ERM model analysis for adaptation to hydrological model errors
Abstract
Hydrological conditions are changed continuously and these phenomenons generate errors on flood forecasting models and will lead to get unrealistic results. Therefore, to overcome these difficulties, a concept called model updating is proposed in hydrological studies. Realtime model updating is one of the challenging processes in hydrological sciences and has not been entirely solved due to lack of knowledge about the future state of the catchment under study. Basically, in terms of flood forecasting process, errors propagated from the rainfallrunoff model are enumerated as the main source of uncertainty in the forecasting model. Hence, to dominate the exciting errors, several methods have been proposed by researchers to update the rainfallrunoff models such as parameter updating, model state updating, and correction on input data. The current study focuses on investigations about the ability of rainfallrunoff model parameters to cope with three types of existing errors, timing, shape and volume as the common errors in hydrological modelling. The new lumped model, the ERM model, has been selected for this study to evaluate its parameters for its use in model updating to cope with the stated errors. Investigation about ten events proves that the ERM model parameters can be updated to cope with the errors without the need to recalibrate the model.
Keywords
Realtime model updating Forecasting errors Concentration time Time to peakIntroduction
Land use/land cover change and climate change have significant influences on catchment hydrological characteristics. The appearance of these phenomena has potential effects on generating unusual flood events and may lead to produce various types of flooding. Study about these natural occurrences and their interaction with hydrological models has been mentioned in the literature with explanation of methods to control and reduce their risks (Arnell 1999; Wilby et al. 1994; Xu 1999; Dibike and Coulibaly 2005; Hagg et al. 2007; Charlton et al. 2006).
Irregular rainfall events have direct linkage with unanticipated floods around a catchment. Consequently, performing precautionary actions like developing a model for realtime flood warning can help reduce flood damages during a flood event significantly. Flood forecasting is an important part of water resource management activities related to flood warning, flood control or reservoir operation (Yang and Michel 2000), but still lots of efforts are required to develop highly accurate models for operational hydrology. Realtime flood forecasting is important for every day operation, management of water control systems and for emergency cases where protection of life and property is concerned (Lardet and Obled 1994). In many countries, flood warning systems come towards the top of the government’s policy priority list (PenningRowsell et al. 2000). Accurate realtime flood forecasting with an adequate lead time can help to confront flood hazards in an efficient time period. Realtime flood forecasting model and an updating technique should be integrated (Yu and Chen 2005) which is applied by operators to predict flood events.
The model updating proposed in this study is model parameter updating by finding a linkage between the model parameter and hydrological conditions as an approach to explore model adaptivity by synthetically generated rainfallrunoff data to test the model’s capability to adapt to the changes. The reason for using the ERM model is due to its clarity, simplicity with a small number of model parameters.
The ERM model has reliable performance when one routing component is used and a brief difference is observed with two routing components. Hence due to the reduced number of calibrated parameters in terms of model updating, the ERM model with one routing component (eight parameters) has been selected. Two scenarios will be discussed in the following sections: updating the model before the forecasting process and preparing the model based on predicted hydrological changes. For the first part, a continuous event is selected and the ERM parameters are calibrated based on the observed runoff, and then ten single events are investigated to update them. Three parameters of the ERM model compatible with the model errors are selected in the updating process. The model updating is carried out just by changing these parameters and the rest of parameters should be kept on their optimum levels. In this way, if the parameters update the model properly, there is no need to recalibrate the model which is not an easy process especially in realtime flood forecasting.
Existing errors in hydrological modelling
Obviously, predicting the hydrological changes and preparing the model parameters based on the new conditions can make the forecasted results more reliable in comparison with modelling by the optimum parameters which are obtained under the calibration conditions. The optimum parameters have been estimated based on a series of recorded data for a certain period of time. However, diversity of hydrological condition at any time make this problem more evident and the model should be adapted based on the new conditions to achieve more reliable flood forecasts.
Catchment conditions for volume error
The source of volume error could be linked with changes in soil moisture content which causes changes in runoff volume in the catchment outlet. These changes may be due to variations in land use/land cover, soil compaction, etc. (Chiew et al. 1995; Fu et al. 2003). Nguyen et al. (1998) did an investigation about the impact of animal grazing on soil physical properties and they confirmed that grazing animals like cattle could change the soil properties and reduce water infiltration into the soil. The changing scale of the soil characteristics may be influenced by the type of animals, pasture cover, stocking rate, grazing duration, soil texture, soil structure and soil organic matter content (McCalla et al. 1984). In addition to the animal grazing, a number of farm activities such as farm machines operation and land tillage are important factors to change the soil density which could alter rainfall infiltration into the soil.
Catchment conditions for timing error
It takes time for the rainfall to reach the river outlet. For a lumped hydrological model such as the ERM model, the timing error would occur if the delay time between rainfall and runoff response from a catchment changes. With different rainfall locations in the catchment (e.g., upper reach, middle reach or lower reach), water arrives at the catchment outlet at different times. If the rainfall is near the catchment outlet, the flow will arrive at the outlet sooner than rainfall far away from the outlet. Hence, selecting a reliable delay time between effective rainfall and the time for starting runoff generation could be helpful to determine the timing errors.
Catchment conditions for shape errors
The shape error is generally linked with catchment concentration time. Many factors can influence the concentration time. For example, different intensities of rainfall may produce different concentration times (raindrops reach the river outlet faster under heavy rainfall intensity than lighter rainfall intensity). Other factors which influence the shape of hydrograph are flow paths with different roughness, slope and length. At the moment, most of lumped rainfallrunoff models assume a fixed catchment concentration time and are unable to overcome the shape error. Hence, assuming a unique concentration time for all storm events around a catchment may cause uncertainties in terms of modelling process. But, by recognising the concentration time (or time to peak) estimated for an event and updating specific parameters of the model, it is possible to update the model according to the new conditions. To overcome difficulties caused by the shape error and update the model to reach a more accurate forecast, the shape of the observed hydrograph is rotated under a certain degree, and then the model parameters are updated based on the new observed hydrograph. The link between the required angle for rotating the hydrograph and the predicted time to peak will be described in the followings sections.
Altered hydrographs to reflect the catchment conditions
The mechanism of model updating in this section is categorised into two classifications: first reducing the modelling errors in simulated runoff hydrograph (before forecasting) and second predicting the possible changes in hydrological conditions and adjusting the model parameters according to the new conditions. For this reason, the observed hydrograph is changed according to the predicted changes and the simulated hydrograph and the model parameters are updated to simulate a runoff hydrograph similar to the altered hydrograph. To cope with each type of error, just one of the ERM parameters is updated. Therefore, instead of recalibrating all of the model parameters, the most effective parameters related to the model errors will be updated. In some cases, the modelling errors could be a mix of all the errors, hence it might be required to update all the three parameters at the same time to cope with all the sources of errors. Consequently, any changes on the system could be carried out by just playing with maximum three parameters. The change of the observed hydrograph to reflect the hydrological changes and selecting the proper parameters of the ERM model will be described in the following sections.
Simulate the volume error conditions
Simulate the timing error conditions
Simulate the shape error conditions
According to Fig. 5, the time to peak has been changed after the rotation process. The relationship between the new time to peak and the rotation angle can be introduced. In the current study, a method has been proposed to adjust the model under the new condition.
The final rotation equation (Eq. (3)) is proposed to derive the required rotation angle, and the time to peak should be estimated before the rotation and after the rotation. In other words, the rotation angle is estimated by the difference between the predicted time to peak and the estimated time to peak before forecasting. Different types of empirical equations have been proposed to estimate the time of concentration and time to peak such as Kirpich (1940), Johnstone and Cross (1949), Haktanir and Sezen (1990) and Fang et al. (2008).
As a wellknown equation in hydrological sciences, the Kinematic wave model was proposed by Morgali and Linsley (1965) to estimate the time of concentration. The equation has been used widely in studies such as McCuen and Spiess (1995), Wong and Chen (1997), etc.
Another empirical equation developed to estimate the catchment concentration time is the Izzard equation (Izzard and Hicks 1946). The study was based on runoff produced by rainfall on a manmade surface such as highway pavement or airfield runway. The Soil Conservation Service (SCS) sheet flow equation was revised based on a modified kinematic wave equation for sheet flow USDA SCS (1986). All the stated equations may be used in the catchments which lack the measured rainfall and runoff data to derive such a relation. Basically, those equations are based on a number of experimental results obtained from specific catchments and under various conditions. Hence, they may have large uncertainties when they are applied to different catchments. As the time to peak is a requirement for the current study, deriving an equation for estimating the time to peak directly by the data obtained from the actual catchment could prevent large errors in the estimation. Hence, in the next section, a particular equation will be derived for the catchment under study.
Developing an empirical equation to estimate time to peak

Estimate the center of storm using tipping bucket gauge records around the catchment;

Derive the time to peak for each storm using effective rainfall and observed runoff hydrograph;

Fit a surface to extract an equation between the time to peak, the maximum effective rainfall and the center of storm.
To start the process, sixty events have been selected from the Brue catchment in derivation of the equation. The events have been selected from different years to cover varieties occurred around the catchment. The ERM model is used to calculate the effective rainfall assigned to each storm event.
Estimating center of storm
For all of the selected events, \(L_{\text{s}}\) should be estimated by the stated equation. The subcatchment area in the equation is selected as the area covered by each rain gauge. The HYREX study used 49 tipping bucket rain gauges around the Brue catchment to collect rainfall data. Based on the recorded data by the rain gauges, the numbers of gauges in service are different from time to time.
To estimate the average areal rainfall and the area covered by each gauge, the Thiessen polygon method is applied. In this term, the catchment area is divided to a series of polygons and each polygon becomes as a subcatchment to estimate \(L_{\text{s}}\). After dividing the catchment area into the polygons, rainfall intensity is estimated for each polygon using the gauges records. Also, the centroids of the polygons are estimated to find the distance of the centroid to the catchment outlet. At the end, \(L_{\text{s}}\) is calculated for the specific rainfall event.
Application of ArcMap to estimate the center of storm
Performance coefficient estimated after surface fitting
Performance coefficient  Value 

\({\text{SSE}}({\text{h}}^{2} )\)  23.45 
\({\text{RMSE}} ({\text{h}})\)  0.647 
\(R^{2}\)  0.85 
As an individual evaluation, the \(R^{2}\) value proves the reliability of the fitted surface. The obtained equation can be considered as the unique equation derived using the Brue catchment data and is useful for estimating the time to peak just for the Brue catchment. However, the proposed method could be used in extracting similar equations in different catchments instead of using empirical equations. In following section, the adaptivity of the ERM model will be discussed and tested by a series of real storm events.
Evaluation of the adaptivity of the ERM
The previous sections described the potential errors in rainfallrunoff models and the simulated hydrograph linked to the hydrological changes for model updating. The simulated hydrograph is used to test the adaptivity of the ERM model by adjusting the model parameters. This process will make the model adapt to the new hydrological conditions and the updated parameters become more reliable to use in flood forecasting.
The model comes with eight parameters which are derived during the model calibration. Also, a parameter called delay time is defined which is the time between the beginning of effective rainfall generation and the starting of runoff generation. This parameter is estimated manually based on trial and error. Among the model parameters, three parameters are selected to cope with volume, shape and timing errors, according to their mechanism and roles in model structure.
\(R^{2}\) and \({\text{RMSE}}\) values calculated by the ERM model for the Brue catchment
\(R^{2}\)  \({\text{RMSE}}\)  

Calibration  0.80  2.2 (m^{3} s^{−1}) 
Validation  0.78  3.2 (m^{3} s^{−1}) 
Calibrated parameters of the ERM obtained for the Brue catchments
Parameter  Description  Units  Est. parameters 

\(C\)  Mass balance  mm^{−1}  0.000915 
\(\tau_{w}\)  Reference drying rate  h  300.02 
\(F\)  Temperature modulation  1/°C  4.28 
\(T_{\text{r}}\)  Reference temperature  °C  1.8 
\(L\)  Soil moisture index threshold  –  0.01 
\(P\)  Power on soil moisture  –  1.16 
\(K\)  Storage constant  h  20.00 
\(X\)  Weighting factor  –  0.31 
To test the ability of the model parameters, eventbased analysis is performed by selecting a number of events from the calibrated continuous hydrograph. Realtime flood forecasting is a shortterm forecasting and updating the model based on an event before the forecasting process will make the forecasting results more trustable. Therefore, the updating process is carried out on individual events. Ten events are extracted for the process of parameter updating.
In the second stage and before the updating process, for each type of error, a parameter of the ERM model is selected. According to the structure of the ERM model, parameter \(C\)(mass balance) is calibrated to ensure that the volume of effective rainfall is equal to the total volume of the observed runoff. Hence, any change in the volume of the observed runoff (as shown in Fig. 3) can be adjusted by regulating the parameter \(C\). Therefore, to overcome the difference between the simulated and observed runoff hydrographs, updating the parameter \(C\) could be helpful to cope with this error.
The Muskingum routing scheme embedded in the ERM structure is based on the continuity equation. Therefore, the effective rainfall is assumed as the inflow into the routing system to generate the outflow. According to the difference between the effective rainfall generation and runoff generation over a catchment, a delay time is defined. The timing error is caused by change in delay time depending on the distance of rainfall event to catchment outlet. Hence, by changing the delay time between the inflow to the catchment (effective rainfall) and catchment response (simulated runoff) in the catchment outlet, the difficulties caused by the timing error (as shown in Fig. 4) could be solved.
Updated parameters for the selected events
Event ID  \(C\)  \(K\)  Delay time (h) 

1  0.00037378  16.2  6 
2  0.00078587  21.3  6 
3  0.00098287  13.4  6 
4  0.00079987  19.1  5 
5  0.000635587  17.5  6 
6  0.00130187  9.5  6 
7  0.00040587  26.2  8 
8  0.00101587  17.4  6 
9  0.000986587  29.4  8 
10  0.0011321  21.8  5 
\(R^{2}\) and \({\text{RMSE}}\) values calculated by ERM model for the selected single events
Event  Start date  End date  \(R^{2} ({\text{sim}})\)  \({\text{RMSE}} ({\text{sim}})({\text{m}}^{3} {\text{s}}^{  1} )\)  \(R^{2} ({\text{upd}})\)  \({\text{RMSE }}({\text{upd}})({\text{m}}^{3} {\text{s}}^{  1} )\) 

1  29/09/93—23:00  02/10/93—17:00  − 1.190  1.509  0.854  0.389 
2  04/10/93—21:00  05/10/93—21:00  0.144  2.636  0.987  0.320 
3  06/10/93—06:00  08/10/93—01:00  0.736  2.580  0.853  1.924 
4  09/10/93—04:00  10/10/93—03:00  0.134  1.689  0.944  0.427 
5  11/10/93—14:00  12/10/93—21:00  − 1.395  3.135  0.563  1.337 
6  12/10/93—22:00  17/10/93—00:00  0.828  7.515  0.926  4.917 
7  09/11/93—20:00  11/11/93—13:00  − 16.224  2.434  0.763  0.285 
8  13/11/93—02:00  15/11/93—02:00  0.605  1.698  0.878  0.941 
9  08/12/93—06:00  09/12/93—13:00  − 3.276  1.344  0.825  0.271 
10  14/12/93—19:00  16/12/93—13:00  0.359  2.248  0.800  1.255 
A comparison between the hydrographs and the performance coefficients proves that by just updating three model parameters, significant improvements could be achieved. This is important in realtime flood forecasting, because it is easier to adjust 1–3 parameters instead of all 9 model parameters. This process helps to classify the flood events by providing a lookup table based on the flood characteristics. In this method, the best parameters are estimated for each flood event and a number of events are analysed to provide a lookup table which is based on a catchment condition. In terms of realtime flood forecasting, after recognising the flood characteristics and weather condition, the best parameter set could be selected from the lookup table for forecasting the shape of the hydrograph. Forecasting the flood hydrograph helps to provide an overview about the events ahead and provides interesting information for hydrologists.
Conclusion
The current paper discusses the prevalent errors on runoff simulation. Occurrence of errors during a simulation process is a considerable concern and hence, the model should be monitored continuously to identify the sources of errors and make efforts to update the model. The problem is highlighted in realtime flood forecasting due to limited time to do the process of model updating. Parameter updating is one of the existing methods to adjust the model before starting the forecasting process. In this study, the ability of the ERM model is investigated to cope with volume, timing and shape errors. In the first part of the study, the probable hydrological changes for the model are illustrated by altering the observed hydrograph. Therefore, by changing the observed hydrograph, the model parameters should be updated to cope with the newly altered hydrographs. Due to the importance of time in process of realtime forecasting and updating, for each error type, one of the ERM model parameter is assigned to cope with a certain error type. By selecting the parameters, their abilities are evaluated on real events. Ten runoff events are selected from a continuous runoff simulation to use in updating process. The selected events are not simulated properly in the progress of continuous simulation. In some cases, the existing one error type (e.g., timing error) is observed and in some cases the existence of all the error types are identified, hence updating all the selected parameters to cope with the errors are required. After implementing the parameter updating on the selected event and calculating the performance coefficients, it is confirmed that the ERM parameters have reliable flexibility to cope with the three possible errors in the simulated hydrographs, without a need to recalibrate the model parameters.
Consequently, the ERM model parameters are updated in two ways: (1) updating the model parameters before starting the forecasting process to reduce difference between the simulated and observed runoff hydrographs; (2) predicting the hydrological changes and alerting the observed hydrograph accordingly, and updating the model parameters based on the new observed hydrograph. Obviously, using the updated parameters is more suitable in new conditions, in comparison with the optimum parameter set. As for the future work, developing a comprehensive model to predict the hydrological changes could be mentioned as a supplementary action to apply in terms of parameter updating. In this way, the amounts of changes are predicted and will be applied to the observed hydrograph and the model will be prepared under new conditions.
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