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 s 2 = 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( on D) = P(A on D) * P(B on D)

UNCERTAINTY DUE TO MISSING DATA P(D) = E X {P(D on X)} =  P(D on A i ) * P(A i ) or  P(D on ) * P( ) etc.

VARIANCE (MISSING DATA)  2 {P(D)} =  {P(D on A i ) - P(D)} 2 * P(A i ) or  {P(D on ) - P(D)} 2 * P( ) etc.

TOTAL UNCERTAINTY Var (Posterior Logit) = Var (Prior Logit) + +  Var (Weights) + Var (Missing Data)

Uncertainty in Logits 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 km 2, weights, contrast (C) with standard deviations (s). In total: 68 deposits on 2945 km 2

Logistic Regression Logit (  i ) =  0 + x i1  1 + x i2  2 + … + x im  m

Newton-Raphson Iteration  (t+1) =  (t) + {X T V(t)X)} -1 X T r(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 N A = Number of unit cells on map layer A n = Number of deposits n A = Number of deposits on map layer A P(d |A) = Probability that unit cell on A contains a deposit X A = Binary random variable for occurrence of deposit in unit cell on A with EX A = P(d |A) = n A / n T = Random variable for number of deposits in study area

Single binary pattern A (~A = not A)   Posterior Probabilities = N A P(d |A) + N ~A P(d |~A) = = N A {n A / N A } + N ~A {n ~A / N ~A } = n  2 (T) = N A 2  2 (X A ) + N ~A 2  2 (X ~A )

Two binary patterns (A and B):   Posterior Probabilities = N AB P(d |AB) + N A~B P(d |A~B) + + N ~AB P(d |~AB) + N ~A~B P(d |~A~B) = = n AB + n A~B + n ~AB + n ~A~B = = n A. n B / n + n A. n ~B / n + n ~A. n B / n + n ~A. n ~B / n = = n A.{n B + n ~B }/ n + n ~A.{n B + n ~B }/ n = n  2 (T) = N AB 2  2 (X AB ) + N A~B 2  2 (X A~B ) + N ~AB 2  2 (X ~AB ) + N ~A~B 2  2 (X ~A~B )

New conditional independence test applied to ocean floor hydrothermal vent example Total number of vents n = 13 3-map layer model predicts (s.d. = 6.45) P(T = N) > 99% (c.l. = 28.03) 5-map layer model predicts (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 IntegrationPotential Evidential Layers (X) Modeling (F) Output Data S Processing DBMS GIS Database GIS Data Preprocessing Interpreting Define correlated patterns using training points Integrated correlated patterns to estimate unknown points Modeling Prediction

Flowing Wells vs. Distance from ORM Spatial Correlation Distance

Flowing Wells vs. Distance From High Slope Zone Spatial Correlation Distance

Flowing Wells vs. Thickness of Drift

Flowing vs. Distance from Thick Drift 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

Linear Regression with Missing Data Y =  0 +  1 x +  b 1 =  (x i -m x )(y i -m y )/  (x i -m x ) 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.

Logistic Regression & Maximum Likelihood P(Y=1|x) =  (x) = e f(x) /{1+ e f(x) } P(Y=0|x) = 1-  (x)  (x i ) =  (x i ) y i {1-  (x i )} 1- y i l(  ) =   (x i )

Bivariate Logistic Regression Logit (  i ) =  0 + x i1  1  = [  0  1 ]

Log Likelihood Function L(  ) = ln{l(  )} = =  [y i  ln{  (x i )}+(1- y i )  ln{1-  (x i )}]

Differentiate with respect to  0 and  1 to obtain likelihood equations:  {y i -  (x i )} = 0  x i {y i -  (x i )} = 0

Total number of discrete events = Sum of estimated probabilities  y i =  p(x i )

Weighted logistic regression convergence experiments (Level of convergence = 0.01) Seafloor Example (N = 13): Unit cell of 0.01 km 2  12.72; km 2  12.97; km 2  Meguma Terrane Example (N = 68) Unit cell of 1 km 2  64.71; 0.1 km 2  67.96