




































 
 

 

47 
© 2020 by the authors; licensee Asian Online Journal Publishing Group 
 

Asian Review of Environmental and Earth Sciences 
Vol. 7, No. 1, 47-54, 2020 

ISSN(E) 2313-8173 / ISSN(P) 2518-0134 
DOI:10.20448/journal.506.2020.71.47.54 

© 2020 by the authors; licensee Asian Online Journal Publishing Group 

  
 

 
 
 
Analysis of Spatial Dependence of Ore-Forming Elements Using Geostatistics and 
Moran Correlogram 

 
Thanh Tien Nguyen    

 

 
( Corresponding Author) 

 
Faculty of Surveying, Mapping and Geographic Information, Hanoi University of Natural Resources and 
Environment, Bac Tu Liem, Hanoi, Vietnam. 

 

 
Abstract 

The spatial dependence of data obtained from the geochemical prospecting process can provide 
useful information for evaluating mineralization potential. This study proposes two approaches to 
study the spatial dependence of ore-forming elements. To reduce the influence of extreme values 
and outliers, a semi-variogram was first used to study spatial variability and degree of spatial 
dependence of geochemical data using Cressie robust semi-variogram estimator. The Moran 
spatial correlogram was then employed to describe spatial heterogeneity and to test for the 
presence of spatial autocorrelation in geochemical data. The Moran‟s I statistics is strongly 
sensitive to positively skewed distribution, therefore, geochemical data were Box-Cox 
transformed before computing spatial correlograms. Results from a case study of Ag and Au 
elements in Jiurui Copper districts (southeast China) have shown that moderate spatial 
dependence was found for both of the Au and Ag variables, the maximum spatial variability was 
20 km for Au and 10 km for Ag, respectively. The degree of spatial dependence among 
geochemical data decreases as distances increase. These findings demonstrate that the spatial 
dependence of ore-forming elements can be effectively measured using geostatistics and Moran 
correlogram. 

 
Keywords: Geostatistics, Cressie semi-variogram, Moran correlogram, Geochemistry, Spatial autocorrelation, Spatial dependence, Spatial 
variability, Jiurui Copper districts (China). 

 
Citation | Thanh Tien Nguyen (2020). Analysis of Spatial 
Dependence of Ore-Forming Elements Using Geostatistics and 
Moran Correlogram. Asian Review of Environmental and Earth 
Sciences, 7(1): 47-54. 
History:  
Received: 30 December 2019 
Revised: 4 February 2020 
Accepted: 6 March 2020 
Published: 10 April 2020 
Licensed: This work is licensed under a Creative Commons 

Attribution 3.0 License  
Publisher:  Asian Online Journal Publishing Group 
 

Acknowledgement: The author thanks the editor and anonymous reviewers 
for their constructive comments, which have greatly improved this 
manuscript, and Dr. Peng Gong (formerly of China University of Geosciences) 
for providing geochemical data. 
Competing Interests: The author declares that there are no conflicts of 
interests regarding the publication of this paper. 
Funding: This study was financially supported by the Yangtze River Delta 
3D Geological Survey Project (grant number 1212011220250).  
Transparency: The author confirms that the manuscript is an honest, 
accurate, and transparent account of the study was reported; that no vital 
features of the study have been omitted; and that any discrepancies from the 
study as planned have been explained. 
Ethical: This study follows all ethical practices during writing.   

 

 

Contents 
1. Introduction ...................................................................................................................................................................................... 48 
2. Description of the Study Area and Data Used ........................................................................................................................... 48 
3. Methodology ..................................................................................................................................................................................... 48 
4. Results and Discussions .................................................................................................................................................................. 51 
5. Conclusions ....................................................................................................................................................................................... 53 
References .............................................................................................................................................................................................. 53 
 

 
 
 

 
 
 
 

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Contribution of this paper to the literature 
This study proposes approaches for assessing spatial dependence of geochemical elements based 
on Geostatistics and Moran correlogram. 

 
1. Introduction 

Geochemists examine the spatial patterns of geochemical elements to understand the mechanisms and that 
control their spatial distribution. Therefore, an understanding of the spatial distribution of geochemical elements 
has important implications. The spatial structures of geochemical elements can be studied using a semi-variogram 
and spatial correlogram. 

Classical semi-variogram was used to study spatial variability and degree of spatial dependence of the data. 
However, the main limitation of this approach is that the sample mean is not a stable estimator of theoretical mean. 
One single outlier or an extreme value can destroy this estimator completely [1]. Besides, the distribution of these 
elements is positively skewed, so the classical semi-variogram estimator is unbiased. For that reason, motivated by 
robust statistics, several types of semi-variogram estimators, based on robust estimation of scale and quantiles have 
been considered. A typical robust semi-variogram proposed by Armstrong and Delfiner [2] is the median of the 
magnitude of increments. Variants proposed include the quantile semi-variogram [3] the “Huberized” semi-
variogram, the jack-knifing [4] and the semi-variogram of order ½ [5]. However, the curves do not satisfy the 
statistical properties of a semi-variogram. Thus, a theoretical curve has to be fitted to the experimental one. Several 
different approaches have been proposed in the literature such as manual fitting [6, 7] automatic fitting including 
least squares using ordinary least squares [8] generalized least squares [9, 10] restricted maximum likelihood 
[11] maximum likelihood [10] Bayesian framework [12] MINQ estimation [13] and robust methods [14]. 

The spatial correlogram has been commonly used to describe spatial structures, spatial heterogeneity in 
quantitative ways and testing for the presence of spatial correlation in data. In addition, for multivariate data, Sokal 
[15] and Oden and Sokal [16] proposed to construct a spatial correlation function based on the Mantel [17] test 
of matrix correlation. BjØrnstad and Falck [18] proposed a spline correlogram, which provides a continuous and 
model-free function for the spatial covariance. The spline correlogram may be seen as a modification of the 
nonparametric covariance function of Hall and co-workers [19, 20]. 

In this study, from the above-discussion, spatial variability and degree of spatial dependence of geochemical 
data are assessed using Cressie robust semi-variogram. The Moran correlogram was then employed to describe 
spatial heterogeneity and to test for the presence of spatial autocorrelation in geochemical data. 

 
2. Description of the Study Area and Data Used 
2.1. Description of the Study Area  

The Jiurui ore district mainly consists of Cu-Au-Mo deposits which are located at the transitional point of the 
arcuate structure of the Middle-Lower Yangtze River metallogenic belt (eastern China) [21, 22]. The developed 
drape fractures in the study area are surrounded by six axial parallel NEE supports, which form a closely linear 
drape belt. Three diggings come from this drape fracture. Fractures are mainly followed by the NWW and NEE 
directions and then followed by NNW and NE-NNE directions. The first two control the distribution of magmatic 
rocks and ore deposits in this area. The last two have a conjugate structure for the first two. Submarine 
sedimentation-related pyrite copper deposits are controlled by a secondary basin of the Yangtze ancient sea basin. 
Intermediate acidic magmatic activities-related hydrothermal deposits are controlled by the NEE direction fault 
structure of Yanshanian, but at the same time controlled by the inherited fault in the NWW direction. 
 
2.2. Data Used 

A total of 1341 composite samples representing about 5364 km2 were collected in the Jiurui ore district at a 
scale of 1:200.000 regional stream sediment survey. The sampling density was 1 composite sample per 4 km2. 
Multi-elements sediment geochemical surveys of streams were carried out in the study area including Ag, As, Au, 
Be, Cd, Cu, Hg, Li, Mn, Mo, Nb, Pb, Sb, Sn, Th, V, W, Y, Zn, Al2O3, CaO, K2O, Na2O, etc., in which Ag and Au are 
ore-forming elements [23]. In addition, three geochemical associations of elements also caused the anomalous area 
in the study areas [24]. There are two metallogenenic series in the study area with a total of 13 known ore 
deposits. 
 

3. Methodology 
3.1. Cressie Semi-Variogram 

Let consider a spatial stochastic process * ( )     +, where   is a fixed subset of       . Assume that this 
stochastic process is ergodic and also satisfies the hypothesis of intrinsic stationarity, the expected (E) and variance 

(Var) values of  ( ) are given by Hoang, et al. [25] in Equations 1 and 2: 

 ( ( ))                         (1) 

   , (   )   ( )-    ( )                (2) 

where:  ( ) is the value of observed variable and   ( ) is the semi-variogram of  ( ) at lag  . This is a very 
simple model that can be used in practice after detrending data [26] or in several cases even used directly. In this 
study, the semi-variogram of order ½ estimator will be used to estimate the robust semi-variogram. Let 

* (  )  (  )     (  )+ be a sample of such a spatial stochastic process. The Cressie robust semi-variogram 
estimator is expressed as the following equation: 

  ̂( )  [
 

  
∑ | (  )   (  )|

   

 ( )

(   )      

]

 

(      
     

  
)⁄            (3) 



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where:  ( )  {(     )                       }   (  ) and  (  ) are values of an observed variable at 

locations i and j,       means that distances may be grouped into distance classes, placing in class   the 

individual distances     that are approximately equal to  . The denominator of Equation 3 corrects for bias under 

gausianity, and: 

  ̂( )  [   {| (  )   (  )|
   

  (     )   ( )}]
  

 ( )
           (4) 

where:  (  ) and  (  ) are defined as in Equation 3,    * + denotes the median of the sequence {| (  )  

 (  )|
   

} and  ( ) corrects for bias [assymptotically,  ( )       ]. The reasoning behind Equation 3 is that, 

for Gaussian data, [ (  )   (  )]
 
 is a chi-squared random variable on one degree of freedom. The power 

transformation that makes this most Gaussian-like is the fourth root [27] namely | (  )   (  )|
   

, the square 

root of the absolute difference. Thus, various location estimators can be applied to {[ (  )   (  )]
 
  (     )  

 ( )}, which, when untransformed and normalized for bias, yield robust semi-variogram estimators. 

A model must be fitted to the sample semi-variogram. The model will provide values of semi-variance for all 
the intermediate distances. The most commonly used models are the spherical model, exponential model, Gaussian 
model, hole effect model, linear model, pure nugget effect model, etc as shown in Figure 1.  
 

 
Figure-1. Commonly used variogram models. 

                                Source: Armstrong and Delfiner [2]. 

 
Semi-variograms may be used to determine whether anisotropy is present in the data set. They may also be 

used to describe anisotropic surfaces or to account for anisotropy in kriging interpolation [24]. A direction of space 

is first chosen (i.e. an angle  ) and a search is then launched for the pairs of sampling points that are within a given 

distance class   in that given direction. There may be only a few such pairs perfectly aligned in the aiming 

direction  , or even none at all, in particular when sampling is not regularly spaced on the map. More pairs of 
points can usually be found by looking within a small neighborhood around the aiming line as shown in Figure 2. 

The neighborhood is determined by an angular tolerance parameter   and a parameter   that sets the tolerance for 

distance classes along the aiming line. For each observed point    in turn, one looks for other points    that are at 

distance   ±   from it. All points found within the search-window are paired with the reference point    included 

in the calculation of semi-variance for distance class   [24]. In Figure 2, from an observed study site   , an aiming 

line is drawn in the direction determined by angle  , the angular tolerance parameter   determines the search zone 

(grey) laterally whereas parameter   sets the tolerance along the aiming line for each distance class  . 
 

 
Figure-2. Search parameters for pairs of points in the semi-variogram. 

                                                      Source: Legendre and Legendre [24]. 

 
Hoang, et al. [25] indicated that the distance at which the variance levels off is referred to as the range (or 

maximum spatial variability), a; beyond that distance, the sampling units are not spatially correlated. The 

parameter for the nugget effect is    and the spatially structured component is represented by   ; the sill,  , is 

equal to       Figure 3. Spatial variability can be investigated using the semi-variogram and the relative nugget 

effect, which is the ratio of the nugget to total semivariance expressed as percentage,    (     ) [24]. A ratio of 



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less than 25% indicates strong spatial dependence, between 25% and 75% indicates moderate spatial dependence, 
and greater than 75% indicates weak spatial dependence [1]. 
 

 
Figure-3. Characteristic features a spherical semivariogram model. 

                           Source: Legendre and Legendre [24], Hoang, et al. [25]. 

 
3.2. Moran Correlogram 

 Spatial correlation of a uni-variate can be measured using spatial correlation statistics such as Moran‟s I or 
Geary‟s C statistics [27]. The Moran's I statistics seems to be the most popularly used [1], that is given: 

 ( )  
 

  

∑ ∑    (    ̅)(    ̅) 
   

 
   

∑ (    ̅)  
   

 (5) 

where:    and    are the values of an observed variable at sites i and j, d is the distance class considered in the 

calculation,    is the sum of the weights    , given    ∑ ∑    
 
     

     ̅ is the sample mean of an observed 

variable, given by:  ̅  ∑
  

 
 
      N stands for the number of observations with     is spatial weight matrix for a 

given distance   and     is the distance between the observations i and j.  

Spatial correlation coefficients can be tested for significance. The tests require that the condition of second-
order stationarity (or a relaxed form of stationarity, intrinsic stationarity). Cliff and Ord [27] describe how to 
compute confidence intervals and test the significance of spatial correlation coefficients. For any normally 

distributed statistic Stat, a confidence interval at significance level α is obtained as follows: 

  (        ⁄ √   (    )               ⁄ √   (    ))      (6) 

In Equation 6, for significance testing with large samples, a one-tailed critical value       at significance level 

α is obtained as follows: 

        √   (    )                                   (7) 

where: value    ⁄  or    in Equation 6 and 7 is found in the table of standard normal deviates. 

It is possible to use this approach because Moran‟s I is asymptotically normally distributed for data sets of 

moderate to large sizes [27]. Under the hypothesis (  ) of the random spatial distribution of the observed values 

  , the expected values (E) of Moran‟s I statistic is given by the following Equation: 

 ( )  
  

   
 (8) 

where:   is defined as in Equation 5. 

It can be seen that, under the null hypothesis,  ( ) approaches 0 as n increases. The variance is computed using 

Equation 9 under a randomization assumption, which simply states that, under   , the observations    are 
independent of their positions in space (second-order stationarity assumption) and, thus, are exchangeable [27]. 

   ( )   (  )  , ( )-  

 
 ,(       )          -    ,( 

   )           -

(   )(   )(   )  
  (

 

   
)
 

 
(9) 

In Equation 9,   ,   , and    can be obtained by the following Equations: 

   
 

 
∑∑(       )

 

 

   

 

   

 (10) 

   
 

 
∑(∑(    ∑   

 

   

)  

 

   

 

   

 (11) 

    ∑(    ̅) 
 

   

 [∑(    ̅) 
 

   

]

 

 (12) 

where:   ,   ,     and     are defined as in Equation 5, and   is number of observations. 

In most cases, tests of spatial correlation are one-tailed. Geological variables are usually positively correlated at 

short distances. To carry out an approximate test of significance, a value of α is selected (e.g α = 0.05) and find    

in a table of the standard normal distribution (e.g               ). Critical values are found as Equation 7 with a 



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correction factor that becomes important when n is small:      √   ( )    (   )   in all cases, using the 

value in the upper tail of the   distribution when testing for positive spatial correlation (e.g               , and 

the value in the lower tail in the opposite case (e.g               ). 

The value taken by the correction factor    depends on the values of n and   . If  (  √ )     

 (    √   )         √   ; otherwise,     . If the test is two-tailed, use      ⁄  to find     and     

before computing critical values. The corrections are based upon simulations reported by Cliff and Ord [27]. Other 
formulas can be found in Cliff and Ord [27] for conducting a test under the assumption of normality, where one 

assumes that the result of   
  from independent draws from a normal population. It has been suggested that when n 

is very small, tests of I should be conducted by permutation [1, 24]. 
 

4. Results and Discussions 
4.1. Spatial Variability Analysis 

Data from Figure 4-a and 4-b demonstrates that the semivariances for the variable Au for the omni-directional 
semi-variogram and directional semi-variograms in four different directions. The experimental semi-variogram 

shows a big nugget effect,   , which accounts for about 57% of the total sill       after fitted with a theoretical 
model as shown in Table 1. A ratio of nugget to total semivariance of 57% between 25% and 75% indicates 
moderate spatial dependence. The nugget effect shows a small-scale variation in the data due to the problem of 
sampling unintensiveness Figure 4-a. However, there was a slightly significant difference in the variation 
according to four different directions. The directional semi-variograms displays the same sill. All curves start at 
about the same point. All curves of directional variograms of Au almost flatten out at 20km and show small 
different total variances Figure 4-b. Similarly, the hole effect appears due to the presence of one or more mineral 
occurrences. 
 

Table-1. Summary table of geostatistical parameters of Au and Ag ore-forming elements. 

Elements Direction 
Nugget 
effect 

(  ) 

Spatially 
structure 

(  ) 

Sill 

(     ) 
Spatial range 

(km) (a) 

Relative 
nugget effect 

   (     ) 

Au omni 0.21 0.16 0.37 35.9 57% 
Ag omni 0.00025 0.00027 0.00052 15.5 48% 

             

 
Figure-4. Semi-Variogram for the variables Au and Ag. 

                

Data from Figures 4-c and 4-d illustrates the semivariances for the variable Ag for the omni-directional 
semivariogram additional to four directional variograms in different directions. Experimental omni-directional 

variogram shows a moderate nugget effect,   , which accounts for about 48% of the total sill       after fitted 
with a theoretical model Table 1. A ratio of nugget to total semivariance of 48% between 25% and 75% indicates 
moderate spatial dependence. The nugget effect of the variable Ag also shows a small-scale variation in the data 
due to the problem of sampling unintensiveness. All curves start at about the same point and they flatten out at 
about 14km. The sampling units are not spatially correlated with no spatial dependence in the variable Ag when 
the distance beyond 15.5km Figures 4-c. Directional semi-variograms vary strongly with directions and big 
different total variance when the distance is more than 14km, especially on the northeast-southwest, northwest-
southeast and north-south directions Figure 4-d. It can be inferred that the hole effect appears due to the presence 
of one or more mineral occurrences. It reflects a tendency for high values to be systematically surrounded by low 
values and vice versa. 

 
 



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4.2. Spatial Structure Analysis 
The surfaces in Figures 5-a and 5-c were made from 1341 sampling points across the surface using a regular 

49×38 grid (2×2 km in a grid). The “height” was noted at each sampling point. The distances among points were 
divided into 60 distance classes. Testing the significance of correlograms with each distance class by randomization 

test with 500 times of permutation at the   = 0.05 level. Spatial correlograms were computed using the „raw‟ data. 
However, the distribution of the “raw” data is strongly right-skewed due to many extreme values and outliers. 
Besides, Moran‟s I statistics is very sensitive to extreme values and asymmetry in the data distributions. To easily 
reach significance in statistical tests, the data were Box-Cox transformed to compute Moran‟s I correlograms. 
Moran‟s I coefficients at distances ranging from 2 km to 34.8 km and their p-values of the Au and Ag variables are 
summarised in Table 2. 
 

Table-2. Summary table of Moran‟s I coefficients and p-values of Au and Ag elements using raw data and Box-Cox transformed data. 

Distance 
(km) 

Au Ag 

Raw data Box-Cox transformed data Raw data 
Box-Cox transformed 

data 

Moran’s I 
coefficient 

p-value 
Moran’s I 

coefficients 
p-value 

Moran’s I 
coefficient 

p-value 
Moran’s I 
coefficient 

p-value 

2.0 0.029 0.018 0.41 0.002 0.11 0.008 0.54 0.002 
3.4 0.021 0.032 0.35 0.002 0.06 0.008 0.40 0.002 
5.1 0.026 0.016 0.30 0.002 0.04 0.006 0.30 0.002 
7.0 0.011 0.05 0.24 0.002 0.01 0.038 0.22 0.002 

9.1 0.007 0.05 0.20 0.002 0.00 0.144 0.16 0.002 
11.1 0.005 0.126 0.17 0.002 0.00 0.118 0.13 0.002 
12.9 0.009 0.022 0.15 0.002 0.00 0.367 0.11 0.002 
14.8 0.005 0.068 0.13 0.002 0.00 0.140 0.10 0.002 
16.9 0.005 0.09 0.12 0.002 0.00 0.204 0.09 0.002 
19.0 0.003 0.116 0.10 0.002 0.01 0.040 0.07 0.002 
21.0 0.005 0.066 0.09 0.002 0.00 0.104 0.06 0.002 
22.8 0.005 0.076 0.07 0.002 0.01 0.036 0.05 0.002 
24.9 0.016 0.004 0.06 0.002 0.00 0.463 0.03 0.002 
26.9 0.003 0.144 0.04 0.002 0.00 0.154 0.03 0.002 
29.0 -0.003 0.246 0.00 0.218 0.00 0.126 0.04 0.002 
30.9 0.007 0.038 0.00 0.210 -0.01 0.036 0.04 0.002 
32.9 0.000 0.253 -0.01 0.092 0.00 0.122 0.03 0.002 
34.8 0.001 0.257 -0.01 0.162 -0.01 0.034 0.02 0.002 

     

Data from Moran‟s I correlogram of the Au variable in Figure 5-b and Table 2 demonstrate that the first 
spatial correlation coefficient of 0.029 indicates a weak positive spatial correlation in the first distance class for the 
„raw‟ data. The second positive and significant spatial correlation coefficient of 0.021 occurs at distances ranging 
from 2 km to 3.4 km. The next positive spatial correlation coefficient was detected at distances from 11.1 km to 
12.9 km. The last positive and significant correlation was found with a coefficient of 0.016. There was no spatial 
correlation (spatial dependence) at distances ranging from 2.0 km to 3.4 km, from 7.0 to 11.1 km, and from 12.9 km 
to 22.8 km, respectively. Besides, negative and significant spatial correlation was detected at distances ranging 
from 50 km to 70 km. 
 

 
Figure-5. The surface of Au: raw data (a) and of transformed data (c) and Moran correlogram: raw data (b) and transformed data (c). 

        

The surface and Moran correlogram for Box-Cox transformed data of the variable Au shown in Figures 5-c 
and 5-d. Data from Table 2 demonstrates that the first spatial correlation coefficient of 0.41 was bigger than that of 
„raw‟ data. This trend is similar to the next spatial correlation coefficients with values of 0.35, 0.30, 0.24, and 0.20 
at distances ranging from 3.4 km to 9.1 km. Spatial correlation coefficients are gradually reduced as distances 
increase. Positive and significant correlation coefficients were also found at distances below 26.9 km, whereas no 



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spatial correlation was detected at distances ranging from 29 km to 34.8 km. Negative and significant spatial 
correlation at distances from 32.9 km to 76 km. 

Data from Moran correlogram for the Ag element in Figure 6-b shows that three positive and significant 
spatial correlation coefficients were found at distances ranging from 2 km to 5.1 km, whereas, no spatial correlation 
was found at distances ranging from 5.1 km to 34.8 km. In addition, there existed negative and significant spatial 
correlation coefficients at distances ranging from 34.8 km to 44 km and no spatial correlation at distances above 44 
km. 

The surface and Moran‟s correlogram for Box-Cox transformed data of the variable Ag in Figures 6-c and 6-d 
show similarities with those obtained from the Au variable, the first spatial correlation coefficient of 0.54 was much 
bigger than that of „raw‟ data and similarly for the next spatial correlation coefficients, 0.40, 0.30, 0.22, 0.16 at 
distances ranging from 3.4 km to 9.1 km. Spatial correlation coefficients were also gradually reduced as distances 
increase. Spatial correlogram in Figure 6-d show that there existed positive and significant correlation at distances 
below 36.9 km and strong spatial correlation at distances below 12.9 km, weak spatial correlation at distances 
between 12.9 km and 36.9 km; no spatial correlation at distances from 30.9 km to 36.9 km, and negative and 
significant at distances from 36.9 km to 74 km. 
 

 
Figure-6. The surface of Ag: raw data (a) and of transformed data (c) and Moran correlogram: raw data (b) and transformed data (c). 

        

5. Conclusions 
In this study, Cressie robust semi-variogram and Moran correlogram have been employed to study the spatial 

dependence of ore-forming elements. Cressie robust semi-variogram was first used to study spatial variability and 
identify the level of spatial dependence in data. The Moran spatial correlogram was then applied to describe spatial 
structure and spatial heterogeneity.  Results from a case study of Ag and Au elements in Jiurui Copper districts 
have shown that moderate spatial dependence was found for both of the Au and Ag variables, the maximum spatial 
variability was 20 km for Au and 10 km for Ag, respectively. An important conclusion can be made is that the 
results were strongly affected by the existence of extreme values and outliers, so it is suggested that the raw data 
should be transformed. In addition, the precision of the spatial correlogram depends on spatial classes (distances), 
whereas, the computational complexity is remarkably increased when spatial classes are decreased.  
 

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