Improved Methods for Ore Deposit Evaluation: Ring Decomposition Method and Its Application

Öz

Ore deposit evaluation techniques by geostatistical simulation were first introduced some 25 years ago, it has not fulfilled its promise as a major tool in the earth sciences. This has been largely due to two main reasons: there are some shortcomings in the method which, although recognized early on by some practitioners, have been slow to be acknowledged and rectified, and alternatively wide usage of kriging methods (there is although a big difference between kriging and simulation). A survey of geostatistical simulation methods is given in Dowd (1992). Amongst proposed methods is Davis’ (1987a) LU (lower and upper) decomposition method and related matrix polynomial approximation method (Davis, 1987b). The LU-matrix (lower and upper) decomposition method of conditional simulation allows fast generation of stochastic processes on small¬ moderate sized grids. The method is simple and based on the LU triangular decomposition of the matrix of covariances between data locations and simulation grid locations (Davis, 1987a; Alabert, 1987). Covariances matrices are symmetric and positive-definite and therefore can be decomposed into the product of a lower and an upper triangular matrix. The advantages of the LU method are that it is simple to implement, performs conditioning simultaneously with simulation. is not limited to particular forms of covariance functions and handles anisotropies. The main drawback of this method is the amount of storage required which, at least in its general form as presented, effectively limits its application to less than 1000 grid locations. When there are many data or when there is a large number of points on which values are to be simulated, the correspondingly large matrices cannot be handled by classical decomposition algorithms. This paper shows how ring decomposition can he used to extend the use of LU decomposition to larger simulations. Ring decomposition can he applied to reduce significantly this memory-size problem, and therefore proposed method can he used for large grid locations. After introducing the mathematical hack ground of ring decomposition method, conditional simulation applications using lower-upper and ring decomposition methods are presented in the study. For the purpose of providing a comparison, simulations on 400 x 400 covariance matrix were performed using both LU decomposition and ring decomposition. The results are shown in Figures 1 and 2. The both methods yield satisfactory simulations. Finally a 1500 x 1500 covariance matrix which is too large for LU decomposition method is solved by ring decomposition and the result is given in Figure 3.

Anahtar Kelimeler

• Ring Decomposition Method
• Geostatistical Simulation
• Covariance
• Ore Deposits

Maden Yataklarının Değerlendirilmesinde Geliştirilmiş Yöntemler: Halka Ayrışım Yöntemi ve Uygulaması

Öz

Bu çalışmada maden yataklarının değerlendirilmesinde kullanılan jeoistatistiksel simülasyon yöntemlerinden "alt ve üst üçgensel matris ayrışım tekniği" ayrıntılı olarak anlatılmaktadır. Kovaryans matrisinin üçgensel analizini kullanarak, orta boyuttaki gridler üzerinde hızlı bir şekilde koşullu simülasyonu gerçekleştiren alt ve üst üçgensel matris tekniğinin algoritmasının büyük boyutlu gridler üzerinde simülasyonu sağlayamadığı gösterilmektedir. Çalışmada bu yönteme alternatif olarak geliştirilen ve halka ayrışım tekniği adı verilen matris analizi alternatif olarak sunulmakta ve önerilen yeni teknik büyük boyutlu gridler üzerinde kullanılabilmektedir. Geliştirilen yeni yöntemin matematiksel temeli verildikten sonra her iki yönteme ilişkin koşullu simülasyon uygulamaları sunulmaktadır. İlk önce yöntemleri birbirleriyle karşılaştırmak amacıyla 400 x 400 kovaryans matrisinden oluşan simülasyon uygulaması alt/üst üçgensel matris tekniği ve halka ayrışım tekniği kullanarak çözülmekte, ardından da alt/üst üçgensel matris tekniğiyle ayrıştırılamayacak boyutlarda olan 1500 x 1500 kovaryans matrisinden oluşan simülasyon, önerilen halka ayrışım tekniğiyle çözülmektedir.

Anahtar Kelimeler

• Halka Ayrışım Yöntemi
• Jeoistatistiksel Simülasyon
• Kovaryans
• Maden Yatakları

Kaynakça

1. Alaberi, F., 1987, The practice of fast conditional simulations through the LU decomposition of the covariance matrix: Mathematical Geology, v. 19, no. 5, 369-386.
2. Anderson, F.W. and Fuller, K.R., 1992, Rings and categories of modules: Second Edition, Graduate texts in mathematics, Springer Verlag New York Inc., no. 13.
3. Davis, M., 1987a, Production of conditional simulations via the LU decomposition of the covariance matrix: Mathematical Geology, v. 19, no. 2,91-98.
4. Davis, M., 1987b, Generating large stochastic simulations-the matrix polynomial approximation method: Mathematical Geology, v. 19, no. 2, 99-108.
5. Deutsch, C.V., and Journel, A.G., 1998, GSLIB, Geostatistical Software Library and User’s Guide: Second Edition, Oxford University Press, New York, 369 s.
6. Dowd, P.A., 1992, A review of recent developments in geostatistics: Computers and Geosciences, v. 17, no. 10, 1481-1500.
7. Dowd, P.A. and Saraç, C., 1993, An extension of the LU decomposition method of simulation, in Geostatistical Simulations, cds: P.A. Dowd and M. Armstrong: Kluwer Academic Publishers, The Netherlands, 23-36.
8. Gomez-Hernandez, J.J. and Srivastava, R.M., 1990, ISIM3D: an ANSI-C three-dimensional multiple indicator conditional simulation program: Computer and Geosciences, v.16, no. 4,395-440.

9. Journel, A.G. and Alabert, F., 1989, Non-Gaussian data expansion in the earth science: Terra Nova, v. 1, 123-134.
10. Journel, A.G. and Alabert, F., 1990, New method for reservoir mapping: Journal of Petroleum Technology, v. 42, no. 2, 212-218.
11. Kasch, F., 1982, Modules and rings: London Mathematical Society, Academic Press Inc., England.
12. Saraç, C. and Tercan, A.E., 1992, A geostatistical case study of the Kizilyuksek-Yataardic chromium orebody: The first international symposium on Eastern Mediterranean Geology, Çukurova University, Adana, Turkey, 219-229.
13. Suro-Perez, V. and Journel, A.G., 1990, Stochastic simulation of lithofacies and improved sequential indicator approach: Proc. Second European Conference on the Mathematics of Oil Recovery (ECMOR), Publ. Technip, Paris, 3-10.
14. Tercan, A.E. and Saraç, C., 2000, Spatial variability of Cr2O3 % in Kizilyuksek-Yataardic chromium deposit (Adana, Turkey): International Geology Review (in press).