114
The results shown that the methodology proposed produces a satisfactory result for the
case study. It is noted that the simulation back transformed mean of the variables reproduces
the original mean of the data very well. However, the samples variances are not well reproduced, +16% for Fe and −14% for Si. The BMEC is a work in progress but it opens a different
approach to build a robust CT with new BMEC’s to be explore.
6 CONCLUSION
The technique based on extracting a covariance table (CT) from a base model to extract covariance (BMEC) was found to obtain good results. The statistical validation of the method and the
speed of obtaining a spatial continuity analysis are good indicators that is possible to substitute
the traditional variogram modeling. Projection pursuit multivariate transform (PPMT) to make
any number of variables to be multivariate Gaussian and uncorrelated, enabling the independent acquisition of CT for each attribute considered, while reproducing the multivariate complexities on the final model. This CT is obtained through a three steps workflow: interpolating
the data set to fill up all grid nodes in a regular grid, auto convolute via FFT algorithm and back
transform to spatial domain. The BMEC represents a simple solution to a complex problem
comparing with previous methods that proposed CT usage. The proposed method does not suppress previous modeling phases, such as choosing stationary domain, but it is a fast automatic
spatial continuity method to compute covariances. For a future work a more complete comparison between the current methodology with the proposal in this paper will be done.
REFERENCES
Barnett, R.M., Manchuk, J.G., and Deutsch, C.V. (2014). Projection Pursuit Multivariate Transform.
Society of Petroleum Engineers, 21(6):337–359.
Bracewell, R.N. and Bracewell, R.N. (1986). The Fourier transform and its applications, volume 31999.
McGraw-Hill New York.
Carvalho, A.R., Carvalho, J.C., Shiguemori, E.H., Da Silva, J.D.S., and Ramos, F.M. (2007). Neural
network based models for the retrieval of methane concentration vertical profiles from remote sensing data. In Anais XIII Simposio Brasileiro de Sensoriamento Remoto. Florianopolis, Brasil, volume
21, page 6.
Chu, J. (1993). Conditional fractal simulation, fast sequential indicator simulation, and interactive variogram modeling. PhD thesis, Stanford University.
Deutsch, C.V. and Journel, A. (1998). Geostatistical software library and users guide. Oxford University
Press, New York.
Johnson, S.G. and Frigo, M. (2008). Implementing FFTs in practice. Fast Fourier Transforms.
Journel, A.G. and Huijbregts, C.J. (1976). Mining geostatistics.
Manchuk, J.G., Barnett, R.M., and Deutsch, C.V. (2017). The reproduction of secondary data
in projection pursuit transformation. Stochastic Environmental Research and Risk Assessment,
31(10):2585–2605.
Öztopal, A. (2006). Artificial neural network approach to spatial estimation of wind velocity data.
Energy Conversion and Management, 47(4):395–406.
Pyrcz, M.J. and Deutsch, C.V. (2006). Spectral corrected semivariogram models. Mathematical Geology,
38(7):891–899.
Rigol, J.P., Jarvis, C.H., and Stuart, N. (2001). Artificial neural networks as a tool for spatial interpolation. International Journal of Geographical Information Science, 15(4):323–343.
Rizzo, D.M. and Dougherty, D.E. (1994). Characterization of aquifer properties using artificial neural
networks: Neural kriging. Water Resources Research, 30(2):483–497.
Verly, G. (1983). The multigaussian approach and its applications to the estimation of local reserves.
Journal of the International Association for Mathematical Geology, 15(2):259–286.
Yao, T. (2004). Reproduction of the mean, variance, and variogram model in spectral simulation. Mathematical geology, 36(4):487–506.
Yao, T. and Journel, A.G. (1998). Automatic modeling of (cross) covariance tables using fast fourier
transform. Mathematical Geology, 30(6):589–615.
Yoo, Y. (2001). Tutorial on fourier theory. Retrieved June, 17:2007.
The results shown that the methodology proposed produces a satisfactory result for the
case study. It is noted that the simulation back transformed mean of the variables reproduces
the original mean of the data very well. However, the samples variances are not well reproduced, +16% for Fe and −14% for Si. The BMEC is a work in progress but it opens a different
approach to build a robust CT with new BMEC’s to be explore.
6 CONCLUSION
The technique based on extracting a covariance table (CT) from a base model to extract covariance (BMEC) was found to obtain good results. The statistical validation of the method and the
speed of obtaining a spatial continuity analysis are good indicators that is possible to substitute
the traditional variogram modeling. Projection pursuit multivariate transform (PPMT) to make
any number of variables to be multivariate Gaussian and uncorrelated, enabling the independent acquisition of CT for each attribute considered, while reproducing the multivariate complexities on the final model. This CT is obtained through a three steps workflow: interpolating
the data set to fill up all grid nodes in a regular grid, auto convolute via FFT algorithm and back
transform to spatial domain. The BMEC represents a simple solution to a complex problem
comparing with previous methods that proposed CT usage. The proposed method does not suppress previous modeling phases, such as choosing stationary domain, but it is a fast automatic
spatial continuity method to compute covariances. For a future work a more complete comparison between the current methodology with the proposal in this paper will be done.
REFERENCES
Barnett, R.M., Manchuk, J.G., and Deutsch, C.V. (2014). Projection Pursuit Multivariate Transform.
Society of Petroleum Engineers, 21(6):337–359.
Bracewell, R.N. and Bracewell, R.N. (1986). The Fourier transform and its applications, volume 31999.
McGraw-Hill New York.
Carvalho, A.R., Carvalho, J.C., Shiguemori, E.H., Da Silva, J.D.S., and Ramos, F.M. (2007). Neural
network based models for the retrieval of methane concentration vertical profiles from remote sensing data. In Anais XIII Simposio Brasileiro de Sensoriamento Remoto. Florianopolis, Brasil, volume
21, page 6.
Chu, J. (1993). Conditional fractal simulation, fast sequential indicator simulation, and interactive variogram modeling. PhD thesis, Stanford University.
Deutsch, C.V. and Journel, A. (1998). Geostatistical software library and users guide. Oxford University
Press, New York.
Johnson, S.G. and Frigo, M. (2008). Implementing FFTs in practice. Fast Fourier Transforms.
Journel, A.G. and Huijbregts, C.J. (1976). Mining geostatistics.
Manchuk, J.G., Barnett, R.M., and Deutsch, C.V. (2017). The reproduction of secondary data
in projection pursuit transformation. Stochastic Environmental Research and Risk Assessment,
31(10):2585–2605.
Öztopal, A. (2006). Artificial neural network approach to spatial estimation of wind velocity data.
Energy Conversion and Management, 47(4):395–406.
Pyrcz, M.J. and Deutsch, C.V. (2006). Spectral corrected semivariogram models. Mathematical Geology,
38(7):891–899.
Rigol, J.P., Jarvis, C.H., and Stuart, N. (2001). Artificial neural networks as a tool for spatial interpolation. International Journal of Geographical Information Science, 15(4):323–343.
Rizzo, D.M. and Dougherty, D.E. (1994). Characterization of aquifer properties using artificial neural
networks: Neural kriging. Water Resources Research, 30(2):483–497.
Verly, G. (1983). The multigaussian approach and its applications to the estimation of local reserves.
Journal of the International Association for Mathematical Geology, 15(2):259–286.
Yao, T. (2004). Reproduction of the mean, variance, and variogram model in spectral simulation. Mathematical geology, 36(4):487–506.
Yao, T. and Journel, A.G. (1998). Automatic modeling of (cross) covariance tables using fast fourier
transform. Mathematical Geology, 30(6):589–615.
Yoo, Y. (2001). Tutorial on fourier theory. Retrieved June, 17:2007.