2. probabilistic mineral resource potential mapping the processing of geo-scientific information for...
Post on 25-Dec-2015
214 Views
Preview:
TRANSCRIPT
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
s2 = 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(<A and B> on D) = P(A on D) * P(B on D)
Table 1. Contingency table for 2x2 conditional independence test
Observed frequencies Expected frequenciesA ~A Sum A ~A Sum
B n AB n ~AB n B B n An B /n n ~An B /n n B
~B n A~B n ~A~B n ~B ~B n An ~B /n n ~An ~B /n n ~B
Sum n A n ~A n Sum n A n ~A n
BBAA
BAABABBA
nnnn
nnnnnX
~~
~~~~2 }{
UNCERTAINTY DUE TO MISSING DATA
P(D) = EX{P(D on X)}
= P(D on Ai) * P(Ai)
or P(D on <Ai and Bk>) * P(<Ai and Bk>)
etc.
VARIANCE (MISSING DATA)
2{P(D)} = {P(D on Ai) - P(D)}2 * P(Ai)
or {P(D on <Ai and Bk>) - P(D)}2 * P(<Ai and Bk>)
etc.
TOTAL UNCERTAINTY
Var (Posterior Logit) = Var (Prior Logit) +
+ Var (Weights) + Var (Missing Data)
Uncertainty inLogits 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 km2, weights, contrast (C) with standard deviations (s). In total: 68 deposits on 2945 km2
deps(+) area(+) deps(-) area(-) deps(0) area(0)1 Anticlines 51 1280 17 1665 0 02 HG contact 33 1030 35 1914 0 03 Goldenville Fm. 63 2016 5 928 0 04 Granite contact 11 383 57 2562 0 05 Kriged As 12 219 56 2725 0 06 Lake geochem. 9 166 15 1597 44 11597 NW linears 14 582 54 2362 0 0
Weight: W+ s(W+) W- s(W-) C s(C) C/s(C)1 0.563 0.143 -0.829 0.244 1.392 0.283 4.9262 0.336 0.177 -0.238 0.171 0.575 0.246 2.3393 0.311 0.128 -1.474 0.448 1.784 0.466 3.8264 0.223 0.306 -0.038 0.134 0.261 0.334 0.7835 0.895 0.297 -0.119 0.135 1.014 0.326 3.1096 1.423 0.343 -0.375 0.259 1.798 0.43 4.1837 0.041 0.271 -0.01 0.138 0.051 0.304 0.169
Logistic Regression
Logit (i) = 0 + xi1 1 + xi2 2 + … + xim m
Newton-Raphson Iteration
(t+1) = (t) + {XTV(t)X)}-1XTr(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
NA = Number of unit cells on map layer A
n = Number of deposits
nA = Number of deposits on map layer A
P(d |A) = Probability that unit cell on A contains a deposit
XA = Binary random variable for occurrence of deposit in unit cell
on A with EXA = P(d |A) = nA / n
T = Random variable for number of deposits in study area
Single binary pattern A (~A = not A)
Posterior Probabilities = NA P(d |A) + N~A P(d |~A) =
= NA {nA / NA} + N~A {n~A / N~A } = n
2(T) = NA 2 2(XA) + N~A
2 2(X~A)
Two binary patterns (A and B):
Posterior Probabilities = NAB P(d |AB) + NA~B P(d |A~B) + + N~AB P(d |~AB) + N~A~B P(d |~A~B) =
= nAB + nA~B + n~AB + n~A~B == nA . nB / n + nA . n~B / n + n~A . nB / n + n~A . n~B / n == nA .{nB + n~B }/ n + n~A .{nB + n~B }/ n = n
2(T) = NAB
2 2(XAB) + NA~B 2 2(XA~B ) + N~AB
2 2(X~AB) + N~A~B 2 2(X~A~B )
Table 3. Estimation of T and s 2 (T ) for 3-layer model; I - age, J - topography, K - rock type; N=100 x Area
IJK Area (km2) P f s (P f ) N IJK P f N IJK2 s 2 (P f )
222 1.344 0.0003 0.0005 0.0403 0.0045212 0.9007 0.0057 0.0065 0.5134 0.3428221 0.4351 0.0008 0.0013 0.0348 0.0032122 0.4187 0.0111 0.0126 0.4648 0.2783112 0.3415 0.1709 0.0907 5.8362 9.5939211 0.2223 0.0154 0.0176 0.3423 0.1531111 0.1771 0.3604 0.153 6.3827 7.3421121 0.1456 0.0297 0.336 0.4324 23.9332
Sum 3.985 14.0470 41.6511
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Posterior probability
Nu
mb
er
of
hy
dro
the
rma
l ve
nts
New conditional independence test applied to ocean floor hydrothermal vent example
Total number of vents n = 13
3-map layer model predicts 14.05 (s.d. = 6.45)
P(T = N) > 99% (c.l. = 28.03)
5-map layer model predicts 37.59 (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
.
.
.Integration Potential
Evidential Layers (X)
Modeling (F) Output Data
S
ProcessingDBMS
GIS Database
GIS Data PreprocessingInterpreting
Define correlated patterns using training points
Integrated correlated patterns to estimate unknown points
Modeling Prediction
Flowing Wells vs. Distance from ORM
-8
-4
0
4
8
12
0 5000 10000 15000
Spatial Correlation
Distance
Flowing Wells vs. Distance From High Slope Zone
-8
-4
0
4
8
0 2000 4000 6000 8000
Spatial Correlation
Distance
Flowing Wells vs. Thickness of Drift
Flowing vs. Distance from Thick Drift
-4
0
4
8
0 5000 10000 15000 20000
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
T=7
P ( X=0 )=1.0A
P ( X=0 )=0.3B
P ( X=0)=0.0C
sA
mA m B mCm
sB
s C
A
1050
BmB= 7.98
=1
CmC =10 .51
=1
mA = 3.45= 0
class area points W+ s(W+) W- s(W-) C s(C) Stud(C)3 2945 684 2940 685 2927 67 -0.0089 0.1236 0.9083 1.0289 -0.9172 1.0363 -0.88516 2891 67 0.0038 0.1236 -0.2243 1.0094 0.228 1.0169 0.22427 2817 67 0.0303 0.1236 -1.098 1.0039 1.1283 1.0115 1.11548 2702 63 0.0097 0.1275 -0.1148 0.4519 0.1245 0.4695 0.26519 2529 60 0.0275 0.1307 -0.1851 0.357 0.2125 0.3802 0.5591
10 2259 53 0.0163 0.139 -0.0557 0.2611 0.072 0.2958 0.243511 1873 46 0.0629 0.1493 -0.12 0.2154 0.1829 0.2621 0.697912 1466 42 0.2213 0.1566 -0.2782 0.1979 0.4995 0.2523 1.979813 1100 32 0.2373 0.1794 -0.1721 0.1683 0.4093 0.246 1.663914 765 23 0.27 0.2117 -0.1142 0.1506 0.3841 0.2598 1.478315 510 20 0.5461 0.2281 -0.1615 0.1458 0.7076 0.2707 2.613816 330 13 0.5488 0.283 -0.0952 0.1363 0.644 0.3141 2.050417 219 12 0.8947 0.2969 -0.1193 0.135 1.014 0.3262 3.108718 150 9 0.9901 0.3438 -0.0915 0.1316 1.0816 0.3681 2.938419 106 5 0.7357 0.4581 -0.0405 0.1274 0.7762 0.4755 1.632520 75 4 0.8593 0.5138 -0.0354 0.1264 0.8947 0.5291 1.69121 57 2 0.4238 0.7198 -0.0104 0.1245 0.4342 0.7305 0.594522 42 2 0.7367 0.7244 -0.0157 0.1245 0.7524 0.735 1.023623 33 2 0.9988 0.7294 -0.019 0.1245 1.0177 0.74 1.375324 26 2 1.231 0.7352 -0.0212 0.1245 1.2523 0.7456 1.679525 20 1 0.7579 1.0249 -0.0079 0.1236 0.7658 1.0323 0.741826 15 1 1.0696 1.0339 -0.0098 0.1236 1.0793 1.0412 1.036627 11 1 1.3758 1.0457 -0.0111 0.1236 1.3869 1.053 1.317128 8 1 1.7329 1.0648 -0.0122 0.1236 1.7451 1.0719 1.62829 5 1 2.2455 1.106 -0.0133 0.1236 2.2588 1.1129 2.029730 3 1 3.0032 1.215 -0.0141 0.1236 3.0173 1.2213 2.470631 1 032 1 0
Linear Regression with Missing Data
Y = 0 + 1 x +
b1 = (xi-mx)(yi-my)/(xi-mx)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.
Coeff. (A) ST.D. (A) Coeff. (B) ST.D. (B) Coeff. (C) ST.D. (C) Coeff. (D) ST.D. (D)0 -9.7317 1.2118 -10.3256 1.2459 -10.5161 1.7714 -10.5879 1.24641 1.1901 0.2995 1.1647 0.2949 1.0535 0.4755 1.1544 0.29512 0.2536 0.2632 0.2182 0.2645 0.6654 0.4863 0.2004 0.2653 1.2055 0.4997 1.2143 0.5002 0.289 0.7249 1.2203 0.50054 0.4899 0.3405 0.4857 0.341 0.7744 0.526 0.4821 0.34165 0.8328 0.3311 0.754 0.3362 0.9305 0.5142 0.7216 0.33756 0.7149 0.3797 1.4876 0.4544 0.9839 0.36897 -0.0065 0.3073 -0.018 0.3078 -0.5948 0.631 -0.0213 0.3081
Logistic Regression& Maximum Likelihood
P(Y=1|x) = (x) = ef(x)/{1+ ef(x)}
P(Y=0|x) = 1-(x)
(xi) = (xi) yi {1-(xi)} 1- yi
l() = (xi)
Bivariate Logistic Regression
Logit (i) = 0 + xi1 1
= [0 1]
Log Likelihood Function
L() = ln{l()} =
= [yiln{(xi)}+(1- yi)ln{1-(xi)}]
Differentiate with respect to 0 and 1 to
obtain likelihood equations:
{yi - (xi)} = 0
xi{yi - (xi)} = 0
Total number of discrete events = Sum of estimated probabilities
yi = p(xi)
Weighted logistic regression convergence experiments (Level of convergence = 0.01)
Seafloor Example (N = 13):
Unit cell of 0.01 km2 12.72;
0.001 km2 12.97; 0.0001 km2 13.00
Meguma Terrane Example (N = 68)
Unit cell of 1 km2 64.71; 0.1 km2 67.96
top related