Constrained Kriging: An Alternative to Predict Global Recoverable Resources

Abstract

In most NI-43-101 resource assessment reports the prediction of global in situ resources is performed by either inverse distance weighting, ordinary kriging (OK) or uniform conditioning (UC). These methods have known drawbacks: OK estimates are oversmoothed, and UC necessitates an additional step to localize resources within panels. An alternative, named constrained kriging (CK), enables to circumvent the smoothing issue of OK by imposing the desired theoretical variance to the interpolated variable. CK is not used in NI-43-101 reports, possibly due to a lack of real application examples and little detailed study of its properties. This paper seeks to fill the gap by comparing the prediction performance for global resources of OK, UC and CK on a synthetic lognormal dataset and two real datasets, the Walker Lake and a gold deposit. Results indicate that CK, although being slightly less precise than OK, provides better predictions of grade-tonnage curves than OK and predictions comparable to UC, a remarkable achievement considering that UC is a widespread nonlinear method specifically designed to predict recovery functions. CK is also shown to provide resource estimates more robust than UC with respect to the variogram model specification. Hence, CK appears as a valuable tool allowing simultaneously to localize resources and easily account for change of support in resources estimation.

Appendix

Recall the system of equations to solve for determining CK weights:

$$\begin{aligned} \mathbf{K} \pmb {\lambda }+ \mu _{\textsf {1}} \mathbf{1} +\mu _{\textsf {2}} \mathbf{K} \pmb {\lambda }= & {} \mathbf{k} \end{aligned}$$

(A.1)

$$\begin{aligned} \mathbf{1'} \pmb {\lambda }= & {} {\textsf {1}} \end{aligned}$$

(A.2)

$$\begin{aligned} \pmb {\lambda '} \mathbf{K} \pmb {\lambda }= & {} \sigma ^2_v \end{aligned}$$

(A.3)

Premultiplying Eq. A.1 by $\mathbf{K}^{-\mathbf{1}}$ and arranging the terms, it follows:

$$\begin{aligned} \pmb {\lambda } = \frac{1}{1+\mu _2} \mathbf{K}^{-\mathbf{1}} \left( \mathbf{k} - \mu _{\textsf {1}} \mathbf{1}\right) = \frac{{\textsf {1}}}{{m_2}} \mathbf{K}^{-\mathbf{1}} \left( \mathbf{k} - \mu _{\textsf {1}} \mathbf{1}\right) \end{aligned}$$

(A.4)

where $m_2=1+\mu _2$. Premultiplying Eq. A.1 by $\mathbf{1' K}^{-\mathbf{1}}$, the following form is written:

$$\begin{aligned} m_2+ \mu _1 s = b \end{aligned}$$

(A.5)

with $b=\mathbf{1'K}^{-\mathbf{1}}{} \mathbf{k}=\mathbf{1'} \pmb {\lambda }_\mathbf{s}$ and $s=\mathbf{1'K}^{-\mathbf{1}}{} \mathbf{1}$. Premultiplying Eq. A.1 by $\pmb {\lambda '}$, the following expression is obtained:

$$\begin{aligned} m_2\sigma ^2_v + \mu _1 = \pmb {\lambda '} \mathbf{k} \end{aligned}$$

(A.6)

Substituting Eq. A.4 in Eq. A.6 leads to the following:

$$\begin{aligned} m_2\sigma ^2_v + \mu _1= & {} \frac{1}{m_2} \left( \mathbf{k'} - \mu _{\textsf {1}} \mathbf{1'}\right) \mathbf{K}^{-\mathbf{1}} \mathbf{k} \end{aligned}$$

(A.7)

$$\begin{aligned}= & {} \frac{1}{m_2} \left( \mathbf {k'} \pmb {\lambda }_s -\mu _1 b \right) \end{aligned}$$

(A.8)

Substituting Eq. A.5 in Eq. A.8, one obtains:

$$\begin{aligned} m_2 = \left[ \frac{\mathbf {k'} \pmb {\lambda }_s - b ^2/s}{\sigma _v^2-1/s} \right] ^{1/2} \end{aligned}$$

(A.9)

which completes the proof.