multifractal modeling of geochemical data: A review

Journal of Geochemical Exploration 164 (2016) 33–41

Contents lists available at ScienceDirect

Journal of Geochemical Exploration journal homepage: www.elsevier.com/locate/jgeoexp

Fractal/multifractal modeling of geochemical data: A review Renguang Zuo ⁎, Jian Wang State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 27 February 2015 Revised 9 April 2015 Accepted 21 April 2015 Available online 1 May 2015 Keywords: Fractal/multifractal Concentration–area fractal model Spectrum–area multifractal model Concentration–distance fractal model Concentration–volume fractal model Singularity

a b s t r a c t Over the past several decades, a wide range of complex structures or phenomena of interest to geologists and geochemists has been quantitatively characterized using fractal/multifractal theory and models. With respect to the application of fractal/multifractal models to geochemical data, the focus has been on how to decompose geochemical populations or quantify the spatial distribution of geochemical data. A variety of fractal/multifractal models for this purpose have been proposed on the basis of the scaling characteristics of geochemical data. These include the concentration–area (C-A) fractal model, concentration–distance (C-D) fractal model, spectrum–area (S-A) multifractal model, multifractal singularity analysis, and the concentration–volume (C-V) fractal model. These fractal models have been widely demonstrated to be useful, as indicated by the increasing number of published papers. In this study, fractal/multifractal modeling of geochemical data including its theory, the way it works, its benefits and limitations, its applications, and the relationships between these models are reviewed. The comparison among of C-A, S-A, and multifractal singularity analysis based on simulated data suggested that mapping singularity technique can enhance and identify weak anomalies caused by buried sources. Future study should focus on how to distinguish the true anomalies associated to mineralization with the false anomalies from a fractal/multifractal perspective. © 2015 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractal/multifractal models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Number–size model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Concentration–area fractal model . . . . . . . . . . . . . . . . . . . . . 2.3. Spectrum–area fractal mode . . . . . . . . . . . . . . . . . . . . . . . 2.4. Concentration–distance fractal model . . . . . . . . . . . . . . . . . . . 2.5. Concentration–volume fractal model . . . . . . . . . . . . . . . . . . . 2.6. Local singularity analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Other parameters related to fractal methods . . . . . . . . . . . . . . . . 3. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Identifying geochemical anomalies . . . . . . . . . . . . . . . . . . . . 3.2. Mapping mineral prospectivity . . . . . . . . . . . . . . . . . . . . . . 3.3. Characterization the vertical distribution of geochemical element concentration 4. Comparison the C-A, S-A and singularity index . . . . . . . . . . . . . . . . . . 5. Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Decomposition of geochemical patterns is a basic task for applied geochemists. With environmental problems becoming increasingly ⁎ Corresponding author. E-mail address: [email protected] (R. Zuo).

http://dx.doi.org/10.1016/j.gexplo.2015.04.010 0375-6742/© 2015 Elsevier B.V. All rights reserved.

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

33 34 35 35 35 35 36 36 36 36 36 37 38 38 39 39 39

important in recent years, discrimination between anthropogenic pollution and natural background is assuming increasing relevance (Albanese et al., 2007; Darnley et al., 1995; Lima et al., 2003, 2005, 2008; Plant et al., 2001). Similarly, how to effectively detect geochemical anomalies from background is one of the major concerns of geochemical exploration, which continues to be a cornerstone to mineral exploration at all scales ranging from regional reconnaissance to local

34

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

exploration (Cohen et al., 2010; Grunsky, 2010). Anomaly patterns, as the end product of either common geological processes over long periods of time or uncommon processes such as ore-forming processes, weathering, human activities and element dispersion from an orebody, are defined simply as geochemical features different from those that usually occur more frequently. These differences consist not only of the frequency and spatial distribution of geochemical data, both of which have been investigated widely in the past several decades (e.g., Agterberg, 2007; Ahrens, 1954; Carranza, 2009; Krige, 1966; Reimann and Filzmoser, 2000; Turcotte, 1986, 1997), but also the geometrical characteristics and scale invariance of geochemical patterns (Afzal et al., 2010, 2011, 2012, 2013a; Agterberg, 2012a,b; Cheng et al., 1994, 1996, 1997, 1999, 2000; Li et al., 2003; Lima et al., 2003; Xu and Cheng, 2001; Zuo et al., 2015). It has been shown that ore elements, especially trace elements, do not follow a normal or lognormal distribution, but instead follow a positively skewed distribution with a long Pareto tail toward high values (e.g., Ahrens, 1957). With respect to the spatial distribution of geochemical data, autocorrelation often exists over a certain spatial range. The geometry of geochemical anomalies, also an important aspect, indicates geological structures. For example, linear anomalies may be associated with underlying faults, while arcuate anomalies may imply intrusive bodies (Cheng et al., 1999). Recent studies of geochemical patterns at different scales have shown that self-similarity or self-affinity are fundamental properties of geochemical data (e.g., Bölviken et al., 1992; Cheng et al., 1994; Zuo et al., 2009a,b). The most effective way to distinguish geochemical anomalies from the background is to adopt a comprehensive technique that combines the properties mentioned above. The typical and most widely used method for detection of geochemical anomalies is setting threshold values, which contain the upper and lower limits of background variations (Hawkes and Webb, 1962). Observations outside of this range are referred to as anomalies, whereas those within background are not. However, traditional methods, including the one mentioned above, exploratory data analysis (Behrens, 1997; Carranza, 2010; Reimann, 2005a,b; Tukey, 1977), and multivariate statistics (Yousefi et al., 2012, 2014; Zuo, 2011a,b; Zuo et al., 2009a,b, 2013), are based on the frequency distribution of geochemical values and, therefore, neglect spatial variation and other potential characteristics that can provide valuable information. Considering the fact that exploration geochemical data are typically spatially dependent, a couple of frequency-space-based methods, such as the inverse distance-weighted

(IDW) and different kriging methods, have been put forward (Krige, 1978; Lam, 1983; Zimmerman et al., 1999). Although these methods acknowledge the spatial dependence of element concentrations, they do not consider that spatial variability is rugged and singular rather than smooth and differentiable. The main attraction of fractal/multifractal theory lies in its ability to quantify irregular and complex phenomena or processes that exhibit similarity over a wide range of scales, which is termed self-similarity (Feder, 1988; Mandelbrot, 1983). Since the concept of fractal was introduced by Mandelbrot in the 1960s, a number of studies were applied to geological processes and phenomena to characterize the spatial distributions of concentrations and the relationship between tonnage and grade of deposits (e.g., Cheng et al., 1994, 2000; Lavallee et al., 1993; Mandelbrot, 1983; Turcotte, 1986, 1997, 2002). With respect to applied geochemistry, various researches are being implemented on the fractal properties of geochemical patterns over different scales, as indicated by the increasing number of published papers (Fig. 1). For example, the number of papers published in the Journal of Geochemical Exploration (JGE) at the five-year scale has increased nearly exponentially as the total number of papers in JGE, Applied Geochemistry (AG), and Geochemistry: Exploration, Environment, Analysis (GEEA). These numbers indicate that fractal/multifractal models have an important role, especially in applications to geochemical exploration. In detail, three important achievements have been made over the past thirty years. The first one is the proposal of a series of fractal/multifractal models used for separating geochemical anomalies from background or for determining baseline concentration in environmental studies. The second is the introduction of the concept of singularity, which enables us to study mineralization from a new nonlinear perspective and provides an effective tool for mapping local and weak anomalies, and the third is the ability to quantify the vertical distribution of geochemical elements using fractal methods. Based on previously published researches, this paper provides an overview of fractal/multifractal modeling of geochemical data, including its theory, the way it works, its benefits and limitations, its applications, and the relationships among various models. 2. Fractal/multifractal models The study of Bölviken et al. (1992) was early to address the importance of fractal models used for geochemical landscape studies and to

Fig. 1. Histogram of the number of papers related with fractal/multifractal modeling of geochemical data published in Journal of Geochemical Exploration (JGE), Applied Geochemistry (AG) and Geochemistry: Exploration, Environment, Analysis (GEEA) during 1991–2014.

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

predict their profound impact on geochemical exploration. Cheng et al. (1994) proposed the concentration-area fractal model (C-A), which represents the first important advancement in fractal/multifractal modeling of geochemical data (Zuo et al., 2012) and is a fundamental technique used frequently for modeling geochemical anomalies (Carranza, 2009). The spectrum-area fractal model (S-A), introduced by Cheng et al. (1999) as a version of the C-A model in the frequency domain, can separate overlapping populations using more than one cutoff value. Based on Mandelbrot’s radial-density law, by replacing density with concentration, Li et al. (2003) proposed the concentration–distance (C-D) fractal model, which is used for discriminating geochemical anomalies from background. Carranza (2009) reviewed the fractal/multifractal methods used to model geochemical data in his book Geochemical Anomaly and Mineral Prospectivity Mapping in GIS and demonstrated the advantages of these models. Afzal et al. (2011) extended the C-A fractal model to 3D and developed the concentration–volume (C-V) fractal model to identify various zones of mineralization. These models have been acknowledged widely as powerful tools for identifying anomalies as well as for determining geochemical baseline in environmental studies (e.g., Afzal et al., 2010, 2011, 2013b; Albanese et al., 2007; Ali et al., 2007; Asadi et al., 2014; Cheng et al., 1994, 1997, 2000, 2010; Deng et al., 2010; Gonçalves et al., 2001; He et al., 2013; Heidari et al., 2013; Jesus et al., 2013; Li et al., 2003; Luz et al., 2014; Panahi et al., 2004; Sadeghi et al., 2012; Sun and Liu, 2014; Wang, G. et al., 2013; Xu and Cheng, 2001; Zuo, 2011a,b, 2014). As for the concept of singularity, Cheng (2007) considered mineralization to be a singular process due to an efficient process of element enrichment. A singularity mapping technique was proposed to characterize the degree of uniqueness of geological features, and it can detect potential targets that are often smoothed by traditional contouring methods (e.g., Arias et al., 2012; Bai et al., 2010; Chen et al., 2015; Cheng, 2007; Cheng and Agterberg, 2009; Liu et al., 2013; Sun et al., 2010; Xiao et al., 2012, 2014; Xie et al., 2007; Zhao et al., 2012, 2013, 2014; Zuo and Cheng, 2008; Zuo et al., 2009a, 2013, 2015). Considering the fact that different points within a small vicinity may have different singularity exponents, Chen et al. (2007) proposed a local singularity iteration algorithm as an improvement of parameter estimation involved in the local singularity analysis. Cheng (2012) proposed a density–area power-law model, attempting to systematically confirm that singularity analysis is effective for recognition of weak geochemical anomalies. Agterberg (2012a, 2012b) noticed that singularity exponents are linearly related to logarithmically transformed element concentration values, which can be used to measure the small-scale nugget effect caused by measurement error and microscopic randomness. Zuo et al. (2015) found that the local singularity index calculated by the original algorithm is influenced by background values and proposed a modified algorithm to overcome these shortcomings. With respect to the third aforementioned important achievement, characterization of the vertical distribution of elements, several parameters, including the box-counting dimension, Hurst exponent, power-law frequency characteristics, etc., usually need to be estimated. A number of studies have demonstrated that the distribution of geochemical elements in borehole exhibit fractal properties, just as the distribution in surficial materials (Monecke et al., 2001; Nazarpour et al., 2014; Sanderson et al., 1994; Wang, G. et al., 2012; Zuo et al., 2009b). These properties can be used to distinguish mineralized zones from those that are not mineralized.

2.1. Number–size model The very important fractal model, i.e., the number–size model (N-S), was firstly proposed by Mandelbrot (1983) to characterize the relationship between the size of objects and the number of objects with size greater than or equal to a given size. Based on this model, several variants have been developed and successfully applied in earth sciences (Agterberg, 1996; Carlson, 1991; Hassanpour and Afzal, 2013;

35

Sanderson et al., 1994; Turcotte, 2002; Zuo et al., 2009b). This model was probably the first one with geochemical relevance. 2.2. Concentration–area fractal model From the multifractal point of view, Cheng et al. (1994) derived the C-A fractal model, which relates the element concentration to the area enclosed by concentration contours by a power-law relation as follows Aðρ ≤ vÞ∝ρ−α 1 ;

Aðρ N vÞ∝ρ−α2

ð1Þ

where A(ρ) denotes the area with concentration values greater than or equal to the contour value ρ, cse is the mathematical symbol for “proportional to”, and α1 and α2 are exponents associated with minimum and maximum singularity, respectively. Two approaches can be used to calculate the enclosed area. One is based on the contour map created by interpolation procedures, and the other is based on superimposing a grid with cells on the study area and calculating the area by means of a boxcounting method. Distinct patterns, each corresponding to a set of similarly shaped contours, can be separated by different straight segments fitted to the values of the contours and enclosed areas on the log–log plot. The slopes of these straight lines can be taken as an estimation of the exponents of the power-law relation in Eq. (1). The optimum threshold for separating geochemical anomalies from background is the concentration value common to both linear relationships on the log–log plot. 2.3. Spectrum–area fractal mode Geochemical patterns in the spatial domain can be considered as superimposed signals of different frequencies. Based on this argument, Cheng et al. (1999) proposed the S-A fractal model to separate geochemical anomalies from background using spectral analysis in the frequency domain combined with the C-A model, which can provide an effective tool for determining an optimum threshold between different patterns on the basis of the scaling property. This fractal model also can be expressed as a power-law relation between the power spectrum ρ and area A(E N ρ) with the power spectrum above the given value ρas follows: ‐2d AðE N ρÞ∝ρ =β ;

ð2Þ

where β is an anisotropic scaling exponent and d represents the overall degree of concentration. Different patterns can be recognized by fitting several straight lines, each with a different slope, to the data pairs of area and power spectrum on the log–log plot. An irregular fractal filter can then be built according to these distinct patterns with background and noise corresponding to low and high power spectrum values being removed. The geochemical anomalies of interest can be obtained by converting the filtered pattern back to the spatial domain (Cheng, 1999a; Cheng et al., 2010; Xu and Cheng, 2001; Zuo, 2011a,b, 2012, 2013). 2.4. Concentration–distance fractal model A geochemical dispersion pattern often involves many sub-patterns at many hierarchical levels, leading to the spatial distribution of element concentration being clustered at different scales (Li et al., 2003). Mandelbrot proposed the radial-density model as an approach to characterize the clustering of point events (Mandelbrot, 1983). Based on this model, Li et al. (2003) developed the concentration–distance (C-D) fractal model by replacing the density with element concentration. This model can directly process original element concentration data, and can avoid the error caused by any interpolation procedure. Determination of the optimum threshold for this model is nearly the same as for the C-A model.

36

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

2.5. Concentration–volume fractal model

2.7. Other parameters related to fractal methods

On the basis of the same idea as the C-A model, the C-V fractal model was proposed by Afzal et al. (2011) to quantify the relationship between element concentration and the accumulative volume with concentration greater than or equal to the given value (Afzal et al., 2011; Delavar et al., 2012; Lin et al., 2013; Sadeghi et al., 2012; Soltani et al., 2014; Sun and Liu, 2014; Wang, G. et al., 2012).

The Hurst exponent (H) was proposed by Hurst (1951) as a measure of long-term dependence, and it has been applied widely in the geosciences (Turcotte, 1997). The expected value of H lies between 0 and 1. A larger H means a stronger persistence of the time series. The rescaled range statistic (R/S) analysis as a measure of how the apparent variability of a series changes with the length of the time-period being considered is one of the most popular methods to calculate the Hurst exponent. The rescaled range is calculated through dividing the range of the values within a portion of the time series by the standard deviation of the values for the whole same portion of the time series. The Hurst exponent can be estimated from the slope of regression line with log (n) versus log(R/S), where n is the length of sub-series. In addition, the box dimension also can be used for measurement of the complexity of geochemical patterns, which can be estimated by the box counting method. Further details on these parameters and models can be found in related literature (e.g., Sanderson et al., 1994; Wang, G. et al., 2012; Zuo et al., 2009b).

2.6. Local singularity analysis Singular processes, including earthquakes, mineralization, etc., usually result in anomalous amounts of energy release or mass accumulation, which are generally confined to a relatively narrow spatial-temporal interval (Cheng, 2007). Singularity is a fundamental property of these complex processes, and they can often be expressed as multifractal distributions. The singularity mapping technique is developed on the basis of the local singularity exponent, which is calculated by assembling a geochemical map at different scales to quantify the scaling characteristics of element concentration or depletion (Cheng, 2007). Within a multifractal context, the singular geochemical distribution can be described by the following power–law relation: α

μ ðAÞ ¼ cA 2

3.1. Identifying geochemical anomalies ð3Þ

where μ(A) denotes the total amount of metal within an area of size A, c is a constant also termed the fractal density, and a is the singularity exponent, which can be estimated using the ratio of the logarithmic transformation of measure μ and area Aas follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi α ¼ logðμ 1 =μ 2 Þ= log A1 =A2

3. Application

ð4Þ

Different cases of α− E (Euclidian dimension) versus zero correspond to different geochemical dispersion patterns, the case with difference value greater than zero indicating element depletion or otherwise enrichment. Cheng (2007) provided a windows-based method for mapping a local singularity, which has been used widely for mineral prospecting. As showed in the Lebesgue decomposition theorem, any complex measure can be decomposed into two components: one is an absolutely continuous measure and the other a singularity. Therefore, for the power-law relation in Eq. (3), the value of c should be a nonsingular component (Chen et al., 2007; Cheng, 2005). In the context of GIS, a point is different from a pixel because a point has no size. Therefore, the singularity exponent based on the raster model represents a local singularity in the sense of small vicinity but not a point-wise singularity, i.e., the estimated value of c still contains the singularity. This kind of knowledge led to the proposal by Chen et al. (2007) of an iterative algorithm to obtain the optimal local singularity exponent, which has been demonstrated to be superior to the non-iterative algorithm originally proposed by Cheng (2007). Considering that the original algorithm used to estimate the local singularity exponent could not directly process a data set containing negative values and that the estimated singularity is influenced by background values, Zuo et al. (2015) developed a modified algorithm to overcome the aforementioned shortcomings. The improvement associated with the new algorithm involves the introduction of a step in which the minimum value within the maximum window for each given location is previously subtracted. This kind of straight processing solves the existing problem and enables the singularity of similar anomalous patterns within different background values to be comparable.

The past several decades saw the emergence of fractal/multifractal models and their utility in the field of applied geochemistry. Particularly, various types of geochemical datasets, including stream and lake sediments, soil, borehole, and lithogeochemistry datasets have been processed to identify geochemical anomalies and discriminate between anthropogenic pollution and natural sources (e.g., Afzal et al., 2010, 2011; Albanese et al., 2007; Asadi et al., 2014; Cheng et al., 1994, 2000, 2010; Deng et al., 2010; Gonçalves et al., 2001; He et al., 2013; Heidari et al., 2013; Jesus et al., 2013; Li et al., 2003; Luz et al., 2014; Panahi et al., 2004; Sadeghi et al., 2012; Sun and Liu, 2014; Wang, W. et al., 2013; Xu and Cheng, 2001; Zuo, 2011a,b, 2014). Many researchers also found that the components decomposed from the original geochemical data often have good correspondence to the geological processes or phenomena. For example, Cheng et al. (1994) applied the C-A fractal model to lithogeochemical data of the Mitchell-Sulphurets precious metal district, British Columbia, and found that different fractal patterns exist inside and outside the potassic, sulfidic, and silicic alteration areas. Afzal et al. (2011) employed the C-V fractal model to separate supergene enrichment and hypogene zones from oxidation zones and barren host rocks, and the zones interpreted on the basis of the C-V fractal model are consistent with the geological settings. In addition, these fractal models are often combined with principal component analysis or factor analysis methods to delineate multi-element association anomalies (e.g., Asadi et al., 2014; Cheng et al., 1997; He et al., 2014; Hosseini et al., 2014; Lin et al., 2013; Panahi et al., 2004; Shamseddin et al., 2014; Zuo, 2011a,b, 2014). New models also can be established based on these models. For instance, Wang et al. (2011) proposed a new fractal model to estimate the reserve. In this model, an orebody can be divided spatially into several parts with different degrees of mineralization via the cutoff values obtained from the C-A fractal model. Furthermore, on the basis of the C-A fractal model, a median C-A model was derived, and the ore tonnage–thickness and the metal tonnage–grade thickness models then were established (Wang, Q. et al., 2012). These new models can be helpful for understanding orebody spatial distribution. Cao and Cheng (2012) combined the S-A fractal model and generalized scale invariance to develop a new method, which can not only separate anomalies from background but can also provide information about anisotropic scale invariance of geochemical patterns.

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

37

Fig. 2. Maps showing simulated geochemical patterns: (a) a regional field created by linearly decreased function; (b) anomalies generated by two sources with a different buried depth; (c) simulated geochemical pattern by superimposing the regional background on the anomalies; (d) a profile demonstrating three simulated maps.

3.2. Mapping mineral prospectivity Cheng (2007) proposed the concept of the singularity exponent to depict the local structure of geochemical patterns, and many case studies have been conducted to demonstrate its utility. Note that some methods, such as principal component analysis, spatially weighted principal component analysis, spatially weighted geographic regression, robust principal component analysis, etc., could be applied to the geochemical data before singularity mapping to integrate information of ore-forming elements (Xiao et al., 2012; Zhao et al, 2012, 2013; Zuo et al., 2015). Considering the difference of elemental mobility in areas with overburden, Xiao et al. (2014) investigated the possibly of mobile elements by means of accumulation coefficient analysis to determine a set of suitable indicator elements for further analysis. This procedure is necessary for identification of weak anomalies from buried mineralization. From the perspective of application, singularity analysis cannot only identify geochemical anomalies but can also map igneous rock bodies if different element associations are analyzed (Zhao et al., 2012). For example, stream sediment geochemical data of K2O, Na2O, SiO2, and Al2O3 can be integrated using principal component analysis and then one component score is processed to map acid igneous rocks using singularity analysis. In addition, on the basis of singularity analysis, new models can be established to assist in geochemical exploration. Considering the advantage that the singularity exponent can identify heterogeneity, Wang, W. et al. (2013) proposed a new model, the fault trace-oriented singularity mapping technique, to characterize anisotropic mineralization-associated geochemical signatures. Furthermore, a tectonic-geochemical exploration model that focuses on faultcontrolled and geochemical halo-associated mineralization was also constructed by Wang, W. et al. (2012). Note that a singularity distribution map derived from a singularity mapping technique requires further analysis for modeling mineral prospectivity, which usually involves setting a series of thresholds to divide the singularity exponents into binary patterns and then calculating

the spatial correlation between these patterns and known deposits and occurrences. These correlation indices, termed Student’s t-values, can be estimated by the weights of evidence method. Further details on this method can be found in related literature (Bonham–Carter, 1994; Cheng et al., 1994). There are also some other applications of singularity mapping techniques that deserve to be mentioned. Local singularity analysis has been proposed in the multifractal context, and it can predict strong local continuity of element concentration values. In comparison with conventional methods, one prominent advantage of this approach is its ability to determine all singularities including positions with extreme values, which often disable statistical methods due to the small sample size problem. Agterberg (2012a) found that estimated singularities are linearly related to logarithmically transformed element concentration values. By means of this relation, the small-scale nugget effect can be measured. This argument has been demonstrated by several cases, including simulated and practical geochemical data. In addition, singularity exponents can be used for interpolation. Conventional methods, such as inverse distance weighting and Kriging, fail to take into account the local properties of geochemical data. The newly proposed multifractal interpolation approach can overcome this drawback by incorporating local singularity into traditional models. In this regard, this method takes into account the local structure and singularity

Table 1 Model and parameters used to simulate anomalies. Anomalies No. 1

No. 2

Model

C Ch

P¼ 3 =2 2 x2 þ y 2 þ h

h/m

Position(m)

4000

80

(400,402)

4000

140

(700,402)

Note: The simulated values range from 1 to 14. P-simulated value; h-buried depth; C-constant; (x,y)-location of center of anomaly source.

38

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

Fig. 3. (a) Log-log plot of concentration versus area with values greater than or equal to the given concentration; (b) anomaly map identified by the C–A fractal model.

in assigning weights for data interpolation. Both one- and twodimensional cases have been used to demonstrate its superiority over conventional methods (Cheng, 1999b, 2000, 2005, 2008). In fact, ordinary moving average techniques can be taken as a special case of the multifractal interpolation method. 3.3. Characterization the vertical distribution of geochemical element concentration Characterization of the vertical distribution of geochemical element concentration plays an important role in economic planning in the mining industry. Cheng (2014) proposed a new non-linear differential equation which assumes the decay rate of concentration is negatively proportional to the concentration itself, with a functional coefficient dependent on vertical distance from the underlying surface of mineralized rocks to quantify the decay behaviour of geochemical concentration of an element in a vertical regolith profile. Many studies have revealed that the distribution of geochemical elements exhibits fractal properties, which can be characterized by fractal or multifractal models. The routine procedure for this purpose involves estimation of several fractal parameters, including the box-counting dimension, Hurst exponent, etc. The box dimension often can be used for quantification of the complexity of phenomena or processes, which usually serves as an indicator to differentiate mineralized and non-mineralized zones (Zuo et al., 2009b). The Hurst exponent is associated with continuity of mineralization, and the larger the Hurst exponent, the better the continuity. In addition, the power-law frequency distribution also plays an important role in characterizing the vertical distribution of geochemical elements. It can reveal different patterns of elemental distribution and provide cutoff values distinguishing these patterns.

Cheng (2012, 2014). In practice, there could be more complex situations which should be further considered. In this paper, the anomaly pattern was generated using a method similar to the gravity forward model (Telford et al., 1990), which exhibited a power-law relation between magnitude of the gravity field and the departure from the geological body. The anomalies were defined as the projection of patterns around sources with different burial depths. The model and detailed parameters used for simulating anomalies can are shown in Table 1 and Fig. 2. The simulated geochemical pattern (Fig. 2c) was created by superimposing the anomalies (Fig. 2b) on the regional background (Fig. 2a). Fig. 2d illustrates the profiles that intersect those patterns in the middle and two anomalous sources with different depth. Obviously, the deeper the source, the weaker the pattern on the surface. Thus, the anomalous pattern corresponding to the second source becomes almost indistinguishable due to the masking effect of the background. Even for the first one, it has been weakened significantly. Three fractal models C-A, S-A, and local singularity analysis will be employed on the simulated geochemical data to detect anomalies with the aid of a Matlab program (Wang and Zuo, 2015). Based on the simulated geochemical pattern, the C-A plot, as illustrated in Fig. 3a, was obtained to reveal the relationship between the threshold value and the number of cells with values greater than or equal to it. Three straight lines can be fit by means of the least square method. Two threshold values can be then obtained and used further to divide the geochemical pattern into three components (Fig. 3b). The left part and the circle within the middle region can be regarded as anomalies. The middle part, with the exception of the circle, can be regarded as moderate anomalies, and the rest can be considered as background. Comparing these results with the anomalies map illustrated in Fig. 2b, only

4. Comparison the C-A, S-A and singularity index Cheng (2014) briefly reviewed the state-of-the-art the vertical distribution of elements in upper regolith over mineral deposits, and proposed a power-law decay function to model the regolith decay trends with increasing distance from the underlying altered rocks or saprocks. Cheng's work indicated that the element concentration is a function of the depth of mineral deposits. For instance, Cheng (2012) showed a geological and geochemical profile in Gejiu Tin district, China (c.f., Fig. 2 in Cheng, 2012). It can be observed that in the east of Gejiu, most of the known deposits with outcropping or shallower ore bodies (c.f., Fig. 2G in Cheng, 2012) have large Sn concentrations; however, in the west, the ore bodies occurred about 1 km below the surface of the Earth, correspond to small Sn concentrations (c.f., Fig. 2F in Cheng, 2012). Following the preceding studies, the C-A, S-A and singularity index were compared based on simulated data, which were created by the superposition of anomalies on a linearly varied background. In this simulated model, we only considered the situation described by

Fig. 4. Log–log plot of power spectrum values and areas.

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

39

Fig. 5. Maps showing background (a) and anomalous (b) map obtained by the S-A model.

the higher anomaly can be detected, which is associated with a large part of background being classified as anomalies. In this regard, the effectiveness of the C-A fractal model depends on the variability of the background. Converting the simulated geochemical data to the frequency domain using fast Fourier transformation (FFT) and applying the C-A model again, the S-A plot can then be obtained (Fig. 4). Two cutoff values can be determined by the same procedure as the C-A fractal model. The filter can be built using the higher cutoff value and further employed on the FFT map. Converting the filtered map back into the spatial domain, two patterns, i.e., background and anomalies, were generated (Fig. 5a and b). Unlike the results obtained by C-A, the S-A can extract anomalies that lie within the varied background. The decomposed background is nearly the same as the originally simulated pattern, with the exception of those locations significantly influenced by the two anomalous sources. As for the anomaly map, both of the sources can be reflected to some degree. However, there are also some symmetrically distributed anomalous areas that have no direct correspondence to potential sources. These areas may be the numerical artifact of the S-A model itself. In this regard, this method has limited capacity to correctly map anomalies weakened by thick overburden. The singularity mapping technique was also used to detect simulated anomalies. Several parameters need to be set prior to the calculation of singularity exponents. A series of square windows was selected with half window lengths ranging from 4 m to 20 m at a 4 m interval. The singularity map (Fig. 6) could then be created. Obviously, two circular regions with relatively low singularity exponents are delineated. The locations and shapes of the regions are highly consistent with the anomalous sources. In addition, there are scarcely any interferences, as indicated by the fact that the singularity exponents of the regional field

area approach 2. However, note that the singularity of the left delineated area is relatively stronger than that of the right one due to their different burial depths. Therefore, the singularity mapping technique is a powerful tool for enhancing weak anomalies and locating anomalous sources from a multifractal perspective. 5. Discussion and conclusions Fractal/multifractal models have provided a new perspective for modeling of geochemical data. These methods also have been proved effectively in practice by many case studies. However, for analyzing geochemical data itself, there are still some unsolved problems. No matter what methods are adopted, the anomalies detected are usually of unequal importance for further exploration. Thus, methods for selecting promising anomalies mathematically are crucial and require additional research. There are few studies on whether fractal/multifractal models could be used to assess anomalies, and to distinguish anomalies associated to mineralization with false anomalies which may be caused by noise, data processing, or other geological processes. In addition, detection of geochemical anomalies in areas with more or less overburden remains the principal challenge. Mechanisms of element dispersion and distribution in the surficial environment should be investigated further with the help of new theoretical approaches, including fractal and multifractal models. This study has mainly presented an overview of fractal/multifractal models used to model geochemical data. These models, including C-A, S-A, the singularity mapping technique, etc., have been used widely to detect anomalies in geochemical exploration and determine the geochemical baseline in environmental studies. The simulated experiment presented here made a comparison of three fractal/mutifractal models, i.e., C-A, S-A, and singularity mapping, indicating that singularity mapping technique can enhance and identify weak anomalies caused by buried sources. Acknowledgments We thank John Carranza, Peyman Afzal and an anonymous reviewer's comments and suggestions, which improve this study. This research benefited from the joint financial support from the National Natural Science Foundation of China (No. 41372007), and the Program for New Century Excellent Talents in University (NCET-13-1016). References

Fig. 6. Map showing the singularity index.

Afzal, P., Khakzad, A., Moarefvand, P., Rashidnejad, O.N., Esfandiari, B., Fadakar, A.Y., 2010. Geochemical anomaly separation by multifractal modeling in Kahang (GorGor) porphyry system, Central Iran. J. Geochem. Explor. 104, 34–46. Afzal, P., Fadakar, A.Y., Khakzad, A., Moarefvand, P., Rashidnejad, O.N., 2011. Delineation of mineralization zones in porphyry Cu deposits by fractal concentration–volume modeling. J. Geochem. Explor. 18, 220–232.

40

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41

Afzal, P., Fadakar, A.Y., Moarefvand, P., Rashidnejad, O.N., Asadi, H.H., 2012. Application of power–spectrum–volume fractal method for detecting hypogene, supergene enrichment, leached and barren zones in Kahang Cu porphyry deposit, Central Iran. J. Geochem. Explor. 112, 131–138. Afzal, P., Harati, H., Fadakar, A.Y., Yasrebi, A.B., 2013a. Application of spectrum–area fractal model to identify of geochemical anomalies based on soil data in Kahang porphyry– type Cu deposit, Iran. Chem. Erde 73, 533–543. Afzal, P., Ahari, H.D., Omran, N.R., Aliyari, F., 2013b. Delineation of gold mineralized zones using concentration–volume fractal model in Qolqoleh gold deposit, NW Iran. Ore Geol. Rev. 55, 125–133. Agterberg, F.P., 1996. Multifractal modeling of the sizes and grades of giant and supergiant deposits. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33, A365. Agterberg, F.P., 2007. Mixtures of multiplicative cascade models in geochemistry. Nonlinear Process. Geophys. 14, 201–209. Agterberg, F.P., 2012a. Multifractals and geostatistics. J. Geochem. Explor. 122, 113–122. Agterberg, F.P., 2012b. Sampling and analysis of chemical element concentration distribution in rock units and orebodies. Nonlinear Process. Geophys. 19, 23–44. Ahrens, L.H., 1954. The lognormal distribution of elements (a fundamental law of geochemistry and its subsidiary). Geochim. Cosmochim. Acta 5, 49–73. Ahrens, L.H., 1957. Lognormal-type distributions-III. Geochim. Cosmochim. Acta 11, 205–212. Albanese, S., De Vivo, B., Lima, A., Cicchella, D., 2007. Geochemical background and baseline values of toxic elements in stream sediments of Campania region (Italy). J. Geochem. Explor. 93, 21–34. Ali, K., Cheng, Q., Chen, Z., 2007. Multifractal power spectrum and singularity analysis for modelling stream sediment geochemical distribution patterns to identify anomalies related to gold mineralization in Yunnan Province, South China. Geochem. Explor. Environ. Anal. 7, 293–301. Arias, M., Gumie, P., Martín-Izard, A., 2012. Multifractal analysis of geochemical anomalies: A tool for assessing prospectivity at the SE border of the Ossa Morena Zone, Variscan Massif (Spain). J. Geochem. Explor. 122, 101–112. Asadi, H.H., Kianpouryan, S., Lu, Y., McCuaig, T.C., 2014. Exploratory data analysis and C-A fractal model applied in mapping multi-element soil anomalies for drilling: A case study from the Sari Gunay epithermal gold deposit, NW Iran. J. Geochem. Explor. 145, 233–241. Bai, J., Porwal, A., Hart, C., Ford, A., Yu, L., 2010. Mapping geochemical singularity using multifractal analysis: Application to anomaly definition on stream sediments data from Funin Sheet, Yunnan, China. J. Geochem. Explor. 104, 1–11. Behrens, J.T., 1997. Principles and Procedures of Exploratory Data Analysis. Psychol. Methods 2, 131–160. Bölviken, B., Stokke, P.R., Feder, J., Jössang, T., 1992. The fractal nature of geochemical landscapes. J. Geochem. Explor. 43, 91–109. Bonham-Carter, G.F., 1994. Geographic information systems for geoscientists: modeling with GIS. Computer Methods in the Geosciences first ed. Pergamon, New York (398 pp.). Cao, L., Cheng, Q., 2012. Quantification of anisotropic scale invariance of geochemical anomalies associated with Sn-Cu mineralization in Gejiu, Yunan Province, China. J. Geochem. Explor. 122, 47–54. Carlson, C.A., 1991. Spatial distribution of ore deposits. Geology 19, 111–114. Carranza, E.J.M., 2009. Geochemical Anomaly and Mineral Prospectivity Mapping in GIS. Handbook of Exploration and Environmental Geochemistry vol. 11. Elsevier, Amsterdam. Carranza, E.J.M., 2010. Mapping of anomalies in continuous and discrete fields of stream sediment geochemical landscapes. Geochem. Explor. Environ. Anal. 10, 171–187. Chen, Z., Cheng, Q., Chen, J., Xie, S., 2007. A novel iterative approach for mapping local singularities from geochemical data. Nonlinear Process. Geophys. 14, 317–324. Chen, G., Cheng, Q., Zuo, R., Liu, T., Xi, Y., 2015. Identifying gravity anomalies caused by granitic intrusions in Nanling mineral district, China: a multifractal perspective. Geophys. Prospect. 63, 256–270. Cheng, Q., 1999a. Spatial and scaling modeling for geochemical anomaly separation. J. Geochem. Explor. 65, 175–194. Cheng, Q., 1999b. Multifractal interpolation. In: Lippard, S.J., Naess, A., Sinding-Larsen, R. (Eds.), Proceedings of the Fifth Annual Conference of the International Association forMathematical Geology, Trondheim, Norway. 6-11 Aug. 1999. Norway University of Science and Technology, Trondheim, pp. 245–250. Cheng, Q., 2000. Interpolation by means of multifractal, kriging and moving average techniques. GeoCanada 2000, Proc. GAC/MAC Meeting, Calgary, AB, Canada [CD-ROM]. 29 May-2 June 2000. Geol. Assoc. Can., St. John’s, NF, Canada. Cheng, Q., 2005. A new model for incorporating spatial association and singularity in interpolation of exploratory data. In: Leuangthong, O., Deutsch, C.V. (Eds.), Geostatistics Banff 2004, Quantitative Geology and Geostatistics 14. Springer Press, Netherlands, pp. 1017–1025. Cheng, Q., 2007. Mapping singularities with stream sediment geochemical data for prediction of undiscovered mineral deposits in Gejiu, Yunnan Province, China. Ore Geol. Rev. 32, 314–324. Cheng, Q., 2008. Modeling local scaling properties for multiscale mapping. Vadose Zone J. 7, 525–532. Cheng, Q., 2012. Singularity theory and methods for mapping geochemical anomalies caused by buried sources and for predicting undiscovered mineral deposits in covered areas. J. Geochem. Explor. 122, 55–70. Cheng, Q., 2014. Vertical distribution of elements in regolith over mineral deposits andimplications for mapping geochemical weak anomalies in covered areas. Geochem. Explor. Environ. Anal. 14, 277–289. Cheng, Q., Agterberg, F.P., 2009. Singularity analysis of ore-mineral and toxic trace elements in stream sediments. Comput. Geosci. 35, 234–244. Cheng, Q., Agterberg, F.P., Ballantyne, S.B., 1994. The separation of geochemical anomalies from background by fractal methods. J. Geochem. Explor. 51, 109–130.

Cheng, Q., Agterberg, F.P., Bonham-Carter, G.F., 1996. A spatial analysis method for geochemical anomaly separation. J. Geochem. Explor. 56, 183–195. Cheng, Q., Bonham-Carter, G.F., Hall, G.E.M., Bajc, A., 1997. Statistical study of trace elements in the soluble organic and amorphous Fe-Mn phases of surficial sediments, Sudbury Basin 1. Multivariate and spatial analysis. J. Geochem. Explor. 59, 27–46. Cheng, Q., Xu, Y., Grunsky, E., 1999. Integrated spatial and spectral analysis for geochemical anomaly separation. In: Lippard, S.J., Naess, A., Sinding-Larsen, R. (Eds.), Proceedings of the Fifth Annual Conference of the International Association for Mathematical Geology, Trondheim, Norway 6–11th August. vol. 1, pp. 87–92. Cheng, Q., Xu, Y., Grunsky, E., 2000. Integrated Spatial and Spectrum Method for Geochemical Anomaly Separation. Nat. Resour. Res. 9, 43–52. Cheng, Q., Xia, Q., Li, W., Zhang, S., Chen, Z., Zuo, R., Wang, W., 2010. Density/area powerlaw models for separating multi-scale anomalies of ore and toxic elements in stream sediments in Gejiu mineral district, Yunnan Province, China. Biogeosciences 7, 3019–3025. Cohen, D.R., Kelley, D.L., Anand, R., Coker, W.B., 2010. Major advances in exploration geochemistry, 1998-2007. Geochem. Explor. Environ. Anal. 10, 3–16. Darnley, A.G., Bjorklund, B., Gustavsson, N., Koval, P.V., Plant, J., Steenfelt, A., Tauchid, T.M., Xie, X.J., 1995. A Global Geochemical Database for environmental and resource management. Recommendations for internationalgeochemical mapping. Earth Sciences Report 19. UNESCO Publishing, Paris. Delavar, S.T., Afzal, P., Borg, G., Rasa, I., Lotfi, M., Omran, N.R., 2012. Delineation of mineralization zones using concentration–volume fractal method in Pb-Zn carbonate hosted deposits. J. Geochem. Explor. 118, 98–110. Deng, J., Wang, Q., Yang, L., Wang, Y., Gong, Q., Liu, H., 2010. Delineation and explanation of geochemical anomalies using fractal models in the Heqing area, Yunnan Province, China. J. Geochem. Explor. 105, 95–105. Feder, J., 1988. Fractals. Plenum Press, New York (283 pp.). Gonçalves, M.A., Mateus, A., Oliveira, V., 2001. Geochemical anomaly separation by multifractal modeling. J. Geochem. Explor. 72, 91–114. Grunsky, E.C., 2010. The interpretation of geochemical survey data. Geochem. Explor. Environ. Anal. 10, 27–74. Hassanpour, Sh., Afzal, P., 2013. Application of concentration-number (C-N) multifractal modelling for geochemical anomaly separation in Haftcheshmeh porphyry system, NW Iran. Arab. J. Geosci. 6, 957–970. Hawkes, H.E., Webb, J.S., 1962. Geochemistry in Mineral Exploration. Harper and Row, New York. He, J., Yao, S., Zhang, Z., You, G., 2013. Complexity and Productivity Differentiation Models of Metallogenic Indicator Elements in Rocks and Supergene Media Around Daijiazhuang Pb-Zn Deposit in Dangchang County, Gansu Province. Nat. Resour. Res. 22, 19–36. He, F., Wang, Z., Fang, C., Wang, L., Geng, X., 2014. Identification and assessment of Sn–polymetallic prospects in the Gejiu western district, Yunnan (China). J. Geochem. Explor. 145, 106–113. Heidari, S.M., Ghaderi, M., Afzal, P., 2013. Delineating mineralized phases based on lithogeochemical data using multifractal model in Touzlar epithermal Au–Ag (Cu) deposit, NW Iran. Appl. Geochem. 31, 119–132. Hosseini, S.A., Afzal, P., Sadeghi, B., Sharmad, T., Shahrokhi, S.V., Farhadinejad, T., 2014. Prospection of Au mineralization based on stream sediments and lithogeochemical data using multifractal modeling in Alut 1:100,000 sheet, NW Iran. Arab. J. Geosci. http://dx.doi.org/10.1007/s12517-014-1436-5. Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770–808. Jesus, A.P., Mateus, A., Gonçalves, M.A., Munhá, J., 2013. Multi-fractal modelling and spatial Cu–soil anomaly analysis along the southern border of the Iberian Terrane in Portugal. J. Geochem. Explor. 126–127, 23–44. Krige, D.G., 1966. A study of gold and uranium distribution patterns in the Klerksdorp goldfield. Geoexploration 4, 43–53. Krige, D.G., 1978. Lognormal de–Wijsiangeostatistics for ore evaluation. South AfricanInstitute Mining and Metallurgy Monograph Series. Geostatistics 1. Lam, N.S., 1983. Spatial interpolation methods: A review. Am. Cartogr. 10, 129–149. Lavallee, D., Lovejoy, S., Schertzer, D., Ladoy, P., 1993. Nonlinear variability and landscape topography: analysis and simulation. Fractals Geogr. 158–192. Li, C., Ma, T., Shi, J., 2003. Application of a fractal method relating concentrations and distances for separation of geochemical anomalies from background. J. Geochem. Explor. 77, 167–175. Lima, A., De, Vivo B., Cicchella, D., Cortini, M., Albanese, S., 2003. Multifractal IDW interpolation and fractal filtering method in environmental studies: an application on regional stream sediments of (Italy), Campania region. Appl. Geochem. 18, 1853–1865. Lima, A., Albanese, S., Cicchella, D., 2005. Geochemical baselines for the radioelements K, U, and Th in the Campania region, Italy: a comparison of stream–sediment geochemistry and gamma-ray surveys. J. Geochem. Explor. 20, 611–625. Lima, A., Plant, J.A., Vivo, B.D., Tarvainen, T., Albanese, S., Cicchella, D., 2008. Interpolation methods for geochemical maps: a comparative study using arsenic data from European stream waters. Geochem. Explor. Environ. Anal. 8, 41–48. Lin, X., Zhang, B., Wang, X., 2013. Application of factor analysis and concentration–volume fractal modeling to delineation of 3D geochemical patterns: a case study of the Jinwozi gold field, NW China. Geochem. Explor. Environ. Anal. 14, 359–367. Liu, Y., Cheng, Q., Xia, Q., Wang, X., 2013. Application of singularity analysis for mineral potential identification using geochemical data – A case study: Nanling W-Sn-Mo polymetallicmetallogenic belt, South China. J. Geochem. Explor. 134, 61–72. Luz, F., Mateus, A., Matos, J.X., Gonçalves, M.A., 2014. Cu- and Zn-Soil Anomalies in the NE Border of the South Portuguese Zone (Iberian Variscides, Portugal) Identified by Multifractal and Geostatistical Analyses. Nat. Resour. Res. 23, 195–215. Mandelbrot, B.B., 1983. The fractal geometry of nature. Freeman, San Francisco (468 pp.).

R. Zuo, J. Wang / Journal of Geochemical Exploration 164 (2016) 33–41 Monecke, T., Gemmell, J.B., Monecke, J., 2001. Fractal distributions of veins in drill core from the Hellyer VHMS deposit, Australia: constraints on the origin and evolution of the mineralizing system. Mineral. Deposita 36, 406–415. Nazarpour, A., Sadeghi, B., Sadeghi, M., 2014. Application of fractal models to characterization and evaluation of vertical distribution of geochemical data in Zarshuran gold deposit, NW Iran. J. Geochem. Explor. 148, 60–70. Panahi, A., Cheng, Q., Bonham-Carter, G.F., 2004. Modelling lake sediment geochemical distribution using principal component, indicator kriging and multifractal powerspectrum analysis: a case study from Gowganda, Ontario. Geochem. Explor. Environ. Anal. 4, 59–70. Plant, J., Smith, D., Smith, B., Williams, L., 2001. Environmental geochemistry at the global scale. Appl. Geochem. 16, 1291–1308. Reimann, C., 2005a. Geochemical mapping: technique or art? Geochem. Explor. Environ. Anal. 5, 359–370. Reimann, C., 2005b. Sub-continental-scale geochemical mapping: sampling, quality control and data analysis issues. Geochem. Explor. Environ. Anal. 5, 311–323. Reimann, C., Filzmoser, P., 2000. Normal and lognormal data distribution in geochemistry: death of a myth. Consequences for the statistical treatment of geochemical and environmental data. Environ. Geol. 39, 1001–1014. Sadeghi, B., Moarefvand, P., Afzal, P., Yasrebi, A.B., Saein, L.D., 2012. Application of fractal models to outline mineralized zones in the Zaghia iron ore deposit, Central Iran. J. Geochem. Explor. 122, 9–19. Sanderson, D.J., Roberts, S., Gumiel, P., 1994. A fractal relationship between vein thickness and gold grade in drill core from La Codosera, Spain. Econ. Geol. 89, 168–173. Shamseddin, M.M., Afzal, P., Gholinejad, M., Yasrebi, A.B., Sadeghi, B., 2014. Delineation of geochemical anomalies using factor analysis and multifractal modeling based on stream sediments data in Sarajeh 1:100,000 sheet, Central Iran. Arab. J. Geosci. 7, 5333–5343. Soltani, F., Afzal, P., Asghari, O., 2014. Delineation of alteration zones based on Sequential Gaussian Simulation and concentration–volume fractal modeling in the hypogene zone of Sungun copper deposit, NW Iran. J. Geochem. Explor. 140, 64–76. Sun, T., Liu, L., 2014. Delineating the complexity of Cu-Mo mineralization in a porphyry intrusion by computational and fractal modeling: A case study of the Chehugou deposit in the Chifeng district, Inner Mongolia, China. J. Geochem. Explor. 144, 128–143. Sun, X., Gong, Q., Wang, Q., Yang, L., Wang, C., Wang, Z., 2010. Application of local singularity model to delineate geochemical anomalies in Xiong'ershan gold and molybdenum ore district, Western Henan province, China. J. Geochem. Explor. 107, 21–29. Telford, W.M., Geldart, L.P., Sheriff, R.E., 1990. Applied geophysics. Cambridge University Press (784 pp.). Tukey, J.W., 1977. Exploratory data analysis. Addison-Wesley, Reading, USA. Turcotte, D.L., 1986. A fractal approach to the relationship between ore grade and tonnage. Econ. Geol. 81, 1528–1532. Turcotte, D.L., 1997. Fractals and chaos in geology and geophysics. second edition. Cambridge University Press (398 pp.). Turcotte, D.L., 2002. Fractals in petrology. Lithos 65, 261–271. Wang, J., Zuo, R., 2015. A MATLAB-based program for processing geochemical data using fractal/multifractal modeling. Earth Sci. Inf. http://dx.doi.org/10.1007/s12145-0150215-5. Wang, Q., Deng, J., Liu, H., Wang, Y., Sun, X., Wan, L., 2011. Fractal models for estimating local reserves with different mineralization qualities and spatial variations. J. Geochem. Explor. 108, 196–208. Wang, G., Carranza, E.J.M., Zuo, R., Hao, Y., Du, Y., Pang, Z., Sun, Y., Qu, J., 2012. Mapping of district-scale potential targets using fractal models. J. Geochem. Explor. 122, 34–46. Wang, Q., Deng, J., Zhao, J., Li, N., Wan, L., 2012. The fractal relationship between orebody tonnage and thickness. J. Geochem. Explor. 122, 4–8. Wang, W., Zhao, J., Cheng, Q., Liu, J., 2012. Tectonic–geochemical exploration modeling for characterizing geo-anomalies in southeastern Yunnan district, China. J. Geochem. Explor. 122, 71–80. Wang, G., Pang, Z., Boisvert, J.B., Hao, Y., Cao, Y., Qu, J., 2013. Quantitative assessment of mineral resources by combining geostatistics and fractal methods in the Tongshan porphyry Cu deposit (China). J. Geochem. Explor. 134, 85–98.

41

Wang, W., Zhao, J., Cheng, Q., 2013. Fault trace-oriented singularity mapping technique to characterize anisotropic geochemical signatures in Gejiu mineral district, China. J. Geochem. Explor. 134, 27–37. Xiao, F., Chen, J., Zhang, Z., Wang, C., Wu, G., Agterberg, F.P., 2012. Singularity mapping and spatially weighted principal component analysis to identify geochemical anomalies associated with Ag and Pb–Zn polymetallic mineralization in Northwest Zhejiang, China. J. Geochem. Explor. 122, 90–100. Xiao, F., Chen, J., Agterberg, F.P., Wang, C., 2014. Element behavior analysis and its implications for geochemical anomaly identification: A case study for porphyry Cu–Mo deposits in Eastern Tianshan, China. J. Geochem. Explor. 145, 1–11. Xie, S., Cheng, Q., Chen, G., Chen, Z., Bao, Z., 2007. Application of local singularity in prospecting potential oil/gas targets. Nonlinear Process. Geophys. 14, 285–292. Xu, Y., Cheng, Q., 2001. A fractal filtering technique for processing regional geochemical maps for mineral exploration. Geochem. Explor. Environ. Anal. 1, 147–156. Yousefi, M., Kamkar-Rouhani, A., Carranza, E.J.M., 2012. Geochemical mineralization probability index (GMPI): A new approach to generate enhanced stream sediment geochemical evidential map for increasing probability of success in mineral potential mapping. J. Geochem. Explor. 115, 24–35. Yousefi, M., Kamkar-Rouhani, A., Carranza, E.J.M., 2014. Application of staged factor analysis and logistic function to create a fuzzy stream sediment geochemical evidence layer for mineral prospectivity. Geochem. Explor. Environ. Anal. 14, 45–58. Zhao, J., Wang, W., Dong, L., Yang, W., Cheng, Q., 2012. Application of geochemical anomaly identification methods in mapping of intermediate and felsic igneous rocks in eastern Tianshan, China. J. Geochem. Explor. 122, 81–89. Zhao, J., Wang, W., Cheng, Q., 2013. Investigation of spatially non-stationary influences of tectono-magmatic processes on Fe mineralization in eastern Tianshan, China with geographically weighted regression. J. Geochem. Explor. 134, 38–50. Zhao, J., Zuo, R., Chen, S., Kreuzer, O.P., 2014. Application of the tectono-geochemistry method to mineral prospectivity mapping: A case study of the Gaosong tinpolymetallic deposit, Gejiu district, SW China. Ore Geol. Rev. http://dx.doi.org/10. 1016/j.oregeorev.2014.09.023. Zimmerman, D., Pavlik, C., Ruggles, A., Armstrong, A.P., 1999. An experimental comparison ofordinary and universal kriging and inverse distance weighting. Math. Geol. 31, 375–390. Zuo, R., 2011a. Decomposing of mixed pattern of arsenic using fractal model in Gangdese belt, Tibet, China. Appl. Geochem. 26, S271–S273. Zuo, R., 2011b. Identifying geochemical anomalies associated with Cu and Pb-Zn skarn mineralization using principal component analysis and spectrum-area fractal modeling in the Gangdese Belt, Tibet (China). J. Geochem. Explor. 111, 13–22. Zuo, R., 2012. Exploring the effects of cell size in geochemical mapping. J. Geochem. Explor. 112, 357–367. Zuo, R., 2014. Identification of geochemical anomalies associated with mineralization in the Fanshan district, Fujian, China. J. Geochem. Explor. 139, 170–176. Zuo, R., Cheng, Q., 2008. Mapping singularities – a technique to identify potential Cu mineral deposits using sediment geochemical data, an example for Tibet, west China. Mineral. Mag. 72, 531–534. Zuo, R., Cheng, Q., Agterberg, F.P., Xia, Q., 2009a. Application of singularity mapping technique to identify local anomalies using stream sediment geochemical data, a case study from Gangdese, Tibet, western China. J. Geochem. Explor. 101, 225–235. Zuo, R., Cheng, Q., Xia, Q., 2009b. Application of fractal models to characterization of vertical distribution of geochemical element concentration. J. Geochem. Explor. 102, 37–43. Zuo, R., Carranza, E.J.M., Cheng, Q., 2012. Fractal/multifractal modelling of geochemical exploration data. J. Geochem. Explor. 122, 1–3. Zuo, R., Xia, Q., Zhang, D., 2013. A comparison study of the C-A and S-A models with singularity analysis to identify geochemical anomalies in covered areas. Appl. Geochem. 33, 165–172. Zuo, R., Wang, J., Chen, G., Yang, M., 2015. Identification of weak anomalies: A multifractal perspective. J. Geochem. Explor. 148, 12–24.