Changes

Several statistical methods can be used to construct a regular grid from control point data. Trend surface analysis is a regression method that fits a power series polynomial to control point data. This method has been used in geological practice to isolate regional trends from sparse control point sets. It is not designed to honor control points, but is used instead to separate regional variation from local variation (for example, separating structural variation from stratigraphic variation). It is often used in exploration, but it has little use in detailed mapping of reservoirs.

Several statistical methods can be used to construct a regular grid from control point data. Trend surface analysis is a regression method that fits a power series polynomial to control point data. This method has been used in geological practice to isolate regional trends from sparse control point sets. It is not designed to honor control points, but is used instead to separate regional variation from local variation (for example, separating structural variation from stratigraphic variation). It is often used in exploration, but it has little use in detailed mapping of reservoirs.

''Kriging'' is a statistical method devised by Krige<ref name=pt08r13>Krige, D. G., 1951, A statistical approach to some mine valuation problems on the Witwatersrand: Journal of Chemical Metallurgy and Mineralogy Society of South Africa, v. 52, n. 6, p. 119–139.</ref> and developed by Matheron,<ref name=pt08r14>Matheron, G., 1971, The theory of regionalized variables and its application: Paris, Les Cahiders du Centre de Morphologie Mathematique, École Nationale Superieur des Mines, Booklet 5, 211 p.</ref> to estimate gold reserves in ore bodies. It has found considerable application as a gridding method in the petroleum industry. The method is based on the theory of the regionalized variable first formulated by Matheron)<ref name=pt08r14 /> and popularized by Clark<ref name=pt08r3>Clark, I., 1979, Practical geostatistics: New York, Elsevier Applied Science Publishers, 129 p.</ref> and Journel and Huijbregts.<ref name=pt08r12>Journel, A. G., Huijbregts, C. J., 1978, Mining geostatistics: New York, Academic Press, 600 p.</ref> Regionalized variable theory breaks spatial variation into three components. The drift is large-scale variation that can be attributed to regional variation, a smaller scale random but spatially correlated part, and still smaller scale random noise. The method uses knowledge of spatial variance of the drift to derive a set of weights for control points that are unbiased statistical estimates. If all of the statistical assumptions are met, it can provide contours that are unbiased estimates. It also provides estimation variance at each grid node. Therefore, the method is statistically superior to the gridding methods discussed earlier. It also strictly honors control points.

''Kriging'' is a statistical method devised by Krige<ref name=pt08r13>Krige, D. G., 1951, A statistical approach to some mine valuation problems on the Witwatersrand: Journal of Chemical Metallurgy and Mineralogy Society of South Africa, v. 52, n. 6, p. 119–139.</ref> and developed by Matheron,<ref name=pt08r14>Matheron, G., 1971, The theory of regionalized variables and its application: Paris, Les Cahiders du Centre de Morphologie Mathematique, École Nationale Superieur des Mines, Booklet 5, 211 p.</ref> to estimate gold reserves in ore bodies. It has found considerable application as a gridding method in the [[petroleum]] industry. The method is based on the theory of the regionalized variable first formulated by Matheron)<ref name=pt08r14 /> and popularized by Clark<ref name=pt08r3>Clark, I., 1979, Practical geostatistics: New York, Elsevier Applied Science Publishers, 129 p.</ref> and Journel and Huijbregts.<ref name=pt08r12>Journel, A. G., Huijbregts, C. J., 1978, Mining geostatistics: New York, Academic Press, 600 p.</ref> Regionalized variable theory breaks spatial variation into three components. The drift is large-scale variation that can be attributed to regional variation, a smaller scale random but spatially correlated part, and still smaller scale random noise. The method uses knowledge of spatial variance of the drift to derive a set of weights for control points that are unbiased statistical estimates. If all of the statistical assumptions are met, it can provide contours that are unbiased estimates. It also provides estimation variance at each grid node. Therefore, the method is statistically superior to the gridding methods discussed earlier. It also strictly honors control points.

Knowledge of the drift function is necessary to use the method to interpolate control point data onto grid nodes. This knowledge is embodied in a function, termed the ''semi-variogram'', that can be estimated for several orientations from geophysical data.<ref name=pt08r15>Olea, R. A., 1975, Optimum mapping techniques using regionalized variable theory: Lawrence, KS, Kansas Geological Survey, Series on Spatial Analysis, n. 2, 137 p</ref> If the semi-variogram cannot be obtained experimentally, it is assumed to be linear or exponential. This assumption can greatly reduce the confidence of estimate thereby defeating the power of the method. Although it is the most complex of the methods discussed here, it has considerable application in reservoir analysis (see [[Correlation and regression analysis]], [[Multivariate data analysis]], and [[Monte Carlo and stochastic simulation methods]]) and [[Reservoir modeling for simulation purposes]] and [[Conducting a reservoir simulation study: an overview]].

Knowledge of the drift function is necessary to use the method to interpolate control point data onto grid nodes. This knowledge is embodied in a function, termed the ''semi-variogram'', that can be estimated for several orientations from geophysical data.<ref name=pt08r15>Olea, R. A., 1975, Optimum mapping techniques using regionalized variable theory: Lawrence, KS, Kansas Geological Survey, Series on Spatial Analysis, n. 2, 137 p</ref> If the semi-variogram cannot be obtained experimentally, it is assumed to be linear or exponential. This assumption can greatly reduce the confidence of estimate thereby defeating the power of the method. Although it is the most complex of the methods discussed here, it has considerable application in reservoir analysis (see [[Correlation and regression analysis]], [[Multivariate data analysis]], and [[Monte Carlo and stochastic simulation methods]]) and [[Reservoir modeling for simulation purposes]] and [[Conducting a reservoir simulation study: an overview]].