MINERAL PROSPECTIVITY MAPPING Alok Porwal Centre of Studies in Resources Engineering, Indian Institute of Technology (Bombay) India

Introduction GIS-based mineral prospectivity mapping Generalized methodology Weights of evidence model Fuzzy model Other models (if time)

Mineral potential maps Garbage In, Garbage Out Good Data In, Good Resource Appraisal Out Mineral potential maps GIS Analyse / Combine Remote Sensing Geophysics Geochemistry Geology Remote Sensing Geophysics Geochemistry Geology GIS Analyse / Combine Mineral potential maps

CONCEPTUAL GIS Systematic GIS-based prospectivity mapping database Mineralization processes Conceptual models Knowledge-base Mappable exploration criteria Spatial proxies Processing Predictor maps Overlay CONCEPTUAL GIS MODEL Continuous-scale favorability map Binary favorability map Validation MINERAL POTENTIAL MAP

GIS Systematic GIS-based prospectivity mapping database Knowledge-base Mineralization processes Conceptual models Knowledge-base Mappable exploration criteria Spatial proxies Processing Predictor maps Overlay GIS MODEL Continuous-scale favorability map Binary favorability map A systematic procedure for mineral prospectivity mapping is needed Proper data set Sound conceptual rationale for the input data - A mathematical model that is compatible with the quality of input data and A robust validation mechanism. There is also a need to know where can one go wrong – that is, where are the uncertainties in the modeling procedure, and what is their nature. Validation MINERAL POTENTIAL MAP

Mineral systems approach (Wyborn et al. 1995) 1. Energy 2. Ligand 3. Source 4. Transport 5. Trap 6. Outflow Mineral System (≤ 500 km) Deposit Halo (≤ 10 km) Deposit (≤ 5 km) Model I Metal source Model II Ligand source Energy (Driving Force) Model III Transporting fluid Fluid Discharge Trap Region No Deposits Deposits focal points of much larger mass flux and energy systems Focus on critical processes that must occur to form a deposit Allows identification of mineralization processes at all scales Not restricted to particular geologic settings/deposit type Amenable to probabilistic analysis

Mineral systems approach : Processes to Proxies 1. Energy 2. Ligand 3. Source 4. Transport 5. Trap 6. Outflow Mineral System (≤ 500 km) Deposit Halo (≤ 10 km) Transporting fluid Deposit (≤ 5 km) Model I Ligand/ brine source Model II Metal source Energy (Driving Force) Model III Fluid Discharge Trap Region Deformation Metamorphism Magmatism Connate brines Magmatic fluids Meteoric fluids Enriched source rocks Magmatic fluids Structures Permeable zones Structures Chemical traps Structures aquifers MAPPABLE CRITERIA Radiometric anomalies, geochemical anomalies, whole-rock geochemistry Fault/shear zones, folds geophysical/ geochemical anomalies, alteration Dilational traps, reactive rocks, geophyiscal/ geochemical anomalies, alteration magnetic/ radiometric/ geochemical anomalies, alteration, structures Metamorphic grade, igneous intrusions, sedimentary thickness Evaporites, Organics, isotopes SPATIAL PROXIES

2-D GIS-based prospectivity mapping Scale dependence of exploration criteria (e.g. orogenic gold system - McCuaig and Beresford, 2009) Broad Regional 2-D GIS-based prospectivity mapping ‘Camp’ Prospect

Model-based mineral potential mapping database Predictor maps Processing Mineralization processes Conceptual models Knowledge-base Mappable exploration criteria Spatial proxies Overlay MODEL Continuous-scale favorability map Binary favorability map A systematic procedure for mineral prospectivity mapping is needed Proper data set Sound conceptual rationale for the input data - A mathematical model that is compatible with the quality of input data and A robust validation mechanism. There is also a need to know where can one go wrong – that is, where are the uncertainties in the modeling procedure, and what is their nature. Validation MINERAL POTENTIAL MAP

strike of nearest fault Creation of candidate layers in GIS: Feature extraction What are the exploration criteria for the mineralisation? solid geology structural geology airborne magnetic gravity Primary data distance to faults Empirical strike of nearest fault rheological contrast reactivity (Fe2+/(Fetot+ Mg + Ca) contrast fav. host rock lithology fav. litho. contact type Conceptual fav. tectonic environment

Proxies to Predictor Maps: Systematic Analysis of Exploration Data 11 Proxies to Predictor Maps: Systematic Analysis of Exploration Data Geology Geochemistry Geophysics Remote Sensing Create candidate layers (Feature extraction based on mineral systems model) GIS Hypothesis testing Select input layers Test/Analyse A systematic procedure for mineral prospectivity mapping is needed Proper data set Sound conceptual rationale for the input data - A mathematical model that is compatible with the quality of input data and A robust validation mechanism. There is also a need to know where can one go wrong – that is, where are the uncertainties in the modeling procedure, and what is their nature. Input layers Derivative predictor maps

12 For each exploration criteria, define a hypothesis in terms of GIS layer(s) - Define an experimental plan - Generate all appropriate layers Redefine hypothesis - Examine each layer’s spatial relationship to the targeted mineral deposits - Quantify results Hypothesis incorrect Hypothesis correct Discard layer Factor identified, keep GIS layer for prospectivity mapping 12

Example: Rheology as an exploration criteria for gold trap GRANITE-GREENSTONE BELTS Ultramafic Felsic Volcanics High-Mg basalts Gneiss Foliated monzogranite Monzogranite Granodiorite Tonalite Dolerite Granophyric dolerite BIF Volcaniclastic/ clastic sedimentary rocks Andesitic Porphyry Dolomite-Qtz rock Tholeiitic basalt 5 3 2 1 4 (Fe x [Fe / Fe+Mg+Ca]) Reactivity Rheological strength 4 1 5 Basalt Granite Komatiite (Provided by David Groves) 13

Experiment 1: Host Rock Rheology 14 Experiment 1: Host Rock Rheology 4 1 5 Basalt Granite Komatiite Hypothesis: More deposits in high rheological strength rocks Expected Results 0.00 0.20 0.40 0.60 0.80 1.00 1 2 3 4 5 Rheology Value Spatial association 14

Experiment 1: Host Rock Rheology 15 Experiment 1: Host Rock Rheology The Results 0.00 0.20 0.40 0.60 0.80 1.00 1 2 3 4 5 Rheology Value Edjudina Kalgoorlie Laverton Leonora Yandal Spatial association Hypothesis rejected! 15

Rheology contrast across geological contacts Experiment 2: Rheological Contrast 4 Hypothesis: More deposits adjacent to lithological contacts that have a higher rheological contrast Basalt 5 Granite 1 Komatiite Expected Results 0.00 0.20 0.40 0.60 0.80 1.00 1 2 3 4 5 Rheology contrast across geological contacts Spatial association 16

Rheology Contrast at lithological contacts Experiment 2: Rheological Contrast The Results 0.00 0.20 0.40 0.60 0.80 1.00 1 2 3 4 Rheology Contrast at lithological contacts Spatial association Edjudina Kalgoorlie Laverton Leonora Yandal Hypothesis rejected! 17

Frequent change in rheology 18 Change in Philosophy Frequent change in rheology Increased fluid flow Deposit localization Region of increased fluid flow 18

Geological contact density weighted by rheology contrast 19 Experiment 3: Rheological contrast density Hypothesis: Expect deposits to be more common in regions of greater density of rheological contrast. Expected Results 0.00 0.20 0.40 0.60 0.80 1.00 1 2 3 4 5 Geological contact density weighted by rheology contrast Spatial association 19

Experiment 3: Rheological contrast density 20 Experiment 3: Rheological contrast density Rheology contrast at lithological contacts Rheology contrast density Kalgoorlie Kalgoorlie LOW HIGH 0 1 2 3 4 20

Rheological Contrast Density 21 Experiment 3: Rheology Contrast Density Results Rheological Contrast Density LOW HIGH Spatial association 0.00 0.20 0.40 0.60 0.80 1.00 Edjudina Kalgoorlie Laverton Leonora Yandal Deposits more common in regions of greater rheology contrast density Select rheology contrast layer for inputting into mineral potential models 21

Model-based mineral potential mapping database Knowledge-base Processing Conceptual models Mineralization processes Predictor maps Overlay Mappable exploration criteria Spatial proxies MODEL Continuous-scale favorability map Binary favorability map A systematic procedure for mineral prospectivity mapping is needed Proper data set Sound conceptual rationale for the input data - A mathematical model that is compatible with the quality of input data and A robust validation mechanism. There is also a need to know where can one go wrong – that is, where are the uncertainties in the modeling procedure, and what is their nature. Validation MINERAL POTENTIAL MAP

∫ Structure of a model for mineral resource potential mapping 23 Integrating function linear or non- linear parameters Output mineral potential map Grey-scale or binary Input predictor maps Categoric or numeric Binary or multi-class

GIS-based mineral resource potential mapping - Modelling approaches 24 GIS-based mineral resource potential mapping - Modelling approaches Exploration datasets with homogenous coverage – required for all models Expert knowledge (a knowledge base) and/or Mineral deposit data Training data Expert knowledge Data-driven Hybrid Knowledge-driven Model parameters estimated from mineral deposits data (Known deposits required) Brownfields exploration Examples - Weights of evidence, Bayesian classifiers, NN, Logistic Regression Model parameters estimated from both mineral deposits data and expert knowledge (Known deposits necessary) Semi-brownfields to brownfields exploration Examples – Neuro-fuzzy systems Model parameters estimated from expert knowledge (Known deposits not necessary) Greenfields exploration Examples – Fuzzy systems; Dempster- Shafer belief theory Knowledge-driven – most sensitive to systemic uncertainties Data-driven – more sensitive to stochastic uncertainties Hybrid – minimize both, yet remain prone to model uncertainties

Neuro-fuzzy; fuzzy WofE 25 Which model is best? Theory Rich Symbolic Artificial Intelligence Any of the methods best: Hybrid Systems Neuro-fuzzy; fuzzy WofE Fuzzy Systems Bayesian Neural Networks/ GA This diagram shows the relationship of neural networks to other techniques in terms of the amount of data available and how well the theory how the output can be predicted from the inputs is understood Symbolic Artificial Intelligence techniques can be used when the relationship between the input parameters and the outputs is well known and available in the form of mathematical formulae or rules. An example of a symbolic AI system is an expert system Statistical techniques are appropriate when there is lots of data but little is known about how to predict the outputs The techniques shown in the middle ground are collectively known as soft- computing techniques. Fuzzy systems attempt to imitate human approximate reasoning with vague or imprecise data NN’s are based on a model of neurons in the brain and deal well with uncertain or error prone data Genetic algorithms are based on an analogy to evolution and survival of the fittest. You take a large variety of possible solutions to a problem and split and recombine parts of these solutions and then test how good these are in order to determine which solutions are selected to form the next generation. That’s how you produce successively better and better solutions. Poor Poor Rich Data Greenfields Brownfields 25

GIS MODELS FOR MINERAL EXPLORATION Probabilistic Model (Weights of Evidence): used in the areas where there are already some known deposits spatial associations of known deposits/oil well with the geological features are used to determine the probability of occurrence of a mineral deposit (or well) in each unit cell of the study area. Fuzzy Model: used in the areas where there are no known mineral deposits each geological feature is assigned a weight based on the expert knowledge, these weights are subsequently combined to determine the probability of occurrence of mineral deposit in each unit cell of the study area.

Weights of Evidence Model for Mineral Prospectivity Mapping

Probabilistic model (Weights of Evidence) What is needed for the WofE calculations? A training point layer – i.e. known mineral deposits; One or more predictor maps in raster format.

PROBABILISTIC MODELS (Weights of Evidence or WofE) Four steps: Convert multiclass maps to binary maps Calculation of prior probability Calculate weights of evidence (conditional probability) for each predictor map Combine weights

Calculation of Prior Probability The probability of the occurrence of the targeted mineral deposit type when no other geological information about the area is available or considered. 1k Study area (S) 1k Target deposits D Assuming- Unit cell size = 1 sq km Each deposit occupies 1 unit cell 10k Total study area = Area (S) = 10 km x 10 km = 100 sq km = 100 unit cells Area where deposits are present = Area (D) = 10 unit cells Prior Probability of occurrence of deposits = P {D} = Area(D)/Area(S)= 10/100 = 0.1 Prior odds of occurrence of deposits = P{D}/(1-P{D}) = 0.1/0.9 = 0.11 10k

Convert multiclass maps into binary maps Define a threshold value, use the threshold for reclassification Multiclass map Binary map

Convert multiclass maps into binary maps How do we define the threshold? Use the distance at which there is maximum spatial association as the threshold !

Convert multiclass maps into binary maps Spatial association – spatial correlation of deposit locations with geological feature. A B C D A B C D 10km 1km 10km 1km Study area (S) Gold Deposit (D)

Convert multiclass maps into binary maps D Which polygon has the highest spatial association with D? More importantly, does any polygon has a positive spatial association with D ??? Positive spatial association – more deposits in a polygon than you would expect if the deposits were randomly distributed. What is the expected distribution of deposits in each polygon, assuming that they were randomly distributed? What is the observed distribution of deposits in each polygon? If observed >> expected; positive association If observed = expected; no association If observed << expected; negative association

Convert multiclass maps into binary maps D OBSERVED DISTRIBUTION Area (A) = n(A) = 25; n(D|A) = 2 Area (B) = n(A) = 21; n(D|B) = 2 Area(C) = n(C) = 7; n(D|C) = 2 Area(D) = n(D) = 47; n(D|D) = 4 Area (S) = n(S) = 100; n(D) = 10

Convert multiclass maps into binary maps D EXPECTED DISTRIBUTION Area (A) = n(A) = 25; n(D|A) = 2.5 Area (B) = n(A) = 21; n(D|B) = 2.1 Area(C) = n(C) = 7; n(D|C) = 0.7 Area(D) = n(D) = 47; n(D|D) = 4.7 (Area (S) = n(S) = 100; n(D) = 10) Expected number of deposits in A = (Area (A)/Area(S))*Total number of deposits

Convert multiclass maps into binary maps D EXPECTED DISTRIBUTION OBSERVED DISTRIBUTION Area (A) = n(A) = 25; n(D|A) = 2.5 Area (B) = n(A) = 21; n(D|B) = 2.1 Area(C) = n(C) = 7; n(D|C) = 0.7 Area(D) = n(D) = 47; n(D|D) = 4.7 (Area (S) = n(S) = 100; n(D) = 10) Area (A) = n(A) = 25; n(D|A) = 2 Area (B) = n(A) = 21; n(D|B) = 2 Area(C) = n(C) = 7; n(D|C) = 2 Area(D) = n(D) = 47; n(D|D) = 4 Area (S) = n(S) = 100; n(D) = 10 Only C has positive association! So, A, B and D are classified as 0; C is classified as 1. Another way of calculating the spatial association : = Observed proportion of deposits/ Expected proportion of deposits = Proportion of deposits in the polygon/Proportion of the area of the polygon = [n(D|A)/n(D)]/[n(A)/n(S)] Positive if this ratio >1 Nil if this ratio = 1 Negative if this ratio is < 1

Convert multiclass maps into binary maps – Line features D 10km 1km 10km 1km Study area (S) Gold Deposit (D)

Convert multiclass maps into binary maps – Line features Distance from the fault No. of pixels No of deposits Ratio (Observed to Expected) 9 1 1.1 21 3 1.4 2 17 0.0 16 1.9 4 14 5 6 7 8 3.3 1 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 1 1 2 3 4 5 6 7 3 2 1 1 2 3 4 5 6 3 2 1 1 2 3 4 5 6 4 3 2 1 1 2 3 4 5 4 3 2 1 1 1 2 3 4 5 4 3 2 1 1 2 3 4 5 4 3 2 1 1km 1 2 3 4 1km Gold Deposit (D)

Convert multiclass maps into binary maps – Line features Calculate observed vs expected distribution of deposits for cumulative distances Distance from the fault No. of pixels Cumul No. of pixels No of deposits Cumul No. of deposits Ratio (Observed to Expected) 9 1 1.1 21 30 3 4 1.3 2 17 47 0.9 16 63 7 14 77 1.2 5 86 1.0 6 92 96 8 99 10 100 1 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 1 1 2 3 4 5 6 7 3 2 1 1 2 3 4 5 6 3 2 1 1 2 3 4 5 6 4 3 2 1 1 2 3 4 5 4 3 2 1 1 1 2 3 4 5 4 3 2 1 1 2 3 4 5 4 3 2 1 1 2 3 4 Gold Deposit (D) =< 1 – positive association (Reclassified into 1) > 1– negative association (Reclassified into 0)

Calculation of Weights of Evidence Weights of evidence ~ quantified spatial associations of the resource with predictor maps Unit cell 1k 10k Study area (S) Deposit locations Predictor feature (B1) Predictor Feature (B2) Objective: To estimate the probability of occurrence of D in each unit cell of the study area Approach: Use BAYES’ THEOREM for updating the prior probability of the occurrence of deposits to posterior probability based on the conditional probabilities (or weights of evidence) of the predictor features.

Calculation of Weights of Evidence Bayes’ theorem: Inference Observation P{D|B} = P{D& B} P{B} = P{D} P{B|D} P{D & B} Posterior probability of D given the presence of B Posterior probability of D given the absence of B D- Well B- Predictor Feature THE BAYES EQUATION ESTIMATES THE PROBABILTY OF A WELL GIVEN THE PREDICTOR FEATURE FROM THE PROBABILITY OF THE FEATURE GIVEN A WELL

+ive weight of evidence (W+) -ive weight of evidence (W-) Calculation of Weights of Evidence Using odds (P/(1-P)) formulation: P{B|D} O{D|B} = O{D} Odds of D given the presence of B P{B|D} P{B|D} O{D|B} = O{D} Odds of D given the absence of B P{B|D} Taking logs on both sides: +ive weight of evidence (W+) P{B|D} Loge (O{D|B}) = Loge(O{D}) + Log of odds of D given the presence of B loge P{B|D} -ive weight of evidence (W-) P{B|D} Loge (O{D|B}) = Loge(O{D}) + loge Log of odds of D given the absence of B P{B|D}

Calculation of contrast Contrast (C) measures the net strength of spatial association between the geological feature and mineral deposits Contrast = W+ – W- + ive Contrast – net positive spatial association -ive Contrast – net negative spatial association zero Contrast – no spatial association Can be used to test spatial associations

WEIGHTS OF EVIDENCE FOR MULTIPLE MAPS

WEIGHTS OF EVIDENCE FOR MULTIPLE MAPS

Log of odds of D given the presence of B1 and B2 Loge (O{D|B1, B2}) = Feature B1 Well D Feature B2 Log of odds of D given the presence of B1 and B2 Loge (O{D|B1, B2}) = Loge(O{D}) + W+B1 + W+B2 Log of odds of D given the absence of B1 and presence B2 Loge (O{D|B1, B2}) = Loge(O{D}) + W-B1 + W+B2 Log of odds of D given the presence of B1 and absence B2 Loge (O{D|B1, B2}) = Loge(O{D}) + W+B1 + W-B2 Loge (O{D|B1, B2}) = Log of odds of D given the absence of B1 and B2 Loge(O{D}) + W-B1 + W-B2 Or in general, for n predictor features, Loge(O{D}) + ∑W+/-Bi i=1 n The sign of W is +ive or -ive, depending on whether the feature is absent or present Loge (O{D|B1, B2, … Bn}) = The odds are converted back to posterior probability using the relation 0 = P/(1+P)

Implementation Loge (O{D|B1, B2}) = Loge(O{D}) + ∑W+/-Bi Calculation of posterior probability (or odds) require: Calculation of pr prob (or odds) of occurrence of wells in the study area Calculation of weights of evidence of all predictor features, i.e, P{B|D} loge W+ = W- = &

Exercise-2 ARAVALLI STUDY AREA Layer – B1 (Fold Axes buffers 1.25k) Layer – B2 (Mafic volcanic rocks) There are 54 base-metal deposits in the study area. Out of these, 35 deposits occur within 1.25 km buffers of fold axes (B1), and 42 deposits are associated with mafic volcanic rocks (B2). The total area in the fold axes buffers is 7132 sq km, while the mafic volcanic rocks occupy 4374 sq km area. If the total study area is 55987 sq km, calculate the following (assuming 1 sq km unit area size): 1. Prior probability and odds of base-metal deposits in the study area; 2. W+B1, W-B1, W+B2, W-B2; 3. Contrast values for B1 and B2; and 4. Posterior Log(odds) and probability of base-metal deposits in the study area, where i. Both B1 and B2 are present; ii. B1 present but B2 absent; iii. B2 present but B1 absent; and iv. Both B1 and B2 are absent

Calculate posterior probability for each cell. EXERCISE Feature B1 Well D Feature B2 Calculate posterior probability for each cell.

Calculation of Prior Probability The probability of the occurrence of the wells when no other information about the area is available or considered. 1k Study area (S) 1k Wells D Assuming- Unit cell size = 1 sq km Each deposit occupies 1 unit cell 10k Total study area = Area (S) = 10 km x 10 km = 100 sq km = 100 unit cells Area where wells are present = Area (D) = 10 unit cells Prior Probability of occurrence of wells = P {D} = Area(D)/Area(S)= 10/100 = 0.1 Prior odds of occurrence of wells = P{D}/(1-P{D}) = 0.1/0.9 = 0.11 10k

Calculation of Weights of Evidence B1 P{B|D} P{B|D} loge W- = W+ = loge B2 = n( )/n(D) = n( )/ Where, P{B|D} B & D D P{B|D} B & D n(D) P{B|D} B & D P{B|D} B & D n(D) S B1 Probabilities are estimated as area (or no. of unit cells) proportions D B1 D B1 D B1 D Basic quantities for estimating weights of evidence Total number of cells in study area: n(S) Total number of cells occupied by wells (D): n(D) Total number of cells occupied by the feature (B): n(B) Total number of cells occupied by both feature and wells : n(B&D) B1 D S B2 D B2 D B2 D B2 D Derivative quantities for estimating weights of evidence Total number of cells not occupied by D: n( ) = n(S) – n(D) Total number of cells not occupied by B: n( ) = n(S) – n(B) Total number of cells occupied by B but not D: n( B & D) = n(B) – n( B & D) Total number of cells occupied by D but not B: n(B & D) = n(D) – n(B & D) Total number of cells occupied by neither D but nor B: n( B & D) = n(S) – n(B) – n(D) + n( B & D) D B2 D B

Exercise 10k S Unit cell size = 1 sq km & each well occupies 1 unit cell B1 10k S n(S) = 100 n(D) = 10 n(B1) = 16 n(B2) = 25 n(B1 & D) = 4 n(B2 & D) = 3 S B1 B2 D D B1 D B1 D B1 D B2 D B2 D B2 D B2 B1 D B2 D Calculate the weights of evidence (W+ and W-) and Contrast values for B1 and B2 = n( )/n(D) = n( )/ = [n(B) – n( )]/[n(S) –n(D)] = n( )/n(D) = [n(D) – n( )]/n(D) = n( )/ = [n(S) – n(B) – n(D) + n( )]/[n(S) –n(D)] B & D B & D P{B|D} n(D) Where, P{B|D} P{B|D} loge W- = W+ = loge

Cells occupied by deposits Solution Cells occupied by deposits 10k S Assuming Unit cell size = 1 sq km and each well occupies 1 unit cell B1 S n(S) = 100 n(D) = 10 n(B1) = 16 n(B2) = 25 n(B1 & D) = 4 n(B2 & D) = 3 S B1 B2 D D B1 D B1 D B1 D B2 D B2 D B2 D 10k B2 B1 D B2 D n(D) = n(S) – n(D) =90 n( ) = n(B2) – n( ) = 22 B2& D n( ) = n(D) – ( ) = 7 B2&D n(D) = n(S) – n(D) =90 n(B2) = n(S) – n(B2) = 75 n(B1) = n(S) – n(B1) = 84 n( ) = n(B1) – n( ) =12 B1& D B1& D n( ) = n(D) – n( ) = 6 B1&D B1& D n( ) = - n(B1) - n(D) + n( ) = 78 = n( )/n(D) = 4/10 = 0.4000 = n( )/ = 12/90 = 0.1333 = n( )/n(D) = 6/10 = 0.6000 = n( )/ = 78/90 = 0.8667 B1& D n(S) B1& D n( ) = - n(B2) - n(D) + n( ) =68 = n( )/n(D) = 3/10 = 0.3000 = n( )/ = 22/90 = 0.2444 = n( )/n(D) = 7/10 = 0.7000 = n( )/ = 68/90 = 0.7555 B2& D n(S) B2&D P{B1|D} B1& D P{B2|D} B2&D P{B1|D} B1&D n(D) P{B2|D} B2& D n(D) P{B1|D} B1&D P{B2|D} B2&D P{B1|D} B1& D n(D) P{B2|D} B2& D n(D) P{B1|D} W+B1 = loge = loge(0.4000/0.1333) = loge(3.0007) = 1.0988 P{B2|D} W+B2 = loge = loge(0.3000/0.2444) = loge(1.2275) = 0.2050 P{B1|D} W-B1 = loge = loge(0.6000/0.8667) = loge(0.6923) = -0.3678 P{B2|D} W-B2 = loge = loge(0.7000/0.7555) = loge(0.9265) = -0.0763 Contrast = W+B1 – W-B1 = 1.0988 – (-0.3678) = 1.4666 Contrast = W+B2 – W-B2 = 0.2050 – (-0.0763) = 0.2812

Exercise - Calculation of Posterior Probability Loge (O{D|B1, B2}) = Loge(O{D}) + W+/-B1 + W+/-B2 Loge(O{D}) = Loge(0.11) = -2.2073 Calculate posterior probability given: 1. Presence of B1 and B2; 2. Presence of B1 and absence of B2; 3. Absence of B1 and presence of B2; 4. Absence of both B1 and B2 S B1 B2 Prior Prb = 0.10 Prior Odds = 0.11

Ground Water potential Map Calculation of Posterior Probability Loge (O{D|B1, B2}) = Loge(O{D}) + W+/-B1 + W+/-B2 Loge(O{D}) = Loge(0.11) = -2.2073 S For the areas where both B1 and B2 are present B1 Loge (O{D|B1, B2}) = -2.2073 + 1.0988 + 0.2050 = -0.8585 O{D|B1, B2} = Antiloge (-0.8585) = 0.4238 P = O/(1+O) = (0.4238)/(1.4238) = 0.2968 For the areas where B1 is present but B2 is absent Loge (O{D|B1, B2}) = -2.2073 + 1.0988 - 0.0763 = -1.1848 O{D|B1, B2} = Antiloge (- 1.1848) = 0.3058 P = O/(1+O) = (0.3058)/(1.3058) = 0.2342 B2 Prior Prb = 0.10 Ground Water potential Map For the areas where B1 is absent but B2 is present Loge (O{D|B1, B2}) = -2.2073 - 0.3678 + 0.2050 = -2.3701 O{D|B1, B2} = Antiloge (-2.3701) = 0.0934 P = O/(1+O) = (0.0934)/(1.0934) = 0.0854 For the areas where both B1 and B2 are absent Loge (O{D|B1, B2}) = -2.2073 - 0.3678 - 0.0763 = -2.6514 O{D|B1, B2} = Antiloge (-2.6514) = 0.0705 P = O/(1+O) = (0.0705)/(1.0705) = 0.0658 Posterior probability 0.2968 0.0854 0.2342 0.0658