The aim of this study is to delineate and identify various mineralized zones
and barren host rocks based on surface and subsurface lithogeochemical data
from the Pulang porphyry copper deposit, southwestern China, utilizing the
number–size (N-S), concentration–volume (C-V) and power-spectrum–volume
(S-V) fractal models. The N-S model reveals three mineralized zones
characterized by Cu thresholds of 0.28 % and 1.45 %:

The definition and delineation of different mineralized zones and non–mineralized wall rocks are the main goal in economic geology and mineral exploration. The investigation of ore mineralogy and paragenetic sequence provides useful data on ore-forming processes in deposits because typical characteristics of various types of ore deposits are reflected by their mineral assemblages (Craig and Vaughan, 1994; White and Hedenquist, 1995). Common methods generally use mineralography, petrography and alteration mineral assemblage analysis to delineate various mineralized zones in porphyry deposits (Beane, 1982; Schwartz, 1947; Sillitoe, 1997; Berger et al., 2008). Lowell (1968) first proposed a conceptual model of the lateral and vertical variations in mineralogy within alteration zones. Some similar models were developed for potassic alteration, which is usually situated in the center and deep parts of porphyry ore deposits, based on this conceptual model (Sillitoe and Gappe, 1984; Cox and Singer, 1986; Melfos et al., 2002). Fluid inclusion and stable isotope studies are other methods used to outline different mineralization phases based on thermometric and isotope element parameters and other geological parameters (e.g., Boyce et al., 2007; Faure et al., 2002; Wilson et al., 2007). Drill hole data and logging information, including mineralographical information, host rock changes and alterations are helpful in delineating mineralization zones. Different geological interpretations could be used to detect zone boundaries, which may also lead to different results because the elemental grade distribution may not be taken into consideration.

Non-Euclidean fractal geometry (Mandelbrot, 1983) is an important branch of nonlinear mathematical sciences and has been applied in various research fields of the geosciences since the 1980s. The relationships between geology, geochemistry and mineralogical settings and spatial information can be researched by methods based on fractal geometry (Afzal et al., 2011; Carranza, 2008, 2009). Bolviken et al. (1992) and Cheng et al. (1994) have shown that geochemical patterns of various elements have fractal dimensions. The concentration–area (C-A) model was proposed by Cheng et al. (1994) to recognize geochemical anomalies from background concentrations and calculate elemental thresholds of different geochemical data. Furthermore, many other fractal models have been proposed and applied in geochemical exploration work, including the number–size (N-S) fractal model proposed by Mandelbrot (1983) and Agterberg (1995), the power-spectrum–area (S-A) fractal model proposed by Cheng (1999), the concentration–distance (C-D) fractal model proposed by Li et al. (2003), the concentration–volume (C-V) fractal model proposed by Afzal et al. (2011) and the power-spectrum–volume (S-V) fractal model proposed by Afzal et al. (2012).

Methods of fractal analysis also illustrate the relationships between geological, geochemical and mineralogical settings and spatial information derived from the analysis of mineral deposit occurrence data (Carranza, 2008, 2010; Carranza et al., 2009; Goncalves et al., 2001). Various geochemical processes can be described based on the differences in fractal dimensions obtained from the analysis of relevant geochemical data. Afzal et al. (2011) considered that the log–log plots obtained by fractal methods are useful tools to delineate different geological populations of geochemical data, and the thresholds could be determined as some breakpoints in those plots.

The application of fractal models to delineate various grades of mineralization zones was dependent on the relationships between the metal grades and volumes (Afzal et al., 2011; Agterberg et al., 1993; Cheng, 2007; Sim et al., 1999; Turcotte, 1986). Afzal et al. (2011, 2012) proposed a C-V and S-V fractal model to delineate different porphyry-Cu mineralized zones and barren host rocks. In this paper, N-S, C-V and S-V fractal models were applied to delineate various mineralized zones and barren host rocks in the Pulang porphyry copper deposit, Yunnan, southwestern China.

The N-S method proposed by Mandelbrot (1983) can be utilized to describe the distribution of geochemical populations (Sadeghi et al., 2012). In this method, geochemical data do not undergo any preprocessing (Mao et al., 2004). This model shows a relationship between desirable attributes (e.g., Cu concentration in this study) and their cumulative number of samples (Sadeghi et al., 2012). A power-law frequency model has been proposed to explain the N-S relationship according to the frequency distribution of elemental concentrations and cumulative number of samples with those attributes (e.g., Li et al., 1994; Sadeghi et al., 2012; Sanderson et al., 1994; Shi and Wang, 1998; Turcotte, 1989, 1996; Zuo et al., 2009).

The N-S model proposed by Mandelbrot (1983) can be expressed as follows:

Afzal et al. (2011) proposed a C-V fractal model
based on the same principle as the C-A model (Cheng et
al., 1994) to analyze the relationship between the concentration of ore
elements and accumulative volume with concentrations greater than or equal
to a given value (Afzal et al., 2011; Zuo and Wang, 2016;
Sadeghi et al., 2012; Soltani et al., 2014; Sun and Liu, 2014; Wang et al., 2011, 2012). This model can be expressed as follows:

Geological map of the Pulang porphyry copper deposit, SW China. Modified after © Yunnan Diqing Nonferrous Metal Co. Ltd. (2009).

Different geochemical patterns in the spatial domain could be seen as layered signals of various frequencies. Cheng (1999) proposed the S-A fractal model to recognize geochemical anomalies from backgrounds utilizing the method of spectrum analysis in the frequency domain according to this argument. This model is combined with a C-A model (Cheng et al., 1994), offering a useful tool to determine an optimum threshold value between various patterns based on the scaling property.

Afzal et al. (2012) proposed the S-V fractal model
to delineate different grades of mineralization zones based on the same
principle as the S-A model proposed by Cheng (1999). The S-V model
was utilized in the frequency domain by applying a fast Fourier
transformation to the assay data. The straight lines obtained by log–log plotting indicate the relationships between the power spectra and relevant
volumes of ore elements. These relationships were utilized to recognize the
hypogene zones and supergene enrichment zones from the barren host rocks and
the leached zone of the deposit. The recognition of various mineralization
zones is based on the power-law relationships between the power spectra and
occupied volumes. The formula is as follows:

Geological 3-D models including lithology, alteration and 3-D
drill hole plot with the legend of each in the Pulang porphyry copper
deposit. Scale is in cubic meters (

By using the method of geostatistical estimation, the drill hole data of elemental concentration values were interpolated to construct a block model of ore element distribution. The power spectrum values can be obtained by 3-D fast Fourier transformation of the ore element grades. The logarithm of all the power spectrum values and accumulative volume values were calculated. Additionally, the log–log plot between power spectrum and volume was drawn according to previously determined values. Then, the filters were constructed on the basis of threshold values obtained by the log–log plot of S-V. Finally, the power spectra were converted back to the space domain by utilizing inverse fast Fourier transformation.

Photographs of alteration and mineralization in the Pulang
porphyry copper deposit, SW China:

The Pulang porphyry copper deposit is situated in the southern end of the
Yidun continental arc, southwestern China (Fig. 1). The continental arc was
produced due to the westward subduction of the Garze–Litang oceanic crust (Deng
et al., 2014b, 2015; Wang et al., 2014). The Pulang ore deposit, one of the
largest porphyry copper deposits in China (Deng et al., 2012, 2014a; Mao et
al., 2012, 2014), is characterized by a typical porphyry-type alteration
zone. The geological characteristics of the deposit, including the
alteration types and their zonation, the geometry of the orebody, the
metallogenic time and the geodynamic settings, have been systematically
researched (Leng et al., 2012; Li et al., 2011, 2013). The deposit consists
of five ore-bearing porphyry bodies covering an area of approximately 9

The statistical characteristics of the residual.

The outcrop strata of the Pulang deposit are dominated by Upper Triassic
Tumugou Formation clastic rocks and andesite and Quaternary sediments (Fig. 1c). The Triassic porphyry intrusions primarily comprise quartz diorite
porphyry, quartz monzonite porphyry, quartz diorite porphyrite and
granodiorite porphyry. The Tumugou Formation strata were intruded by the
quartz diorite porphyry with an age of

The porphyry-type alteration zones transition from early potassium-silicate alteration through quartz sericite alteration to propylitization, upward and outward from the core of the quartz monzonite porphyry (Fig. 4). The wall rocks near the porphyries were mostly changed into hornfels. Systematic drilling has demonstrated that the potassium-silicate and quartz-sericite zones host the main orebodies, constituting the core of mineralized zones. The propylitic zones and hornfels only develop weak mineralization. The orebodies occur mainly in potassium silicate and quartz sericite and occur as veins in the propylitic zones and hornfels. The major rock types in the deposit are quartz monzonite porphyry, quartz diorite porphyrite, granite diorite porphyry, quartz diorite porphyry and hornfels (Fig. 2). Metallic minerals mainly include pyrite, chalcopyrite with a small amount of molybdenite and pyrrhotite (Fig. 3).

Based on the geological data (which include the collar coordinates of each
drill hole, azimuth and dip (orientation) as well as lithology and mineralogy) recorded
from 130 drill holes in the Pulang deposit, 20 492 lithogeochemical samples
were collected at 2

Cross section along exploration line 0 in the Pulang porphyry copper deposit, SW China. Modified after Wang et al. (2012).

Histograms of

The experimental semivariogram (omni-directional) of Cu data in the Pulang deposit.

The accuracy of the spatial interpolation analysis is verified by comparing the difference between the measured values and the predicted values to select the best variogram model. To test the variogram model, the cross-validation method was used to determine whether the parameters of the variogram model were correct. The distribution of the residual is normal (Fig. 7), and the mean error between the actual and estimated Cu grades is equal to 0 (Table 1). This result indicates that this model is reasonable and that the variogram parameters used for estimating the Cu grade are unbiased.

The cross-validation results:

The obtained block models were used as inputs to the fractal models. The
Pulang deposit was modeled by

The N-S model was applied to the Cu data (Fig. 8). The selection of breakpoints as threshold values is an objective decision because geochemical populations are defined by different line segments in the N-S log–log plot. The straight fitted lines were obtained based on least-square regression (Agterberg et al., 1996; Spalla et al., 2010). In other words, the intensity of element enrichment is depicted by each slope of the line segments in the N-S log–log plots (Afzal et al., 2010; Bai et al., 2010).

N-S log–log plot for Cu concentrations in the Pulang deposit.

Based on the classification of the 3-D model of Cu data and the thresholds obtained from the N-S fractal model (Table 2), highly mineralized zones are situated in the southern and central parts of the Pulang deposit and coincide with the potassium-silicate alterations. However, small and highly mineralized zones are located in the central parts of the Pulang deposit (Fig. 9). Moderately mineralized zones occur along a northwest–southeast trend and correlate with the phyllic zones. Weakly mineralized zones and barren host rocks are situated in the marginal parts of the area.

Threshold concentrations obtained by using a N-S model based on the copper percentage (Cu %) in the Pulang deposit.

Zones in the Pulang deposit based on thresholds defined from
the N-S fractal model of Cu data:

The occupied volumes corresponding to the Cu grades were computed to obtain the C-V model according to the 3-D model of the Pulang deposit. Through the obtained C-V log–log plot, the threshold values of the Cu grades were determined (Fig. 10). These results indicate the power-law relationship between Cu grade and volume. According to these results (Table 3), the low-concentration zones exist in many parts of the deposit and occur along a northwest–southeast trend. Moderately and highly mineralized zones are situated in several parts of the central deposit and to the south of the deposit (Fig. 11).

Threshold concentrations obtained by using a C-V model based on copper percentage (Cu %) in the Pulang deposit.

C-V log–log plot for Cu concentrations in the Pulang deposit.

Zones in the Pulang deposit based on thresholds defined
from the C-V fractal model of Cu data:

Based on the geological data (which include the collar coordinates of each drill hole, azimuth and dip (orientation) as well as lithology and mineralogy) recorded from 130 drill holes in the deposit, a 3-D model and block model of the distribution of Cu in the Pulang deposit were constructed with ordinary kriging using GEOVIA Surpac software.

The power spectrum (

S-V log–log plot for Cu concentrations in the Pulang deposit.

Zones in the Pulang deposit based on thresholds defined
from the S-V fractal model of Cu data:

Ranges of the power spectrum (PS) for different mineralization zones in the Pulang deposit.

Alteration models have a key role in zone delineation and in presenting geological models, as described by Lowell and Guilbert (1970). The potassic and phyllic alterations control major mineralization within supergene enrichment and hypogene zones according to these models. Models of Cu mineralization zones derived via fractal models can be compared with geological data to validate the results of analysis in different porphyry Cu deposits. The results of the fractal modeling of the Pulang deposit were compared with the 3-D geological model of the deposit constructed by using GEOVIA Surpac and drill hole data (Fig. 2). Moreover, the results obtained from these fractal models were controlled by mineralogical investigations.

Matrix for comparing performance of fractal modeling
results with the geological model.

Carranza (2011) has illustrated an analysis for the calculation of spatial correlations between two binary datasets, especially mathematical and geological models. An intersection operation between the mineralization zones obtained from fractal models and the different alteration zones in the geological model was performed to derive the amount of voxels corresponding to each of the classes of overlap zones (Table 5). Using the obtained numbers of voxels, the Type I error (T1E), Type II error (T2E) and overall accuracy (OA) of the fractal model were estimated with respect to different alteration zones and the geological data (Carranza, 2011). The OAs of the fractal models of the mineralized zones were compared as follows.

A comparison between highly mineralized zones based on the fractal models
and potassic alteration zones resulting from the 3-D geological model shows
that there is a similarity among these fractal models (Fig. 14). The overall
accuracies for the C-V, N-S and S-V models are 0.50, 0.51 and 0.52,
respectively (Table 6), which indicate that the S-V model gives better
results for identifying highly mineralized zones in the deposit. The number
of overlapped voxels (

Overall accuracy (OA), Type I and Type II errors (T1E and T2E, respectively) with respect to potassic alteration zone, resulting from the geological model and threshold values of Cu obtained through C-V, N-S and S-V fractal modeling.

Overall accuracy (OA), Type I and Type II errors (T1E and T2E, respectively) with respect to phyllic alteration zone, resulting from the geological model and threshold values of Cu obtained through C-V, N-S and S-V fractal modeling.

Results of XRF analysis of samples collected from different mineralized zones in the Pulang porphyry copper deposit.

Highly mineralized zones in the Pulang deposit:

Moderately mineralized zones in the Pulang deposit:

Chalcopyrite content in several samples based on
mineralographical study:

A comparison between phyllic alteration zones resulting from the 3-D geological model and moderately and weakly mineralized zones from the fractal modeling shows that the overall accuracies of the C-V, N-S and S-V fractal models with respect to phyllic alteration zones of the geological model are 0.59, 0.54 and 0.56, respectively. The overall accuracy of moderately and weakly mineralized zones obtained from C-V modeling is higher than that of mineralized zones obtained from N-S and S-V modeling (Table 7). On the other hand, moderately mineralized zones defined by C-V modeling overlap with phyllic zones defined by the 3-D geological model (Fig. 15). However, the results of the C-V model are more accurate than those of the N-S and S-V models with respect to the phyllic zones defined by the 3-D geological model.

It could be considered that there are spatial correlations between different modeled Cu zones and geological features such as alterations and mineralogy. Several samples were collected from different drill holes in different grades of mineralization zones of the Pulang deposit to validate the results of the fractal models. These samples were analyzed by microscopic identification and XRF (X-ray fluorescence spectrometry). The PL-B82 sample was collected from the drill hole situated in a high-grade mineralization zone and includes a high chalcopyrite content and some molybdenite (Fig. 16a). The PL-B62 sample was collected from the drill hole situated in a moderate-grade mineralization zone and includes a low chalcopyrite content and some pyrrhotite in the polished section (Fig. 16b). The PL-B74 sample was collected from the drill hole located in a weakly mineralized zone with lower chalcopyrite content and some pyrrhotite (Fig. 16c and d). The results obtained from the mineralogy, microscopic identification and drill hole scanning by XRF of these samples indicate that the Cu concentrations are 1.80 %, 1.32 % and 0.41 % in the PL-B82, PL-B62 and PL-B74 samples, respectively (Table 8).

In many cases, drill hole logging is dealing with the lack of proper diagnosis of geological phenomena, which can undermine the delineation of mineralized zones because it depends on the subjective interpretation of individual loggers, and no two loggers provide the same interpretations. However, conventional geological modeling based on drill hole data is fundamentally important for understanding the orebody spatial structure. Grades of ore elements are not determined by conventional methods of geological ore modeling, while the variation in ore grades in a mineral deposit is an obvious and salient feature. Given the problems mentioned above, using a series of newly established methods based on mathematical analyses such as fractal modeling seems to be inevitable.

In this paper, the number–size (N-S), concentration–volume (C-V) and power
spectrum-volume (S-V) fractal models were used to delineate and recognize
various Cu mineralized zones of the Pulang porphyry copper deposit in the
southern end of the Yidun continental arc, southwestern China. All these
fractal models reveal that high-grade Cu mineralized zones are situated in
the central and southern parts of the deposit. The Cu threshold values of
highly mineralized zones are 1.45 % and 1.88 % based on the N-S and C-V
fractal models. The Cu threshold of supergene enrichment zones is 1.33 %
based on the S-V fractal model. The models of moderately mineralized zones
contain 0.28 %–1.45 % Cu according to the N-S model and 1.48 %–1.88 % Cu
according to the C-V model. The hypogene zones contain 0.23 %–1.33 % Cu
according to the S-V model. The N-S model reveals weakly mineralized zones
and barren host rocks containing

The comparison between highly mineralized zones based on the fractal models
and potassic zones resulting from the 3-D geological model illustrates that
the S-V fractal model is better than the N-S and C-V model because the
number of overlapped voxels (

A comparison between phyllic alteration zones obtained from the 3-D geological model and moderate-grade mineralization zones obtained from the fractal models indicates that the OA values of the C-V, N-S and S-V fractal methods in reference to the phyllic alteration zones of the geological model are 0.59, 0.54 and 0.56, respectively. The overall accuracy of the moderately and weakly mineralized zones obtained from C-V modeling is higher than the mineralized zones obtained from N-S and S-V modeling (Table 7).

According to the correlation between the results driven by fractal modeling and geological logging from drill holes in the Pulang porphyry copper deposit, high-grade mineralization zones generated by fractal models, especially the S-V model, have a better correlation with potassic alteration zones resulting from the 3-D geological model than from the N-S and C-V models. The highly and moderately mineralized zones obtained from the fractal models are both situated in the southern and central parts of the Pulang deposit and coincide with potassic and phyllic alteration zones. There is a better relationship between the moderately and weakly mineralized zones derived by the C-V model and the phyllic alteration zones from the 3-D geological model than those derived by the N-S and S-V models.

The underlying data is confidential information and therefore cannot be made publicly accessible.

XW is the major contributor to this paper, including writing this paper, data processing and so on. QX provided many critical reviews and constructive suggestions. TL, SL, YL, LK and ZC participated in the field investigation. LW provided parts of the raw data.

The authors declare that they have no conflict of interest.

This research was supported by the National Key R&D Program of China (2016YFC0600508). The authors thank Tao Dong, Haijun Yu, Qiwu Shen, Zhipeng Li, Baosheng Shi and Jinhong Yang for supporting in field investigation and providing parts of the raw data.

This research has been supported by the National Key R&D Program of China: The three-dimensional prediction model of deep minerals and virtual reality (grant no. 2016YFC0600508).

This paper was edited by Behzad Ghanbarian and reviewed by two anonymous referees.