Assessment of Experimental Semivariogram Uncertainty in the Presence of a Polynomial Drift

Abstract

The semivariogram, which measures the spatial variability between experimental data, is generally used as a structural input in all two-point geostatistical procedures. However, in most geoscience applications, experimental semivariograms are usually computed from a limited number of sparsely spaced measurements, which results in uncertainty associated with the semivariance values estimated for a specified number of lags. More importantly, considering a spatial variable modelled by a nonstationary random field, uncertainty is not only in the experimental semivariogram of the residuals, but also in the coefficients of the drift model estimated from the available experimental data. Therefore, when assessing the reliability of an experimental semivariogram (or estimated semivariances) in the nonstationary case, both aforementioned uncertainties should be taken into account. The aim of this paper is to extend the “Generalised Bootstrap” procedure to the nonstationary model by propagating the uncertainty associated with the estimated drift coefficients into the uncertainty in the experimental semivariogram of the residuals. The proposed methodology is demonstrated in a case study using abundant geophysical measurements characterised by a nonstationary random function. Two scenarios are evaluated in the case study: (1) it is assumed that the drift coefficients can be estimated without any uncertainty, and (2) uncertainty of the drift coefficients is taken into account. We have explained the methodology that allows to assess the uncertainty of the semivariogram lag estimates in the presence of the drift in the mean. Considering the second scenario, uncertainty is obviously larger than the case where uncertainty of the drift in the mean is ignored. This evaluation should be considered in applications where the data is often rather limited, such as subsurface hydrology (i.e. porosity, transmissivity), soil science (i.e. heavy metal content, soil moisture) and mining (i.e. scoping or pre-feasibility stage of the project). In fact, modern geostatistics should provide not only the semivariogram estimates but also estimation of its uncertainty.

Appendix A. The Steps of the Generalised Bootstrap Algorithm for the Spatial Variable Modelled by a Nonstationary Random Function

The workflow of the generalised bootstrap procedure adapted to the semivariogram uncertainty assessment in the nonstationary case is described as follows:

• 1. Consider n spatial data $\{z({\mathbf{u }}_i);\;i=1,\ldots ,n\}$ modelled by a nonstationary random function. First, model the drift $m({\mathbf{u }})$ as a low-order polynomial (i.e. the drift order 1 or linear model) and estimate its coefficients $\hat{\varvec{\upbeta }}=[{\hat{\beta }}_{1},{\hat{\beta }}_{2},{\hat{\beta }}_{3}]$ by the ML approach.

• 2. Generate L sets of drift $\hat{\varvec{\upbeta }}_{l}=[{\hat{\beta }}^{l}_{1},{\hat{\beta }}^{l}_{2},{\hat{\beta }}^{l}_{3}]$, $\{l=1,\ldots ,L\}$ with different likelihoods by discretising the parameter space of the drift coefficients estimated by the ML approach.

• 3. Select P sets of plausible drift models (or as defined by its coefficients) $\hat{\varvec{\upbeta }}_{l}=[{\hat{\beta }}^{l}_{1},{\hat{\beta }}^{l}_{2},{\hat{\beta }}^{l}_{3}]$ with $\{l=1,\ldots ,P<L\}$ within the $95\%$ CR using the likelihood ratio statistic.

• 4. Calculate P corresponding sets of residual data $\{r^{p}({\mathbf{u }}_i), i=1,\ldots ,n\;\text {and}\;p=1,\ldots ,P\}$.

• 5. For each set, transform residual data $r^{p}({\mathbf{u }}_i)$ to normal (Gaussian) scores $y^{p}({\mathbf{u }}_{i})=\varphi (r^{p}({\mathbf{u }}_{i}))$ with mean 0 and variance 1.

• 6. For each set, model the experimental semivariogram ${\hat{\gamma }}_{p}({\mathbf{h}})$ of the Gaussian residuals $\{y^{p}({\mathbf{u }}_i),i=1,\ldots ,n\;\text {and}\;p=1,\ldots ,P\}$ and obtain the covariance model $C_{p}({\mathbf{h}})$.

• 7. Calculate the $n\times n$ variance-covariance matrix $C_{p}$ from the covariance model $C_{p}({\mathbf{h}})$ of each set of the Gaussian residual data $y^{p}({\mathbf{u }}_i)$.

• 8. Perform a decomposition of the covariance matrix $C_{p}({\mathbf{h}})$ by the LU factorisation as the product of a lower triangular matrix and upper triangular matrix (which is the transpose of the lower triangular), $C_{p}({\mathbf{h}})={\mathbf{LL}} ^T$, where ${\mathbf{L }}$ is an $n\times n$ lower triangular matrix.

• 9. Generate a set of n independent, identically distributed (idd) samples as ${\tilde{y}}^{p}({\mathbf{u }}_{i})={\mathbf{L }}^{-1}\;y^{p}({\mathbf{u }}_{i})$.

• 10. Check the decorrelation of each set of idd residual data ${\tilde{y}}^{p}({\mathbf{u }}_{i})$. The null hypothesis is true when the data ${\tilde{y}}^{p}({\mathbf{u }}_{i})$ are uncorrelated.

• 11. Create a bootstrap resample ${\tilde{y}}^{*(p)}({\mathbf{u }}_{i})$ from the decorrelated data ${\tilde{y}}^{p}({\mathbf{u }}_{i})$.

• 12. The Gaussian resamples that are correlated are obtained as $y^{*(p)}({\mathbf{u }}_{i})={\mathbf{L }}\;{{\tilde{y}}^{*(p)}({\mathbf{u }}_{i})}$.

• 13. The normal scores of each set of resamples are backtransformed by the inverse transformation function:

  $r^{*(p)}({\mathbf{u }}_{i})= \varphi (y^{*(p)}({\mathbf{u }}_{i}))^{-1}$

• 14. Repeat steps $11-13$ to create 2000 bootstrap resamples $\{r^{*(p)}_{\alpha }({\mathbf{u }}_i)\;i=1,\ldots ,n\;;\;p=1,\ldots ,P\;\text {and}\;\alpha =1,\ldots ,2000\}$ for each set of residual data.

• 15. Combine all resamples $r^{*(p)}_{\alpha }({\mathbf{u }}_i)$ according to the ratio between the probability of the particular drift coefficients and the probability of the drift coefficients estimated by the ML approach (see Section ).

• 16. Estimate the experimental semivariogram ${\hat{\gamma }}^{*(p)}_{\alpha }({\mathbf{h}})$ of the selected residual resamples from step 15.

• 17. Calculate bootstrap histograms of the estimated semivariance values for each lag $\{{\hat{\gamma }}^{*(p)}_{\alpha }(h_j);\;j=1,\ldots ,J\}$ from the pooled set of models and quantify the uncertainty as CIs (i.e. $68\%$, $90\%$ and $95\%$) for the $j{\text {th}}$ lag.

• 18. End.