < Q 10 36 and 46 and 12 January 2011, flood discharges were precisely fore- 10 < Q 20 17 and 34 casted using Discharge Interval method at each hour. The 20 < Q 30 10 and 36 rising limb, peak and the falling limb were accurately fore- 30 < Q 40 16 and 26 casted for this case. At the second peak discharge, the 40 < Q 50 10 and 16 Discharge Interval method forecast showed a much faster 50 < Q 60 32 and 45 rising and falling limb with a higher peak prediction than Q > 60 2 and 45 the observed. 937 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Figure 8 | Observed discharge with forecasts from Discharge Interval method for Dec 2012 event.

In the second flood event scenario of January 2012, discharge intervals. However, as the results already Discharge Interval method could not forecast the flood showed a good agreement with the validation of three event with enough accuracy at any of the three gauge additional events at three gauges, totally different parameter points. The first flood hydrograph has a duration of 2 sets are not expected. In any case, by increasing the ensem- days, considering its time of rising, peak and retention ble members, e.g., 100 runs, it might produce a different set time. The second peak discharge in this event had a dur- of parameters, which may slightly improve our results. A ation of nearly 4 days. The Ensemble-based forecast drastic increase is not expected since the validation already model as well as the Discharge Interval forecast model showed similarity to that of the Ensemble method. showed, to some extent, better performance at forecasting the second peak than the first. However, the highest dis- Assessing the forecast quality charge value was underpredicted at both peak flows at Kauerndorf and Ködnitz but overestimated at Unterzettlitz. To check the forecast quality, a likelihood table is pre- The forecast of the falling limb of this flood event showed sented considering the forecasted discharge using both better accuracy at all the three gauges. Ensemble method as well as our proposed Discharge Inter- In May 2013, the observed discharge peak was found to val method. For this purpose, data at Ködnitz gauge was be up to 50 m3/s, 120 m3/s and 180 m3/s at the three gauges, considered (Table 3). The table was formed considering respectively. The entire flood duration was for 11 days. Like yes/no forecasts and yes/no observed discharge consider- the other events, it also contained two peak discharges. Both ation after dividing the forecasted discharge range into forecast models underestimated the first peak but overesti- several subgroups/discharge interval thresholds. Hind- mated the second at both Kauerndorf and Ködnitz, but casted discharge data and observed discharge at each overestimated at Unterzettlitz. Although, in all the cases, hour were checked if they fall within a given threshold the Discharge Interval showed similar characteristics fore- bin and a data table was prepared considering a ‘yes’ if cast range to that of the 90% interval of the Ensemble the data are below the given threshold and a ‘no’ if the method. data are over the threshold. When both forecast and In the case a different flood event was used as a selec- observed data show ‘yes’, that is considered as a hit (a) tion event of best pair, there could have been a marginal andwhenbothshow‘no’,thatisconsideredasanothersuc- change in the selection of parameter sets at different cess of the forecast. 938 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Figure 9 | Hindcasting three validation events using Ensemble method and the proposed Discharge Interval method. The three flood events are Jan 2011 (left), Jan 2012 (middle) and May 2013 (right). The first, third and the fifth row indicate results from Ensemble method and the second, fourth and sixth row show the Discharge Interval method. The dashed line at each case indicates the observed discharge and the grey shades are indicators of forecast results.

After the likelihood table construction, hit ratio (also The results are represented as Receiver Operating known as the probability of detection–yes or PODy) and Characteristics (ROC) (Mason & Graham ; Marzban false alarm ratio (also known as the probability of false ; Liguori et al. ; Codo & Rico-Ramirez ), detection or POFD) are counted using the following which describe the occurrence of hit and false alarm rate equations for every discharge threshold considered: for a defined event, helping in crucial decision-making issues (Mason & Graham ). a Hit ratio ¼ (1) a þ b The area under the curve is a good indicator of the fore- b cast quality and applied in many studies to assess the False alarm ratio ¼ (2) b þ d goodness of numerical forecasts (Murphy & Winkler ; 939 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Table 3 | Likelihood table for calculating a forecast category may happen in some cases based on the observed data and the interval chosen in ROC curve construction. For example, Observed Observed Threshold discharge ¼ x discharge x discharge > x for a particular discharge interval, if observation and the

Forecast discharge xa b 95th percentile fall in the same interval whereas the 5th per- Forecast discharge > xc d centile remains below, the 95th percentile would contribute to higher ROC area. It may change for another case where the 5th percentile and the observation may fall in the same cat- Zhang & Casey ; Mullen & Buizza ). A good fore- egory keeping the 95th percentile outside. In that case, the 5th cast is indicated by a higher area under the curve, which is percentile will contribute to higher ROC area. For these closer to 1. If a forecast can predict a predefined event per- reasons, the ROC curves may intersect each other. Either of fectly, the corresponding event would calculate a PODy ¼ 1 the percentiles may not necessarily be better than the other. and POFD ¼ 0, and the point would be located at the left top The same condition applies to the upper and the lower corner of the curve. The first diagonal line represents the bound forecasts of the Discharge Interval method. In line of ‘No skill’. Whenever a point is located below the diag- Table 4, all the average area under the ROC curve is presented. onal line, it is termed as worse scenario than no prediction. For the Ensemble method, the average area from 5th and 95th The computational method is not parametric without percentile are calculated and for Discharge Interval method, suggesting any hypothesis regarding the probability distri- the averaged area between upper bound ROC and the lower bution of the forecast. The ROC curve characteristic, bound ROC is considered. however, is dependent on the number of discrete values Comparing the two forecast bands, we can see that the used in creating the curve (Mason & Graham ). ROC curve areas of both forecast methods are adjacent to In this work, the forecast quality was assessed using the each other. The flood events of January 2012, December 5th and 95th percentile forecast data, selected from the 2012 and May 2013 forecasts showed high predictability Ensemble method discharge results. The Discharge Interval and had ROC curve area between 0.83 and 0.95. The Janu- method provides two discharge ensembles considering the ary 2011 event had the least forecast area under the curve. upper and the lower bound of the forecasted discharge, These results also show that the predictability of discharge and both ensembles were used to construct the ROC decreases slightly with lead in both forecast methods. curves. The predefined discharge intervals were chosen at Table 4 also shows the fact that the forecast data pro- an interval of 5 m3/s, starting from 0. Figure 10 shows one vided by the Discharge Interval method are similar to that example of ROC curves from hindcast results of all the of the Ensemble method. The forecast band provided by four flood events each using the four different forecast this method can be comparable to the 90th percentile inter- results with 4 hours of forecast lead. val of the Ensemble method. This indicates that the The Ensemble method and the Discharge Interval method Discharge Interval method forecasts the discharges con- both showed almost similar forecast quality (see Figure 10). siderably faster as well as with reliable accuracy. ROC curve band created by the Discharge Interval method Forecasting using Discharge Interval method takes less was narrower, which also indicates a narrower flood forecast than 4 minutes to run a 12-hour forecast simulation set range. However, out of the four events, the January 2011 using a desktop with 3 GHz Intel processors and 16 GB of event had the worst forecast quality in both methods. Forecasts RAM. This indicates a seven times speed-up when compared of the other three events showed a better performance. All the to the simulation of Ensemble method with 50 members. forecast data showed less hit ratio (PODy) with higher false alarm (POFD) when forecasting small discharges. At forecast- ing higher discharges, the PODy was much closer to 1 for CONCLUSIONS January 2012, December 2012 and May 2013 events. During December 2012 and May 2013 events, the 5th and 95th per- In this work, a new flood discharge forecast methodology has centile ensembles’ ROC curves intersect each other. This been proposed to produce reliable real-time flood forecast. 940 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Figure 10 | ROC curve from the four hindcasted events at 4 hours lead time using 90th percentile of Ensemble method (EN band) and band of Discharge Interval method (DI band).

This is based on optimised Bayesian Model Averaged method The new method was implemented within the Flood- and termed as Discharge Interval method. This method uses Evac tool developed for real-time flood forecasting. The previously hindcasted flood discharge uncertainty ensembles tool considers both input and parameter uncertainty in and chooses a pair of ensemble members which can bracket simulating flood scenario ensembles. It creates spatial rain- maximum observations within a given discharge range. Later fall distributions using geospatial simulations and the it uses the parameter sets of the selected ensemble members uncertainty of parameters considering multi-Ensemble to forecast a flood of the same range. The pair creates an based simulations and provides the combinations of scen- upper and lower limit of the flood discharge forecast. arios to a hydrological model to generate flood discharge 941 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

Table 4 | Average area under the ROC curve for all the four events with both forecasting methods and different forecast leads at Ködnitz

Forecast lead time (hr)

Hindcast event 123456789101112

Jan 2011 EN method 0.72 0.65 0.64 0.64 0.61 0.59 0.58 0.55 0.53 0.52 0.51 0.52 DI method 0.67 0.69 0.68 0.70 0.73 0.73 0.70 0.69 0.67 0.72 0.66 0.63 Jan 2012 EN method 0.83 0.84 0.93 0.94 0.95 0.96 0.95 0.94 0.96 0.93 0.91 0.91 DI method 0.78 0.84 0.88 0.87 0.88 0.92 0.89 0.93 0.91 0.91 0.92 0.93 Dec 2012 EN method 0.87 0.87 0.91 0.91 0.92 0.92 0.93 0.94 0.95 0.95 0.96 0.96 DI method 0.89 0.89 0.91 0.92 0.88 0.88 0.86 0.83 0.82 0.82 0.81 0.81 May 2013 EN method 0.84 0.87 0.88 0.88 0.90 0.90 0.90 0.90 0.91 0.91 0.91 0.90 DI method 0.88 0.87 0.88 0.90 0.89 0.87 0.87 0.87 0.87 0.86 0.86 0.86

EN refers to Ensemble; DI refers to Discharge Interval method.

ensembles. The tool was applied to hindcast flood at the city of the training event/ensemble member selection event. of Kulmbach located at the catchment of the Upper Main The method considers that the hydrological response of river. the catchment remains unchanged during the other fore- In this case study, the tool effectively hindcasted four casts. For this reason, it is recommended to retrain the historical flood events and was found useful to predict model after a high flood event. flood discharge with 12 hours of lead time using both Ensemble method and Discharge Interval method. Uncer- tainty analysis was done from the Ensemble method ACKNOWLEDGEMENTS results. Analysis showed that the uncertainty in the fore- casted discharge increases with the forecast lead time. The The development of the FloodEvac tool is funded by the forecast showed a narrow scatter plot spread for up to 4 German Federal Ministry of Education and Research fi hours lead time. The forecast within 5 to 6 hours lead (BMBF, FKZ 13N13196 (TU Munich)). The rst, second showed a slightly wider uncertainty band which would and sixth authors would also like to acknowledge the ’ give approximately 7.5 hours of preparation time for QUICS project funded by the European Union s Seventh taking the necessary preventive actions by the city council. Framework Programme for research, technological The forecast quality of the proposed Discharge Interval development and demonstration under grant agreement fi method was compared with results from Ensemble-based no. 607000 and Project UID/MAR/04292/2019, nanced forecast simulations. The comparison was done using ROC by MEC (Portuguese Ministry of Education and Science) curves. ROC curves of the Ensemble method and Discharge as well as the ESF (European Social Fund), under the Interval method showed that the latter methodology could programs POPH/QREN (Human Potential Operational. deliver qualitatively similar forecast data when compared Very special thanks to the Bavarian Water Authority and with the results provided by the former. Still, the time Bavarian Environment Agency in Hof for providing us required for forecasting was considerably small. However, with the quality data to conduct the research. this method is unable to produce confidence interval band estimations in the forecast scenarios. REFERENCES The proposed Discharge Interval method is very fast indicating an increase of speed of seven times as it runs Barbetta, S., Coccia, G., Moramarco, T., Brocca, L. & Todini, E. only a pair of simulations for the forecast. However, the  The multi temporal/multi-model approach to predictive method is dependent on the proper hydrological response uncertainty assessment in real-time flood forecasting. Journal 942 M. N. A. Beg et al. | Discharge Interval method for uncertain flood forecasts: city of Kulmbach Journal of Hydroinformatics | 21.5 | 2019

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First received 20 November 2018; accepted in revised form 20 June 2019. Available online 18 July 2019

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