ࡱ> ikh @?bjbj >V؝؝4E8,,@X:P:@@@@@@@$CRWE:@-:@4g@H#H#H#@@H#@H#H#@<l>. ضpc8! <?t}@0@=\+FD"+F0l>l>+F> PH#lP:@:@ "X PROMOTING MATHEMATICAL CREATIVITY USING GEOSTATISTICS FOR ENGINEERING STUDENTS IN GEOSCIENCES FIELDS JOS QUINTN CUADOR GIL Abstract: Methodological considerations about the geostatistical methods are presented in this paper. For geosciences engineering students is important to take in advance interpolation methods. The geostatistics is the most popular method, which have been widely debated in the last twenty years. Geostatistics was developed for mining and more precisely for the grade estimation of ore deposits. More generally, they are coming from natural sciences, where it is very difficult or even impossible to build deterministic models describing a continuous spatiotemporal phenomenon either because it is too complex, or because there is not enough information available. The characterization of the spatial variables using geostatistics has two main steep: the structural analysis to obtain semivariogram models and the estimation by Kriging. The solution of real problems allows to promoting mathematical creativity in the students when they apply the methodology and have to take decisions. Key words: Creative Semivariograms Analysis, Creative Estimation and Simulation Process, Grade Estimation, Kriging. INTRODUCTION Not all the problems found in the world can be explained by an exact or precise form using determinist way. There are several problems with spatial behaviour, for which the solution can only be related with probabilistic laws. For this reason, particularly, the students of geosciences specialities needs to take in advance estimation methods. A classic example of this situation is the estimation grades in ore deposit for which the Geostatistics was formalized by G. Matheron in 1962 (Matheron and Kleingeld, 1987). The Geostatistic is defined as the application of the theory of Random Function to the recognition and estimation of natural phenomenon (Journel and Huijbregts, 1978). Nowadays, the Geostatistics is widely used in other field on earth sciences as petrol industry, environment studies, etc (Isaaks and Srivastava, 1989). It provides a wide variety of tools for spatial data analysis and combines the empirical conceptual ideas that are implicitly subject to degrees of uncertainty with the rigor of mathematics and formal statistical analysis (Yarus and Chambers, 2006). The main objective of this paper is to present a brief introduction to the spatial problem analysis, focus in the mathematical creativity in the students. BASIC GEOESTATISTICAL STAGES To characterize spatiotemporal phenomenon some samples must be taken from the study area. The information obtained is compound by spatial coordinates and variables, the last is called regionalized variable, because it is always joined with its coordinates. The stationary behaviour of the data must be verified to apply the geostatistic analysis. The first steep is the knowledge of the problem. All the elements that supplies knowledge of the problem to solve must be studies. The exploratory data analysis must be carried out to verify the data stationarity, which is performed in different ways: visualizing the distribution of the data, calculating basics statistics, searches for trend and looks for outlier values. The data stationary ensures that your estimates are valid. The second steep is the structural analysis: the semivariogram plays a key role in this steep; it provides an experimental spatial variability of the data, which will be taken into account in the estimations process. Anisotropies or others spatial behaviour can be revealed by directional semivariogram analysis. After, the feting of theoretical and authorized semivariograms is performed on the experimental one, giving the final model of spatial variability and correlation. This model must be validated through the cross validation method, which is a way to check the model and search neighbourhood assumptions used in the next process. The third steep is the estimation or simulation. The estimation is performed by Kriging having different variants for all the practical problems than can be found in the real world. Within a probabilistic framework, Kriging attempts to minimize the estimation error variance, in this sense; it uses a linear combination of surrounding sampled values to make predictions. To make such predictions weights are assigned to each surrounding sampled data. Kriging allows to derive weights that result in optimal and unbiased estimates. In other hand, the estimation map obtained using Kriging has not the same semivariogram and variance as the original data. In this sense, the simulation allows us to come up with theoretically an infinite number of realizations of the map each of ones has the same semivariogram and variance of the original data. That is, while the estimation provides a soft representation of the phenomenon studied, the simulations reproduce the dispersion of the original data. THE KRIGING ESTIMATOR Kriging is the family of estimators proposed by geostatistics, which is defined as Best Linear Unbiased Estimator. There are different variants according to the data behaviour: the linear geostatistic with Ordinary Kriging and Simple Kriging for stationary data, the non stationary geostatistic with the Universal Kriging for non stationary data, the non linear geostatistic with Lognormal Kriging, Indicator Kriging, between others, for data that must be transformed, the multivariate geostatistic with Cokriging, Collocated Cokriging, Kriging with External Drift for multivariate studies. THE GEOSTATISTIC SIMULATION Local and global estimation are often insufficient in the characterization of spatial phenomenon. For mining engineer as well as other specialist in geosciences fields, it is often essential to be able to predict the variation of the characteristics of the variables studies (Journel and Huijbregts, 1978). The most important idea is that the simulation has the same value at the experimental data location and the same dispersion as the real data; the first two experimentally moments, mean and covariance or semivariogram, as well as the histogram are similar between real and simulated values. One of the most popular algorithm is the Sequential Gaussian Simulation which is an efficient method widely used in the mining industry and other fields. THE MOST CRUCIAL PROBLEM There are two crucial problems in the geostatistical estimation or simulation process: 1) the semivariogram analysis and 2) the search of nearest neighbour definition. The first is a problem widely debated nowadays, the semivariograms calculation, the feting of theoretical semivariograms, process that close with the validation of the feting preformed. The most important moment is the decision of the feting of theoretical and authorized semivariograms. The model used later in the estimation or simulation process is obtained here. The second is to find the search neighbourhood definition, the influential neighbouring points in estimating the value for a grid location. In the estimation or simulation process is handled any number of nearby samples. The weight assigned to the samples is distributed bigger from the nearby estimating location to the smaller in the border of the search area. For isotropic situation the search area is a circle defined by the range and in the anisotropic situation is an ellipse defined by de ranges in mayor and minor continuity direction. The weight assigned to the sample must close zero or smaller in the border but if the search area is not the correct or screen effect is present maybe possible a negative values for weight and consequently in the estimation values. This situation produces a false or not real estimation value. To avoid negative weights and obtain an adequate distribution of weights is the most crucial problems. The solution is to test different configuration in some location inside the estimation area until to find an adequate weight distribution. THE GEOSTATISTICAL SOFTWARE Some software has been created including geostatistical tools. Here, we present two of them, in our opinion, the most important: the Isatis from the Centre of Geostatistics, commercialized by Geovariances and the Geostatistical Library, GsLib, from Stanford University. Isatis is a geostatistical package presenting all the major methodologies that geostatistics can offer to the industry. It has been jointly developed at the Centre de Gostatistique de l`Ecole des Mines de Paris and Geovariances in Fontainebleau, France, and benefits from the numerous years of experiences of both contributors in the geostatistical domain. It is a powerful geostatistical tool dedicated to all sectors of natural resources, containing all the geostatistical methods validated by the research community. Isatis offers statistical tools to analyze and better understand your data and makes possible more accurate estimates together with confidence intervals. Further risk analysis may be based on simulations. Isatis integrates a wide range of geostatistical methods. Geostatistical Library (GsLib) is a collection of programs for the most specific geostatistic applications written in Fortran programming language and distributed free in source code (Deutsch and Journel, 1998). The most program operations are separated into short, easily understood, program modules. The Gslib has not a visual environment, it works in MSDOS console. The input parameters are specified in a parameter file with an own format. After the execution of any program, with data and parameter file joined, the processing results are obtained in an output file which can be interpreted individually. For this reason the GsLib could be erroneously rejected by some specialists, we say erroneously because it is the best free tool for Geostatistical calculations. Supplementary, a didactical software is presented in this work, which allow to present methodological consideration in the geostatistical study, over all, to proof some data sets to analyze the influence of the spatial configuration in the weight values for the estimation, to study and to understand the influence of the screen effect for different spatial data configuration. The above aspect is considered as the most crucial problem of the strategy search in the study and application of the geostatistic estimator and simulation. PROMOTING MATHEMATICAL CREATIVITY USING GEOSTATISTICS When the students solve problem in real cases, several situations can be found, for which it is necessary to apply the theoretical and methodological geostatistical concepts adequately. Consequently they have to formulate a correct interpretation of the results. The main objective in this moment is to take decisions according to the appropriate solution of the problems under study, for example: the verification of the stationarity of the data, the more precise semivariograms models obtained in the structural analysis, the selection of the adequate Kriging variant to apply and the appropriate search for neighbourhood definition to perform the interpolation method selected. This way, students have to develop creative abilities in the employment and interpretation of these tools, promoting mathematical creativity in the geosciences field. CONCLUSIONS AND FUTURE WORK The practical application of the Geostatistical tools provides the students the most useful way to characterize regionalized variables in the field of the geosciences. Real data from different practical problems must be analysed with different objectives. The most important geostatistical stages can be enumerated as: the well-known of the problem, the structural analysis and the estimation or the simulation. When estimations or simulations are obtained the results are interpreted in order to solve the objectives proposed. There are some crucial problems in all this process, one of them is the definition of the search of nearest neighbours. An adequate analysis of the neighbourhood search provides the correct estimation. The strategy search can be implemented in all geostatistical software with different form; the most important is to know its philosophy in order to perform the best possible search definition. Beside this work, we are working in a web site with the objective to provide didactical material to study these interesting and important tools. REFERENCES Deutsch, C. and Journel, A. (1998) GSLIB: Geostatistical Software Library and Users Guide, Second Edition, Oxford Univ.Press, 369p. Isaaks, E.H. and Srivastava, R.M. (1989), Applied Geostatistics, New York Oxford, Oxford University Press, 561p. Journel, A. and C.Huijbregts (1978) Mining Geostatistics, Acad.Press, NY, 600p. Matheron, G. and Kleingeld, W.J. (1987). The Evolution of Geostatistics. APCOM 87. Proceedings of the Twentieth International Symposium on the Application of Computers and Mathematics in the Mineral Industries. Volume 3. Geostatistics. Johannesburg, SAIMM, pp. 9-12. Yarus, J.M. and Chambers, R.L. (2006). Practical Geostatistics - An Armchair Overview for Petroleum Reservoir Engineers, Distinguished Author Series, Society of Petroleum Engineers. ABOUT THE AUTHOR Jos Quintn Cuador Gil, Ph.D. Department of Informatics, Faculty of Informatics and Telecommunications University of Pinar del Ro, 272 Mart str., 20100, Pinar del Ro, Cuba Phone: 53.48.779660, -mails:  HYPERLINK "mailto:emily@ami.ru.acad.bg" cuador@info.upr.edu.cu     Promoting Mathematical Creativity Using Geostatistics for Engineering Students in Geosciences Fields Jos Quintn Cuador Gil PAGE 228 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 PAGE 229 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 224 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 fh\ ] h 34GcfgzLb ,'H'0ŷxxœūūgūœœœœ hjhjCJPJaJnHtHhjhjCJaJmH sH hjCJaJmH sH hjhj5CJaJhjhjCJ\aJhjhj6CJaJhjhj56CJaJhjhjCJaJhjhj5CJ aJ $hjhjCJ$OJQJaJ$mH sH  hjhjOJQJaJmH sH %)Pfh] 4g<gdj$d7$8$H$\$a$gdjd\$gdy d\$`gdj $d\$a$gdj$d\$]^a$gdy $d\$a$gdjd\$gdj"=?gKLb +','H'V(f+l.000 4 $d\$a$gdj $<a$gdj $d\$a$gdj$<7$8$H$a$gdj$d7$8$H$\$a$gdj <7$8$H$gdj$d7$8$H$\$a$gdj00 4 44*4V8W8b888888899&9(9W9X9c9d9h9j9|999999::,;.;l;<<÷䯣䗏rrbhjhj5CJaJmH sH hjhjB*CJaJph# hjhj56mH sH hjhj5hjhj5mH sH hjhj6CJaJhjCJaJhjhj5CJaJhjhjCJaJmH sH hjhjmH sH  hjhjhjhjCJaJhjhj5CJaJmH sH % 44*4V8W8b88X99:k;l;};;;\<$a$gdj $xxa$gdj$07$8$H$^`0a$gdj$0^`0a$gdj$0^`0a$gdj $xa$gdj`gdj $d\$a$gdjxgdj $<a$gdj<<<<<== ="=$=(=*=.=0=4=6=:=h====>>>>> >ۦyyjfWOfh95OJQJhjh95CJOJQJaJh95h kh95CJOJQJaJhy5OJQJaJmH sH #hfh955OJQJaJmH sH hpjhpUhjh kCJaJ#hjhj0J5CJaJmH sH .jhjhj5CJUaJmH sH hjhj5CJaJmH sH (jhjhj5CJUaJmH sH \<"=&=(=,=.=2=4=8=:==>>>>> >->>>>$'&#$+D,a$gdY$hh]h`ha$gd*$y&#$+D,gdY$a$gd k$a$$a$gdj$a$gdj >!>'>(>+>,>->2>6>8>v>>>>>>>>>>>>??n?p?q?w?x?{?|?}?ٹكtfXfh-Ch9556CJaJh-Ch9556CJ aJ h;h95CJOJQJaJh950JCJOJQJaJh95'h;h956CJOJQJaJmHsHh;h956CJOJQJaJh956CJOJQJaJ%hy0JCJOJQJaJmHnHu hAoIh950JCJOJQJaJ)jhAoIh950JCJOJQJUaJ>>>>??L?o?p?}?~?????$a$gdj$ 9r  n!&#$+Dq]a$gd+y&#$+D,gdY$ 9r  ]a$gdN$  ]a$gdN$a$gd+}?~?????????мิhjh kCJaJhph95'h;h956CJOJQJaJmHsHh;h956CJOJQJaJh956CJOJQJaJ#h;h950J6CJOJQJaJ < 00&P 1hP:py. 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