Joint Rock Coefficient Estimation Based on Hausdorff Dimension ()

Dalibor Martišek^{}

Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic.

**DOI: **10.4236/apm.2017.711037
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Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic.

The strength of rock structures strongly depends inter alia on surface irregularities of rock joints. These irregularities are characterized by a coefficient of joint roughness. For its estimation, visual comparison is often used. This is rather a subjective method, therefore, fully computerized image recognition procedures were proposed. However, many of them contain imperfections, some of them even mathematical nonsenses and their application can be very dangerous in technical practice. In this paper, we recommend mathematically correct method of fully automatic estimation of the joint roughness coefficient. This method requires only the Barton profiles as a standard.

Keywords

Hausdorff Dimension, Self-Similarity, Self-Affinity, Box Counting Method, Power Function Method, Barton Profile, JRC Index

Share and Cite:

Martišek, D. (2017) Joint Rock Coefficient Estimation Based on Hausdorff Dimension. *Advances in Pure Mathematics*, **7**, 615-640. doi: 10.4236/apm.2017.711037.

1. Introduction

A shape of geological discontinuities plays an important role in influencing the stability of rock masses. Many approaches have been used for its determination. The method of Barton and Choubey (1977) is well known in geotechnical practice. These authors introduced the method which is able to calculate the shear strength $\tau $ of rock joints as

$\tau ={\sigma}_{n}\cdot \mathrm{tan}\left({\phi}_{r}+JRC\cdot \mathrm{log}\frac{JCS}{{\sigma}_{n}}\right)$ (1)

where JRC is the joint roughness coefficient, JCS is the joint compressive strength, ${\phi}_{r}$ is the residual friction angle, and ${\sigma}_{n}$ is the normal stress.

The method of Barton and Choubey [1] is well known in geotechnical practice-a visual comparison a fracture rock surface to be analysed with the standard Barton profiles is preferred way for determining JRC values.

A quick and easy estimate is probably one of the main reasons for this preference. However, this method is very subjective. Therefore, objective methods for JRC estimation are searched―see [2] [3] [4] [5] [6] for example. Unfortunately, some published papers contain many inaccuracies and even mathematical nonsenses. Application of some published “indicators of similarity” may be very dangerous in civil engineering. We refer to some of them and we recommend a mathematically correct method of fully automatic estimation of the JRC index in the following text.

2. Some Errors of Present Methods Based on Fractal Dimension

As was said in Introduction, subjective visual comparison a fracture rock surface to be analyzed with the standard Barton profiles (see Figure 1) is preferred way for determining JRC values. Objective methods for JRC are searched, unfortunately, many of them are incorrect.

Many researchers believe that the surface roughness of rock joints needs to be characterized using scale invariant parameters such as fractal parameters. Several researchers have suggested using the fractal dimension to quantify rock joint roughness (see [7] - [13] for example).

In [14] , there is “derived” a “direct relationship” between the JRC index and fractal dimension D

$JRC\approx 50\cdot \left(D-1\right)$ (2)

However, it is a nonsense as the following example illustrates.

Figure 1. Standard Barton roughness profiles and their joint rock coefficients.

Example: A fractal dimension is namely affine invariant, i.e. each bijective affine transformation of the profile has the same dimension as an original. The profile $p\left(x\right)$ in Figure 2 was generated as a fractional Brownian motion and for every x is $P\left(x\right)=4\cdot p\left(x\right)$ . To easily determine the dimension of the resulting fractal, a random number must be generated by the Gaussian distribution $N\left(0;1\right)$ and the i-th iteration step variations ${\sigma}_{i}$ have to be adjusted in accordance with

${\sigma}_{i}^{2}=\frac{{\sigma}_{0}^{2}}{{2}^{2Hi}}$ (3)

where $H\in \langle 0;1\rangle $ is so called Hurst exponent.

Due to affine invariance, both profiles have the same dimension ( $D=1.5$ ) and should have the same roughness therefore. This is evidently not true. Moreover, JRC of both profiles is $JRC\approx 25$ according to (1). This is also not true.

In [14] [15] , another „direct relationship” between dimension and JRC was published:

$JRC=-0.87804+37.7844\cdot \left(\frac{D-1}{0.015}\right)-16.9304\cdot {\left(\frac{D-1}{0.015}\right)}^{2}$ (4)

This relationship is often cited (see [16] [17] [18] [19] for example) but it is quite false. Equation (4) gives a totally nonsensical results for Barton profiles as will be shown in 2.5 (see the last column of Table 3).

Table 1. Hausdorff dimension and grid measure of the original Koch curve A and its affine representation B estimated by box-counting method and power-function method. Used affinity is $\left[x;y\right]\to \left[x;2y\right]$ .

Figure 2. The profile $p\left(x\right)$ was generated as a fractional Brownian motion. Due to affine invariance, the profiles $p\left(x\right)$ ; $P\left(x\right)$ have the same dimension but evidently different roughness.

We can often read that for computing of fractal dimension, it is necessary to decide whether the object is self-similar or self-affine (see [20] [21] [22] [23] ). It is said that “the computation of fractal dimensions of self-affine fractals requires modified computational methods” [14] [20] and their dimensions D have to be computed by others methods than the dimension of self-similar curves. Alleged reason is “different scaling” in the x-and y-directions which changes its dimensionality (see [14] for example). However, it is a deep mistake. One example for all: The curve B in Figure 3 contains two its copies with the same scaling (red and pink). These copies required “non-modified” method. However, the same curve contains two copies with different scaling (green and blue). These copies required “modified” method. Can we use the modified or the non-modified method for its dimensionality estimation?

3. Hausdorff Measure and Hausdorff Dimension

Hausdorff defined the first dimension that allows non-integer values. Hausdorff s-dimensional outer measure of a set A is defined as

${}^{\ast}H{}^{\left(s\right)}\left(A\right)=\underset{n\to \infty}{\mathrm{lim}}\underset{A\subseteq \cup {A}_{n;i}}{\mathrm{inf}}\left\{\underset{i\in I}{{\displaystyle \sum}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{s}|diam\text{\hspace{0.05em}}{A}_{n;i}\le \frac{1}{n}\right\}$ (5)

where I is an at most countable index set. Restriction of ${H}^{\left(s\right)}$ to the sets measurable with ${}^{\ast}H{}^{\left(s\right)}$ (H-measurable sets) is called Hausdorff s-dimensional measure of the set A. The number

${D}_{H}\left(A\right)=\mathrm{sup}\left\{d\in {\mathbb{R}}_{0}^{+}\cup \left\{\infty \right\}|{H}^{\left(d\right)}\left(A\right)=\infty \right\}=\mathrm{inf}\left\{d\in {\mathbb{R}}_{0}^{+}\cup \left\{\infty \right\}|{H}^{\left(d\right)}\left(A\right)=0\right\}$ (6)

is called Hausdorff dimension of the set A.

Mandelbrot [24] defined fractal as a set which Hausdorff dimension is sharply greater than the topologic dimension. Ever after several dimension which allows non-integer values was defined (see [25] for example). Each of them is called the fractal dimension.

For estimation of the Hausdorff dimension of sets which are constructed on digital devices, so called grid measure and grid dimension are used. The grid s-dimensional outer measure is defined as

${}^{\ast}G{}^{\left(s\right)}\left(A\right)=\underset{n\to \infty}{\mathrm{lim}}\underset{A\subseteq \cup {A}_{n;i}}{\mathrm{inf}}\left\{\underset{i\in I}{{\displaystyle \sum}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{s}|diam\text{\hspace{0.05em}}{A}_{n;i}=\frac{1}{n}\right\}$ (7)

(a) (b)

Figure 3. “Different scaling” in x- and y-directions can change the self-similar set (a) to self-affine set (b). However, “different scaling” in x- and y-directions can change the self-affine (b) to self-similar set (a).

Its restriction ${G}^{\left(s\right)}$ to the sets measurable with ${}^{\ast}G{}^{\left(s\right)}$ (G-measurable set) is called the grid measure. The grid dimension (G-dimension) of the set A is defined as

${D}_{G}\left(A\right)=\mathrm{sup}\left\{d\in {\mathbb{R}}_{0}^{+}\cup \left\{\infty \right\}|{G}^{\left(d\right)}\left(A\right)=\infty \right\}=\mathrm{inf}\left\{d\in {\mathbb{R}}_{0}^{+}\cup \left\{\infty \right\}|{G}^{\left(d\right)}\left(A\right)=0\right\}$ (8)

The G-dimension is suitable for digital data and since the limit condition
$n\to \infty $ in Formula (7) cannot be realized, the limit is omitted and the Formula (7)

${}^{\ast}G{}^{\left(s\right)}\left(A\right)\approx \underset{A\subseteq \cup {A}_{n;i}}{\mathrm{inf}}\left\{\underset{i\in I}{{\displaystyle \sum}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{s}|diam\text{\hspace{0.05em}}{A}_{n;i}=\frac{1}{n}\right\}$ (9)

4. Box Counting and Power-Function Method

For the infimum to be computed in (9), only those sets ${A}_{n;i}$ are taken to the union $\cup {A}_{n;i}$ for which ${A}_{n;i}\cap A\ne \varnothing $ . Due to the fact that only bounded sets (or more precisely their approximations containing finite elements) can be represented in the computer, the system $\left\{{A}_{n;i}\right\}$ is always finite. Let us denote its cardinality by $N\left(n\right)$ . The measured approximations are always G-measurable. In software implementations of the measurement, used metric is a square metric, where the diameter of a square is equal to its side. According to Formula (9) we obtain for measure in Hausdorff dimension

${G}^{\left(D\right)}\left(A\right)\approx \underset{i=1}{\overset{N\left(n\right)}{{\displaystyle \sum}}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{D}=\underset{i=1}{\overset{N\left(n\right)}{{\displaystyle \sum}}}\frac{1}{{n}^{D}}=N\left(n\right)\cdot {n}^{-D}$ (10)

Therefore,

$N\left(n\right)\approx {G}^{\left(D\right)}\cdot {n}^{D}$ (11)

Applying the logarithm on both sides of the approximate equality (11) we obtain

$\mathrm{ln}N\left(n\right)\approx D\cdot \mathrm{ln}n+\mathrm{ln}{G}^{\left(D\right)}$ (12)

Measuring with a specified n, a $N\left(n\right)$ is obtained for a covering of the measured set. The values D and ${G}^{\left(D\right)}$ are calculated by fitting the straight line in the form (12) using the least square method. It is evident that for a high enough n we can calculate D as

$D\approx \frac{\mathrm{ln}N\left(n\right)}{\mathrm{ln}n}$ (13)

and even define the dimension as the limit of that fraction, i.e.

${D}_{B}=\underset{n\to \infty}{\mathrm{lim}}\frac{\mathrm{ln}N\left(n\right)}{\mathrm{ln}n}$ (14)

This dimension and the method for its measurement are known as the box counting.

Note that n is the reciprocal value of the diameter of covering sets, which is often marked as $\epsilon $ . Therefore, if we denote the cardinality $N\left(n\right)$ of the covering of the set to be measured as $N\left(\epsilon \right)$ , we obtain

$N\left(\epsilon \right)\approx {G}^{\left(D\right)}\left(A\right)\cdot {\epsilon}^{-D}$ (15)

from (11)

$\mathrm{ln}N\left(\epsilon \right)\approx -D\cdot \mathrm{ln}\epsilon +\mathrm{ln}{G}^{\left(D\right)}\left(A\right)$ (16)

from (12) or

${D}_{B}=\underset{\epsilon \to 0+}{\mathrm{lim}}\frac{\mathrm{ln}N\left(\epsilon \right)}{\mathrm{ln}{\epsilon}^{-1}}$ (17)

from (14) respectively.

To calculate this dimension for the fractal F, it is necessary to insert this fractal into an evenly spaced grid and count how many squares (2D case) or boxes (3D case) are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm.

It is possible to shown that the theoretically defined box counting dimension (14) is equal to the Hausdorff dimension―see Formula (8). A problem is that the dimension (14) has to be estimated with the least-square method form the linear function (12). If we denote the power function (11) as $f\left(x\right)=G\cdot {x}^{D}$ , sum of its residues is ${R}_{n}=\text{\Sigma}{\left[N\left(n\right)-f\left(n\right)\right]}^{2}$ , while sum of residues for the linear function (12) is ${R}_{n}^{*}=\text{\Sigma}{\left[\mathrm{ln}N\left(n\right)-\mathrm{ln}f\left(n\right)\right]}^{2}$ . Of course ${R}_{n}^{*}\ll {R}_{n}$ . Thus the box-counting method systematically overestimates residues of low values and underestimates residues of its high values. Moreover, negative values of the difference $\mathrm{ln}N\left(n\right)-\mathrm{ln}f\left(n\right)$ have lower weights than positive values. This somehow lowers the tangent of the straight line as thus the value of the estimated dimension.

This problem can be overcome by searching for the power function (11) instead of the linear function (12). The least square method requires in this case minimization of the function

$f\left(N,D\right)=\underset{n}{{\displaystyle \sum}}{\left(N-G\cdot {n}^{D}\right)}^{2}$ (18)

This leads to the equation

$\underset{n}{{\displaystyle \sum}}\left(N{n}^{D}\mathrm{ln}n\right)\underset{n}{{\displaystyle \sum}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{n}^{2D}-\underset{n}{{\displaystyle \sum}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}N{n}^{D}\underset{n}{{\displaystyle \sum}}\left({n}^{2D}\mathrm{ln}n\right)=0$ (19)

This equation is then solved numerically―see [26] for more information.

5. Self-Similarity and Self-Affinity

Many technical papers describe the fractals. We can read that the fractals can be either self-similar or self-affine and the original box counting method is a self-similar method and it provides accurate results only for self-similar profiles. Natural rock joint profiles are self-affine, therefore, the box-counting method is not useable for their fractal dimension―see [27] for example. However, these affirmations are very inaccurately. It is said that self-affine curves, in contrast to self-similar ones, are not identically scaled in x- and y-directions (see [14] [20] [21] [22] for example). This “definition” is unprofessional and very narrow (restricted). Many others fractals are self-affine.

A self-affine fractal is any fractal F, for which there exist affine mappings ${\phi}_{i};i=1;2;\cdots ;n$ so it holds

$F={\displaystyle \underset{i=1}{\overset{n}{\cup}}{\phi}_{i}\left(F\right)}={\displaystyle \underset{i=1}{\overset{n}{\cup}}{F}_{i}}$ (20)

If all affinities ${\phi}_{i}$ are the similarities then the self-affine fractal is concurrently self-similar. It means that the self-similarity is a special case of the self-affinity, i.e. each self-similar set is self-affine concurrently.

In Euclidean space, each affinity is given by

${X}^{\prime}={F}_{i}\cdot X+{v}_{i}$ (21)

where ${F}_{i}$ is any square matrix and $v$ is any vector. If

${F}_{i}\cdot {F}_{i}^{\text{T}}={\lambda}^{2}\cdot I$ (22)

(where $I$ is the identity matrix) then the affinity is called the similarity, number $\lambda $ is its coefficient. Except self-similar and self-affine fractals, there exist sets which are neither self-similar nor self-affine (Mandelbrot set for example).

According of the definition of the Hausdorff dimension is

$0<{H}^{\left(D\right)}\left(A\right)=\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\left\{\underset{k\in I}{{\displaystyle \sum}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{s}|diam\text{\hspace{0.05em}}{A}_{n;i}\le \frac{1}{n}\right\}<\infty $ (23)

If a set A is self-similar and ${\lambda}_{1};{\lambda}_{2};\cdots ;{\lambda}_{p}$ are coefficients of the similarities ${\phi}_{i}$ from (20), and for each $i\ne j$ is ${H}^{\left(D\right)}\left({\phi}_{i}\left(A\right)\cap {\phi}_{j}\left(A\right)\right)=0$ then

$\begin{array}{c}{H}^{\left(D\right)}\left(A\right)=\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\left\{\underset{k\in I}{{\displaystyle \sum}}{\left(diam\text{\hspace{0.05em}}{A}_{n;i}\right)}^{D}|diam\text{\hspace{0.05em}}{A}_{n;i}\le \frac{1}{n}\right\}\\ =\underset{{H}^{\left(D\right)}\left(A\right)}{\underset{\ufe38}{\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}{\left(\text{diam}A\right)}^{D}}}\underset{i=1}{\overset{p}{{\displaystyle \sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\lambda}_{i}^{D}\end{array}$ (24)

It means that

${H}^{\left(D\right)}\left(A\right)={H}^{\left(D\right)}\left(A\right)\cdot \underset{i=1}{\overset{p}{{\displaystyle \sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\lambda}_{i}^{D}$ (25)

${\lambda}_{1}^{D}+{\lambda}_{2}^{D}+\cdots +{\lambda}_{p}^{D}=1$ (26)

In case of ${\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{p}=\lambda $ is

${\lambda}_{1}^{D}+{\lambda}_{2}^{D}+\cdots +{\lambda}_{p}^{D}=p\cdot {\lambda}^{D}=1\Rightarrow D=\frac{\mathrm{ln}p}{\mathrm{ln}\frac{1}{\lambda}}$ (27)

Example: the Koch curve is self-similar with four copies of itself, each scaled by the factor one third, its dimension is $D=\frac{\mathrm{ln}4}{\mathrm{ln}3}$ . The Sierpinski triangle or

Sierpinski square are also self-similar with three copies scaled by one half or

eight copies scaled by one third respectively, their dimensions are $D=\frac{\mathrm{ln}3}{\mathrm{ln}2}$ or $D=\frac{\mathrm{ln}8}{\mathrm{ln}3}$ respectively.

For H-measure of any fractal, the H-measure of its covering $\text{C}={\displaystyle {\cup}_{k}{A}_{nk}}$ is determinative. This covering consists of cubes in case of square metric. The measure of self-affine fractals (19) is equal to the sum of measures its affine copies ${\phi}_{i}$ . Each affinity ${\phi}_{i}$ transforms a cubical covering C to the set of parallelepipeds ${\phi}_{i}\left(\text{C}\right)$ . We have to find how the volume of cube will change by its transform to the parallelepiped.

Each cube is given by orthonormal vectors $a=\left({a}_{1};{a}_{2};{a}_{3}\right)$ ; $b=\left({b}_{1};{b}_{2};{b}_{3}\right)$ ; $c=\left({c}_{1};{c}_{2};{c}_{3}\right)$ which are transform to linearly independent vectors ${a}^{\prime}=\left({{a}^{\prime}}_{1};{{a}^{\prime}}_{2};{{a}^{\prime}}_{3}\right)$ ; ${b}^{\prime}=\left({{b}^{\prime}}_{1};{{b}^{\prime}}_{2};{{b}^{\prime}}_{3}\right)$ ; ${c}^{\prime}=\left({{c}^{\prime}}_{1};{{c}^{\prime}}_{2};{{c}^{\prime}}_{3}\right)$ by bijective affinity ${\phi}_{i}$ , i.e.

${{a}^{\prime}}^{\text{T}}={F}_{i}\cdot {a}^{\text{T}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{b}^{\prime}}^{\text{T}}={F}_{i}\cdot {b}^{\text{T}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{c}^{\prime}}^{\text{T}}={F}_{i}\cdot {c}^{\text{T}}$ (28)

or

${a}^{\prime}=a\cdot {F}_{i}^{\text{T}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}^{\prime}=b\cdot {F}_{i}^{\text{T}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}^{\prime}={F}_{i}^{\text{T}}\cdot {c}^{\text{T}}$ (29)

where ${F}_{i}$ is the matrix of the affinty ${\phi}_{i}$ . This implies

$\left({{a}^{\prime}}_{1};{{a}^{\prime}}_{2};{{a}^{\prime}}_{3}\right)=\left({a}_{1};{a}_{2};{a}_{3}\right)\cdot \left(\begin{array}{ccc}{z}_{11}& {z}_{21}& {z}_{31}\\ {z}_{12}& {z}_{22}& {z}_{32}\\ {z}_{13}& {z}_{23}& {z}_{33}\end{array}\right)=\left(\left(a;{f}_{1}\right);\left(a;{f}_{2}\right);\left(a;{f}_{3}\right)\right)$ (30)

By analogy

$\left({{b}^{\prime}}_{1};{{b}^{\prime}}_{2};{{b}^{\prime}}_{3}\right)=\left(\left(b;{f}_{1}\right);\left(b;{f}_{2}\right);\left(b;{f}_{3}\right)\right)$ (31)

$\left({{c}^{\prime}}_{1};{{c}^{\prime}}_{2};{{c}^{\prime}}_{3}\right)=\left(\left(c;{f}_{1}\right);\left(c;{f}_{2}\right);\left(c;{f}_{3}\right)\right)$ (32)

Volume of the parallelepiped which is given by vectors ${a}^{\prime};{b}^{\prime};{c}^{\prime}$ is equal to the scalar triple product. Therefore, we obtain

$\begin{array}{c}\left|\begin{array}{ccc}{{a}^{\prime}}_{1}& {{a}^{\prime}}_{2}& {{a}^{\prime}}_{3}\\ {{b}^{\prime}}_{1}& {{b}^{\prime}}_{2}& {{b}^{\prime}}_{3}\\ {{c}^{\prime}}_{1}& {{c}^{\prime}}_{2}& {{c}^{\prime}}_{3}\end{array}\right|=\left|\begin{array}{ccc}\left(a;{f}_{1}\right)& \left(a;{f}_{2}\right)& \left(a;{f}_{3}\right)\\ \left(b;{f}_{1}\right)& \left(b;{f}_{2}\right)& \left(b;{f}_{3}\right)\\ \left(c;{f}_{1}\right)& \left(c;{f}_{2}\right)& \left(c;{f}_{3}\right)\end{array}\right|\\ =\left|\begin{array}{ccc}{a}_{1}& {a}_{2}& {a}_{3}\\ {b}_{1}& {b}_{2}& {b}_{3}\\ {c}_{1}& {c}_{2}& {c}_{3}\end{array}\right|\cdot \left|\begin{array}{ccc}{f}_{11}& {f}_{21}& {f}_{31}\\ {f}_{12}& {f}_{22}& {f}_{32}\\ {f}_{13}& {f}_{23}& {f}_{33}\end{array}\right|\end{array}$ (33)

from (30), (31), (32). Therefore

$V\left({a}^{\prime};{b}^{\prime};{c}^{\prime}\right)=\left|\mathrm{det}{F}_{i}^{\text{T}}\right|\cdot V\left(a;b;c\right)=\left|\mathrm{det}{F}_{i}\right|\cdot V\left(a;b;c\right)$ (34)

It is possible to obtain

$S\left({a}^{\prime};{b}^{\prime}\right)=\mathrm{det}{F}_{i}\cdot S\left(a;b\right)$

for the area of the parallelogram in two-dimensional space. The diameter of cube (or square) which have the same volume (or area) in square metric is

$\text{diam}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{{A}^{\prime}}_{nk}=\sqrt[3]{\left|\mathrm{det}{F}_{i}\right|\cdot V\left(a;b;c\right)}=\sqrt[3]{\left|\mathrm{det}{F}_{i}\right|}\cdot \text{diam}\text{\hspace{0.05em}}{A}_{nk}$ (35)

or

$\text{diam}\text{\hspace{0.05em}}{{A}^{\prime}}_{nk}=\sqrt{\left|\mathrm{det}{F}_{i}\right|\cdot S\left(a;b\right)}=\sqrt{\left|\mathrm{det}{F}_{i}\right|}\cdot \text{diam}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{A}_{nk}$ (36)

respectively.

For the H-measure of a self-affine fractal A which contains p affine copies of itself, we obtain

$\begin{array}{c}{H}^{\left(D\right)}\left(A\right)=\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\left\{\underset{k\in I}{{\displaystyle \sum}}{\left(\text{diam}\text{\hspace{0.05em}}{A}_{n;i}\right)}^{D}|\text{diam}\text{\hspace{0.05em}}{A}_{nk}\le \frac{1}{n}\right\}\\ =\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\left\{\underset{k\in I}{{\displaystyle \sum}}{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\cdot \text{diam}\text{\hspace{0.05em}}{A}_{nk}\right)}^{D}|\text{diam}\text{\hspace{0.05em}}{A}_{nk}\le \frac{1}{n}\right\}\\ =\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\left\{\underset{i=1}{\overset{p}{{\displaystyle \sum}}}\left[{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}\underset{k\in I}{{\displaystyle \sum}}{\left(\text{diam}\text{\hspace{0.05em}}{A}_{nk}\right)}^{D}\right]|\text{diam}\text{\hspace{0.05em}}{A}_{nk}\le \frac{1}{n}\right\}\\ =\underset{{H}^{\left(D\right)}\left(A\right)}{\underset{\ufe38}{\underset{A\subseteq {{{\displaystyle \cup}}^{\text{}}}_{k}{A}_{nk}}{\mathrm{inf}}\underset{k\in I}{{\displaystyle \sum}}{\left(\text{diam}{A}_{nk}\right)}^{D}}}\underset{i=1}{\overset{p}{{\displaystyle \sum}}}{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}\\ ={H}^{\left(D\right)}\left(A\right)\underset{i=1}{\overset{p}{{\displaystyle \sum}}}{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}\end{array}$

i.e.

${H}^{\left(D\right)}\left(A\right)={H}^{\left(D\right)}\left(A\right)\underset{i=1}{\overset{p}{{\displaystyle \sum}}}{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}$ (37)

Therefore

$\underset{i=1}{\overset{p}{{\displaystyle \sum}}}{\left(\sqrt{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}=1$ (38)

and

$\underset{i=1}{\overset{p}{{\displaystyle \sum}}}{\left(\sqrt[3]{\left|\mathrm{det}{F}_{i}\right|}\right)}^{D}=1$ (39)

in three-dimensional space by analogy.

6. Experiments with Approximations of Theoretical Sets

According to [27] , the original box counting methods are the self-similar methods and they provide accurate results only for self-similar profiles. Problems are supposedly encountered when self-similar methods are used in the calculation of fractal dimensions for the self-affine objects. However, this is incorrect. The box counting method gives accurate or inaccurate results in case of self-similarity or self-affinity in the same way. However, the power function method does not suffer by any systematic error and is more precise as was said in the previous section.

Bijective affine transform (scaling in one direction for example) changes the measure of the transformed set but it does not change its dimensionality. For these measurement, the same methods may be used (without any modification). We estimated the dimensionality and measure of the Koch curve (see Figure 3), i.e. namely the self-similar original A and further its self-affine scaling $B:\left[x;y\right]\to \left[x;2y\right]$ . The box counting method and power function method with the same parameters are used in both cases. Both curves have been generated as image with resolution $4096\times 4096$ pixels. Results of these measurement are summarised in Table 1 and graphically represented in Figure 4 (box counting) and Figure 5 (power function). For both curves (self-similar and self-affine), approximately the same dimension has been measured ( $D\approx 1.26$ ) using two methods (box counting and power function) with the same parameters ( $\epsilon =5,10,\cdots ,50$ ) and withouth any modification. Theoretical dimension is

$D=\frac{\mathrm{ln}4}{\mathrm{ln}3}=2.261859\cdots $ in both cases.

Straightlines (in case of the box counting) or power curves (in case of the power function) differs in shifting (extension) in vertical direction only. By this shifting (extension), set measure in corresponding fractal dimension is determined. Remind that the measure is the length in case of
$D=1$ which is measured in linear micrometers (μm^{1}) for example. In case of
$D=2$ , the measure is caled the area which is measured in square micrometers (μm^{2}) for

example. In case of Koch curve which dimension is $D=\frac{\mathrm{ln}4}{\mathrm{ln}3}$ , we must measure in micrometers powered by $D=\frac{\mathrm{ln}4}{\mathrm{ln}3}$ . If we presume that the pixel is a square with a side 1 μm then the measure of the curve A is

Figure 4. Box-counting dimension and box-counting measure of the Koch curve A and its affine representation B. Used affinity is $\left[x;y\right]\to \left[x;2y\right]$ .

Figure 5. Hausdorff dimension and grid measure of the Koch curve A and its affine representation B. Power-function method, used affinity is $\left[x;y\right]\to \left[x;2y\right]$ .

${G}^{\left(D\right)}\left(A\right)\approx \mathrm{exp}\left(10.4184\right)\approx 33470\text{\hspace{0.17em}}\mu {\text{m}}^{D}$ (40)

according to box counting method and

${G}^{\left(D\right)}\left(A\right)\approx 33365\text{\hspace{0.17em}}\mu {\text{m}}^{D}$ (41)

according to power function method. For the affine representation B of the curve A, these values are

${G}^{\left(D\right)}\left(B\right)\approx \mathrm{exp}\left(10.8730\right)\approx 52733\text{\hspace{0.17em}}\mu {\text{m}}^{D}$ (42)

according to box counting method and

${G}^{\left(D\right)}\left(B\right)\approx 52739\text{\hspace{0.17em}}\mu {\text{m}}^{D}$ (43)

according to power function method.

For testing of the power function method, following fractals has been chosen: Koch curve, Sierpinski triangle and Sierpinski square (see previous example). The subsequent set (triangle) is constructed as the union of three affine copies of itself, matrices of the affinities―see Equation (21)―are

${F}_{1}=\left(\begin{array}{cc}0.36& 0.48\\ 0.48& -0.36\end{array}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{2}=\left(\begin{array}{cc}0.36& -0.48\\ -0.48& -0.36\end{array}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{3}=\left(\begin{array}{cc}-0.28& 0\\ 0& 0.28\end{array}\right)$ (44)

(vectors ${v}_{i}$ are irrelevant for its dimension) then it implies from (38)

${\left(\sqrt{\left|\mathrm{det}{F}_{1}\right|}\right)}^{D}+{\left(\sqrt{\left|\mathrm{det}{F}_{2}\right|}\right)}^{D}+{\left(\sqrt{\left|\mathrm{det}{F}_{3}\right|}\right)}^{D}=1$ (45)

in our case

${\left(\sqrt{0.36}\right)}^{D}+{\left(\sqrt{0.36}\right)}^{D}+{\left(\sqrt{0.0784}\right)}^{D}=1$ (46)

and $D=1.622339\cdots $

This triangle is self-affine, however it also holds

${F}_{1}{F}_{1}^{\text{T}}={F}_{2}{F}_{2}^{\text{T}}={0.6}^{2}\cdot I;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{3}{F}_{3}^{\text{T}}={0.28}^{2}\cdot I$ (47)

It means that this fractal is not only self-affine but also self-similar. It consists of two contractions with ${\lambda}_{1}={\lambda}_{2}=0.36$ and one contraction with ${\lambda}_{3}=0.28$ . Therefore, we can also use Equation (26)

${0.6}^{D}+{0.6}^{D}+{0.28}^{D}=1$ (48)

It is the same equation as (46) and gives the same result. This fractal is illustrated in Figure 6 on the left and it is called “subsequent triangle” in Table 2.

As the next fractal, a self-affine set is constructed (see Figure 6 on the right, it is named as “self-affine square” in Table 2). It contains five affine copies of itself, matrices of the affinities are

${F}_{1}={F}_{2}={F}_{3}={F}_{4}={F}_{5}=\left(\begin{array}{cc}\frac{1}{3}& 0\\ 0& \frac{1}{2}\end{array}\right)$ (49)

Figure 6. The self-affine triangle and self-affine square.

Table 2. The theoretical dimension of some self-similar and self-affine fractals and the dimension estimated by the power function method.

its dimension is

$\underset{i=1}{\overset{5}{{\displaystyle \sum}}}{\left(\sqrt{\left|\mathrm{det}{M}_{i}\right|}\right)}^{D}=1\Rightarrow 5\cdot {\left(\sqrt{\frac{1}{6}}\right)}^{D}=1\Rightarrow D=2\cdot \frac{\mathrm{ln}5}{\mathrm{ln}6}\approx 1.796488\cdots $ (50)

Sixth set in Table 2 is the Barnsley fern (see Figure 7 on the left), it has the affinity matrices

${F}_{1}=\left(\begin{array}{cc}0.01& 0\\ 0& 0.2\end{array}\right);\text{\hspace{0.17em}}{F}_{2}=\left(\begin{array}{cc}0.2& 0.2\\ -0.3& 0.2\end{array}\right);\text{\hspace{0.17em}}{F}_{3}=\left(\begin{array}{cc}-0.1& 0.3\\ 0.3& 0.2\end{array}\right);\text{\hspace{0.17em}}{F}_{4}=\left(\begin{array}{cc}0.83& -0.05\\ 0.05& 0.83\end{array}\right)$ (51)

According to (37) its dimension is $D=1.764625\cdots $

Seveth tested fractal is a tree (see Figure 7 in the middle) with matrices

$\begin{array}{l}{F}_{1}=\left(\begin{array}{cc}0.195& -0.488\\ 0.344& 0.443\end{array}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{2}=\left(\begin{array}{cc}0.462& 0.414\\ -0.252& 0.361\end{array}\right);\\ {F}_{3}=\left(\begin{array}{cc}-0.058& -0.070\\ 0.453& -0.111\end{array}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{4}=\left(\begin{array}{cc}-0.637& 0\\ 0& 0.501\end{array}\right)\end{array}$ (52)

and dimension $D=1.816162\cdots $

The last fractal-“sea horse” has matrices

${F}_{1}=\left(\begin{array}{cc}0.8& 0.3\\ -0.3& 0.8\end{array}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{2}=\left(\begin{array}{cc}-0.3& 0.3\\ -0.4& -0.3\end{array}\right)$ (53)

and dimension $D=1.796166\cdots $ (see Figure 7 on the right).

In Table 2, we can compare these theoretical dimensions of previous eight sets with the dimension which was estimated by the power function method. Data was generated by the IFS method (original resolution $4096\times 4096$ pixels). It is clear that the results of this method are sufficiently precise for both types of fractals.

7. Estimation of Hausdorff Dimension of Barton Profiles

Some authors alerts, that any fractal dimension itself cannot be used for roughness modelling (see [7] [28] [29] [30] for example). It is also clear from the example in Section 2 and from Figure 2. We illustrate this fact also in the case of the Barton Profile.

Figure 7. The self-affine fractals―fern, tree and sea horse.

In Figure 8, we can see the original of fourth Barton profile (K) and its scalings $L:\left[x;y\right]\to \left[x;5y\right]$ ; $M:\left[x;y\right]\to \left[x;10y\right]$ . These three profiles have been measured by the power function method with the same parameters.

These measurement are graphically represented in Figure 9. For all three profiles, approximately the same dimension has been measured.

8. JRC Estimators

As is clear from previous text, JRC depends not only on the fractal dimension, but also on its statistical variability. Remember that the important variability characteristics are:

The square root of average of the squared differences from the mean, i.e.

$\sigma \left(n\right)=\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum}}}{\left({x}_{i}-\stackrel{\xaf}{x}\right)}^{2}}$ (54)

where n is the number of elements of the set, ${x}_{i}$ are its elements and $\stackrel{\xaf}{x}$ is arithmetic mean (standard deviation) and the arithmetic mean of absolute values of differences between elements of statistical sets and their arithmetic mean, i.e.

$\rho \left(n\right)=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum}}}\left|{x}_{i}-\stackrel{\xaf}{x}\right|$ (55)

Figure 8. The fourth Barton profile and its affine copies.

Figure 9. Hausdorff dimension and grid measures of the fourth Barton profile (K) and its affine representations $K,M$ . Power-function method, used affinities are $\left[x;y\right]\to \left[x;5y\right]$ or $\left[x;y\right]\to \left[x;10y\right]$ respectively.

(average deviation).

The Hurst exponent is directly related to the fractal dimension, which measures the smoothness of a surface, or, in our case, the smoothness of a rock profiles. The relationship between the fractal dimension D and the Hurst exponent H, is given by

$H=n+1-D$ (56)

where n is the topological dimension of the measured set (see (59) for example). The Equation (56)―see [31] or [32] for proof―enables to compare the roughness in different topological dimensions and also to compare the standard Barton 2D profile with the real 3D profiles to be measured. Therefore, a roughness estimator can be designed to be able to determine the JRC in different topological dimensions, i.e. the JRC of fractal curves and the JRC of fractal surfaces as well. Therefore, it works with the Hurst exponent for which values is $H\in \langle 0;1\rangle $ in both cases instead the fractal dimension for which is $D\in \langle 1;2\rangle $ in case of the fractal curves and $D\in \langle 2;3\rangle $ in case of the fractal surfaces.

The JRC is given not only by the Hurst exponent but also by heights of curve or surface irregularities. These irregularities can be quantified using the standard deviation (54) or average deviation (55).

Increasing irregularities heights denotes increasing of the JRC and conversely. Therefore, the standard deviation (53) or average deviation (55) must be placed to numerator of expression to be found. Thus, corresponding formulas are:

${E}_{\sigma}=\frac{\sigma}{H}$ (57)

(standard deviation estimator)

${E}_{\rho}=\frac{\rho}{H}$ (58)

(average deviation estimator), $\sigma $ and $\rho $ are given by (54), (55).

For JRC estimation of any profile or surface, so called characteristic functions $JR{C}_{\sigma}\left({E}_{\sigma}\right)$ ; and $JR{C}_{\rho}\left({E}_{\rho}\right)$ have been constructed. Each of them has been designed to pass through the origin of the coordinate system (if surface variability is equal to zero then surface is completely smooth horizontal plane, Hurst exponent is equal to one and $JRC=0$ ). Each of them must be non-negative and increasing (as the JRC). Each of them must describe a dependence of the JRC on ${E}_{\sigma}$ or ${E}_{\rho}$ respectively and has been found using of the least squares method.

9. Estimation of the Characteristic Functions

In this section, Hausdorff dimension of all standard Barton profiles has been estimated using power function method and values of ${E}_{\sigma}$ ; ${E}_{\rho}$ for the standard Barton profiles have been measured. Results of these measurements are summarized in Table 3.

For JRC estimation of any profile or surface, so called characteristic functions $JR{C}_{\sigma}\left({E}_{\sigma}\right)$ ; and $JR{C}_{\rho}\left({E}_{\rho}\right)$ have been constructed. Each of them has been de-

Table 3. Hausdorff dimensions, Hurst exponents, standard deviations, average deviations, standard deviation estimators and average deviation estimators of the standard Barton profiles. In the last column, “JRC” assigned to corresponding dimension by present used and often cited expression (4).

signed to pass through the origin of the coordinate system (if surface variability is equal to zero then surface is completely smooth horizontal plane, Hurst exponent is equal to one and $JRC=0$ ). Each of them must be non-negative and increasing (as the JRC). Each of them must describe a dependence of the JRC on ${E}_{\sigma}$ or ${E}_{\rho}$ respectively and has been found using of the least squares method.

Equations of these functions are

$JR{C}_{\sigma}\left({E}_{\sigma}\right)=9.186\cdot {E}_{\sigma}^{0.651}$ (59)

(see Figure 10)

$JR{C}_{\rho}\left({E}_{\rho}\right)=10.095\cdot {E}_{\rho}^{0.612}$ (60)

(see Figure 11).

10. Estimation of the JRC Index of Real Samples

All geological data used in this paper has been acquired by prof. Tomáš Ficker from the Faculty of Civil Engineering of our university. All the samples are specimens of limestone (locality Brno-Hády, Czech Republic). All processing and visualization of these data have been made by original author’s software. For more information of these reconstructions and visualizations see [33] [34] [35] [36] .

In this section, limestone surfaces in Figure 12 have been used for testing.

If we presume that the surface is isotropic, i.e. its joint roughness coefficient is not dependent on the direction, one JRC may be assigned to 3D surface. In this case, the surface is covered by diminishing cubes (thickening spatial grid) for estimation of the Hausdorff dimension using power function method according to (21). There is $D\in \left(2;3\right)$ , $n=3$ and $H\in \left(0;1\right)$ in expression (56) which serves for JRC estimation. This JRC we call the global JRC.

Figure 10. The JRC as function of standard deviation estimator ${E}_{\sigma}$ ―see (59).

Figure 11. The JRC as function of average deviation estimator ${E}_{\rho}$ ―see (60).

However, the JRC may have different values along different orientations on a rock surface. In this case, we can choose the direction of the JRC estimation. The profile curve is generated for selected direction and its Hausdorff dimension is measured by power function method according to (11). There is $D\in \left(1;2\right)$ , $n=2$ in expression (56) and also $H\in \left(0;1\right)$ in expressions (57), (58) which serves for JRC estimation. This JRC we call the directional JRC.

The global JRC has been estimated for the samples $A;B;C;D$ from Figure 12. For each of them, thirty six directions have been chosen for estimation of the directional JRC. These directions are illustrated in Figure 13 for the sample D.

(A) (B) (C) (D)

Figure 12. The limestone samples under tests. 3D reconstruction from the series of partially focused images (see [33] [34] [35] [36] for more information).

Figure 13. Directions that were used for estimation of the directional JRCs of the samples $A;B;C;D$ . The profiles marked as blue on the left are illustrated on the right.

Results of these measurements are summarized in Table 4 and Table 5 and graphically represented in Figures 14-19. In Figure 14 we can see dimension estimation of the profile with direction 0˚ on the sample A, the profile with direction 90˚ on the sample B, the profile with direction 100˚ on the sample C and the profile with direction 200˚ on the sample D using box counting method. In Figure 15, there are illustrated estimation of the same profiles using power function method.

In the second last row of Table 2 and Table 3, averages of Hausdorff dimensions, Hurst exponents, standard deviations, average deviations and JRCs in individual directions are stated. In the last row of Table 4 and Table 5, the Hausdorff dimension, Hurst exponent, standard deviation, average deviation and JRCs measured over the whole surface are stated. In Figures 16-19.

Table 4. Hausdorff dimensions estimated by power function method, Hurst exponents, standard deviations, average deviations and directional JRCs of the samples $A,B$ . Averaged values on these quantities are in the second last row, corresponding values of the global JRCs are in the last row.

Table 5. Hausdorff dimensions estimated by power function method, Hurst exponents, standard deviations, average deviations and directional JRCs of samples $C,D$ . Averaged values on these quantities are in the second last row, corresponding values of 3D surface are in the last row.

Figure 14. Graphical representation of profile dimension estimation: sample A, direction 0˚, sample B, direction 90˚, sample C direction 100˚, sample D, direction 200˚ (box counting method).

Figure 15. Graphical representation of profile dimension estimation: sample A, direction 0˚, sample B, direction 90˚, sample C direction 100˚, sample D, direction 200˚ (power function method).

1) Directional $JR{C}_{\sigma}$ is marked as red solid

2) Directional $JR{C}_{\rho}$ is marked as green solid

3) Average of directional $JR{C}_{\sigma}$ is marked as red dashed

4) Average of directional $JR{C}_{\rho}$ is marked as green dashed

5) Global $JR{C}_{\sigma}$ is marked as blue

6) Global $JR{C}_{\rho}$ is marked as dark pink

Figure 16. Directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , average of directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , global (3) $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ of the sample A.

Figure 17. Directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , average of directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , global (3D) $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ of the sample B.

Figure 18. Directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , average of directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , global (3D) $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ of the sample C.

Figure 19. Directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , average of the directional $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ , global (3D) $JR{C}_{\sigma}$ and $JR{C}_{\rho}$ of the sample D.

11. Conclusions

This article showed that the fractal dimension does not dependent on scaling. Therefore, there exists no direct relationship between the fractal dimension and JRC, any fractal dimension itself cannot be used for roughness modelling. JRC depends not only on the fractal dimension, but also on other variables. In this paper, statistical variability of the surface has been used. Increasing irregularities heights denote increasing of the JRC and conversely. Therefore, the standard deviation or average deviation must be placed to numerator of the JRC estimator.

The JRC estimator is designed to be able to determine the JRC in different topological dimensions, i.e. the JRC of fractal curves and the JRC of fractal surfaces as well. Therefore, Hurst exponent was used instead the fractal dimension. Increasing dimension denotes increasing roughness and decreasing Hurst exponent. Conversely-decreasing dimension denotes decreasing roughness and increasing Hurst exponent. For this reason, Hurst exponent must be placed to denominator of the JRC is estimator.

The estimator enables fully automatic estimation of the isotropic (global) joint roughness coefficient (this assumes independence on the direction) and also anisotropic (directional) joint roughness coefficient (which value depends on the direction). In case of the isotropic JRC, the estimator works with whole surface which is topologically two-dimensional, in case of the anisotropic JRC, the estimator works in chosen direction, i.e. with topologically one-dimensional profile. The average of the anisotropic JRC estimated for 360˚ with step 10˚ is approximately equal to the isotropic (global) JRC.

Acknowledgements

This work was supported by the Project LO1202 by financial means from the Ministry of Education, Youth and Sports under the National Sustainability Programme I.

The author thanks to prof. Tomáš Ficker from the Faculty of Civil Engineering of Brno University of Technology for the provided data.

Conflicts of Interest

The authors declare no conflicts of interest.

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