Incorporating Likelihood information into Multiobjective Calibration of Conceptual Rainfall- Runoff Models

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1 International Congress on Environmental Modelling and Software Brigham Young University BYU ScholarsArchive th International Congress on Environmental Modelling and Software - Barcelona, Catalonia, Spain - July Jul st, : AM Incorporating Likelihood information into Multiobjective Calibration of Conceptual Rainfall- Runoff Models Alireza Azemi Andrew H. C. Chan Xin Yao Follow this and additional works at: Azemi, Alireza; Chan, Andrew H. C.; and Yao, Xin, "Incorporating Likelihood information into Multi-objective Calibration of Conceptual Rainfall-Runoff Models" (). International Congress on Environmental Modelling and Software.. This Event is brought to you for free and open access by the Civil and Environmental Engineering at BYU ScholarsArchive. It has been accepted for inclusion in International Congress on Environmental Modelling and Software by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 iemss : International Congress on Environmental Modelling and Software Intelligent Environmental Decision Support Systems th Biennial Meeting of iemss, M. Sànchez-Marrè, J. Béjar, J. Comas, A. Rizzoli and G. Guariso (Eds.) International Environmental Modelling and Software Society (iemss), Incorporating Likelihood information into Multi-objective Calibration of Conceptual Rainfall-Runoff Models Alireza azemi a, Andrew H. C. Chan a, Xin Yao b a Department of Civil Engineering, The University of Birmingham, Birmingham, UK, B TT (a.nazemi@cs.bham.ac.uk, a.h.chan@bham.ac.uk ) b School of Computer Science, The University of Birmingham, Birmingham, UK, B TT (x.yao@cs.bham.ac.uk) Abstract: In this paper, the incorporation of parametric likelihood information into NSGA- II algorithm has been attempted in order to preserve solutions with more overall likelihood. The crowded comparison operator in NSGA-II, which is used to select the potential solutions for the next generation, is substituted by likelihood comparison operator which includes the consideration of the likelihood information about the potential solutions rather than their distance from each other. As a result the potential solution with higher overall likelihood measure has more chance to be selected in the next generation. Three different scenarios for the estimation of overall likelihood measure are presented. The modified algorithm is used for calibration of two different conceptual rainfall-runoff models in USA MOPEX catchments. The results show that the new modification results to different searching process which can be compared with NSGA-II from different perspectives. Keywords: Rainfall-Runoff models; Multi-objective calibration; NSGA-II; Parametric Likelihood, Likelihood comparison operator.. I TRODUCTIO The conceptual modelling paradigm assumes that rainfall-runoff process can be described by a simple sequential structure formed from a soil moisture accounting unit which converts the total rainfall into effective rainfall, followed by a routing unit which translates the amount of effective rainfall to its corresponding runoff quantity. All the conceptual models have some internal parameters which should be approximated using the observed measurements. Although the majority of previous calibration studies have been focused on single objective optimization for the calibration of conceptual rainfall-runoff models, recent investigations suggested that single objective function, no matter how carefully chosen, is often inadequate to properly represent all of the characteristics of the observed data and simulated values [Vrugt et al., a]. Also, Boyle et al. [] showed the existence of different response modes on catchment s hydrograph, during the wetting up and drainage periods. It can be also shown that there are different response modes during high and low flows in both the wetting and drainage periods. These observations have opened new research direction toward multi-objective calibration of rainfall-runoff models, in which each particular part of the hydrograph can be described by a separate objective function. Considering more than two conflicting objective functions in the search process, it may result in a set of solutions which might be quite diverse in both parametric and objective spaces. However, many of the solutions might not be useful in practice because of the considerable amount of sensitivity to small parametric changes. This behaviour, in many cases, is a disadvantage particularly in dealing with real world modelling problems, such as rainfall-runoff process, which contains several sources of uncertainties that can affect the performance of overall model output [Nazemi et al., a]. Looking at this problem from another perspective, it can be questioned that if it is more promising to use other aspects of

3 the solutions such as parametric likelihood in the search process which may be able to give more chance to more likely solutions to be included in the final calibration results. In this paper, an intuitive framework is introduced to incorporate likelihood information into the multi-objective calibration of rainfall-runoff models.. MULTI-OBJECTIVE CALIBRATIO OF RAI FALL-RU OFF MODELS Let assume that the observed hydrograph and the model structure are Qobs andξ, respectively. By considering the model parameter set θ, the simulated hydrograph Q( ξ θ ) can be calculated. In the case of multi-objective calibration, the difference between Q Q( ξ θ ) is measured by a set of M objective functions { f, f,..., f M } and calibration can be described as: min{ f, f,..., f θ Θ M } obs and The result of this optimization problem will be a set of Pareto parametric valuesφ in which there is no solution better than the other regarding to all M objective functions. From the optimization point of view, the task of multi-objective optimization is not only assigning the best parametric values regarding each objective function; but also locating all * possible Pareto optimal solutions and store them in the non-dominated setφ. Before applying multi-objective optimization framework to the calibration of rainfall-runoff models, a set of objective functions should be assigned. One possible approach would be to divide the whole hydrograph into different response modes and try to find a behavioural error measure related to each segment. We applied a heuristic segmentation approach introduced by Wagener and Wheater [] which uses the gradient of the hydrograph and an additional threshold as segmentation criteria to divide the whole hydrograph into different response modes. The flow gradient separates the hydrograph into wetting up and drainage periods in a way that positive gradient shows the rainfall periods, i.e. wetting up times and negative gradient shows the drainage periods. A threshold is used to separate period of high and low flow, which is the mean flow for wetting periods and % of the mean flow for drainage periods. In order to quantify the behaviour of the model for each segment, the Root Mean Squared Error (RMSE) in each mode was used. For instance for Driven High flows (FDH), the following error measure can be defined. nfdh Qi Qˆ ( i) () RMSE i FDH= =, i FDH nfdh Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a multi-objective evolutionary algorithm introduced by Deb et al. []. NSGA-II has been used in rainfall-runoff model calibration such as Tang et al [] and Nazemi et al [b]. In brief, NSGA-II can be considered as an ordinary evolutionary multi-objective algorithm, coupled with three new operators. The first operator is the fast non-dominated sort which can efficiently rank a set of solutions based on the multi-objective dominancy principle. The second operator is the crowding distance calculation operator which gives each solution a distant measure indicating its diversity and it is designed in a way that gives ultimate amount of distant to the edges of the Pareto-front. The third operator, i.e. the crowded comparison operator, is a selection rule designed in a way that not only consider the rank of the solutions but also select the solutions with more diversity distant measures based on the information obtained from crowding distance calculation. By this rule, the diversity of solution would be preserved during the search process. The algorithm will be run for a number of generations and the best non-dominating front will be saved in an archive. The solutions in this archive are the non-dominated solutions of the calibration. Although these solutions have the greatest importance from the theoretical point of view, they might be unsuitable for the implementation of the model in practice. Comprehensive () * 7

4 preliminary study done by authors showed that, in most of the cases, the usage of the whole Pareto set is practically impossible for model implementation because of the high variation in the model output and the diversity of non-dominated solutions in the parametric space. As a result, a solution or a region of solutions should be picked. However, the parametric selection is not an easy task due to the existence of different parametric and objective clusters that can be linked to each other in a quite irregular way. It means that similar (close) solutions in either space can map to different (distant) solutions in the other space, which shows large amount of multi-objective multi-modality in the models landscape.. I CORPORATI G LIKELIHOOD I FORMATIO I TO MULTI-OBJECTIVE CALIBRATIO OF RAI FALL-RU OFF MODELS As suggested by Vrugt et al [b], in Bayesian statistics, the parametric solutions are treated as probabilistic variables having joint probability density functions, which show the belief about the parametric solutions in the light of measured observation. Considering the model parameter as θ and the measured observation as y, the probability density function pθ ( y) is proportional to the product of likelihood function and the initial probability density function. Beven and Binley (99) proposed the imposition of a uniform distribution on the feasible parametric space as the prior density function. Assuming a noninformative prior distribution, it has been shown that the posterior probability density ( t) function p( θ y) is proportional to the sum square error of the model in the form of: ( t) ( t) p( y) ( e( ) i) i= () θ θ The above formulation can be easily extended to each segment of the hydrograph. As an (t) illustration, the posterior density distribution of parameter set θ in the light of observed driven high flow (FDH) segment of the hydrograph can be described as: FDH FDH t t ( ) ( ) () p( θ y FDH ) ( e( θ ) i) i = The crowded comparison operator of NSGA-II is an intuitive rule: A solution i dominates another solution j, if and only if the solution i has a better rank or both solutions have the same rank but the solution i has a higher crowding distance measure. Based on Deb et al [], this rule can be formulated as following: i p n j if ( i rank < jrank ) () Or [( i rank = jrank ) and ( i dist > jdist )] This formulation can be easily reconstructed in order to preserve the solutions with higher likelihood. Considering ilikelihood and j Likelihood as the representatives of the overall likelihood estimations for solutions i and j, respectively; the Likelihood Comparison Operator can be defined as following: i p n j if ( i rank < jrank ) Or i = j ) and () [( rank rank ( i Likelihood > jlikelihood )] For each solution, the row measures of total and segmental posterior density measure be calculated using Equation and Equation respectively. It should be noted that these equations allocate a lower posterior measures to more likely solutions; as a result, in order to express the more likely solutions with higher measures of likelihood, the row measures can be considered with a minus sign. In order to eliminate the effect of different scales, all likelihood measures should be normalized to the same scale, i.e., [,] using the maximum and minimum of the minus row posterior density measures regarding each segment. Therefore, considering m as the number of segments considered and n as the number of solutions in each selecting pool, it would be a matrix L indicating the relative likelihood n m

5 measures of the solutions in a particular selecting pool. Here, three different scenarios are suggested in order to estimate the overall likelihood measure for each solution. Scenario I. The maximum likelihood operator: In this scenario, the overall likelihood for each solution i can be estimated using following equation: max( U m i Likelihood = { i }) (7) k k= Scenario II. The average likelihood operator: In this interpretation, the overall likelihood can be estimated using following equation: m i Likelihood = ( i k ) () m k= Scenario III. The minimum likelihood operator: The overall measure can be described as min( U m i Likelihood = { i k }) (9) k= By applying one of these scenarios to estimate the overall likelihood measure, the crowded comparison operator of NSGA-II (Equation ) can be replaced by the likelihood comparison operator (Equation ) in order to preserve the solutions with higher amount of overall likelihood measure. The new modification can be considered as the Likelihood nondominated Sorting Genetic Algorithm II or LNSGA-II.. CASE STUDIES A D MODELS APPLIED. MOPEX catchments The Model Parameter Estimation Experiment (MOPEX) is an international project aimed at developing enhanced techniques for the a priori estimation of parameters in hydrologic models and in land surface parameterization schemes of atmospheric models [Duan et al. ]. In this study we considered daily time series data for USA catchments provided by MOPEX. These catchments can represent different hydrological mechanism. The key physical characteristics of the catchments used in this study are shown in Table. Table. The variation of key physical properties of twenty four USA MOPEX catchments used in this study ( Area (km ) Main channel slope (m/km) Stream length (km) Mean basin elevation (m) Area of storage (%) Forest area (%) Mean annual rainfall (mm) Models applied The conceptual structures used here are based on the Rainfall Runoff Modelling Toolbox (RRMT) [Wagener et al., ]. Both structures have a soil moisture accounting unit and a routing module. The two soil moisture accounting units have been used here are the catchment wetness index [Jakeman and Hornberger, 99] and the probability distribution of stores [Moore, 999]. These units have been combined with a two parallel linear reservoirs representing quick and slow responses of the system producing two conceptual models. Here, we labelled the two models used as model and Model. Both models have five free parameters. Table and Table summarize the free model parameters and their feasible parametric space. Table. Description and feasible interval for free parameters of Model applied in this study [Wagener et al., ] Model Feasible Definition parameter interval M- Soil moisture accounting parameter, time constant of the catchment losses [ ] M- Soil moisture accounting parameter, modulation factor [ ] M- Routing parameter, time constant of quick flow reservoir [ ] M- Routing parameter, time constant of slow flow reservoir [ ] 9

6 M- Routing parameter, fraction of flow through quick reservoir [ ] Table. Description and feasible interval for free parameters of Model applied in this study [Wagener et al., ] Model Feasible Definition parameter interval M- Soil moisture accounting parameter, maximum storage capacity [ ] M- Soil moisture accounting parameter, shape of Pareto distribution [.] M- Routing parameter, time constant of quick flow reservoir [ ] M- Routing parameter, time constant of slow flow reservoir [ ] M- Routing parameter, fraction of flow through quick reservoir [ ]. Calibration framework Three scenarios of LNSGA-II were introduced along with original NSGA-II algorithm for calibration of free parameters of Model and Model in MOPEX catchments. For each catchment, five years daily data have been considered in the calibration. Scatter crossover, Gaussian mutation and binary tournament selection have been fixed as the genetic operators in this experimental study. For the calibration of Model, five objective functions have been used including total and segmental RMSE. Also the same values for internal parameters have been used for NSGA-II and LNSGA-II scenarios. For Model, probability of crossover, probability of mutation, number of population and number of generations were assumed to be.,., and respectively. In the case of Model only four segmental error measures were used and the internal parameters were assumed to be.,., and for the probability of crossover, probability of mutation, number of population and number of generations, respectively. In order to compare the objective optimality of solutions, the non-dominated solutions resulted from applied algorithms were compared with the single objective calibration solution of SCEM [Vrugt et al., b]. If the archived solutions can reach relatively to the same level of optimality that single objective calibration does, it can be confirmed that the multi-objective sampler is at least as optimal as the single objective optimization but more time efficient because a set of objective functions were considered simultaneously. Considering the results of single objective optimization achieved by SCEM as the benchmark for the comparison, the level FS of optimality can be defined as in which F S and F M are the objective function FM convergence achieved by the single and multi-objective procedures, respectively.. RESULTS A D DISCUSSIO S Table shows the average level of optimality achieved by considered algorithms in MOPEX catchments using applied models in the case of Model, it is possible for LNSGA-II scenarios to converge to some optimal objective function values that can dominate the results of NSGA-II or even single objective solutions assigned by SCEM. In the case of Model, it can be concluded that LNSGA-II scenarios are not as successful as they were in the case of Model ; however, it is still possible for LNSGA-II scenarios to result in better level of objective function optimality in comparison to NSGA-II and SCEM for some of the MOPEX catchments. Considering the relationship between the Pareto fronts achieved from LNSGA-II scenarios and NSGA-II algorithm, three different outcomes can be observed and they are shown in Figure. In the first case, the LNSGA-II front can dominate the solutions achieved from NSGA-II. In the second one, LNSGA-II and NSGA-II can converge to more-or-less same front whereas in the third situation, the LNSGA-II solutions were dominated by NSGA-II. Based on this observation, it can be concluded that the applying likelihood comparison operator can result in an entirely different searching mechanism which can result in dominating, similar, or dominated solutions when compared to the NSGA-II solutions. Figure shows the histograms for the number of archived calibration solutions assigned by NSGA-II and scenarios of LNSGA-II considered in the MOPEX catchments. Analysing Figure, it can be concluded that for the LNSGA-II algorithm the diversity preservation

7 capability is lower than NSGA-II. This characteristic can be addressed by fundamental difference between the crowded comparison operator and the likelihood comparison operator for which the aim is to preserve the more likely solutions rather than providing a spread-out Pareto front. However, as Figure shows regarding the calibration of Model, LNSGA-II can converge quite fast toward the neighbourhood of the non-dominated solutions. This property can have a particular importance in the calibration of rainfallrunoff models in which in some cases, each model simulation can be quite computationally expensive. Perhaps, the advantage of this fast convergence can be further highlighted for the regionalization purposes in which a model should be calibrated using several catchments. However, it should be noted that there is a trade-off between speed and robustness of convergence in LNSGA-II. The same observation can be made for Model. Table. The average level of optimality achieved by considered algorithms in MOPEX catchments using applied models Model Calibration method RMSE RMSE(FDH) RMSE(FDL) RMSE(FNQ) RMSE(FNS) NSGA-II 9.7% 9.% 9.% 9.% 9.% Model LNSGA-II, S.% 9.%.% 97.9% 9.7% LNSGA-II, S 99.% 9.%.7% 97.% 9.7% LNSGA-II, S 99.% 9.%.9% 9.% 9.% NSGA-II % 9.77% 99.9% 9.% Model LNSGA-II, S % 99.7% 99.7% 9.7% LNSGA-II, S % 9.% 99.% 9.% LNSGA-II, S % 9.7% 99.9%.9% RMSE(FNQ)..7. RMSE(FNQ)... RMSE(FNQ) RMSE(FDH). RMSE(FDH)... RMSE(FDH) Figure. Different conditions between the Pareto front achieved by NSGA-II and LNSGA-II scenarios. Dots are representing NSGA-II solutions and the results of LNSGA- II scenarios have been plotted using stars, circles and triangular respectively. Left) Results of Model in catchment ID 7; Middle) Results of Model in catchment ID 7; Right) Results of Model in catchment ID 77 NSGA-II 7 LNSGA-II, First scenario 7 LNSGA-II, Second scenario 7 LNSGA-II, Third scenario.. O c c ure n c e... 7 NSGA-II LNSGA-II, First Scenario (a) LNSGA-II, Second Scenario LNSGA -II, Third Scenario Oc cu re nc e (b)

8 Figure. Comparison among the numbers of archived solutions for the applied models in MOPEX catchments using NSGA-II and LNSGA-II scenarios; (a) Model (b) Model Using LNSGA-II, a more compact and clustered solutions in the parametric space can result which makes the parameter selection less complex. As an illustration, Figure compares the MDS plots [Buja and Swayne, ] of parametric solutions resulted from NSGA-II and three scenarios using LNSGA-II for the calibration of Model in the catchment ID 77. As can be observed, NSGA-II results in wide spread solutions in the parametric space; whereas the solutions are more compact and clustered in MDS plots for LNSGA-II scenarios. RMSE(FDH) RMSE(FDL) RMSE(FNQ) RMSE(FNS).. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x.. Function evaluation x Figure. Comparison among the speed of convergence related to calibration of Model in MOPEX catchments using three different scenarios of LNSGA-II. The first row indicates the number of required function evaluations in the first scenario and the other two rows are related to second and third scenarios, respectively. (a) (b) (c) (d) Figure. Comparison among MDS plots of non-dominated solutions resulted from calibration of Model in catchment ID 77 using NSGA-II and three different scenarios for LNSGA-II; (a) NSGA-II (b) LNSGA-II, first scenario (c) LNSGA-II, second scenario (d) LNSGA-II, third scenario

9 . CO CLUSIO S Applying the whole Pareto calibration solutions of rainfall-runoff models is practically impossible due to the considerable amount of diversity in the assigned solutions of NSGA- II. This paper has explored the possibility of tailoring NSGA-II in order to capture more likely solutions instead of providing spread out Pareto optimal solutions. The results have shown that the solutions assigned by the tailored algorithm, i.e. LNSGA-II, contain less diversity (due to the less number of archived questions) and can locate more compact solutions in the parametric space (due to the results of MDS plots). In addition, changing the selection rule establishes a new searching mechanism which can result in a better or worse Pareto-front in comparison to the result of NSGA-II. However, it was found that LNSGA-II scenarios can converge much faster than NSGA-II. REFERE CES Beven, K. J. and Binley, A. M. The future of distributed models: model calibration and uncertainty prediction. Hydrological Processes,, pp. 79-9, 99. Boyle, D. P., Gupta, H. V., and Sorooshian S., Toward improved calibration of hydrological models: Combination the strengths of manual and automatic methods, Water Resources Research, VOL. (), -7,. Buja, A. and Swayne, D. Visualization Methodology for Multidimensional Scaling, Journal of Classification, (9), pp 7, Deb, K., Pratap, A., Agarwal, S., Mayarivan, T., Fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, VOL., NO., pp.-9,. Duan, Q., Schaake, J., Andreassian, V., et al., Model Parameter Estimation Experiment (MOPEX): An Overview of Science Strategy and Major Results from the Second and Third Workshops, J. Hydrol.,, vol., nos., pp. 7. Jakeman, A. J., Hornberger, G. M., How much complexity is warranted in rainfall-runoff model, Water Resources Research, VOL. 9, NO., pp. 7-9, 99. Moore, R.J., Real-time flood forecasting systems: Perspectives and prospects. In: Floods and landslides: Integrated risk assessment (Eds. R. Casale and C. Margottini), Springer, Berlin, Nazemi, A.-R., Yao, x., Chan A. H., " Extracting a Set of Robust Pareto-Optimal Parameters for Hydrologic Models using NSGA-II and SCEM, in the proceeding of IEEE 7th World Congress on Computational Intelligence, Vancouver, Canada, July. Nazemi, A.-R., Yao, X., Chan A. H., " Multi-objective calibration of conceptual rainfallrunoff models using NSGA-II, in the proceeding of 7th International Conference on Hydroinformatics, Nice, France, September. Tang, Y., Reed, P., Wagener, T., How effective and efficient are multi-objective evolutionary algorithms at hydrologic model calibration? Hydrology and Earth Sciences Discussion, (), pp -,. Vrugt, J. A., Gupta H. V., Bastidas, L. A., Bouten, W., and Sorooshian, S., Effective and efficient algorithm for multi-objective optimization of hydrologic models, Water Resources Research, VOL. 9 (),. Vrugt, J. A., Gupta H. V., Bouten, W., and Sorooshian, S., A shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters, Water Resources Research, VOL. 9 (),. Wagener, T., Lee, M. J., Wheater, H. S., Rainfall-runoff modelling toolbox user manual (RRMT v.), Civil Engineering and Environmental Sciences Department, Imperial College, London,. Wagener, T., Wheater, H. S., On the evaluation of conceptual rainfall-runoff models using multiple-objectives and dynamic identifiability analysis, In Littlewood, I. (ed.), Continuous river flow simulation: methods, applications and uncertainty, British Hydrological Society, Occasional paper, No., Wallingford, UK, pp.-,.