a hybrid fuzzy weights-of-evidence model for mineral1

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Natural Resources Research, Vol. 15, No. 1, March 2006 ( C 2006)DOI: 10.1007/s11053-006-9012-7

A Hybrid Fuzzy Weights-of-Evidence Model for Mineral

Potential Mapping

Alok Porwal, 1,2 Emmanuel John M. Carranza, 1,4 and Martin Hale 1,3

Received 30 June 2005; accepted 9 August 2005Published online: 4 November 2006

This paper describes a hybrid fuzzy weights-of-evidence (WofE) model for mineral potentialmapping that generates fuzzy predictor patterns based on (a) knowledge-based fuzzy mem-bership values and (b) data-based conditional probabilities. The fuzzy membership values arecalculated using a knowledge-driven logistic membership function, which provides a frame-work for treating systemic uncertainty and also facilitates the use of multiclass predictor mapsin the modeling procedure. The fuzzy predictor patterns are combined using Bayes’ rule ina log-linear form (under an assumption of conditional independence) to update the priorprobability of target deposit-type occurrence in every unique combination of predictor pat-terns. The hybrid fuzzy WofE model is applied to a regional-scale mapping of base-metaldeposit potential in the south-central part of the Aravalli metallogenic province (westernIndia). The output map of fuzzy posterior probabilities of base-metal deposit occurrence isclassied subsequently to delineate zones with high-favorability, moderate favorability, andlow-favorability for occurrence of base-metal deposits. An analysis of the favorability mapindicates (a) signicant improvement of probability of base-metal deposit occurrence in thehigh-favorability and moderate-favorability zones and (b) signicant deterioration of proba-bility of base-metal deposit occurrence in the low-favorability zones. The results demonstrateusefulness of the hybrid fuzzy WofE model in representation and in integration of evidential

features to map relative potential for mineral deposit occurrence.KEY WORDS: Fuzzymembership, conditionalprobability, base-metal deposits, GIS, Aravalli province.

INTRODUCTION

In the data-driven weights-of-evidence (WofE)approach to mineral potential mapping, the spa-tial association of a predictor pattern B with tar-get mineral deposits D is quantied in terms of apair of weights for that predictor pattern. The pairof weights is calculated from the conditional prob-

1International Institute for Geo-information Science and EarthObservation (ITC), Enschede, The Netherlands.

2Department of Mines and Geology, Govt. of Rajasthan,Udaipur, India.

3Faculty of Geoscience, Utrecht University, Utrecht, TheNetherlands.

4To whom correspondence should be addressed at ITC,Hengelosestraat 99, 7500 AA Enschede, The Netherlands;e-mail: [email protected].

abilities of B , given the presence or absence of D(Agterberg, Bonham-Carter, and Wright, 1990;Bonham-Carter and Agterberg, 1990):

W + = logeP [ B | D ]P [ B | ¯ D ]

andW − = loge

P [ ¯ B | D ]P [ ¯ B | ¯ D ] .

From the preceding equations, it follows that the pairof weights of a predictor pattern is determinable,if, and only if, it contains at least one training de-posit (i.e., log e of zero is indeterminable). More-over, robustness of conditional probability of a pre-dictor pattern estimated using the WofE methoddecreases with a decrease in number of training de-posits contained by the predictor pattern (Agterberg,

11520-7439/06/0300-0001/0 C 2006 International Association for Mathematical Geology


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A Hybrid Fuzzy Weights-of-Evidence Model for Mineral Potential Mapping 3

10, whereas the least favorable pattern is ranked 1. Inthis scheme, the rank of a pattern is its class weight .As much as possible, the patterns are ranked at equalintervals. This method of allotting weights by rank-ing is simple, as most experts are likely to agree uponthe favorability rank of a predictor pattern but arelikely to disagree upon its favorability weight. Simi-larly, based on their importance with respect to per-tinent recognition criteria, all predictor maps are as-signed map weights according to the same procedurefor assigning class weights as described above. How-ever, in the situation of assigning map weights, pre-dictor maps are ranked in the scale of 5 to 10, insteadof 1 to 10. This is because we consider a rank below5 to represent low or insignicant importance and amap would not have been considered a predictor if it is not important in indicating potential for mineraloccurrence.

Using the fuzzied predictor maps, we derive afuzzy WofE model as follows. Fuzzy log e posteriorodds of D , given X i(i = 1 to n ) predictor maps withmulti-class x ij patterns, each predictor being repre-sented by ˜ A i (i = 1 to n ) fuzzy sets, can be estimatedas:

loge O[D | X i] = loge O[D ] +n

i= 1

r

j = 1

W µ Aij , (4)

where W µ Aij is the fuzzy WofE of xij pattern in X i predictor map and is estimated as (Cheng andAgterberg, 1999 ):

W µ A i( xij ) =

logeµ A i ( xij )P [ A i1|D ] + { 1 − µ A i ( xij )}P [ A i2|D ]µ A i ( xij )P [ A i1| ¯ D ] + { 1 − µ A i ( xij )}P [ A i2| ¯ D ]

,

(5)

where P [ A i1|D ] and P [ A i2|D ] are, respectively, theconditional probabilities of two crisp sets A i1 and

A i2, given the presence of D , whereas P [ A i1| ¯ D ] andP [ A i2| ¯ D ] are, respectively, the conditional probabil-ities of the two crisp sets A i1 and A i2, given the ab-sence of D . The two crisp sets A i1 and A i2 (i = 1 ton) in the fuzzy set ˜ A i (i = 1 to n ) are dened here asfollows:

A i1 = { xij |µ A ij = MAX( µ A ij )} and

A i2 = { xij |µ A ij = MIN( µ A ij )}. (6)

Fuzzy posterior probability can be calculated fromEquation (4):

P [D | X i] =eloge O[D | X i]

1 + eloge O [D | X i]. (7)

The following equations for estimating variance of the fuzzy posterior probabilities because of missingpatterns, miss-assigned patterns, and fuzzy member-ship values are adapted from Cheng and Agterberg(1999). The variance of the fuzzy posterior probabil-ities because of X k missing patterns is estimated as:

σ 2k (P [D | X i]) = { P [D | X k] − P [D ]}2P [ X k]

+ { P [D | X k] − P [D ]}2P [ X k]. (8)

In X l predictor map, the variance of the fuzzy poste-rior probabilities due to miss-assigned patterns from

X l to X l or vice versa is estimated as:

σ 2 X l (P [D | X i]) = { P [D | X k] − P [D | X k]}2P [D ],

andσ 2 X l (P [D | X i]) = { P [D | X k] − P [D | X k]}

2P [ ¯ D ].(9)

The variance of the fuzzy posterior probabilities dueto the fuzzy membership values, µ ˜ A i , is estimatedas:

σ 2µ A i (P [D | X i]) =2µ A i (1 − µ A i )

p [µ A i ] p [ X i]P [ X i]

× σ 2 X l (P [D | X i]) + σ 2

X l (P [D | X i]) .

(10)

APPLICATION TO BASE-METALDEPOSIT POTENTIAL MAPPING IN

ARAVALLI PROVINCE, WESTERN INDIAThe study area forms a part of the Aravalli met-

allogenic province in the state of Rajasthan, westernIndia (Fig. 1). It measures about 34,000 km 2 and islocated between latitudes 23 ◦ 30 N and 26 ◦ N and be-tween longitudes 73 ◦ E and 75 ◦ E.

The geology of the province is character-ized by two major fold belts, viz., the Palaeo-Mesoproterozoic Aravalli Fold Belt and the Meso-Neoproterozoic Delhi Fold Belt, which are ingrainedin a reworked basement complex named the BandedGneissic Complex that contains incontrovertibleArchaean components. This three-fold tectonostrati-graphic classication of the province, which was pro-posed rst by Heron ( 1953), remains the basic frame-work of reference for all subsequent studies (RajaRao and others, 1971; Roy, 1988; Roy and others,1993; Gupta and others, 1997). In an attempt to ex-plain the evolution of the province in the frameworkof Proterozoic Wilson cycles and plate tectonics,Deb and Sarkar ( 1990) and Sugden, Deb, and


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4 Porwal, Carranza, and Hale

Figure 1. Location map of study area in state of Rajasthan, India. Small black circles are locations of base-metal deposits.

Windley ( 1990) divided the province into a num-ber of tectonic domains based on lithogeochemicaland structural considerations. Our work on geophys-ical data also indicates that the province is com-prised of subparallel linearly disposed belts (Fig.2), each having distinct geophysical and tectoniccharacteristics, coinciding broadly with the tectonicdomains proposed by Sugden, Deb, and Windley(1990).

The Aravalli province holds substantial reservesof base-metal sulde deposits, particularly, lead andzinc. The economically viable lead-zinc reserves inthe province stand at 130 million tonnes with 2.2%

Pb and 9.2% Zn with an additional 30 million tonnesof possible resources in producing mines and de-posits under detailed exploration (Haldar, 2001).These form the entire economically viable lead-zincresource-base of the country.

A vast majority of the lead-zinc sulde depositsof the province are contained in the study area, whichis considered one of the prime exploration targetareas for base-metals in the country. The deposits

in the area are hosted by the supracrustal rocksof the Bhilwara, Aravalli, and South Delhi belts(Fig. 2). Major concentrations of mineralization inthe Bhilwara belt occur in the Rampura-Agucha de-posit and in the Pur-Banera and Bethumni-Dariba-Bhinder mineralized zones. Rampura-Agucha is aworld-class Zn-Pb-(Ag) deposit with the highestcombined metal grade (about 15%) of all base-metaldeposits in India. In the Bethumni-Dariba-Bhinderzone, Zn–Pb–(Cu) deposits are located in a 17 km-long belt extending from Bethumni in the north toDariba in the south, with a lean pyrite zone far-ther south nearby Bhinder. The Pur-Banera is a

lean polymetallic zone with several small deposits.In the Aravalli belt, low-grade Cu, Au, and U min-eralizations occur in basal sequences, whereas Zn–Pb deposits occur in upper sequences of the Za-war zone. The South Delhi belt hosts small depositsof Cu–(Zn) in the Basantgarh area. The details of base-metal metallogenesis and nature of mineraliza-tion of occur in Porwal, Carranza, and Hale ( 2003a,2003b).


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Figure 2. Generalized geological map of south-central Aravalli province showing study area for fuzzy WofE modeling(demarcated in heavy black) and important mineralized zones (outlined in white). White circles are locations of base-metal deposits.

Recognition Criteria and Predictor Maps

Porwal, Carranza, and Hale ( 2003a, 2003b) iden-tied the following regional-scale recognition crite-ria for base-metal deposit occurrence in the province:(1) host rock lithology, (2) stratigraphic position, (3)(paleo-)sedimentary environment, (4) association of synsedimentary mac volcanic rocks, (5) proximity

to regional tectonic discontinuities, and (6) proximityto favorable structures. Each recognition criterion isrepresented by at least one predictor map preparedfrom available spatial data sets.

A regional-scale GIS was compiled by digitiz-ing the lithostratigraphic map (Gupta and others,1995a), the structural map (Gupta and others, 1995b)and the map of total magnetic eld intensity (GSI,


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1981). The locations of 54 known base-metal de-posits were compiled from various sources. The dig-itized maps in a vector format were converted intogrid format for subsequent operations. Based on theresolution of the original maps used for creatingthe GIS database, a grid cell size of 250 m × 250 mwas considered appropriate for representing thesmallest feature in the original analogue maps. Thegrid maps were processed, interpreted, and reclas-sied, as described by Porwal, Carranza, and Hale(2003b), to create predictor maps. Of the eightpredictor maps we prepared, seven are multiclass(lithologies, stratigraphic groups, sedimentary en-vironments, buffered regional lineaments, bufferedNE-trending lineaments, buffered NW-trending lin-eaments, and buffered fold axes) and one is binary(mac igneous rocks).

Pairwise Test of Conditional Independence . Allpairs of predictor maps were tested for conditionalindependence (CI) with respect to the locationsof base-metal deposits using the χ 2 test (Bonham-Carter and Agterberg, 1990). The test, although notaccurate in determining CI, because it is gives low es-timates of correlations (see Singer and Kouda, 1999,for a detailed discussion), can be used for identify-ing pairs of predictor maps that lack CI with respectto target variable. The results indicate that predic-tor map of sedimentary environments and predictormap of lithologies signicantly lack CI with respectto the locations of base-metal deposits (Table 1). The

predictor map of sedimentary environments was not

considered further in the hybrid fuzzy WofE model-ing because the sedimentary environments were in-ferred mainly from available geoscience information,whereas the lithologies were mapped in the eld. Af-ter combining the predictor maps, we also performedan overall test of CI (see next).

Estimation of Fuzzy WofE of Predictor Patterns

For each pattern in a predictor map, estima-tion of fuzzy WofE via Equation (5) requires esti-mates of (a) fuzzy membership values and (b) condi-tional probabilities (given the presence or absence of a base-metal deposit) of map patterns with the high-est and the lowest fuzzy membership values [Eq. (6)].

Fuzzy membership values were estimatedusing the logistic membership function dened inEquation (2). The values used for the variables band a in Equation (2) were 50 and 0.1, respectively,which yield a curve that is symmetrical about theinexion point at (50, 0.5), where 50 is the class scoreand 0.5 is the corresponding knowledge-driven fuzzymembership value. The class scores were calculatedfrom class weights and map weights [Equation (3)],which were assigned subjectively using the rank-ing procedure described earlier. The assignmentof subjective ranks to map patterns was basedon our experience of base-metal exploration in theprovince, informal discussions with mining geologists

of M/s Hindustan Zinc Ltd. (a Government of India

Table 1. Pair-Wise χ 2 Test for Conditional Independence among Input Predictor Maps

Predictor maps LithologiesSedimentaryenvironments

Mac Igneousrocks

Bufferedregional

lineamentsBuffered NW

lineamentsBuffered NElineaments

Bufferedfold axes

Stratigraphy 49.02 (56) 57.66 (48) 6.36 (8) 0.74 (8) 0.96 (8) 0.89 (8) 1.91 (8)74.468 65.171 15.507 15.507 15.507 15.507 15.507

Lithologies 68.42∗ (42) 3.01 (7) 0.85 (7) 6.44 (7) 7.28 (7) 1.19 (7) 58.124 14.067 14.067 14.067 14.067 14.067

Sedimentary environments 4.96 (6) 0.11 (6) 2.53 (6) 7.09 (6) 1.49 (6)12.529 12.529 12.529 12.529 12.529

Mac igneous rocks 0.46 (1) 0.01 (1) 0.07 (1) 0.03 (1) 3.84 3.84 3.84 3.84

Buffered regional lineaments 0.002 (1) 0.03 (1) 0.38 (1) 3.84 3.84 3.84

Buffered NW lineaments 1.53 (1) 3.83 (1) 3.84 3.84

Buffered NE lineaments 0.32 (1) 3.84

Note. Values in bold and italics are calculated and tabulated χ 2 values (at 0.05% signicance level), respectively. Figures in parenthesesare degrees of freedom ( = (v − 1)(u − 1), where v and u are number of classes in two predictor maps).∗Null hypothesis of conditional independence is rejected at 95% condence level.


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enterprise, which owns most of the large basemetaldeposits in the province), and recommendations of various authors, especially Deb ( 1999) and Sarkar(2000). The detailed rationale for ranking of thepredictor maps and individual map patterns are inPorwal, Carranza, and Hale ( 2003b).

Conditional probabilities of individual patternsin a predictor map, given the presence or absence of a base-metal deposit, were estimated by using a ran-domly selected set of 23 training points (or ‘known’deposits) out of the 54 base-metal deposits. The es-timations were performed using the gridded predic-tor maps and taking 1 km 2 as the unit cell size.Thus, the estimated prior probability is (23/34000 ≈ )0.0007. The remaining 31 base-metal deposits werelater used as validation points (or ‘unknown’ de-posits). The class weights, map weights, class scores ,knowledge-driven fuzzy membership values and con-ditional probabilities of the individual patterns ineach of the predictor maps are given in Table 2.Based on the estimated fuzzy membership values andconditional probabilities, Equations (5) and (6) wereused to estimate fuzzy WofE of every pattern in eachpredictor map (Table 2).

Combining Predictor Maps

The predictor maps were combined by gen-erating a unique conditions map (Bonham-Carter

and Agterberg, 1990; Kemp, Bonham-Carter, andRaines, 1999). A unique condition is formed from theunique combination of at least two patterns in dif-ferent predictor maps (Kemp, Bonham-Carter, andRaines, 1999). The attribute table (unique conditionstable) associated with a unique conditions grid hasone record (or row) per unique condition and oneor more elds (or columns) containing attribute data(variables).

The multiclass predictor maps were combinedusing digital overlay, which resulted in a map with20258 unique conditions. The fuzzy posterior prob-ability for each unique condition was derived fromestimates of the fuzzy WofE of predictor patternsand the prior probability of base-metal deposits[Eqs. (4) and (7)]. Figure 3 shows the spatial dis-tribution of fuzzy posterior probabilities of base-metal occurrence in the study area. The variance of the fuzzy posterior probabilities due to missing pat-terns, to miss-assigned patterns, and to fuzzy mem-bership values were calculated using Equations (8),(9), and (10), respectively. The spatial distribution

of total variance of the fuzzy posterior probabilitiesare because of missing patterns and miss-assignedpatterns is shown in Figure 4A. The spatial distri-bution of variance of the fuzzy posterior probabili-ties because of fuzzy membership values is shown inFigure 4B.

Overall Test of Conditional Independence . Weapplied the new omnibus test (NOT) (Agterberg andCheng, 2002; Thiart, Bonham-Carter, and Agterberg,2003) to determine violation or nonviolation of CI as-sumption in the nal fuzzy posterior probability map.The NOT statistic is ratio of the difference betweenthe predicted number of (training) deposits and theobserved number of (training) deposits divided bythe standard deviation of predicted number of (train-ing) deposits. Values of NOT are assumed to ap-proximate a standard normal distribution such thatprobability that the difference between the predictednumber of (training) deposits and the observed num-ber of (training) deposits is statistically greater thanzero can be estimated (from statistical tables) in or-der to judge whether to accept or reject the null hy-pothesis of CI.

We used total estimated variance of the fuzzyposterior probabilities [i.e., sum of estimated vari-ance using Equations (8), (9), and (10)] to estimatestandard deviation of the predicted number of (train-ing) deposits. The NOT test showed that the nalfuzzy posterior probability map signicantly violatesCI assumption. However, as suggested by Pan and

Harris ( 2000), violation of CI assumption can be ig-nored if, and only if, posterior probabilities are in-terpreted in a relative sense only and not in ab-solute sense (see further in the discussion section).Therefore, we interpreted estimates of fuzzy poste-rior probabilities in a relative sense only in order toclassify zones in the study area with high-favorability,moderate-favorability, and low-favorability for base-metal deposit occurrence.

Reclassication of Fuzzy Posterior Probability Map

It is cumbersome to interpret the fuzzy pos-terior probability map (Fig. 3) for selecting targetareas for base-metal exploration, because it showsthe fuzzy posterior probability of base-metal de-posits in a continuous gray-scale from the least fa-vorable (fuzzy posterior probability ∼0) to the mostfavorable (fuzzy posterior probability ∼0.5). More-over, a fuzzy posterior probability cannot be in-terpreted in the absolute sense of probability per


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Table 2. Map Weights, Class Weights, Fuzzy Membership Values, Conditional Probabilities, and Fuzzy WofE of Patternsin Predictor Maps

Predictor Map/PatternMap

weightClass

weightClassscore

Fuzzymembership P [ X i |D ] P [ X i | ¯ D ]

FuzzyWofE

Predictor map of lithotogiesDolomite/dolomitic-marble 10 10 100 0.99 0.1739 0.022 1.8265Calc-silicates 10 9 90 0.98 0.3043 0.0118 1.6288Graphitic meta-pelites 10 8 80 0.95 0.3043 0.0096 1.1837Magnetite quartzite 10 7 70 0.88 0.2174 0.0002 0.5244Meta-basites 10 6 60 0.73 0.0000 0.0213 − 0.3280Calc-schist/gneiss 10 5 50 0.5 0.0000 0.0467 − 1.2640Qzite-Arkose-Conglomerate 10 4 40 0.27 0.0000 0.0839 − 2.2360Migmatites; gneisses 10 2 20 0.05 0.0000 0.2092 − 4.1740Unrelated to base-metals 10 1 10 0.02 0.0000 0.5935 − 5.1200

Predictor map of stratigraphiesRajpura-Dariba group 9 10 90 0.98 0.2609 0.0044 3.0684Pur-Banera group 9 9 81 0.96 0.5217 0.0117 2.5593Debari group 9 8 72 0.09 0.0870 0.0545 1.7249Nathdwara group 9 7 63 0.79 0.0000 0.0036 0.9093Phulad Ophiolites group 9 6 54 0.60 0.0000 0.0083 0.0153Udaipur group 9 5 45 0.38 0.0870 0.0961 − 0.8700Jharol group 9 4 36 0.20 0.0000 0.1528 − 1.7620Sandmata Complex 9 3 27 0.09 0.0435 0.1005 − 2.6880Mangalwar Complex 9 2 18 0.04 0.0000 0.1925 − 3.5510Unrelated to base-metals 9 1 9 0.02 0.0000 0.3738 − 4.2650

Predictor map of mac Igneous rocksBasic metavolcanic rocks 8 10 80 0.95 0.7391 0.1265 1.4737Unrelated to base-metals 8 1 8 0.01 0.2609 0.8731 − 1.1810

Predictor map buffered regional magnetic lineaments0–2 Km 8 10 80 0.9526 0.2609 0.1897 0.24552–4 Km 8 8 64 0.8022 0.4348 0.1633 0.00804–6 Km 8 6 48 0.4502 0.2174 0.1336 − 0.62206–8 Km 8 4 32 0.1419 0.0435 0.1066 − 1.41008–10 Km 8 2 16 0.0323 0.0000 0.083 − 1.8430> 10 Km 8 1 8 0.0148 0.0435 0.3235 − 1.9290

Predictor map of buffered fold axes0–0.5 Km 7 10 70 0.8808 0.6957 0.1128 1.77410.5–1 Km 7 9 63 0.7858 0.1739 0.1026 1.73031–1.5 Km 7 8 56 0.6457 0.0435 0.0908 1.64761.5–2 Km 7 7 49 0.4750 0.0000 0.0787 1.49922–2.5 Km 7 6 42 0.3100 0.0000 0.0699 1.25422.5–3 Km 7 5 35 0.1824 0.0000 0.061 0.89113–3.5 Km 7 4 28 0.0998 0.0000 0.0523 0.41353.5–4 Km 7 3 21 0.0522 0.0000 0.0466 − 0.15404–4.5 Km 7 2 14 0.0266 0.0000 0.0415 − 0.78304.5–5 Km 7 1 7 0.0134 0.0000 0.0385 − 1.4440> 5 Km 7 1 7 0.0134 0.087 0.3051 − 1.4440

Predictor map of NW-trending lineaments0–1.5 Km 6 10 60 0.7311 0.4783 0.2430 0.31421.5–3 Km 6 8 48 0.4502 0.2609 0.1822 − 0.1780

3–4.5 Km 6 6 36 0.1978 0.1739 0.1169 −

0.85804.5–6 Km 6 4 24 0.0691 0.0000 0.0793 − 1.45606–7.5 Km 6 2 12 0.0219 0.0435 0.0574 − 1.7950> 7.5 Km 6 1 6 0.0121 0.0435 0.3209 − 1.8810

Predictor map of NE-trending lineaments0–1.5 Km 6 10 60 0.7311 0.3913 0.1890 0.32571.5–3 Km 6 8 48 0.4502 0.2609 0.1678 − 0.14703–4.5 Km 6 6 36 0.1978 0.1304 0.1360 − 0.68404.5–6 Km 6 4 24 0.0691 0.087 0.1056 − 1.04906–7.5 Km 6 2 12 0.0219 0.0435 0.0841 − 1.2110> 7.5 Km 6 1 6 0.0121 0.0870 0.3171 − 1.2470


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Figure 3. Continuous grey-scale map of fuzzy posterior probabili-ties, which range from zero (white) to 0.484 (black).

se (see discussion section next; Singer and Kouda,1999; Pan and Harris, 2000) in terms of makingdecisions regarding selection of exploration targets.We therefore reclassied the fuzzy posterior prob-ability map (Fig. 3) into a ternary map of rela-tive favorability for base-metal deposit occurrence(Fig. 5). Threshold fuzzy posterior probability val-ues to differentiate between high-favorability andmoderate-favorability zones and between moderate-favorability and low-favorability zones were deter-mined using the graphical procedure described byPorwal, Carranza, and Hale ( 2003a). The cumula-tive fuzzy posterior probability values (rearrangedin a descending order) were plotted against the per-centage of cumulative area (Fig. 6). Two inectionpoints were identied along the curve at which theslope changes from steep to moderate and from mod-erate to almost at. The fuzzy posterior probabil-ity value corresponding to the lower inexion point(0.0046) was used as a threshold value to distinguish

between high-favorability and moderate-favorabilityzones, whereas the fuzzy posterior probability valuecorresponding to the upper inexion point (0.0012)was used as a threshold value to distinguish betweenmoderate-favorability’ and low-favorability zones.The threshold fuzzy posterior probability value of 0.0046 means that fuzzy posterior probabilities inhigh-favorability zones range from about 6.6 to atleast 700 times greater than the estimated prior prob-ability (0.0007). The threshold fuzzy posterior prob-ability value of 0.0012 means that fuzzy posteriorprobabilities in low-favorability zones range fromabout zero to at most 1.7 times greater than the es-timated prior probability. Fuzzy posterior probabili-ties in moderate-favorability zones, therefore, rangefrom about 1.7 to about 6.6 times greater than the es-timated prior probability.

Validation of Favorability Map

The favorability map was validated by overlay-ing the 23 deposit training points and 31 depositvalidation points on the favorability map (Fig. 5).Table 3 shows the distribution of deposit valida-tion points and deposit training points in the favor-ability map. In the favorability map, (a) the high-favorability zones occupy about 6% of the study areaand contain about 74% and about 96%, respectively,of the deposit validation points and deposit training

points, (b) the moderate-favorability zones occupyabout 4% of the study area and contain about 13% of the deposit validation points and 13% of the deposittraining points, and (c) the low-favorability zones oc-cupy about 90% of the study area and contain about13% and about 4%, respectively, of the deposit vali-dation points and deposit training points.

Brown and others ( 2000) use the following prob-ability ratio for expressing the quality of a favorabil-ity map:

p (D | A ) p (D )

=n(D A )/ n(D total )

n( A )/ n(T ) , (11)

where n(D A ) is thenumber of deposits in favorabilityclass A , n(D total ) is the total number of deposits, n( A )is the area of favorability class A , n (T ) is the totalarea, p (D | A ) is the posterior probability of a depositgiven favorability class A and p(D ) is the prior prob-ability of a deposit. Probability ratios higher than 1indicate increase of probability of mineral depositoccurrence, whereas probability ratios less than 1indicate decrease of probability of mineral deposit


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Figure 4. Continuous gray-scale maps of variances of fuzzy posterior probabilities due (A) to missing patterns and tomiss-assigned patterns [variance ranges from zero (white) to 0.2148 (black)] and (B) to fuzzy membership values [varianceranges from zero (white) to 0.0001 (black)].

occurrence. Probability ratios close to 1 indicate noimprovement of probability of mineral depositoccurrence. An efcient predictive model shoulddelineate favorability zones with probability ratioshigher or lower than 1.

To validate the favorability map (Fig. 5) usingEquation (11), prior probability ( n(D total )/n(T )) isre-estimated using all known base-metal deposit oc-currences (i.e., n(D total ) = 54), posterior probability(n(D A )/n( A )) is re-calculated for each favorabilityzone ( A ), and then the probability ratio in each fa-vorability zone is calculated. The reason for using

all the 54 known base-metal deposit occurrences inEquation (11) is to determine whether the favora-bility map, derived by using 23 training deposits, isuseful in guiding exploration towards zones with po-tential for ‘undiscovered’ (i.e., validation) deposits.The results of validation (Table 3) show that the (re-)estimated prior probability (0.0016) of base-metaldeposit occurrence (a) increases to a posterior prob-ability of 0.0226 in the high-favorability zones, (b)

increases to a posterior probability of 0.0027 in themoderate-favorability’ zones, and (c) decreases to aposterior probability of 0.0002 in the low-favorabilityzones. Consequently, there is a distinct contrast inmagnitude of the probability ratios in the differentfavorability zones. These imply that the favorabilitymap (Fig. 3) would be useful in guiding further base-metal exploration in the province.

DISCUSSION

The knowledge-based logistic function used inthe hybrid fuzzy WofE model provides a frameworkfor dealing with systemic uncertainties in a exibleand consistent way. The function uses map weightsand class weights to derive fuzzy membership val-ues of predictor patterns [Eqs. (2) and (3)]. The mapweight , which is similar in concept to the ‘condencevalue’ of Knox-Robinson ( 2000), is assigned on thebasis of (a) the delity and precision of a predictor


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Figure 5. Favorability map generated by reclassication of fuzzyposterior probability map shown in Figure 3. Triangles are “train-ing” base-metal deposits andcircles are“validation” basemetal de-posits.

map and (b) the relative importance of the recog-nition criteria represented by a predictor map. Themap weight therefore provides a method for treat-ing systemic uncertainty, which generally arises from(a) imprecision in mapping of predictor patterns, (b)involvement of heuristics in generation of one ormore predictor patterns (for example, several predic-

Figure 6. Plot of cumulative fuzzy posterior probability versus cu-mulative percent of study area.

tor patterns in the present study were based on inter-pretations of total magnetic eld intensity data), and(c) an unknown contribution of different genetic fac-tors, and, hence, of predictor patterns which repre-sent them, in spatial localization of mineral deposits.

The S-shaped logistic membership functiontransforms linearly distributed class scores to log-arithmically distributed fuzzy membership values,so the differences in fuzzy membership values arelarger in the central part of the curve than alongits tails, as discussed by Porwal, Carranza, and Hale(2003b). The function therefore separates unfavor-able patterns from favorable patterns with less uncer-tainty, although among favorable (and unfavorable)

Table 3. Validation of Favorability Map

Validation based on size of study area, training deposits,and validation deposits

Validation based on Equation (11) and all 54known deposits

Zone% of study area

34,000 km2)no. (and %) of

training depositsno. (and %) of

validation depositsPrior

probabilityPosterior

probabilityProbability

ratio

High-favorability 5.9 22 (95.6) 23 (74.2) 0.0016 0.0226 14.13Moderate-favorability 4.3 0 (0.0) 4 (12.9) 0.0016 0.0027 1.69Low-favorability 89.8 1 (4.4) 4 (12.9) 0.0016 0.0001 0.06


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patterns, the separation is more uncertain (Porwal,Carranza, and Hale, 2003a). In a spatial domain,this results in a well-dened separation of high-favorability zones from low-favorability zones.

Cheng and Agterberg ( 1999) used a linear fuzzymembership function based on an index of spa-tial association turned contrast (Bonham-Carter andAgterberg, 1990) for estimating fuzzy membershipvalues in their data-driven fuzzy WofE model. Al-though the use of contrast for calculating fuzzy mem-bership values incorporates a purely data-driven ap-proach in the modeling procedure, it is suitable formulticlass predictor maps if, and only if, each pat-tern contains at least one known deposit, becausethe contrast value ( = W + − W − ) of a predictor pat-tern is estimated from a pair of weights, which aredeterminable if, and only if, the pattern contains atleast one known deposit. Therefore, the fuzzy WofEof a predictor pattern in a data-driven fuzzy WofEmodel may become indeterminable if the patterndoes not contain known deposits. On the other hand,the knowledge-based logistic function [Equation (2)]used in the present hybrid fuzzy WofE model canbe applied to derive fuzzy membership of a predic-tor pattern even if it does not contain a known de-posit. The calculation of the fuzzy WofE of a predic-tor pattern in the present hybrid fuzzy WofE model[Eqs. (5) and (6)] requires the conditional probabili-ties of only the patterns with the highest and the low-est fuzzy membership values, given the presence or

absence of a deposit. The fuzzy WofE of a predictorpattern in the present hybrid fuzzy WofE model is,however, indeterminable if, and only if, neither thepattern with the highest fuzzy membership value northe pattern with the lowest fuzzy membership valuecontains a known deposit, which can happen rarely.At the least, the pattern with the highest fuzzy mem-bership value always contains at least one knowndeposit. Consequently, hybrid fuzzy WofE modelingcan be usedmoreconveniently with multipass predic-tor maps, even if there are few known deposits avail-able. For the same reason, the hybrid fuzzy WofEmodel is applicable to mineral potential mapping inpoorly of explored provinces containing few knownmineral deposits.

The fuzzy WofE model [Eq. (4)] uses Bayes’rule under an assumption of conditional indepen-dence for combining fuzzy WofE to derive fuzzy pos-terior probabilities. The linear nature of the fuzzyWofE model, however, entails that it is highly sen-sitive to violation of the assumption of CI among twoor more predictor maps with respect to the target

variable. As discussed by Singer and Kouda ( 1999),the assumption of CI may be difcult to validate us-ing a χ 2 test. Moreover, our previous work on theapplication of the ordinary WofE method (Porwal,Carranza, and Hale, 2003a) indicates that even if Kolmogorov-Smirnov and χ 2 tests for goodness-of-t (Agterberg, Bonham-Carter, and Wright, 1990;Bonham-Carter and Agterberg, 1990) return a statis-tically signicant goodness-of-t between observedand expected frequencies of deposits, the omnibustest (Bonham-Carter, 1994) may indicate conditionaldependence between two or more input predictorpatterns (Porwal, Carranza, and Hale, 2003a; alsosee Singer and Kouda, 1999). Here, we applied thenew omnibus test (Agterberg and Cheng, 2002) anddetermined signicant violation of CI assumption inthe nal fuzzy posterior probability map. Therefore,in practice, the possibility of conditional dependenceamong two or more predictor patterns with respectto locations of target deposit-type cannot be ruledout completely. As a result, fuzzy posterior probabil-ities calculated using Bayes’ rule in general, are arti-cially inated and, hence, cannot be interpreted inan absolute sense for decision-making. Moreover, afuzzy posterior probability is an updated prior prob-ability of a deposit-type, given the presence of a num-ber of input predictor patterns, and can be acceptedin an absolute sense, if, and only if, it is assumedthat the input predictor patterns adequately repre-sent all geologic processes that were responsible for

the spatial localization of the deposit-type. Such anassumption is never justied in practice. In addition,there is always an uncertainty (because of missingpatterns, miss-assigned patterns and fuzzy member-ship values) associated with fuzzy posterior proba-bility values. However, the effect of over-estimationof fuzzy posterior probability might be mitigated if the exploration targets are selected on the basis of relative favorability rather than absolute fuzzy pos-terior probabilities, as suggested by Pan and Harris(2000) for the ordinary WofE approach. Accord-ingly, we interpret the fuzzy posterior probabilitiesin relative terms only in order to classify zones withhigh-favorability, moderate-favorability, and lowfa-vorability for base-metal deposit occurrence.

The validation of the hybrid fuzzy WofE modelof favorability for base-metal occurrence indicatesthat, relative to estimated prior probability (us-ing either the 23 training deposits or all the54 known deposits), (a) the probability of base-metal deposit occurrence is signicantly enhancedin the high-favorability zones, (b) the probability of


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base-metal deposit occurrence is slightly enhanced inthe moderate-favorability zones, and (c) the proba-bility of base-metal deposit occurrence is signicantlyreduced in the low-favorability zones. A comparisonbetween total uncertainty due to missing and miss-assigned patterns and uncertainty due to fuzzy mem-bership values (Fig. 4) shows that the former is lowerin all parts of the study area. This indicates that if the ‘missing pattern’ areas are assigned appropriatefuzzy membership values then the total uncertaintyin fuzzy posterior probabilities can be reduced, assuggested by Cheng and Agterberg ( 1999).

CONCLUSIONS

(1) The use of a knowledge-based fuzzy mem-bership function in the hybrid fuzzy WofEapproach facilitates representation of spatialevidence of mineral deposit occurrence asmulticlass predictor maps.

(2) In practice, the possibility of conditional de-pendence among two or more input predic-tor maps with respect to target variable can-not be ruled out completely. As a result,estimates of fuzzy posterior probabilitiesgenerally are articially inated. These es-timates of posterior probabilities, therefore,cannot be strictly applied in absolute terms.However, they can be used to classify areas interms of relative favorability for occurrence

of target deposit-type.(3) The hybrid fuzzy WofE approach is demon-

strated to be useful in demarcating prospec-tive ground for mineral exploration. In thisstudy, the hybrid fuzzy WofE approach wasused to demarcate high-favorability zones forbase-metal deposit occurrence in the south-central Aravalli province based on regional-scale predictor maps. The same approach,however, requires further testing in model-ing of larger-scale predictor maps to demar-cate specic target zones within predicted

prospective ground.

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