Geoscience Reference
In-Depth Information
9
Recoverable Resources: Probabilistic
Estimation
Abstract
The block estimates made by conventional inverse distance or kriging techniques have no
reliable measure of uncertainty attached to them. The approach presented in this chapter
consists of directly predicting the variability/uncertainty in the mining block grades based
on a probability distribution model. The limitations and assumptions supporting these mod-
els are summarized, as well as some of the most important issues regarding the estimation
of point and block distributions.
9.1
Conditional Distributions
The parametric approach to building probabilistic models
was developed early in geostatistics (Matheron
1971
,
1973
;
Marcotte and David
1985
). The methods are based on as-
suming a multivariate or bivariate distribution for the RF
model
{
Conditional probability functions are alternatives to the esti-
mation of a single point or block grade. The original data are
interpreted to provide a conditional distribution function that
is updated locally to obtain posterior probability distribution
functions at each unsampled point/block location. This func-
tion is represented as a cumulative conditional distribution
function, or ccdf, and describes the range of possible values
that the estimate can take. The ccdf is written as
}
. This assumption entails that all ccdfs
are fully specified by a limited number of parameters.
The most common parametric approach is the Gaussian-
based methods, of which the log-normal transformation can
be considered a special case. Since Gaussian distributions
are uniquely characterized by its mean and variance, the
problem of determining the ccdf model
Z
(
u
),
u
∈
A
{
}
Prob
Z
( )
u
≤
z
|( )
n
{
}
becomes one of estimating the two parameters of the model.
Specific examples are multi-Gaussian kriging; Disjunctive
Kriging; Uniform Conditioning; and Lognormal kriging.
The limitation of the parametric approach lies as much in
establishing the appropriateness of the model, as in estimat-
ing its parameters.
Non-parametric transformations derive in methods that
do not make strong univariate assumptions about the distri-
bution; rather, they directly estimate a number of probabili-
ties of the
F z
( ,
u
|( ))
n
=
Prob
Z
( )
u
≤
z
|( )
n
where “
|
(
n
)” means conditional to the nearby information
used to derive the ccdf. This function contains all informa-
tion that may be available about the unknown location. Basic
distribution parameters that can be extracted are: E-type or
average estimates; probabilities of the grade exceeding criti-
cal thresholds; probabilities of the grade being within certain
thresholds; and so on.
{
}
≤
u
function, and then interpo-
late these to obtain the full ccdf. All variants of indicator
kriging fall into this category, including probability kriging.
The non-parametric methods rely heavily on the quantity
and quality of data available. The modeling process is more
time consuming because it requires inference of more spa-
tial continuity models. There must be sufficient information
within the stationary domain to implement a robust and reli-
able non-parametric estimation.
Prob
Z
( )
z
|( )
n
Nonlinear Transforms
The first step to understand the
methods used to estimate probability functions is to under-
stand nonlinear geostatistics. All nonlinear kriging algo-
rithms are actually linear kriging (SK or OK) applied to
specific nonlinear transforms of the original data. The nonlin-
ear transform used specifies the nonlinear kriging algorithm
considered. The transforms lead to methods that are classified
as parametric or non parametric.
