indian institute of technology bombay mineral prospectivity mapping alok porwal centre of studies in...
TRANSCRIPT
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
MINERAL PROSPECTIVITY MAPPING
Alok Porwal
Centre of Studies in Resources Engineering,
Indian Institute of Technology (Bombay) India
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
• Introduction
GIS-based mineral prospectivity mapping
Generalized methodology
• Weights of evidence model
• Fuzzy model
• Other models (if time)
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
Remote SensingGeophysics
Geochemistry
Geology
Garbage In,
Garbage Out
Mineral potential maps
GIS
Analyse / Combine
Good Data In, Good
Resource Appraisal Out
Mineral potential maps
GIS
Analyse / Combine
Remote SensingGeophysics
Geochemistry
Geology
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
database
Predictor maps
Continuous-scale favorability map
Binary favorability map
MINERAL POTENTIAL MAP
MODEL
Mineralization processes
Conceptual models
Knowledge-base
Mappable exploration criteria
Spatial proxies
Processing
Overlay
Validation
Systematic GIS-based prospectivity mapping
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
database
Predictor maps
Continuous-scale favorability map
Binary favorability map
MINERAL POTENTIAL MAP
MODEL
Mineralization processes
Conceptual models
Knowledge-base
Mappable exploration criteria
Spatial proxies
Processing
Overlay
Validation
Systematic GIS-based prospectivity mapping
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
• 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
Fluid Discharge
Ligand source
Metal source
Model I
Model II
Model III
Trap Region
Energy(Driving Force)
No Deposits
Mineral System(≤ 500 km) Deposit Halo
Deposit(≤ 10 km)
(≤ 5 km)
1. Energy 2. Ligand 3. Source 4. Transport 5. Trap 6. Outflow
Transporting fluid
Mineral systems approach (Wyborn et al. 1995)
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
Fluid Discharge
Metal source
Model I
Model II
Model III
Trap Region
Energy(Driving Force)
Mineral System
(≤ 500 km)
Deposit Halo
Deposit(≤ 10 km)
(≤ 5 km)
1. Energy 2. Ligand 3. Source 4. Transport 5. Trap 6. Outflow
Transporting fluid
Mineral systems approach :Processes to Proxies
Deformation Metamorphism
Magmatism
Connate brinesMagmatic fluidsMeteoric fluids
Enriched source rocks
Magmatic fluids
Structures Permeable
zones
Structures Chemical
traps
Structures aquifers
MA
PPA
BLE
C
RIT
ER
IA
Metamorphic grade, igneous
intrusions, sedimentary
thickness
Evaporites, Organics, isotopes
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
SPA
TIA
L PR
OX
IES
Ligand/brine
source
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Indian Institute of Technology Bombay
Broad Regiona
l
Prospect
Scale dependence of exploration criteria
(e.g. orogenic gold system - McCuaig and Beresford, 2009)
‘Camp’ 2-D GIS-based prospectivity mapping
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
Continuous-scale favorability map
Binary favorability map
MINERAL POTENTIAL MAP
MODEL
Mineralization processes
Conceptual models
Knowledge-base
Mappable exploration criteria
Spatial proxies
Overlay
Validation
database
Predictor maps
Processing
Model-based mineral potential mapping
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay Primary
data
solid geology
structural geology
airborne magnetic
gravity
reactivity (Fe2+/(Fetot+ Mg + Ca) contrast
distance to faults
strike of nearest fault
rheological contrast
Empirical
fav. tectonic environment
fav. host rock lithology
Conceptualfav. litho. contact type
Creation of candidate layers in GIS: Feature extraction What are the exploration criteria for the mineralisation?
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
11Proxies to Predictor Maps: Systematic
Analysis of Exploration Data
GIS
geology(1-of-n coded)
magnetic anomalygamma-ray channels total countdistance to nearest faultKThU
input layer hiddenlayer outputlayer
outputinternal biases = 1
Test/Analyse
Derivative predictor maps
Remote SensingGeophysicsGeochemistry Geology
• Hypothesis testing• Select input layers
• Create candidate layers (Feature extraction based on mineral systems model)
• Input layers
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Indian Institute of Technology Bombay
12
For each exploration criteria, define a hypothesis in terms of GIS layer(s)
- Generate all appropriate layers
- Examine each layer’s spatial relationship to the targeted mineral deposits- Quantify results
- Define an experimental plan
Hypothesis incorrect Hypothesis correct
Factor identified,
keep GIS layer for
prospectivity mapping
Discard layer
Red
efi
ne h
yp
oth
esis
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Indian Institute of Technology Bombay
Example: Rheology as an exploration criteria for gold trap
4
1
5Basalt
Granite
Komatiite
GRANITE-GREENSTONE
BELTS
UltramaficFelsic
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
0
4
1 2 3 4 5
Reacti
vit
y
(Fe x
[Fe /
Fe+
Mg
+C
a])
Rheological strength
(Provided by David Groves)
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
14Experiment 1: Host Rock Rheology
Expected Results
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4 5Rheology Value
Sp
ati
al associa
tion
Hypothesis: More deposits in high rheological strength rocks
4
1
5Basalt
Granite
Komatiite
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Indian Institute of Technology Bombay
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
Sp
ati
al associa
tion
Hypothesis rejected!
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay Hypothesis: More deposits adjacent to
lithological contacts that have a higher rheological contrast
Experiment 2: Rheological Contrast
Expected Results
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4 5Rheology contrast across geological contacts
Sp
ati
al associa
tion
4
1
5Basalt
Granite
Komatiite
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
The Results
0.00
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4
Rheology Contrast at lithological contactsSp
ati
al associa
tion
Edjudina
Kalgoorlie
Laverton
Leonora
Yandal
Experiment 2: Rheological Contrast
Hypothesis rejected!
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Indian Institute of Technology Bombay
18
Change in Philosophy
Frequent change
in rheology
Increased fluid flow
Deposit localization
Region of increased fluid
flow
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Indian Institute of Technology Bombay
19
Hypothesis: Expect deposits to be more common in regions of greater density of rheological contrast.
Experiment 3: Rheological contrast density
Expected Results
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4 5Geological contact density weighted by rheology contrast
Sp
ati
al associa
tion
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
20
0 1 2 3 4
Rheology contrast at lithological contacts Rheology contrast density
LOW HIGH
Experiment 3: Rheological contrast density
Kalgoorlie
Kalgoorlie
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
21
Edjudina
Kalgoorlie
Laverton
Leonora
Yandal
Experiment 3: Rheology Contrast Density
Results
Rheological Contrast DensityLOW HIGHS
pati
al associa
tion
0.00
0.20
0.40
0.60
0.80
1.00
• Deposits more common in regions of greater rheology contrast density
• Select rheology contrast layer for inputting into mineral potential models
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Indian Institute of Technology Bombay
22
Mineralization processes
Conceptual models
database
Predictor maps
Continuous-scale favorability map
Binary favorability map
MINERAL POTENTIAL MAP
Knowledge-base
MODEL
Mappable exploration criteria
Spatial proxies
Processing
Overlay
Validation
Model-based mineral potential mapping
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
23
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
Indian Institute of Technology Bombay
Indian Institute of Technology Bombay
24GIS-based mineral resource potential
mapping - Modelling approaches
Exploration datasets with homogenous coverage – required for all models Expert knowledge (a knowledge base) and/or
Model parameters estimated from mineral deposits data(Known deposits required)Brownfields explorationExamples - Weights of evidence, Bayesian classifiers, NN, Logistic Regression
Data-driven Knowledge-driven
Hybrid
Mineral deposit data
Model parameters estimated from both mineral deposits dataand expert knowledge (Known deposits necessary)Semi-brownfields to brownfields explorationExamples – Neuro-fuzzy systems
Training data Expert knowledge
Model parameters estimated from expert knowledge (Known deposits not necessary)Greenfields explorationExamples – Fuzzy systems; Dempster-Shafer belief theory
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Indian Institute of Technology Bombay
25
Theory
RichSymbolic
Artificial
Intelligence
Fuzzy
Systems
Neural
Networks/
GA
Poor
Poor Rich
Bayesian
Any of the methods
best: Hybrid Systems
Neuro-fuzzy; fuzzy WofE
Which model is best?
Greenfields Brownfields
Data
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26
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
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27
Weights of Evidence Model for Mineral Prospectivity Mapping
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28
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.
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29
PROBABILISTIC MODELS (Weights of Evidence or WofE)
Four steps:1. Convert multiclass maps to binary maps2. Calculation of prior probability3. Calculate weights of evidence (conditional probability) for
each predictor map4. Combine weights
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30
• The probability of the occurrence of the targeted mineral deposit type when no other geological information about the area is available or considered.
Study area (S)
Target deposits D
Assuming- 1. Unit cell size = 1 sq km2. Each deposit occupies 1 unit cell
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 = 0.1110k
10k
1k
1k
Calculation of Prior Probability
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31
Convert multiclass maps into binary maps
• Define a threshold value, use the threshold for reclassification
Multiclass map
Binary map
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• 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
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• Spatial association – spatial correlation of deposit locations with geological feature.
A
BC
D
A
BC
D
10km
10km
1km
1km
Gold Deposit (D) Study area (S)
Convert multiclass maps into binary maps
Indian Institute of Technology Bombay A
B C
D
Which polygon has the highest spatial association with D?More importantly, does any polygon has a positive spatial association with D ???
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?
Positive spatial association – more deposits in a polygon than you would expect if the deposits were randomly distributed.
If observed >> expected; positive associationIf observed = expected; no association If observed << expected; negative association
Convert multiclass maps into binary maps
Indian Institute of Technology Bombay
A
B C
D
Area (A) = n(A) = 25; n(D|A) = 2Area (B) = n(A) = 21; n(D|B) = 2Area(C) = n(C) = 7; n(D|C) = 2 Area(D) = n(D) = 47; n(D|D) = 4Area (S) = n(S) = 100; n(D) = 10
OBSERVED DISTRIBUTION
Convert multiclass maps into binary maps
Indian Institute of Technology Bombay
A
B C
DArea (A) = n(A) = 25; n(D|A) = 2.5Area (B) = n(A) = 21; n(D|B) = 2.1Area(C) = n(C) = 7; n(D|C) = 0.7Area(D) = n(D) = 47; n(D|D) = 4.7(Area (S) = n(S) = 100; n(D) = 10)
EXPECTED DISTRIBUTION
Expected number of deposits in A = (Area (A)/Area(S))*Total number of deposits
Convert multiclass maps into binary maps
Indian Institute of Technology Bombay
A
B C
D
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 DISTRIBUTIONArea (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
OBSERVED DISTRIBUTION
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
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LA
BC
D
10km
10km
1km
1km
Gold Deposit (D) Study area (S)
Convert multiclass maps into binary maps – Line features
Indian Institute of Technology Bombay
1km
1km
Gold Deposit (D)
10
0
000
0
000
11
112
11
1
1111
111
111
1
11
2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 2 3 4 5 6 7 8
2 3 4 5 6 7 2 3 4 5 6 2 3 4 5 6
2 3 4 5 2 3 4 2 3 4 2 3 4
3 2 3 2 4 3 2 4 3 2
5 4 3 2 5 4 3 2
Distance from
the fault
No. of
pixels
No of deposit
s
Ratio (Observe
d to Expected
)
0 9 1 1.1
1 21 3 1.4
2 17 0 0.0
3 16 3 1.9
4 14 2 1.4
5 9 0 0.0
6 6 0 0.0
7 4 0 0.0
8 3 1 3.3
9 1 0 0.0
Convert multiclass maps into binary maps – Line features
Indian Institute of Technology Bombay
• Calculate observed vs expected distribution of deposits for cumulative distances
Gold Deposit (D)
10
0
000
0
000
11
112
11
1
1111
111
111
1
11
2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 2 3 4 5 6 7 8
2 3 4 5 6 7 2 3 4 5 6 2 3 4 5 6
2 3 4 5 2 3 4 2 3 4 2 3 4
3 2 3 2 4 3 2 4 3 2
5 4 3 2 5 4 3 2
Distance from the fault
No. of pixels
Cumul No. of pixels
No of deposits
Cumul No. of deposits
Ratio (Observed to Expected)
0 9 9 1 1 1.1
1 21 30 3 4 1.3
2 17 47 0 4 0.9
3 16 63 3 7 1.1
4 14 77 2 9 1.2
5 9 86 0 9 1.0
6 6 92 0 9 1.0
7 4 96 0 9 0.9
8 3 99 1 10 1.0
9 1 100 0 10 1.0
=< 1 – positive association (Reclassified into 1)> 1– negative association (Reclassified into 0)
Convert multiclass maps into binary maps – Line features
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Weights of evidence ~ quantified spatial associations of the resource with predictor maps
Calculation of Weights of Evidence
Study area (S)
10k
10k
Unit cell
1k1k
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.
Deposit locationsPredictor feature (B1)
Predictor Feature (B2)
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42
P{D|B} = P{D& B}
P{B}= P{D} P{B|D}
P{B}
P{D|B} = P{D & B}
P{B}= P{D} P{B|D}
P{B}
Posterior probability of D given the presence of B
Posterior probability of D given the absence of B
Bayes’ theorem:
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
ObservationInference
Calculation of Weights of Evidence
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43
Using odds (P/(1-P)) formulation:
O{D|B} = O{D} P{B|D}
P{B|D}Odds of D given the presence of B
O{D|B} = O{D} P{B|D}
P{B|D}Odds of D given the absence of B
Taking logs on both sides:
Loge (O{D|B}) = Loge(O{D}) + Log of odds of D given the presence of BP{B|D}
P{B|D}loge
Loge (O{D|B}) = Loge(O{D}) + Log of odds of D given the absence of BP{B|D}
P{B|D}loge
+ive weight of evidence (W+)
-ive weight of evidence (W-)
Calculation of Weights of Evidence
Indian Institute of Technology Bombay
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
44
Calculation of contrast
Indian Institute of Technology Bombay
2121
2
2
1
121
2
2
1
1
21
21
2
2
1
1
2
2
1
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WWDOddsDBBOdds
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DBPDOddsDBBOdds
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DBPDOdds
DBPDBP
DBPDBPDOdds
BPDBP
BPDBP
DP
BPDBP
BPDBP
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BBPDBBP
DP
BBPDBBP
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BBDP
BBDPDBBOdds
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DBP
BP
DBPDPBBDP
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DBP
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DBPDBBP
BBP
DBBPDPBBDP
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DBPDPBDP
WEIGHTS OF EVIDENCE FOR MULTIPLE MAPS
Indian Institute of Technology Bombay
2121
2121
2121
2121
)()/&(log(
)()/&(log(
)()/&(log(
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WEIGHTS OF EVIDENCE FOR MULTIPLE MAPS
Indian Institute of Technology Bombay
47
Loge (O{D|B1, B2})
=
Loge(O{D}) + W+B1 + W+B2
Loge (O{D|B1, B2})
=
Loge(O{D}) + W-B1 + W+B2
Loge (O{D|B1, B2})
=
Loge(O{D}) + W+B1 + W-B2
Loge (O{D|B1, B2})
=
Loge(O{D}) + W-B1 + W-B2
Log of odds of D given the presence of B1 and B2
Log of odds of D given the absence of B1 and presence B2
Log of odds of D given the presence of B1 and absence B2
Log of odds of D given the absence of B1 and B2
Loge (O{D|B1, B2, … Bn})
=
Loge(O{D}) + ∑W+/-Bii=1
n
Or in general, for n predictor features,
The sign of W is +ive or -ive, depending on whether the feature is absent or present
The odds are converted back to posterior probability using the relation 0 = P/(1+P)
Feature B2
Feature B1
Well D
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48
Loge (O{D|B1, B2})
=
Loge(O{D}) + ∑W+/-Bii=1
n
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,
Implementation
P{B|D}
P{B|D}log
e
P{B|D}
P{B|D}log
e
W+ =
W- =
&
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49
Exercise-2Layer – B1 (Fold Axes buffers
1.25k)
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
Layer – B2 (Mafic volcanic rocks)
ARAVALLI STUDY AREA
Indian Institute of Technology Bombay
EXERCISE
Feature B2
Feature B1
Well D
Calculate posterior probability for each cell.
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51
• The probability of the occurrence of the wells when no other information about the area is available or considered.
Study area (S)
Wells D
Assuming- 1. Unit cell size = 1 sq km2. Each deposit occupies 1 unit cell
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
10k
1k
1k
Calculation of Prior Probability
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Basic quantities for estimating weights of evidenceTotal 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)
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
B
Probabilities are estimated as area (or no. of unit cells) proportions
Calculation of Weights of Evidence
P{B|D}
P{B|D}log
e
W+ = P{B|D}
P{B|D}log
e
W- =
= n( )/n(D)
= n( )/
= n( )/n(D)
= n( )/
B & D
B & DP{B|D}
P{B|D} B & D
P{B|D}
P{B|D} B & D
n(D)
n(D)
Where,
B1
D
B2
B2 D B2 D
B2 D
B2 D
B1 D B1 D
B1 D
B1 D
B1
S
D
B2
S
D
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53
Exercise
B2 D B2 D
B2 D
B2 DB1 D B1 D
B1 D
B1 D
10k
10k
B1
B2
S
B1
SD
B2
S
D
Unit cell size = 1 sq km & each well occupies 1 unit cell
n(S) = 100
n(D) = 10
n(B1) = 16
n(B2) = 25
n(B1 & D) = 4
n(B2 & D) = 3
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 & DP{B|D}
P{B|D} B & D
P{B|D}
P{B|D} B & D
n(D)
n(D)
Where,
B & D
B & D
B & D
P{B|D}
P{B|D}log
e
W+ = P{B|D}
P{B|D}log
e
W- =
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Solution
B2 D B2 D
B2 D
B2 DB1 D B1 D
B1 D
B1 D
10k
10k
B1
B2
S
B1
S
DB
2
S
D
W-B1 = loge = loge(0.6000/0.8667)
= loge(0.6923) = -0.3678
= 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
B1&D P{B1|D}P{B1|D}
B1&D
P{B1|D}
P{B1|D}
B1& D
n(D)n(D)
W+B1 = loge = loge(0.4000/0.1333)
= loge(3.0007) = 1.0988
P{B1|D}P{B1|D}
P{B1|D}P{B1|D}
Contrast = W+B1 – W-B1 = 1.0988 – (-0.3678) = 1.4666
W-B2 = loge = loge(0.7000/0.7555)
= loge(0.9265) = -0.0763
= 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
B2& DP{B2|D}
P{B2|D}
B2&D
P{B2|D}
P{B2|D}
B2& D
n(D)
n(D)
W+B2 = loge = loge(0.3000/0.2444)
= loge(1.2275) = 0.2050
P{B2|D}P{B2|D}
P{B2|D}P{B2|D}
Contrast = W+B2 – W-B2 = 0.2050 – (-0.0763) = 0.2812
Assuming Unit cell size = 1 sq km and each well occupies 1 unit cell
n(S) = 100
n(D) = 10
n(B1) = 16
n(B2) = 25
n(B1 & D) = 4
n(B2 & D) = 3
n( ) = - n(B2) - n(D) + n( ) =68
B2& D n(S) B2&D
n( ) = n(B2) – n( ) = 22
B2& D
n( ) = n(D) – ( ) = 7
B2& D
B2& D B2&
D
n(D) = n(S) – n(D) =90
n(B2) = n(S) – n(B2) = 75
Cells occupied by
deposits
n( ) = n(B1) – n( ) =12
B1& D
n( ) = n(D) – n( ) = 6
B1&D
B1& DB1& D
n(D) = n(S) – n(D) =90
n(B1) = n(S) – n(B1) = 84
n( ) = - n(B1) - n(D) + n( ) = 78
B1& D n(S) B1& D
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Loge (O{D|B1, B2})
=
Loge(O{D}) + W+/-B1 + W+/-B2
Loge(O{D}) = Loge(0.11) = -2.2073
Exercise - Calculation of Posterior Probability
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
B1
B2
S
Prior Prb = 0.10
Prior Odds = 0.11
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Loge (O{D|B1, B2})
=
Loge(O{D}) + W+/-B1 + W+/-B2
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 both B1 and B2 are present
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
For the areas where B1 is present but B2 is absent
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
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
For the areas where both B1 and B2 are absent
For the areas where B1 is absent but B2 is present
Loge(O{D}) = Loge(0.11) = -2.2073
Posterior probability
0.2968
0.2342
0.0854
0.0658
Ground Water potential Map
Calculation of Posterior Probability
B1
B2
S
Prior Prb = 0.10