|
|
|
| AN IMPROVED METHOD OF GOODNESS-OF-FIT TEST FOR FISHER DISTRIBUTION TO DISCONTINUITY ORIENTATIONS |
| ZHENG Jun1，2，3，DENG Jianhui2，WEI Jinbing2 |
| (1. College of Civil Engineering and Architecture，Zhejiang University，Hangzhou，Zhejiang 310058，China；2. State Key Laboratory of Hydraulics and Mountain River Engineering，Sichuan University，Chengdu，Sichuan 610065，China；3. Rock Mass Modeling and Computational Rock Mechanics Laboratories，University of Arizona，Tucson 85721，USA) |
|
|
|
|
Abstract Fisher distribution is the most commonly used function of probability density for discontinuity orientations. Before using Fisher distribution to describe the discontinuity orientations，a goodness-of-fit test for them should be performed. The traditional method of goodness-of-fit tests for fisher distribution to discontinuity orientations was firstly reviewed in this paper and an improved method was then proposed. The core idea of the improved method was to adjust those orientations extending beyond the edge of the upper hemisphere projection(OEBEUHP) during the test. Three examples were given. The data of examples 1 and 2 were generated using the Monte Carlo simulation technique，which theoretically followed Fisher distributions. A comparison of the test results from the proposed and the traditional methods was made to verify the proposed method. The data of Example 3 was the application of the proposed method to a mine slope of an open pit. The results showed that the test results from the traditional method might be wrong because of the influence of the OEBEUHP. The correct test results were always obtained with the proposed method since the OEBEUHP were adjusted(such as in examples 1 and 3). The proposed method was therefore suggested to be used while in preforming a goodness-of-fit test to Fisher distribution of discontinuity orientations.
|
|
Received: 16 June 2014
|
|
|
|
| [1] ZHANG L. Engineering Properties of Rocks[M]. Elsevier Science，Amsterdam，Netherlands，2005，304.
[2] ISRM. Suggested methods for the quantitative description of discontinuities in rock masses. International society for rock mechanics，commission on standardization of laboratory and field tests[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1978，15：319–368.
[3] LIN C T，AMADEI B，JUNG J，et al. Extensions of discontinuous deformation analysis for jointed rock masses[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1996，33：671–694.
[4] WU J，OHNISHI Y，NISHIYAMA S. Simulation of the mechanical behavior of inclined jointed rock masses during tunnel construction using discontinuous deformation analysis(DDA)[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41：731–743.
[5] KULATILAKE PHSW，HE W，UM J，et al. A physical model study of jointed rock mass strength under uniaxial compressive loading[J]. International Journal of Rock Mechanics and Mining Sciences，1997，34：165.e1–165.e15.
[6] SINGH M，RAO SK. Empirical methods to estimate the strength of jointed rock masses[J]. Engineering Geology，2005，77：127–137.
[7] BAGHBANAN A，JING L. Hydraulic properties of fractured rock masses with correlated fracture length and aperture[J]. International Journal of Rock Mechanics and Mining Sciences，2007，44：704–719.
[8] LARSEN B，GRUNNALEITE I，GUDMUNDSSON A. How fracture systems affect permeability development in shallow-water carbonate rocks：an example from the Gargano Peninsula，Italy[J]. Journal of Structural Geology，2010，32：1 212–1 230.
[9] TAI T W，HUANG T H. A constitutive model for the deformation of a rock mass containing sets of ubiquitous joints[J]. International Journal of Rock Mechanics and Mining Sciences，2009，46：521–530.
[10] LIU D A，WANG S J，LI LY. Investigation of fracture behavior during rock mass failure[J]. International Journal of Rock Mechanics and Mining Sciences，2000，37：489–497.
[11] ZHENG J，KULATILAKE PHSW，SHU B，et al. Probabilistic block theory analysis for a rock slope at an open pit mine in USA[J]. Computers and Geotechnics，2014，61：254–265.
[12] ZHENG J， KULATILAKE PHSW，DENG J. Development of a probabilistic block theory analysis procedure and its application to a rock slope at a hydropower station in China[J]. Engineering Geology，2015：http：// dx.doi.org/10.1016/j.enggeo.2015.01.010.
[13] PITEAU D R. Geological factors significant to the stability of slopes cut in rock[C]. In symposium on planning open pit mines，South African Institute of Mining and Metallurgy，Johannesburg，1970，33–53.
[14] PITEAU D R. Characterizing and extrapolating rock joint properties in engineering practice[J]. Rock Mechanics，1973，supplement 2：5–31.
[15] YE J， ZHANG Y， SUN J，et al. Correction of the probabilistic density function of discontinuities spacing considering the statistical error based on negative exponential distribution[J]. Journal of Structural Geology，2012，40：17–28.
[16] KULATILAKE PHSW. Fitting Fisher distributions to discontinuity orientation data[J]. Journal of Geological Education，1985，33：266–269.
[17] KULATILAKE PHSW，WU T H，WATHUGALA D N. Probabilistic modelling of joint orientation[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1990，14：325–350.
[18] PRIEST S D，HUDSON J A. Discontinuity spacings in rock[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1976，13：135–148.
[19] HUDSON J A，PRIEST S D. Discontinuity frequency in rock masses[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1983，20：73–89.
[20] PRIEST S D. Determination of discontinuity size distributions from scanline data[J]. Rock Mechanics Rock Engineering，2004，37(5)：347–368.
[21] ZHANG L，EINSTEIN H H. Estimating the mean trace length of rock discontinuities[J]. Rock Mechanics Rock Engineering，1998，31(4)：217–235.
[22] FISHER S R. Dispersion on a sphere[J]. Proceedings of Royal Society A：Mathematical，Physical and engineering Sciences，1953，217：295–305.
[23] PRIEST S D. Discontinuity analysis for rock engineering[M]. Chapman and Hall，London，1993，473.
[24] KEMENY J，POST R. Estimating three-dimensional rock discontinuity orientation from digital mages of fracture traces[J]. Computer Geosciences， 2003，29：65–77.
[25] ROCSCIENCE DIPS，2014. Available at：http：//www.rocscience.com/ products/1/Dips，[Date accessed：1 June 2014].
[26] BINGHAM C. Distributions on the sphere and on the projective plane[Ph. D. Thesis][D]. Yale University，USA，1964：86.
[27] BINGHAM C. An antipodally symmetric distribution on the sphere[J]. The Annals of Statistics，1974，2：1 201–1 225.
[28] KULATILAKE PHSW. Bivariate normal distribution fitting on discontinuity orientation clusters[J]. Mathematical geology，1986，18：181–195.
[29] RUBINSTEIN R Y，KROESE D P. Simulation and the Monte Carlo Method[M]. 2nd Edition. Wiley-Interscience，Hoboken，NJ，2007：372.
[30] WATSON GS，IRVING E. Statistical methods in rock magnetism[J]. Royal Astronomical Society Geophysical Supplement，1957，7(6)：289–300.
[31] 盛 骤，谢式千，潘承毅. 概率论与数理统计(第四版)[M]. 北京：高等教育出版社，2008：198–205.(SHENG Zhou，XIE Shiqian，PAN Chenyi. Probability Theory and Mathematical Statistics[M]. 4th Edition. Beijing：Higher Education Press，2008：198–205.(in Chinese))
[32] ZHENG J，DENG J，YANG X，et al. An improved Monte Carlo simulation method for discontinuity orientations based on Fisher distribution and its program implementation[J]. Computers and Geotechnics，2014，61：266–276.
[33] GOODMAN R E，SHI G. Block theory and its application to rock engineering[M]. Englewood Cliffs：Prentice-Hall，1985：338.
[34] WATSON G S. The statistics of orientation data[J]. The Journal of Geology，1966，74(5)：786–797.
[35] MARDIA K V. Statistics of directional data[M]. Academic Press：London and New York，1972：357.
[36] YU Q. Analyses for fluid flow and solute transport in discrete fracture network[Ph. D. Thesis][D]. Kyoto University，Japan，2000：161.
[37] FISHER NI，LEWIS T，EMBLETON BJJ. Statistical analysis of spherical data[M]. Cambridge University press，Cambridge，London，1987：329. |
|
|
|
2015, Vol. 34 
