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Emmanuel John M. Carranza
Mines and Geosciences Bureau, Regional Office No. 5, Legazpi City, Philippines
International Institute for Aerospace Survey and Earth Sciences, Delft, The Netherlands
E-mail: carranza@itc.nl
Martin Hale
International Institute for Aerospace Survey and Earth Sciences, Delft, The Netherlands
A probabilistic approach to mineral potential mapping is demonstrated in the Baguio gold district, Philippines. Using Bayes' rule, two probabilities can be computed that a binary predictor pattern contains a mineral occurrence. The natural logarithms (loge) of these probabilities represent the weight for the pattern present, W+, and the weight for the pattern absent, W-. The contrast, , provides a measure of the spatial correlation between a binary pattern and a set of occurrence points. In this study, the Studentized C [C/s(C)] served as the basis for converting multi-class 'proximity maps' into binary predictor patterns. The loge of the posterior odds of a mineral occurrence given the presence/absence of a predictor pattern is then obtained by adding the weights of each binary pattern and the loge of the prior odds. The result is a map of posterior probabilities whose magnitude represent mineral potential. Combining the binary maps requires that these are conditionally independent from one another with respect to the mineral occurrences. Conditional independence is tested by pairwise chi-square tests and by overall comparison of predicted and observed number of mineral occurrences.
The datasets used are geological map, map of faults/fractures and large-scale and small-scale gold occurrences data. The predictive maps based on each occurrence dataset show that 79% and 70% of the known large-scale and small-scale gold occurrences, respectively, fall within zones of high potential. Cross-comparison of each predictive map with the other gold occurrence dataset indicate that at least 56% of the gold occurrence points fall within zones of high potential. The probabilistic approach to mineral potential mapping is effective for the Baguio district in that the predictive maps depict several zones of high potential that contain or are close to the known gold occurrences.
Key words: Bayesian probability, weights of evidence, spatial correlation, mineral potential mapping, Baguio gold district (Philippines)
The geology of any given area is probably the single most important indicator of its mineral potential. In well-explored areas, the qualitative knowledge of the spatial correlations of known mineral deposits with the different geological features is the basis of most exploration programs. However, as the spatial correlation of mineral occurrences with geological features varies from place to place, a qualitative knowledge alone is inadequate for finding new deposits. A quantitative knowledge of the spatial correlation between known mineral occurrences and the different geological features present is equally important in mineral exploration.
Previous work on quantitative methods for mapping mineral potential, based on known mineral occurrences, predominantly used regression techniques (e.g., Chung and Agterberg, 1980; Harris, 1984). The known mineral occurrences in a region are used to develop a multivariate signature for mineralization, expressed as a vector of coefficients for the geological predictor variables. The coefficients are calculated using least squares regression; the resulting equation is used to generate regression scores whose magnitude reflect mineral potential. Regression techniques, however, are weak for a number of reasons. One is that they invariably assume that the relationship of the dependent variable (i.e., location of known mineral occurrences) to the predictor variables (i.e., geological features) is linear, which is not always valid. Another weakness of regression techniques is that, no assumption of, and consequently no test for, conditional independence between the predictor variables is required.
An alternative approach to mapping mineral potential that avoids the limitations associated with regression techniques is to use Bayes' rule (e.g., Bonham-Carter et al., 1988; Agterberg et al., 1990). The Bayesian approach to the problem of combining multiple predictor variables uses a probability framework, that is, the idea of unconditional (prior) and conditional (posterior) probability. Starting with a prior probability of mineral deposits occurring in a unit area, a posterior probability is calculated based on the weights of evidence for the presence and absence of a predictor variable (Bonham-Carter et al., 1989). The weights of evidences for all predictor variables are combined in order to estimate the conditional probability of mineral occurrence given the presence and absence of all the binary predictor variables. Combining the weights of evidences of the different binary predictor maps requires an assumption that the input maps are conditionally independent. The application of a test for conditional independence of each pair of predictor maps with respect to the known mineral occurrences can lead to the rejection of some input maps.
In this paper, we present an application of the Bayesian approach of combining binary patterns of geological features for mapping gold potential in the Baguio mineral district of the Philippines. We use two sets of mineral occurrences data, locations of small-scale workings for gold by local people and locations of gold deposits that either have been mined on a large-scale or have been explored extensively for development. We use two sets of mineral occurrences data to evaluate the resulting mineral potential maps with respect to each dataset. In addition, we intend to show the usefulness of the occurrences of small-scale mine workings in the probabilistic mapping of mineral potential. In the last two decades, at least two major gold deposits in the Philippines have been discovered based on occurrences of small-scale workings. Because of the general lack of exploration data other than geological data in most parts of the Philippines, except in known mineral districts, the probabilistic mapping of mineral potential we present here is constrained to the predictor geological variables. We use a geographic information system (GIS) of geological and mineral occurrences data for the Baguio district to examine empirically the spatial correlation between the geological variables and the large-scale and small-scale gold occurrences. Probabilistic models are then developed for predicting gold potential using the spatial relationships between the geological variables and the known gold occurrences.
The following formulation of the Bayesian probability model as applied to mineral potential mapping is synthesized from Bonham-Carter (1994) and Bonham-Carter et al., (1989). Following Bayes' rule, there are two posterior or conditional probabilities of a mineral occurrence: one given the presence of a predictor pattern and the other given the absence of a predictor pattern. Thus,
(Eq. 1) and
(Eq. 2),
where D represents mineral occurrence, is predictor pattern j present and is predictor pattern j absent. Expressed as posterior odds, these equations respectively become:
(Eq. 3) and
(Eq. 4),
where and are respectively the posterior odds of a mineral occurrence given the presence and absence of a predictor pattern, O{D} is the prior odds of a mineral occurrence. The natural logarithm of the ratio is the positive weight of evidence, W+, when a binary predictor pattern is present. The natural logarithm of the ratio is the negative weight of evidence, W-, when a binary predictor pattern is absent. Thus, it can be shown that
(Eq. 5) and
(Eq. 6).
The variances of the weights can be calculated by the following expressions (Bishop et al., 1975):
(Eq. 7) and
(Eq. 8).
Now suppose there are two binary predictor patterns, and . From probability theory, it can be shown that the conditional probability of a mineral occurrence given the presence of two predictor patterns is
(Eq. 9).
If and are conditionally independent with respect to a set of mineral occurrence points, it means that the following relationship is satisfied:
(Eq. 10).
This allows Eq. 9 to be simplified, thus:
(Eq. 11).
Equation 11 is like Equation 1, except that multiplication factors for two maps are used to update the prior probability to give the posterior probability. Using the odds formulation, it can be shown that:
(Eq. 12),
(Eq. 13),
(Eq. 14), and
(Eq. 15).
Similarly, if more than two binary predictor maps are used, they can be added provided they are also conditionally independent of one another with respect to the mineral occurrence points. Thus, with (j=1,2,...,n) binary predictor maps, the natural logarithm of the posterior odds are:
(Eq. 16),
where the superscript k is positive (+) or negative (-) if the binary predictor pattern is present or absent, respectively. The posterior odds can then be converted to posterior probabilities, indicating favourability for mineral potential, using P=O/(1+O).
For each binary predictor map, the contrast gives a useful measure of spatial correlation with the mineral occurrence points (Bonham-Carter et al., 1989). For a positive spatial correlation, C is positive; for a negative correlation, C is negative. The standard deviation of C is calculated as . The Studentized value of C, calculated as the ratio of C to its standard deviation, C/s(C), serves as an informal test that C is significantly different from zero or if the contrast is likely to be 'real' (Bonham-Carter, 1994).
The district (Fig. 1) is underlain by five major lithologic units. The Pugo Fm is a sequence of metavolcanic and metasedimentary rocks of Cretaceous to Eocene age. Unconformably overlying the Pugo Fm is the Zigzag Fm. It consists largely of marine sedimentary rocks of Early to Middle Miocene in age based on diagnostic foraminiferal assemblage (Balce et al., 1980). However, it is intruded by andesite porphyry dated 15.0 ( 1.6 Ma (Wolfe, 1981) which imply a pre-Middle Miocene age for the formation. Mitchell and Leach (1991) consider the Zigzag Fm to be largely Late Eocene but may include rocks of Early Miocene age. The Kennon Fm of Middle Miocene age conformably overlies the Zigzag Fm (Balce et al., 1980). It is made up of limestones occurring in a discontinuous north-trending belt west of the mining district. Unconformably overlying all formations in the Baguio district is the Klondyke Fm. Most workers agree that its age may be fixed as Late Miocene (Balce et al., 1980; Wolfe, 1988; Mitchell and Leach, 1991). The Klondyke Fm includes a wide range of lithologies and sedimentary facies, but most are clastic and almost entirely andesitic in composition. The other major lithologic unit in the district is the Agno Batholith; it is predominantly composed of hornblende quartz diorite. Radiometric dating indicates that the different rocks in the batholith were intruded in several phases. Wolfe (1981) cited an average 27 Ma for the earlier phases and a range of 12-15 Ma for the later phases. More recent workers contend that the Agno Batholith is intrusive mostly into the Pugo Fm (Balce et al., 1980) but earlier workers also believed that it also intruded the Zigzag Fm (Peña, 1970; Sawkins et al., 1979). The Zigzag and Pugo Formations have also been intruded by younger intrusive complexes that vary in age from Late Miocene to Pleistocene (Mitchell and Leach, 1991; Cooke et al., 1996).
Figure 1. Simplified geologic map of the Baguio mineral district (modified
after MMAJ; 1977; Balce et al., 1980; MGB, 1995). Numbered squares
are large-scale gold occurrences referred to in Table 1. Small circles are
small-scale gold occurrences. Curvi-linear features are faults/fractures.
Most of the gold in the Baguio district comes from epithermal systems that are confined to a north-trending zone about 7 km wide east of Baguio City (Fernandez and Damasco, 1979; Mitchell and Leach, 1991). The epithermal gold deposits are confined to quartz veins, commonly fracture-controlled, and stockworks. Most productive veins trend northeasterly to easterly although there are some productive northwesterly trending veins (Mitchell and Leach, 1991). The auriferous veins are mostly hosted by the Zigzag and Pugo Formations and the Agno Batholith. Almost all known productive veins are found in the western fringe of the contact zone between the Agno Batholith and the intruded rocks. Wolfe (1988) suggested that the Agno plutons are the source of the precious metals. More recent workers claim that the epithermal gold mineralization is related to the younger intrusive complexes (Mitchell and Leach, 1991; Cooke et al., 1996). Mitchell and Leach (1991) proposed that the age of the epithermal mineralization could be late Miocene to Pliocene.
The datasets used are geological map, map of faults/fractures, large-scale and small-scale gold occurrences data. The boundaries of lithologic units and the fault/fractures were hand-digitized into vector (polygon) format. The large-scale gold occurrences data (Mitchell and Leach, 1991; MGB-CAR, 1992; MMAJ, 1996) were stored in a spreadsheet database with rows for each occurrence and columns for their spatial and non-spatial attributes (Table 1). The locations of small-scale gold occurrences were hand-digitized from paper maps (MGB-CAR, 1992).
Table 1. Large-scale gold occurrences in the Baguio mineral district.
No. | Gold Mines/Prospects | ore reserve/status | Au grade | Reference |
---|---|---|---|---|
1 | Acupan | >200t, prod. since 1915 | 6.1 g/t | Cooke et al., 1996 |
2 | Antamok | >300 t, prod. since 1907 | 4-5 g/t | Mitchell & Leach, 1991 |
3 | Baco | 0.36 Mt | 7-17 g/t | Yumul, 1980; Mitchell & Leach, 1991 |
4 | Baguio gold | depleted | up to 29 g/t | Yumul, 1980; Mitchell & Leach, 1991 |
5 | Belle | explored deposit | - | Mitchell & Leach, 1991 |
6 | Cal Horr | depleted | 2.5 g/t | Mitchell & Leach, 1991 |
7 | Camp 7 | 0.026 Mt | 30.5 g/t | Yumul, 1980 |
8 | Capunga | explored deposit | 03-13 g/t | Yumul, 1980 |
9 | Chico | 0.689 Mt, closed since 1977 | 4.3 g/t | Yumul, 1980 |
10 | Demonstration | explored deposit | - | Yumul, 1980 |
11 | Gold Fields | explored deposit | - | Yumul, 1980 |
12 | Itogon | 4.123 Mt | 4.1 g/t | Yumul, 1980 |
13 | Kelly | - | 4 g/t | Mitchell & Leach, 1991 |
14 | Keystone | - | 2-3 g/t | Mitchell & Leach, 1991 |
15 | King Solomon | explored deposit | - | Yumul, 1980 |
16 | Macawiwili | 1.552 Mt | 11 g/t | Yumul, 1980 |
17 | Nagawa | explored deposit | 2 g/t | Yumul, 1980 |
18 | Omico | - | 10 g/t | Mitchell& Leach, 1991. |
19 | Sierra Oro | depleted | 6-13 g/t | Mitchell & Leach, 1991 |
The data input and succeeding map operations and analyses were carried out using ILWIS (Integrated Land and Water Information Systems), a GIS software developed by the International Institute for Aerospace Survey and Earth Sciences in the Netherlands. In ILWIS, spatial data analysis is carried out in the raster mode. Calculation of the weights of the binary predictors patterns involves crossing the raster maps of binary patterns with the raster maps of gold occurrence points. The procedure for extracting the binary predictor maps used in the probabilistic mapping of gold potential is discussed below.
A pixel size (or unit cell) of 100 m by 100 m is used in rasterizing the input vector maps for the creation and the computation of the weights of the binary predictor maps. This pixel size is chosen to ensure that only one mineral occurrence is present in any give pixel; it is also a realistic size for a mineralized area. The rasterized lithologic map was reclassified into a binary map of favourable 'lithologies' by assigning a score of 1 (i.e., presence) to the Zigzag and Pugo Formations and a score of 0 (i.e., absence) to the other units. From the vector map of lithologic contacts, the outlines of the Agno Batholith and the younger intrusive complexes were extracted into two separate vector maps. Likewise, from the vector map of faults/fractures, northeasterly and northwesterly trending faults/fractures were extracted into two separate vector maps. These vector maps were rasterized and 'corridor' maps showing relative distances away from these geological features were generated.
Because the distance corridor maps have multiple classes, we need to select the distance for which the spatial correlation between the mineral occurrences and the geological features is optimal in order to convert them into binary maps. The optimum cutoff distance is chosen by calculating and examining the contrast (C) at successive cumulative distance intervals. The highest C usually indicates the optimum cutoff distance at which the predictive power of the binary pattern is maximized (Bonham-Carter et al., 1988). However, in cases where there are only a small number of occurrence points or small areas, such as the present case, the uncertainty of the weights could be large so that C is meaningless (Bonham-Carter, 1994). In the present case, the Studentized C was useful for choosing the cutoff distance because it serves as a measure of the certainty (and uncertainty) of the contrast.
Shown in Figure 2a is the variation in the contrast for cumulative distances from the Agno Batholith contact with respect to large-scale gold occurrences. The contrast is highest at 2000 m; 18 of the 19 gold occurrences are present within this distance. However, because not all the gold deposits in the district are genetically (or spatially) associated with the Agno Batholith (Mitchell and Leach, 1991; Cooke et al., 1996), 2000 m could not be the optimum cutoff distance. The Studentized C indicates that the optimum cutoff distance is 1000 meters. Fifteen of the 19 gold occurrences are present within 1000 meters; the other occurrences are probably spatially correlated with other geological features. In the case of the small-scale gold occurrences, the contrast is highest for 2500 meters (Figure 2b); 62 of the 63 gold occurrences are present within this distance. However, Studentized C indicates that the optimum cutoff distance is 750 meters; 43 of the 63 small-scale occurrences are present within this distance.
Figure 2. Variation in contrast for cumulative distances away from Agno Batholith contacts with
respect to (a) large-scale and (b) small-scale gold occurrences.
Shown in Figure 3a is the variation in the contrast for cumulative distances from the outlines of the younger intrusives with respect to the large-scale gold occurrences. The contrast is highest for 5250 meters wherein 18 of the 19 gold occurrences are present. The Studentized C is highest for 2750 meters wherein 15 of the 19 gold occurrences are present. Using either of these distances as the optimum cutoff would mean that almost all the large-scale gold occurrences are spatially correlated with the younger intrusive complexes, which is untrue. On the Studentized C curve, there is another peak at 750 meters. The number of gold occurrences within this distance is 6, which is about the same number (4) of gold occurrences present outside the cutoff distance of 1000 meters from the Agno Batholith (see above). The optimum cutoff distance from the younger intrusives with respect to the large-scale gold occurrences is therefore set at 750 meters. The case is different when the small-scale gold occurrences are used (Figure 3b). The contrast is maximum for 4500 meters wherein 62 of the 63 gold occurrences are present. According to the Studentized C, however, the optimum cutoff distance from the younger intrusives with respect to the small-scale gold occurrences is at 1750 meters wherein 44 of the 63 gold occurrences are present.
Figure 3. Variation of contrast for cumulative distances away from young intrusives contacts
with respect to (a) large-scale and (b) small-scale gold occurrences.
The variation of contrast for cumulative distances from northeasterly trending faults/fractures with respect to the large-scale gold occurrences is given in Figure 4a. The contrast is maximum for 1100 meters wherein 18 of the 19 gold occurrences are present. On the other hand, the Studentized C indicates that the optimum cutoff distance is 400 meters wherein 13 of the 19 gold occurrences are present. Using the small-scale gold occurrences, C is maximum for 1400 meters wherein 62 of the 63 gold occurrences are present (Figure 4b). The Studentized C, however, indicates that the optimum cutoff distance from the northeasterly trending lineament is 600 meters wherein 50 of the 63 gold occurrences are present.
Figure 4. Variation of contrast for cumulative distances away from NE-trending faults/fractures
with respect to (a) large-scale and (b) small-scale gold occurrences.
The variation of contrast for cumulative distances from northwesterly trending faults/fractures with respect to the large-scale gold occurrences is given in Figure 5a. For all successive cumulative distances away from the northwesterly trending faults/fractures, C is negative. Nonetheless, both C and Studentized C are highest for 300 meters wherein 5 of the 19 gold occurrences are present. This suggests that a few of the large-scale gold occurrences tend to have spatial correlation with northwesterly trending faults/fractures.. With respect to the small-scale gold occurrences, the contrasts for the successive cumulative distances from the northwesterly trending faults/fractures are either positive or negative (Figure 5b). The maximum value of C and Studentized C is at 1400 meters wherein 58 of the 63 gold occurrences are in the pattern. However, there are two other peaks on either the C and Studentized C curves; one at 500 meters and the other at 1000 meters. The former is interesting because it is at this distance when C first becomes positive and becomes negative again 200 meters farther. The optimum cutoff distance from the northwesterly trending faults/fractures with respect to the small-scale gold occurrences is therefore set at 500 meters.
Figure 5. Variation of contrast for cumulative distances away from NW-trending faults/fractures
with respect to (a) large-scale and (b) small-scale gold occurrences.
We now have two sets of five binary predictor maps that can be integrated to map the gold potential in the Baguio district based on the large-scale and small-scale gold occurrences (Figs. 6 and 7, respectively). The weights, contrasts and their standard deviations for each of the binary predictor maps are summarized in Tables 2 and 3. Prior to combining the binary predictor maps with the Bayesian probability model described above, the assumption of conditional independence between all pairs of maps is tested.
Table 2. Weights of binary predictor patterns with respect to large-scale gold occurrences.
Binary Predictor Map | W+ | s(W+) | W- | s(W-) | C | s(C) | C/s(C) | |
---|---|---|---|---|---|---|---|---|
Pugo & Zigzag Fms. | 0.256 | 0.302 | -0.271 | 0.354 | 0.527 | 0.465 | 1.134 | |
Agno Batholith contact (1 km) | 0.617 | 0.258 | -1.003 | 0.500 | 1.621 | 0.563 | 2.880 | |
young intrusives contact (0.75km) | 1.247 | 0.409 | -0.284 | 0.277 | 1.531 | 0.494 | 3.101 | |
NE faults/fractures (0.4 km) | 0.676 | 0.277 | -0.725 | 0.408 | 1.401 | 0.494 | 2.837 | |
NW faults/fractures (0.3 km) | -0.018 | 0.447 | 0.006 | 0.267 | -0.024 | 0.521 | -0.046 |
Figure 6. Binary geological patterns and their weights with respect
to the large-scale gold occurrences.
Table 3. Weights of binary predictor patterns with respect to small-scale gold occurrences.
Binary Predictor Map | W+ | s(W+) | W- | s(W-) | C | s(C) | C/s(C) | |
---|---|---|---|---|---|---|---|---|
Pugo & Zigzag Fms. | 0.186 | 0.172 | -0.182 | 0.186 | 0.368 | 0.253 | 1.455 | |
Agno Batholith contact (0.75 km) | 0.678 | 0.153 | -0.722 | 0.224 | 1.401 | 0.271 | 5.171 | |
Young intrusives contact (1.75km) | 1.146 | 0.151 | -0.948 | 0.229 | 2.093 | 0.275 | 7.619 | |
NE faults/fractures (0.6 km) | 0.467 | 0.142 | -0.890 | 0.277 | 1.358 | 0.311 | 4.358 | |
NW faults/fractures (0.5 km) | 0.063 | 0.193 | -0.045 | 0.167 | 0.108 | 0.255 | 0.425 |
Figure 7. Binary geological patterns and their weights with respect
to the small-scale gold occurrences.
Equation 10 above defines the relationship if two binary patterns are conditionally independent with respect to a set of mineral occurrence points. Algebraic manipulation will show that Equation 10 is equivalent to the following equation:
(Eq. 17).
The left-hand side of the equation is the observed number of deposits in the overlap region where both binary patterns B1 and B2 are present. The right-hand side is the expected number of deposits in this overlap zone. The relationship leads to a contingency table calculation for testing the conditional independence of two binary patterns (Table 4). The four cells in the table correspond to the four overlap conditions between two binary patterns where mineral occurrence points are present. The conditional independence is tested by calculating as follows:
(Eq. 18).
Because the mineral occurrences are considered as points, or small unit cells, the resulting values of are unaffected by the units of area measurement. The calculated value can then be compared with tabled values of with one degree of freedom (Bonham-Carter, 1994). An example for using the contingency table for testing conditional independence is shown in Table 5.
Table 4. Contingency table for testing conditional independence.
The four values in the table are either the expected or calculated
values according to Eq. 17, or the observed values measured
from the maps. (Adopted from Bonham-Carter, 1994).
B1 Present | B1 Absent | Total | |
B2 Present | |||
B2 Absent | |||
Total |
Table 5. Example of using contingency table for testing conditional
independence between the binary map of Pugo/Zigzag Fms (B1) and
the binary map of 'distance to' Agno Batholith contact (B2). Values
in bold are observed on the map, those in brackets are the expected
values using right-hand of Eq. 17. For this example, calculated chi-
square value is 3.68. Tabled chi-square value at the 95% significance
level with 1 degree of freedom is 3.8; thus, the assumption of condi-
tional independence between the two binary maps is not rejected at
95% significance level.
B1 Present | B1 Absent | Total | |
B2 Present | 7 (8.7) | 4 (2.3) | 11 |
B2 Absent | 8 (6.3) | 0 (1.7) | 8 |
Total | 15 | 4 | 19 |
With respect to the large-scale gold occurrences, all the pairs of binary geological factors are conditionally independent (Table 6). With respect to the small-scale gold occurrences, all pairs of binary maps, except the pair Agno Batholith contact - young intrusives contact, show conditional independence (Table 7). The effect of this conditional dependence between the pair is evaluated below.
Table 6. Calculated chi-square values for testing conditional independence between all pairs of 5 binary maps with respect to the large-scale gold occurrences. With 1 degree of freedom and a probability level of 98%, tabled chi-square value is 5.4. For all pairs of binary maps, the null hypothesis of conditional independence is not rejected at the 98% probability level.
Binary Map | Agno Batholith contact | Young intrusives contact | NE-trending faults/fractures | NW-trending faults/fractures |
Pugo & Zigzag Fms. | 3.68 | 2.33 | 0.28 | 0.01 |
Agno Batholith contact | - | 4.42 | 0.80 | 0.00 |
Young intrusives contact | - | - | 1.38 | 0.22 |
NE-trending faults/fractures | - | - | - | 0.22 |
Table 7. Calculated chi-square values for testing conditional independence between all pairs of 5 binary maps with respect to the large-scale gold occurrences. With 1 degree of freedom and a probability level of 98%, tabled chi-square value is 5.4. Values in bold indicate pairs for which the null hypothesis of conditional independence is not rejected at the 98% probability level.
Binary Map | Agno Batholith contact | Young intrusives contact | NE-trending faults/fractures | NW-trending faults/fractures |
Pugo & Zigzag Fms. | 3.03 | 2.29 | 1.59 | 5.12 |
Agno Batholith contact | - | 8.58 | 0.34 | 0.61 |
Young intrusives contact | - | - | 0.00 | 4.58 |
NE-trending faults/fractures | - | - | - | 2.62 |
Finally, the binary predictor maps are assigned the weights (Tables 2 and 3) and are combined according to Equation 16. Using the large-scale gold occurrences, the prior probability P{D}=19/42039=0.00045 and natural logarithm of O{D}= -7.7. Using the small-scale gold occurrences, the prior probability P{D} is 63/42039=0.0015 and natural logarithm of O{D}=-6.5. After combining the binary predictor maps, the predicted number of occurrences can be calculated as the sum of the products of the number of pixels, N{A}, and their posterior probabilities, P, for all pixels on the map, thus
(Eq. 19),
where there are k=1,2,...,m pixels on the map. If the predicted number of occurrences is larger than 15% of the observed occurrences, then the assumption of conditional independence is seriously violated (Bonham-Carter, 1994). Problematic maps should then be removed from the analysis.
In mapping the gold potential, we equate the degree potential to the magnitude of the ratio of posterior probability to prior probability (Pposterior/Pprior). Thus, if the ratio is less than 1 (i.e., Pposterior < Pprior) then potential is low, if the ratio ranges from 1 to 5 then potential is high and if the ratio is greater than 5 then potential is very high.
The resulting map of posterior probabilities after combining the binary predictor patterns whose weights were calculated with respect to the large-scale gold occurrences is shown in Figure 8. Crossing this map with the map of large-scale gold occurrences reveals that 4 of the 19 large-scale occurrences are in low potential zones, 14 are in high potential zones and 1 is in high potential zones (Table 8). These mean that 79% of the 19 large-scale occurrences are in zones of high to very high potential. The predicted number of large-scale gold occurrences based on Equation 19 is 21, only about 11% larger than the observed occurrences. Thus, the input binary predictor patterns are conditionally independent as already indicated in Table 5.
Figure 8. Final map of gold potential based on binary predictor patterns
whose weights were calculated based on the large-scale gold occurrences.
Many of the small-scale occurrences are in or near zones of high to very high potential (Table 8). Crossing the raster map of gold potential and the raster map of small-scale occurrences indicates that about 56% of the small-scale occurrences are in zones of high to very high potential.
Table 8. Inventory of gold occurrences in zones of different potentials based on the large-scale and small-scale gold occurrences datasets.
Pposterior/Pprior | Gold Potential | Potential map based on large-scale gold occurrences | Potential map based on small-scale gold occurrences | ||||||
large-scale gold occurrences | small-scale gold occurrences | large-scale gold occurrences | small-scale gold occurrences | ||||||
no. | % | no. | % | no. | % | no. | % | ||
<1 | low | 4 | 21 | 28 | 44 | 8 | 42 | 19 | 30 |
1-5 | high | 14 | 79 | 29 | 56 | 9 | 58 | 14 | 70 |
>5 | very high | 1 | 6 | 2 | 30 |
Figure 9 shows the resulting map of posterior probabilities after combining the binary predictor patterns whose weights were calculated with respect to the small-scale occurrences. Many of the small-scale occurrences are in zones with high to very high potential (Table 8); 14 small-scale occurrences are in zones with high potential and 30 small-scale occurrences are in zones of very high potential. Only 19 of the small-scale occurrences are in zones of low potential. These show that about 70% of the small-scale occurrences are in zones of high to very high potential. The predicted number of small-scale gold occurrences based on Equation 19 is 73, or only about 6% larger than the observed occurrences. This indicates that the assumption of conditional independence is not seriously violated although one of the map pairs shows conditional dependence (Table 7).
Figure 9. Final map of gold potential based on binary predictor patterns
whose weights were calculated based on the small-scale gold occurrences.
Many of the large-scale occurrences are in or near the zones of high to very high potential (Table 8). Crossing the raster map of gold potential with the raster map of large-scale occurrences reveals that about 58% of the known large-scale occurrences are in zones of high to very high potential.
As shown above, the Studentized C was useful in the reclassification of distance corridor maps into binary predictor patterns. Bonham-Carter et al., (1988) stated that the maximum value of C will, under normal conditions, give the cutoff distance. They did not state, however, what the 'normal' conditions are. We propose two possibilities why the maximum value of C alone was not useful in determining the cutoff. One possibility is that the gold occurrences in the district are genetically and spatially associated with at least two magmatic regimes. The other possibility is that the mineral occurrence points are spatially concentrated within a north-south trending zone. In the study area of Bonham-Carter et al., (1988), gold mineralization is associated with one magmatic regime and that the mineral occurrences are more or less evenly spatially distributed throughout the area studied.
The magnitude of Studentized C (Tables 2 and 3) also reveals that the input binary maps of relative importance for predicting both known large-scale and small-scale gold occurrences are (1) proximity to Late Miocene - Pleistocene intrusive complexes, (2) proximity to the Late Oligocene - Early Miocene Agno Batholith, (3) proximity to NE-trending fractures/faults, (4) presence of the Zigzag and Pugo Formations and (5) proximity to NW-trending fractures/faults. These quantifications are agreeable with the observations of previous workers. Based on the values of Studentized C, the proximity to the Agno Batholith contact is the most significant predictor of the large-scale gold occurrences. On the other hand, the proximity to the young intrusives contact is the most significant predictor of the small-scale gold occurrences. These findings remain to be proven.
Bonham-Carter et al. (1989) indicated that if the value of Studentized C greater than 1.96 then the value of C is statistically different from zero and therefore statistically significant. This means that the spatial correlations of the Zigzag/Pugo Formations and the proximity to NW-trending faults/fractures with the mineral occurrence points are not statistically significant. We did not reject the binary maps of these geological features in the analysis because gold mineralization in the district is known to be associated with them. Judging from the magnitude of their weights (Tables 2 and 3), removal of these binary patterns from the analysis would have little effect on the resulting map of posterior probabilities.
The test for conditional independence shows that chi-square value for the pair Agno Batholith contact and young intrusives contact are the highest with respect to both large-scale and small-scale gold occurrences. This tendency of conditional dependence between the two patterns may be explained by two geological possibilities. One is that that the young intrusives are later magmatic differentiates of the Agno Batholith; the other is that the younger intrusives may have caused remobilization of pre-existing gold mineralization related to the batholith. If there is conditional dependence between the two geological features, it is likely that the younger intrusives are conditionally dependent on the Agno Batholith. Removal of either binary pattern of these geological features from the analysis would be 'geologically' incorrect although statistically correct. Combining these two geologic features is also incorrect as they are associated with different episodes of epithermal gold mineralization. Incidentally, comparison of the predicted number and observed number of small-scale gold occurrences indicated that conditional independence is not violated.
It is interesting to note that the resulting maps of posterior probabilities based on the two sets of mineral occurrences data are similar (Figures 8 and 9). It is also interesting to note that when the resulting predictive maps are cross-compared with the mineral occurrence points, the percentages of mineral occurrence points falling into the predicted zones of gold potential are similar. These observations imply that each set of mineral occurrence points may be used to predict the location of the other set of mineral occurrence points. This is possible only because both sets of mineral occurrences are genetically and spatially associated with similar geological features. This further implies that the locations of small-scale mineral occurrences are practically important in guiding exploration to search for potentially more economic deposits of similar genesis.
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