A non-stationary geostatistical approach to multigaussian kriging for local reserve estimation

Abstract

Multigaussian kriging technique has many applications in mining, soil science, environmental science and other fields. Particularly, in the local reserve estimation of a mineral deposit, multigaussian kriging is employed to derive panel-wise tonnages by predicting conditional probability of block grades. Additionally, integration of a suitable change of support model is also required to estimate the functions of the variables with larger support than that of the samples. However, under the assumption of strict stationarity, the grade distributions and important recovery functions are estimated by multigaussian kriging using samples within a supposedly spatial homogeneous domain. Conventionally, the underlying random function model is required to be stationary in order to carry out the inference on ore grade distribution and relevant statistics. In reality, conventional stationary model often fails to represent complicated geological structure. Traditionally, the simple stationary model neither considers the obvious changes in local means and variances, nor is it able to replicate spatial continuity of the deposit and hence produces unreliable outcomes. This study deals with the theoretical design of a non-stationary multigaussian kriging model allowing change of support and its application in the mineral reserve estimation scenario. Local multivariate distributions are assumed here to be strictly stationary in the neighborhood of the panels. The local cumulative distribution function and related statistics with respect to the panels are estimated using a distance kernel approach. A rigorous investigation through simulation experiments is performed to analyze the relevance of the developed model followed by a case study on a copper deposit.

Appendix

This appendix aims at interpreting the variables used in the equations of this paper.

Z(x):

grade variable at location x

$x_i$, $i=1(1)n$:

spatial locations

h :

spatial distance

$z_i$, $i=1(1)n$:

grade values

$P_\alpha$,$\alpha =1(1)K$:

$\alpha ^{th}$ panel

$\epsilon$ :

background constant

w :

kernel bandwidth

z(x):

realization of the grade variable Z(x)

$z_q(P)$ :

local sample data quantile

$y_q$ :

standard Gaussian quantile

$H_p(y)$ :

Hermite polynomial of order p

$Z_P(v)$ :

grade of SMU v under panel P

$Y_P(v)$ :

standard normal transform of $Z_P(v)$

$Z_P(x)$ :

grade at sample point x under panel P

$Y_P(x)$ :

standard normal transform of $Z_P(x)$

$\phi (P)$ :

local Gaussian anamorphosis function

$\phi _k(P), \ k=0,1,2, \ldots$ :

local anamorphosis coefficient

r(P):

local change of support coefficient

$\rho _P(v_i,v_j)$ :

covariance between $Y_P(v_i)$ and $Y_P(v_j)$

$y_c$ :

Gaussian transformed cut-off of $z_c$

$\sigma ^{SK}_{vP}$ :

local simple kriging variance

g(t):

pdf of standard Gaussian variable T