Keywords

1 INTRODUCTION

The weights of evidence (WofE) is one of the most popular methods for mapping mineral prospectivity, which use the Bayesian theory of conditional probability to quantify spatial associations between evidence layers (or geological factors) and known mineral occurrences. It combines the prior probability of mineral occurrence with the conditional probability of mineral occurrence for each evidential layer using Bayes’ rule to derive posterior probabilities of mineral occurrence [1, 2].

In WofE modeling, a map pattern consisting of points (e.g., mineral deposits on 1:200,000 map) is related to one or more map patterns that are continuous in that they assume values at all points in the study area. Suppose that a study area is digitized as a number (n) of pixels and that X i (i = 1, 2,…, p) are a number of binary explanatory variables used to predict a dichotomous random variable Y representing the presence (Y k =1) or the absence (Y k = 0) of mineralization at the k-th pixel. Provided that n is very large, we can redefine the situation in terms of binary sets B i corresponding to the X i , and a set D corresponding to Y. In most WofE applications, the B i are binary with or without missing data, but the method also can be used with multi-state explanatory variables [3].

WofE is based on the idea of prior and posterior probability; the former is the probability that a terrain unit contains an event before considering any existing predictor variables (the favorable conditioning factors); the latter is an “adjustment” or response of the prior probability, taking into account the evidence from one or more spatial patterns. Positive and negative weights (W + and W -) are initially calculated; the magnitude of these weights depends on the measured association between the response variable and each class of each predictor variable. The difference between W + and W − defines the contrast (C=W +- W -), an overall measure of the degree of the spatial association between each class of the predictor variables and the response variable. The MORPAS (Metal Ore Resource Prediction and Assessment System) that provides tools for weights of evidence, logistic regression, fuzzy logic, and neural networks based on MAPGIS, has been used in this study as a useful tool for automatically calculating the mentioned parameters [4].

2 OVERVIEW OF WEIGHTS OF EVIDENCE

When there is a single pattern B, the odds O(D|B) (i.e., O = p/(l - p)) for occurrence of mineralization if B is present is given by the ratio of the following two expressions of Bayes’s rule:

$$ P\left(D\Big|B\right)=\frac{P\left(B\Big|D\right)P(D)}{P(B)},P\left(\overline{D}\Big|B\right)=\frac{P\left(B\Big|\overline{D}\right)P\left(\overline{D}\right)}{P(B)} $$
(81.1)

where the set \( \overline{D} \) represents the complement of D. Consequently,

$$ \mathrm{In}\kern2em O\left(D\Big|B\right)= In\kern0.5em O(D)+{W}^{+} $$
(81.2)

where the positive weight for the presence of B is

$$ {W}^{+}=\mathrm{In}\frac{P\left(B\Big|D\right)}{P\left(B\Big|\overline{D}\right)} $$
(81.3)

The negative weight for the absence of B is

$$ {W}^{-}=\mathrm{In}\frac{P\left(\overline{B}\Big|D\right)}{P\left(\overline{B}\Big|\overline{D}\right)} $$
(81.4)

When p = n, there are two relations:

$$ P\left(D\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)=\frac{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i\Big|D}\right)P(D)}{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)}P\left(\overline{D}\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)=\frac{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i\Big|D}\right)P\left(\overline{D}\right)}{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)} $$
(81.5)

Conditional independence of D with respect to B 1 , B 2 , …,B n implies that

$$ P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i\Big|D}\right)={\displaystyle \prod_{i=1}^nP\left({B}_i\Big|D\right)}P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i\Big|\overline{D}}\right)={\displaystyle \prod_{i=1}^nP\left({B}_i\Big|\overline{D}\right)} $$
(81.6)
$$ \mathrm{Consequently},\kern1em P\left(D\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)=P(D)\frac{{\displaystyle \prod_{i=1}^nP\left({B}_i\Big|D\right)}}{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)}P\left(\overline{D}\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)=P\left(\overline{D}\right)\frac{{\displaystyle \prod_{i=1}^nP\left({B}_i\Big|\overline{D}\right)}}{P\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)} $$
(81.7)

From these two equations it follows that

$$ \frac{P\left(\overline{D}\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)}{P\left(\overline{D}\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)}=\frac{P(D){\displaystyle \prod_{i=1}^nP\left({B}_i\Big|\overline{D}\right)}}{P\left(\overline{D}\right){\displaystyle \prod_{i=1}^nP\left({B}_i\Big|\overline{D}\right)}} $$
(81.8)

This expression is equivalent to

$$ \mathrm{In}\kern0.5em O\left(D\Big|{\displaystyle \underset{i=1}{\overset{n}{\cap }}{B}_i}\right)=\mathrm{In}\kern0.5em O(D)+{\displaystyle \sum_{i=1}^n{W}_i^{+}} $$
(81.9)

The log-odds on the left side of this expression is also known as the posterior logit. It is simply the sum of the prior logit and the weights of n map layers. The posterior probability follows from the posterior logit. Similar expressions apply when either one or both patterns are absent [5].

3 EXAMPLE OF APPLICATION

WofE modeling of mineral potential involves a three-stage process: (a) estimation of prior probability (P prior) of prospect occurrence, (b) estimation of weights to be assigned to presence and absence of spatial evidence with respect to the prospects, and (c) updating of P prior by using these weights to estimate the posterior probabilities (P posterior).

Based on regional geological background analysis relationship and between minerals and geological background, the study regions are divided into 1755 grid cells with 5km×5km and the eleven geological variables (or evidential layers) are obtained by 1:200 000 geological maps and related minerals distribution maps(or layers). The weights for each geological variable (or evidential layer) are implemented in a GIS environment using MORPAS and MAPGIS (see Table 81.1).Based on proximity analysis, the posterior probability map and the five metallogenic prospective regions of Cu deposits are worked out by tectonism, stratum, rock, fractures, geological abnormity, minerogenetic abnormity, the values of posterior probability and the measures of the association of training sites (Fig. 81.1).

Table 81.1 Weights for each geological variable (or evidential layer)
Full size table
Fig. 81.1
figure 1

Posterior probability map and five metallogenic prospective regions of Cu deposits

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4 CONCLUSIONS

The study regions are divided into 1755 grid cells with 5km×5km and the eleven geological variables (or evidential layers) are obtained by 1:200 000 geological maps and related minerals distribution maps and based on regional geological background analysis relationship and between minerals and geological background. The weights for each geological variable (or evidential layer) are implemented in a GIS environment using MORPAS and MAPGIS. The weights of evidence model is considered as reasonable and delineates permissive regions for Cu deposits. The weights of evidence model has the additional characteristics that it is well defined, reproducible, objective, and provides a quantitative measure of confidence.