To test further the robustness and to generalize, the ICE algorithm was tested on dif-ferent models and difdif-ferent catchments
7.5.1 Result from physically based model
The ICE algorithm was developed using conceptual model. To make more general pur-pose, it was used in a complex physically based model like WaSim-ETH. A brief expla-nation of the application of the ICE algorithm for WaSim-ETH is given in this section;
for more details please refer to combine work withLiang (2010). For details about the WaSim-ETH model refer to Chapter 4. The study area chosen for this work is located in the Rems catchment (for more details, refer to chapter 4). The data for the period 1992-1993 was chosen for the model calibration. Year 1992 was used for model warm up.
A events were selected based on discharge using ICE algorithm. An example of event selected from calibration period is given in figure7.6. It is very noticeable here that low flow can also be chosen as a critical event. The TOPMODEL approach of WaSim-ETH model was used in this chapter. The nine most sensitive parameters of soil modules were calibrated using the ROPE algorithm for two cases:
• Case 1 Using whole data set
• Case 2 Using carefully selected events
7.5 Applications of the ICE Algorithm
Figure 7.6: Critical events selected from year 1993 from Rems catchments
Para. initial min max mean std.
m case 1
0.0001-0.2 0.006 0.080 0.035 0.022
case 2 0.0004 0.086 0.040 0.022
Tkor
case 1
0.1-500 6.71 52.4 29.5 11.3
case 2 7.29 78.8 44.5 20.1
Kkor case 1
0-5000 38.7 922.7 455.9 205.1
case 2 80.7 946.9 476.0 212.2
kD case 1
0-500 31.6 53.2 42.7 4.25
case 2 39.6 61.3 49.4 2.49
shmx case 1
0-100 3.66 57.0 32.5 13.2
case 2 9.71 85.7 46.3 21.8
kH
case 1
0-1000 29.9 254.1 141.1 53.9
case 2 41.3 454.7 228.5 99.8
Pgrz case 1
0-100 1.520 23.4 13.2 5.50
case 2 0.806 43.3 21.7 9.66
rk case 1
0-1.0 0.211 0.846 0.546 0.174
case 2 0.112 0.944 0.536 0.212
cmlt case 1
0-1.0 0.217 0.943 0.551 0.172
case 2 0.060 0.878 0.487 0.209
Table 7.7: Statistics of the best 10 % parameter sets of WaSim-ETH, calibrated by ROPE al-gorithm
97
Table 7.7 shows the statistical result of the best 10 % calibrated parameter set. It is very clear from table that model parameters obtained using both cases (as mentioned above) is very different. In order to see the difference of parameter properties obtained during calibration for both cases, the distribution of each parameter value is plotted (fig 7.7). Here, one can see that, parameter distribution for each case is different. Some of the parameters, like Kd, the surface flow recession constant, are more structured in case 2 than case 1. This shows that this parameter is more identifiable in case 2.
When we look carefully at the model structure, this parameter is active during peak flows, and event selected by ICE algorithm were related to peaks. So it is obvious that these parameter get a better chance to adjust during calibration in case 2. Very similar features can be seen in other parameters. To compare the structure of the distribution of parameter obtained in both the cases, the statistical entropy was introduced. Smaller entropy means more structure in distribution. Statistical entropy is given by:
S =−X
pilnpi (7.3)
where i is the class of discretization andpi is the corresponding probability of occurrence in that class. Statistical entropy for all the parameters of WaSim-ETH model is given in table7.8. The entropy values for parametersm,rk and cmltcalibrated using case 1 are smaller, which implies that these parameters are more concentrated in one or two certain classes, while those calibrated using case 2 are more scattered. On the other hand, parameters Tkor, kD, shmx, kH and Pgrz have a lower statistical entropy values in case 2. This shows that these parameters as more structured in this case where we selected events by the ICE algorithm. This gives more identifiability to these parameters.
ForKkor, the entropy values in both cases are more or less the same, which means the identifiability of this parameter on certain values doesn’t change much. However, it doesn’t mean these certain values will be the same since entropy only represents the scattering degree of a sample, but not why at which point of the sample is scattered.
Para. m Tkor Kkor kD shmx kH Pgrz rk cmlt case 1 1.74 1.97 2.04 1.05 2.12 1.98 1.99 1.80 1.91 case 2 2.04 1.76 2.07 0.52 1.77 1.72 1.81 2.03 2.09 Table 7.8: Statistical entropy for different parameters of WaSim-ETH
The calibrated WaSim-ETH model was validated for another time period (1994-1996).
Table 7.9 shows the calibration and validation results. The comparison for both the cases were made based on several objective functions as given in table. For different objective functions, the performance of the parameter sets obtained by case 1 is slightly better than those by case 2 in calibration period. However, if we look for the validation period, the result obtained by both the cases is nearly the same. The mean Nash Suttcliff coefficient for year 1994, 1995, 1996 is 0.93, 0.78 and 0.88, respectively, in case 1 and is very similar in case 2. Calibration using both the cases is therefore, nearly the same. To aid the visual appraisal of the results, the hydrograph for the calibration and validation period is plotted in figure 7.8. ROPE calibration does not give single parameter set,
7.5 Applications of the ICE Algorithm
Figure 7.7: Distributions of parameters calibrated in both cases
99
instead giving parameter space, hence confidence bound for hydrograph is plotted in figure7.8. As we can seen from figure, in both the cases, dynamic of hydrograph is same in calibration period except at certain peaks (e.g. at days around 330 to 360). During validation for year 1994 to 1996, the result in case 1 and case 2 is nearly equal. This clearly indicates that the event selected by ICE algorithm is also suitable for calibrating a very complex physically based model.
Obj. func. calib93 valid94 valid95 valid96
case 1 case 2 case 1 case 2 case 1 case 2 case 1 case 2 NS
min 0.833 0.833 0.924 0.924 0.738 0.779 0.869 0.868 max 0.851 0.841 0.951 0.947 0.816 0.816 0.913 0.914 mean 0.837 0.834 0.932 0.925 0.786 0.781 0.884 0.871 stdv 0.006 0.001 0.010 0.004 0.012 0.006 0.016 0.008 logNS
min 0.761 0.761 0.709 0.709 0.723 0.736 0.594 0.612 max 0.808 0.789 0.775 0.768 0.792 0.785 0.672 0.664 mean 0.774 0.762 0.734 0.712 0.754 0.739 0.631 0.615 stdv 0.018 0.005 0.028 0.012 0.020 0.010 0.022 0.010 RMSE
min 0.573 0.593 0.397 0.411 0.405 0.404 0.218 0.217 max 0.606 0.606 0.493 0.494 0.483 0.443 0.267 0.268 mean 0.600 0.606 0.464 0.490 0.435 0.441 0.251 0.265 stdv 0.010 0.003 0.035 0.015 0.013 0.006 0.017 0.009 logRMSE
min 0.292 0.306 0.316 0.321 0.285 0.289 0.310 0.313 max 0.326 0.326 0.360 0.360 0.328 0.320 0.344 0.337 mean 0.316 0.325 0.344 0.358 0.309 0.319 0.328 0.335 stdv 0.013 0.004 0.018 0.008 0.013 0.006 0.010 0.005 Table 7.9: Statistics for calibration and validation of WaSim-ETH (best 10 % performance)
7.5.2 Result from data driven model
The ICE algorithm was also applied in improving the training of Artificial Neural Net-works (data driven model for fitting), an application that relies on information-rich data.
As the name suggests, data driven models (DDM) try to infer the behavior of a given system from the data presented for model training/calibration. Hence, the input data used for training should cover the entire range of inputs that the system is likely to experience and data of all the relevant variables should be used. Some modellers (Maier and Dandy,2000; Bowden et al., 2002;Anctil et al., 2004;Bowden et al.,2005; Leahy et al.,2008) feel that the DDMs have the ability to determine which model inputs are critical and so a large amount of input data is given to the models, at times without any pre-processing. This approach has many disadvantages: more time and effort is needed to train the model and frequently one may end up at a locally optimal solution.
A number of studies have been carried out in the past to determine the inputs to the data driven models (Gaweda et al., 2001; Bowden et al., 2002; Anctil et al., 2004; Bowden
7.5 Applications of the ICE Algorithm
Figure 7.8: Calibration and validation with 90 % confidence
101
et al.,2005; May et al., 2008;Fernando et al., 2009). In data driven modeling (Fuzzy, ANN) there is no rigorous criteria that exists for input selection (Gaweda et al.,2001).
Commonly used methods involve taking a time series model to determine the inputs for a DDM. A review of relevant studies was provided by Maier and Dandy (2000).
Regarding the length of the data series, a common assumption is that the use of a longer time series of data will result in better training. This is because a longer series may contain different kinds of events and this may improve the training of DDMs. However, experience shows that a longer time series does not necessarily mean more information because there can be many repetitions of similar type of information (Wagener et al., 2003). In such cases, one may not necessarily get a better trained model despite wasting a lot of computational time and may over-fit the series (Fernando et al.,2009;Gaweda et al.,2001).
From the above discussion, it is can be concluded that the training of a DDM could be improved if the data of the events that are “rich” in information are used. Here the term
“rich” denotes the data with very high information content. Use of this term is based on the fact that some data epochs contain more information about the system than others.
Available input data can be pre-processed to leave out the data which does not contain any new information. This is important in training a DDM because these critical events mainly influence the training process and the calculation of weights.
In this study, geometrical properties of data were used to identify critical events from the long time series of data. Identification of critical events (ICE) algorithm was used to identify the critical events from the data series. An artificial neural network (ANN), which is a DDM approach, was trained on the critical events identified by the identifi-cation of critical events (ICE) algorithm. To test the robustness of the ICE algorithm for identification of critical events, random selection of events was performed and ANN was also trained on randomly selected events. The result was compared with the ANN trained on whole data.