usefulness and general applicability of these models ... · PDF fileThe Tank Model With its structure consisting of a set of linear ... Sample s points at random inTZ Compute the function

Application of a genetic algorithm for calibration

and structural modification of Tank Model

L.S. Diniz, R.S.S. Gois, V.S. Srinivasan

Department of Civil Engineering, CCT, UFPb, Campus II,

Cam? fWa/ JOJ, J& 700-970, Campma GraWe, fB, jgraz/7

Abstract

In this paper a Genetic Algorithm developed by Duan et al (1992) has been usedto evaluate and modify the structure of "Tank Model" for two different basins inthe northeastern Brazil for daily and monthly data. The algorithm entitled"Shuffled Complex Evolution - SCE" is based on the formulation and naturalevolution of multiple initial complexes (parameter sets). It combines the powerof the Simplex Method with the concepts of controlled random search andextraction of genetic operators generating shuffled complexes of the parametervalues. The Tank Model With its structure consisting of a set of linearreservoirs in series or parallel with side and bottom passages has been found togive good results for the semi-arid basins in the northeast of Brazil. Theapplication of the genetic algorithm resulted in three different versions of theTank Model for humid and semi-arid conditions with monthly and daily data.

1 Introduction

The rainfall-runoff models constitute a very important tool in the evaluation ofthe water resources of a hydrologic basin. Thus any planning process thatenvisages either the utilization of the available water resources or the control ofthe floods must rely on an appropriate rainfall-runoff model. Conceptualrainfall-runoff models (CRR) are designed to approximate within theirstructures the general physical mechanisms which govern the hydrologic cycle.Among the more widely used ones the major component is the soil moistureaccounting system (Duan et aP).

The soil moisture accounting part in the conceptual models is generallyrepresented by interconnected subsystems, each representing some phase of thehydrologic process. The nature of the functions and the extent of the details thattranslate the physical process into a mathematical representation defines thedegree of sophistication and realism of the model employed (Duan et aP). The

Transactions on Ecology and the Environment vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

12 Hydraulic Engineering Software

usefulness and general applicability of these models, however, depends on twomajor factors. The first one is the identification of the best values of theparameters of the model and the second is the closeness of the physicalprocesses in the specific basin with the generalized structure if the model. Theformer is carried out through a process of calibration of the model in which thevalues of the parameters are adjusted such that the values calculated from themodel are as close as possible to the observed values. The latter is checked bymeans of a validation process in which the results predicted by a calibratedmodel are compared with the observed values. If the deviations are not withinthe limits of acceptable variation, it must be concluded that the structure of themodel is incompatible with the physical processes within the basin.

It is important to note that any conclusion about the adequacy of thestructure of the model depends on "knowing" or finding the true optimumvalues of the parameters during the calibration process. This process, in essence,tries to minimize an objective function related to the differences betweenobserved and calculated flows and the procedure adopted may be an automaticsearch procedure or a mannual trial and error method. The second one, ofcourse, is not only cumbersome and slow , but also that the quality of the finalresults are uncertain. The automated search procedures while methodically seekto obtain the optimum value or the minimum value of the objective function,several difficulties in their application have been reported in the literature(Johnston & Pilgrinf, Gan & Burges*). The most common problems reportedbeing the difficulty in obtaining a unique set of parameters suggesting that theseprocesses get locked up in local optima and don't find the global minimum.

Holland* first proposed the use of Genetic Algorithms as a searchprocedure based on the mechanics of natural selection and natural genetics andsubsequently they have been used with successes by several researchers in thecalibration of CRR models (Duan et aP,Wang , Liong et af). This paper dealswith the application of the Genetic Algorithm denominated Shuffled ComplexEvolution (SCE-UA) developed by Duan et aP for the calibration of "TankModel" (Sugawara ) for two basins in the north east of Brazil. The paperdescribes the structural modifications that could be effected in the modelmotivated by the robustness of the Genetic Algorithm.

2 The SCE-UA algorithm

Based on concepts drawn from principles of natural biological evolution, Duanet aP developed and proposed the SCE-UA algorithm. The procedure is aglobal optimization strategy that combines the strength of the simplex method(Nelder & Mead®) with the concepts of controlled random search, competitiveevolution and complex shuffling. The essence of the method is as follows (Duanet al>).

The process starts with a population of V points sampled randomly inthe 'n' dimensional space of the model parameters. The population is partitionedinto several communities, each containing 2n + 1 points. Each community ismade to evolve based on a statistical reproduction process that uses the simplex


Hydraulic Engineering Software 13

process to proceed in the direction of improvement. At periodic stages of theevolution of the complexes, new random values are introduced that correspondto "mutation". The entire population is shuffled at regular intervals to shareinformation and begin a new search procedure. As the process evolves, theentire population tends to converge to the neighborhood of global optimum ifthe initial population of points (each representing a fixed set of values of the nparameters) is sufficiently large .

The SCE-UA Algorithm consists of a principal sequence in which thesearch and shuffling processes are carried out and a complementary sequence inwhich individual communities evolve into new "generations" on a competitivebasis (CCE Algorithm). The flow charts proposed by Duan et aP for the twoAlgorithms are shown in Figures 1 and 2.

Input: n = dimension, p = number of complexesm = number of points in each complex

Compute: sample size s = p . m

No

Sample s points at random inTZCompute the function value at each point

Sort the points in order of increasingfunction value. Store them in D.

Partition D into p complexes of m pointsD = (A*.k=l p)

Evolve each complex A*, k = l,...,p

TReplace A* ,k=l,. . m,in toD

Convergencecriteria satisfied ?

Figure 1 - Flowchart of the SCE-UA algorithm(after Duan et al, 1992.)



Given dimension n complex A, and number of points min A. select q, a, ft, where 2<=q<=m, a>=l, ft> = 1. Set t =1

Assign a triangular probability distribution to A:Pt = 2(m+l-I) / m(m+l), Set t = 1,... ,m

Select q points from A according to p,. Store them in Band their relative positions in A in L. Set J =1.

Sort B and L in order of increasing function value.Compute the centroid of ui,...,Uq_i, and let Uq be the worst point in B

iComputer r = 2g - Uq (reflection step)!

NoGenerate a point z atrandom in H. Set r=z.

Replace B into A according to L and sort Ain order of increasing function value.

Yes Return to SCE

Fig. 2. Flowchart of the CCE strategy of the SCE-UA algorithm(after Duan et al, 1992.)



The competitiveness of the better points of the communities is secured byassigning a linear weightage or probability according to the relative position ofthe point in the community ranked sequentially from best to worst. The methodof direct search by Simplex procedure is applied to the sub-communities inorder to evolve better generations. The procedure, according to the authors, isnot only effective in finding the global optimum but also quite efficientcompared with other global search methods.

3 The Tank Model

Several types of rainfall-runoff models may be found in the literature withvarying degrees of complexities and sophistication. The Tank Model conceivedby Sugawara in 1961 is one of the relatively simple ones and consists of a seriesof tanks or reservoirs arranged vertically with one bottom orifice and one ormore side orifices. The discharges through these orifices are proportional to thestorage volume above the level of the outlet. The discharge coefficients and thelevels at which the side orifices are located become model parameters that needto be optimized. The series of vertical tanks correspond to the different storagestrata of the river basin and as such the number of these tanks should be reallyassociated with the upper and lower soil strata as well as the presence of thesaturated ground water zones . In its most common arrangement as shown inFigure 3 (see section 5), three tanks are considered to represent the stratamentioned earlier.

The side orifices contribute to the basin runoff and the bottom orificestransfer flow from one stratum to another. The simple structure of the modelmakes it relatively easy to calibrate by a trial and error procedure and has beensuccessfully employed for modeling some of the river basins in the north east ofBrazil (Gois & Suzukf). However, when the number of parameters to becalibrated becomes large, the procedure gets tedious and the results becomeunsatisfactory. The employment of a reliable procedure to obtain optimumparameter values would be highly desirable and the application of a GeneticAlgorithm for calibration of the model parameters is one of the few alternativesavailable for global search. Thus, the chosen algorithm, SCE-UA, was used withthe Tank Model to obtain the parameter values for two river basins in the northeast of Brazil.

3.1 Characteristics of the River Basins

Most of the north east region of Brazil is typically characterized by a narrowhumid zone along the coast and a semiarid zone in the interior. Even theprincipal rivers of the interior are predominantly intermittent and the coastalrivers are perennial. Two basins were chosen to serve as representative ones foreach of the two zones. The Mumuaba river basin has a drainage area of about129 knf and is located in the coastal region of the state of Paraiba. The annualprecipitation is of the order of 1500 mm. Most of the rains are concentrated inthe period February-July. The basin has some parts of the coastal forest but the


1 6 Hydraulic Engineering Software

vegetation is mostly comprised of sugar cane and other cultivated cropsThe other basin chosen was that of the river Salobro with a drainage

area of about 16 knf located in the interior region of the state of Pernambuco.This is a sub-basin of the Riacho do Navio basin and has am annual rainfall ofabout 600 mm. The basin is located typically in the semiarid region where thesparse and intense rainfall is concentrated in about a period of about 3 months(March-May) and the rest of the year is hot and dry. The river Salobro dries upsoon after the rainy period indicating the absence of the base flow in the basin.

4 Application of the genetic algorithm SCE-UA

The flow charts of the algorithms presented in Figures 1 and 2 were coded inFORTRAN and as a first-step the program was tested by verifying theconvergence of the parameters to a pre-chosen set. For this purpose, a 3 yearrainfall (72-74) and evaporation data were used as input to the Tank Model anda synthetic sequence of runoff data was generated on a daily basis utilizing afixed but separate sets of parameters for each of the basins. These parametersbecome the optimum values to be reached in the search procedure by theGenetic Algorithm. Three different objective functions were employed in thesearch process. They were:

F(l) = Z(Qo-Qc): (1)o + Qc)] (2)

(3)

in which, Qo and Qc are the observed and calculated daily discharges. For thetest runs, the synthetic flows generated were considered as the observed flows.

A range was defined for each parameter such that the values could beconsidered realistic and the pre-chosen values would lie within the range. Withinthis parameter space an initial population was selected equal to the product ofthe number of complexes (p = 2) and the number of points in each complex (m= 2n + 1, where n is the number of parameters). For each of the objectivefunctions and each of the two basins, 100 runs were realized with the Algorithmstarting each time with a completely new and random set of points in theparameter space. The performance and efficiency of the Genetic Algorithm wasevaluated by the criteria of the number of failures (NF) to reach the trueparameter values and the number of times the objective function was evaluated(NFE). Table 1 shows the results obtained for the two basins.

The difference in the number of function evaluations in the two basins isdue to the different number of parameters involved in the two cases. Therobustness of the Algorithm is seen by the 100 % success rate in all cases. Theefficiency is quite evident from the relatively low number of function evaluationscompared to the number of parameters involved, 10 in the case of Mamuabaand 7 in the case of Salobro. With the results of the Table 1, it was concludedthat the SCE Algorithm does produce the globally optimum values if the rangeof the parameters is sufficiently large.



Table 1 - Number of failures and function evaluations in 100 trialsof the SCE-UA Algorithm.

Obj.Mamua

FunctionF(l)F(2)F(3)

NF000

iba basinNFE245022702120

SalobroNF000

basinNFE1030950990

Once the reliability of the Algorithm was established, the next step wasto utilize the natural flow data for the calibration of the Tank Model. Theprocedure was carried out for daily data as well as monthly data. With the useof 2 years' data for Mamuaba and one year data for Salobro with daily intervals,the optimum parameter values obtained in both the cases were totallyunsatisfactory when the simulated and observed hydrographs were compared.The results were not much different with the use of monthly data for a period of4 years (72-75). This led to the conclusion that either the observed flow datawere not reliable or the Tank Model with its structure of 3 reservoirs (figure 3)was not realistic in terms of the physical process in the basin. While the flowdata derived from rating curves of ephemeral rivers like that of Salobro may beconsidered less reliable, the flow data from the humid basin of Mamuabacouldn't be considered to be of poor quality. Hence, it was decided to verify ifsome appropriate changes in the model representation could be made to obtain asatisfactory calibration.

5 Structural modification of the Tank Model

In an attempt to establish a physical structure that would correspond to thenatural processes in the basins chosen, several alterations were tried consistingof mainly in the alteration of the number of tanks and the number of sideorifices in the uppermost reservoir. For each of the changes the parameters wereoptimized and the results were evaluated by the comparison of observed and thegenerated hydrographs. The final result that was satisfactory proved to be threedifferent arrangements of the tanks and side outlets. For daily simulation, it wasfound that the humid and the semiarid regions require totally different versionsand the monthly simulations could be satisfactory realized with only one typeof arrangement for both the regions. The features of these versions are shown infigures 4, 5 and 6.

For the Mamuaba basin, it was found that a system of 3 reservoirs wasneeded with the uppermost tank consisting of two sub sections where thestorage essentially satisfies the evapo-transpiration demands (Figure 4). Thestorage water in the uppermost tank and the primary storage (PS) in the sub-section are used up by evapo-transpiration at potential rates and when these areexhausted the secondary storage (SS) in the lower sub-section is used to satisfythe demand at the capillary rising rate to the upper level. Whenever the primarystorage is saturated, the excess is partly transmitted to the secondary storageand partly transformed into runoff through the side orifices. This arrangement



has 10 parameters in all excluding the factors that determine the rate ofinterchange of flow in the sub-sections.

The structure for daily flow simulation for the semi arid Salobro basinwas found to consist of only one reservoir with 3 side outlets and one bottomoutlet (Figure 5). The lateral flows correspond to direct runoff, interflow andsub-surface runoff while the bottom flow corresponds to losses due to deeppercolation. The evapo-transpiration is satisfied at the potential rate directlyfrom the storage in the reservoir. This arrangement needed 7 parameters.

Si

L1 —

hi

I— qsi

C

1 ^tT~^

qbi

(3)

qbi

(5) (6)

Explanation of symbols: Basic equations of theTank Modelai, a2, ... lateral discharge parameters qs = a (S - h);bi, b2,... bottom discharge parameters qb = b S;hi, 1%2,... storage level control parameters AS/At = (P - qs - qb - Ev)Si, 82, ... current storage level parameters P = precipitationqsi, qs2, ..lateral discharges Ev = evapotranpirationqbi, qb2,...bottom discharges AS = storage variation in the period

Figures 3, 4, 5, 6. Normal and modified configurations of the Tank Model



The monthly version for the two basins resulted in a sequence of two tanks withthe upper one having two side outlets and one bottom outlet (Figure 6). Thelower tank with one side outlet and one bottom outlet corresponds to delayedsub-surface flow and deep percolation loss. The monthly evaporation had to beseparately evaluated for dry and wet months by means of pan coefficients Ksand Ku chosen by the limiting monthly precipitation LLP. In the model Ks, Kuand LLP were optimized as parameters. Thus the monthly version of the TankModel presented 11 parameters in all.

6 Evaluation of the results

A comparison of the hydrograph generated by using the calibrated parametersand the observed hydrograph showed that the structural modificationsmentioned in the earlier section really improved the model performance in amost significant way. This improvement further reinforces the ability of theGenetic Algorithm used in identifying the optimum values of the parameters.

In order to evaluate the calibration results obtained for the daily data ofMamuaba basin, the data for the year 1974 was used to verify the modelperformance. The parameters were obtained using the data for the years 1972-1973. The evaluation of the results was done by means of the conventionalstatistics of the data like mean, standard deviation, the maximum and theminimum values. Two additional parameters were also used in the evaluation.They are a measure of the bias, designated FBI AS = I (Q™ - CW/Q™, whereQ™ and Qmo are the mean values of the simulated and observed flows during aspecific period. (Calibration or validation period). The other measure utilizedwas the linear coefficient of correlation between the observed and the calculatedflows. Table 2 shows the summary of the results including the optimumparameter values, obtained for the Mamuaba basin. As for the Salobro basin,except for 1974 the available data were either too unreliable or had too manymissing data to attempt a reliable validation.

Table 2 - Results of the calibration and validation of the Tank Model for dailyflows in the Mamuaba basin

ai0,065

Mean (nStd. DmMaximuMinimaPBIASR(%)

&20,003

am)nation (irini (mm)m (mm)

Optiias

0,08034

0,004

St

tm)

obser1,51,5

10,30,4

nized Paibi

0,282

atistics o:Calibra

ved3029

-0,0294,4

•ameter vb2

0,279

~evaluatktionsimulat1,491,4610,220,26

7[

aluesbs hi

0,005 88,72SS114,6

m

edValidation

observed1,641,308,370,60

simu]1,31,C6,60,3

-0,15989,8

PS42,7

ated8667



7 Conclusion

The main interest in the present study was to verify the ability of the GeneticAlgorithm SCE-UA to identify the optimum values of the parameters of arainfall-runoff model. The application of the algorithm to calibrate the TankModel for two basins in the north east of Brazil not only proved to be highlysuccessful but also showed that the SCE algorithm in particular and GeneticAlgoritms in general are highly useful in identifying the appropriate structure ofthe model as well.

References

1. Duan, Q., Sorooshian, S., Gupta, V.K. Effective and efficient globaloptimization for conceptual rainfall-runoff models, Water resourcesresearch, v.28, n.4, p.1015-1031, 1992.

2. Johnston, P.R., and Pilgrim, D., Parameter optimization forwatersheds model, Water Resources Research, pp. 477-486, 1976.

3. Gan, T.Y., and Burgers, S.J., An assessment of a conceptual rainfall-runoff models ability to represent the dynamic of small hypotheticalcatchments, 1, Models, model properties, and experimental design,Water Resources Research, pp. 1595-1604, 1990a

4. Holland, J.H., Adaptation in natural and artificial Systems,University of Michigan Press, Ann Arbor, 1975

5. Wang, Q. I., The genetic algorithm and its application to calibratingconceptual rainfall-runoff models, Water Resources Research, 27(9),pp.2427-1471, 1991.

6. Liong, S., Chan, W.T., ShreeeRam, J., Peak-flow forecasting withgenetic algorithm and SWMM, Journal of Hydraulic Engineering.613-617, 1995.

7. Sugawara, M., Automatic calibration of the Tank Model,Hydrological science bulletin, v.24, n.3, p.375-358, 1979.

8. Nelder, J.A. and Mead, R., A simplex method for functionminimization, Comput. journal, 7(4), pp.303-313, 1965.

9. Gois, R.S.S., Suzuki, K. Runoff characteristics of small rives innortheast Brazil, in: proc. 29th Japanese conference on hydraulics,Tokyo, Japan, 1987.


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usefulness and general applicability of these models ... · PDF fileThe Tank Model With its structure consisting of a set of linear ... Sample s points at random inTZ Compute the function