Tectonophysics 522–523 (2012) 176–186

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Tectonophysics

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Quantification of the spatial distribution of mineral phases and grains in rock using a2-D multiple-area density map technique

Sungshil Kim, Jin-Han Ree ⁎Department of Earth and Environmental Sciences, Korea University, Seoul 136-701, South Korea

⁎ Corresponding author. Tel.: +82 2 3290 3175.E-mail addresses: neogeo94@korea.ac.kr (S. Kim), re

0040-1951/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.tecto.2011.11.025

a b s t r a c t

a r t i c l e i n f o

Article history:Received 24 March 2011Received in revised form 12 November 2011Accepted 26 November 2011Available online 7 December 2011

Keywords:Rock fabricsSpatial distributionClusteringMultiple-area density mapGrain growthPinning effect

The most widely used method for quantitative analyses of the spatial distribution of grains in a rock body isthe nearest-neighbor method (a position-based method), which recognizes the grain center as the position ofthe grain without consideration of grain size or shape. However, the spatial distribution of grains isinfluenced by their size and shape, as well as their position. Here, we propose a multiple-area density map(MADM) method to quantify the spatial distribution of phases or grains. The method is based on imageanalysis and simultaneously considers the position, size, shape, and proportion of grains. The MADMconstructed by overlaying density maps of grain areas with a set of template size produces a normalizedstandard deviation (NSD) value of phase area density, which represents the degree to which it is spatiallyclustered. The NSD value is used to quantify the distribution of the phase, without considering individualgrains. To identify the spatial distribution of individual grains, the method employs the clustering index(CI), which is calculated from the ratio of NSD to the reciprocal of the number of grains in a log–log plot.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

A key factor in interpreting the origin and evolution of rocks is therock fabric, which may be defined as the spatial and geometricalconfigurations of its components (Gaillot et al., 1999; Hobbs et al.,1976). Previous studies have proposed methods for quantifying thegrain size and shape (e.g. Heilbronner and Keulen, 2006; Higgins,2000; Panozzo, 1983), whereas relatively few studies have soughtto quantify the spatial distribution of mineral phases in the rockmass (e.g., Higgins, 2000). Such studies have included analyses ofthe spatial distribution of phases in the context of constraining thenucleation processes of various phases in igneous and metamorphicrocks (Jerram et al., 1996, 2003; Philpotts et al., 1998; Saltzer et al.,2001), the identification of phases in equilibrium with each other(Morishita and Obata, 1995), and assessing the effect of one phase onthe static or dynamic grain growth of another phase in metamorphicrocks (Herwegh and Berger, 2004; Song and Ree, 2007). However,existing methods for quantifying the spatial distribution of a givenphase do not provide a unique solution, since the quantification isdependent on several factors that are not always taken into account,including clustering, fraction, size, shape, and shape-preferredorientation.

In this paper, we introduce a new method for quantifying thespatial distribution of a given phase in a rock body using 2-D image

ejh@korea.ac.kr (J.-H. Ree).

rights reserved.

analysis. Although the method does not provide a unique solution(i.e., it does not consider all of the controlling factors simultaneously),it is simpler than existing approaches in that it extracts a simple indexas a reference parameter that is utilized to consider other controllingfactors.

2. Summary of previous methods

In a study of the spatial distribution of a given phase, it is essentialto define the positions of individual grains, which may be done bylocating the center of each grain. The most frequently used method insuch studies is the ‘distance-to-nearest-neighbor’ method developedoriginally in ecology (e.g., Clark and Evans, 1954). In this method, thevalue ‘Big R’ is defined as the ratio of the observed mean nearest-neighbor distance to the expected mean nearest-neighbor distance foran ideally random distribution. However, the effect of local clustersand the size, shape, and shape-preferred orientation of individual grainsis not considered in the ‘distance-to-nearest-neighbor’ method, asdemonstrated by Song and Ree (2007).

Jerram et al. (1996) refined the ‘distance-to-nearest-neighbor’method by introducing a variable for the phase volume fraction, inaddition to the R-value; however, the refined method does notaddress the problems outlined above. Jerram and Cheadle (2000)introduced the method of complete linkage hierarchical clusteranalysis (CLHCA), which involves the construction of a dendrogramin which the distances between individual grains or local clustersare represented as height. Although CLHCA effectively expresses

Fig. 1. (a) Example of the ASCII raster grid file format. Here, 0 represents non-target phase and 1 and 2 indicate target phase. (b) The bitmap image from the grid file of (a).

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the effect of local clustering on the spatial distribution, it does notreflect the effect of size and shape of individual grains.

To consider the effect of grain size and shape, Jerram and Cheadle(2000) developed the method of density linkage cluster analysis(DLCA), in which the spatial distribution of grains is defined bytheir boundaries rather than their centers. DLCA recognizes adjacentgrains or clusters within a certain radius from the boundary of agrain; however, the results generated by DLCA depend on the radiusof the ‘searching circle’.

The wavelet transform technique is an image analysis techniquewhich extracts information on the orientation and spacing of clustersin ordered structures (Darrozes et al., 1997; Gaillot et al., 1997, 1999).Although this method is similar to the method proposed here, in thatboth methods calculate spatial parameters through image analysis atmultiple scales, their technique focuses mainly on the orientation ofclusters rather than the spatial distribution of grains within a givenarea.

3. Method

The proposed method uses the ASCII raster grid file format forinput and output data. This data structure is similar to digital images,such as bitmaps, which store information pertaining to the colorscheme in each pixel. Another example of a similar file type is thegrid file format for a digital elevation map (DEM), which containstopological information, such as the elevation of the land surface, ineach grid cell (Trauth, 2007). The ASCII raster grid file format is

Fig. 2. (a) A template to calculate area density of objects on the input grid. The calculated detemplate in order to make area density map (ADM). (b) The constructed ADM from (a).

text-based and straightforward to handle; consequently, it is oftenused as a basic data structure for numerical computer simulationsor digital image analyses. In the proposed method, the input gridfile contains information pertaining to the phase of the materialpresent at each grid point. This information is used to distinguish aspecific phase from other phases or voids (Fig. 1), and is defined aseither target phase cells or non-target phase cells. Non-target phasecells represent all areas not considered after designating targetcells; thus, they include both voids and the other phase(s) present.

We define an ‘object’ as an area consisting of grid cells of the targetphase. This can be either a single grain or a grain aggregate of thetarget phase. In other words, a single quartz grain cannot be distin-guished via this definition from a polycrystalline quartz ribbon, becausethe grid cells of the data structure are defined by two types of phaseinformation only; however, it is possible to distinguish between agrain aggregate and a grain by including additional information, suchas average grain size or other data set to designate individual grains.

The term “area density” is used here to represent the degree ofobject dispersion (or clustering) in a given area. The area density isthe ratio of the area of the target phase cells within a template tothat of the template (Fig. 2a). The template has a constant size andshape for each calculation of area density across the grid. To calculatethe area density, the center of the template is located at each cell inturn, the area of the target phase within the template is measured,and the ratio is calculated. The resultant area density values areallocated for each cell in an output grid file referred to as an “areadensity map (ADM)” (Fig. 2b). To carry out the density calculations,

nsity value within the template is allocated at the cell that is located in the center of the

Fig. 3. (a) Single ADMs. The density contrast of an ADM depends on the used template size. (b) 3-D topography of a compiled multiple area density map (MADM) from a set ofsingle ADMs. This compiled MADM can also be represented as 2-D map as in (a).

Fig. 4. (a) and (b) Example of 3-D area density map with two objects. Input image isshown as a square box in the top-right side. Samples of (a) and (b) have the samesize and shape of objects but different distance between them. (c) Profiles of twoMADMs which are cut in the middle of the samples of (a) in broken line and (b) inline. Notice that the area density curve of the sample (b) having a closer distancebetween the objects shows a higher amplitude than that of (a).

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we assume that all of the marginal sides of the grid are connectedwith their opposite sides, resulting in a repetition of the pattern inthe same manner as decorative tiles are repeated in architecture.This “tiled boundary effect” allows the method to calculate acomplete density value for the grid edges, which would otherwisebe missing data.

The calculated area density value depends on the size and shape ofthe template used. The smaller the template, the more abrupt thechange in the calculated area density from one cell to the next,enabling small variations in area density to be distinguished. However,the larger the template, the easier it is to represent the relationshipbetween grains, since the method can only express relationshipsbetween grains when they are both within the same template.

To offset these limitations related to the dependence of the calcu-lated area density on the template size, a set of ADMs is producedusing multiple templates, each with a different size. Then, thesemaps are combined into a multiple-ADM (MADM) by calculatingthe average density value at each grid cell from the set of ADMs(Fig. 3). The maximum possible size of a given template is the sameas the dimensions of the input grid, and its minimum size should bethe diameter of the smallest grain present, to enable grains of thatsize to be distinguished. The size interval between each of thetemplates should be constant. The number of templates used to

Fig. 5. Maximum standard deviation (SD) curve estimated using ideally clusteredobjects (i.e. circular aggregate) for a given fraction of object. Open circles representthe maximum SD of representative samples.

Fig. 6. Discrimination diagram of spatial distribution using clustering index (CI).NSD: normalized standard deviation of area density of objects. np: number of objects.See text for the details. Fig. 8. (a) Box plots of normalized density (ND) for samples in Fig. 7. The distribution of

ND values is highly asymmetric in clustered samples. (b) Standard deviation (SD) andaverage value of area density for samples of (a). The average value of area density is thesame for all the samples since they have the same fraction of grains.

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calculate the MADM can vary, but 10 ADMs per MADM has beenfound to be appropriate: the use of more than 10 ADMs results inexcessive calculation time, while the use of too few ADMs results indiscontinuous and abrupt changes in density on the resultantMADM. When comparing the clustering degree of two or moresamples, the same parameters and the same template sizes shouldbe used, since a change in the diameter of the template can result inchanges to the calculated density on a map.

Fig. 7. Samples for verification of the multiple-area-density-map (MADM) method. The size(SD) of MADM is shown both in 2-D and 3-D in each sample.

The MADM has two useful characteristics. First, the density valuesof the MADM increase in proportion to the concentration of the targetobjects. In other words, the MADM represents the spatial distributionof the objects. Second, if samples contain the same fraction of a targetphase, the average density values of their MADMs are equal to thefraction of the target phase regardless of its spatial distribution.

, shape and fraction of grains are the same in all of the samples. The standard deviation

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Fig. 4a and b shows samples with two grains (or objects) of thesame size and shape; however, the distance between the grains isdifferent in each case. Fig. 4c shows the density curve profiles, takenfor the center of the samples, for the ADMs for both Fig. 4a and b.The density curve of the sample with a shorter distance betweengrains (Fig. 4b) has a higher amplitude than that of the sample witha longer distance between grains (Fig. 4a). The distance betweengrains in this example is the only dependent parameter causing thedifference in the density value. Therefore, the increased amplitudeis caused by the tighter clustering of the grains. When objects aredispersed, the amplitude of the density is smaller; it approaches theaverage density value as maximum dispersal is approached. There-fore, the spatial distribution of objects can be represented by thedeviation of density values from the average density for the map.The smaller the deviation of density values from the average density,the greater the dispersion of objects, and vice versa.

4. Clustering index using MADM

As explained above, the proposed method calculates the standarddeviation (SD) of the density values for a MADM. This densitycontrast represents the spatial distribution of the objects; however,the SD is dependent on the fraction as well as the spatial distributionof the objects. Therefore, the SDs of the MADMs containing differentfractions of objects cannot be compared with one another in an effortto determine their relative spatial distributions. Thus, we need afraction-independent index to quantify the spatial distribution.

If the objects are perfectly dispersed, a constant density value willbe obtained for all areas of the MADM and the SD will have a value ofzero (the minimum value). In contrast, the SD will achieve itsmaximum value for a given fraction when the objects are perfectlyclustered; this will be achieved when the clustered aggregate, likethe template, is circular in shape (in 2-D, or spherical in 3-D). Thus,all of the calculated SDs from the MADM will fall between themaximum and minimum values for that fraction. Once the maximumSD for a given fraction of objects is defined, the normalized standarddeviation (NSD) is produced by dividing the SD of the MADM by themaximum SD. This is a fraction-independent value that enablescomparison between samples containing different fractions of objects.

The calculated maximum SDs are plotted against their fractions inFig. 5. The best-fit curve of the maxima shows a bell shape with itspeak value at a fraction of about 0.45. For fractions above ~0.6, thebest-fit curve deviates somewhat from a symmetrical bell shape

Fig. 9. Clustering index (CI) value for samples of Fig. 7 in a log–log plot of normalizedstandard deviation (NSD) vs. reciprocal of number of grains (np). CI decreases withincreasing the degree of clustering.

(dashed portion of the line in Fig. 5); this deviation is due to theabove-mentioned “tiled boundary effect” of the proposed method.

Although the NSD quantitatively defines the spatial distribution ofobjects, it does not include any information on the frequency ofindividual grains (i.e., the number of grains) or their size. In studiesof natural rocks, the spatial distribution of individual grains may bemore meaningful than that of a given phase (Higgins, 2006). Toextract the spatial distribution of individual grains from the calculatedNSD, we need information on the size of the grains, because the spatialdistribution (or clustering) of the grains is dependent on their size andsize distribution. For example, if each of the two objects shown inFig. 4a represents a single grain, the two grains are not stronglyclustered. In contrast, if each of the objects is an aggregate consisting

Fig. 10. Clustering index (CI) plot for samples with various grain size, fraction andspatial distribution. (a) Calculated CI of the samples is plotted on the discriminationdiagram of spatial distribution. (b) Sample images used for (a) are shown on thediscrimination diagram. The detailed parameters of the samples are shown in Table 1.

Table 2Analyzed results of 2D grain growth simulations.

Sample MCS dg ng m

Sample_006 6000 39.7 100 0.44010,000 41.8 86 0.45840,000 46.2 66 0.492

Sample_007 6000 42.7 91 0.46510,000 46.1 80 0.49120,000 48.9 73 0.511

Sample_008 6000 39.3 99 0.43710,000 42.0 82 0.46030,000 46.3 67 0.493

Sample_020 6000 55.5 52 0.33710,000 65.2 38 0.391

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of smaller grains, the grains show strong local clustering. Therefore, weintroduce a new index, the clustering index (CI), which takes intoaccount the effect of local clustering on the bulk spatial distribution.The CI is the ratio of NSD to the reciprocal of the number of grains in alog–log plot:

CI ¼ log NSD

log 1=np

� � ¼ − log NSDlog np

ð1Þ

where np is the number of objects. The reciprocal of np represents theaverage size of the individual grains relative to the total area (fraction)of a target phase. Thus, 1/np has a value between 0 and 1, since it is theratio of the average ‘relative’ area of the individual grains to the totalarea of the objects, assuming a uniform size distribution of the grains.Fig. 6 shows the degree of clustering of the grains (ordered, random,and clustered) based on CI values (the slope of the lines in the plot). Alow CI indicates a high degree of clustering, and vice versa. The CI lineof 0.75 indicates the maximum possible value for CI; a uniform (orordered) distribution lies along (or close to) this line. The CI for arandom distribution ranges from 0.75 to 0.5; the dashed line in Fig. 6indicates the minimum boundary of random distribution; for CI valuesbelow 0.5 the distribution is clustered.

5. Examples

5.1. Verification of the method using simple patterns

To test the proposed method, we first applied it to six arbitrarilydefined samples (Fig. 7). For this test we set the size, number,shape, and fraction of the grains to be the same for each of the samples,leaving only the grain distribution as a variable; this illustrates thequantification of the distribution pattern. In the resultant MADMs,the samples with a clustered distribution show both an increase inSDs and density contrast with higher clustering, while those sampleswith a dispersed distribution of grains (uniform, ordered, andrandom) show lower SDs and density contrasts.

Fig. 8 shows the normalized and average density values for thesesamples using box plots. The deviation of density increases withincreased clustering. The average density for these samples is constant,regardless of the distribution pattern of the samples, because it is equalto the fraction of the grains in the samples. The CIs of the samples areshown in Fig. 9. The uniform (CI=0.732) and ordered (CI=0.725)distributions are both close to the maximum boundary for CI. Therandom distribution (CI=0.643) falls within the index range for arandom distribution. The clustered samples (CI=0.436, 0.281, and0.161) fall within the range for a clustered distribution.

A second test was conducted using a larger pool of samples so thatmore than one of the attributes could be varied. The CI values forrepresentative samples from this large test pool are shown inFig. 10. The particle properties of the samples for this test are listed

Table 1Analyzed results of the samples in Fig. 10.

Sample fp dp np Std. Dev. NSD CI

Sample_006 0.052 10.8 91 0.0165 0.210 0.35Sample_007 0.052 10.8 91 0.0058 0.075 0.58Sample_008 0.052 10.8 91 0.0270 0.344 0.24Sample_020 0.051 20.4 25 0.0101 0.132 0.63Sample_021 0.052 28.6 13 0.0167 0.213 0.60Sample_023 0.051 20.5 25 0.0407 0.526 0.20Sample_024 0.052 28.4 13 0.0516 0.663 0.16Sample_028* 0.051 20.5 25 0.0088 0.113 0.68Sample_029* 0.051 20.5 25 0.0407 0.526 0.20

*Samples with anisotropic grain shape. fp: fraction of particles, dp: diameter of particles,np: number of particles, Std. Dev.: standard deviation, NSD: normalized standarddeviation, CI: clustering index.

in Table 1. Samples 006, 007, and 008 form a set with the same size,shape, and fraction of grains, but have a different spatial distribution.Other sets with the same properties for their grains, but differentspatial distributions, include the set comprising samples 020 and023, and the set comprising samples 021 and 024. Samples 028*and 029* are composed of needle-shaped grains; their grain fractionand spatial distribution are the same as those of 020 and 023, respec-tively. As seen in Fig. 10a, the samples with needle-shaped grainstend to have a higher CI value than their counterparts with circulargrains, although the difference in CI is not significant. The CI valuesfor the samples with the same fraction and distribution, but differentgrain shapes, tend to become higher with increasing dispersion ofgrains, suggesting that the effect of grain shape on the spatial distri-bution becomes more significant when the grains are more widelydistributed.

5.2. Analysis of the pinning effect of second-phase particles

Impurities, or second-phase particles, are important for thestabilization of the grain size of the main phase during dynamic andstatic recrystallization, and thus for the rheology of rocks and metals(e.g., De Bresser et al., 1998, 2001; Etheridge and Wilkie, 1979; Evanset al., 2001; Herwegh and Berger, 2004; Olgaard, 1990; Song and Ree,2007). Most of these previous studies focused on the effect caused bythe fraction of second-phase particles present; however, the disper-sion of second-phase particles is also an important factor in terms ofthe recrystallized grain size of the main phase (Olgaard and Evans,1986; Song and Ree, 2007; Weygand et al., 2000). In this section,we test the proposed method in quantifying the effect of second-phase particle dispersion on the recrstallized grain size of the mainphase, using a conventional computer simulation and a natural rocksample.

We used the Mesoscale Microstructure Simulation Project (MMSP)to simulate grain growth of the main (matrix) phase using MonteCarlo simulations (Gruber et al., 2009) and employed the method ofMiyake (2001) tomodel the pinning effect of the second-phase particlesduring static grain growth of the main phase. The size of each input

30,000 91.8 20 0.506Sample_021 6000 58.3 46 0.242

10,000 74.9 29 0.32670,000 117.9 11 0.480

Sample_023 6000 52.3 57 0.31610,000 59.8 44 0.36170,000 74.2 21 0.434

Sample_024 6000 55.9 51 0.22810,000 64.1 37 0.27480,000 91.6 12 0.395

Sample_028* 6000 52.7 60 0.31910,000 59.5 45 0.36040,000 85.6 23 0.482

Sample_029* 6000 52.1 56 0.31510,000 63.3 38 0.38160,000 79.0 19 0.455

MCS: Mote Carlo steps, dg: diameter of grains, ng: number of grains,m: geometric factorof particles.

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matrix is 500×500 cells. The data from the samples shown in Fig. 10were used for the second-phase particles during the growth simulations.The initial matrix grain size (of one or two pixels) was smaller than thatof the second-phase particles in all samples. The growth simulation wasrun for each sample until the size of matrix grains stabilized (amaximum of 100,000 Monte Carlo Steps (MCS)).

When impurities or second-phase particles are present duringgrain growth, the relationship between the maximum size of matrixgrains and the properties of the second-phase particles (Zener drag)can be expressed as follows:

dmax ¼ Cdpf m

ð2Þ

where dmax is the stable matrix grain size; C is an empiricallydetermined constant that depends on the models or experimentmaterial; dp and f are the size and fraction of the second-phaseparticles, respectively; and m is the geometric factor of the particles(Evans et al., 2001; Herwegh and Berger, 2004; Olgaard and Evans,1986). The geometric factor (m) has a value ranging from 1/2 to 1/3depending on the position of the particles on the boundaries ofmatrix grains. Computer simulations indicate that the ideal value form is 1/2 when the size of the matrix grains has stabilized (Miyake,2001; Srolovitz et al., 1984; Weygand et al., 2000).

Fig. 11. Histograms representing size distribution of matrix grains in the samples simulatestage. The detailed parameters of the samples are shown in Table 2.

Table 2 summarizes the results of the growth simulations, showingthe average size (dg) and number (ng) of matrix grains, and the valueof m at 6000 and 10,000 steps, as well as at the step corresponding tothe final stabilization of matrix grain size for each sample. The matrixgrains are considered to have stabilized when their average grain sizeand size distribution no longer fluctuate with increasing MCS, despitesome residual fluctuation for individual grains. The m values of thesamples are generally close to 0.5 when they have reached a stabilizedstate (see below for a discussion of the two exceptions, 023 and 024).We have grouped samples with the same fraction and grain shape ofsecond-phase particles (e.g., samples 006, 007, and 008). Withineach group, the spatial distribution is the unique factor controllingthe pinning effect.

The samples with clustered second-phase particles (i.e., with lowCI in Table 1) show two distinctive features. First, it takes a longertime (MCS in Table 2) for the growth of the matrix grains to reachthe stable state than it does for samples with dispersed particles.For the clustered samples with a very low fraction of second-phaseparticles, such as samples 023 and 024, their m value is smaller than0.5 and the matrix grains do not reach the stable state at all, evenwhen using a high MCS. Second, there is a bimodal pattern to thesize distribution of the matrix grains in the samples with clusteredsecond-phase particles, whereas that of the samples with distributedparticles shows a normal (bell-shaped) distribution (Fig. 11).

d at 6000 (early) and 10,000 (middle) MCS (Monte Carlo steps) and at the stabilized

Table 3Parameters and analyzed results of the quartzite.

Sample fp dp np SD NSD dg ng CI Z m

Area_01 0.016 40.66 67 0.0041 0.139 163.8 217 0.47 2.48×103 0.34Area_02 0.034 40.26 134 0.0092 0.170 136.2 315 0.36 1.19×103 0.36Area_03 0.079 49.24 199 0.0141 0.130 115.8 424 0.38 6.27×102 0.34Area_04 0.026 43.51 92 0.0054 0.125 164.6 217 0.46 1.69×103 0.36Area_05 0.037 46.80 110 0.0078 0.131 144.7 274 0.43 1.25×103 0.34Area_06 0.055 43.75 182 0.0123 0.150 130.9 327 0.37 7.94×102 0.38

fp: fraction of particles, dp: diameter of particles (unit: μm), np: number of particles, SD:standard deviation, NSD: normalized standard deviation, dg: diameter of grains (unit: μm),ng: numberof grains, CI: clustering index, Z: Zener parameter,m: geometric factor of particles.

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The effect of the second-phase particle shape on the grain growthof the matrix grains is significant; samples 028* and 029*, with theirneedle-shaped particles, show a smaller size distribution for matrixgrains than do the equivalent samples (020 and 023) with roundsecond-phase particles. This finding indicates the occurrence oflimited matrix grain growth when the second-phase particles areneedle-shaped, even though the CI values of the particles are similarto that of dispersed grains. In summary, the simulations of graingrowth show that the spatial distribution and shape of second-phaseparticles as well as their fraction are important factors governingtheir pinning effect on grain growth of the main phase.

To determine if the present 2-D simulations of grain growth withimpurities are applicable to natural rocks, we analyzed a micaceousquartzite using the proposed method. The quartzite experiencedpost-tectonic thermal metamorphism (for a detailed description ofthe sample, see Park et al., 2001). We selected six areas within thesample for analysis, each with a different fraction and distributionpattern of muscovite impurities and microstructural properties(Fig. 12). The results, including the Zener parameter (Herwegh andBerger, 2004), are listed in Table 3. Assuming a value of 1 for theZener drag constant (C) yields a geometric factor (m) of about 1/3 foreach area (Table 3). This result implies that the matrix quartz grainsreached a stable state, since their m values are similar regardless ofthe spatial distribution and fraction of muscovite.

Fig. 13 shows the size distribution of the main phase, quartz, ineach area. With increasing fraction of muscovite, or decreasingZener parameter, the quartz grains show a narrower size distributionand a higher frequency, indicating that the pinning effect is controlledmainly by the fraction of secondary particles (muscovite). Areas 02and 05 have the same muscovite fraction, but different CI values.

Fig. 12. (a) Photomicrograph of the quartzite sample used for the analy

The size distribution of area 02, which has a lower CI value, is moreasymmetrical than that for area 05, and it has a higher frequency ofsmall grains (Fig. 14). However, the difference in size distributionsbetween these areas is not as significant as that obtained in thesimulations above, possibly due to complications arising frommeasuring objects which appear to be generally needle-shaped in2-D but are actually plate-shaped in 3-D.

5.3. Phase distribution analysis of a poly-phase rock

Jerram and Cheadle (2000) analyzed the spatial distribution ofmajor minerals in a granulite using the CLHCA method, which isbased on the grain-center distribution technique. They interpreted arandom distribution for pyroxene and scapolite, but clustering forsphene. They attributed the clustering to restrictions on sphenegrowth imposed by pyroxene and scapolite grains. We analyzed theimages of Jerram and Cheadle (2000) using the proposed method

sis. (b) Manual traces of grain boundaries and the analyzed areas.

Fig. 13. Histograms representing size distributions of quartz grains in analyzed areas. fp: fraction of mica. CI: clustering index. Z: Zener parameter.

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(Fig. 15), yielding CI values for pyroxene and scapolite of 0.27 and0.25, respectively, indicating that these grains are strongly clustered(Table 4). The sphene grains, in contrast, are weakly clustered witha CI of 0.42. Thus, our results conflict with those obtained by theCLHCA method, probably because the proposed method focuses onthe proximity of grains rather than their positions. This point isdiscussed in the following section.

6. Discussion

Analytical methods for calculating the spatial distribution of aphase can be classified into two types: position- and proximity-based methods. The position-based methods use the coordinates ofan object's center to define its position without consideration of itssize or shape. These methods include the nearest-neighbor (or BigR; Clark and Evans, 1954; Jerram et al., 1996) and CLHCA (Jerram

Fig. 14. Comparison of the size distribution for Areas 2 and 5. The size distribution ofArea 2 is more asymmetric when compared to that of Area 5.

and Cheadle, 2000) methods. Proximity-based methods are basedon the density of objects in a certain area, or the distance betweenthe boundaries of the objects; therefore, they consider the size andshape as well as the position of the objects. These methods includeDLCA (density linkage cluster analysis; Jerram and Cheadle, 2000),wavelet methods (Darrozes et al., 1997; Gaillot et al., 1997, 1999),and the proposed MADM method. Position-based methods are usedto extract a simple index of the spatial distribution, which makesthem useful in studies of crystal nucleation, such as in a magmabody. Proximity-based methods are not appropriate for generating asimple index of spatial distribution, because of the complicationsarising from the controlling factors (fraction, and size and shape ofobjects). However, these complications give the proximity-basedmethods an advantage in studies of metamorphic and/or deformedrocks (Higgins, 1996). Thus, if the purpose of study is an understandingof the spatial occupancy of grains, the MADM method will give betterresults than the CLHCA method, as shown for the quartzite samplediscussed above and illustrated in Fig. 13.

The MADM method differs from other proximity-based methodsin that it can produce a single index for the spatial distribution of aphase. The MADM method consists of several analysis steps, andeach step produces its own value for the spatial distribution. Dependingon the purpose of analysis, the appropriate step and correspondingindex value can be chosen. For example, when the identification ofindividual grains is not important, the analysis may be stopped oncethe NSD value has been obtained; however, if grain clustering isrelevant, then the individual grains should be identified and the CIvalue should be used for this purpose.

In proximity-based methods, a sample with a high fraction of agiven phase will be more strongly clustered than a sample with alower fraction. In other words, the spatial distribution of a phase isstrongly dependent on its fraction in proximity-based methods. Inthe MADM method, however, the NSD value is independent of thefraction of the phase because it is normalized by the maximum SDfor a perfectly clustered sample for a given fraction of the phase.

In theMADMmethod, CI values are ideal for recognizing individualgrains in samples with a unimodal grain-size distribution; they canalso be used for samples with a uniform grain-size distribution.

Fig. 15. Separated phase images and constructed MADM for the sample used in Jerram and Cheadle (2000). (a) pyroxene, (b) scapolite, (c) sphene, and (d) manually traced imageof the sample.

Table 4Analyzed results of the sample of Fig. 15.

Mineral Fraction SD NSD CI

Sphene 0.05 0.010 0.141 0.42Pyroxene 0.57 0.055 0.222 0.27Scapolite 0.49 0.063 0.244 0.25

SD: standard deviation, NSD: normalized standard deviation, CI: clustering index.

185S. Kim, J.-H. Ree / Tectonophysics 522–523 (2012) 176–186

However, it would be problematic to obtain a CI value for sampleswith a bimodal grain-size distribution.

The MADM method, as described above, does not consider theshape of individual grains or grain aggregates. For example, the twosamples in Fig. 16 have the same size, shape, and fraction of grains;they also have the same CI value, despite the alignment of the grainaggregates in Fig. 16a. This result arises because the shape of thetemplate of the MADM method is circular (isotropic); however, ifwe use an anisotropic template (e.g., ellipsoidal), the CI values of

Fig. 16. Effect of preferred orientation of aggregates on spatial distribution for layered (a) anobjects. (c) Box plots for density values of the two samples.

the two samples will be different, reflecting the shape-preferredorientation of grain aggregates.

We tested the validity of anisotropic templates using samples 020,023, 028*, and 029* (see Fig. 10 for their shapes and distributions).The anisotropic template chosen is ellipsoidal with an axial ratio of1:2. The long axis of the ellipsoidal template was set to horizontalfor some tests and to vertical for others. The NSD values calculatedfor both long-axis orientations of the elliptical template, and thecircular template, are shown in Table 5. For sample 020, whichshows a random distribution with no shape-preferred orientation ofgrains or grain aggregates, the NSD values of each of the threetemplates are similar to one another. In contrast, for sample 028*,which shows a random distribution with a shape-preferred orienta-tion of individual needle-shaped grains, the NSD value of the ellipticaltemplate with a horizontal long-axis oriented parallel to the orienta-tion of the grains, is higher than that of the other templates. Thus, anappropriately shaped anisotropic template can be used for a betterquantification of the spatial distribution for samples with grains orgrain aggregates that show a shape-preferred orientation. If samples

d locally clustered (b) samples. Both samples have the same size, shape and fraction of

Table 5Analyzed results with anisotropic templates for some samples in Fig. 10.

Sample 020 Sample 023 Sample 028* Sample 029*

NSD CIR 0.0101 0.0407 0.0088 0.0407HOR 0.0106 0.0517 0.0103 0.0517VER 0.0105 0.0281 0.0079 0.0279

HOR/CIR 105% 127% 118% 127%VER/CIR 103% 69% 91% 69%HOR/VER 101% 184% 130% 185%

NSD: normalized standard deviation, CIR: circular template, HOR: elliptical templatewith horizontal long axis, VER: elliptical template with vertical long axis.

186 S. Kim, J.-H. Ree / Tectonophysics 522–523 (2012) 176–186

have a layered configuration, however, the result of anisotropictemplate is not much different from that of isotropic template asshown in samples 023 and 029* (Table 5 and Fig. 10). Finally, theMADM method may also be extended to a 3-D analysis by using acubic-shaped grid file and a spherical template.

7. Conclusions

The multiple area density map (MADM) method, a proximity-based method, produces quantitative indices for the spatial distribu-tion of a phase based on an overlay analysis of a sample image. Thenormalized standard deviation (NSD) calculated from this imageanalysis can represent the spatial distribution of a specific phase,but it does not identify individual grains within grain aggregates.The clustering index (CI) characterizes the spatial distribution of theindividual grains within grain aggregates by using their averagegrain size and the NSD. The CI enables us to distinguish betweenclustered (CIb0.5) and random (CI between 0.5 and 0.75) distribu-tions of a phase. The CI value decreases with increasing degree ofclustering.

Computer simulations that examined the pinning effect of second-phase particles on the grain growth of a main phase reveal that thespatial distribution of particles, as well as their fraction and size, cansignificantly influence the growth of the main phase grains. Inparticular, lower CI values (i.e., increased clustering) of particlesresults in an increase of the stabilization time of the grain growth ofthe matrix phase with an asymmetrical size distribution. Theapplication of the proposed method to a muscovite-bearing quartziteyielded similar results to those of the simulations. The software forour 2-D multiple-area density map method is available at GoogleCode (http://code.google.com/p/multiple-area-density-map).

Acknowledgments

We thank Paul Bons and an anonymous reviewer for constructivecomments and Fabrizio Storti for editorial handling. We also appreciatethe invaluable advice by Jason Gruber on the use of MesoscaleMicrostructure Simulation Project (MMSP) codes (http://matforge.org/cmu/wiki/mmsp). This work was supported by the NationalResearch Foundation of Korea fund 2010-0024206 to Ree.

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