Natural Resources Research ( 2018) https://doi.org/10.1007/s11053-018-9374-7

Original Paper

New Insights into Element Distribution Patterns in Geochemistry: A Perspective from Fractal Density Yue Liu,1,2,3,6 Qiuming Cheng,4,5,6,7 and Kefa Zhou1,2,3 Received 24 December 2017; accepted 9 February 2018

Multifractal features of element concentrations in the EarthÕs crust have demonstrated to be closely associated with multiple probability distributions such as normal, lognormal and power law. However, traditional understanding of geochemical distribution satisfying normal, lognormal or power-law models still faces a serious problem in adjusting theoretical statistics with the empirical distribution. Given that the differences among different geochemical distribution populations may have considerable effects on the target estimation, a new perspective from the singularity of fractal density is adopted to investigate mixed geochemical distribution patterns within frequency and space domains. In the framework of fractal geometry, ordinary density such as volume density (e.g., g/cm3 and kg/m3) described in Euclidean space can be considered as a special case of the fractal density (e.g., g/cma and kg/ma). According to the nature of fractal density, geochemical information obtained from Euclidean geometry may not sufficiently reflect inherent geochemical features, because some information might be hidden within fractal geometry that can be only revealed by means of a set of fractional dimensions. In the present study, stream sediment geochemical data collected from west Tianshan region, Xinjiang (China), were used to explore element distribution patterns in the EarthÕs crust based on a fractal density model. Four elements Cu, Zn, K and Na were selected to study the differences between minor and major elements in terms of their geochemical distribution patterns. The results strongly suggest that element distribution patterns can be well revealed and interpreted by means of a fractal density model and related statistical and multifractal parameters. KEY WORDS: Element distribution patterns, Fractal density, Singularity analysis, Multifractal theory, West Tianshan region.

1

State Key Laboratory of Desert and Oasis Ecology, Xinjiang Institute of Ecology and Geography, Chinese Academy of ¨ ru¨mqi 830011, Xinjiang, China. Sciences, U 2 Xinjiang Research Centre for Mineral Resources, Chinese ¨ ru¨mqi 830011, Xinjiang, China. Academy of Sciences, U 3 Xinjiang Key Laboratory of Mineral Resources and Digital ¨ ru¨mqi 830011, Xinjiang, China. Geology, U 4 State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Wuhan 430074, China. 5 State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Beijing 100083, China. 6 Department of Earth and Space Science and Engineering, York University, Toronto M3J 1P3, Canada. 7 To whom correspondence should be addressed; e-mail: [email protected]

INTRODUCTION The question of whether element concentrations in the EarthÕs crust follow some kind of universal geochemical law has been investigated for several decades since Ahrens (1953, 1954) pointed out that the frequency distribution of element concentration is not normal but lognormal (e.g., Oertel 1969; Bølviken et al. 1992; Allegre and Lewin 1995; Reimann and Filzmoser 2000; Rantitsch 2001; Monecke et al. 2005; Cheng 2007; Agterberg 2007, 2015). It should be noted in AhrensÕ paper (1953,

 2018 International Association for Mathematical Geosciences

Liu, Cheng and Zhou 1954) the samples were collected from specific rock types (e.g., granite, igneous rock), which possibly resulted in enhancing homogeneity of the elements. It can be readily understood why element concentration values show a greater tendency to a lognormal distribution, because this operation may restrict the variability of rock types, and these relatively high or low values may be arbitrarily eliminated. Another explanation for most data tending to normality or lognormality lies in the central limit theorem (CLT) in mathematical statistics, and geologically is caused by the increasing randomness in geological/geochemical processes. Ondrick and Griffiths (1969) examined frequency distributions of major, minor and trace elements by using normal and lognormal models, indicating that most distributions of elements fail to follow strictly hypothetical distributions. Sampling of heterogeneous materials and the presence of anomalous dispersion patterns overlain on previous distribution patterns can cause skewed or even multimodal distributions (Allegre and Lewin 1995). Although logarithmic transformation has been widely used to improve the efficiency of mean estimation, there are still a number of outliers expressed as positive skewness (Link and Koch 1975; Govett et al. 1975). Given that the differences between normal/lognormal and tailed distributions may result in considerable effects on target estimation, it is meaningful to determine whether it is possible to quantify the behavior of these tailed data, and if so, what parameters can be used to describe these tailed distributions. Since the concept of fractal/multifractal analysis was proposed by Mandelbrot in the 1970s (Mandelbrot 1975, 1983), fractal/multifractal analysis has been widely used to characterize variability of many natural phenomena. Meanwhile, it provides a new perspective for geologists exploring element distribution patterns, because geochemical dispersion patterns commonly exhibit strong nonlinearity and large variations over a wide range of spatial scales. Probability distribution patterns of geochemical dispersion such as normal, lognormal and power law have been demonstrated to associate with multifractal distribution, because the heavy skewed or tailed distributions can be characterized by scaleinvariant power-law behavior. For example, Turcotte (1986) claimed that tonnage and mean grade relationship for economic ore deposits show a power-law relationship; Cheng et al. (1994) proposed a concentration–area method to characterize the power-law relationship between element con-

centration values and cumulative area on a log–log plot; and Allegre and Lewin (1995) interpreted empirical frequency distributions as normal/multimodal distributions, and fractal/multifractal distributions. It is well known that geochemical compositions carry relative information (Egozcue et al. 2003; Pawlowsky-Glahn and Buccianti 2011; Filzmoser et al. 2011; Carranza 2017). A perspective from compositional theory indicates that multivariate analysis integrated with log-ratio transformed methods allows for better understanding the dynamic mechanisms of geochemical systems by interpreting the frequency distribution patterns of geochemical compositions (Buccianti 2015; Buccianti et al. 2017). Recent studies also indicate that the theory of multiplicative cascade processes plays a significant role in explaining genetic mechanism of geochemical distributions (Agterberg 2007, 2015; Cheng 2012, 2014, 2017a; Buccianti et al. 2017). Although previous scholars have done much on exploring element distribution patterns in geochemistry, some questions are not well addressed. For example, how to quantitatively characterize and separate different geochemical distributions, and their spatial patterns? A single fractal dimension cannot sufficiently describe complex geological processes or natural phenomena because of heterogeneous characteristics, while a multifractal is composed of a set of spatially intertwined fractals over a series of physical space that is frequently characterized by scale invariance, self-similarity, multiscale, variability and heterogeneity (Halsey et al. 1986; Cheng 1999; Agterberg 2012). From the perspective of multifractals, Cheng (2016, 2017b) proposed the concept of fractal density to characterize extreme natural phenomena such as flood, earthquake and heat flow of the mid-ocean ridge. According to the properties of fractal density, ordinary density such as volume density (e.g., g/cm3 and kg/m3) described in Euclidean geometry can be considered as a special case of the fractal density (e.g., g/cma and kg/ma). Based on fractal density and power-law models, Chen and Cheng (2017) studied singularity properties of crustal physicochemical features using the German KTB main hole data, and Cheng (2017b) discussed the relationships of global zircon U–Pb ages with the growth of the continental crust and the development of supercontinents. For discussion of geochemical distribution patterns, previous studies mainly focused on studying frequency distribution patterns, whereas geographic

New Insights into Element Distribution Patterns in Geochemistry information of a sample (e.g., rocks, stream sediments, soils) collected from a specific location is commonly ignored. Actually, insights to spatial distribution patterns of geochemical data are equally important to frequency distribution patterns. To some degree, the spatial distribution pattern is more important in practical applications. For example, delineation of geochemical anomalies for mineral exploration and separation of potential heavy metal contamination baselines are always required to examine geochemical anomalies that usually possess different spatial patterns. Additionally, the frequency distribution of measured elements cannot be meaningfully interpreted without considering whether the spatial distribution in the measurements is stochastic or deterministic. The use of a fractal density model and relevant multifractal parameters combined with geographical information systems (GIS) provides an effective means for investigating element distribution patterns within the frequency and space domains. Variations in stream sediment geochemical data can be influenced by various processes (e.g., diagenesis, mineralization, chemical weathering, transport, deposition and erosion, and anthropogenic activity) that could result in a complex catchment basin(s) system in which numerous factors (e.g., parent material, climate, organisms, topography and heavy metals) may interact and mix together. According to Buccianti and Zuo (2016), a complex catchment basin system could experience interdependent feedback mechanisms with transition to more stable interactions. Therefore, sampling from catchment basin(s) system presents significant challenges to developing a clear understanding of the nature of these surface materials, although catchment basin analysis methods have been developed to reduce these influences so that geochemical background and anomaly can be identified accurately (Carranza and Hale 1997; Carranza 2010a, b; Yousefi et al. 2013; Abdolmaleki et al. 2014; Lancianese and Dinelli 2016; Nezhad et al. 2017; Cracknell and de Caritat 2017). In the present study, stream sediment samples collected from the west Tianshan region of Xinjiang (China) were used to demonstrate the application of fractal density model in modeling element distribution patterns on the EarthÕs surface. According to Carranza (2010a), the application of continuous or discrete field modeling of geochemical landscapes depends largely on the mapping scale of

exploration and/or environmental geochemical surveys, and the sampling density. Considering the scale of the study area, sampling density and the purpose of the present study, continuous field modeling of stream sediment geochemical data was adopted to study element distribution patterns within a complex geologic evolution system or catchment basins system. Four elements, Cu, Zn, K and Na, were selected to investigate the differences between minor and major elements in terms of their distribution patterns within a given regional-scale study area where many Cu occurrences are present.

METHODS Fractal Density Model Density is a fundamental property of mass or energy that usually refers to a measure of how much of mass or energy is within a fixed amount of space. The most commonly used units are the units of mass over volume (e.g., g/cm3, kg/m3) or energy over volume (J/cm3, w/l3). Mathematically, density is defined as mass divided by volume: q¼

mðvÞ v

ð1Þ

where q is the density, m is the mass, and v is the volume. For a homogeneous object, density has the same numerical value as its mass concentration. However, if the object has heterogeneous properties, density varies between different regions of the object, which can be calculated using the derivative of the mass over volume: q¼

dmðvÞ mðvÞ ¼ lim v!0 v dv

ð2Þ

It should be noted that the above definition is correct just under the condition that the object belongs to Euclidean geometry, implying changing the amount of a substance does not change its density. However, from the perspective of fractals and multifractals, traditional Euclidean geometry is insufficient to measure irregular geometry or uneven distributed compositions. Recently, Cheng (2016, 2017b) proposed the concept of fractal density (qa) to characterize irregular geometry by adding a parameter a to Eq. 2, expressed as:

Liu, Cheng and Zhou mðvÞ v!0 va=3

qa ¼ lim

ð3Þ

The fractal density qa can be considered as the generalization of ordinary density q, and a is the fractal dimension expressed by continuous numerical value. Therefore, the fractal density defined in Eq. 3 has the units of mass ratio to a fractal set of a dimensions such as kg/ma or g/cma. A special case of qa is the integral dimension with a = 3, corresponding to ordinary density q. According to Eqs. 2 and 3, the relationship between ordinary density and fractal density can be determined by: qðvÞ ¼ qa v½1a=3

ð4Þ

Consequently, ordinary density follows a power-law relationship with volume and has the following properties (Cheng 2007; Cheng and Agterberg 2009): 8 < 0; if a[3 lim q 1; if a\3 ð5Þ v!0 : qa ; if a ¼ 3 According to Eqs. 3 and 4, fractal density can be generalized to other situations expressed by a more general model: qðeÞ ¼ qa e½Ea ¼ qa eDa

ð6Þ

where E is the Euclidian dimension (E = 1, 2, 3), and e is the linear size of an E-dimensional set. E = 1, 2, 3 represents linear, areal and volumetric fractal density, respectively. The singularity index Da can measure the degree of heterogeneity or singularity of fractal density distribution; thus, Da = 0 suggests the nonsingular property of an object with qa = q, while Da „ 0 suggests the heterogeneity of the object. On a log–log plot, the relationship between q(e) and e can be fitted by least squares method to determine the slope (E  a) and the intercept qa.

Multifractal Analysis Several methods have been implemented for multifractal modeling, among which the moment method is the most commonly used for characterizing multifractal structures and related spatial statistics (Halsey et al. 1986; Cheng 1999; Xie and Bao 2004). The moment method involves three functions: mass exponent function s(q), Coarse Holder expo-

nent or singularity exponent a(q) and fractal spectrum f(a). The mass exponent function s(q) is estimated first from the partition function: log vðeÞ ¼ lim sðqÞ ¼ lim e!0 log e

log

NðeÞ P i¼1

lqi ðeÞ

log e

ð7Þ

e!0

where e represents box size, N(e) is number of boxes, v(e) is the partition function obtained from different moments q ( ¥
ð8Þ

where / denotes proportionality. The functions a and f(a) can be obtained by Legendre transformation expressed as (Evertsz and Mandelbrot 1992): aðqÞ ¼

dsðqÞ ; dq

f ðaÞ ¼ aðqÞq  sðqÞ

ð9Þ

The singularity spectrum curve is a concave shape extending over a finite range [amin, amax], which is determined by plotting of f(a) vs. a. The amin and amax denote the minimum and maximum values of the singularity exponent, respectively. The width of the singularity spectrum, Da = amax  amin, reflects the degree of heterogeneity. The larger the Da, the stronger the multifractality is and the more heterogeneous the object is.

Singularity–Quantile Method Local singularity analysis is an efficient technique for geochemical anomaly identification due to its capacity of distinguishing different geochemical patterns within a sample population (Cheng 2007). Geochemical distribution patterns characterized by singularity indices in the frequency domain may follow normal, lognormal and multifractal distributions. Specifically, most singularity indices with a  2 satisfy normal or lognormal distribution, whereas singularity indices with a>2 and a<2 might follow fractal/multifractal distributions (Cheng 2007; Cheng and Agterberg 2009). Recently, a singularity–

New Insights into Element Distribution Patterns in Geochemistry quantile (S-Q) method was proposed to separate geochemical anomalies related to mineralization, which provides excellent performance for separating multiple geochemical populations based on integrating singularity analysis and quantile–quantile plot analysis (Liu et al. 2017). In this study, we attempt to apply the S-Q method to explore element distribution patterns based on a fractal density model. The frequency distribution patterns of singularity indices can be characterized by plotting singularity index quantiles vs. standard normal quantiles. As indicated by Figure 1, the x- and yaxes are represented by standard normal quantiles and singularity index quantiles, respectively. If a frequency distribution is normally distributed, data will be fitted by a straight line. From a statistical point of view, most values of a  2 approximately obey lognormal distribution as wells normal distribution, because these data are located near the normal reference line and are obtained from the intercept of the plot of qa(e) vs. e on the log–log scale. In order to limit these values of a  2, we set a 99% confidence interval of singularity indices and select suitable percentile intervals such as 15th and 85th, to determine the normal reference line and residual fitting curve. The polynomial curve can be fitted by total a-values (Fig. 1). Using these three equations, two intersection points or thresholds [(x1, y1), (x2, y2)] can be solved above and below the normal reference line, respectively. Therefore, hybrid distribution patterns of singularity indices can be separated into three groups. Then frequencydistributed singularity indices are converted back to the spatial domain for visual representation of different patterns. The S-Q method belongs to nonparametric statistics that do not rely on the assumption of normality, thus less rigorous conditions are imposed on the scales of measurement.

CASE STUDY AND DATASET The west Tianshan region located in the southern margin of the Central Asian Orogenic Belt (CAOB) has undergone multiple tectonothermal events during the Paleozoic (Fig. 2a) (Xiao et al. 2008; Gao et al. 2009), and hosts several porphyrytype Cu (Mo, Zn) deposits/occurrences (Zhang et al. 2010; Zhao et al. 2014). The region is principally composed of high mountains and a series of catchment systems on the landscape (Fig. 2b). The topo-

graphic elevation varies from around 1500–4500 m. The dense drainage and snow water in the area contribute to the transportation of stream sediments that lead to the propagation of geochemical elements in a catchment basin system. Stream sediment samples were collected at the bottom of riverbeds, near the waterlines, at dried-up riverbeds or near paleo-channels, which represent drainage basins measuring a total of over 70,000 km2 across the west Tianshan region. A total of 1874 samples were collected, and concentrations of 39 minor elements and major oxides were analyzed, including Cr, Nb, P, Pb, Th, Ti, Y, Zr, Ba, Be, Co, Cu, La, Li, Mn, Ni, Sr, V, Zn, Ag, B, Sn, Cd, Au, As, Sb, Bi, Hg, U, F, W, Mo, Fe2O3, K2O, Al2O3, SiO2, CaO, MgO and Na2O (Xie et al. 1997; Liu et al. 2016), among which Cu, Zn, K2O and Na2O were selected and used in this study to explore element distribution patterns within frequency and space domains based on a fractal density model.

RESULTS AND DISCUSSION Fractal Concentration Density and Model Fitting Here, a fractal density model was generalized to characterize geochemical structures of stream sediments that are controlled by universal element distribution patterns in the EarthÕs crust. First, spatial distribution maps of raw geochemical data are created by inverse distance weighted (IDW) interpolation with 1 km 9 1 km grid cells. Figure 3a and d shows ordinary density of minor elements (Cu and Zn) and major elements (K and Na) whose units are defined as c/e20 that is characterized in Euclidean space (where c denotes average element concentration measured in ppm for Cu and Zn, and in percentage for K and Na; e20 denotes the measurement of unit size e0 in 2-dimensional Euclidean geometry). The box-gliding algorithm complied in MATLAB code was used to estimate fractal density of Cu, Zn, K and Na that is determined by the intercept qa on the log–log plot. Figure 3e and h shows fractal density of individual elements characterized by continuous fractional dimensions with the unit of c/ ea0 (where c is average element concentration measured in ppm for Cu and Zn, and in percentage for K and Na, ea0 is the measurement of unit size e0 in a-dimensional fractal geometry). Obviously, the fractal density model has reshaped the spatial distribution patterns of geochemistry data by means of

Liu, Cheng and Zhou

Figure 1. Illustration of S-Q method in frequency domain.

fractional dimensions as shown in Figure 3e and h. The results indicate that the variabilities of Cu and Zn are greater than those of K and Na by comparison of individual fractal density and ordinary density that can be observed by their maximum and minimum values (Fig. 3). There are similar spatial distribution patterns between fractal density and ordinary density of individual elements, although the numerical differences may be significant as indicated by Figure 3a vs. e and b vs. f. It can be presumed that geochemical information that cannot be characterized by ordinary density could be revealed by fractal density by transformation of geochemical data from Euclidean geometry to fractal geometry with a set of fractional dimensions. For instance, the maximum value of Cu fractal density is 387.624 expressed in fractal geometry, while the maximum value of Cu ordinary density is 193.36 expressed in Euclidean geometry, implying that a fractal density model is better to reveal extreme values than ordinary density, and thus more efficient for characterizing geochemical heterogeneity. Moreover, Figure 3 suggests that a fractal density model is more suitable for modeling minor element distribution patterns, instead of major elements in terms of characterizing the degree of element enrichment or depletion. In addition, the fractal density model has the ability to overcome the influence of the smoothing effect (e.g., moving average) of interpolations such as IDW and multifractal IDW methods as indicated by Figure 3, showing a better performance in characterizing extreme values compared to ordinary density. Similar

to local singularity analysis (Cheng 2007), it can be concluded that a fractal density model is suitable to depict these extreme geo-processes in the crust. In order to examine the relationships between fractal density and singularity index of individual elements, the power model was applied to fit the observed data, as indicated by Figure 4. The results indicate that better power-law models are determined by Cu and Zn according to the R2 (square of correlation coefficient R) values of 0.7339 and 0.7488, respectively (Fig. 4a and b). In contrast, the R2 values for K and Na of 0.5491 and 0.3057, respectively, indicate lower correlation between fractal density values and singularity indices (Fig. 4c and d). Figure 4a and b shows that power-law models with high R2 values are caused by a small amount of extreme data on both tails of the curves. Figure 5 shows log-transformed fractal density of individual elements plotted against singularity index, which is fitted by a linear model. There appear to be good linear relationships between the log-transformed fractal density and singularity index. Overall, the R2 values indicate that the power-law model is more suitable for fitting the relationship between fractal density and singularity index because of high correlation in comparison with the linear model. However, a slightly better linear model than powerlaw model can be observed based on the R2 values of 0.3429 and 0.3057 shown in Figures 5d and 4d, respectively, implying non-significant influence of extreme values on Na concentration values. As stated above, the R2 values indicate that the power-law model commonly provides better results,

New Insights into Element Distribution Patterns in Geochemistry

Figure 2. (a) Schematic map showing the position of west Tianshan region. (b) Sampling sites of geochemical data overlain on a shaded relief map.

especially for minor elements Cu and Zn, in terms of fitting the relationships between fractal concentration density and singularity index. However, major elements such as K generally do not fit power-law distributions because of lower correlation as indicated in Figure 4c and d, which are more likely

tending to be normal or lognormal distribution. Power-law distribution often refers to heavy-tailed distribution or Pareto distribution. Previous studies demonstrate that power-law distribution is closely associated with lognormal distribution because both are generated from multiplicative cascade processes,

Liu, Cheng and Zhou

Figure 3. Ordinary density and fractal density of Cu, Zn, K and Na, on the left expressed as ordinary density with unit size of c/e20 (Fig. 3a–d) and to the right expressed as fractal density with unit size of c/ea0 (Fig. 3e–h).

New Insights into Element Distribution Patterns in Geochemistry

Figure 4. Power model to fit the fractal density of (a) Cu, (b )Zn, c K, d Na.

although lognormal distribution tends to be determined by relatively homogeneous multiplicative interactions (Lovejoy and Schertzer 2007; Agterberg 2007; Cheng 2012; van Rooij et al. 2013). Korvin (1992) provided a heuristic exposition of the idea that self-similarity can result in a power-law relationship. By investigation of episodic growth of the continental crust, Cheng (2017a) explained the generation of power-law distribution with singularities based on phase transition, self-organized criticality and multiplicative cascade processes, which provided deep understanding of mantle convection mechanisms and the causation between the U–Pb age peaks and avalanches of events in the evolution

of the EarthÕs crust. Nevertheless, it should be noted that the power-law distribution of element concentration is not a decisive condition for producing multifractal distribution.

Multifractal Spectrum Analysis of Fractal Density Here, multifractal structures of geochemical distribution patterns were characterized by using a continuous spectrum of scaling exponents. The multifractal spectra and its parameters provide valuable information in the investigation of geochemical distribution patterns that enable us to

Liu, Cheng and Zhou

Figure 5. Power model to fit the fractal density after log-transformation of (a) Cu, (b) Zn, (c) K, (d) Na.

examine the local scaling properties of individual elements (Xie and Bao 2004; Panahi and Cheng 2004; Xie et al. 2010). The height of the spectrum f(q) corresponds to the dimension of these scaling exponents with smaller f(q) values meaning rare events. The width of multifractal spectrum f(a) and singularity exponents Da = amax  amin suggests the strength of singularity or the degree of irregularity of element concentration enrichment around a local space. In this study, the partition function of different q values ranging from  4 to 4 with an interval of 0.1 was constructed. The multifractal spectra of individual elements are shown in Figure 6 by plotting the fractal spectrum f(q) against the singularity

exponent a(q). It can be observed that multifractal spectrum curves for Cu and Zn have greater deviation toward the left, compared to the deviations to the right, while K and Na possess opposite characteristics, implying that minor elements have different geochemical distribution patterns from major elements (Fig. 6). The strength of the singularity also indicates that Cu and Zn have stronger heterogeneity and irregularity in comparison with K and Na. The largest Da value of 0.9271 for Cu appears to agree with the geological setting that largescale Cu mineralization experienced complex geological/geochemical processes in the study area, and the degree of Cu enrichment or depletion is significantly influenced by geological/geochemical pro-

New Insights into Element Distribution Patterns in Geochemistry

Figure 6. Singularity spectrum of fractal density of (a) Cu, (b) Zn, (c) K, (d) Na.

cesses. In contrast, the Da values for K and Na indicate that major elements do not possess strong heterogeneous scale-invariant characteristics over a wide range of spatial scales (Fig. 6c and d). The nonsymmetric shape can reflect different element enrichment/depletion patterns (Xia et al. 2004). The wider the left branch determined by a<2, the greater the singularity is, which implies a highly localized element enrichment pattern, or otherwise if the right branch is determined by a>2. The multifractal spectrum has proved to be an important indicator in characterizing geochemical behaviors for the sake of mineral exploration (Xie et al. 2010; Arias et al. 2012). According to the properties of the singularity index, as described in sections fractal density model and singularity–quantile method,

normal/lognormal distributed data are located around f(a) = 2. It can be readily observed that major elements K and Na appear to be close to a normal/lognormal distribution because of the narrower intervals of singularity exponents with [1.9467, 2.0843] and [1.9270, 2.2447], respectively, by comparison with the singularity exponents of Cu and Zn.

Frequency-Based Geochemical Pattern Analysis Fractal density of element concentration was used to generate singularity maps based on the local singularity analysis technique. It should be noted that singularity index a-values are not necessary the same as singularity exponents that were described in

Liu, Cheng and Zhou

Figure 7. Singularity index quantiles vs. standard normal quantiles of (a) Cu, (b) Zn, (c) K2O, (d) Na2O.

the preceding section multifractal spectrum analysis of fractal density, because they are constrained by different parameters in singularity analysis and multifractal spectra analysis. Without any assumptions about data distribution, the S-Q method was used to produce a graphical data expression relying on the inherent geochemical structure, which allows direct detection of deviations from normality or lognormality, because data with normal/lognormal distribution are exhibited around a straight line on the S-Q plot. Figure 7 shows the S-Q method performed on singularity indices of fractal density of individual elements in the frequency domain. The distribution

patterns of singularity indices were clearly distinguished by the S-Q method, and thus mixed geochemical distribution patterns were separated into three subsets according to two thresholds. The avalues near the normal reference line are constrained by two fitting lines, which are supposed to follow a lognormal distribution including normality because singularity indices are derived from the log– log plot. Theoretically, normally distributed geochemical data also satisfy lognormal distribution, because these partially skewed geochemical data that do not follow normal distribution can be adjusted to some degree by log-transformation, whereas data with lognormal distribution necessarily

New Insights into Element Distribution Patterns in Geochemistry fit a normal distribution. The a-values above and below the fitting lines are defined as upper- and lower-truncated tail distributions, respectively, but they do not limit the multifractal distribution that will be discussed in detail in the following paragraphs. Because the S-Q method provides critical threshold estimation for separating element concentration values into three different subsets within a larger population, these thresholds may correspond to unusual physicochemical attributes caused by drastically changing geological phenomena such as lithostratigraphic contact, local mineralization, weathering product, fluid phase, mineral alteration phase, fault orientation or special rock mass. In practice, these critical thresholds are of great significance for geochemical anomaly separation, which have been successfully applied for chromitite exploration in the Western Junggar region, China, where chromite anomalies delineated by critical thresholds are mainly caused by ophiolite me´lange and related weathering products (Liu et al. 2017). According to the thresholds determined by a<2, a  2 and a>2, the fractal density of individual elements was further separated into three groups. Then, the relationships between fractal density and singularity index were examined by scatter plot analysis and further employed to characterize the frequency distribution patterns of individual elements. As indicated in Figure 8, the power-law relationship of the fractal density of Zn with singularity index has the strongest correlation coefficient of R2 = 0.837 under the condition of a<2 (Fig. 8d); the second best power-law fitting model occurs for the relationship between the fractal density of Cu and the singularity index as indicated by the correlation coefficient of R2 = 0.7687 (Fig. 8a). High correlation expressed by the power-law model between fractal density and singularity index of a<2 for minor elements (Cu and Zn) implies that element distribution patterns with multifractal structures can be estimated from local singularity analysis. It can be concluded that the fractal density of minor elements constrained by singularity indices less than 2 obeys a multifractal distribution. However, fractal density values defined by a>2 appear not to follow a multifractal distribution because of relatively low correlation R2, which will be interpreted with the aid of histograms of Cu and Zn. Figure 8g and j shows that major elements K and Na have weak power-law relationships between fractal density and singularity index when the singularity indices are less than 2, especially for Na

which shows that almost no correlation exists as indicated by the R2 value of 0.0056 (Fig. 8j). Therefore, the fractal density of major elements defined by a<2 cannot be defined as having a multifractal distribution. The same behaviors can be found in Figure 8i and m under the condition of a>2. According to Fig. 1, it is hard to explain why these data deviate from the normal reference line and what distribution pattern they follow. We consider that these data follow a lognormal distribution but have a different mean or dataset mode from the data defined by a  2. This question will be given detailed explanation by means of their histograms in the following paragraphs. Under the condition of a  2, the linear model was used to fit the relationships between fractal concentration density and singularity index of individual elements. The R2 are 0.1726, 0.1398, 0.2213 and 0.1533 for Cu, Zn, K and Na, respectively, shown in Figure 8b, e, h and k, indicating weak correlation between fractal density and singularity index for major elements, implying that they are more likely to follow a random distribution or lognormal/normal distribution. It can be also observed that most data tend to be concentrated around the fitting line that may be explained by CLT that the average of an increasing number of independent variables tends toward a normal distribution under the premise of an unconstrained hypothesis. Histograms combining statistical parameters such as mean, standard deviation (Std), kurtosis and skewness can provide a powerful tool for graphical and statistical characterization of geochemical distribution patterns. Histograms of fractal density of individual elements were plotted under the condition of a>2 and a<2, and then a fitted normal distribution line was superimposed as shown in Figure 9. Several statistical parameters including maximum (Max), minimum (Min), mean, Std, kurtosis and skewness were calculated, among which kurtosis and skewness are important indicators for justifying whether the data follow normal or other distributions. Kurtosis is a measure of whether the peak of the distribution is sharper or flatter than a normal distribution (Taylor 2008). Theoretically, a kurtosis value equal to 3 indicates a standard normal distribution, and kurtosis value greater than 3 indicates more outliers rather than following a normal distribution, and the reverse for kurtosis value less than 3. As shown in Figure 9a and c, the histograms of Cu(a<2) and Zn(a<2) depict very high kurtosis

Liu, Cheng and Zhou

Figure 8. Frequency distribution patterns of fractal density with (a) aCu<2, (b) aCu  2, (c) aCu>2, (d) aZn<2, (e) aZn  2, (f) aZn>2, (g) aK<2, (h) aK  2, (i) aK>2, (j) aNa<2, (k) aNa  2, (m) aNa>2.

New Insights into Element Distribution Patterns in Geochemistry

Figure 8. continued.

Liu, Cheng and Zhou

Figure 9. Histograms of fractal density of (a) aCu<2, (b) aCu>2, (c) aZn<2, (d) aZn>2, (e) aK<2, (f) ak>2, (g) aNa<2, (h) aNa>2, superimposed with a fitted normal distribution line.

values of 30.5819 and 40.9148, respectively, which are supposed to be leptokurtic, implying that fractal density distributions of Cu(a<2) and Zn(a<2) produce more extreme values than a normal distribution. Figure 9e and f shows that histograms of K(a<2) and K(a>2) have kurtosis values slightly higher than 3, meaning that only a few extreme values exist in the samples and exhibit an approximately normal distribution. Figure 9g and h shows that the kurtosis values of fractal density distributions of Na(a<2) and Na(a>2) are less than 3, which are supposed to be platykurtic, implying that both distributions generate fewer and less extreme

values than a normal distribution. It also appears that a bimodal distribution curve may occur if the samples were split by the dividing line in Figure 10g, indicating the existence of two types of lognormal distributions. The skewness was applied to measure the asymmetry of data distribution (Groeneveld and Meeden 1984), and to further examine whether a fractal density distribution follows normal or other distributions. Theoretically, skewness equal to 0 indicates a standard normal distribution. Commonly, the abundance patterns of minor elements in rocks and minerals exhibit positive skewness with a longer

New Insights into Element Distribution Patterns in Geochemistry

Figure 9. continued.

tail on the right, indicating that element concentration distribution pattern is asymmetric due to a few extreme values that highly deviate from the data mean, median or mode. The distribution diagrams for Cu(a<2), Cu(a>2), Zn(a<2), Zn(a>2), K(a<2) and Na(a>2) in Figure 9 exhibit rightskewed patterns, while K(a>2) and Na(a<2) exhibit left-skewed patterns, as indicated by the skewness values. Note that the histograms of Cu(a<2) and Zn(a<2) have significant asymmetry with long-right spread tails indicated by skewness values of 4.0668 and 4.7934, respectively, implying that both variables do not satisfy a normal distribution (Fig. 9a and c). However, the histograms of

fractal density of other variables tend to show symmetric concentration distributions, especially K and Na, implying that these variables are approximately normally distributed under the condition of a>2 and a<2. It is noted that they closely follow a lognormal distribution, including the fractal density of Cu(a> 2) and Zn(a>2), because these variables are constrained by singularity indices that deviate from the normal reference line as shown in the log–log plot (Fig. 1). The fractal concentration density of minor and major elements defined by a>2 does not exhibit a significant multifractal distribution, but has an approximate lognormal distribution due to the

Liu, Cheng and Zhou

Figure 10. Singularity maps with continuous values of (a) Cu, (b) Zn, (c) K, (d) Na.

existence of extreme values. This phenomenon has been interpreted to mean that these extremely small values cannot be detected using present techniques, and thus are constrained by the detection limit of laboratory analysis (Ma et al. 2014). Commonly, values below detection limit are set to some fixed values instead of the true measured values, which is why no extreme or very small values exist at the lower end of the plot. Another problem might be that geochemical data are compositional following internal proportionality among geochemical compositions (Aitchison 1986), and minimum values are constrained in real space, unlike maximum values that are infinite at the upper end of the plot. Much research work has been done on the relationships among normal/lognormal, fractal/multifractal, power-law, Pareto or other distributions related to geochemical elements in the EarthÕs crust (Turcotte 1986; Bølviken et al. 1992; Reimann and Filzmoser 2000; Cheng 2007; Liu et al. 2013; Agterberg 2017). Recently, the frequency distribution patterns (e.g., normal, lognormal, power laws, multifractality) of major and minor elements in ba-

saltic volcanic glasses were explored based on logratio transformation and principal components analysis, indicating the presence of highly fragmented geochemical processes in the spatial and temporal domain (Buccianti 2015). A new idea using isometric log-ratio coordinates representing weathering chemical reactions to investigate the frequency distribution patterns has been developed, which allows to characterize power law and lognormal distributions and capture the behavior of nonlinear systems where the presence of fractal/multifractal structures is due to complex dynamic mechanisms of geochemical systems (Buccianti and Zuo 2016). From another perspective, Allegre and Lewin (1995) proposed a unified theory based on differentiation-mixing operators to interpret the formation of element distribution patterns in the EarthÕs materials, describing that the differentiation operator results in fractal distributions, whereas the mixing one produces normal distributions. Based on such knowledge, Buccianti (2015) considered that the sequential binary partition (SBP) approach (Egozcue and Pawlowsky-Glahn 2005) may be able to capture the proportional effect

New Insights into Element Distribution Patterns in Geochemistry

Figure 11. Singularity map with categorical values based on S-Q analysis of (a) Cu, (b) Zn, (c) K, (d) Na.

of element partition processes that characterize crystallization and melting phenomena in solid geological materials. For our studies, multifractal distribution is the result of a series of fractionation processes during a more complex history with multiple episodes of remobilization, which can be characterized by a<2, while multimodal distributions of normality and lognormality are derived from the sum of a few distributions that have different mean or modal values (Sinclair 1991; Stanley2006), which can be characterized by a  2 and a>2, and a special case can be observed in the histogram of Na(a>2) exhibiting a bimodal pattern (Fig. 9g).

Space-Based Geochemical Patterns Analysis Frequency-based S-Q analysis can reveal element distribution patterns based on the ranking of singularity indices, but it is unable to detect the heterogeneous and homogeneous geochemical structures in space. In order to better spatially visu-

alize multiple geochemical distribution patterns that are characterized in the frequency domain, singularity indices of individual elements calculated from local singularity analysis were first expressed as spatial patterns with continuous values, as shown in Figure 10. According to the maximum and minimum values indicated by the legends of Figure 10, minor elements Cu and Zn possess stronger heterogeneous characteristics, compared to major elements K and Na. The patterns show that the degree of element enrichment is inversely proportional to the singularity index. Areas with high degree of singularity are characterized by singularity indices a<2, while singularity indices a>2 show that areas with high degree of singularity represent element depletion. After classification of singularity maps into three groups based on the thresholds (Fig. 11), the area percentage of individual elements with singularity indices defined by a<2, a  2 and a>2 were calculated, as shown in Table 1. It can be observed that the areas defined by a  2 account for more than 70% of the total area, which are commonly considered as background.

Liu, Cheng and Zhou Table 1. Area percentage of singularity indices of four elements defined by a<2, a  2 and a>2 Elements

Cu Zn K Na

Area percentage of singularity indices a<2 (%)

a  2 (%)

a>2 (%)

8.8 8.6 11.4 6.5

73.9 74.3 71.4 76.3

17.3 17.1 17.2 17.2

However, the areas defined by a<2 and a>2 occupy a small fraction of the total study area and are commonly considered as anomalous (Table 1). Figure 11 shows that normal/lognormal or random distributed background zones with fractal dimensions close to 2 occupy the majority of areas defined by a  2, while element enrichment and depletion zones, respectively, defined by a<2 and a>2, scatter around the study area. The distribution patterns of fractal density of individual elements defined by a<2, a  2 and a<2 are depicted in Figure 11. As stated above, the degree of element enrichment is inversely proportional to the singularity index, and minor elements follow multifractal distributions under the condition of a<2, and further, the relationship between fractal density and the singularity index can be determined by the correlation coefficient R. Therefore, the variability of fractal density with the singularity index can be explained by geological complexity or singularity across the study area. For example, high correlation coefficient R = 0.8768 for Cu(a<2) implies that larger fractal density values correspond with stronger geological complexity or more intense degrees of element enrichment (Fig. 12a). A similar situation explains Zn(a<2) as indicated by Figure 12d with correlation coefficient R = 0.9148. It should be noted that Figure 12a and d may reflect the variation of fractal density with the singularity index under the condition of a<2, but they cannot be applied to the variation of all fractal density values with singularity indices as shown in Figure 3e and h. The variations in fractal density of major elements K(a<2) and Na(a<2) tend to be randomly or normally distributed in space because of the small correlation coefficient R2 (Figs. 8g, j and 9e, g), implying that these very high or very low fractal density values do not reflect complex geological features and geochemical anomalies. The remaining figures including Figure 12b, c, e, f, h, i, k and m also

display a similar behavior, which approximates the normal/lognormal distribution according to the above description, meaning that large or small fractal density values cannot reflect inherent geological complexity and the degree of element enrichment/depletion. The results further indicate that geological complexity or singularity is not only characterized by larger fractal density values of minor elements (e.g., Cu and Zn), but also by relative small values, which are determined by where these values occur as indicated by Figure 12a and d, such as the interval from 9.9571 to 387.62 of Cu fractal density with a<2, because they possess multifractal structures across a set of spatial scales in the study area. Extreme values can be caused by complex geological events or processes that commonly occur at such places with dramatic physicochemical changes that result in mineralization and complex geological bodies. Therefore, extreme values are commonly very high or very low concentration values, but not necessarily all the high or low values. Detailed understanding of why extreme values occur at some places needs to be combined with identification of the geological setting. Relative to frequency-based geochemical pattern analysis, space-based geochemical pattern analysis is equally important because it provides a powerful tool for investigating the variations of different element distribution patterns with sample sites within an observed spatial scale. Thus, it is of considerable importance in the practice of such fields as mineral exploration and environmental assessment. Recent studies have considered spacebased distribution patterns of singularity indices to delineate geochemical anomalies associated with rare-earth-element mineralization and chromite mineralization (Liu et al. 2014, 2017). The areas depicted by singularity indices less than 2 are favorable for the formation of ore deposits. For environmental data, fractal density of some ele-

New Insights into Element Distribution Patterns in Geochemistry

Figure 12. Spatial distribution patterns of fractal density of (a) aCu<2, (b) aCu  2, (c) aCu>2, (d) aZn<2, (e) aZn  2, (f) aZn>2, (g) aK<2, (h) aK  2, (i) aK>2, (j) aNa<2, (k) aNa  2, (m) aNa>2.

Liu, Cheng and Zhou

Figure 12. continued.

ments characterized by singularity indices less than 2 may be indicative of a pollution source.

CONCLUSIONS 1. In this study, geochemical distribution patterns of minor/major elements were explored from the perspective of fractal density. We

consider that ordinary density of element concentration represents simplified approximations of geochemical systems, whereas fractal density reflects the inherent properties of geochemical systems. Fractal density can be obtained by transformation of ordinary density into fractal space by means of local singularity analysis. Our studies indicate that the fractal density analysis for

New Insights into Element Distribution Patterns in Geochemistry minor elements (e.g., Cu and Zn) possesses excellent performance in revealing extreme values that cannot be characterized by ordinary density in 2-dimensional Euclidean space, but the method is not efficient for major elements (e.g., K and Na). 2. Element distribution patterns of fractal density of individual elements were investigated geographically and statistically by means of their multiple distribution patterns including power-law, multifractal, tailed truncation, normal and lognormal distributions. Geochemical distribution patterns defined by a<2, a  2 and a>2 were compared between minor elements (e.g., Cu and Zn) and major elements (e.g., K and Na). Our studies support the idea that fractal density values of minor elements defined by a<2 tend toward multifractal distributions displaying a complex heterogeneous pattern, the values defined by a  2 tend toward normal/lognormal distributions, and the values defined by a>2 tend toward lognormal distributions. For major elements, the fractal density values tend toward lognormal distributions under the condition of a<2 and a>2, while the values defined by a  2 tend toward normal/lognormal distributions. It can be concluded from the spatial-based fractal density distribution patterns of minor elements that the degree of element enrichment or geological complexity is proportional to the fractal density values under the condition of a<2. 3. The S-Q method provides a powerful tool for separation of different geochemical distribution patterns that are originally mixed and coexisting in a myriad of fractional dimensions over a wide range of scales in space. Different visualization methods and statistical parameters such as histogram, multifractal spectrum, skewness and kurtosis were employed to examine geochemical distribution patterns. Our studies demonstrate that graphic inspection of the fractal density and singularity indices by means of statistical and geographical displays to investigate element distribution patterns in the EarthÕs crust are far better suited for estimating the thresholds, geological complexity, geochemical anomaly and background variation. The primary purpose for

geochemical pattern identification is to reveal potential patterns that control the distribution of the individual elements in the EarthÕs crust, which are critical for such fields as assessment of contaminated land, mineral exploration and environmental regulation.

ACKNOWLEDGMENTS Editor-in-Chief Dr. John Carranza is warmly thanked for efficient editorial handling and his helpful review of the manuscript, and special thanks to Dr. Antonella Buccianti and an anonymous reviewer for their valuable comments. This work is jointly funded by the CAS ‘‘Light of West China’’ program (2015-XBQN-B-23), and the National Natural Science Foundation of China (Nos: 41702356, U1503291, 41430320).

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