Chapter 5. A Simplification of Weights of Evidence using a Density Function and Fuzzy Distributions; Geothermal Systems, Nevada

Transcription

1 Chapter 5 A Simplification of Weights of Evidence using a Density Function and Fuzzy Distributions; Geothermal Systems, Nevada Mark F. Coolbaugh Great Basin Center for Geothermal Energy, Mackay School of Mines, Department of Geological Sciences, MS 172, University of Nevada, Reno, NV, USA, Richard L. Bedell Arthur Brandt Laboratory for Exploration Geophysics, Mackay School of Mines, Department of Geological Sciences, MS 172, University of Nevada, Reno, NV, USA, and AuEx Ventures Inc., 940 Matley Lane, Suite 17, Reno, NV, USA, Abstract Probability modelling in the geosciences has been dominated by binary weights-of-evidence techniques. However, the reduction of each layer of data to a binary yes or no designation is not considered adequate by all geoscientists. Alternate methods define gradational weights over a range of values, as is done with fuzzy logic modelling. The problem with fuzzy logic is that there is no unbiased way to combine layers of evidence, and the relationship between fuzzy weights and probability distributions is undefined. A new technique for estimating weights over a range of evidence values is proposed. A density function is defined as the fraction of the total training sites occurring within a given histogram bin of an evidence layer's data distribution, divided by the fraction of the total study area within that bin. If the total training site area is small relative to the total study area, as is often the case, then the natural log of this density function approximates a weight of evidence. Evidence layers represented by the density function can be added together in log-transformed space in a manner analogous to the posterior logit calculation in weights of evidence. Using geothermal systems in Nevada, USA, as an example, the density function model correlates remarkably well with binary weights of evidence, suggesting not only that the density method can mimic data-driven probabilistic methods, but also that binary weights of evidence is a robust technique in its own right. However, the density function method yields more detailed favourability rankings than binary weights of evidence, potentially advantageous in "vectoring in" toward exploration targets. The density method also offers ease of calculation; minimal expert guidance is used to smooth statistical noise that can make determination of multiclass weights of evidence difficult. Résumé En sciences de la Terre, le domaine de la modélisation des probabilités a été dominé par des techniques de pondération de d'informations probantes binaires. Cependant, la réduction de chaque couche d'information en données binaire (présence ou absence) est insatisfaisante pour nombre de géoscientifiques. D'autres méthodes utilisent des échelles de 115

2 GIS FOR THE EARTH SCIENCES pondération appliquée à une gamme de valeurs, comme c'est le cas dans la modélisation par logique floue. Le problème avec les méthodes par logique floue, c'est qu'il n'existe pas de façon impartiale de combiner les couches d'information, et que la relation entre les pondérations floues et les distributions de probabilités sont non-définies. Une nouvelle technique d'estimation de la pondération par gamme d'indices est proposée. Une fonction de densité est définie comme étant la fraction du nombre total de sites de référence de la population des classes d'un histogramme de la distribution des indices d'une couche d'information probante, divisée par la fraction de la surface totale étudiée correspondant à cette population. Si la surface de tous les sites de référence est petite par rapport à la surface de la zone étudiée, comme c'est souvent le cas, alors le log naturel de cette fonction de densité équivaut à peu près à la pondération de l'information probante. Les couches d'indices représentées par la fonction de densité peuvent alors être additionnées dans un espace transformé logarithmiquement, comme on le ferait dans un calcul logit a posteriori d'une pondération d'information probante. En prenant des systèmes géothermaux dans l'état de Nevada aux États-Unis comme exemples, la fonction de densité affiche une remarquable corrélation avec la pondération d'information probante binaire, permettant de croire non seulement que la méthode par fonction de densité peut reproduire les résultats de méthodes probabilistes à base de données, mais aussi que la méthode binaire par pondération d'information probante est elle-même une technique robuste. Cela dit, la méthode par fonction de densité produit donne des niveaux de favorabilité plus nombreux que la méthode binaire de pondération d'information probante, ce qui peut s'avérer intéressant au moment d'établir des cibles d'exploration par vectorisation. La méthode par fonction de densité présente l'avantage d'un calcul facile et ne requiert une assistance experte que pour lisser le bruit de fond statistique, dans les cas où il s'avérerait difficile de déterminer une pondération de l'information probante multiclasse. INTRODUCTION The choice of which method to use in building a predictive model depends, in part, on the acceptance and understanding of those that will use it. In reality, the model is only one component of the decision-making process (Figure 1). The conceptualization of a problem is derived from human understanding, and decisions to implement a program in a legal or financial context (usually both) involve individuals. In the mining industry, these individuals typically include a spectrum of managers and data experts, who must be comfortable with the data going in and how it is represented in the model. Individual experts must be convinced that the data layers they are most familiar with are adequately represented in the model. In the authors' experiences working with surveys, academics, and mining companies, acceptance of binary input is difficult for some data specialists. Although binary patterns can work well, as in the example of distinguishing between anomalous and background values in geochemistry (Bonham-Carter et al., 1988), some experts prefer more continuous data distributions that more intuitively and quantitatively represent the data. In contrast, managers are often more concerned with understanding the entire modelling method, especially if the modelling results do not fit preconceived notions. Therefore, for acceptance in a larger decision-making context, it is advantageous for the modelling process to be as simple as possible, and advantageous for the model to fully represent data distributions. Much previous modelling using Geographic Information Systems (GIS) has invoked simple Boolean overlay of evidence and addition of evidence layers, with or without an additional expert weighting (Pan and Harris, 2000; Drury, 1993). These methods, although overly simplistic, are intuitive and allow a consensus in the decision making process. These methods do not consider the essential concepts of prior and posterior probability which are central to Bayesian methods. Figure 1. Probability modelling is a component of decision-making. Individuals derive the exploration concept based on their interaction with the data and the probability model. The decision to proceed is a management decision and managers need to grasp the modelling concept. The manager in turn is influenced on an individual level by data experts who will argue against an over-simplification of their data as a binary input layer. Binary weights of evidence (WofE) offers a more rigorous and objective statistical approach that has found many applications in modelling real world situations (Agterberg, 1989; Bonham-Carter, 1996; Knox-Robinson, 2000; Raines and Bonham-Carter, this volume; Mihalasky, this volume). A major benefit of WofE is the unbiased, statistically derived weight it provides for individual layers of data (evidence maps). However, WofE is perceived by some users as both an oversimplification because of its typically binary input, and yet overly complex in mathematics. Multiclass WofE offers bet- 116

3 SPECIAL PUBLICATION 44 ter representation of data distributions, but statistical noise can sometimes limit the effective use of multiclass weights. Fuzzy logic methods offer gradational weighting schemes for individual data layers and relatively simple arithmetic operations for combining evidence layers into predictive models. With fuzzy logic, when data layers are assembled, it is done with the opinion of an expert. Expert input helps ensure the appropriate use of data, but the advantages of unbiased statistical weighting are lost. Bedell (2000) recently presented an unbiased method of combining fuzzy membership functions from diverse evidence layers using geothermal systems as an example. His weighting scheme is based on comparing the cumulative frequency distribution of a data layer to the data layer intersected with training points (mineral deposits or occurrences). An F-Factor was used for normalizing areas under the cumulative frequency curves to compare the original data with that intersecting the known geothermal systems. This method minimizes expert bias, but the weights are not derived in a formal probabilistic context. Few examples are available where researchers have merged the statistical rigor of WofE with the gradational weighting schemes afforded by fuzzy logic. One example is provided by Cheng and Agterberg (1999), who proposed a "fuzzy weights of evidence" method in which membership functions are scaled to cumulative contrast. In this paper, we present a "density function" method that is similar to the approach of Cheng and Agterberg (1999) in that it provides a multiple-class weighting scheme for evidence layers. This new technique emphasizes simplicity of calculation and interpretation, and although the method strictly speaking belongs to "fuzzy logic", it's predictions track closely with, and correlate strongly with, WofE. The suggested methods are simple and therefore can readily be accepted by group decision. They can be easily implemented on any GIS platform without program development. The use of multiclass weighting helps maintain the integrity and detail of individual data distributions in the model. The objective of this paper is to show that a simple model using the ratios of normalized geothermal systems over normalized areas (a density factor) provides a robust method integrating most of the benefits of WofE and all of the benefits of fuzzy logic. GEOTHERMAL GIS A GIS of active geothermal systems in Nevada, USA, recently constructed by Coolbaugh et al. (2002), was used to evaluate the density function method and compare it to binary WofE. The use of the geothermal case study provided an opportunity to explore multiclass weighting techniques for evidence layers using a diverse dataset whose predictive potential had already been explored using other techniques. Electric power-producing geothermal systems in Nevada are unlike most others in the world because they are not closely associated with young silicic volcanism (Koenig and McNitt, 1983). Instead, geothermal systems in Nevada (termed "extensional-type" geothermal systems by Wisian et al. (1999)) are associated with areas of high crustal heat flow and active extensional faulting (Koenig and McNitt, 1983; Wisian et al., 1999). Many disparate types of evidence can help signal the location of an "extensional-type" system, making this problem well suited for statistical analysis within a GIS. In this study, seven evidence layers were used. A brief description of each evidence layer follows; further discussion can be found in Coolbaugh et al. (2002). Geothermal training sites are also discussed below. Regional Heat Flux Geothermal systems are known to correlate with regions of high heat flux (Sass et al., 1971; Wisian et al., 1999). High heat flow brings more thermal energy close to the earth's surface where it can heat circulating meteoric fluids. A regional digital heat flow map was provided by David Blackwell of Southern Methodist University (SMU). Blackwell (1983) discusses some of the methods and considerations used to construct the heat flow map; that map and further discussions are posted at the SMU geothermal website located at Young Faults In Nevada and elsewhere in the world, geothermal systems are associated with areas of active faulting (Koenig and McNitt, 1983; Wisian et al., 1999; Bowen, 1989) because fault and fracture systems are the principal means by which meteoric fluids penetrate deeply into the crust. Also in Nevada, northeast-trending young faults are more closely associated with high-temperature geothermal activity than northwest-trending young faults, a relationship described earlier by Rowan and Wetlaufer (1981) and Koenig and McNitt (1983) and confirmed by Coolbaugh et al. (2002) using spatial statistics. This, in turn, has been linked to extensional crustal strain in Nevada (Blewitt et al., 2002; Coolbaugh et al., 2002), which for much of the state is directed northwesterly (as determined by a strain net measured by GPS stations (Bennett et al., 1998)). For the current analysis, a map of late Pleistocene and younger faults produced by Dohrenwend et al. (1996), and assembled into a digital database by Raines et al. (1996), was used. A buffered map showing the distance to the nearest northeast-trending young fault was created for input into the model. Depth to Water Table The depth to the water table can be considered as the geothermal equivalent of "outcrop" in mineral exploration. Hot springs are more likely to occur where the depth to the water table is shallow (Koenig and McNitt, 1983). Subsurface geothermal systems are more likely to be discovered in areas where hot springs are present at the surface. This is believed to be the reason why deep water tables negatively correlate with known geothermal systems in Nevada (Coolbaugh et al., 2002). Future exploration in areas with deeper water tables might locate more geothermal systems. A map of depth to the water table was generated using water well information from the United States Geological Survey (USGS) National Water Information System (NWIS) database ( gov/nv/nwis/gwlevels). Groundwater Geochemistry Geothermal fluids often contain high concentrations of certain metals, including boron, lithium, and arsenic (Ellis and Mahon, 1977; White et al., 1976; Ballantyne and Moore, 1988) compared to most groundwater. Anomalous concentrations of these metals in ground- 117

4 GIS FOR THE EARTH SCIENCES water can therefore be an indicator of geothermal activity. Boron concentrations in groundwater were found by Coolbaugh et al., (2002) to correlate best with geothermal activity in Nevada compared to other dissolved metals; consequently a map of groundwater boron concentrations (derived from the NWIS database) was used as a predictive (evidence) layer in the model. Young Volcanics Even though extensional geothermal systems in Nevada are not directly related to active volcanism, volcanic rocks 1.5 Ma or younger (mostly mafic in composition) are preferentially associated with geothermal activity in the state (Coolbaugh et al., 2002). Two possible explanations for this correlation are: 1) recent volcanism is restricted to areas of active crustal extension where hydrothermal fluids might circulate to greater depth, and 2) if sufficiently young, the volcanic rocks may indicate areas of high heat-flow at depth. An age-date database compiled by Mark Mihalasky and incorporated into the Great Basin Digital Database (Raines et al., 1996) was used to create a buffered map of the distance to the nearest young volcanics in the state. Earthquakes Earthquakes reveal areas of active faulting where pathways for deeply circulating hydrothermal fluids could be present. Maps of earthquake density were derived by adding all earthquake magnitudes greater than 4.0 within a radius of 40 km for each cell in the model. A threshold magnitude of 4.0 was selected because lower magnitude quakes may not be detected in all portions of the state as a result of limitations in the distribution of observatories (Diane DePolo, personal communication, 2001). Earthquake data were taken from catalogues maintained by the Nevada Seismological Laboratory at the website catalog/. Paleozoic Carbonates In Nevada, deep aquifers occur in thick sequences of Paleozoic carbonate rocks in the eastern third of the state (Harrill and Prudic, 1998). Relatively few geothermal systems having high temperatures are known in areas underlain by the carbonate rocks; it is believed that the deep aquifers trap and entrain rising thermal fluids, preventing them from reaching the surface where they could be observed (Sass et al., 1971). An interpretive map of Paleozoic carbonate rocks, partially projected beneath Cenozoic cover (Ludington et al., 1996), was obtained in digital format from The Nevada Bureau of Mines and Geology. Training Sites Fifty-nine geothermal systems and hot springs in Nevada comprise the training set used for modelling. These sites represent all known occurrences where geothermal fluids have been measured, or are estimated to have reached, a temperature of 100 C or hotter. Temperatures were either measured directly or estimated using geothermometer calculations based on fluid chemical compositions (Mariner et al., 1982; Coolbaugh et al., 2002). Sources of data used to assemble the set of training points include Garside (1994), Shevenell et al. (2000), and a western U.S. geothermal database maintained at SMU (Richards and Blackwell, 2002; smu.edu/geothermal/). WEIGHTS-OF-EVIDENCE MODEL A binary WofE model was constructed using the same training sites and evidence layers used to build the density function model. ArcView 3.2a software was used for modelling computations, and Arc-SDM, an ArcView extension, was used for WofE calculations. Arc-SDM is a spatial data-modelling package developed by the Geological Survey of Canada (GSC) and the United States Geological Survey (Kemp et al., 2001). Weights-of-evidence analysis is based on Bayes' Theorem, which assumes conditional independence between evidence maps. The application of WofE analysis to mineral deposit modelling has been described previously (Bonham-Carter et al., 1988; Bonham- Carter, 1996, ch. 9; Wright, 1996) and is not discussed further here. Raines et al. (2000) and Raines and Bonham-Carter (this volume) provide a concise description of the application of WofE analysis to mineral deposit exploration using ArcView software. For each of the seven evidence layers in the model, a binary map was produced indicating areas favourable and unfavourable for the occurrence of geothermal systems. The optimal binary pattern for each map was determined by either maximizing the WofE cumulative contrast statistic (C), or by maximizing the Studentized contrast statistic (SC, see below). The contrast is defined as the absolute difference between positive (W + ) and negative (W - ) weights of evidence (Wright, 1996, p. 110; Bonham-Carter, 1996, p ): C = W + - W - (1) The Studentized contrast is a measure of confidence, and is defined as the ratio of the contrast divided by its standard deviation (Bonham-Carter, 1996, p. 323): SC = C / S contrast = (W + - W - ) / {sqrt[s 2 (W + ) + s 2 (W - )]} (2) For evidence layers with a maximum contrast greater than 2, the maximum contrast was used to determine the binary pattern. For evidence layers where the maximum contrast was less than 2, the maximum Studentized contrast was used instead, to help insure statistical significance. Positive weights, negative weights, contrast, and Studentized contrasts for each of the seven evidence layers are listed in Table 1. The unit area assigned to each geothermal system was 9 km 2, which corresponds to the approximate average lateral dimensions of a geothermal system, based on the distribution of drillholes in wellexplored systems. The study area encompasses the entire state of Nevada in the United States. A favourability map of known geothermal systems in Nevada (Figure 2a) was made by combining the statistically-derived weights of the seven evidence layers using a posterior logit equation (Bonham-Carter, 1996, equation 9-33). The posterior logit was chosen for plotting "favourability" instead of the more customary posterior probability, because the former yielded superior amounts of colour-scaled detail on the map, and was the only representation of 118

5 SPECIAL PUBLICATION 44 Table 1. Statistics for the seven evidence layers used to build the binary weights of evidence model. The columns titled Additional Classes list the number of additional weights potentially justified for each layer, using the arguments presented in the section on Statistical Significance Standard Additional Additional Positive Negative Student Deviation Classes Classes Evidence Layer Weight Weight Contrast Contrast of Contrast (z score = 2) (z score = 1) Young Volcanics Water Table Depth Regional Heat Flux Paleozoic Carbonates NE-trending Young Faults Boron in Groundwater Earthquakes Sum: Figure 2a. Favourability map for geothermal systems in Nevada, based on the posterior logit of binary weights of evidence. Higher favourabilities in northwestern Nevada are due in part to higher heat flow, presence of northeast-trending young faults, and a lack of deep carbonate aquifers, compared to the remainder of the state. The colour version of this figure shows background topographic shading. Figure 2b. Favourability map of geothermal systems in Nevada based on the density function. Because a multiclass instead of binary weighting scheme was employed, more gradations in favourability ranking are available compared to binary weights of evidence (Figure 2a). This helps define the Humboldt structural zone (Figure 2a). The colour version of this figure shows the variations in favourability in more detail. The colour version also features background topographic shading. 119

6 GIS FOR THE EARTH SCIENCES weights that could compete with the detailed colour-rankings provided by the density function method discussed below. A possible explanation is provided by the fact that posterior probabilities in this study approximate a log-normal distribution, and plotting on a log-scale (such as that afforded by the posterior logit equation) should yield a better spread of rankings. This is crudely analogous to plotting log-normal distributed geochemical data using contour intervals that progress logarithmically instead of arithmetically. An overall conditional independence (C.I.) index of 0.76, based on the method of Bonham-Carter (Bonham-Carter, 1996, p. 315), indicates some dependencies among the data and suggests the total number of geothermal systems is over-predicted by roughly 30%. It is for this reason that the posterior probability map (Figure 2a) was labelled a "favourability map" instead of a "posterior probability" map. Since the primary objective of this paper was to compare modelling methods, conditional dependency issues were not considered further because they will affect both the WofE model and the density function model (discussed below) similarly. DENSITY FUNCTION MODEL The density function (DF) measures the "density" or "frequency" of training sites (known geothermal systems) within a given map pattern, and is defined as the fraction of training sites falling in a given map pattern divided by the fraction of the study area covered by that same pattern: DF i,j = [ (N i /N t ) / (A i /A t ) ] j (3) where DF i,j = the density function value for the ith pattern on evidence map j, N i = number of deposits or geothermal systems associated with pattern i, N t = total number of deposits, A i = area of the pattern i (on the evidence map j), and A t = total area of the study. If a multiple number of mutually exclusive i patterns are defined for a given evidence layer j, each pattern may have a different density of training points. Defined in this manner, the area-weighted average value of the density function (DF) for a given evidence layer over an entire study area will always equal 1. Values of DF greater than 1 indicate a favourable or positive association between a pattern and the training sites, while values less than one indicate an unfavourable or negative association. If portions of the study area are lacking evidence (no information), a neutral value of 1 can be assigned. When expressed in log form, values for the density function are usually very similar to a weight of evidence. In fact, when the total area of training sites is small relative to the study area and small relative to the area of a given map pattern, the similarity between DF (in log form) and the weight of evidence can be demonstrated mathematically. In WofE analysis (using the nomenclature of Bonham-Carter et al. (1988)), a positive weight can be defined as follows: W j + = ln [ P(J D) / P(J 9 D) ] (4) where W j + equals the positive weight of evidence, P(J D) equals the probability of map pattern j being present given a training site is present, and P(J 9 D) equals the probability of map pattern j being present given a training site is not present. When weights are defined for multiple classes on an evidence layer, the equation for a positive weight (4) can be generalized so it represents all weights for an evidence layer, regardless of whether they are positive or negative: W i,j = ln [ P(J i D) / P(J i 9 D) ] (5) where in this case W i,j equals the weight (which could be either positive or negative) for the "i"th pattern for evidence map j, P(J i D) equals the probability of given map pattern J i being present given a training site is present, and P(J i 9 D) equals the probability of given map pattern J i being present, given a training site is not present. For the numerator of equation (5), using the definition of P(J i D) it can be shown that: P(J i D) = (N i /N t ) j (6) For the denominator, following the terminology of Bonham-Carter et al. (1988, p. 1588): P(J i 9 D) = [(A i! A di ) / (A t! A dt )] j (7) where A di = the total area of deposits in pattern i and A dt = the total area of deposits in the study area. When the area of a deposit is very small, both A di and A dt are likely to be very small relative to A i (the total area of the pattern i) and A t (the total study area). In that case: P(J i 9 D) = [(A i! A di ) / (A t! A dt )] j = ~ (A i / A t ) j (8) With substitution of equations (6) and (8), equation (5) becomes: W i,j = ~ ln [ (N i/n t ) / (A i /A t ) ] j = ~ ln (DF i,j) (9) It is important to emphasize the simplicity of this method, given the basic assumption. If the total spatial area of the training sites is small (i.e., the area of active geothermal systems relative to the entire State of Nevada or relative to a given map pattern), then the log-transformed density function can serve as a simplified proxy for weights of evidence. Smoothed Density Function For those layers in the geothermal model represented by real numbers (i.e., ratio or non-categorical data), DF values were found to change systematically over the range of data values, supporting the argument for a multiple, instead of binary, weighting scheme. As an example, Figure 3 shows how DF increases systematically with increasing heat flux. The higher the regional heat flux at a given location in Nevada, the more likely a geothermal system (with a temperature $100 C) will occur. This relationship was previously documented by Wisian et al. (1999) using a similar density calculation, though they did not integrate the information with other evidence layers in a GIS or produce predictive maps. Figure 3 was produced by reclassifying a real-number, interpolated grid map of heat flux values into a map with a finite number of categories of progressively increasing heat flux values. Each category can be considered as a histogram bin with a corresponding unique map pattern, for which a DF value can be calculated. Note that this is a categorical graph, not a cumulative graph. Also worthy of note is that for each of the evidence layers, WofE and density function statistics proved to be nearly identical, as shown in Figure 4 for the case of heat flux. 120

7 SPECIAL PUBLICATION 44 Figure 3. Density function plot for regional heat flux. Raw density function (DF) values were calculated directly from the data using equation (3). Heat flux values were subdivided into 21 histograms, each with an equal spacing of 2.87 heat flux units (mwm -2 ): the mid-point of each bin was used to plot DF values. A smoothed curve was fitted to the raw DF data, and smoothed DF values were recalculated for each histogram bin. Some histogram bins at the low end of the heat flux scale were grouped together, so that 16 different smoothed DF values were used for modelling. The fraction of the total area covered by each histogram pattern is also indicated. Figure 4. Log-transformed raw DF values compared to multiclass weights of evidence for the regional heat flux evidence layer. At the scale of the plot, density function and weights-of-evidence statistics are virtually identical (irresolvable). 121

8 GIS FOR THE EARTH SCIENCES Individual DF values on Figure 3 appear erratic. This "noise" is caused by use of a large number of bins (21 total), causing a small number of training points to fall into some categories. Measurement noise could be reduced by using coarser bin spacings, but at a sacrifice of resolution at the high flux end of the scale. Alternatively, a smooth curve could be fitted to the data using algorithms and/or expert guidance, and in this paper for simplicity we used "expert guidance", as shown in the heat flux example (Figure 3), where interpretation of a steadily increasing DF is straightforward. The method presented here is not the only one available, and other methods, such as linear regression, or bin size equalization using the number of training points (J. Harris, personal communication, 2003) could yield equivalent results, and more research in this area is warranted. In any case, a proper balance between bin size and curve fitting should yield optimal results. Raw and smoothed DF values were calculated for the five other real-number-based evidence layers (Figures 5 to 9) used in the modelling. In each case, DF values progress from low to high values as the correlative evidence either decreases, as in the case of the depth to the water table (Figure 5), buffer distance to northeasttrending faults (Figure 6), and buffer distance to young volcanics (Figure 7), or increases, as in the case of the concentration of boron in groundwater (Figure 8) and the sum of earthquake magnitudes (Figure 9). A potential difficulty can arise in assigning a DF value when no training sites fall within a given histogram bin or group of bins, as is the case in Figure 3 where heat flux is less than 67 mwm -2. In these situations calculation of WofE statistics is not possible. We have dealt with this issue in several ways. In some cases (Figures 3 and 5) we have chosen to arbitrarily assign "1/2" training site to the group of histogram bins less than 67 mwm -2 and calculate the resulting density factor (Figure 3). This procedure is grossly analogous to a practice in geochemistry, where one may assign a value of one-half the detection limit to elements not detected in analysis (Amor et al., 1998). In other cases, we have expanded the histogram bin size to include neighboring training points (Figures 6 and 8) or used curve smoothing techniques (Figures 7 and 9). These methods were chosen for their simplicity and expediency, but other methods of dealing with missing data could be developed. Logarithmic Addition The fact that log-transformed DF values are approximations of weights of evidence suggests that the posterior logit equation (adapted from Equation (10); Bonham-Carter, 1996, equation 9-33; Wright, 1996, equation 5-9) can be used as a statistically-based guide for combining DF values from each evidence layer into a single predictive model: Posterior logit = prior logit (D) + E j (W i,j ) (10) An equivalent expression (without the prior logit) for adding density functions (DFs) is as follows, where LnDF T = final predictive value for the DF model: LnDF T = E j [ ln (DF i,j ) ] (11) Figure 5. Density function plot for depth to water table. A crude logarithmic relationship between water table depth and the density function underscores the impact the water table has on influencing the discovery of known geothermal systems. A total of 20 different smoothed density function values were used to represent this layer. 122

9 SPECIAL PUBLICATION 44 Figure 6. Density function plot for young northeast-trending faults. Raw DF values become more erratic or noisy when individual histogram bins contain a small percentage of the total area, as is the case when the distance to a fault is short. In these instances, more interpretation of a smoothed DF is necessary: that interpretation can rely on the overall relationships depicted by the graph. Six different DF values were used to model the favourability of the fault layer. Figure 7. Density function plot for young volcanics. Three separate regions of the graph can be distinguished. Within 10 km of a young volcanic vent, high DF values indicate a strong correlation between young volcanism and geothermal activity (to be expected). Between 10 and 90 km, young volcanics define broader regional zones of moderate geothermal favourability. Beyond 90 km, young volcanics do not appear to have a positive influence on geothermal activity. Eight smoothed DF values were used for modelling. 123

10 GIS FOR THE EARTH SCIENCES Figure 8. Density function plot for boron in groundwater. Similar to the young volcanic layer, three separate regions of the graph can be distinguished: strong favourability where boron concentrations are greater than 10 ppm, moderate favourability where concentrations are greater than 200 ppb, and no positive influence when boron concentrations are less than 100 ppb. More smoothing of the DF is required when the histogram bins have small map areas, as is the case on both extremes of the graph. The boron DF curve was represented with six different smoothed values. Figure 9. DF plot for earthquakes. Because of statistical noise, only one DF value was used to represent earthquake favourability when the earthquake Richter magnitude sum exceeded 15 (log earthquake sum $1.2). The relationship between geothermal activity and earthquakes is somewhat unusual and there is evidence that the strongest earthquakes may not correlate as well with geothermal activity as moderate earthquakes. This is because larger earthquakes tend to occur in regions of compressive stress whereas moderate earthquakes are more likely to occur in extensional environments that favour the formation of geothermal systems. Therefore, an equally valid "expert" interpretation of smoothed DF values would be to decrease DF values at the high end (left side) of the earthquake magnitude sum scale. 124

11 SPECIAL PUBLICATION 44 LnDF T is simply the sum of the log-transformed DF factors for each evidence layer. Because the smoothed DF values, as presented herein, are fuzzy transforms and not strictly representative of probability, it was not considered necessary to add an equivalent expression for the prior probability. The prior probability is constant over the entire study area, and consequently its exclusion in the formulae will not affect the ability to discriminate favourable from unfavourable areas on a final predictive map. This keeps equation (11) simple, and avoids the necessity of assigning a cell size to training sites (which is required in WofE analysis and is often estimated in a qualitative and somewhat arbitrary manner). If desired, LnDF T could be expressed in exponential form, just as posterior probability and posterior odds represent exponential forms of the posterior logit in WofE analysis (Bonham-Carter, 1994; Wright, 1996). However, in the geothermal case study the final predictive variable (posterior probability for WofE) has a highly skewed distribution, spanning multiple orders of magnitude, and is more easily displayed on maps and graphs in log-space. It was considered advantageous to maintain LnDF T in log form, analogous to the posterior logit in WofE. STATISTICAL SIGNIFICANCE A question arises as to whether or not multiclass weightings with DF provide a statistically significant improvement in weights classification compared to binary WofE. In other words, does a multiclass weighting function provide additional useful information, or does it only add noise to a predictive model? A partial, qualitative answer can be obtained by examining the DF graphs themselves (Figures 3 and 5 to 9). A progressive gradational increase or decrease in the DF suggests that the data could support more than a binary classification. But a somewhat more quantitative assessment can be made by examination of the Studentized contrast statistic defined in equation (2), whose form is similar to the expression used to test for significant difference between two means (Miller and Freund, 1965, p. 166). In our case, the two means correspond to the positive and negative binary weights from WofE, and the SC can be used to approximate a z-score for testing significance. Used in this manner, if the z-score does not meet the required level of confidence, binary weighting is not statistically justified and the evidence layer is not useful. This is a qualitative test only, because not all the conditions necessary for formal statistical testing are met (Bonham-Carter, 1994, p. 323). In any case, if the z-score is much greater than the minimum value for confidence in a binary layer, a ternary or higher weighting scheme may be justified. A general formulae for estimating the number of additional weights (beyond two) could be written as follows: N = ( SC / z )! 1 (12) where N = the number of additional weights (beyond two) justified, and z = the required level of confidence, expressed as a z-score (user determined). For this approach to work best, the Studentized contrast should represent the maximum Studentized contrast definable from the data. For evidence layers derived from real-number grids, the maximum studentized contrast can be estimated using cumulative studentized contrast curves. Table 1 lists the number of potential additional weights (beyond a binary layer) for each of the six real-number-based evidence layers in the geothermal model using the above criteria. Using a z-score of 2, a total of 4.5 additional classifications could be justified: that number increases to 16 if a z-score of 1 is chosen. Increasing the number of weights by 4 or 16 may not sound like much, but it has a drastic effect on the number of possible outcomes on the predictive map. With binary WofE and seven evidence layers, a total of 128 (2 7 ) predictive levels (number of entries in a unique conditions table) are possible. With 4.5 additional weights, the number of unique conditions jumps to 796 and with 16 additional weights, the number increases to nearly 22,000. This equates to a finer resolution of favourability in the final predictive model. The actual number of weights used in the fuzzy DF model was much greater than the minimum number statistically justified with the argument above, and ranged from five weights (earthquake layer) to 20 weights (groundwater layer). (For the Paleozoic carbonate rock layer, which is based on a categorical geologic map, binary weighting was used.) The concept was to use a gradational scale of weighting for each evidence map, while realizing that any point on the DF curve has an intrinsic error associated with it. This ensures that the full resolution of data is available for predicting geothermal systems on the final map, and that the ability to resolve, or vector towards, anomalies is maximized. At the same time, when an anomaly on the final map is investigated in detail, it is important to be aware of the statistical significance each anomaly has. There are many factors affecting the statistical significance of anomalies on the final map (and on most other types of predictive maps), most of which are not taken into account with the measures of uncertainty available in WofE. These include original assay or measurement errors, data entry errors, insufficient data, interpolation errors, and resolution and registration problems. Because of this error multiplicity, the significance of individual anomalies on a final predictive map can best be assessed by examining each anomaly, identifying it's component parts (i.e., determining which evidence layers are contributing to an anomaly), and considering the reliability and quality of the data and interpretations leading to anomaly definition. There is no substitute for familiarity with the data and its relevance to (in this case) geothermal systems. A GIS is well suited for such a qualitative review because all the individual data layers are co-registered and many of their attributes are readily accessible. DISCUSSION OF RESULTS A predictive map of geothermal systems using the DF method (LnDF T -- Equation (11)) is shown in Figure 2b. As would be expected, the DF map appears to show a more detailed favourability ranking than the WofE map (Figure 2a) (the reader is encouraged to inspect coloured versions of these figures on the CD that accompanies this volume). Both figures define a northeast-trending zone of high geothermal favourability following the Humboldt structural zone (Figure 2a), a region in central and northern Nevada characterized by northeast-trending Quaternary faults, geomorphic lineaments, anomalous heat flow, and associated high-temperature geothermal systems (Rowan and Wetlaufer, 1981; Sass et al., 1980). Although the Humboldt structural zone has previously been recognized from lineament and geothermal studies (see references above), the new predictive maps nevertheless outline the favoura- 125

12 GIS FOR THE EARTH SCIENCES bility of that zone better than any previously published map, whether it be a map of heat flow, young faults, lineaments, or geothermal occurrences. Because of it's finer resolution of favourability, the LnDF T map (Figure 2b), appears to outline this zone with greater clarity than the binary WofE map (Figure 2a). The overall predictive capabilities of the DF and binary WofE maps are remarkably similar. Both methods produce very similar cumulative area-training site rankings (Figure 10) and both methods predict their original 59 geothermal training points equally well (Figure 10). The Pearson's correlation coefficient between the two maps is 0.91 (r 2 = 0.83). Also interesting is the fact that both methods do an even better job ranking an inclusive subset of 10 geothermal systems currently producing electric power in Nevada (Figure 10). Apparently the most economic (and generally the highest temperature) geothermal systems result from the superposition of a multitude of favourable factors, while a less-restrictive class of geothermal systems with maximum temperatures $100 C (but not necessarily producing electrical energy) occur in broader zones where some, but not all, favourable factors may be present. Differences between the DF and WofE maps do exist and tend to be greatest in the intermediate ranges, similar to observations of other predictive techniques (Knox-Robinson, 2000); these differences are illustrated in Figure 11, which is a map created by subtraction of the WofE grid from the DF grid. These differences are apparent along the Humboldt lineament (Figure 2a), where higher intermediate rankings of the density function serve to outline the lineament better and enhance vectoring toward favourable regions (this is seen best on the colour version of Figure 11). Another distinctive feature of the difference map, when viewed in detail (Figure 12), is the presence of multiple rings or bulls-eyes surrounding areas of high and low favourability. This ringing is a natural result of the finer subdivisions of favourability ranking in the DF map relative to the binary WofE map. CONCLUSIONS Similarities between the DF and WofE maps should suggest to managers and explorationists that WofE is an effective technique in its own right. Binary WofE predicts geothermal systems well in spite of converting evidence layers into yes/no, favourable/unfavourable images (although the binary nature of the WofE maps may have been more evident if fewer than seven evidence maps were used). The WofE method also has the advantages of mathematical rigour and providing measures of uncertainty. However, similarities in the two maps also help support and justify the DF approach, which mimics the statistically rigorous WofE method well even though Figure 10. Cumulative area-classification curves compare the prediction of binary weights of evidence to the density function, based on the ability to classify training points and geothermal power plants. Curves displaced furthest to the right on the graph indicate better predictive characteristics, because fewer geothermal systems would be found in portions of the state with lower favourability rankings. Binary weights of evidence and the density function appear to predict equally well, based on their classification of 59 training sites and a subset of 10 geothermal systems producing electrical energy. However, for both methods, power-producing geothermal systems are predicted better than are the set of geothermal training points. This suggests that not all training points are created equal, and that power-producing geothermal systems (generally the highest temperature systems) are more dependent on the conjunction of favourable geologic conditions. 126

13 SPECIAL PUBLICATION 44 Figure 11. Difference between the density function (LnDF T ) and weights-of-evidence (posterior logit) favourability maps. The two component maps were rescaled to equal ranges prior to subtraction. Red colours indicate where density function values are higher, blue colours indicate where weights-of-evidence values are higher. Greater differences tend to occur in the intermediate value ranges. Topographic shading is present in the background. 127

14 GIS FOR THE EARTH SCIENCES Figure 12. Difference between the density function and weights-of-evidence maps in west-central Nevada. A pronounced ringing or bullseye effect surrounds individual areas of higher favourability (Figures 2a, 2b) and is caused by the finer subdivision of favourability rankings in the density function relative to binary weights of evidence. Topographic shading is present in the background. "fuzzy interpolation" is used to interpret smoothed gradational density functions. The advantage of the density method is threefold: it is simple, relatively unbiased, and provides a high density of information for each evidence layer. It provides a practical alternative in predictive modelling by stepping carefully outside a purely probabilistic framework. In addition, DF modelling can be done with two simple ratios in a spreadsheet. Such simplicity is of paramount importance in many work environments. The multiclass weighting scheme of DF utilizes the entire data distribution (divided into multiple classes) for each evidence layer; this in turn gives confidence to data experts that the modelling method takes full advantage of the available information. The weighting methods employed are simple and understandable, which can make it easier for managers and groups to trust and rely on model predictions and reach a consensus on how to act on the results. The more detailed favourability rankings afforded by the 128

15 SPECIAL PUBLICATION 44 DF method can make it easier to "vector in" and identify anomalies of potential interest. Not every anomaly may be statistically significant, but a GIS-based review of the causative sources of each anomaly, conducted quickly and efficiently using the cross-referenced and co-located data available on computer, can help determine which anomalies are worthy of follow-up examination. The mathematics used to calculate densities and combine evidence layers remove much of the potential "expert" bias that characterizes fuzzy logic methods. Density function weights are similar to weights calculated from WofE; in fact, when training site areas are small relative to the areas of evidence patterns and the total study area, as is often the case in mineral exploration programs, the density function calculation represents a simplification of the WofE formulae. Similar to WofE, when some areas of an evidence map do not contain information, a neutral weighting value can be assigned. The density method described herein differs from multiclass WofE because the raw density values calculated for individual histogram bins are smoothed by drawing lines or curves linking adjacent densities on graphs. This is considered necessary because the subdivision of an evidence layer into multiple histogram bins (multiple patterns) increases the error or uncertainty associated with the density calculation, especially when the number of training sites per bin drops to a low number. A similar difficulty affects the accuracy of weight calculations for multiclass WofE. Smoothed density values offer a method of minimizing these errors for real-number based grids (ratio data), by relying on information from adjacent histogram bins. In this study, we have used "expert guidance" to fit curves to multiclass data, but other more statistically based methods could be developed and employed, such as linear regression. ACKNOWLEDGMENTS This manuscript has benefited from numerous insightful suggestions and comments from Gary Raines of the USGS. Jim Taranik, as director of the Great Basin Center for Geothermal Energy, provided key support and encouragement necessary to pursue this study. Mark Mihalasky (USGS) supplied the initial geothermal GIS and databases along with his enthusiastic approval for expanding the scope of work, and Lisa Shevenell of the Nevada Bureau of Mines and Geology made important contributions of geothermal databases and interpretations of geothermometer calculations. Don Sawatzky (Great Basin Center for Geothermal Energy) provided advice on structural databases and developed LinAnl, an ArcView extension for statistical analysis of fault strikes used in this study. Suggestions and comments from several anonymous reviewers and Jeff Harris significantly improved the flow, presentation, and rigour of the text, for which we are grateful. 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