Joint Rock Coefficient Estimation Based on Hausdorff Dimension

Advances in Pure Mathematics, 2017, 7, 615-640 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 DOI: 10.4236/apm.2017.711037 Nov. 15, 2017 615 Advances in Pure Mathematics Joint Rock Coefficient Estimation Based on Hausdorff Dimension Dalibor Martišek Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic Abstract The strength of rock structures strongly depends inter alia on surface irreg ularities 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 coeff icient. 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 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 τ of rock joints as tan log nr n JCS JRC τσ ϕ σ  = +⋅   ⋅  (1) where JRC is the joint roughness coefficient, JCS is the joint compressive strength, r ϕ is the residual friction angle, and n σ is the normal stress. The method of Barton and Choubey [1] is well known in geotechnical pracHow to cite this paper: Martišek, D. (201 7) Joint Rock Coefficient Estimation Based on Hausdorff Dimension . Advances in Pure Mathematics , 7 , 615-640. https: //doi.org/10.4236/apm.2017.711037 Received: September 11, 2017 Accepted: November 12, 2017 Published: November 15, 2017 Copyright © 201 7 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access D. Martišek DOI: 10.4236/apm.2017.711037 616 Advances in Pure Mathematics tice-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 ( ) 50 1JRC D ≈⋅− (2) However, it is a nonsense as the following example illustrates. Figure 1. Standard Barton roughness profiles and their joint rock coefficients. D. Martišek DOI: 10.4236/apm.2017.711037 617 Advances in Pure Mathematics 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 ( ) px in Figure 2 was generated as a fractional Brownian motion and for every x is ( ) ( ) 4Px px= ⋅ . To easily determine the dimension of the resulting fractal, a random number must be generated by the Gaussian distribution ( ) 0;1N and the i -th iteration step variations i σ have to be adjusted in accordance with 2 20 2 2 iHi σ σ = (3) where 0;1H∈ is so called Hurst exponent. Due to affine invariance, both profiles have the same dimension ( 1.5D= ) and should have the same roughness therefore. This is evidently not true. Moreover, JRC of both profiles is 25JRC ≈ according to (1). This is also not true. In [14] [15], another „direct relationship” between dimension and JRC was published: 2 11 0.87804 37.7844 16.9304 0.015 0.015 DD JRC −−   =−+⋅ −⋅     (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 [] [ ] ; ;2 xy x y→ . Koch curve A Koch curve B ε N ( ε ) ln( ε ) ln(p) ε N ( ε ) ln( ε ) ln(p) 5 4322 1.6094 8.3715 5 32474 1.6094 10.3882 10 20449 2.3026 9.9257 10 13364 2.3026 9.5003 15 12296 2.7081 9.4170 15 7978 2.7081 8.9844 20 8701 2.9957 9.0712 20 5470 2.9957 8.6070 25 6512 3.2189 8.7814 25 4222 3.2189 8.3481 30 5201 3.4012 8.5566 30 3351 3.4012 8.1170 35 4250 3.5553 8.3547 35 2718 3.5553 7.9077 40 3585 3.6889 8.1845 40 2316 3.6889 7.7476 45 3049 3.8067 8.0226 45 2014 3.8067 7.6079 50 2674 3.9120 7.8913 50 1743 3.9120 7.4634 Power. f. Box-counting Power. f. Box-counting Dim. 1.2621 1.2662 1.2627 1.2599 Meas. 33365 exp (10.4184) =33470 52739 exp (10.8730) =52733 D. Martišek DOI: 10.4236/apm.2017.711037 618 Advances in Pure Mathematics Figure 2. The profile ( ) px was generated as a fractional Brownian motion. Due to affine invariance, the profiles ( ) px ; ( ) Px 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 ( ) ( ) ( ) ; ;; 1 lim inf ni s sni ni n AA iI H A diamA diamA n ∗ →∞ ⊆ ∈  = ≤   ∑  (5) where I is an at most countable index set. Restriction of ( ) s H to the sets measurable with ( ) s H ∗ ( H -measurable sets) is called Hausdorff s -dimensional measure of the set A . The number ( ) { } ( ) ( ) { } { } ( ) ( ) { } 00 sup inf 0 d Hd DA d HA d HA ++ =∈∞ =∞=∈∞ = (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 ( ) ( ) ( ) ;;; 1 lim infni s sni ni n AA iI G A diamA diamA n ∗ →∞ ⊆ ∈  = =   ∑  (7) D. Martišek DOI: 10.4236/apm.2017.711037 619 Advances in Pure Mathematics (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 ( ) s G to the sets measurable with ( ) s G ∗ ( G -measurable set) is called the grid measure. The grid dimension ( G -dimension) of the set A is defined as ( ) { } ( ) ( ) { } { } ( ) ( ) { } 00 sup inf 0 dd G DA d GA d GA ++ =∈∞ =∞=∈∞ = (8) The G -dimension is suitable for digital data and since the limit condition n→∞ in Formula (7) cannot be realized, the limit is omitted and the Formula (7) is replaced with the approximate equality ( ) ( ) ( ) ;;; 1 inf ni s sni ni AA iI G A diamA diamA n ∗ ⊆∈  ≈=   ∑  (9) 4. Box Counting and Power-Function Method For the infimum to be computed in (9), only those sets ; ni A are taken to the union ;ni A for which ; ni AA≠∅ . Due to the fact that only bounded sets (or more precisely their approximations containing finite elements) can be represented in the computer, the system { } ;ni A is always finite. Let us denote its cardinality by ( ) Nn . 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 ( ) ( ) ( ) ( ) ( ) ( ) ; 11 1 Nn Nn D DD ni D ii G A diamA N n n n − = = ≈==⋅ ∑∑ (10) Therefore, ( ) ( ) DD Nn G n≈ ⋅ (11) Applying the logarithm on both sides of the approximate equality (11) we obtain ( ) ( ) ln ln ln D Nn D n G≈⋅ + (12) Measuring with a specified n , a ( ) Nn is obtained for a covering of the measured set. The values D and ( ) D G 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. Martišek DOI: 10.4236/apm.2017.711037 620 Advances in Pure Mathematics ( ) ln ln Nn Dn ≈ (13) and even define the dimension as the limit of that fraction, i.e. ( ) ln lim ln Bn Nn Dn →∞ = (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 ε . Therefore, if we denote the cardinality ( ) Nn of the covering of the set to be measured as ( ) N ε , we obtain ( ) () () DD N GA εε − ≈⋅ (15) from (11) ( ) ( ) ( ) ln ln ln D N D GA εε ≈− ⋅ + (16) from (12) or ( ) 1 0 ln lim ln B N D ε ε ε − →+ = (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 ( ) D fx Gx= ⋅ , sum of its residues is ( ) ( ) 2 Σ n R Nn fn= −   , while sum of residues for the linear function (12) is ( ) ( ) 2 * Σln ln n R Nn fn= −   . Of course * nn RR . Thus the box-counting method systematically overestimates residues of low values and underestimates residues of its high values. Moreover, negative values of the difference ( ) ( ) ln lnNn fn− 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 () ( ) 2 ,D n f ND N Gn= −⋅ ∑ (18) This leads to the equation ( ) ( ) 22 ln ln 0 D D DD n n nn Nn n n Nn n n−= ∑ ∑ ∑∑ (19) This equation is then solved numerically—see [26] for more information. D. Martišek DOI: 10.4236/apm.2017.711037 621 Advances in Pure Mathematics 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 ; 1; 2; ; iin ϕ = so it holds ( ) 11 nn ii ii F FF ϕ = = = =  (20) If all affinities i ϕ 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 ii XX⋅ ′= +Fv (21) where i F is any square matrix and v is any vector. If T2 ii λ = ⋅⋅FF I (22) (where I is the identity matrix) then the affinity is called the similarity, number λ 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 ( ) ( ) () ;; 1 0 inf nk k s ni ni AA D kI H A diamA diamA n ⊆∈  < = ≤ <∞   ∑  (23) If a set A is self-similar and 12 ;;; p λλ λ  are coefficients of the similarities i ϕ from (20), and for each ij ≠ is ( ) ( ) ( ) ( ) 0 Dij HAA ϕϕ = then () ( ) ( ) ( ) ( ) ( ) ;; 1 1 inf inf diam nk k nk D k D ni ni AAkI p DD i AA HA D i H A diamA diamA n A λ ⊆∈ ⊆=  = ≤   = ∑ ∑    (24) It means that ( ) ( ) ( ) ( ) 1 pD i i DD HAHA λ = = ⋅ ∑ (25) 12 1 DD D p λλ λ + ++ = (26) In case of 12 p λλ λ λ = = = = is D. Martišek DOI: 10.4236/apm.2017.711037 622 Advances in Pure Mathematics 12 ln 11 ln DD D D p p pD λλ λ λ λ + ++ =⋅ =⇒= (27) Example: the Koch curve is self-similar with four copies of itself, each scaled by the factor one third, its dimension is ln4 ln3 D= . 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 ln3 ln2 D= or ln8 ln3 D= respectively. For H -measure of any fractal, the H -measure of its covering C nk k A =  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 i ϕ . Each affinity i ϕ transforms a cubical covering C to the set of parallelepipeds ( ) C i ϕ . We have to find how the volume of cube will change by its transform to the parallelepiped. Each cube is given by orthonormal vectors ( ) 123 ;;aaa=a ; ( ) 123 ;;bbb=b ; ( ) 123 ;; ccc=c which are transform to linearly independent vectors ( ) 123 ;;aaa ′ ′′′ =a ; ( ) 123 ;;bbb ′ ′′′ =b ; ( ) 123 ;; ccc ′ ′′′ =c by bijective affinity i ϕ , i.e. T TT TT T ;; iii ′′′ = = ⋅ =⋅⋅a Fa b Fb c Fc (28) or T T TT ;; i ii ′′′ =⋅=⋅=⋅aaFbbFcFc (29) where i F is the matrix of the affinty i ϕ . This implies ( ) ( ) () ( ) ( ) () 11 21 31 1 2 3 1 2 3 12 22 32 1 2 3 13 23 33 ;; ;; ; ;; ;; zzz aaa aaa z z z zzz   ′′′ =⋅=    af af af (30) By analogy ( ) ( ) ( ) ( ) ( ) 123 1 2 3 ;; ; ;; ;;bbb ′′′=bf bf bf (31) ( ) ( ) ( ) ( ) ( ) 123 1 2 3 ;; ; ;; ;;ccc ′′′=cf cf cf (32) Volume of the parallelepiped which is given by vectors ;; ′′′ abc is equal to the scalar triple product. Therefore, we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 123 1 123 1 123 1 1 2 3 11 21 31 1 2 3 12 22 32 1 2 3 13 23 23 23 23 33 ;;; ;;; ;;; aaa bbb ccc aaa f f f bbb f f f ccc f f f ′′′ ′′′ = ′′′ = ⋅ af af af bf bf bf cf cf cf (33) from (30), (31), (32). Therefore ( ) ( ) ( ) T ; ; det ; ; det ; ; ii V VV ′′′ =⋅=⋅a b c F abc F abc (34) D. Martišek DOI: 10.4236/apm.2017.711037 623 Advances in Pure Mathematics It is possible to obtain ( ) ( ) ; det ; i SS ′= ⋅ ′ a b F ab 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 ( ) 33 diam det ; ; det diam nk i i nk AV A ′=⋅=⋅F abc F (35) or ( ) diam det ; det diam nk i i nk AS A ′= ⋅= ⋅F ab F (36) respectively. For the H -measure of a self-affine fractal A which contains p affine copies of itself, we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ; 1 1 1 1 inf diam diam 1 inf det diam diam 1 inf det diam diam inf diam det d nk k nk k nk k nk k D D n i nk AAkI D i nk nk AAkI pDD i nk nk AAi kI pD D D D nk i AA kI i HA p i HA A An AA n AA n A HA ⊆∈ ⊆∈ ⊆= ∈ ⊆∈= =  = ≤    =⋅≤     = ≤     = = ∑ ∑ ∑∑ ∑∑ ∑ F F F      ( ) et D i F i.e. ( ) ( ) ( ) ( ) ( ) 1 det DD pD i i HAHA = = ∑F (37) Therefore ( ) 1 det 1 pD i i= = ∑F (38) and ( ) 3 1 det 1 pD i i= = ∑ F (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. D. Martišek DOI: 10.4236/apm.2017.711037 630 Advances in Pure Mathematics 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). No Dim Hurst σ ρ E σ E ρ JRC JRC (2) 1 1.023 0.977 0.081 0.030 0.083 0.031 1 17.253 2 1.147 0.853 0.159 0.124 0.187 0.146 3 −1256.586 3 1.192 0.808 0.334 0.272 0.414 0.337 5 −2291.114 4 1.241 0.759 0.498 0.397 0.656 0.524 7 −3764.184 5 1.286 0.714 0.646 0.552 0.904 0.773 9 −5435.295 6 1.314 0.686 0.894 0.760 1.303 1.108 11 −6628.901 7 1.335 0.665 1.134 0.933 1.705 1.403 13 −7601.534 8 1.357 0.643 1.416 1.186 2.202 1.844 15 −8691.665 9 1.365 0.635 1.641 1.460 2.585 2.299 17 −9106.136 10 1.398 0.602 1.910 1.693 3.172 2.813 19 −10917.635 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 0JRC = ). Each of them must be non-negative and increasing (as the JRC ). Each of them must describe a dependence of the JRC on E σ or E ρ respectively and has been found using of the least squares method. Equations of these functions are ( ) 0.651 9.186JRC E E σσ σ = ⋅ (59) (see Figure 10) ( ) 0.612 10.095JRC E E ρρ ρ = ⋅ (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 3 D 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 ( ) 2;3D∈ , 3n = and ( ) 0;1H∈ in expression (56) which serves for JRC estimation. This JRC we call the global JRC . D. Martišek DOI: 10.4236/apm.2017.711037 631 Advances in Pure Mathematics Figure 10. The JRC as function of standard deviation estimator E σ —see (59). Figure 11. The JRC as function of average deviation estimator E ρ —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 ( ) 1; 2D∈ , 2n = in expression (56) and also ( ) 0;1H∈ 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 ;;;ABCD 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 . D. Martišek DOI: 10.4236/apm.2017.711037 632 Advances in Pure Mathematics (A) (B) (C) (D) Figure 12. The limestone samples under tests. 3 D 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 ;;;ABCD . 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. D. Martišek DOI: 10.4236/apm.2017.711037 633 Advances in Pure Mathematics Table 4. Hausdorff dimensions estimated by power function method, Hurst exponents, standard deviations, average deviations and directional JRCs of the samples ,AB . Averaged values on these quantities are in the second last row, corresponding values of the global JRCs are in the last row. Angle Sample A Sample B Dim Hurst σ ρ JRC σ JRC ρ Dim Hurst σ ρ JRC σ JRC ρ 0˚ 1.330 0.670 1.028 1.076 5.523 5.481 1.280 0.720 1.405 1.519 6.450 6.464 10˚ 1.381 0.619 0.960 1.037 5.560 5.620 1.269 0.731 1.484 1.543 6.622 6.469 20˚ 1.448 0.552 0.934 0.989 5.886 5.856 1.195 0.805 1.706 1.854 6.806 6.818 30˚ 1.395 0.605 0.914 0.971 5.468 5.479 1.250 0.750 1.894 2.040 7.630 7.545 40˚ 1.373 0.627 0.808 0.869 4.931 5.011 1.250 0.750 1.893 2.133 7.624 7.751 50˚ 1.403 0.597 0.698 0.668 4.632 4.403 1.237 0.763 1.480 1.941 6.425 7.242 60˚ 1.337 0.663 0.656 0.650 4.156 4.060 1.262 0.738 1.348 1.586 6.183 6.539 70˚ 1.316 0.684 0.551 0.538 3.638 3,553 1.250 0.750 1.141 1.328 5.492 5.814 80˚ 1.218 0.782 0.584 0.562 3.463 3.363 1.214 0.786 1.385 1.416 6.040 5.875 90˚ 1.277 0.723 0.603 0.556 3.722 3.504 1.268 0.732 1.141 1.204 5.577 5.558 100˚ 1.448 0.552 0.505 0.502 3.951 3.879 1.299 0.701 0.807 1.255 4.586 5.854 110˚ 1.367 0.633 0.581 0.604 3.959 3.996 1.264 0.736 0.775 1.303 4.328 5.813 120˚ 1.376 0.624 0.613 0.694 4.139 4.385 1.381 0.619 0.803 1.063 4.954 5.708 130˚ 1.260 0.740 0.918 0.914 4.811 4.674 1.262 0.738 0.838 1.262 4.546 5.692 140˚ 1.289 0.711 1.074 1.081 5.466 5.299 1.466 0.534 0.936 1.066 6.018 6.252 150˚ 1.262 0.738 1.211 1.231 5.768 5.607 1.313 0.687 1.248 1.322 6.163 6.114 160˚ 1.296 0.704 1.213 1.212 5.952 5.713 1.350 0.650 1.522 1.500 7.265 6.828 170˚ 1.244 0.756 1.202 1.214 5.651 5.479 1.316 0.684 1.620 1.705 7.314 7.152 180˚ 1.306 0.694 1.036 0.983 5.424 5.075 1.268 0.732 2.141 2.344 8.385 8.327 190˚ 1.286 0.714 0.933 0.915 4.974 4.775 1.334 0.666 2.241 2.427 9.189 9.012 200˚ 1.279 0.721 0.930 0.868 4.934 4.601 1.249 0.751 2.541 2.683 9.220 8.903 210˚ 1.246 0.754 0.889 0.828 4.653 4.349 1.233 0.767 2.777 2.902 9.629 9.213 220˚ 1.283 0.717 0.717 0.708 4.185 4.079 1.143 0.857 3.139 3.692 9.701 9.970 230˚ 1.334 0.666 0.669 0.600 4.196 3.858 1.141 0.859 3.653 4.028 10.689 10.498 240˚ 1.305 0.695 0.576 0.554 3.705 3.580 1.066 0.934 4.256 4.565 11.179 10.766 250˚ 1.326 0.674 0.541 0.510 3.629 3.470 1.184 0.816 4.395 4.850 12.460 12.124 260˚ 1.291 0.709 0.474 0.443 3.222 3.090 1.084 0.916 4.730 5.288 12.126 11.913 270˚ 1.273 0.727 0.488 0.434 3.234 3.006 1.179 0.821 4.651 5.155 12.878 12.538 280˚ 1.368 0.632 0.480 0.426 3.500 3.234 1.107 0.893 4.713 5.053 12.296 11.768 290˚ 1.350 0.650 0.511 0.474 3.581 3.393 1.094 0.906 4.410 4.679 11.664 11.130 300˚ 1.350 0.650 0.541 0.525 3.715 3.610 1.180 0.820 4.010 4.349 11.708 11.318 310˚ 1.348 0.652 0.631 0.636 4.098 4.048 1.116 0.884 3.858 4.270 10.871 10.690 320˚ 1.359 0.641 0.684 0.673 4.364 4.232 1.124 0.876 3.633 4.060 10.521 10.428 330˚ 1.320 0.680 0.819 0.781 4.723 4.471 1.165 0.835 3.412 3.833 10.421 10.367 340˚ 1.351 0.649 0.879 0.868 5.092 4.903 1.218 0.782 2.421 2.281 8.700 7.867 350˚ 1.340 0.660 0.998 0.996 5.471 5.276 1.280 0.720 1.418 1.804 6.491 7.177 Aver. 1.326 0.674 0.774 0.766 4.538 4.400 1.228 0.772 2.441 2.720 8.370 8.424 3D 2.312 0.688 0.783 0.761 4.549 4.369 2.205 0.795 2.349 2.468 8.443 8.174 D. Martišek DOI: 10.4236/apm.2017.711037 634 Advances in Pure Mathematics Table 5. Hausdorff dimensions estimated by power function method, Hurst exponents, standard deviations, average deviations and directional JRCs of samples , CD . Averaged values on these quantities are in the second last row, corresponding values of 3 D surface are in the last row. Angle Sample C Sample D Dim Hurst σ ρ JRC σ JRC ρ Dim Hurst σ ρ JRC σ JRC ρ 0˚ 1.318 0.682 3.994 4.156 13.149 12.306 1.087 0.913 2.765 3.232 8.579 8.852 10˚ 1.459 0.541 3.319 3.689 13.564 13.185 1.109 0.891 3.404 3.933 9.970 10.119 20˚ 1.494 0.506 2.909 3.495 12.992 13.275 1.161 0.839 3.511 4.508 10.578 11.403 30˚ 1.455 0.545 3.500 3.411 13.968 12.514 1.157 0.843 3.517 4.607 10.555 11.518 40˚ 1.401 0.599 3.888 4.073 14.063 13.158 1.156 0.844 4.151 4.693 11.743 11.640 50˚ 1.372 0.628 4.749 4.507 15.526 13.598 1.150 0.850 4.427 4.776 12.192 11.718 60˚ 1.399 0.601 4.283 4.656 14.945 14.246 1.117 0.883 4.449 4.821 11.932 11.514 70˚ 1.346 0.654 4.284 4.504 14.137 13.253 1.116 0.884 4.057 4.713 11.232 11.349 80˚ 1.293 0.707 4.260 4.166 13.402 12.064 1.105 0.895 3.730 4.286 10.549 10.631 90˚ 1.415 0.585 3.120 3.356 12.385 11.872 1.132 0.868 2.953 3.397 9.248 9.406 100˚ 1.481 0.519 2.142 2.488 10.491 10.645 1.094 0.906 4.227 4.147 11.351 10.346 110˚ 1.385 0.615 1.939 2.104 8.809 8.670 1.086 0.914 5.082 4.872 12.721 11.350 120˚ 1.396 0.604 1.865 1.846 8.689 8.099 1.103 0.897 5.948 6.555 14.253 13.741 130˚ 1.419 0.581 1.783 1.809 8.656 8.190 1.047 0.953 7.430 10.987 15.831 18.124 140˚ 1.474 0.526 1.609 1.819 8.636 8.729 1.142 0.858 8.048 11.253 17.850 19.603 150˚ 1.398 0.602 1.705 1.797 8.220 7.986 1.059 0.941 9.160 12.247 18.281 19.509 160˚ 1.338 0.662 2.288 2.540 9.345 9.295 1.062 0.938 8.338 10.554 17.234 17.857 170˚ 1.357 0.643 2.630 2.903 10.420 10.256 1.061 0.939 6.999 8.614 15.376 15.776 180˚ 1.270 0.730 3.976 4.447 12.550 12.307 1.050 0.950 5.728 7.242 13.408 14.104 190˚ 1.279 0.721 4.487 4.769 13.679 12.933 1.082 0.918 6.110 6.520 14.293 13.510 200˚ 1.256 0.744 3.871 3.862 12.179 11.164 1.076 0.924 6.594 7.103 14.948 14.170 210˚ 1.341 0.659 2.901 2.715 10.938 9.709 1.104 0.896 7.429 9.604 16.480 17.344 220˚ 1.389 0.611 2.006 1.724 9.038 7.712 1.044 0.956 9.084 12.200 17.998 19.277 230˚ 1.464 0.536 1.476 1.475 8.069 7.599 1.059 0.941 9.968 12.504 19.312 19.758 240˚ 1.293 0.707 2.257 2.359 8.878 8.539 1.047 0.953 9.273 12.537 18.282 19.641 250˚ 1.359 0.641 2.206 2.253 9.322 8.813 1.061 0.939 6.954 7.819 15.318 14.881 260˚ 1.299 0.701 2.556 3.251 9.680 10.432 1.126 0.874 4.916 4.973 12.810 11.803 270˚ 1.258 0.742 2.704 3.393 9.675 10.343 1.156 0.844 3.479 2.549 10.476 8.038 280˚ 1.269 0.731 3.062 3.612 10.589 10.841 1.168 0.832 2.007 1.992 7.402 6.981 290˚ 1.283 0.717 3.271 4.590 11.189 12.685 1.203 0.797 1.653 1.454 6.711 5.918 300˚ 1.229 0.771 4.576 6.858 13.262 15.480 1.223 0.777 2.021 2.052 7.771 7.407 310˚ 1.273 0.727 5.372 7.117 15.294 16.415 1.219 0.781 2.294 2.474 8.410 8.274 320˚ 1.255 0.745 5.775 7.138 15.770 16.198 1.131 0.869 2.798 3.073 8.924 8.845 330˚ 1.224 0.776 5.668 6.636 15.178 15.120 1.154 0.846 2.435 2.700 8.303 8.315 340˚ 1.298 0.702 4.931 5.576 14.793 14.452 1.111 0.889 2.514 2.735 8.205 8.128 350˚ 1.376 0.624 4.124 4.400 14.221 13.445 1.089 0.911 2.406 2.584 7.849 7.736 Aver. 1.350 0.650 3.319 3.708 11.825 11.542 1.112 0.888 4.996 5.953 12.399 12.461 3D 2.398 0.602 3.356 3.612 12.739 12.193 2.081 0.919 5.144 5.487 12.775 12.159 D. Martišek DOI: 10.4236/apm.2017.711037 635 Advances in Pure Mathematics 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 JRC σ is marked as red solid 2) Directional JRC ρ is marked as green solid 3) Average of directional JRC σ is marked as red dashed 4) Average of directional JRC ρ is marked as green dashed 5) Global JRC σ is marked as blue 6) Global JRC ρ is marked as dark pink D. Martišek DOI: 10.4236/apm.2017.711037 636 Advances in Pure Mathematics Figure 16. Directional JRC σ and JRC ρ , average of directional JRC σ and JRC ρ , global (3) JRC σ and JRC ρ of the sample A . Figure 17. Directional JRC σ and JRC ρ , average of directional JRC σ and JRC ρ , global (3 D ) JRC σ and JRC ρ of the sample B . D. Martišek DOI: 10.4236/apm.2017.711037 637 Advances in Pure Mathematics Figure 18. Directional JRC σ and JRC ρ , average of directional JRC σ and JRC ρ , global (3 D ) JRC σ and JRC ρ of the sample C . Figure 19. Directional JRC σ and JRC ρ , average of the directional JRC σ and JRC ρ , global (3 D ) JRC σ and JRC ρ of the sample D . D. Martišek DOI: 10.4236/apm.2017.711037 638 Advances in Pure Mathematics 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. References [1] Barton, N. and Choubey, V. (1977): The Shear Strength of Rock Joints in Theory and Practice. Rock Mechanics , 10, 1-65. https://doi.org/10.1007/BF01261801 [2] Tse, R. and Cruden, D.M. (1979) Estimating Joint Roughness Coefficients. 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