Quantitative characterization of rock material for modelling of microstructure

QUANTITATIVE CHARACTERIZATION OF ROCK MATERIAL, FOR MODELLING OF MICROSTRUCTURE

M. Taborda Duarte, P.-A. Lindqvist, S.Q. Kou, K. Miskovsky Department of Civil and Rock Engineering

Luleå University of Technology SE-97187 Luleå - Sweden

ABSTRACT

Quantitative characterization is performed to model rock microstructure, by means of statistical modelling. The microstructural modelling is new and consists in the selection of the appropriate descriptors to characterize the rock, to quantify them through analysis of images taken with the polarizing microscope and thereafter to find the appropriate model among twelve statistical distributions. This approach was successfully applied to a fine even grained granite using grain size, grain shape and grain orientation of the microstructure but other rocks can be characterized. Concerning the orientation of the microstructure only Normal and Logistic distributions are appropriate models. Between the two distributions, Logistic distribution with parameters a=-3.17 and b=24.7 lead to the best fitting with data histogram. Concerning grain size and grain shape of the microstructure, Exponential distribution, Laplace distribution, Poisson distribution and Geometric distribution are not appropriate models. Beta distribution requires a pre treatment of the data and therefore does not fit the data. Among the remaining six distributions, Weibull distribution is the best model with b=75.7, c=1.20 as the seed parameters. Shape is best modelled by Weibull distribution b=65.0 and c=3.40.

Quantitative characterization of the rock is important for the modelling and therefore the understanding of the rock material.

Key words: microstructure, statistical modelling, image analysis, microscope 1. Introduction

Characterization of rock material by means of microstructural modelling

presented in this paper is new, and was developed to model rock material. Liu et

al. (2002b) have used the microscopic image directly to incorporate the rock

microstructure. Results were very satisfactory; however a rock characterization

approach is required. The major advantage of this modelling approach is that the

rock material can be reproduced and specify by the seed parameters of the

statistical distributions. This opens the way to investigate and deeper understand

of intrinsic heterogeneity of the rock. The final goal is to incorporate it into a rock

breakage code so that different rock types can be distinguished and their

mechanical response under different processes may be simulated. As local

variations of the microstructure are known to influence considerably rock

mechanical response, microstructural modelling for rock characterization is an

important step to understand rock mechanical response. In fact, several industries

dealing with rock material, such as rock cutting, drilling, crushing and milling can

benefit from the approach presented here and the analysis of the micro heterogeneity presented in future work.

Shortly, in this paper in series, the microstructure is simplified, quantified and modelled. In order to this, the microstructure of fine even grained granite is seen as matrix where mineral grains are the main constituents. Besides that mineral grains are responsible for rock hardness, the grain boundaries and microcracks that exist in the granite are discontinuities which determine the granite strength and lead the material to fail. Hard rock can be reproduced when the properties of the matrix are appropriately modelled. Among the several properties of this microstructure, main sources of weaknesses in the hard rock are in focus. Once the simplification is performed, quantified by means of image analysis of the grain size, grain shape and grain orientation. Thereafter, by means of statistical distributions and its seed parameters, the microstructure is modelled. Thus, it is possible to reproduce the network of the boundaries, the main source of weaknesses. In the future the microstructural modelling approach will be used with the goal of analysing the heterogeneity and deformation present in microscopic scale.

2. Micro structural Modelling

Rock appears as a mosaic of tightly and interlocked grain profiles regardless of the scale of observation. Both in thin sections when observed in the microscope as well as in polished sections in hand specimen, it is possible to observe projections of the geometrical characteristics of the crystal sections, including their apparent shapes, orientation and distributions. In this paper we are concern with the arrangement of the mineral grains named microstructure, which is the structure of the rock material when observed in the microscope. The polarizing microscope is the only one, among microscopes in which it is possible to observe the network of mineral grains. This microscope differs from the microscope for the metallurgist or the biologist, in that the light is polarized in order to vibrate at two perpendicular directions (Jones, 1987). The optical properties of the minerals are explored in this microscope and a trained observer is able to identify them.

Bard (1980) have summarized and classified the principal textures of igneous and metamorphic rocks. Similar classification of the rock microstructures and a specific methodology to analyse the rock microstructure does not exist.

An to specify different rocks despite their similar mineralogical composition is to quantitatively characterize their microstructure, which is the goal of this paper.

Modelling of microstructure is done by through image analysis observations of thin sections. This requires having appropriate descriptors of the microstructure, having quantified them by semi automatic analysis of microscopic images and thereafter to model them by probabilistic distributions. To use appropriate distributions and to determine their seed parameters is the main goal of the micro structural modelling. Twelve distributions will be tested to fit the distributions of measurements of the descriptors of the microstructure of a fine and even grained granite. Selected of descriptors to characterize the hard rock crystal matrix are grain size, grain shape, grain orientation. Micro cracks are going to be modelled in the future.

The fine and even grained granite is, when observed in hand scale, an

homogeneous rock. When observed under the polarizing microscope at twenty

five times amplification (Fig.1), a thin section of the granite shows an imbricate microstructure. Therefore one can say that rock is homogeneous in hand scale but its microstructure might also show some variability, which is going to be evaluated in future work. With the help of a digital camera installed in the polarizing microscope which is connected to the computer, four microscopic images from a thin section are taken, which do not match, nor do they overlap (Fig.1).

2.1 Simplification of the microstructure

Several simplifications are necessary to reduce the complexity of the microstructure of the fine and even grained granite. They are the basis for consider the results of the microstructural modelling as representative of the analysed rock material. In the sequel, it is assumed that:

(1) One thin section gives representative information about the rock. As the rock is homogeneous in the hand scale often one thin section is required to petrographic examination. The reasonability of this assumption is investigated in future work in progress;

(2) Grain minerals are homogenous and heterogeneity within the grain is not considered;

(3) Thin section of the rock is representative of the real 3D microstructure and no stereological considerations are made. Stereology deals with the interpretation of the three-dimensional structures by means of the two-dimensional sections or projections (Nemati, 2000). This assumption is meaningful as the rock considered is fine even grained. This means that planar thin sections are representative of 3D structure.

These introduce errors which simplifications give rise to error in the future crystal matrix model, if compared with reality but the error is considered to be meaningless, and therefore the modelling approach acquires the most meaningful characteristics of the microstructure.

Figure 1- Four microscopic images of the fine and even grained granite.

2.2 Image Analysis of microstructure

Microscopic images taken with polarizing light are used to study the distribution of grain boundaries while microscopic images taken with parallel light are appropriate to study the distribution of micro cracks and weak areas. Four characteristics emerge due to the optical properties of the mineral: a) different tones and patterns can be observed for the same mineral type, depending the orientation of the crystallographic axis relatively to the cutting plane of the thin section; (b) the boundaries of the small grains are seldom clearly observed because of grain intergrowth; (c) within the mineral bounds the mineral shows a non homogeneous surface, (d) dark minerals occur due to optical extinction and are not possible to be identified mineralogically. The stated characteristics of the microscopic image make the segmentation of the network of grains impossible with known segmentation algorithms. Important code development was done in the early eighties as shown in Fabri (1984) but was not turned into commercial software.

Figure 2 - Simplified microstructure of the fine and even grained granite, after image analysis. Digits (not legible) correspond to identified and analysed grains.

There is correspondence between each one of images in Figure 1 and Figure 2. All the microscopic images come from the same thin section.

Today, image analysis software is not appropriate to deal with microscopic images, because of the complexity of the geological images just described.

Therefore, automatic image analysis requires the development of specific routines.

However in this paper, the boundaries were marked by hand in the original image and then saved in digital file and thereafter the descriptors where measured automatically. Fig. 2 shows four digital files of the granite after segmentation of the microstructure.

2.3 Descriptors of Microstructure

A descriptor is a parameter used to quantify a morphological property of the microstructure. There are a large variety of descriptors (Russ, 1995) and methods (Brandom and Kaplan, 1999) that can be used. The selection of the descriptor depends on the characteristics of the microstructure and the goal of the modelling.

In case of the microstructural modelling, it is necessary to first isolate the properties that influence the rock failure depending on the rock type to be considered, and thereafter to select the appropriate way to measure them, in a word, its descriptor. Thus we can expect important modifications in the microstructural modelling as more rocks are investigated to be specified by the CMM. For instance, porosity is for non hard rocks the kernel for the breakage. In the hard rock microstructure interesting elements are the ones that will lead the material to fail, as the mineral bounds and the microcracks. Grain boundaries are weaknesses in the hard rock matrix as they are discontinuities in a continuum medium of hard minerals phases and are, moreover, the place of where microcracks and voids tend to be concentrated.

Besides the interest in rebuilding the network of boundaries for the CMM model the grain size shape and orientation are by themselves properties that other authors have selected in order to investigate the influence of the microstructure in the mechanical properties. It is known that grain size of rocks has a strong influence on the material strength. To investigate grain size distribution is similar to evaluate the grain boundaries because long boundaries come from large grains.

Orientation is also an important property of the hard rock microstructures because

it accounts for the existence of the anisotropy. Microcracks are accepted as a

determinant factor in the failure of rock and will be modelled in future work.

For the hard rock material, as the one investigated in this paper, the descriptors are parameters that are used to measure the grain size, grain shape and grain orientation. Feret length (FL) is one of the descriptors of the grain size. It is defined as the length of the best fit rectangle on the orientation of a particle. Feret width (FW) is the width of the best fit rectangle on the orientation of the particle.

Feret elongation (FE) is the descriptor of grain shape and is the ratio of FL by FW. Particle orientation (θ) is the angle in degrees between the axis that passes through the centre of the gravity of the particle and follows the main direction of the particle and the horizontal axis. The three descriptors are shown in Figure 3.

Fig. 3 – Descriptor of size (FL), shape (FE=FL/FW) and orientation (θ).

Descriptors are acquired by image analysis (Particle, 2001) of four digital images of the simplified microstructure of one thin section (Fig.2), which sums a total of 428 mineral grains measured for size, shape and orientation. Measurements are done in pixels units, because the goal of the study is to gather the seed parameters of the probabilistic distribution rather than the magnitude of the measurements itself. Results of these measurements are going to be used in the statistical modelling of the microstructure. Some basic statistics of the descriptors are presented in table 1.

Table 1 –Some basic statistics of the microstructure descriptors Thin Section

N=428

Feret Length (mm)

Feret Elongation

Orientation (degrees) Mean

Min.

Max.

Std. Dev.

0,030 0,003 0,191 0,027

58.4 5.6 99.8 19.3

-3.2 -89.1 89 44.8

As the goal of this paper is to use the statistical modelling to quantify the rock microstructure, as presented later on in this paper, the measurements of size are grouped in 10 classes to build the histogram. Than, the statistical function is fitted to. Here we are dealing with the size measured by means of the descriptor Feret length of mineral grain or crystals of non fragmented rock material. However the use of grain size distribution is a very widely used way to describe the size of the particles which result from fragmentation of rock material. However, the way that the size measurements of mineral grains obtained in the microstructural modelling can be modified in order to be similar to particle size distribution as shown in Fig.4. Size measurements were presented in a log normal-normal graph and the measurements in microns were classified according with the bins 1, 2,10,24, 36,75,125,250 as united stated norms for size curves. It is worth to state that this type of size distribution curve obtained by the quantification of the microstructure

Feret Length Feret Width

ө

have additional interest for liberation studies besides the statistical modelling approach.

0 10 20 30 40 50 60 70 80 90 100

1 10 100 1000

Grain Size (µ)

P er ce nta ge a nd C umu la tiv e P er ce nta ge

Simple Histogram Cumulative Histogram

Figure 4- Grain size of the hard crystal rock matrix.

3 Statistical Modelling of the Microstructure

Statistical modelling of microstructure consists of the determination of the appropriate statistical distribution and its seed parameters in order to be able to quantitatively evaluate the microstructure for the rock material model. The results of measurements of descriptors can be presented in the form of histograms, to which density probability function is adjusted. This is a crucial part for the statistical modelling of the microstructure. Twelve statistical distributions are evaluated, namely the density probability function of Poisson, Beta, Exponential, Extreme, Gamma, Geometric, Laplace, Logistic, Lognormal, Normal, Poisson, Rayleigh and Weibull distributions. The definition of these statistical distribution functions and their seed parameters are presented in Table 2. Among the twelve statistical distributions tested in this paper, Weibull is the most interesting one because of the theory behind it, called weakest link theory (Weibull,1939). Briefly this theory states that the strength of material is determined by the occurrence of small defects within the material.

Table 2- Density probability functions of the statistical distributions, their definition, seed parameters and correct range of data and parameters.

Density Probability Function Definition Beta Distribution (x,ν,ω)

ν, ω shape parameters f(x) = Γ(ν+ω)/(Γ(ν)Γ(ω)) x

(1-x)

for 0 < x < 1, ν > 0, ω > 0

Exponential Distribution (x,λ)

λ is a scale parameter f(x) = λe

0 < x < ∞ , λ > 0 Extreme Distribution (x,a,b)

a, b location and scale parameter

f(x) = 1/b e

e

-∞ < x < ∞, b > 0 Gamma Distribution (x,c)

b, c scale and shape parameter f(x) = (x/b)

e

[1/bΓ(c)]

0 ≤ x, b > 0, c > 0,

Geometric Distribution (x,p) f(x) = p(1-p)

p probability of event Laplace Distribution (x,a,b) a, b mean and scale

f(x) = 1/(2b)e

-∞<x<∞

Logistic Distribution (x,a,b) a, b mean and scale

f(x) = (1/b)e

[1+e

]

-∞<x<∞, 0<b

Lognormal Distribution(x,µ,σ) µ, σ scale and shape

f(x) = 1/[xσ(2π)

] e^{ -[log(x)-µ]

/2σ

} 0 < x < ∞ , µ > 0, σ > 0

Normal Distribution (µ,σ) µ, σ mean and std

f(x) = 1/[(2π)

σ] e^{-1/2 [(x-µ)

/σ]

} -∞ < x < ∞

Poisson Distribution (x,λ)

λ is the mean f(x) = (λ

e

)/x!

for x≥0, λ>0 Rayleigh Distribution (x,b)

b scale parameter

f(x) = x/b

e^ [-(x

/2b

)]

0 ≤ x < ∞, b > 0 Weibull Distribution(x,b,c,θ)

b, c and θ are scale, shape, location f (x)= c/b{[(x-θ)/b]^(c-1)}e^{-[(x-θ)/b]^c}

for θ < x, b > 0, c > 0

The distribution parameter presented in table 2, as ν, ω, λ, a, b, c, p, µ,σ, are used to describe the model the microstructure of rock material while x refers to one of the three descriptors previously selected. Γ refers to gamma function. Here the Weibull distribution have θ=0, named the two parameters Weibull distribution.

After fitting the data each one of the theoretical distribution defined in Table 2, the question that came out is which one is the best. Goodness of fit tests are used to test if a distribution is appropriate to describe the data or even to evaluate if the seed parameters obtained by maximum likelihood method are representative for the population and are not only representative for the evaluated sample. Beside Kolmogorov-Smironov test other analytical tests exist to evaluate how good the theoretical distribution is, such a χ

test or Anderson-Darling. However, they do not replace the graphical evaluation or the by probability-plots. Probability plots can also be used for the same effect of histogram with density distributions. More on statistical distributions, maximum likelihood method, goodness of fit tests of Kolmogorov-Smirnof test and χ

test and statistical significance value (p) can be found in Devore (1995). In this paper the selection of the best distributions is done by comparison of the quality of the fitting of the histogram graph with each one of the twelve statistical distributions. Kolmogorov-Smirnof test is made to assure that the distributions are statistically significant.

3.1 Distribution Fitting

Results of fitting the theoretical density probability functions to the histogram

with 10 classes of the measurements of the microstructure descriptors are

summarized in Table 3. The seed parameters are obtained by the maximum

likelihood method (StatSoft, 2001). In order to evaluate if a probability

distribution is significant of the population, from which a sample was evaluated,

the Kolmogorov-Smirnov (K-S) test with confidence level of 99% is performed

and results partly shown in Table 3. In case of lack of statistical confidence in the

reliability of the distribution found, this is presented with p>0.1 in the same table

3. Invalid range of values (IRV) occurs when the data does not respects the

correct range defined in table 2.

Table 3- Seed parameters of the density function of twelve statistical distributions that fit the histogram of the descriptors. Results of goodness of fit test of Kolmogorov-Smirnov and its p-value are given. IRV stands for invalid range of values.

Descriptor Dist,K-S p-value

x= FL (Size) x= FE (Shape) x=Ө Orientation

Beta Distribution (x,υ,ω) IRV IRV IRV

Exponential Dist.(x,λ) K-S test p-value

(x,0.0141) p<0.01

(x,0.0171) p<0.01

IRV Extreme Dist. (x,a,b)

K-S test p-value

(x,45.0,39.0) p<0.01

(x,48.6,19.2) p<0.01

IRV Gamma Dist. (x/b,c)

K-S test p-value

(x/47.1,1.50) p>0.1

(x/7.88 ,7.4) p<0.01

IRV Geometric Dist. (x,p)

K-S test p-value

(x,0.0139) p<0.01

(x,0.0168) p<0.01

IRV Laplace Dist. (x,a,b)

K-S test p-value

(x,70.8,46.3) p<0.01

(x,58.4,13.6) p<0.01

IRV Logistic Dist. (x,a,b)

K-S test p-value

(x,70.8,36.1) p<0.01

(x,58.4,10.6) p<0.01

(x,-3.2,24.7) p<0.01 Lognormal Dist. (x,µ,σ)

K-S test p-value

(x,3.90,0.88) p<0.01

(x,4.0,0.41) p<0.01

IRV Normal Dist. (µ,σ)

K-S test p-value

(x,70.8,65.5) p<0.01

(x,58.4,19.3) p≥0.1

(x,-3.2,44.8) p<0.05 Poisson Dist. (x,λ)

K-S test p-value

(x,70.8) p<0.01

(x,58.4) p<0.01

IRV Rayleigh Dist. (x,b)

K-S test p-value

(x,68.2) p<0.01

(x,43.4) p<0.01

IRV Weibull Dist. (x,b,c,θ)

K-S test p-value

(x,75.7,1.2,0) p<0.01

(x,65.0,3.4,0) p<0.01

IRV

Beta distribution, is not appropriate for the three descriptors because it requires that the data fall in the range of 0<x<1, and the data evaluated for FL, FE and Ө fall outside of the valid range (IRV shown in table 3). Similar case of limitations of statistical modelling, because of the range of values, occurs with the orientation descriptor, which negative values do not enable the use of many distributions (Table 3). The way to overcome the problem is to measure or convert the negative angles in the direction confectioned positive. This would enable to model with the other statistical distributions. In terms of statistical significance level very good results are obtained by using the microstructure modelling approach presented in this paper. The results reported in table 3 shows that most of the statistical distributions used to fit the data are significant for the population. This is caused by the large sample size, 428 data to draw a histogram. Only Normal distribution for Feret elongation, Gamma distribution for Feret length and Normal distribution for orientation are not representative for the population with 90% confidence level (p>0.1) based on Kolmogorov-Smirnov test results. Therefore these distributions for these descriptors are to avoid. All the other statistical distributions are statistically significative. Selection of the most appropriate among the twelve is made by means of graphical assessment of how the curve fit the histogram data.

The best four statistical models, among the evaluated twelve previously defined

(Table 2), are presented in Fig. 5-7, while the remaining seven statistical distributions are presented in appendix. Beta distributions and other non valid distributions for the orientation descriptor are not shown in the appendix. The legend of Fig. 5 to Fig. 7 gives the seed parameters required for the statistical modelling following the notation of Table 2.

-50 0 50 100 150 200 250 300 350 400 450 500

Feret Lengh (pixels) 0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

47%

51%

56%

Percent of obs

1. Normal (70.76,65.5),p<0.01 2.Gamma (1086,1.5), p<0.01 3. Weibull (75.7,1.20), p<0.01 4.Logistic (70.8,36.1203), p<0.01

1 4 2

3

Fig. 5- The two best fitted distributions of size descriptor including Weibull and Normal distributions.

Fig. 5 shows that both Logistic and Normal distributions are not appropriate to model the size as it models the existence of negative values that do not make physical sense. On the other hand Weibull and Gamma fit very well the histogram of the measurements. As these two statistical distributions have a similar curve, we propose Weibull distribution due to reasons previously mentioned. The microstructure of the fine and even grained granite can be modelled by means of the Weibull (75.7, 1.20) according with the definition of the Weibull distribution in table 2.

-10 0 10 20 30 40 50 60 70 80 90 100 110

Feret Elong 0%

5%

9%

14%

19%

23%

28%

Percent of obs

1.Logistic (58.4,10.6).p<0.01 2.Normal (58.4, 19.3), p<0.01 3. Weibull (65.0,3.4),p<0.01 4. Gamma (8514,7.4),p<0.01

1

2 3

4

Fig. 6- The two best fitted distributions for shape descriptor,

including Weibull and Normal distributions.

Fig. 6 shows that Logistic, Normal, Weibull and Gamma distributions fit reasonable well the data measurements of the shape of the microstructural grains.

However Normal and Weibull have a similar curve that has an almost perfect fit to histogram shape. For the same reason stated for size distribution in Fig.5, the Weibull distribution is selected. The microstructure of the fine and even grained granite can be modelled by means of the Weibull (65.0, 3.40) according with definition of the Weibull distribution in table 2.

-120 -100

-80 -60

-40 -20

0 20

40 60

80 100

346.53 Orientation (degrees)

0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of obs

1. Logistic (-3.17,24.7), p<0.01 2. Normal (-3.17,44.7), p <0.01 3.Weibull IRV 4.Gamma IRV 1

2

Fig. 7- The best two fitted distributions, more Weibull distribution for orientation descriptor.

Fig. 7 shows that Weibull and Gamma distributions can not be used to model orientation because of the negative data values. Normal and Logistic distributions very reasonable fit to the measurements of the orientation of the microstructure.

Between these two distributions, Logistic is the most appropriate one as it better show the trend of the data to the orientation. Therefore, the orientation descriptor of the fine and even grained granite can be modelled by means of the Logistic distribution (-3.17, 24.7) according with definition of the Logistic distribution presented in the table 2.

4 Rock Material Model (MM) by means of microstructural modelling

The basis for the development of the microstructural modelling approach is to build the RMM. This model should account for the heterogeneity and therefore enable to specify different rocks. This is very promising. MM can reproduce the imbricated microstructure of mineral grains. Monte Carlo method can be used to simulate the descriptors of the grain size, grain shape and grain orientation of the microstructure through the seed parameters of the statistical distributions that model them. Thus, the RMM is the network of boundaries with the microcracks on the top of them. Modelling of microcracks will be considered in future work.

To combine the RMM with R-T

(rock and tool interaction simulation code) is

the future application of the microstructural modelling approach. This means, to

incorporate the rock material model into a mechanical model in order to be able to predict breakage of different rock types. R-T

, is a numerical simulation tool to predict the breakage of the heterogeneous rock material under different processes (Liu, 2002a). However, it is necessary to investigate which mechanical properties should be assigned to weaknesses modelled with the RMM, as the mechanical properties of the minerals are known but the mechanical properties of the weaknesses such as grain boundaries and microcracks are unknown.

5 Conclusions

In order to specify different rock material, a new and promising approach was presented, named micro structural modelling. This requires having appropriate descriptors of the microstructure, to quantify them by semi automatic analysis of microscopic images and thereafter to model them by statistical distributions. To determine the appropriate distribution and its seed parameter among twelve different statistical distributions is the kernel of micro structural approach for material modelling.

The micro structure of the fine even grained granite was characterized through the grain size shape and orientation descriptors. Concerning grain orientation only Normal and Logistic distributions are appropriate to model properly the data.

Concerning size and shape of the grains of the micro structure the distributions Exponential distribution, Laplace distribution, Poisson distribution, and Geometric distribution are not appropriate model. Beta distribution requires a pre treatment of the data and therefore it was not considered. Normal distribution Logistic distribution, Raleigh distribution, Extreme distribution Weibull distribution and Gamma distribution are reasonable data models. Among these six distributions, Weibull distribution and Logistics distribution showed to be the most appropriate models.

By means of the seed parameters the microstructure can be reproduced.

Concerning size, Weibull distribution is the best model with (75.7, 1.20) as the seed parameters. Shape is best modelled by Weibull distribution (65.0, 3.40) while orientation is best modelled by the Logistic distribution (-3.17, 24.7). The parameters refer to definition presented in the paper.

Acknowledgements

Research presented in this paper is part of the research program Rock Fragmentation Design financed by LKAB´s Foundation for the Promotion of Research and Education at Luleå University of Technology, Trelleborg AB´s Research and Education Foundation at Luleå University of Technology, Arne S.

Lundberg´s Foundation for Applied Geoscience at Luleå University of Technology, School of Postgraduate Studies in Mining and Ore Processing supported by The Knowledge Foundation and The Foundation for Technology Transfer, Luleå. The support from these organisations is greatly appreciated.

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edition, Boca Raton

Fla.: CRC

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Handl 1939; 151; 544

-50 0 50 100 150 200 250 300 350 400 450 500 Feret Length (pixels)

0%

12%

23%

35%

47%

58%

70%

82%

Percent of observations

Exponential (0.014), K-S p<0.01 -50 0 50 100 150 200 250 300 350 400 450 500

Feret Size (pixels) 0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

47%

51%

Percent of observations

Extreme (45.0,38.9), K-S p<0.01

-50 0 50 100 150 200 250 300 350 400 450 500 Feret Size (pixels)

0%

47%

93%

140%

187%

234%

280%

Percent of observations

Poisson (70.8), K-S p<0.01

-50 0 50 100 150 200 250 300 350 400 450 500 Feret Size (pixels)

0%

12%

23%

35%

47%

58%

70%

82%

Percent of obs

Lognormal (3.9, 0.88), K-S p<0.01

-50 0 50 100 150 200 250 300 350 400 450 500 Feret lenght (pixels)

0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

47%

51%

Percent of observations

Rayleigh (68.2) , K-S p<0.01

-50 0 50 100 150 200 250 300 350 400 450 500 Feret Length (pixels)

0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

47%

51%

56%

Percent of observations

Laplace (70.8,36.1), K-S p<0.01 -50 0 50 100 150 200 250 300 350 400 450 500

Feret length (pixels) 0%

12%

23%

35%

47%

58%

70%

82%

Percent of observations

Geometric (0.0139), K-S p<0.01

Appendix Fitting of the density probabilistic functions on Feret Length data histograms

-10 0 10 20 30 40 50 60 70 80 90 100 110 Feret Shape

0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of observations

Rayleigh (43.5), K-S p>0.01

-10 0 10 20 30 40 50 60 70 80 90 100 110 Feret Shape

0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of observations

lognormal (4.0, 0.40), K-S p<0.1 6,0 15,4 24,8 34,2 43,6 53,0 62,4 71,8 81,2 90,6 100,0

Feret Shape 0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of observations

Exponential (0,0171) , K-S p< 0.001

6 15 25 34 44 53 62 72 81 91 100

Feret Shape 0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of observations Extreme (48.6,19.2)

K-S p<0.01

-10 0 10 20 30 40 50 60 70 80 90 100 110

Feret Shape 0%

2%

5%

7%

9%

12%

14%

16%

19%

21%

23%

Percent of observations

Geometric (0,0168), K-S p<0,01

-10 0 10 20 30 40 50 60 70 80 90 100 110

Feret Shape 0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

47%

51%

56%

Percent of observations

Poisson (58.4), K-S p<0.01

Fitting of the density probabilistic functions on Feret Shape data histograms

-10 0 10 20 30 40 50 60 70 80 90 100 110

Feret Shape 0%

5%

9%

14%

19%

23%

28%

33%

37%

42%

Percent of observations

Laplace (58.4,13.6), K-S p<0.01