Non-stationary Geostatistical Modeling Based on Distance Weighted Statistics and Distributions

Abstract

A common assumption in geostatistics is that the underlying joint distribution of possible values of a geological attribute at different locations is stationary within a homogeneous domain. This joint distribution is commonly modeled as multi-Gaussian, with correlations defined by a stationary covariance function. This results in attribute maps that fail to reproduce local changes in the mean, in the variance and, particularly, in the spatial continuity. The proposed alternative is to build local distributions, variograms, and correlograms. These are inferred by weighting the samples depending on their distance to selected locations. The local distributions are locally transformed into Gaussian distributions embedding information on the local histogram. The distance weighted experimental variograms and correlograms are able to adapt to local changes in the direction and range of spatial continuity. The automatically fitted local variogram models and the local Gaussian transformation parameters are used in spatial estimation algorithms assuming local stationarity. The resulting maps are rich in nonstationary spatial features. The proposed process implies a higher computational effort than traditional stationary techniques, but if data availability allows for a reliable inference of the local distributions and statistics, a higher accuracy of estimates can be achieved.

Appendices

Appendix A: Locally Weighted Statistics

Weighted local univariate statistics have been previously presented as geographically weighted statistics (Brunsdon et al. 2002). For a set of calibrated weights $\hat{\omega} (\mathbf{u}_{\alpha} ;\mathbf{o})$, α=1,…,n, anchored at point o, the local mean and variance are obtained from

(A.1)

and

(A.2)

Given a separation vector h, the locally weighted variograms are estimated by (Machuca-Mory and Deutsch 2008; Harris et al. 2010)

(A.3)

where ω′(u _α ,u _α +h;o) are the standardized 2-point distance weights and N(h) is the number of pairs separated by h. The local experimental covariance can be estimated as

$$ \hat{C}(\mathbf{h};\mathbf{o}) = \sum _{\alpha = 1}^{N(\mathbf{h})} \omega'(\mathbf{u}_{\alpha} ,\mathbf{u}_{\alpha} + \mathbf{h};\mathbf{o}) \bigl[ z( \mathbf{u}_{\alpha} ) - \hat{m}_{ - \mathbf{h}}(\mathbf{o}) \bigr] \bigl[ z( \mathbf{u}_{\alpha} + \mathbf{h}) - \hat{m}_{\mathbf{ + h}}(\mathbf{o}) \bigr].\mbox{\qquad} $$

(A.4)

The tail and head local means are obtained from

$$ \everymath{\displaystyle }\begin{array}{l} \hat{m}_{\mathbf{ - h}}(\mathbf{o}) = \sum_{\alpha = 1}^{N(\mathbf{h})} \omega'(\mathbf{u}_{\alpha} ,\mathbf{u}_{\alpha} + \mathbf{h};\mathbf{o}) \cdot z(\mathbf{u}_{\alpha} ) , \\[5pt] \hat{m}_{\mathbf{ + h}}(\mathbf{o}) = \sum_{\alpha = 1}^{N(\mathbf{h})} \omega'(\mathbf{u}_{\alpha} ,\mathbf{u}_{\alpha} + \mathbf{h};\mathbf{o}) \cdot z(\mathbf{u}_{\alpha} + \mathbf{h}) . \end{array} $$

(A.5)

And the local experimental correlogram is estimated as

(A.6)

The required local tail and head variances are estimated by

$$ \everymath{\displaystyle }\begin{array} {l} \hat{\sigma}_{ - \mathbf{h}}^{2}( \mathbf{o}) = \sum_{\alpha = 1}^{N(\mathbf{h})} \omega'(\mathbf{u}_{\alpha} ,\mathbf{u}_{\alpha} + \mathbf{h};\mathbf{o}) \cdot \bigl[ z(\mathbf{u}_{\alpha} ) - \hat{m}_{\mathbf{ - h}}(\mathbf{o}) \bigr]^{2} , \\[5pt] \hat{\sigma}_{ + \mathbf{h}}^{2}(\mathbf{o}) = \sum _{\alpha = 1}^{N(\mathbf{h})} \omega'(\mathbf{u}_{\alpha} ,\mathbf{u}_{\alpha} + \mathbf{h};\mathbf{o}) \cdot \bigl[ z( \mathbf{u}_{\alpha} + \mathbf{h}) - \hat{m}_{\mathbf{ + h}}(\mathbf{o}) \bigr]^{2}. \end{array} $$

(A.7)

Appendix B: Hermite Model of Local Gaussian Transformation

Given a number of quantiles z _p (o) of the local distribution, the local Gaussian transformation function is approximated by (Journel and Huijbregts 1978; Wackernagel 2003)

(B.1)

H _q [y _p ] is the Hermite polynomial of order q, and the corresponding local coefficients ϕ _q (o) are obtained by

(B.2)

Depending on the number of samples and the complexity of the cdf, between 50 to 200 local quantiles may suffice for the modeling of the local Gaussian transformation function. The expansion into Hermite polynomials allows for the approximation of the local variance (Rivoirard 1994)

(B.3)

In practice, expansions of degree 20 and 40 are commonly used (Vann and Sans 1996; Wackernagel 2003, p. 247).

Appendix C: Locally Stationary Simple Kriging Estimator and Variance

The locally-stationary simple kriging variance is expressed as

(C.1)

C _Z (0;o) is equivalent to the location-dependent variance. The locally stationary simple kriging estimator is

(C.2)

with m(o)as the local mean.