# Scale Effect and Anisotropy Analyzed for Neutrosophic Numbers of Rock Joint Roughness Coefficient Based on Neutrosophic Statistics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}). Then, Zhang et al. [3] improved the root mean square (Z

_{2}) by considering the inclination angle, amplitude of asperities, and their directions, and then introduced a new roughness index (λ) by using the modified root mean square (Z

_{2}’) to calculate JRC values. To quantify the anisotropic roughness of joint surfaces effectively, a variogram function and a new index were proposed by Chen et al. [4] based on the digital image processing technique, and then they also studied the scale effect by calculating the JRC values of different sample lengths [5]. However, all of these traditional methods do not consider the uncertainties of JRC values in real rock engineering practice.

## 2. Basic Concepts and Neutrosophic Statistical Algorithm of NNs

^{L}, I

^{U}] is indeterminacy. It is clear that the NN can express the determinate and/or indeterminate information. Here is a numerical example. A NN is z = 5 + 6I for I $\in $ [0, 0.3]. Then, the NN is z $\in $ [5, 6.8] for I $\in $ [0, 0.3] and its possible range/interval is z = [5, 6.8], where its determinate part is 5 and its indeterminate part is 6I. For the numerical example, z = 5 + 6I for I $\in $ [0, 0.3] can be also expressed as another form z = 5 + 3I for I $\in $ [0, 0.6]. Therefore, we can specify some suitable interval range [I

^{L}, I

^{U}] for the indeterminacy I according to the different applied demands to adapt the actual representation. In fact, NN is a changeable interval number depending on the indeterminacy I $\in $ [I

^{L}, I

^{U}].

_{1}= 1 + 2I, z

_{2}= 2 + 3I, z

_{3}= 3 + 4I, and z

_{4}= 4 + 5I for I $\in $ [0, 0.2], then the average value of these four neutrosophic numbers can be obtained by the following calculational steps:

_{1}= a

_{1}+ b

_{1}I and z

_{2}= a

_{2}+ b

_{2}I for I $\in $ [I

^{L}, I

^{U}]. Then, Ye et al. [12] proposed their basic operations:

_{i}= a

_{i}+ b

_{i}I (i = 1, 2,…, n) be a group of NNs for I $\in $ [I

^{L}, I

^{U}], then their neutrosophic average value and standard deviation can be calculated by the following neutrosophic statistical algorithm:

**Step 1:**Calculate the neutrosophic average value of a

_{i}(i = 1, 2,…, n):

**Step 2:**Calculate the average value of b

_{i}(i = 1, 2,…, n):

**Step 3:**Obtain the neutrosophic average value:

**Step 4:**Get the differences between z

_{i}(i = 1, 2,…, n) and $\overline{z}$:

**Step 5:**Calculate the square of all the differences between z

_{i}(i = 1, 2,…, n) and $\overline{z}$:

**Step 6:**Calculate the neutrosophic standard deviation:

## 3. JRC Values and JRC-NNs in an Actual Case

_{ij}and standard deviations σ

_{ij}(i = 1, 2,…, 24; j = 1, 2,…, 10) of actually measured data in different sample lengths and different measurement orientations, which are shown in Table 1.

_{ij}= a

_{ij}+ b

_{ij}I (i = 1, 2,…, 24; j = 1, 2,…, 10) to express the JRC values in each orientation θ and in each sample length L. Various NNs of the JRC values are indicated by the real numbers of a

_{ij}and b

_{ij}in z

_{ij}(i = 1, 2,…, 24; j = 1, 2,…, 10). For convenient neutrosophic statistical analysis, the indeterminacy I is specified as the unified form I $\in $ [0, 1] in all the JRC-NNs. Thus, there is z

_{ij}= a

_{ij}+ b

_{ij}I = μ

_{ij}− σ

_{ij}+ 2 σ

_{ij}I (i = 1, 2,…, 24; j = 1, 2,…, 10), where a

_{ij}= μ

_{ij}− σ

_{ij}is the lower bound of the JRC value and z

_{ij}may choose a robust range/confidence interval [μ

_{ij}− σ

_{ij}, μ

_{ij}+ σ

_{ij}] for the symmetry about the average value μ

_{ij}(see the references [10,11] in detail), and then based on μ

_{ij}and σ

_{ij}in Table 1 a

_{ij}and b

_{ij}in z

_{ij}(i = 1, 2,…, 24; j =1, 2,…, 10) are shown in Table 2. For example, when θ = 0° and L = 10 cm for i = 1 and j = 1, we can obtain from Table 2 that the JRC-NN is z

_{11}= 8.3040 + 4.4771I for I $\in $ [0, 1].

## 4. Scale Effect Analysis in Different Sample Lengths Based on the Neutrosophic Statistical Algorithm

**Step 1:**By Equation (2), calculate the average value of the determinate parts a

_{i}

_{1}(i = 1, 2,…, 24) in the JRC-NNs corresponding to the first column as follows:

**Step 2:**By Equation (3), calculate the average value of the indeterminate coefficients b

_{i}

_{1}(i = 1, 2,…, 24) in the JRC-NNs:

**Step 3:**By Equation (4), obtain the neutrosophic average value of the JRC-NNs in the first column:

**Step 4:**By Equation (5), calculate the differences between z

_{i}

_{1}(i = 1, 2,…, 24) and ${\overline{z}}_{1}$:

**Step 5:**By Equation (6), calculate the square of all the differences:

**Step 6:**By Equation (7), calculate the neutrosophic standard deviation:

## 5. Anisotropic Analysis in Different Measurement Orientations Based on the Neutrosophic Statistical Algorithm

**Step 1:**By Equation (2), calculate the average value of the determinate parts a

_{1j}(j = 1, 2,…, 10) of the JRC-NNs in the first row (i = 1) as follows:

**Step 2:**By Equation (3), calculate the average value of b

_{1j}(j = 1, 2,…, 10) in the indeterminate parts of the JRC-NNs:

**Step 3:**By Equation (4), get the neutrosophic average value of the JRC-NNs in the first row:

**Step 4:**By Equation (5), calculate the differences between z

_{1j}(j = 1, 2,…, 10) and ${\overline{z}}_{1}$:

**Step 5:**By Equation (6), calculate the square of these differences:

**Step 6:**By Equation (7), calculate the neutrosophic standard deviation:

## 6. Conclusion Remarks

- (1)
- The neutrosophic statistical analysis method without fitting functions is more feasible and more reasonable than the existing method [10].
- (2)
- The neutrosophic statistical analysis method based on the neutrosophic average values and neutrosophic standard deviations of JRC-NNs can retain much more information and reflect the scale effect and anisotropic characteristics of JRC values in detail.
- (3)
- The presented neutrosophic statistical algorithm can analyze the scale effect and the anisotropy of JRC-NNs (JRC values) directly and effectively so as to reduce the information distortion.
- (4)
- The presented neutrosophic statistical algorithm based on the neutrosophic statistical averages and standard deviations of JRC-NNs is more convenient and simpler than the existing curve fitting and derivative analysis of JRC-NN functions in [10].
- (5)
- The presented neutrosophic statistical algorithm can overcome the insufficiencies of the existing method in the fitting and analysis process [10].
- (6)

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**The average values μ

_{ij}and standard deviations σ

_{ij}of actually measured data in different sample lengths L and different measurement orientations θ.

L | 10 cm | 20 cm | 30 cm | 40 cm | 50 cm | 60 cm | 70 cm | 80 cm | 90 cm | 100 cm | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

θ | μ_{i1} | σ_{i1} | μ_{i2} | σ_{i2} | μ_{i3} | σ_{i3} | μ_{i4} | σ_{i4} | μ_{i5} | σ_{i5} | μ_{i6} | σ_{i6} | μ_{i7} | σ_{i7} | μ_{i8} | σ_{i8} | μ_{i9} | σ_{i9} | μ_{i10} | σ_{i10} |

0° | 10.5425 | 2.2385 | 9.6532 | 1.7162 | 9.2733 | 1.5227 | 8.9745 | 1.7092 | 8.8222 | 1.6230 | 8.8016 | 1.6069 | 8.6815 | 1.6066 | 8.6009 | 1.5043 | 8.5681 | 1.3465 | 8.4630 | 1.2806 |

15° | 10.7111 | 2.2392 | 9.9679 | 1.7379 | 9.3433 | 1.5555 | 9.2708 | 1.2743 | 9.2299 | 1.2850 | 8.9729 | 1.3071 | 8.8332 | 1.1706 | 8.5868 | 0.9413 | 8.3604 | 0.7673 | 8.1404 | 0.6372 |

30° | 10.5943 | 2.3528 | 9.9289 | 2.0286 | 9.5715 | 1.6665 | 9.1209 | 1.4207 | 9.0920 | 1.4119 | 8.6006 | 0.9899 | 8.7596 | 1.1489 | 8.5713 | 1.0776 | 8.2927 | 1.0128 | 8.1041 | 0.9664 |

45° | 9.9244 | 2.3120 | 9.2005 | 1.7237 | 9.0081 | 1.6464 | 8.5078 | 1.1376 | 8.3336 | 1.431 | 8.6237 | 1.3427 | 8.3262 | 1.2184 | 8.0768 | 1.2717 | 7.8458 | 1.2096 | 7.5734 | 1.1294 |

60° | 9.0253 | 2.4592 | 8.4047 | 1.9813 | 7.8836 | 1.8199 | 7.7941 | 1.8829 | 7.1873 | 1.167 | 8.2678 | 1.7830 | 7.3595 | 1.5956 | 7.1381 | 1.4082 | 6.8722 | 1.2178 | 6.7131 | 0.9627 |

75° | 7.9352 | 2.1063 | 7.4604 | 1.7756 | 6.7725 | 1.4153 | 6.3056 | 1.0241 | 6.5446 | 1.2140 | 6.4993 | 1.3108 | 6.2440 | 1.1208 | 6.0933 | 0.9171 | 5.9499 | 0.7311 | 5.8317 | 0.5855 |

90° | 7.0467 | 2.4054 | 6.6915 | 1.8482 | 6.3378 | 1.4743 | 5.9993 | 1.1700 | 6.1481 | 1.1920 | 6.0893 | 1.1850 | 5.9543 | 1.1021 | 5.8932 | 0.9630 | 5.8259 | 0.9181 | 5.8219 | 0.8355 |

105° | 7.7766 | 2.4105 | 7.2221 | 1.7560 | 6.6770 | 1.2608 | 6.2318 | 0.985 | 6.4634 | 1.2288 | 6.4609 | 1.5029 | 6.1670 | 1.3236 | 5.9923 | 1.1016 | 5.8903 | 0.9868 | 5.8359 | 0.8479 |

120° | 9.1324 | 2.3250 | 8.5206 | 1.8963 | 8.1998 | 1.5792 | 7.9671 | 1.4094 | 7.3207 | 1.0418 | 7.8245 | 1.1807 | 7.2472 | 1.0637 | 7.0649 | 0.9507 | 6.8537 | 0.8122 | 6.6909 | 0.7715 |

135° | 9.2258 | 1.9104 | 8.5670 | 1.5412 | 8.0898 | 1.3452 | 7.8194 | 0.9910 | 7.3735 | 0.9848 | 7.6660 | 1.2845 | 7.3846 | 1.1608 | 7.0872 | 1.1589 | 6.9154 | 1.0345 | 6.7586 | 0.9157 |

150° | 10.4673 | 2.4365 | 9.5650 | 1.9065 | 8.9102 | 1.6863 | 8.9059 | 1.4562 | 8.3930 | 1.1855 | 8.8162 | 1.5870 | 8.2064 | 1.3432 | 8.0153 | 1.1287 | 7.6556 | 1.0101 | 7.4443 | 0.9080 |

165° | 10.6035 | 2.2090 | 9.9647 | 1.6606 | 9.5320 | 1.5695 | 8.8760 | 1.5994 | 8.6121 | 1.4899 | 8.6463 | 1.5942 | 8.3931 | 1.3637 | 8.1107 | 1.2203 | 7.9051 | 1.0893 | 7.7175 | 1.0050 |

180° | 9.8501 | 2.1439 | 9.0984 | 1.8556 | 8.7574 | 1.7300 | 8.6002 | 1.6753 | 8.2973 | 1.5862 | 8.1266 | 1.6278 | 7.9647 | 1.4864 | 7.8981 | 1.3395 | 7.8338 | 1.1935 | 7.8291 | 1.0616 |

195° | 9.9383 | 2.2254 | 9.2299 | 1.8331 | 8.6781 | 1.6791 | 8.7993 | 1.4556 | 8.5308 | 1.5551 | 8.1016 | 1.5598 | 7.9219 | 1.2559 | 7.6562 | 0.9674 | 7.4610 | 0.8060 | 7.3131 | 0.7402 |

210° | 9.5903 | 1.9444 | 8.9414 | 1.5298 | 8.6532 | 1.6227 | 8.2601 | 1.5626 | 8.2065 | 1.5438 | 7.3828 | 1.2507 | 7.7527 | 1.2989 | 7.5050 | 1.1484 | 7.2495 | 1.0876 | 7.0479 | 0.9558 |

225° | 8.9167 | 1.9764 | 8.2550 | 1.4256 | 8.1330 | 1.4751 | 7.7012 | 1.2124 | 7.6798 | 1.4502 | 7.4365 | 1.1748 | 7.3183 | 1.2086 | 7.1309 | 1.2749 | 6.8652 | 1.2190 | 6.6742 | 1.1571 |

240° | 7.8582 | 1.8456 | 7.3032 | 1.4385 | 6.8241 | 1.1626 | 6.7427 | 1.2022 | 6.3250 | 0.8971 | 6.8181 | 1.1123 | 6.3526 | 1.0430 | 6.1521 | 0.9953 | 5.9138 | 0.8906 | 5.7515 | 0.7329 |

255° | 7.2166 | 1.9341 | 6.8638 | 1.3901 | 6.3349 | 1.2705 | 6.1050 | 1.0350 | 6.0333 | 0.9671 | 6.0693 | 1.1394 | 5.8924 | 0.9417 | 5.7122 | 0.8153 | 5.7803 | 0.8598 | 5.3946 | 0.5627 |

270° | 6.8025 | 2.1165 | 6.3123 | 1.6374 | 6.0061 | 1.3786 | 5.8815 | 1.3700 | 5.7871 | 1.1783 | 5.9707 | 1.2858 | 5.8530 | 1.2711 | 5.7376 | 1.1886 | 5.8259 | 0.9181 | 5.5856 | 1.0273 |

285° | 7.0061 | 1.5474 | 6.4941 | 1.1183 | 6.1107 | 0.9586 | 5.8455 | 0.9821 | 5.7563 | 0.9033 | 6.0606 | 1.3603 | 5.8403 | 1.2714 | 5.6386 | 1.1359 | 5.4716 | 1.0374 | 5.3629 | 0.9501 |

300° | 8.4720 | 1.7448 | 7.8124 | 1.3531 | 7.5303 | 1.2127 | 7.2813 | 1.0247 | 6.9533 | 1.1089 | 7.0673 | 0.8880 | 6.8002 | 0.9202 | 6.6414 | 0.8727 | 6.4460 | 0.8434 | 6.3104 | 0.7904 |

315° | 10.1428 | 2.4790 | 9.4554 | 2.1149 | 8.9644 | 1.7308 | 8.5698 | 1.4949 | 8.1224 | 1.4089 | 8.6863 | 1.5162 | 8.3659 | 1.5934 | 7.6582 | 1.3811 | 7.4641 | 1.1563 | 7.3537 | 1.0960 |

330° | 9.8295 | 2.2844 | 9.0011 | 1.6139 | 8.3261 | 1.6005 | 8.3290 | 1.3232 | 7.8712 | 1.2376 | 8.0526 | 1.2755 | 7.9134 | 1.1209 | 7.6498 | 1.0157 | 7.3466 | 0.9740 | 7.0927 | 0.9342 |

345° | 9.6831 | 2.0192 | 9.1761 | 1.6305 | 8.7732 | 1.1686 | 8.4741 | 1.1887 | 7.8597 | 1.1436 | 7.8485 | 1.0332 | 7.7270 | 1.0174 | 7.4667 | 0.9254 | 7.1781 | 0.821 | 7.0038 | 0.7346 |

**Table 2.**The values of a

_{ij}and b

_{ij}in JRC neutrosophic numbers (JRC-NNs) z

_{ij}(i = 1, 2,…, 24; j =1, 2,…, 10) for each orientation θ and each sample length L.

L | 10 cm | 20 cm | 30 cm | 40 cm | 50 cm | 60 cm | 70 cm | 80 cm | 90 cm | 100 cm | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

θ | a_{i1} | b_{i1} | a_{i2} | b_{i2} | a_{i3} | b_{i3} | a_{i4} | b_{i4} | a_{i5} | b_{i5} | a_{i6} | b_{i6} | a_{i7} | b_{i7} | a_{i8} | b_{i8} | a_{i9} | b_{i9} | a_{i10} | b_{i10} |

0° | 8.3040 | 4.4771 | 7.9370 | 3.4325 | 7.7506 | 3.0454 | 7.2653 | 3.4184 | 7.1992 | 3.2459 | 7.1947 | 3.2138 | 7.0750 | 3.2132 | 7.0966 | 3.0085 | 7.2216 | 2.6930 | 7.1824 | 2.5612 |

15° | 8.4719 | 4.4784 | 8.2300 | 3.4759 | 7.7878 | 3.1110 | 7.9964 | 2.5487 | 7.9449 | 2.5700 | 7.6657 | 2.6142 | 7.6627 | 2.3412 | 7.6456 | 1.8825 | 7.5931 | 1.5347 | 7.5032 | 1.2745 |

30° | 8.2415 | 4.7057 | 7.9003 | 4.0572 | 7.9051 | 3.3330 | 7.7002 | 2.8414 | 7.6801 | 2.8239 | 7.6107 | 1.9798 | 7.6107 | 2.2977 | 7.4938 | 2.1552 | 7.2799 | 2.0256 | 7.1377 | 1.9328 |

45° | 7.6124 | 4.6240 | 7.4768 | 3.4474 | 7.3616 | 3.2929 | 7.3701 | 2.2753 | 6.9018 | 2.8636 | 7.2810 | 2.6853 | 7.1078 | 2.4369 | 6.8051 | 2.5434 | 6.6362 | 2.4192 | 6.4440 | 2.2589 |

60° | 6.5660 | 4.9185 | 6.4234 | 3.9627 | 6.0638 | 3.6397 | 5.9112 | 3.7658 | 6.0203 | 2.3341 | 6.4848 | 3.5660 | 5.7639 | 3.1912 | 5.7299 | 2.8163 | 5.6544 | 2.4355 | 5.7504 | 1.9253 |

75° | 5.8289 | 4.2126 | 5.6847 | 3.5513 | 5.3573 | 2.8306 | 5.2815 | 2.0483 | 5.3307 | 2.4279 | 5.1885 | 2.6216 | 5.1232 | 2.2416 | 5.1762 | 1.8342 | 5.2188 | 1.4622 | 5.2462 | 1.1710 |

90° | 4.6413 | 4.8108 | 4.8432 | 3.6965 | 4.8635 | 2.9486 | 4.8293 | 2.3399 | 4.9561 | 2.3841 | 4.9043 | 2.3701 | 4.8522 | 2.2043 | 4.9302 | 1.9260 | 4.9078 | 1.8362 | 4.9865 | 1.6709 |

105° | 5.3661 | 4.821 | 5.4661 | 3.5119 | 5.4162 | 2.5216 | 5.2460 | 1.9717 | 5.2346 | 2.4576 | 4.9580 | 3.0058 | 3.0054 | 2.6472 | 4.8907 | 2.2031 | 4.9034 | 1.9737 | 4.9881 | 1.6957 |

120° | 6.8074 | 4.6500 | 6.6243 | 3.7926 | 6.6206 | 3.1584 | 6.5577 | 2.8188 | 6.2789 | 2.0837 | 6.6438 | 2.3614 | 6.1834 | 2.1274 | 6.1142 | 1.9014 | 6.0415 | 1.6243 | 5.9194 | 1.5430 |

135° | 7.3153 | 3.8208 | 7.0258 | 3.0824 | 6.7446 | 2.6904 | 6.8283 | 1.9821 | 6.3887 | 1.9696 | 6.3815 | 2.5690 | 6.2238 | 2.3216 | 5.9283 | 2.3178 | 5.8810 | 2.0689 | 5.8429 | 1.8314 |

150° | 8.0308 | 4.8731 | 7.6585 | 3.8130 | 7.2240 | 3.3725 | 7.4497 | 2.9125 | 7.2075 | 2.3710 | 7.2292 | 7.2291 | 6.8633 | 2.6863 | 6.8866 | 2.2573 | 6.6454 | 2.0203 | 6.5363 | 1.8161 |

165° | 8.3945 | 4.4180 | 8.3040 | 3.3213 | 7.9625 | 3.1391 | 7.2766 | 3.1988 | 7.1222 | 2.9799 | 7.0521 | 3.1884 | 7.0294 | 2.7274 | 6.8904 | 2.4406 | 6.8158 | 2.1787 | 6.7124 | 2.0101 |

180° | 7.7062 | 4.2877 | 7.2427 | 3.7113 | 7.0273 | 3.4601 | 6.9249 | 3.3506 | 6.7111 | 3.1724 | 6.4988 | 3.2556 | 6.4782 | 2.9729 | 6.5586 | 2.6790 | 6.6403 | 2.3871 | 6.7675 | 2.1232 |

195° | 7.7130 | 4.4507 | 7.3968 | 3.6661 | 6.9990 | 3.3583 | 7.3437 | 2.9113 | 6.9757 | 3.1102 | 6.5419 | 3.1195 | 6.6660 | 2.5119 | 6.6888 | 1.9348 | 6.6550 | 1.6120 | 6.5729 | 1.4803 |

210° | 7.6459 | 3.8887 | 7.4116 | 3.0596 | 7.0305 | 3.2453 | 6.6975 | 3.1252 | 6.6628 | 3.0875 | 6.1321 | 2.5014 | 6.4538 | 2.5977 | 6.3566 | 2.2967 | 6.1619 | 2.1752 | 6.0921 | 1.9116 |

225° | 6.9402 | 3.9529 | 6.8294 | 2.8512 | 6.6580 | 2.9502 | 6.4888 | 2.4248 | 6.2296 | 2.9004 | 6.2617 | 2.3495 | 6.1097 | 2.4172 | 5.8560 | 2.5498 | 5.6462 | 2.4379 | 5.5170 | 2.3143 |

240° | 6.0125 | 3.6913 | 5.8648 | 2.8769 | 5.6615 | 2.3252 | 5.5405 | 2.4044 | 5.4280 | 1.7941 | 5.7058 | 2.2246 | 5.3096 | 2.0861 | 5.1568 | 1.9906 | 5.0231 | 1.7812 | 5.0186 | 1.4658 |

255° | 5.2825 | 3.8683 | 5.4738 | 2.7801 | 5.0644 | 2.5410 | 5.0700 | 2.0701 | 5.0662 | 1.9343 | 4.9300 | 2.2788 | 4.9507 | 1.8834 | 4.8968 | 1.6307 | 4.9204 | 1.7197 | 4.8319 | 1.1253 |

270° | 4.6859 | 4.2330 | 4.6748 | 3.2749 | 4.6275 | 2.7571 | 4.5115 | 2.7401 | 4.6088 | 2.3565 | 4.6849 | 2.5716 | 4.5820 | 2.5422 | 4.5490 | 2.3772 | 4.9078 | 1.8362 | 4.5584 | 2.0545 |

285° | 5.4587 | 3.0948 | 5.3757 | 2.2367 | 5.1521 | 1.9172 | 4.8634 | 1.9642 | 4.8530 | 1.8066 | 4.7003 | 2.7205 | 4.5688 | 2.5429 | 4.5027 | 2.2719 | 4.4341 | 2.0749 | 4.4128 | 1.9002 |

300° | 6.7272 | 3.4897 | 6.4594 | 2.7061 | 6.3176 | 2.4254 | 6.2566 | 2.0494 | 5.8444 | 2.2178 | 6.1793 | 1.7760 | 5.8800 | 1.8404 | 5.7687 | 1.7453 | 5.6025 | 1.6869 | 5.5200 | 1.5808 |

315° | 7.6638 | 4.9579 | 7.3405 | 4.2297 | 7.2336 | 3.4616 | 7.0749 | 2.9898 | 6.7135 | 2.8178 | 7.1701 | 3.0324 | 6.7725 | 3.1868 | 6.2771 | 2.7622 | 6.3079 | 2.3125 | 6.2577 | 2.1921 |

330° | 7.5450 | 4.5689 | 7.3872 | 3.2277 | 6.7256 | 3.2009 | 7.0058 | 2.6464 | 6.6335 | 2.4751 | 6.7770 | 2.5510 | 6.7925 | 2.2418 | 6.6340 | 2.0314 | 6.3726 | 1.9480 | 6.1586 | 1.8684 |

345° | 7.6639 | 4.0383 | 7.5456 | 3.2610 | 7.6046 | 2.3372 | 7.2854 | 2.3774 | 6.7161 | 2.2872 | 6.8153 | 2.0664 | 6.7096 | 2.0348 | 6.5413 | 1.8508 | 6.3570 | 1.6421 | 6.2692 | 1.4692 |

**Table 3.**The neutrosophic average values and standard deviations of JRC-NNs in different sample lengths.

Sample Length L | Average Value | Standard Deviation ${\mathit{\sigma}}_{\mathit{z}\mathit{j}}$ | ||
---|---|---|---|---|

${\overline{\mathit{a}}}_{\mathit{j}}$ | ${\overline{\mathit{b}}}_{\mathit{j}}$ | ${\overline{\mathit{z}}}_{\mathit{j}}$ (I $\in $ [0, 1]) | ||

10 cm | 6.9427 | 4.3055 | [6.9427, 11.2482] | [1.0866, 1.4375] |

20 cm | 6.7740 | 3.3761 | [6.7740, 10.1501] | [0.9894, 1.3176] |

30 cm | 6.5483 | 2.9609 | [6.5483, 9.5092] | [0.9878, 1.3073] |

40 cm | 6.4490 | 2.6322 | [6.4490, 9.0812] | [0.9607, 1.3257] |

50 cm | 6.2795 | 2.5196 | [6.2795, 8.7991] | [0.8988, 1.2243] |

60 cm | 6.2913 | 2.6582 | [6.2913, 8.9495] | [0.8594, 1.1493] |

70 cm | 6.1505 | 2.4706 | [6.1505, 8.6211] | [0.8711, 1.1260] |

80 cm | 6.0573 | 2.2253 | [6.0573, 8.2826] | [0.8352, 1.0883] |

90 cm | 5.9928 | 1.9952 | [5.9928, 7.9880] | [0.7960, 1.0300] |

100 cm | 5.9261 | 1.7990 | [5.9261, 7.7251] | [0.7644, 1.0553] |

Orientation θ | Average Value | Standard Deviation ${\mathit{\sigma}}_{\mathit{z}\mathit{i}}$ | ||
---|---|---|---|---|

${\overline{\mathit{a}}}_{\mathit{i}}$ | ${\overline{\mathit{b}}}_{\mathit{i}}$ | ${\overline{\mathit{z}}}_{\mathit{i}}$ (I $\in $ [0, 1]) | ||

0° | 7.4226 | 3.2309 | [7.4226, 10.6535] | [0.3844, 0.8420] |

15° | 7.8501 | 2.5831 | [7.8501, 10.4332] | [0.2843, 1.1698] |

30° | 7.6560 | 2.8152 | [7.6560, 10.4712] | [0.3013, 1.1842] |

45° | 7.0997 | 2.8847 | [7.0997, 9.9844] | [0.3385, 0.9850] |

60° | 6.0368 | 3.2555 | [6.0368, 9.2923] | [0.3130, 1.1182] |

75° | 5.3436 | 2.4401 | [5.3436, 7.7837] | [0.2130, 1.0704] |

90° | 4.8714 | 2.6187 | [4.8714, 7.4901] | [0.0907, 0.8406] |

105° | 5.1312 | 2.6809 | [5.1312, 7.8121] | [0.1902, 1.0122] |

120° | 6.3791 | 2.6061 | [6.3791, 8.9852] | [0.2789, 1.2189] |

135° | 6.4560 | 2.4654 | [6.4560, 8.9214] | [0.4636, 1.0067] |

150° | 7.1731 | 2.9296 | [7.1731, 10.1027] | [0.4368, 1.2946] |

165° | 7.3560 | 2.9602 | [7.3560, 10.3162] | [0.5843, 1.1961] |

180° | 6.8556 | 3.1400 | [6.8556, 9.9956] | [0.3554, 0.9170] |

195° | 6.9553 | 2.8155 | [6.9553, 9.7708] | [0.3640, 1.2298] |

210° | 6.6645 | 2.7889 | [6.6645, 9.4534] | [0.5157, 1.0531] |

225° | 6.2537 | 2.7148 | [6.2537, 8.9685] | [0.4522, 0.8612] |

240° | 5.4721 | 2.2640 | [5.4721, 7.7361] | [0.3255, 0.9058] |

255° | 5.0487 | 2.1831 | [5.0487, 7.2318] | [0.1818, 0.8701] |

270° | 4.6391 | 2.6743 | [4.6391, 7.3134] | [0.1003, 0.6426] |

285° | 4.8322 | 2.2530 | [4.8322, 7.0852] | [0.3335, 0.6340] |

300° | 6.0556 | 2.1518 | [6.0556, 8.2074] | [0.3653, 0.9042] |

315° | 6.8812 | 3.1943 | [6.8812, 10.0755] | [0.4565, 1.2396] |

330° | 6.8032 | 2.6760 | [6.8032, 9.4792] | [0.3983, 1.1377] |

345° | 6.9508 | 2.3364 | [6.9508, 9.2872] | [0.5018, 1.1878] |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, J.; Ye, J.; Du, S.
Scale Effect and Anisotropy Analyzed for Neutrosophic Numbers of Rock Joint Roughness Coefficient Based on Neutrosophic Statistics. *Symmetry* **2017**, *9*, 208.
https://doi.org/10.3390/sym9100208

**AMA Style**

Chen J, Ye J, Du S.
Scale Effect and Anisotropy Analyzed for Neutrosophic Numbers of Rock Joint Roughness Coefficient Based on Neutrosophic Statistics. *Symmetry*. 2017; 9(10):208.
https://doi.org/10.3390/sym9100208

**Chicago/Turabian Style**

Chen, Jiqian, Jun Ye, and Shigui Du.
2017. "Scale Effect and Anisotropy Analyzed for Neutrosophic Numbers of Rock Joint Roughness Coefficient Based on Neutrosophic Statistics" *Symmetry* 9, no. 10: 208.
https://doi.org/10.3390/sym9100208