1. Indian Institute of Technology Bombay 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 deposits data Model parameters estimated from mineral deposits data (Known deposits required) Brownfields exploration Examples - Weights of evidence, Bayesian classifiers, NN, Logistic Regression Data-driven Knowledge-driven Hybrid  Mineral deposit data mineral deposits data Model parameters estimated from both mineral deposits data expert knowledge and expert knowledge (Known deposits necessary) Semi-brownfields to brownfields exploration Examples – Neuro-fuzzy systems Training data Expert knowledge expert knowledge Model parameters estimated from expert knowledge (Known deposits not necessary) Greenfields exploration Examples – Fuzzy systems; Dempster- Shafer belief theory

2. Indian Institute of Technology Bombay 2 Structure of a model for mineral resource potential mapping ∫ Integrating function linear or non- linear parameters Input predictor maps Categoric or numeric Binary or multi-class Output mineral potential map Grey-scale or binary

3. Indian Institute of Technology Bombay 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. GIS MODELS FOR MINERAL EXPLORATION

4. Indian Institute of Technology Bombay Fuzzy Inference Systems for Mineral Prospectivity Mapping

5. Indian Institute of Technology Bombay

6. Introduction Fuzzy logic: A way to represent imprecision in logic and approximate reasoning A way to make use of natural language in logic Humans say things like : “If it is cool and dry, I will walk faster on my morning walk” "If it is overcast and warm and humid, it will rain heavily" Linguistic variables: Speed: {Fast, slow} Temp: {freezing, cool, warm, hot} Cloud Cover: {overcast, partly cloudy, sunny} Humid: {high, average, low} Rain: {Heavy, moderate, light}

7. Indian Institute of Technology Bombay Crisp (Traditional) Variables Crisp variables represent precise quantities: Temp = 36 deg C Humidity = 70% Some thing like: –If the cloud cover is 90% and temperature is 40 degrees C and humidity is 70%, then the rainfall would be 30 mm. –If the temperature is 25 deg C, I will walk at 20 km/hr on my morning walk.

8. Indian Institute of Technology Bombay Fuzzy Sets Extension of Classical Sets Not just a membership value of in the set and out the set, 1 and 0 –but partial membership value, between 1 and 0

9. Indian Institute of Technology Bombay Example: Height Tall people: say taller than or equal to 6 feet –6’, 6’1”, 6’3” feet are members of this set –5’11.9” are not members of this set - Is that reasonable? (measurement can be inaccurate and/or imprecise )

10. Indian Institute of Technology Bombay Example: Weekend days

11. Indian Institute of Technology Bombay In fuzzy logic, the truth of any statement becomes a matter of degree. Q: Is Saturday a weekend day? A: 1 (yes, or true) Q: Is Tuesday a weekend day? A: 0 (no, or false) Q: Is Friday a weekend day? A: 0.8 (for the most part yes, but not completely) Q: Is Sunday a weekend day? A: 0.95 (yes, but not quite as much as Saturday). BINARY FUZZY

12. Indian Institute of Technology Bombay Example: Season BINARY FUZZY SummerRainyAutumnWinterSummerRainyAutumnWinter

13. Indian Institute of Technology Bombay MEMBERSHIP FUNCTION A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. Example: Membership function of a set of tall people Crisp set Fuzzy set:

14. Indian Institute of Technology Bombay Example: Height

15. Indian Institute of Technology Bombay Membership Function A = {x | x > 6} A = {x, µ A (x) | x ∈ X} A membership function must vary between 0 and 1. The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency. A classical set might be expressed as A fuzzy set is an extension of a classical set. If X is the universe of discourse and its elements are denoted by x, then a fuzzy set A in X is defined as a set of ordered pairs. µ A (x) is called the membership function (or MF) of x in A. The membership function maps each element of X to a membership value between 0 and 1.

16. Indian Institute of Technology Bombay Membership functions piece-wise linear functions the Gaussian distribution function the sigmoid curve quadratic and cubic polynomial curves

17. Indian Institute of Technology Bombay Membership Functions For each variable value a different membership function is required Temp: {Freezing, Cool, Warm, Hot} 520 30 40 Temp O C

18. Indian Institute of Technology Bombay Fuzzy Operators How do we use fuzzy membership functions in predicate logic? Fuzzy logic Connectives: –Fuzzy Conjunction,  –Fuzzy Disjunction,  Operate on degrees of membership in fuzzy sets

19. Indian Institute of Technology Bombay Fuzzy Disjunction (= OR Operator) A  B max(A, B) A  B = C "Quality C is the disjunction of Quality A and B" (A  B = C)  (C = 0.75)

20. Indian Institute of Technology Bombay Fuzzy Conjunction (=AND Operator) A  B min(A, B) A  B = C "Quality C is the conjunction of Quality A and B" (A  B = C)  (C = 0.375)

21. Indian Institute of Technology Bombay Example: Fuzzy Conjunction Calculate A  B given that A is.4 and B is 20

22. Indian Institute of Technology Bombay Example: Fuzzy Conjunction Calculate A  B given that A is.4 and B is 20 Determine degrees of membership.

23. Indian Institute of Technology Bombay Subjective assignment of fuzzy membership values It is also possible to assign the fuzzy membership values subjectively (without using a membership function) LithotypeMembership value Granite0.2 Dolerite0.7 Magnetite quartzite 0.9 Diorite0.4 Distance to a fault Membership value 0 – 1 km0.9 1 – 2 km0.7 2 – 3 km0.5 3 – 4 km0.3 4 – 5 km0.1 > 5 km0.001

24. Indian Institute of Technology Bombay Fuzzy if-then rules Fuzzy if-then rule statements are used to formulate the conditional statements that comprise fuzzy logic. A single fuzzy if-then rule assumes the form: if x is A then y is B where A and B are linguistic values defined by fuzzy sets on the ranges X and Y, respectively. The if-part of the rule "x is A" is called the antecedent or premise The then-part of the rule "y is B" is called the consequent or conclusion.

25. Indian Institute of Technology Bombay if x is A then y is B For example: If FeO is high then gold potential is average “High” is represented as a number between 0 and 1, and so the antecedent is an interpretation that returns a single number between 0 and 1. “Average” is represented as a fuzzy set, and so the consequent is an assignment that assigns the entire fuzzy set B to the output variable y. Fuzzy if-then rules

26. Indian Institute of Technology Bombay The input to an if-then rule is the current value for the input variable (in this case, FeO) and the output is an entire fuzzy set (in this case, gold potential). The consequent specifies a fuzzy set be assigned to the output. The implication function then modifies that fuzzy set to the degree specified by the antecedent. The most common ways to modify the output fuzzy set are truncation using the min function, where the fuzzy set is truncated. This set has to be defuzzified, assigning one value to the output. 40%50%60%70% 0 1 1 tonne 1000 tonnes 0 1 0.3 Fuzzy if-then rules FeO Gold

27. Indian Institute of Technology Bombay Fuzzy if then rules can be quite complex: IF sky is gray AND wind is strong AND humidity is high AND temperature low, THEN rainfall will be heavy. IF Granite is Proximal AND Fault is Proximal AND FeO is high AND SiO 2 is low, THEN Gold potential is high. Several fuzzy if-then rules are combined to generate a Fuzzy Inference System Fuzzy if-then rules & inference systems

28. Indian Institute of Technology Bombay Fuzzy Inference System Step 1: Identify the factors that control a system: For example: Formation of a deposit (or mineral potential of an area) depends on the following factors: 1.FeO content of the rocks 2.SiO 2 content of the rocks 3.Closeness (or proximity) to granite 4.Closeness (or proximity) to faults Step 2: Identify the variables for each of the factor: 1. FeO content: {high, average, low} 2. SiO 2 content: {high, average, low} 3. Proximity to granite :{proximal, intermediate, distal} 4. Proximity to faults: {proximal, intermediate, distal} Step 3: Identify the output variable of the system Mineral potential: {high, average, low}

29. Indian Institute of Technology Bombay Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable 30%50%70% 0 1 High 30%50%70% 0 1 Average FeO 40%55%70% 0 1 High 40%55%70% 0 1 Average SiO 2 30%50%70% 0 1 Low 40%55%70% 0 1 Low

30. Indian Institute of Technology Bombay Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable (Contd.) Granite 0 km10 km20km 0 1 Proximal 0 1 Intermediate 0 km10 km20km 0 1 Distal 0 km10 km20km Fault 0 km5 km20km 0 1 Proximal 0 1 Intermediate 0 km5 km20km 0 1 Distal 0 km5 km10km

31. Indian Institute of Technology Bombay Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable (Contd.) Mineral potential 0 1 Low 0 1 Average 0 1 High 0t 100t 1000t

32. Indian Institute of Technology Bombay Fuzzy Inference System Step 5: Develop a set of fuzzy if-then rules to explain the behavior of the system Rule1: IF FeO is high AND SiO 2 is low AND Granite is proximal AND Fault is proximal, THEN mineral potential is high. Rule 2: IF FeO is average AND SiO 2 is high AND Granite is intermediate AND Fault is proximal, THEN mineral potential is average. Rule 3: IF FeO is low AND SiO 2 is high AND Granite is distal AND Fault is distal, THEN mineral potential is low.

33. Indian Institute of Technology Bombay Input data preparation Step 6: Rasterize the input predictor maps, combine them, and generate feature vectors Input feature vector [3, 8, 33, 800] MgO% Rock type FeO% Distance to Fault

34. Indian Institute of Technology Bombay Step 7: Represent fuzzy if-then rules in terms of membership functions 1: IF FeO is high & SiO 2 is low & Granite is prox & Fault is prox, THEN metal is highImplication (Max) 0 1 0 1 = 2: IF FeO is aver & SiO 2 is high & Granite is interm & Fault is prox, THEN metal is aver 30% 50% 70% 0 1 40% 55% 70% 0 km 10 km 20km 0 km 5 km 10km 0t 100t 1000t 3: IF FeO is low & SiO 2 is high & Granite is dist & Fault is dist, THEN metal is low FeO = 60%SiO 2 = 60% Granite = 5 km Fault = 1 km Metal = ? 0t 100t 1000t = =

35. Indian Institute of Technology Bombay Step 8: Combine outputs of each rule Rule 1: Rule 2: Rule 3: Aggregate (Max) + + = Defuzzify (Find centroid) 125 tonnes metal Formula for centroid