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Structure of a model for mineral resource potential mapping

Structure of a model for mineral resource potential mapping. ∫. 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. 1.

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Structure of a model for mineral resource potential mapping

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  1. Structure of a model for mineral resource potential mapping ∫ 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 1

  2. 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 Expert knowledge Training data Knowledge-driven Hybrid Data-driven 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 mineral deposits data (Known deposits required) Brownfields exploration Examples - Weights of evidence, Bayesian classifiers, NN, Logistic Regression Model parameters estimated from expert knowledge (Known deposits not necessary) Greenfields exploration Examples – Fuzzy systems; Dempster-Shafer belief theory

  3. 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.

  4. Fuzzy Inference Systems for Mineral Prospectivity Mapping

  5. Artificial intelligence!

  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 overcastand 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. 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. Crisp (Traditional) Variables

  8. 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 Fuzzy Sets

  9. 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 ) Example: Height

  10. Example: Weekend days

  11. FUZZY BINARY

  12. Example: Season Summer Rainy Autumn Winter Summer Rainy Autumn Winter FUZZY BINARY

  13. 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. Example: Height

  15. Membership Function 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. µA(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. µA(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. Membership functions • piece-wise linear functions • the Gaussian distribution function • the sigmoid curve • quadratic and cubic polynomial curves

  17. For each variable value a different membership function is required Temp: {Freezing, Cool, Warm, Hot} Membership Functions 5 20 30 40 Temp OC

  18. 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 Fuzzy Operators

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

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

  21. Calculate AB given that A is .4 and B is 20 Example: Fuzzy Conjunction

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

  23. Subjective assignment of fuzzy membership values • It is also possible to assign the fuzzy membership values subjectively (without using a membership function)

  24. 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: ifx 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. Fuzzy if-then rules if x is A then y is B For example: IfFeOishighthen goldpotentialisaverage “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.

  26. Fuzzy if-then rules 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. Theimplication 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. 1 1 0.3 0 0 40% 50% 60% 70% 1 tonne 1000 tonnes Gold FeO

  27. Fuzzy if-then rules & inference systems Fuzzy if then rules can be quite complex: IFsky is gray AND wind is strong ANDhumidity is high ANDtemperature low, THEN rainfall will be heavy. IF Granite is Proximal AND Fault is Proximal ANDFeOis high ANDSiO2 is low, THEN Gold potential is high. Several fuzzy if-then rules are combined to generate a Fuzzy Inference System

  28. 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: • FeO content of the rocks • SiO2 content of the rocks • Closeness (or proximity) to granite • Closeness (or proximity) to faults • Step 2: Identify the variables for each of the factor: • 1. FeO content: {high, average, low} • 2. SiO2content: {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. Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable High High Low Average Average Low 1 1 1 1 1 1 FeO 30% 50% 70% 0 0 0 0 0 0 40% 55% 70% 30% 50% 70% 30% 50% 70% 40% 55% 70% SiO2 40% 55% 70%

  30. Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable (Contd.) Proximal Proximal Intermediate Distal Intermediate Distal 1 1 1 1 1 1 Granite 0 km 5 km 10km 0 km 10 km 20km 0 km 5 km 20km 0 km 10 km 20km 0 km 5 km 20km 0 km 10 km 20km 0 0 0 0 0 0 Fault

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

  32. Fuzzy Inference System Step 5: Develop a set of fuzzy if-then rules to explain the behavior of the system Rule1: IFFeOis high ANDSiO2 is low ANDGranite is proximal AND Fault is proximal, THEN mineral potential is high. Rule 2: IFFeOis average ANDSiO2 is high ANDGranite is intermediate AND Fault is proximal, THEN mineral potential is average. Rule 3: IFFeOis low ANDSiO2 is high ANDGranite is distal AND Fault is distal, THEN mineral potential is low.

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

  34. Step 7: Represent fuzzy if-then rules in terms of membership functions 1: IFFeOis high & SiO2 is low & Granite isprox& Fault isprox, THEN metal is high Implication (Max) 1 = 1 1 0 2: IFFeOis aver & SiO2 is high & Granite isinterm& Fault isprox, THEN metal is aver = 30% 50% 70% 0 km 10 km 20km 0 0 40% 55% 70% 3: IFFeOis low & SiO2 is high & Granite is dist & Fault is dist, THEN metal is low = 0t 100t 1000t 0 km 5 km 10km 0t 100t 1000t FeO = 60% SiO2 = 60% Metal = ? Granite = 5 km Fault = 1 km

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

  36. Weights of Evidence Model for Mineral Prospectivity Mapping

  37. 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.

  38. 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

  39. The probability of the occurrence of the targeted mineral deposit type when no other geological information about the area is available or considered. Calculation of Prior Probability 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

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

  41. 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 !

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

  43. Convert multiclass maps into binary maps Which polygon has the highest spatial association with D? More importantly, does any polygon has a positive spatial association with D ??? A Positive spatial association – more deposits in a polygon than you would expect if the deposits were randomly distributed. D C B What is the expecteddistribution 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

  44. Convert multiclass maps into binary maps 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 A D C B

  45. Convert multiclass maps into binary maps EXPECTED DISTRIBUTION A 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) D C B Expected number of deposits in A = (Area (A)/Area(S))*Total number of deposits

  46. Convert multiclass maps into binary maps EXPECTED DISTRIBUTION 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 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) A Only C has positive association! So, A, B and D are classified as 0; C is classified as 1. D • 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 C B

  47. Convert multiclass maps into binary maps – Line features A L 10km D C B 1km 10km 1km Study area (S) Gold Deposit (D)

  48. Convert multiclass maps into binary maps – Line features 1 1 2 3 4 5 6 7 8 9 1 0 2 3 4 5 6 7 8 1 1 1 0 2 3 4 5 6 7 8 2 1 2 3 4 5 6 7 0 1 0 3 2 1 2 3 4 5 6 1 3 2 1 1 0 2 3 4 5 6 4 3 2 1 1 0 2 3 4 5 4 3 2 1 1 1 2 3 4 0 5 4 3 2 0 1 1 2 3 4 1km 1 0 5 4 3 2 1 2 3 4 1km Gold Deposit (D)

  49. Convert multiclass maps into binary maps – Line features • Calculate observed vs expected distribution of deposits for cumulative distances 1 1 2 3 4 5 6 7 8 9 1 0 2 3 4 5 6 7 8 1 1 1 0 2 3 4 5 6 7 8 2 1 2 3 4 5 6 7 0 1 0 3 2 1 2 3 4 5 6 1 3 2 1 1 0 2 3 4 5 6 4 3 2 1 1 0 2 3 4 5 4 3 2 1 1 1 2 3 4 0 5 4 3 2 0 1 1 2 3 4 1 0 5 4 3 2 1 2 3 4 Gold Deposit (D) =< 4 – positive association (Reclassified into 1) >4 – negative association (Reclassified into 0)

  50. 1k 1k Unit cell Target deposits Geological Feature (B1) Geological Feature (B2) Calculation of Weights of Evidence Weights of evidence ~ quantified spatial associations of deposits with geological features Study area (S) 10k 10k 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 mineral deposit to posterior probability based on the conditional probabilities (or weights of evidence) of the geological features.

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