Uncertainty Assessment over any Volume without Simulation: Revisiting Multi-Gaussian Kriging

Abstract

Assessing spatial uncertainty over an arbitrary volume is usually done by generating multiple simulations of the random function and averaging the property over each realization to build its uncertainty distribution. However, this is a cumbersome process for practitioners, as they need to compute and process a large number of realizations. Multi-Gaussian kriging provides a simpler alternative, by directly computing the conditional probability density functions of the random variables. In this work, we revisit the multi-Gaussian framework and present the implementation details to determine the conditional distribution at any support, by numerical integration of the conditional probabilities, using an importance sampling approach. We demonstrate the use of this approach and assess its accuracy in the lognormal and exponential cases with synthetic data. We also apply it to a real three-dimensional mining case, where the uncertainty over scheduled production volumes is determined. The ability to assess this uncertainty may prove valuable, as it enables schedule changes to be made in a mining setting in order to ensure the smooth running of downstream processes.

Appendices

Appendix A: Inferring the Point-Support Conditional Distribution in the Lognormal Case

Let $Z\sim Logn(\mu ,\sigma ^2)$. Then, its cumulative distribution function is defined by

$$\begin{aligned} {F}_{Z}(z)=G\Bigg (\frac{\mathrm{ln}(z)-\mu }{\sigma }\Bigg ). \end{aligned}$$

It follows that the anamorphosis function is

$$\begin{aligned} y=\phi ^{-1}(z)=\frac{\mathrm{ln}(z)-\mu }{\sigma },\quad z=\phi (y)=e^\mu \cdot e^{\sigma y}, \end{aligned}$$

(17)

and

$$\begin{aligned} \phi ^{'}(y)=\sigma \cdot e^\mu \cdot e^{\sigma y}=\sigma \phi (y), \end{aligned}$$

by noting that $\phi ^{'}[\phi ^{-1}(z)]=\sigma \phi [\phi ^{-1}(z)]=\sigma z$. Then, the moments can be computed using Eq. (6)

$$\begin{aligned} \mathbb {E}[Z^{n}(\mathbf{u} )|\mathrm{data}]= & {} \int _{-\infty }^{\infty } \phi ^{n}(t) \cdot g_{y_{SK}}^{\sigma _{SK}^2} (t)\,\mathrm{d}t\nonumber \\= & {} e^{n\mu }\int _{-\infty }^{\infty } e^{n\sigma t} \cdot g_{y_{SK}}^{\sigma _{SK}^2} (t)\,\mathrm{d}t\nonumber \\= & {} e^{n\mu +n \sigma y_{SK}+{\frac{1}{2}}n^{2}\sigma ^{2}\sigma _{SK}^{2}}. \end{aligned}$$

(18)

Fig. 19

Example of probability distributions at a certain pair of points $Z_1$ and $Z_2$ given a lognormal prior distribution (left), and their bivariate behavior when the correlation of their Gaussian transformations is $\rho =0.6$ (right)

Furthermore, the conditioned local distribution, given by Eq. (4), is

$$\begin{aligned} f_{Z|\mathrm{data}}(z)= & {} \frac{1}{\phi ^{'}(\phi ^{-1}(z))}\cdot g_{y_{SK}}^{\sigma _{SK}^2}(\phi ^{-1}(z))\\= & {} \frac{1}{\smash {z\sigma \sigma _{SK}}\sqrt{2\pi }} \cdot e^{-\frac{(\mathrm{ln}(z)-\mu -\sigma y_{SK})^2}{2\sigma ^2\sigma _{SK}^2}}. \end{aligned}$$

This result corresponds to a lognormal distribution with parameters $\mu +\sigma \cdot y_{SK}$ and $\sigma ^2 \cdot \sigma _{SK}^2$ for the mean and variance, respectively. Therefore, the local distribution at a certain point $\mathbf{u}$ conditioned by the data, which presents a lognormal prior distribution, preserves the lognormality.

This derivation allows one to directly see that the variance is proportional to the square of the mean, which is known as the proportional effect (Matheron 1974a).

Following the same procedure, by extending Eq. (13) to the N-dimensional case, we can compute the local conditional distribution of the vector of random variables $[Z(\mathbf{u} _1),\ldots ,Z(\mathbf{u} _N)]^T$, each one following a $Logn(\mu ,\sigma ^2)$ prior distribution.

Here, $[Y(\mathbf{u} _1),\ldots ,Y(\mathbf{u} _N)]^T = (\mathrm{ln}[Z(\mathbf{u} _1)],\ldots ,\mathrm{ln}[Z(\mathbf{u} _N)])^T$ has an N-dimensional normal distribution with mean vector ${{\varvec{\mu }}}= [\mu +\sigma \cdot y_{SK}(\mathbf{u} _1),\ldots ,\mu +\sigma \cdot y_{SK}(\mathbf{u} _N)]^T$ and covariance matrix ${{\varvec{\Sigma }}}=({\varvec{\Sigma }})_{ij}$, $i,j \in \{1,\ldots ,N\}$, such that $({{\varvec{\Sigma }}})_{ij} = ({\varvec{\Sigma }})_{ji}= \sigma ^2 \cdot \sigma _{SK}(\mathbf{u} _i,\mathbf{u} _j)$ and $({{\varvec{\Sigma }}})_{ii} = \sigma ^2 \cdot \sigma _{SK}^2(\mathbf{u} _i)$, $i \in \{1,\ldots ,N\}$, given the data.

In Fig. 19, some possible posterior distributions and their bivariate behavior are presented.

Appendix B: Inferring the Point-Support Conditional Distribution in the Exponential Case

Let $Z\sim Exp(\lambda )$. Then its probability density function is given by $f_{Z}(z)=\lambda e^{-\lambda z}$, $z \ge 0$, and the cdf is

$$\begin{aligned} {F}_{Z}(z)=1-e^{-\lambda z}, \quad z \ge 0. \end{aligned}$$

It follows that the anamorphosis function is given by

$$\begin{aligned} z=\phi (y)=-\frac{1}{\lambda }\cdot \mathrm{ln}(1-G(y)). \end{aligned}$$

(19)

From here we obtain

$$\begin{aligned} \phi ^{'}(y)=\frac{1}{\lambda }\cdot \frac{1}{1-G(y)}\cdot g(y). \end{aligned}$$

Fig. 20

Example of probability distributions at a certain pair of points $Z_1$ and $Z_2$ given an exponential prior distribution (left), and their bivariate behavior when the correlation of their Gaussian transformations is $\rho =0.6$ (right)

We will not attempt to find the back-transformed conditional probability distribution, as we did with the lognormal case. However, we provide numerical results of the posterior distributions (Fig. 20). We note that the posterior distributions do not preserve the exponential characteristic. Instead, results are similar to a lognormal distribution.