2. Probabilistic Mineral Resource Potential Mapping

• The processing of geo-scientific information for the purpose of estimating probabilities of occurrence for various types of mineral deposits was made easier when Geographic Information Systems became available. Weights-of-Evidence modeling and logistic regression are examples of GIS implementations.

Weights of Evidence (WofE)

BAYES’ RULE

P(D on A) = P(D and A)/P(A)

P(A on D) = P(A and D)/P(D)

P(D on A) = P(A on D) * P(D)/P(A)

ODDS & LOGITS

O = P/(1-P); P = O/(1+O); logit = ln O

ln O(D on A) = W+(A) + ln O(D)

W+(A) = ln {P(A on D)/P(A not on D)}

VARIANCE OF WEIGHT

s2 = n-1 (A and D) + n-1 (A and not D)

Negative Weight & Contrast

W-(A) = W+(not A)

Contrast: C = W+(A) - W-(A)

PRESENT, ABSENT or MISSING

add W+, W- or 0

to prior logit

TWO or MORE LAYERS

Add Weight(s) assuming

Conditional Independence

P(<A and B> on D) = P(A on D) * P(B on D)

Table 1. Contingency table for 2x2 conditional independence test

Observed frequencies Expected frequenciesA ~A Sum A ~A Sum

B n AB n ~AB n B B n An B /n n ~An B /n n B

~B n A~B n ~A~B n ~B ~B n An ~B /n n ~An ~B /n n ~B

Sum n A n ~A n Sum n A n ~A n

BBAA

BAABABBA

nnnn

nnnnnX

~~

~~~~2 }{

UNCERTAINTY DUE TO MISSING DATA

P(D) = EX{P(D on X)}

= P(D on Ai) * P(Ai)

or P(D on <Ai and Bk>) * P(<Ai and Bk>)

etc.

VARIANCE (MISSING DATA)

2{P(D)} = {P(D on Ai) - P(D)}2 * P(Ai)

or {P(D on <Ai and Bk>) - P(D)}2 * P(<Ai and Bk>)

etc.

TOTAL UNCERTAINTY

Var (Posterior Logit) = Var (Prior Logit) +

+ Var (Weights) + Var (Missing Data)

Uncertainty inLogits and Probabilities

D {Logit (P)} = 1/P(1-P)

(P) ~ P(1-P) Logit (P)}

Meguma Terrain Example

Table 1. Number of gold deposits, area in km2, weights, contrast (C) with standard deviations (s). In total: 68 deposits on 2945 km2

deps(+) area(+) deps(-) area(-) deps(0) area(0)1 Anticlines 51 1280 17 1665 0 02 HG contact 33 1030 35 1914 0 03 Goldenville Fm. 63 2016 5 928 0 04 Granite contact 11 383 57 2562 0 05 Kriged As 12 219 56 2725 0 06 Lake geochem. 9 166 15 1597 44 11597 NW linears 14 582 54 2362 0 0

Weight: W+ s(W+) W- s(W-) C s(C) C/s(C)1 0.563 0.143 -0.829 0.244 1.392 0.283 4.9262 0.336 0.177 -0.238 0.171 0.575 0.246 2.3393 0.311 0.128 -1.474 0.448 1.784 0.466 3.8264 0.223 0.306 -0.038 0.134 0.261 0.334 0.7835 0.895 0.297 -0.119 0.135 1.014 0.326 3.1096 1.423 0.343 -0.375 0.259 1.798 0.43 4.1837 0.041 0.271 -0.01 0.138 0.051 0.304 0.169

Logistic Regression

Logit (i) = 0 + xi1 1 + xi2 2 + … + xim m

Newton-Raphson Iteration

(t+1) = (t) + {XTV(t)X)}-1XTr(t), t = 1, 2, …

r(t) = y(t) - p(t)

Seafloor Example

NEW CONDITIONAL INDEPENDENCE TEST FOR

WEIGHTS OF EVIDENCE METHOD

Definitions

N = Number of unit cells

NA = Number of unit cells on map layer A

n = Number of deposits

nA = Number of deposits on map layer A

P(d |A) = Probability that unit cell on A contains a deposit

XA = Binary random variable for occurrence of deposit in unit cell

on A with EXA = P(d |A) = nA / n

T = Random variable for number of deposits in study area

Single binary pattern A (~A = not A)

Posterior Probabilities = NA P(d |A) + N~A P(d |~A) =

= NA {nA / NA} + N~A {n~A / N~A } = n

2(T) = NA 2 2(XA) + N~A

2 2(X~A)

Two binary patterns (A and B):

Posterior Probabilities = NAB P(d |AB) + NA~B P(d |A~B) + + N~AB P(d |~AB) + N~A~B P(d |~A~B) =

= nAB + nA~B + n~AB + n~A~B == nA . nB / n + nA . n~B / n + n~A . nB / n + n~A . n~B / n == nA .{nB + n~B }/ n + n~A .{nB + n~B }/ n = n

2(T) = NAB

2 2(XAB) + NA~B 2 2(XA~B ) + N~AB

2 2(X~AB) + N~A~B 2 2(X~A~B )

Table 3. Estimation of T and s 2 (T ) for 3-layer model; I - age, J - topography, K - rock type; N=100 x Area

IJK Area (km2) P f s (P f ) N IJK P f N IJK2 s 2 (P f )

222 1.344 0.0003 0.0005 0.0403 0.0045212 0.9007 0.0057 0.0065 0.5134 0.3428221 0.4351 0.0008 0.0013 0.0348 0.0032122 0.4187 0.0111 0.0126 0.4648 0.2783112 0.3415 0.1709 0.0907 5.8362 9.5939211 0.2223 0.0154 0.0176 0.3423 0.1531111 0.1771 0.3604 0.153 6.3827 7.3421121 0.1456 0.0297 0.336 0.4324 23.9332

Sum 3.985 14.0470 41.6511

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Posterior probability

Nu

mb

er

of

hy

dro

the

rma

l ve

nts

New conditional independence test applied to ocean floor hydrothermal vent example

Total number of vents n = 13

3-map layer model predicts 14.05 (s.d. = 6.45)

P(T = N) > 99% (c.l. = 28.03)

5-map layer model predicts 37.59 (s.d. = 10.47)

P(T > N) > 99% (c.l. = 37.40)

Application of Weights of Evidence Method for Assessment of Flowing Wells in the Greater Toronto Area,

Canada

By Qiuming Cheng,Natural Resources Research, vol. 13,

no. 2, June 2004

ORM Study Area and Surficial Geology of Southern Ontario

Oak Ridges Moraine

Digital Elevation Model Southern Ontario

DEM and Location of ORM

Geology of ORM•

Flowing Wells and Springs

Spatial Decision Support System (SDSS)GIS Data Integration for Prediction

Aquifers

Drift Thickness

Slope

Lithology

.

.

.Integration Potential

Evidential Layers (X)

Modeling (F) Output Data

S

ProcessingDBMS

GIS Database

GIS Data PreprocessingInterpreting

Define correlated patterns using training points

Integrated correlated patterns to estimate unknown points

Modeling Prediction

Flowing Wells vs. Distance from ORM

-8

-4

0

4

8

12

0 5000 10000 15000

Spatial Correlation

Distance

Flowing Wells vs. Distance From High Slope Zone

-8

-4

0

4

8

0 2000 4000 6000 8000

Spatial Correlation

Distance

Flowing Wells vs. Thickness of Drift

Flowing vs. Distance from Thick Drift

-4

0

4

8

0 5000 10000 15000 20000

Spatial Correlation

Distance

Posterior Probability Map calculated by Arc-WofE from buffer zones around ORM and steep slope zones

Mapping potential groundwater discharges using Multivariate Logistic Regression

Modelling Uncertainty in Weights due to Kriging variance

T=7

P ( X=0 )=1.0A

P ( X=0 )=0.3B

P ( X=0)=0.0C

sA

mA m B mCm

sB

s C

A

1050

BmB= 7.98

=1

CmC =10 .51

=1

mA = 3.45= 0

class area points W+ s(W+) W- s(W-) C s(C) Stud(C)3 2945 684 2940 685 2927 67 -0.0089 0.1236 0.9083 1.0289 -0.9172 1.0363 -0.88516 2891 67 0.0038 0.1236 -0.2243 1.0094 0.228 1.0169 0.22427 2817 67 0.0303 0.1236 -1.098 1.0039 1.1283 1.0115 1.11548 2702 63 0.0097 0.1275 -0.1148 0.4519 0.1245 0.4695 0.26519 2529 60 0.0275 0.1307 -0.1851 0.357 0.2125 0.3802 0.5591

10 2259 53 0.0163 0.139 -0.0557 0.2611 0.072 0.2958 0.243511 1873 46 0.0629 0.1493 -0.12 0.2154 0.1829 0.2621 0.697912 1466 42 0.2213 0.1566 -0.2782 0.1979 0.4995 0.2523 1.979813 1100 32 0.2373 0.1794 -0.1721 0.1683 0.4093 0.246 1.663914 765 23 0.27 0.2117 -0.1142 0.1506 0.3841 0.2598 1.478315 510 20 0.5461 0.2281 -0.1615 0.1458 0.7076 0.2707 2.613816 330 13 0.5488 0.283 -0.0952 0.1363 0.644 0.3141 2.050417 219 12 0.8947 0.2969 -0.1193 0.135 1.014 0.3262 3.108718 150 9 0.9901 0.3438 -0.0915 0.1316 1.0816 0.3681 2.938419 106 5 0.7357 0.4581 -0.0405 0.1274 0.7762 0.4755 1.632520 75 4 0.8593 0.5138 -0.0354 0.1264 0.8947 0.5291 1.69121 57 2 0.4238 0.7198 -0.0104 0.1245 0.4342 0.7305 0.594522 42 2 0.7367 0.7244 -0.0157 0.1245 0.7524 0.735 1.023623 33 2 0.9988 0.7294 -0.019 0.1245 1.0177 0.74 1.375324 26 2 1.231 0.7352 -0.0212 0.1245 1.2523 0.7456 1.679525 20 1 0.7579 1.0249 -0.0079 0.1236 0.7658 1.0323 0.741826 15 1 1.0696 1.0339 -0.0098 0.1236 1.0793 1.0412 1.036627 11 1 1.3758 1.0457 -0.0111 0.1236 1.3869 1.053 1.317128 8 1 1.7329 1.0648 -0.0122 0.1236 1.7451 1.0719 1.62829 5 1 2.2455 1.106 -0.0133 0.1236 2.2588 1.1129 2.029730 3 1 3.0032 1.215 -0.0141 0.1236 3.0173 1.2213 2.470631 1 032 1 0

Linear Regression with Missing Data

Y = 0 + 1 x +

b1 = (xi-mx)(yi-my)/(xi-mx)2

Table 2. Comparison of 4 logistic regression solutions: A. Layer deleted; B. Absences set to 0; C. Cells deleted; D. Use of Weighted Mean.

Coeff. (A) ST.D. (A) Coeff. (B) ST.D. (B) Coeff. (C) ST.D. (C) Coeff. (D) ST.D. (D)0 -9.7317 1.2118 -10.3256 1.2459 -10.5161 1.7714 -10.5879 1.24641 1.1901 0.2995 1.1647 0.2949 1.0535 0.4755 1.1544 0.29512 0.2536 0.2632 0.2182 0.2645 0.6654 0.4863 0.2004 0.2653 1.2055 0.4997 1.2143 0.5002 0.289 0.7249 1.2203 0.50054 0.4899 0.3405 0.4857 0.341 0.7744 0.526 0.4821 0.34165 0.8328 0.3311 0.754 0.3362 0.9305 0.5142 0.7216 0.33756 0.7149 0.3797 1.4876 0.4544 0.9839 0.36897 -0.0065 0.3073 -0.018 0.3078 -0.5948 0.631 -0.0213 0.3081

Logistic Regression& Maximum Likelihood

P(Y=1|x) = (x) = ef(x)/{1+ ef(x)}

P(Y=0|x) = 1-(x)

(xi) = (xi) yi {1-(xi)} 1- yi

l() = (xi)

Bivariate Logistic Regression

Logit (i) = 0 + xi1 1

= [0 1]

Log Likelihood Function

L() = ln{l()} =

= [yiln{(xi)}+(1- yi)ln{1-(xi)}]

Differentiate with respect to 0 and 1 to

obtain likelihood equations:

{yi - (xi)} = 0

xi{yi - (xi)} = 0

Total number of discrete events = Sum of estimated probabilities

yi = p(xi)

Weighted logistic regression convergence experiments (Level of convergence = 0.01)

Seafloor Example (N = 13):

Unit cell of 0.01 km2 12.72;

0.001 km2 12.97; 0.0001 km2 13.00

Meguma Terrane Example (N = 68)

Unit cell of 1 km2 64.71; 0.1 km2 67.96