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A nonlinear rainfall–runoff model embedded with an automated calibration method – Part 2: The automated calibration method
Journal of Hydrology (2007) 341, 196– 206
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jhydrol
A nonlinear rainfall–runoff model embedded with an automated calibration method – Part 2: The automated calibration method Gwo-Fong Lin *, Chun-Ming Wang Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan Received 25 May 2006; received in revised form 4 March 2007; accepted 4 May 2007
KEYWORDS Automated calibration; Rainfall–runoff modeling; Global optimization; Fitness evaluation; Genetic algorithms; Sigma truncation
The purpose of this paper is to develop an automated calibration method for the nonlinear computational units cascaded (NCUC) model. The simple genetic algorithm (SGA), a popular and robust optimization technique, is introduced in this paper as the basis of the automated calibration method. Therefore, the way to transform the model calibration problem into the optimization problem is first proposed. The general scheme to appropriately arrange the parameters of the NCUC model is then developed, so that the chromosomes of the SGA can be properly constructed. Two performance criterion functions, which are frequently used to evaluate the performance of the rainfall–runoff modeling, are adopted in this paper as the objective function to calibrate the NCUC model. Since the SGA imposes two restrictions on the fitness values, the key of the proposed automated calibration method is the evaluation of the fitness values. The methods to evaluate the fitness values according to the two objective functions are both given in this paper. With the proposed automated calibration method, high-quality parameters of the NCUC model can be obtained without modelers’ subjective interventions. ª 2007 Elsevier B.V. All rights reserved.
Summary
Introduction Many rainfall–runoff models have been developed by hydrologists to model the rainfall–runoff process. Among these models, the conceptual models are frequently used. The parameters of conceptual models, in general, cannot be ob* Corresponding author. Tel.: +886 2 3366 4368; fax: +886 2 2363 1558. E-mail address: [email protected] (G.-F. Lin).
tained from the measurable quantities of watersheds, but can be calibrated using the available rainfall and runoff records. The accuracy of the conceptual models greatly depends on appropriate parameters. Therefore, the model calibration is a necessary procedure for the conceptual models. Model calibration is often performed by either manual adjustment or by the computer-based automated calibration techniques. Manual calibration is often done by trail and error, and thus only the experienced hydrologists can obtain feasible parameters by manual calibration. Besides, the
0022-1694/$ - see front matter ª 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2007.05.008
A nonlinear rainfall–runoff model embedded with an automated calibration method modeling result is often not as good as that of the ANN-based model. For example, Franchini et al. (1996) used the TOPMODEL (Beven, 2001), which is a set of conceptual tools based on the approximate hydrological theories, to model the rainfall–runoff process of the Sieve basin in the central Italy. The model parameters were obtained using manual calibration, and the performance of the model (compared with that of the ANN-based model) is just acceptable. Because the performance of the conceptual rainfall–runoff models highly depends on proper parameters, the crucial problem encountered by modelers is to select the appropriate parameters that make the model closely characterize the rainfall–runoff process of the study watershed. The calibration of a conceptual rainfall–runoff model, in fact, is to search a set of model parameters so that the simulated runoff is in good agreement with the observed. The calibration of conceptual rainfall–runoff models is difficult and time consuming, so modelers require a simple and effective automated calibration method to estimate the proper parameters. In the recent years, with the advances of the computer technology, the job of the model calibration is preferably done using computers. However, modelers still need an efficient computer-based method to effectively calibrate the conceptual models. In this paper, a popular, simple and powerful technique, the simple genetic algorithm (SGA), which is a kind of genetic algorithms (GAs), is adopted as basis of the automated calibration method. The GAs are widely used in many studies of the optimization of water resources planning (Cai et al., 2001; Erickson et al., 2002) and the calibration of model parameters (Cooper et al., 1997; Wang, 1997). The concept of the GAs is quite simple and is stated in the later section. ‘‘Survival of the fittest’’, the principle of evolution, is the prime thought of the GAs. The GAs search the solution space for many feasible solutions of the studied problem simultaneously using a stochastic and parallel approach. One advantage of the GAs over gradient-based nonlinear method is that the GAs can search good approximate solutions, even when the model functions are nondifferentiable or discontinuous (Reeves, 1997). With a well setup and sufficient iterations, the GAs mature quickly and the optimal solutions can then be obtained. For facilitating the usability, an effective and efficient method to calibrate the NCUC model without modeler’s subjective intervention is necessary. The purpose of this paper is to develop a SGA based method that can automatically calibrate the NCUC model according to the desired objective function. Since the GAs are for the optimization problems, the method that transforms the model calibration problem into the optimization problem is developed. Hence the GAs can be applied to calibrate the NCUC model. Then necessary transformations of the objective functions are made in order to calibrate the NCUC model using the GAs. Therefore, for the engineering projects of different purposes the proposed automated calibration method can be employed to calibrate the NCUC model using different objective functions.
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descriptions of the SGA can be found in Goldberg (1989). In this paper, only a concise introduction of SGA is provided so that the readers can easily understand the methodology proposed herein. The capabilities of the natural creatures adapting to the environment exist implicitly in the chromosomes of their DNAs. The survival pressure among natural creatures increases with the increasing proliferations of the natural creatures and the changes of the environment. New creatures are produced by the mating of their parents. They receive certain parts of the DNAs that are from their parents respectively. In other words, they inherit certain features of their parents that give them the probability to adapt to the environment. Only the natural creatures that are adaptive to the environment can survive. The essentials of the evolutions of nature are adopted to form the fundamentals of the GAs. A chromosome that denotes a feasible solution to the studied problem is the building block of the GAs. A chromosome is composed of several parameters that are coded using a certain coding strategy. One popular coding strategy is the binary coding because it can preserve more information for the GAs (Goldberg, 1989). A certain number of chromosomes form the population of the GAs. The population is initialized randomly and then updated continuously in further evolutions. The capability of a chromosome adapting to the environment is called the fitness. The fitness of a chromosome is calculated using the objective function that is selected depending on the users’ requirements. There are many variations of the GAs, while the important features are common. The three operators, reproduction, crossover and mutation, that all manipulate the population form the basics of the GAs. Among the population, the chromosomes that have larger fitness have larger possibilities to be selected by the reproduction operator for the mating of new chromosomes. Only the better chromosomes that represent better solutions to the studied problem are preserved for generating probably better solutions in the further generations. Then these better chromosomes exchange the useful information contained in themselves by using the crossover operator. Through the exchanges of these chromosomes, probably better new chromosomes are generated. For preventing from falling to the local optimums and directing the GAs to the solution space that may have not been searched, the mutation operator alters randomly certain genes (i.e., the basic element of a chromosome) of the chromosomes of the population with a very small probability. With a sufficient number of iterations till the maturation of the GAs, the global optimal solutions can then be obtained.
Calibration of the NCUC model using the SGA Transformation of the model calibration problem into the optimization problem The simulated runoff of the NCUC model is represented by
Genetic algorithms (GAs)
b ðc; hÞ ¼ NCUCj ðPÞ Q c;h
There are many variations of the GAs. SGA, one of the variations of the GAs, is used in this paper. Comprehensive
b ðc; hÞ is the simulated runoff generated by the where Q NCUC model of pattern c with parameters set h, NCUCjc,h()
ð1Þ
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represents the NCUC model of pattern c with parameters set h, and P is the areal average rainfall that is the input of the NCUC model. As indicated in Part 1 of these serial papers (Lin and Wang, 2007), the configurations and the number of the NCUs that are cascaded in the NCUC model are referred to as the pattern of the NCUC model. For objectively quantifying the accuracy of the NCUC model of pattern c with parameters set h, the following equation is used: b ðc; hÞÞ ps ¼ PCFðQ ; Q
ð2Þ
where PCF(Æ,Æ) is the performance criterion function, ps is the performance score, and Q is the observed runoff. Various performance criterion functions have been proposed to evaluate the performance of the rainfall–runoff modeling for different purposes. The calibration problem is transformed into the optimization problem by treating the performance criterion function (Eq. (2)) as the objective function. Then the calibration of the NCUC model is equivalent to optimizing the objective: b ðc; hÞÞ minimize or maximize OFðQ ; Q
ð3Þ
subject to the constraints: EISi P 0; i ¼ 1; 2; . . . ; m 0 6 CVCij 6 1; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ni
ð4Þ ð5Þ
SCVij P 0; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ni 0 6 CSVi 6 1; i ¼ 1; 2; . . . ; m 0 6 CISVi 6 1; i ¼ 1; 2; . . . ; m
ð6Þ ð7Þ ð8Þ
where OF(Æ, Æ) is the objective function that is the same as the performance criterion function of Eq. (2), EISi is the ele-
Figure 1
vation of the initial storage of the ith NCU, CVCij is the jth composite vent coefficient of the ith NCU, SCVij is the sill of the jth composite vent of the ith NCU, CSVi is the coefficient of single vent of the ith NCU, CISVi is the coefficient of the intermediate single vent of the ith NCU, m is the number of the NCUs contained in the NCUC model, and ni is the number of the composite vents of the ith NCU. The aforementioned constraints are the ranges of the values of the NCUC model’s parameters. With this transformation, the parameters set that optimizes the objective function subject to the constraints is the optimal parameters set for the NCUC model. In this paper, the terms ‘objective function’ and ‘performance criterion function’ are considered as equivalent. For the objective function that indicates the better performance using a smaller value of the performance score, the optimal parameters set is obtained by minimizing the objective function. Otherwise, the optimal parameters set is obtained by maximizing the objective function.
Construction of the chromosomes There are four different types of NCUs. Modelers can freely select the configuration of an NCU as well as the number of the NCUs cascaded in the NCUC model. Obviously, the number of patterns of the NCUC model is infinite. Therefore, a general method for the construction of the chromosomes used in the SGA is crucial. The parameters of an NCU can be divided into two categories, namely, intrinsic and optional. The intrinsic parameters include the elevation of the initial storage (EIS), the coefficient of the single vent (CSV) and the coefficient of the intermediate single vent (CISV). The number
Layouts of the partial chromosomes representing the parameters of the corresponding NCUs.
A nonlinear rainfall–runoff model embedded with an automated calibration method
Figure 2
199
Layouts of the chromosomes representing the parameters of the NCUC model with the pattern {D1, C2}.
of the intrinsic parameters of a specific NUC is fixed. For example, the intrinsic parameter of the A-NCU is only the EIS, and the intrinsic parameters of the D-NCU are the EIS, CSV and CISV. The optional parameters are the parameters of the composite vents. A composite vent has two parameters, the composite vent coefficient (CVC) and the sill of composite vents (SCV). These two parameters certainly appear pairwisely. An NCU is perhaps only a unit of the NCUC model, so the parameters of an NCU form a part of the chromosome. The method to arrange the parameters in a chromosome is given in Fig. 1. In this figure, (a), (b), (c), and (d) are the layouts of the A-, B-, C-, and D-NCUs, respectively, and each grid represents a gene of the corresponding chromosome. The binary coding (Goldberg, 1989) is used to encode the NCUC model’s parameters in this paper and hence a gene that represents a parameter of the NCUC model is a binary string. The parameters of an NCU represented by the binary-coded genes are denoted by the corresponding symbols above the genes. For the four different NCUs, the intrinsic parameters are all placed in the beginning positions of a chromosome and then the optional parameters are concatenated successively. The number n in Fig. 1 is the number of the composite vents. With the method to arrange the parameters in a chromosome, the construction of the chromosomes of an NCUC model can then be easily achieved. The chromosomes of an NCUC are simply generated by concatenating the partial chromosomes of the NCUs sequentially. For example, the chromosome of the NCUC model with the pattern {D1, C2} is shown in Fig. 2. Readers can refer to Part 1 of these serial papers for more information regarding the notation {D1, C2}. With the aforementioned strategy for the construction of the chromosomes, the automated calibration method can thus be applied to calibrate the NCUC model of arbitrary patterns.
Ranges of the values and the precision of the parameters In fact, the restrictions on the NCUC model’s parameters (see Eqs. (4)–(8)) are not actually the constraints. They
are merely the upper and the lower bounds of the values of the parameters. With these restrictions, the solution space of the calibration problem of the NCUC model can be more easily defined. Nevertheless, the upper bounds of parameters EIS and SCV are not defined. The problem can be easily solved by enforcing very large values to the upper bounds of these two parameters. Because the EIS implies the water content of a watershed, it is not less than zero. And the water content of a watershed is not a very large value either, since the capability of a watershed to store water is limited. However a large value of the EIS is possible. Therefore, if an extremely large value is assigned to be the upper bound of the EIS, the SGA will search the optimal value of the EIS automatically within the artificially selected range. And the probability of the EIS being a large value would be kept. The strategy to select the upper bound of the EIS is also applied to choose the upper bound of the SCV. An extremely large value is picked to be the upper bound of the SCV as well, and the cause of such a choice is identical to that of the selection of the EIS’s upper bound. A binary string represents a gene of the chromosome in SGA. The lengths (the number of the digits of the binary string) of the genes affect the efficiency of the calibration and the precision of the calibration result. The precision of a gene representing a parameter is defined as follows: p¼
Umax Umin 2l 1
ð9Þ
where p is the precision of the parameter, Umax is the upper bound of the parameter, Umin is the lower bound of the parameter, and l is the length of binary string of the gene that represents the parameter. The precision of a parameter represents the minimum altered step of the parameter during the calibration processes. A high precision results from a long binary string, but a long binary string causes inefficiency of the calibration process. On the other hand, using a short binary string reduces the accuracy of the modeling, but the efficiency of the calibration process is increased. Modelers should carefully consider the lengths of the binary strings that represent the model parameters according to their requirements.
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Objective functions and the evaluation of the fitness values of chromosomes Most behaviors of the SGA are controlled by the fitness values of the chromosomes. The fitness value of a chromosome implies the degree of the chromosome’s goodness regarding the calibration. The evaluation of the fitness value of a chromosome is based on the objective function. However, the values of the fitness should be nonnegative, and the better chromosome should be represented by a larger value of fitness (Goldberg, 1989). To comply with these two restrictions, the values of the objective functions should be transformed properly into the fitness values. Hydrological engineering projects of different purposes require different objective functions to calibrate the rainfall–runoff model. In this paper, two frequently used objective functions are adopted for the calibration of the NCUC model. One is the simple least square (SLS): SLS ¼
r h i2 X b ðtÞ Q ðtÞ Q
ð10Þ
t¼1
b ðtÞ is the observed runoff at time t, Q(t) is the simwhere Q ulated runoff at time t, and r is the number of the time steps. For the SLS objective function, the set of model parameters that minimizes the SLS is the optimal solution. The SLS is nonnegative. To comply with the other restriction on the fitness of the SGA, when the fitness is evaluated using the SLS as the objective function, the fitness of a chromosome is calculated by f¼
1 SLS
where f 0 is the scaled fitness, lEC and rEC are respectively the average and standard deviation of the EC of all the chromosomes in the population, c is a small integer, and x; x > 0 NðxÞ ¼ ð14Þ 0; otherwise
Procedures of the automated calibration method The procedures of the automated calibration method for the NCUC model are illustrated in Fig. 3. In the initialization step, modelers determine the pattern of the NCUC model and the parameters of the SGA that control SGA’s behaviors. Four parameters of the SGA should be selected appropriately, so that the result of the calibration is desirable. The four parameters are the crossover probability, mutation probability, the population size, and the number of generations. A good result of the calibration using the SGA requires a large crossover probability, a small mutation probability, a moderate population size and a sufficient number of generations. Like the behavior of the natural creatures, the crossover probability is set as a value close to unity and hence crossover is not performed on all chromosomes. A small value close to zero is selected for the mutation probability, which is similar to the behavior of the natural creatures. If a large mutation probability is chosen, SGA will be a random search method that is not desirable. The population
ð11Þ
where f is the fitness of a chromosome. Using this method, a large SLS is transformed into a small f, so a worse chromosome has a smaller fitness value. On the other hand, a small SLS is transformed into a large f, so a better chromosome has a larger fitness value. The other objective function used to calibrate the NCUC model is the efficiency coefficient (EC) (Nash and Sutcliffe, 1970): Pr
t¼1 EC ¼ 1 P r
b ðtÞ2 ½Q ðtÞ Q
t¼1 ½Q ðtÞ
Q 2
ð12Þ
where Q is the average observed runoff. A larger EC implies the corresponding parameters set is a better one and otherwise the parameters set is not desirable. This characteristic of the efficiency coefficient satisfies the requirement of the SGA that the better solution has a larger fitness value. When the EC is used as the objective function to calibrate the NCUC model, the model parameters that maximize the EC are the optimal solution. Nonetheless, once the numerator of the second term on the right-hand side of Eq. (12) is larger than its denominator, the EC is negative. SGA does not allow the fitness value being negative. For solving this problem, the sigma truncation scaling technique (Dumitrescu, 2000) is introduced to transform the EC values into the scaled fitness values: f 0 ¼ NðEC ðlEC crEC ÞÞ
ð13Þ
Figure 3 Flowchart of the SGA-based automated calibration method for the NCUC model.
A nonlinear rainfall–runoff model embedded with an automated calibration method size should be a moderate value to provide adequate diversity of the initial chromosomes while the computational burden is limited. The number of generations should be large enough so that the performance of the calibrated parameters is satisfactory. After the initialization, the fitness of each chromosome in the population is evaluated using the corresponding method proposed above. Following the step of the ‘‘evaluate fitness’’ (see Fig. 3), the three operators of SGA are applied to the population sequentially until all chromosomes are processed. The terminal condition of the calibration in these serial papers is a predetermined number of generations. After the terminal condition is satisfied, the optimal model parameters, whose fitness value is the largest among the population, can be obtained.
Application The study area is the Fei-Tsui Reservoir Watershed in northern Taiwan. Information regarding the Fei-Tsui Reservoir Watershed can be found in Part 1 of these serial papers. Four storm events are selected for the modeling in this paper. As shown in Table 1, two events are used for calibration and the remaining events are used for validation. Storm events for the calibration are used to estimate the proper parameters of the NCUC model, and those for the validation are employed to verify the reliability of the calibrated model parameters. The input of the NCUC model, the areal average rainfall, is estimated using the Thiessen method (Chow et al., 1988). As aforementioned, the objective function is equivalent to the performance criterion function. Two frequently used performance criterion function, simple least square (SLS) and coefficient of efficiency (EC), are adopted as the objective functions for calibrating the NCUC model. Various patterns of the NCUC model have been examined. Four different and representative patterns of the NCUC model are picked for demonstrations (Table 2), and the corresponding parameters are calibrated according to the two different objective functions, SLS and EC, respectively. It should be noted that the calibration type in this paper is single-objective. That is, during the calibration stage only one objective function is used. The parameters of the SGA, the basis of the automated calibration method, are set as follows: the population size is 100, the probability of crossover is 0.8, the probability of mutation is 0.001, and the number of the evolution generations is 30,000. The upper bounds of all the EISs (elevation of the initial sill of an NCU) and the SCVs (the sill of a composite vent of an NCU) are set as 100 mm. In the automated calibration method, the lengths of all the binary strings that represent the model parame-
Table 1
201
ters are 10 digits. Therefore, the precision of the coefficients of the vents (including the single vents and the composite vents) is 9.78 · 104 and the precision of the EISs and the SCVs is 0.098 mm. The modeling results of the NCUC model with four different patterns are given in Figs. 4–7.
Results and discussion The versatility of the NCUC model and the optimal model pattern Sajikumar and Thandaveswara (1999) proposed two important characteristics, the accuracy and the versatility, for evaluating the performance of the rainfall–runoff model. The accuracy of the NCUC model is discussed in Part 1 of these serial papers. The term ‘‘versatility’’ represents that the rainfall–runoff model can perform well even if its performance is evaluated using the criterion different from the objective function used during the calibration process. Five frequently used performance criteria are adopted herein for evaluating the performance of the four patterns of the NCUC model. They are (1) error of peak runoff, (2) error of total runoff volume, (3) coefficient of determination, (4) root mean square error, and (5) efficiency coefficient. Readers can refer to Part 1 of these serial papers for more information regarding these performance criterion functions. As indicated in Part 1 of these serial papers, the EC and the R2 are two appropriate criteria to evaluate the accuracy of a rainfall–runoff model. These two criteria are referred to as the accuracy indices in this paper. The modeling performance of the four patterns of the NCUC model that are calibrated by minimizing the SLS is listed in Table 3. The values of the modeling performance that are evaluated using the EC fall within the range between 0.94 and 0.96. The modeling performance that is evaluated using the R2 is in the range between 0.94 and 0.97. During the calibration stage, the best values of the accuracy indices are attained
Table 2
Four patterns of the NCUC model
Symbol
Pattern
Number of parameters
I II III IV
{C2, A2} {D3, D3, A3} {C3, D3, C3, A3} {C2, D2, D2, D2, A2}
11 25 32 32
Storm events used
Number
Date (yyyy/mm/dd)
Duration (h)
Peak runoff (m3/s)
Name of typhoon
Remark
1 2 3 4
1992/08/26 1992/09/20 1994/08/19 1994/08/31
137 97 96 64
970.43 922.00 718.30 1449.33
Polly Ted Fred Gladys
Validation Validation Calibration Calibration
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Figure 4 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C2, A2} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
by Pattern III. Moreover, during the validation stage, the best values of the accuracy indices are also achieved by Pattern III. The modeling performance of the four patterns of the NCUC model that are calibrated based on the EC is given in Table 4. The EC falls within the range between 0.89 and 0.96, and the R2 are in the range between 0.90 and 0.96. As shown in Table 4, during the calibration stage, Pattern IV gives both the best EC and R2. During the validation stage, Pattern IV also yields the best EC and R2. In Tables 3 and 4, it can be found that the accuracy indices of the four patterns of the NCUC model are good during the calibration stage. However, during the validation stage, differences can be found from the accuracy indices of the four patterns of the NCUC model. A good rainfall–runoff model should perform well during the calibration as well as during the validation. Under such a consideration, the optimal model is selected according to the accuracy indices during the calibration and the validation. For the NCUC model calibrated by minimizing the SLS, the optimal model pattern is chosen as Pattern III. For the NCUC models calibrated by maximizing the EC, during the validation stage
the performance of Pattern IV is very close to that of Pattern III, but the performance of Pattern IV during the calibration stage is better than that of Pattern III. Therefore, for the NCUC models calibrated by maximizing the EC, the optimal model is Pattern IV. Versatility is another important credibility of a rainfall– runoff model. However, a model must perform well in accuracy firstly and then the evaluation of the performance using other performance criteria is considerable. Therefore, the values of performance criteria of all model patterns are also given in Tables 3 and 4, but only the versatilities of the optimal model patterns are discussed in this paper. It can be found that the values of the four performance criteria of the optimal model patterns are outstanding. This fact indicates that the NCUC model is a versatile model. The characteristics of different watersheds are definitely different. Therefore, the NCUC model provides the flexibility of the selection of model patterns, so modelers can freely select the model pattern that is the most suitable for a watershed. However, modelers should examine various patterns of the NCUC model and hence the optimal model pattern can be selected.
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Figure 5 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {D3, D3, A3} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
Since the number of the patterns of the NCUC model is infinite, the corresponding automated calibration method must be able to be applied to calibrate all possible patterns of the NCUC model so that the optimal model pattern can then be selected. From Tables 3 and 4, it is found that the automated calibration method can be applied to calibrate the four patterns of the NCUC model. In fact, various patterns of the NCUC model have been calibrated using the automated calibration method. Therefore, it can be said that the automated calibration method can well calibrate the NCUC model of any pattern. The key of the automated calibration method to calibrate the NCUC model of any pattern is the strategy for constructing the chromosomes.
The efficiency and parameters setting of the automated calibration method Since the modeling results (Tables 3 and 4) show that the parameters of the NCUC models are appropriate, the effectiveness of the automated calibration method is proven. The efficiency is another important issue of the automated
calibration method. For the evaluation of the efficiency of the automated calibration method, the time consumed during the calibration stage (Tc) is a practical index. However, it should be emphasized that Tc is not a universal measure because it varies with various factors, such as the level of the computer, the programming techniques and the lengths of the binary strings used in the automated calibration method, etc. In this paper, a desktop PC is used to implement the applications. For model Patterns I, II, III, and IV, the CPU time consumed during the calibration stage are 300, 410, 720, and 720 s, respectively. Obviously, the more parameters of the NCUC model need to be estimated, the more time for the calibration is consumed. From the results, it can be said that the Tc of the proposed automated calibration method falls within a tolerable range and hence the efficiency of the proposed automated calibration method is acceptable. Another aspect for the evaluation of the efficiency of the automated calibration method is also given as follows: e¼
sp tp
ð15Þ
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Figure 6 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C3, D3, C3, A3} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
where e is the ratio of the sp to the tp, sp is the number of the searched points, and the tp is the number of the whole points contained in the solution space. The solution space is a bounded hyperspace formed by the entire feasible solutions of the calibration problem. Each axis of the solution space represents a parameter of the NCUC model currently calibrated, and each feasible solution is a specific point of the solution space. As aforementioned, the lengths of the binary strings used to represent the parameters of the NCUC model are all 10 digits. This fact indicates that every parameter of the four patterns of the NCUC model used in this paper has 210 possible values during the calibration stage. The numbers of parameters possessed by the four patterns of the NCUC model are given in Table 2. Therefore, for Patterns I, II, III, and IV, the numbers of the entire feasible solutions (tp) are 2110, 2250, 2320 and 2320, respectively. In Eq. (15), the number of the searched points (sp) is the number of the feasible solutions that are searched by the automated calibration method. Since a chromosome represents one feasible solution and the population size in this paper is 100, the maximum numbers of the searched points (sp) for all the modeling scenarios are all 3 · 106. Thus, the ratios, e, of Patterns I, II, III, and IV are 2.31 · 1027,
1.69 · 1069, 1.40 · 1090, and 1.40 · 1090, respectively. It can be found that the optimal parameters are obtained by searching only an amazingly small amount of the feasible solutions within the solution space. From this viewpoint, it can be said that the efficiency of the automated calibration method is marvelous.
Conclusions In this paper, a SGA-based automated calibration method has been developed to calibrate the NCUC models of arbitrary patterns, so the function of the NCUC model is thus completed. Since SGA has been proven a robust, effective and efficient algorithm for various optimization problems, it is suitable to apply the SGA to calibrate the NCUC model with a slight modification. The evaluation of the fitness values is the only modification that is required to apply the SGA to the calibration. The methods that can appropriately evaluate the fitness values based on the two frequently used objective functions are developed in this paper. One of the advantages of the proposed automated calibration method is that it hardly requires subjective interventions of the modelers, and hence modelers can obtain a set of
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Figure 7 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C2, D2, D2, D2, A2} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
Table 3 SLS Pattern
I II III IV
Table 4 EC Pattern
I II III IV
Modeling performance of the four patterns of the NCUC model that are calibrated by minimizing the objective function Calibration
Validation 3
2
EC
RMSE (m /s)
EV (%)
EQp (%)
R
0.94 0.95 0.96 0.96
58.22 53.71 43.60 54.23
9.63 7.85 8.90 0.28
15.94 14.45 11.32 15.05
0.94 0.97 0.97 0.96
EC
RMSE (m3/s)
EV (%)
EQp(%)
R2
0.91 0.89 0.94 0.91
70.46 76.25 58.35 70.00
0.51 1.21 1.68 4.90
2.0 1.10 4.85 2.45
0.91 0.87 0.93 0.91
Modeling performance of the four patterns of the NCUC model that are calibrated by maximizing the objective function Calibration
Validation 3
2
EC
RMSE (m /s)
EV (%)
EQp (%)
R
EC
RMSE (m3/s)
EV (%)
EQp (%)
R2
0.89 0.95 0.92 0.96
76.91 53.71 67.08 36.15
7.25 7.85 0.06 0.88
22.28 14.45 7.26 9.24
0.90 0.96 0.96 0.95
0.91 0.89 0.93 0.93
71.35 76.25 60.16 61.38
1.50 1.21 3.06 5.71
10.13 1.10 9.78 6.69
0.91 0.90 0.92 0.92
206 high quality parameters without understanding too many details of the NCUC model. Only the values of the upper bounds of the EIS and the SCV require the interventions of modelers and the simple strategy for setting the two bounds are also proposed in this paper. In conclusion, with the automated calibration method, modelers can obtain a set of high quality parameters of the NCUC model. Theoretically, the automated calibration method for the NCUC model can be applied to arbitrary objective functions. However, for conforming to the restrictions on the fitness of SGA, the values of objective functions should be properly transformed, so that the fitness can be evaluated directly using the transformed objective functions. In this paper, four different patterns of the NCUC model are applied to an actual watershed, and two objective functions are respectively used to calibrate the model. The simulated hydrographs are in excellent agreement with the observed. The capability of the automated calibration method is thus proven. The automated calibration method is also able to calibrate arbitrary patterns of the NCUC model. The strategy for constructing the chromosomes is the solution to calibrate the NCUC model of any patterns. With such a solution, the selection of the optimal pattern of the NCUC model is thus practicable. The comparison shows that the accuracy of the NCUC model significantly outperforms that of the conventional models. Comparison of the NCUC model with the ANN based rainfall–runoff models are made as well. The NCUC model performs as well as the ANN based models, but the NCUC model does not have the shortcomings in the ANN based models. Hence, one can conclude that the NCUC model embedded with the automated calibration method is superior to the conventional conceptual models and ANN based models and it is an outstanding, efficient and effective tool for modeling the rainfall–runoff process.
G.-F. Lin, C.-M. Wang
References Beven, K.J., 2001. Rainfall–Runoff Modelling: the Primer. John Wiley, New York. Cai, X., McKinney, D.C., Lasdon, L.S., 2001. Solving nonlinear water management models using a combined genetic algorithm and linear programming approach. Advances in Water Resources 24 (6), 667–676. Chow, V.T., Maidment, D.R., Mays, L.W., 1988. Applied Hydrology. McGraw-Hill, New York. Cooper, V.A., Nguyen, V.T.V., Nicell, J.A., 1997. Evaluation of global optimization methods for conceptual rainfall–runoff model calibration. Water Science and Technology 36 (5), 53–60. Dumitrescu, D., 2000. Evolutionary computationCRC Press International Series on Computational Intelligence. CRC Press, Boca Raton, FL. Erickson, M., Mayer, A., Horn, J., 2002. Multi-objective optimal design of groundwater remediation systems: application of the niched Pareto genetic algorithm (NPGA). Advances in Water Resources 25 (1), 51–65. Franchini, M., Wendling, J., Obled, C., Todini, E., 1996. Physical interpretation and sensitivity analysis of the TOPMODEL. Journal of Hydrology 175 (1–4), 293–338. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Massachusetts. Lin, G.F., Wang, C.M., 2007. A nonlinear rainfall–runoff model embedded with an automated calibration method. Part 1. The model. Journal of Hydrology, doi:10.1016/j.jhydrol.2007. 05.009. Nash, J.E., Sutcliffe, J.V., 1970. River flow forecasting through conceptual models. Part 1: A discussion of principles. Journal of Hydrology 10 (3), 282–290. Reeves, C., 1997. Genetic algorithms for the operations researcher. INFORMS Journal on Computing 9 (3), 231–250. Sajikumar, N., Thandaveswara, B.S., 1999. A non-linear rainfall– runoff model using an artificial neural network. Journal of Hydrology 216 (1–2), 32–55. Wang, Q.J., 1997. Using genetic algorithms to optimise model parameters. Environmental Modeling & Software 12 (1), 27– 34.
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A nonlinear rainfall–runoff model embedded with an automated calibration method – Part 2: The automated calibration method Gwo-Fong Lin *, Chun-Ming Wang Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan Received 25 May 2006; received in revised form 4 March 2007; accepted 4 May 2007
KEYWORDS Automated calibration; Rainfall–runoff modeling; Global optimization; Fitness evaluation; Genetic algorithms; Sigma truncation
The purpose of this paper is to develop an automated calibration method for the nonlinear computational units cascaded (NCUC) model. The simple genetic algorithm (SGA), a popular and robust optimization technique, is introduced in this paper as the basis of the automated calibration method. Therefore, the way to transform the model calibration problem into the optimization problem is first proposed. The general scheme to appropriately arrange the parameters of the NCUC model is then developed, so that the chromosomes of the SGA can be properly constructed. Two performance criterion functions, which are frequently used to evaluate the performance of the rainfall–runoff modeling, are adopted in this paper as the objective function to calibrate the NCUC model. Since the SGA imposes two restrictions on the fitness values, the key of the proposed automated calibration method is the evaluation of the fitness values. The methods to evaluate the fitness values according to the two objective functions are both given in this paper. With the proposed automated calibration method, high-quality parameters of the NCUC model can be obtained without modelers’ subjective interventions. ª 2007 Elsevier B.V. All rights reserved.
Summary
Introduction Many rainfall–runoff models have been developed by hydrologists to model the rainfall–runoff process. Among these models, the conceptual models are frequently used. The parameters of conceptual models, in general, cannot be ob* Corresponding author. Tel.: +886 2 3366 4368; fax: +886 2 2363 1558. E-mail address: [email protected] (G.-F. Lin).
tained from the measurable quantities of watersheds, but can be calibrated using the available rainfall and runoff records. The accuracy of the conceptual models greatly depends on appropriate parameters. Therefore, the model calibration is a necessary procedure for the conceptual models. Model calibration is often performed by either manual adjustment or by the computer-based automated calibration techniques. Manual calibration is often done by trail and error, and thus only the experienced hydrologists can obtain feasible parameters by manual calibration. Besides, the
0022-1694/$ - see front matter ª 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2007.05.008
A nonlinear rainfall–runoff model embedded with an automated calibration method modeling result is often not as good as that of the ANN-based model. For example, Franchini et al. (1996) used the TOPMODEL (Beven, 2001), which is a set of conceptual tools based on the approximate hydrological theories, to model the rainfall–runoff process of the Sieve basin in the central Italy. The model parameters were obtained using manual calibration, and the performance of the model (compared with that of the ANN-based model) is just acceptable. Because the performance of the conceptual rainfall–runoff models highly depends on proper parameters, the crucial problem encountered by modelers is to select the appropriate parameters that make the model closely characterize the rainfall–runoff process of the study watershed. The calibration of a conceptual rainfall–runoff model, in fact, is to search a set of model parameters so that the simulated runoff is in good agreement with the observed. The calibration of conceptual rainfall–runoff models is difficult and time consuming, so modelers require a simple and effective automated calibration method to estimate the proper parameters. In the recent years, with the advances of the computer technology, the job of the model calibration is preferably done using computers. However, modelers still need an efficient computer-based method to effectively calibrate the conceptual models. In this paper, a popular, simple and powerful technique, the simple genetic algorithm (SGA), which is a kind of genetic algorithms (GAs), is adopted as basis of the automated calibration method. The GAs are widely used in many studies of the optimization of water resources planning (Cai et al., 2001; Erickson et al., 2002) and the calibration of model parameters (Cooper et al., 1997; Wang, 1997). The concept of the GAs is quite simple and is stated in the later section. ‘‘Survival of the fittest’’, the principle of evolution, is the prime thought of the GAs. The GAs search the solution space for many feasible solutions of the studied problem simultaneously using a stochastic and parallel approach. One advantage of the GAs over gradient-based nonlinear method is that the GAs can search good approximate solutions, even when the model functions are nondifferentiable or discontinuous (Reeves, 1997). With a well setup and sufficient iterations, the GAs mature quickly and the optimal solutions can then be obtained. For facilitating the usability, an effective and efficient method to calibrate the NCUC model without modeler’s subjective intervention is necessary. The purpose of this paper is to develop a SGA based method that can automatically calibrate the NCUC model according to the desired objective function. Since the GAs are for the optimization problems, the method that transforms the model calibration problem into the optimization problem is developed. Hence the GAs can be applied to calibrate the NCUC model. Then necessary transformations of the objective functions are made in order to calibrate the NCUC model using the GAs. Therefore, for the engineering projects of different purposes the proposed automated calibration method can be employed to calibrate the NCUC model using different objective functions.
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descriptions of the SGA can be found in Goldberg (1989). In this paper, only a concise introduction of SGA is provided so that the readers can easily understand the methodology proposed herein. The capabilities of the natural creatures adapting to the environment exist implicitly in the chromosomes of their DNAs. The survival pressure among natural creatures increases with the increasing proliferations of the natural creatures and the changes of the environment. New creatures are produced by the mating of their parents. They receive certain parts of the DNAs that are from their parents respectively. In other words, they inherit certain features of their parents that give them the probability to adapt to the environment. Only the natural creatures that are adaptive to the environment can survive. The essentials of the evolutions of nature are adopted to form the fundamentals of the GAs. A chromosome that denotes a feasible solution to the studied problem is the building block of the GAs. A chromosome is composed of several parameters that are coded using a certain coding strategy. One popular coding strategy is the binary coding because it can preserve more information for the GAs (Goldberg, 1989). A certain number of chromosomes form the population of the GAs. The population is initialized randomly and then updated continuously in further evolutions. The capability of a chromosome adapting to the environment is called the fitness. The fitness of a chromosome is calculated using the objective function that is selected depending on the users’ requirements. There are many variations of the GAs, while the important features are common. The three operators, reproduction, crossover and mutation, that all manipulate the population form the basics of the GAs. Among the population, the chromosomes that have larger fitness have larger possibilities to be selected by the reproduction operator for the mating of new chromosomes. Only the better chromosomes that represent better solutions to the studied problem are preserved for generating probably better solutions in the further generations. Then these better chromosomes exchange the useful information contained in themselves by using the crossover operator. Through the exchanges of these chromosomes, probably better new chromosomes are generated. For preventing from falling to the local optimums and directing the GAs to the solution space that may have not been searched, the mutation operator alters randomly certain genes (i.e., the basic element of a chromosome) of the chromosomes of the population with a very small probability. With a sufficient number of iterations till the maturation of the GAs, the global optimal solutions can then be obtained.
Calibration of the NCUC model using the SGA Transformation of the model calibration problem into the optimization problem The simulated runoff of the NCUC model is represented by
Genetic algorithms (GAs)
b ðc; hÞ ¼ NCUCj ðPÞ Q c;h
There are many variations of the GAs. SGA, one of the variations of the GAs, is used in this paper. Comprehensive
b ðc; hÞ is the simulated runoff generated by the where Q NCUC model of pattern c with parameters set h, NCUCjc,h()
ð1Þ
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represents the NCUC model of pattern c with parameters set h, and P is the areal average rainfall that is the input of the NCUC model. As indicated in Part 1 of these serial papers (Lin and Wang, 2007), the configurations and the number of the NCUs that are cascaded in the NCUC model are referred to as the pattern of the NCUC model. For objectively quantifying the accuracy of the NCUC model of pattern c with parameters set h, the following equation is used: b ðc; hÞÞ ps ¼ PCFðQ ; Q
ð2Þ
where PCF(Æ,Æ) is the performance criterion function, ps is the performance score, and Q is the observed runoff. Various performance criterion functions have been proposed to evaluate the performance of the rainfall–runoff modeling for different purposes. The calibration problem is transformed into the optimization problem by treating the performance criterion function (Eq. (2)) as the objective function. Then the calibration of the NCUC model is equivalent to optimizing the objective: b ðc; hÞÞ minimize or maximize OFðQ ; Q
ð3Þ
subject to the constraints: EISi P 0; i ¼ 1; 2; . . . ; m 0 6 CVCij 6 1; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ni
ð4Þ ð5Þ
SCVij P 0; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ni 0 6 CSVi 6 1; i ¼ 1; 2; . . . ; m 0 6 CISVi 6 1; i ¼ 1; 2; . . . ; m
ð6Þ ð7Þ ð8Þ
where OF(Æ, Æ) is the objective function that is the same as the performance criterion function of Eq. (2), EISi is the ele-
Figure 1
vation of the initial storage of the ith NCU, CVCij is the jth composite vent coefficient of the ith NCU, SCVij is the sill of the jth composite vent of the ith NCU, CSVi is the coefficient of single vent of the ith NCU, CISVi is the coefficient of the intermediate single vent of the ith NCU, m is the number of the NCUs contained in the NCUC model, and ni is the number of the composite vents of the ith NCU. The aforementioned constraints are the ranges of the values of the NCUC model’s parameters. With this transformation, the parameters set that optimizes the objective function subject to the constraints is the optimal parameters set for the NCUC model. In this paper, the terms ‘objective function’ and ‘performance criterion function’ are considered as equivalent. For the objective function that indicates the better performance using a smaller value of the performance score, the optimal parameters set is obtained by minimizing the objective function. Otherwise, the optimal parameters set is obtained by maximizing the objective function.
Construction of the chromosomes There are four different types of NCUs. Modelers can freely select the configuration of an NCU as well as the number of the NCUs cascaded in the NCUC model. Obviously, the number of patterns of the NCUC model is infinite. Therefore, a general method for the construction of the chromosomes used in the SGA is crucial. The parameters of an NCU can be divided into two categories, namely, intrinsic and optional. The intrinsic parameters include the elevation of the initial storage (EIS), the coefficient of the single vent (CSV) and the coefficient of the intermediate single vent (CISV). The number
Layouts of the partial chromosomes representing the parameters of the corresponding NCUs.
A nonlinear rainfall–runoff model embedded with an automated calibration method
Figure 2
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Layouts of the chromosomes representing the parameters of the NCUC model with the pattern {D1, C2}.
of the intrinsic parameters of a specific NUC is fixed. For example, the intrinsic parameter of the A-NCU is only the EIS, and the intrinsic parameters of the D-NCU are the EIS, CSV and CISV. The optional parameters are the parameters of the composite vents. A composite vent has two parameters, the composite vent coefficient (CVC) and the sill of composite vents (SCV). These two parameters certainly appear pairwisely. An NCU is perhaps only a unit of the NCUC model, so the parameters of an NCU form a part of the chromosome. The method to arrange the parameters in a chromosome is given in Fig. 1. In this figure, (a), (b), (c), and (d) are the layouts of the A-, B-, C-, and D-NCUs, respectively, and each grid represents a gene of the corresponding chromosome. The binary coding (Goldberg, 1989) is used to encode the NCUC model’s parameters in this paper and hence a gene that represents a parameter of the NCUC model is a binary string. The parameters of an NCU represented by the binary-coded genes are denoted by the corresponding symbols above the genes. For the four different NCUs, the intrinsic parameters are all placed in the beginning positions of a chromosome and then the optional parameters are concatenated successively. The number n in Fig. 1 is the number of the composite vents. With the method to arrange the parameters in a chromosome, the construction of the chromosomes of an NCUC model can then be easily achieved. The chromosomes of an NCUC are simply generated by concatenating the partial chromosomes of the NCUs sequentially. For example, the chromosome of the NCUC model with the pattern {D1, C2} is shown in Fig. 2. Readers can refer to Part 1 of these serial papers for more information regarding the notation {D1, C2}. With the aforementioned strategy for the construction of the chromosomes, the automated calibration method can thus be applied to calibrate the NCUC model of arbitrary patterns.
Ranges of the values and the precision of the parameters In fact, the restrictions on the NCUC model’s parameters (see Eqs. (4)–(8)) are not actually the constraints. They
are merely the upper and the lower bounds of the values of the parameters. With these restrictions, the solution space of the calibration problem of the NCUC model can be more easily defined. Nevertheless, the upper bounds of parameters EIS and SCV are not defined. The problem can be easily solved by enforcing very large values to the upper bounds of these two parameters. Because the EIS implies the water content of a watershed, it is not less than zero. And the water content of a watershed is not a very large value either, since the capability of a watershed to store water is limited. However a large value of the EIS is possible. Therefore, if an extremely large value is assigned to be the upper bound of the EIS, the SGA will search the optimal value of the EIS automatically within the artificially selected range. And the probability of the EIS being a large value would be kept. The strategy to select the upper bound of the EIS is also applied to choose the upper bound of the SCV. An extremely large value is picked to be the upper bound of the SCV as well, and the cause of such a choice is identical to that of the selection of the EIS’s upper bound. A binary string represents a gene of the chromosome in SGA. The lengths (the number of the digits of the binary string) of the genes affect the efficiency of the calibration and the precision of the calibration result. The precision of a gene representing a parameter is defined as follows: p¼
Umax Umin 2l 1
ð9Þ
where p is the precision of the parameter, Umax is the upper bound of the parameter, Umin is the lower bound of the parameter, and l is the length of binary string of the gene that represents the parameter. The precision of a parameter represents the minimum altered step of the parameter during the calibration processes. A high precision results from a long binary string, but a long binary string causes inefficiency of the calibration process. On the other hand, using a short binary string reduces the accuracy of the modeling, but the efficiency of the calibration process is increased. Modelers should carefully consider the lengths of the binary strings that represent the model parameters according to their requirements.
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Objective functions and the evaluation of the fitness values of chromosomes Most behaviors of the SGA are controlled by the fitness values of the chromosomes. The fitness value of a chromosome implies the degree of the chromosome’s goodness regarding the calibration. The evaluation of the fitness value of a chromosome is based on the objective function. However, the values of the fitness should be nonnegative, and the better chromosome should be represented by a larger value of fitness (Goldberg, 1989). To comply with these two restrictions, the values of the objective functions should be transformed properly into the fitness values. Hydrological engineering projects of different purposes require different objective functions to calibrate the rainfall–runoff model. In this paper, two frequently used objective functions are adopted for the calibration of the NCUC model. One is the simple least square (SLS): SLS ¼
r h i2 X b ðtÞ Q ðtÞ Q
ð10Þ
t¼1
b ðtÞ is the observed runoff at time t, Q(t) is the simwhere Q ulated runoff at time t, and r is the number of the time steps. For the SLS objective function, the set of model parameters that minimizes the SLS is the optimal solution. The SLS is nonnegative. To comply with the other restriction on the fitness of the SGA, when the fitness is evaluated using the SLS as the objective function, the fitness of a chromosome is calculated by f¼
1 SLS
where f 0 is the scaled fitness, lEC and rEC are respectively the average and standard deviation of the EC of all the chromosomes in the population, c is a small integer, and x; x > 0 NðxÞ ¼ ð14Þ 0; otherwise
Procedures of the automated calibration method The procedures of the automated calibration method for the NCUC model are illustrated in Fig. 3. In the initialization step, modelers determine the pattern of the NCUC model and the parameters of the SGA that control SGA’s behaviors. Four parameters of the SGA should be selected appropriately, so that the result of the calibration is desirable. The four parameters are the crossover probability, mutation probability, the population size, and the number of generations. A good result of the calibration using the SGA requires a large crossover probability, a small mutation probability, a moderate population size and a sufficient number of generations. Like the behavior of the natural creatures, the crossover probability is set as a value close to unity and hence crossover is not performed on all chromosomes. A small value close to zero is selected for the mutation probability, which is similar to the behavior of the natural creatures. If a large mutation probability is chosen, SGA will be a random search method that is not desirable. The population
ð11Þ
where f is the fitness of a chromosome. Using this method, a large SLS is transformed into a small f, so a worse chromosome has a smaller fitness value. On the other hand, a small SLS is transformed into a large f, so a better chromosome has a larger fitness value. The other objective function used to calibrate the NCUC model is the efficiency coefficient (EC) (Nash and Sutcliffe, 1970): Pr
t¼1 EC ¼ 1 P r
b ðtÞ2 ½Q ðtÞ Q
t¼1 ½Q ðtÞ
Q 2
ð12Þ
where Q is the average observed runoff. A larger EC implies the corresponding parameters set is a better one and otherwise the parameters set is not desirable. This characteristic of the efficiency coefficient satisfies the requirement of the SGA that the better solution has a larger fitness value. When the EC is used as the objective function to calibrate the NCUC model, the model parameters that maximize the EC are the optimal solution. Nonetheless, once the numerator of the second term on the right-hand side of Eq. (12) is larger than its denominator, the EC is negative. SGA does not allow the fitness value being negative. For solving this problem, the sigma truncation scaling technique (Dumitrescu, 2000) is introduced to transform the EC values into the scaled fitness values: f 0 ¼ NðEC ðlEC crEC ÞÞ
ð13Þ
Figure 3 Flowchart of the SGA-based automated calibration method for the NCUC model.
A nonlinear rainfall–runoff model embedded with an automated calibration method size should be a moderate value to provide adequate diversity of the initial chromosomes while the computational burden is limited. The number of generations should be large enough so that the performance of the calibrated parameters is satisfactory. After the initialization, the fitness of each chromosome in the population is evaluated using the corresponding method proposed above. Following the step of the ‘‘evaluate fitness’’ (see Fig. 3), the three operators of SGA are applied to the population sequentially until all chromosomes are processed. The terminal condition of the calibration in these serial papers is a predetermined number of generations. After the terminal condition is satisfied, the optimal model parameters, whose fitness value is the largest among the population, can be obtained.
Application The study area is the Fei-Tsui Reservoir Watershed in northern Taiwan. Information regarding the Fei-Tsui Reservoir Watershed can be found in Part 1 of these serial papers. Four storm events are selected for the modeling in this paper. As shown in Table 1, two events are used for calibration and the remaining events are used for validation. Storm events for the calibration are used to estimate the proper parameters of the NCUC model, and those for the validation are employed to verify the reliability of the calibrated model parameters. The input of the NCUC model, the areal average rainfall, is estimated using the Thiessen method (Chow et al., 1988). As aforementioned, the objective function is equivalent to the performance criterion function. Two frequently used performance criterion function, simple least square (SLS) and coefficient of efficiency (EC), are adopted as the objective functions for calibrating the NCUC model. Various patterns of the NCUC model have been examined. Four different and representative patterns of the NCUC model are picked for demonstrations (Table 2), and the corresponding parameters are calibrated according to the two different objective functions, SLS and EC, respectively. It should be noted that the calibration type in this paper is single-objective. That is, during the calibration stage only one objective function is used. The parameters of the SGA, the basis of the automated calibration method, are set as follows: the population size is 100, the probability of crossover is 0.8, the probability of mutation is 0.001, and the number of the evolution generations is 30,000. The upper bounds of all the EISs (elevation of the initial sill of an NCU) and the SCVs (the sill of a composite vent of an NCU) are set as 100 mm. In the automated calibration method, the lengths of all the binary strings that represent the model parame-
Table 1
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ters are 10 digits. Therefore, the precision of the coefficients of the vents (including the single vents and the composite vents) is 9.78 · 104 and the precision of the EISs and the SCVs is 0.098 mm. The modeling results of the NCUC model with four different patterns are given in Figs. 4–7.
Results and discussion The versatility of the NCUC model and the optimal model pattern Sajikumar and Thandaveswara (1999) proposed two important characteristics, the accuracy and the versatility, for evaluating the performance of the rainfall–runoff model. The accuracy of the NCUC model is discussed in Part 1 of these serial papers. The term ‘‘versatility’’ represents that the rainfall–runoff model can perform well even if its performance is evaluated using the criterion different from the objective function used during the calibration process. Five frequently used performance criteria are adopted herein for evaluating the performance of the four patterns of the NCUC model. They are (1) error of peak runoff, (2) error of total runoff volume, (3) coefficient of determination, (4) root mean square error, and (5) efficiency coefficient. Readers can refer to Part 1 of these serial papers for more information regarding these performance criterion functions. As indicated in Part 1 of these serial papers, the EC and the R2 are two appropriate criteria to evaluate the accuracy of a rainfall–runoff model. These two criteria are referred to as the accuracy indices in this paper. The modeling performance of the four patterns of the NCUC model that are calibrated by minimizing the SLS is listed in Table 3. The values of the modeling performance that are evaluated using the EC fall within the range between 0.94 and 0.96. The modeling performance that is evaluated using the R2 is in the range between 0.94 and 0.97. During the calibration stage, the best values of the accuracy indices are attained
Table 2
Four patterns of the NCUC model
Symbol
Pattern
Number of parameters
I II III IV
{C2, A2} {D3, D3, A3} {C3, D3, C3, A3} {C2, D2, D2, D2, A2}
11 25 32 32
Storm events used
Number
Date (yyyy/mm/dd)
Duration (h)
Peak runoff (m3/s)
Name of typhoon
Remark
1 2 3 4
1992/08/26 1992/09/20 1994/08/19 1994/08/31
137 97 96 64
970.43 922.00 718.30 1449.33
Polly Ted Fred Gladys
Validation Validation Calibration Calibration
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Figure 4 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C2, A2} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
by Pattern III. Moreover, during the validation stage, the best values of the accuracy indices are also achieved by Pattern III. The modeling performance of the four patterns of the NCUC model that are calibrated based on the EC is given in Table 4. The EC falls within the range between 0.89 and 0.96, and the R2 are in the range between 0.90 and 0.96. As shown in Table 4, during the calibration stage, Pattern IV gives both the best EC and R2. During the validation stage, Pattern IV also yields the best EC and R2. In Tables 3 and 4, it can be found that the accuracy indices of the four patterns of the NCUC model are good during the calibration stage. However, during the validation stage, differences can be found from the accuracy indices of the four patterns of the NCUC model. A good rainfall–runoff model should perform well during the calibration as well as during the validation. Under such a consideration, the optimal model is selected according to the accuracy indices during the calibration and the validation. For the NCUC model calibrated by minimizing the SLS, the optimal model pattern is chosen as Pattern III. For the NCUC models calibrated by maximizing the EC, during the validation stage
the performance of Pattern IV is very close to that of Pattern III, but the performance of Pattern IV during the calibration stage is better than that of Pattern III. Therefore, for the NCUC models calibrated by maximizing the EC, the optimal model is Pattern IV. Versatility is another important credibility of a rainfall– runoff model. However, a model must perform well in accuracy firstly and then the evaluation of the performance using other performance criteria is considerable. Therefore, the values of performance criteria of all model patterns are also given in Tables 3 and 4, but only the versatilities of the optimal model patterns are discussed in this paper. It can be found that the values of the four performance criteria of the optimal model patterns are outstanding. This fact indicates that the NCUC model is a versatile model. The characteristics of different watersheds are definitely different. Therefore, the NCUC model provides the flexibility of the selection of model patterns, so modelers can freely select the model pattern that is the most suitable for a watershed. However, modelers should examine various patterns of the NCUC model and hence the optimal model pattern can be selected.
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Figure 5 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {D3, D3, A3} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
Since the number of the patterns of the NCUC model is infinite, the corresponding automated calibration method must be able to be applied to calibrate all possible patterns of the NCUC model so that the optimal model pattern can then be selected. From Tables 3 and 4, it is found that the automated calibration method can be applied to calibrate the four patterns of the NCUC model. In fact, various patterns of the NCUC model have been calibrated using the automated calibration method. Therefore, it can be said that the automated calibration method can well calibrate the NCUC model of any pattern. The key of the automated calibration method to calibrate the NCUC model of any pattern is the strategy for constructing the chromosomes.
The efficiency and parameters setting of the automated calibration method Since the modeling results (Tables 3 and 4) show that the parameters of the NCUC models are appropriate, the effectiveness of the automated calibration method is proven. The efficiency is another important issue of the automated
calibration method. For the evaluation of the efficiency of the automated calibration method, the time consumed during the calibration stage (Tc) is a practical index. However, it should be emphasized that Tc is not a universal measure because it varies with various factors, such as the level of the computer, the programming techniques and the lengths of the binary strings used in the automated calibration method, etc. In this paper, a desktop PC is used to implement the applications. For model Patterns I, II, III, and IV, the CPU time consumed during the calibration stage are 300, 410, 720, and 720 s, respectively. Obviously, the more parameters of the NCUC model need to be estimated, the more time for the calibration is consumed. From the results, it can be said that the Tc of the proposed automated calibration method falls within a tolerable range and hence the efficiency of the proposed automated calibration method is acceptable. Another aspect for the evaluation of the efficiency of the automated calibration method is also given as follows: e¼
sp tp
ð15Þ
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Figure 6 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C3, D3, C3, A3} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
where e is the ratio of the sp to the tp, sp is the number of the searched points, and the tp is the number of the whole points contained in the solution space. The solution space is a bounded hyperspace formed by the entire feasible solutions of the calibration problem. Each axis of the solution space represents a parameter of the NCUC model currently calibrated, and each feasible solution is a specific point of the solution space. As aforementioned, the lengths of the binary strings used to represent the parameters of the NCUC model are all 10 digits. This fact indicates that every parameter of the four patterns of the NCUC model used in this paper has 210 possible values during the calibration stage. The numbers of parameters possessed by the four patterns of the NCUC model are given in Table 2. Therefore, for Patterns I, II, III, and IV, the numbers of the entire feasible solutions (tp) are 2110, 2250, 2320 and 2320, respectively. In Eq. (15), the number of the searched points (sp) is the number of the feasible solutions that are searched by the automated calibration method. Since a chromosome represents one feasible solution and the population size in this paper is 100, the maximum numbers of the searched points (sp) for all the modeling scenarios are all 3 · 106. Thus, the ratios, e, of Patterns I, II, III, and IV are 2.31 · 1027,
1.69 · 1069, 1.40 · 1090, and 1.40 · 1090, respectively. It can be found that the optimal parameters are obtained by searching only an amazingly small amount of the feasible solutions within the solution space. From this viewpoint, it can be said that the efficiency of the automated calibration method is marvelous.
Conclusions In this paper, a SGA-based automated calibration method has been developed to calibrate the NCUC models of arbitrary patterns, so the function of the NCUC model is thus completed. Since SGA has been proven a robust, effective and efficient algorithm for various optimization problems, it is suitable to apply the SGA to calibrate the NCUC model with a slight modification. The evaluation of the fitness values is the only modification that is required to apply the SGA to the calibration. The methods that can appropriately evaluate the fitness values based on the two frequently used objective functions are developed in this paper. One of the advantages of the proposed automated calibration method is that it hardly requires subjective interventions of the modelers, and hence modelers can obtain a set of
A nonlinear rainfall–runoff model embedded with an automated calibration method
205
Figure 7 Comparison of the observed and simulated runoff during the calibration and the validation stages for the {C2, D2, D2, D2, A2} NCUC model calibrated using objective functions: (a) SLS and (b) EC.
Table 3 SLS Pattern
I II III IV
Table 4 EC Pattern
I II III IV
Modeling performance of the four patterns of the NCUC model that are calibrated by minimizing the objective function Calibration
Validation 3
2
EC
RMSE (m /s)
EV (%)
EQp (%)
R
0.94 0.95 0.96 0.96
58.22 53.71 43.60 54.23
9.63 7.85 8.90 0.28
15.94 14.45 11.32 15.05
0.94 0.97 0.97 0.96
EC
RMSE (m3/s)
EV (%)
EQp(%)
R2
0.91 0.89 0.94 0.91
70.46 76.25 58.35 70.00
0.51 1.21 1.68 4.90
2.0 1.10 4.85 2.45
0.91 0.87 0.93 0.91
Modeling performance of the four patterns of the NCUC model that are calibrated by maximizing the objective function Calibration
Validation 3
2
EC
RMSE (m /s)
EV (%)
EQp (%)
R
EC
RMSE (m3/s)
EV (%)
EQp (%)
R2
0.89 0.95 0.92 0.96
76.91 53.71 67.08 36.15
7.25 7.85 0.06 0.88
22.28 14.45 7.26 9.24
0.90 0.96 0.96 0.95
0.91 0.89 0.93 0.93
71.35 76.25 60.16 61.38
1.50 1.21 3.06 5.71
10.13 1.10 9.78 6.69
0.91 0.90 0.92 0.92
206 high quality parameters without understanding too many details of the NCUC model. Only the values of the upper bounds of the EIS and the SCV require the interventions of modelers and the simple strategy for setting the two bounds are also proposed in this paper. In conclusion, with the automated calibration method, modelers can obtain a set of high quality parameters of the NCUC model. Theoretically, the automated calibration method for the NCUC model can be applied to arbitrary objective functions. However, for conforming to the restrictions on the fitness of SGA, the values of objective functions should be properly transformed, so that the fitness can be evaluated directly using the transformed objective functions. In this paper, four different patterns of the NCUC model are applied to an actual watershed, and two objective functions are respectively used to calibrate the model. The simulated hydrographs are in excellent agreement with the observed. The capability of the automated calibration method is thus proven. The automated calibration method is also able to calibrate arbitrary patterns of the NCUC model. The strategy for constructing the chromosomes is the solution to calibrate the NCUC model of any patterns. With such a solution, the selection of the optimal pattern of the NCUC model is thus practicable. The comparison shows that the accuracy of the NCUC model significantly outperforms that of the conventional models. Comparison of the NCUC model with the ANN based rainfall–runoff models are made as well. The NCUC model performs as well as the ANN based models, but the NCUC model does not have the shortcomings in the ANN based models. Hence, one can conclude that the NCUC model embedded with the automated calibration method is superior to the conventional conceptual models and ANN based models and it is an outstanding, efficient and effective tool for modeling the rainfall–runoff process.
G.-F. Lin, C.-M. Wang
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