Underground water quality model inversion of genetic algorithm

Clobal Geology, 12(3): 164-167(2009)doi:10. 3969/j. isn. 1673-9736. 2009.03.07Article D: 1673-9736(2009 )03-0164-04Underground water quality model inversionof genetic algorithmMA Rujjie' and LI Xin21. College of Mathemais, Jjlin Uniersity, Changchun 130026, China2. College of Construcion Eninering, Jilin Uniersiy, Changchun 130026, ChinaAbstract: The underground walter quality model with non-linear inversion problem is ill. posed, and boils downto solving the minimum of nonlinear function. Genetie algorithms are adopted in a number of individuals ofgroups by iterative search to find the optimal solution of the problem, the encoding strings as its operational ob-jective, and achieving the iterative calculations by the genetic operators. It is an effective method of inverseproblems of groundwater, with incomparable advantages and practical significances.Key words: underground water; quality model; inversion; genetie algorithm; genetic operatornetic algorithm in the 1970s De Jong carried out nu-Introductionmerical pure numerical function optimization computa-Groundwater quality model is based on thetional experiments on the computer, and on the basisgroundwater resources model and the problems of theof a series of studies , summarized by Goldberg in thegroundwater quality, Thus, it is inevitable to invert1980s, the basic framework of genetic algorithm wasthe hydrogeological parameter. And the fact is thatformed. It mimics the theory of survival of the fttestdifferent geohydrologic condition may produce similarand genetic variation. Through simulating Darwin'sor the same water head, and the geohyrologic condi-principle ( survival of the fttest) incent good struc-tion cannot be always determined merely by observedture; and by simulating Mendel's theory of the genetichydrological materials. Take the dynamic observationvariation in the iterative process to maintain the exist-of the natural groundwater or the observational data ofing structure, as well as to seek a better structure.the pumping test into consideration and in turm under-Genetic Algorithm has a very strong implicit parallel-stand the problems of the geological conditions, whichism and global search capability. When dealing withis called inverse problem or indirect problem.some large-scale and highly nonlinear problem, it hasFrom the perspective of math , inverse problem isunique advantage. Genetic algorithm is a kind of nu-model identification problem- -optimal objective func-merical algorithm, which is of wide application range,tion. Inversion from observed values to the parametershigh efficiency and with the capability of global opti-of media destined to this inversion problem is n-lin-mization numerical algorthm. It can efectively dealear and il-posed.withInction optimizationGenetie algrithm was founded in 1962 by Prof.pr中国煤化inaon numHolland from Michigan University. On the basis of ge-ical:TYHCNMH G avantages of r0Recrired 24 April, 2009; acepted 10 June, 2009, Underground water quality model inversion of genetic algorithm165bustness, parallel and high eficiency.and computer technology.Genetic algorithm has three basic elements :enco-1 Mathematic modelded mode , genetic operators and ftness function.Let's suppose that there are N wells in the regionEncoding of which is to convert the solutions ofG (Zhou et al. , 2000) made by the closed curve T ,the problem into coded strings to simulate the way thethe distrbution as shown in Fig. 1, and among themBiological chromosomes genetic computing for geneticm wells are extraction wells,operations; operator is a series of operations , the sim-ulation of individual survival environment; fitnessfunction is measuring function, the criteria for evalua-P0tion of the viability of the individual. Apart from the)(x,y)three fundamental elements, the genetic algorithm alsoinvolves some parameters of controlling factors, name-ly group size, crossover probability ，mutation proba-Fig.1 Distribution of wellsbility, genetic algebra and so on. Genetic algorithmbasic idea is to abstract the biological evolutionNe wells are observation wells, m + Ne = N; iprocess and then described as reproduction , crossoverwell centroid coordinates are (x;，y;), Well Radius isand mutation operator three operators. Genetic algo-rci, Well Perimeter areT; To (x, y;) as the center ofrithm can effectively deal with multivariate and thea circle, to rci for the radius ，this circular domain re-solving of complex function relation optimization prob-corded as Kci (i=1, 2, .., N); also assumed thatlems, due to genetic algorithm not required differenti-the N; extraction wells has three wells of given produc-able, parallel and widely applicable. The generaltion volumes and heads ha(a=1,2,.*,N ) on thestructure of genetic algorithm model and its applica-wall; has N; exploration wells on the wall only giventions are as fllows:head ha(a=N; +1,N+2,-,N, +N,) ,here N; +N;(1)Genetic encoding: Will solve the problem in=m, three observation wells centroid given the well-which each variable is seen as a gene, according tohead value h(x,y) (i=m+1,.. ,m +Ne) ,and setthe type of variables and range , and select the appro-valueh;(x, y, t) to the border r , in the region Gpriate digit binary code to be encoded separately ,given the initial head hq(x, y).called the genetic code , such as: x=[a，a2, a3,Known leakage recharge intensity E (Xu,2003 )as].recharge q, initial value ho , exterior boundary's head(2) fitness function: Genetic algorithm when inhl, Partial wall boundary head h。and flow rate Q2(asearch, substantially without using external informna-=1,2, .. m), part of the observation head h。(ation, only appropriate value function as the basis ,=m+1,... ,m +Ne)，to reverse hydrogeology param-Zhu ( 1997) used species of each individual's fitness toeters of the mathematics model.carry out the search. According to the functional rela-tionship between problem solving and gene encoding2 Genetic algorithm stepsrules calculate all individuals' the fitness function thatGenetic algorithm is based on the principle of bi-is :ological evolution, a global optimization algorithm ,F, =f(x;, y;,z) i = 1,2,**,ndrawing on the Biological natural selection and Genet-中国煤化Todel, the objectiveic evolution mechanism to develop a global optimiza-fu.MH.CNMH Gas taken as partoftion of adaptive probabilistic search algorithm, thethe genetc algontnm Inaviaual ntness function, em-product of combination of biological genetic technologybedded in the solving process of fitness function, real-Ma R.J. and Ii X.izing the coupling of genetice algorithm and the numeri-optimal individuals in the next generation, such as s0-cal method, the coupling mode isphisticated, then the end of the solution process,In the formulas:M-size of a group, that is, thewhen received by the individual and the species is tonumber of chromosomes; Np-the number of calculat-solve the problem of the optimal solution ( Georg,ing points; Fitness (i)-i individual fitness1990; Allgower & Ceorg, 1990). If mature, thenThe establishment of species: biology live in na-end the solving process of the population. Then theture in the form of species group. A species P(t) hasindividuals and species obtained is the optimal solu-N individuals: P(t) =(A, A2, .,. An), (Ortega &tion of problem solving.Rheinboldt, 1970; Allgower & Georg, 1983). As a3Applicationstarting point for the initial evolution species P(0)can be generated randomly or by other means. Gener-The initial populations are selected from aboutally adopt randomly generated approach to produce1 000 randomly solution individuals according to theirpopulation scale pop-size chromosomes individuals.fitness function values quality, and it is better to setThat can avoid search some points not necessary, sothe following parameters through the calculation val-actually equal to search for more points, peculiar toue: population size m = 100，the probability of cross-the genetic algorithm of implicit parallelism.computing P =0. 7, the probability of mutation calcu-Reproduction: Select individuals from species plation pc =0.033, a length of string r=4.(t) copy to species P(t+1). Each individual repro-duction probability of the breeding opportunity is con-Table 1 Some known prior conditions for the parameterstrolled by the frility probablity Ps. The value of Ps1 m.d'depends on each individual's fitness function Fi; InAreaTrue valueInitialMinimum Upper limitthe process of copying will operate crossover and mu-10s40150tation. (He, 2003, Garcia & Gould, 1980). The500250200700choice of cross is controlled by crossover probability5000Pm. Using the mutation probability, the formula is :20002500017030000Pm (h+1)= Pm (h)-[0.3-Pm (1)]m,;100014007501700in the formula, h stands for genetic algebra ，kms ison behalf of the largest genetic algebra, pm(1) repre-Table 2 Optimal results of GA inversionsents the first-generation mutation probabilityand Pm/ m.d'(k) the k-th-generation mutation probability. As soonObjective function EIIas the reproduction, crossover and mutation of speciesOptimal rsuls 9.99 500 001 500 002 1999 89100.0P(1+1) is completed, species P(t + 1) to replace00505000 2000 1000species P(t), which achieves a generation of breed-ing.Stralegy selection: Evaluate the species P(t +After 100 times of iteration, the solution remains1)，testing the speed of evolution and convergencestable, increasing the number of iterations has nodegrees, to determine whether the evolution sophisti-effect on the results, and at this time it has been simi-cated. If immature, then continue with each genera-lar to the optimal solution, the time spent a total of 4tion reproduction and evolution, so that the quality of中国煤化工individual species has been optimized-Maintain theMYHCNMHGoptimal strategy ，relention of m optimal individuals inthe previous generation, and the rest select pop-size mAlthough the genetic algorithm to achieve optimalUnderground water quality model inversion of genetic algorithm167over 100 times iteration, but only four seconds to run ,York: Plenium Press, 472-498.indicating that genetic algorithm is a optimizationHeDK, WangF L, Zhang C M.2003. Establishment of Pamethod of a high degree of robustness and an efectiverameters of genetic algorithm Based on Uniform Design.Jounal of Northesern Uniersity, 24(5): 409411. (inway to solve the groundwater inverse problem, withChinese with English abstract)the advantages other traditional inversion methods canOrtega J M, Rheinboldt W C. 1970. leraive solution of non-not compare with and practical significance.linear equations in Several Variables. New York: Aca-Referencesdemic Press, 1-572.XuQ, Chen R Q, Cuan Y L, et al. The shortest path analyisAllgower E L, Georg K.1983. Predictor-Corrector and simpli-based on genetic algorithms. Jounal of East China Geo-cial methods for approximating fixed points and zero pointslogical Institute, 2003, 26(2): 168-172. ( in Chineseof nonlinear mappings in mathematical programning. Ber-with English abstract)lin: Spinger, 163-184.ZhouZ H. Chen zQ, ChenS F.2000. Research of field theo-GareiaC B, Gould F J. 1980. Relations between several pathry based adaptive resonance neural network. Joumnal offollowing algorithms and local global Newton methods. SI-Nanjing Unirersity: Natural Seiences, 36(2) 140-147.AM Review ,22(3) : 263-274.(in Chinee with Engish abstract)GeorgK. 1990. A note on step size control for numerical curveZhu H s.1997. Features and pplications of muli-goup ge-following. Proceeding of the NATO advanced research in-netic algorihms. Systems Engineering Theory & Practice,stitute on homotopy methods and global convergence. New78-85. ( in Chinese with English abstract)中国煤化工MYHCNMHG