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ARMA 15-493 Investigation of Roughness Algorithms Applied to JRC Profiles for Assessment of Weathering McGough, M. University of Florida, Gainesville, FL, USA Kimes, L., Harris, A., Kreidl, O.P., and Hudyma, N. University of North Florida, Jacksonville, FL, USA Copyright 2015 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 49th US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 28 June1 July 2015. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: Although JRC profiles are commonly used for shear strength assessment of rock joints, they can also serve as a base model for weathering assessment. In this study, JRC profiles were characterized using eight promising roughness algorithms from diverse fields such as geomorphology, rock mechanics and signal processing. Roughness indices of digitized JRC profiles were assessed at three different sampling intervals to observe how scale impacts their correlation with JRC. Plots of JRC versus roughness indices exhibit one of two general shapes: either a generally (but sometimes monotonically increasing) curve with occasional small peaks at JRC 6-8 and JRC 12-14 or a bimodal shape that peaks consistently at JRC 10-12 and JRC 14-16 followed by a sudden drop in roughness. Z2 and Mean Absolute Angle show the highest linearity of the eight algorithms. Z2 has been used numerous times to assess JRC profiles but Mean Absolute Angle is a new and promising roughness algorithm. Standard Deviation (σ) and Signal Energy (Es) demonstrate remarkable consistency between sampling intervals but their relative non-linearity with JRC suggests they are insufficient in quantifying roughness. A new JRC relationship, based on Z2 and intercept of the log-log plot of Z2 versus sampling interval, is presented. 1. INTRODUCTION Roughness and associated shear strength of discontinuities has long been an important aspect of rock engineering. One of the first studies relating normal force and joint roughness to shear behavior was conducted by Patton [1]. As this research progressed, joint roughness became an input into rock mass classification systems. All students, practitioners, and researchers in the field of rock mechanics are familiar with the joint roughness coefficient profiles (JRC) developed by Barton and Choubey [2]. This seminal tool has been instrumental in the development of roughness and shear strength assessment of discontinuities. Quantifying roughness using various algorithms to compute roughness indices (RI) is far from a novel idea and has been explored very soon after the publication of the JRC profiles. However, agreement is seldom found in regards to the proper roughness index [3, 4, 5, and 6], discretization methods such as sampling interval selection [4, 7, 8, and 9], and even the statistical nature of the JRC profiles and other rough surfaces [10 and 11], proving that the task can be deceptively complex. As such, the particulars of proper JRC assessment are still being openly discussed. roughness Characterization of rock roughness is not limited to shear strength determinations. The joint roughness coefficient profiles serve as an inspiration to researchers attempting to assess the weathering state of rock. It is well documented that rock surface weathering leads to increased rock surface roughness [5]. Figure 1 contains results from surface laser scans of weathered limestone core specimens from Florida. The rough weathered surface of the specimens is undulating and elevation changes due to weathering range from zero to approximately 16 mm. The similarity between JRC values and weathered specimen surfaces is very apparent. Weathering state is currently assessed using visual and descriptive methods. A study utilizing geo-professionals to visually assess weathering states showed weathering state was more difficult to assess in fine grained than coarse grained rock and as weathering increased the weathering classifications become more divergent [12]. The assessment of weathering must be a quantifiable measure rather than a descriptive measure. Unfortunately roughness is an arbitrary measure; it is scale dependent and for its use in quantifying weathering, there is no single satisfactory scale [13]. As a precursor to weathering quantification using surface roughness, eight promising roughness algorithms will be assessed using the joint roughness profiles. 2 1 L dy Z2 dx L 0 dx 1 N 1 yi 1 y1 N 1 i 1 x i 1 x i 2 (1) where L is the nominal length of the profile, or the total length along the x-axis over which the profile spans (the nominal length of a perfectly smooth profile would be equal to the sum of the distance between all adjacent points), the sampling interval is xi+1–xi (referred to as Δx), and N is the total number of samples. The data points in the JRC profiles are evenly-spaced so that L=ΔxN. It is important to note that Z2 tends to increase as sampling interval decreases [4, 8, and 9], which may indicate consideration for high-frequency roughness that was not accounted for at higher sampling intervals. Grasselli claims that Z2 is not an appropriate measure for non-stationary roughness [14]. 2.2. Structure Function (SF) The structure function was also explored by Tse and Cruden in the same paper as the Z2 and yielded very similar results, though it is apparently far less popular as a roughness index [3]. Instead of emphasizing the rate of change in the x and y directions, the structure function represents the squared difference between adjacent height samples separated by a finite lag Δx. The structure function is described by: SF f x f x x L 2 (2) 0 Fig. 1. Surface roughness of three weathered limestone specimens 2.3. Sinuosity (S) 2. ROUGHNESS ALGORITHMS The most expedient method of obtaining a single assessment of the quantitative roughness of a linear set of data is through the use of a roughness index formula or roughness algorithm. The following methods are borrowed from several disciplines including geomorphology, rock mechanics, and signal processing: Z2, Structure Function (SF), Sinuosity (S), Mean Absolute Angle (θA), Standard Deviation (σ), Semivariance (γv), Signal Energy (Es), and Fractal Dimension (D). Each of the algorithms is described below. 2.1. Z2 The root-mean-square (RMS) of the first derivative of a profile, also known as the Z2, was popularized for JRC study in a paper by Tse and Cruden who found that the measure had a strong correlation with the JRC profiles [3]. The Z2 describes the magnitude of the incremental rate of change of a profile and is given by: The total length of a path divided by the straight-line length separating its two end points describes the sinuosity of that curve. Like Z2, it takes the horizontal distance between adjacent points into account, which is of no consequence for evenly-spaced data. The method is mathematically equivalent to the chain method common in soil science as a roughness assessment [15]. The sinuosity is prized for its simplicity and is described by: S 1 N 1 L i 1 x i1 x i yi1 yi 2 2 (3) where L describes the total nominal length of the profile. Sinuosity increases invariably as sampling interval decreases. It has been demonstrated that profiles with clearly differing degrees of surface roughness can possess the same sinuosity value [5]. 2.4. Mean Absolute Angle (θA) The average angular difference between adjacent points on a curve describes the mean absolute angle. Assuming the profile has no inherent slope, this value is one method of representing the irregularity between adjacent heights in a profile. A 1 N 1 1 yi 1 yi tan x x N i 1 i i 1 Es (4) This measure is highly sampling interval-dependent, typically yielding low values at small sampling intervals and high values at large intervals [5]. Large-scale waviness also causes spuriously high values with this measure. 2.5. Standard Deviation (σ) A simple, ubiquitous description of the average deviation of a distribution from its mean, the standard deviation is not commonly used for rock roughness assessment as it does not incorporate spatial separation. The “deviogram” is a graphical method of describing both the scale and magnitude of roughness using the standard deviation [5]. The deviogram is distinct from the variance-based variogram in that it automatically disregards the trend from a set of data [5]. This is of limited usefulness since data detrending is a simple task for computerized data. 2.6. Semivariance (γv) Used to generate the “variogram,” a multi-scale representation of the spatial variability of a set [16], the semivariance describes the average squared variation in height between pairs. Much like the sinuosity, mean absolute angle, and Z2, the semivariance is spatially dependent rather than point-by-point, meaning it represents the variation in intensity between adjacent points rather than taking a global average of intensity [17]. The semivariance is given by: Nh 1 2 v yi yi h 2 N h i 1 (2) where h represents a fixed lag that is varied to alter the scale at which measurements are taken. In this study the scaling alterations are performed on the data itself so h is assumed to be equal to one in all cases. The semivariance does not take the magnitude of sampling interval into account. 2.7. Signal Energy (Es) The energy contained in a changing electrical signal contains a finite amount of energy. When this signal is referenced from its mean, this quantity of energy is roughly equivalent to the amount of “activity” present within a signal. The signal energy Es is given by: 2 1 N 2 x t dt yi L i 1 (4) The mathematical definition is derived from Parseval’s Theorem, which relates the signal energy to the integral of the square of the Fourier transform of the signal. The signal energy is popular for the study of signals in many electrical engineering applications. 2.8. Fractal Dimension (D) The fractal dimension, when introduced in 1975 by Benoit B. Mandelbrot, signified a startling departure from Euclidean geometry that enabled mathematicians to describe the dimensionality of a surface as a fraction [18]. A fundamental trait of fractal surfaces is “selfsimilarity,” or the property of a surface being similar in structure irrespective of the scale of observation. At first glance it would seem that a measure capable of representing the roughness of a surface irrespective of the scale at which it is examined would be a perfect match for joint profiles. However, it has been demonstrated that the JRC profiles are not self-similar, but self-affine, indicating that scaling in the x and y directions must be done separately and with different ratios [19, 20]. To do this one must obtain the Hurst exponent through one of several possible means, but the resultant fractal dimension has still been found to be insufficient to the purpose of characterizing roughness [9, 11, and 21]. Nonetheless, it is still a method in widespread use for roughness characterization, attesting to its relative effectiveness as a scale-independent roughness index. Unlike the aforementioned indices, no single agreed algorithm exists for fractal dimension computation, but several methods including the divider method, variogram method, spectral method, and roughnesslength method yield fairly similar results [11]. The roughness-length method [20] was selected for this study, as it does not require downsampling (divider method) [6] or the Fourier transform (spectral method) and is not dependent on the semivariance (variogram method). To calculate D using the roughness-length method, the RMS roughness of several windows of length w along the profile must be averaged according to: RMS w 1 nw nw j1 1 m j2 mj y iw j i yJ (2) where nw is the number of windows, j is the window index, wj is each window’s length, mj is the number of points within each window, and y J is the mean within window wj. The variables yi and y J denote values that are derived from the residuals of a linear regression local to the window j. These local windows prevent spuriously large estimations of roughness that may be the result of windowing a profile containing large-scale curvature. The value of D using this method is dependent on the slope of the log-log plot of RMS roughness against window length w according to D = 2 - . differences were found between PNG and BMP images used for the analyses. 3. JRC PROFILE DIGITIZATION The ten JRC profiles were scanned into MATLAB from a PDF version of the JRC profiles found on the Rocscience website [22]. The figure contains a scale that was used in the digitization process. Each of the profiles was scanned at the native resolution of the document and saved in separate image files using both PNG and BMP formats. The ten images were individually inspected using a high-pass filter and edited for any stray noise in the white space, which for all but one profile proved unnecessary. A threshold was applied to the images to yield black and white images in which pixels along the profile were set equal to one and white space set equal to zero. Individual pixels were referenced by pairs of row m and column n, indicating the indices of those pixels (Fig. 2). The pixel length in real units was then derived from the scale provided within the JRC figure. These black-and-white images were then converted into a pair of one-dimensional vectors containing the coordinates xi and yi corresponding to points along the profile. The MATLAB code scanned each profile one column at a time, averaging the locations of pixels containing ones along each column and recording this location into the corresponding column of a vector. The yi values (corresponding to heights whereas xi indicates horizontal location) are subtracted from the mean height yi in order to reference all height values relative to the mean. An average profile length of 693 pixels was obtained using this method. The digitization procedure is shown in Fig. 2. 4. RESULTS AND DISCUSSION With the JRC profiles successfully digitized, the eight roughness algorithms were applied to the profiles to produce roughness indices. To investigate the influence of sampling interval, roughness indices were computed using sample sets with sampling intervals Δx equivalent to the original (0.0142 cm), half (0.0284 cm), and a quarter (0.0568 cm) of the original image resolution. The roughness indices were then plotted as a function of JRC values to determine the distribution of the indices values. Linear regression was also performed on the data, using both the linear value and log10 value of the roughness indices. The log10 values were utilized to investigate any potential improvements in the linear regression coefficients. It should be noted that no Fig. 2. Digitization technique employed 4.1. Roughness Indices as a Function of JRC Plots of the roughness indices against the original JRC values are presented in Fig. 3, using computed roughness indices, and Fig. 4, using log10 of the roughness indices. The plots exhibit one of two general shapes: either a generally, but sometimes monotonically, increasing curve with occasional small peaks at JRC 6-8 and JRC 12-14 or a bimodal shape that peaks consistently at JRC 10-12 and JRC 14-16 followed by a sudden drop in roughness. Roughness indices for Z2, Structure Function (SF), Sinuosity (S), Mean Absolute Angle (θA), and Semivariance (γv), fall into the first category. The Standard Deviation (σ) and Signal Energy (Es) fall into the second category. The fractal dimension does not fit into either pattern. 4.2. Effect of Sampling Frequency and Regression Analyses This study confirms previously noted concerns about the impact of sampling interval on roughness assessment [4, 7, 8, and 9]. At all three sampling intervals, none of the roughness indices rise monotonically with JRC. The 0.0568 cm sampling interval yields equal or greater correlation than the other two scales using most of the roughness indices, as shown in Tables 1 and 2. Fig. 3. Roughness indices and JRC values at three different sampling frequencies. Fig. 4. Log10 roughness indices and JRC values at three different sampling frequencies on a log10 scale. Table 1. Linear Correlations of Roughness Indices Table 2. Log10 Correlations of Roughness Indices Index Z2 SF S θA σ γv Es D a Correlation Coefficient Δx=0.0142 cm Δx=0.0284 cm Δx=0.0568 cm 0.9186 0.9373 0.9578 0.8913 0.8936 0.8983 0.9029 0.9057 0.9130 0.9518 0.9692 0.9716 0.8003 0.8006 0.7998 0.8937 0.8954 0.9007 0.7093 0.7098 0.7092 -0.7371 -0.8279 -0.6991 -0.2159 -0.1806 0.2839 Index Z2 SF S θA σ γv Es D a Correlation Coefficient Δx=0.0142 cm Δx=0.0284 cm Δx=0.0568 cm 0.9384 0.9610 0.9797 0.9377 0.9607 0.9793 0.9065 0.9097 0.9170 0.9693 0.9734 0.9723 0.8574 0.8574 0.8567 0.9384 0.9610 0.9797 0.8579 0.8580 0.8578 -0.7210 -0.8230 -0.6819 -0.2031 -0.2000 0.1856 Among all of these roughness indices, Z2 and Mean Absolute Angle (θA) clearly stand out on linearity, achieving a maximum correlation of up to 0.958 and 0.972, respectively, at the 0.0568 cm interval. The two exhibit very similar results, including an anomalously small estimation of JRC 8-10 at the 0.0568 cm sampling interval. Interestingly, the poorest fit using Z2 and Mean Absolute Angle (θA) occurred at the original resolution of the image (0.0142 cm) with a correlation coefficient of 0.919 and 0.952, respectively. It is well-established that Z2 has a high correlation with JRC [3, 9, and 21] but mean absolute angle has seldom been used to assess JRC roughness. Z2, the results of calculating the root-mean-square of the first derivative of a set of data, indicates the magnitude of the rate of change in a profile. The mean absolute angle, an indication of the average difference in angle between adjacent points, similarly describes variation on a scale comparable with the sampling interval. Both consequentially emphasize rapid changes on a small scale more so than slower, larger scale changes. Therefore, an otherwise smooth profile with large-scale roughness will yield a small Z2 and Mean Absolute Angle (θA) at small sampling intervals compared to a flat profile with small-scale roughness. In this sense, the Z2 and Mean Absolute Angle (θA) may be more appropriate measures of roughness at scales that compare with the sampling interval chosen. 4.3. Fractal Dimension In agreement with findings presented in previous studies, the Fractal Dimension (D) does not increase monotonically with JRC [6, 11]. Not only does it demonstrate some of the poorest correlation (between 0.828 and -0.699) with JRC of the methods explored here, the overall trend is towards decreasing D with increasing JRC, which defies conventional logic regarding roughness. It has been demonstrated [6] that a, the intercept of a regression applied to the plot used to generate D, provides a reasonable estimate of JRC roughness on natural rock as compared to the Tse and Cruden [3] method. The plot is only non-flattening within a certain range so it is important to apply the regression to this range in order to generate reliable a values. However, this range can fluctuate, which may introduce inconsistencies between different data sets. Hsiung et al. [11] suspected that Fractal Dimension (D) corresponds to shorter wavelengths (secondary asperities) that contribute less to overall shear strength than a, which corresponds to longer wavelengths (primary asperities). Unlike the Tse & Cruden [3] method, it does not yield negative values among the sample of rocks explored. This was verified by Jang et al. [6] using both the divider method and variogram method for fractal dimension estimation and although their predictions agree within the range investigated, the plot of a versus JRC is concave down for the divider method and concave up for the variogram method. Calculating a at the three scales investigated here with the roughness-length method instead yields a very poor correlation (between -0.216 and 0.284) with JRC at all three scales. Coefficient a, using the roughness-length method, was explored by Rahman et al. [20] but not for JRC estimation. Fig. 5 contains a plot of intercept value a and JRC at the three scales used in this study. Fig. 5. Intercept a of fractal dimension versus JRC value 4.4. Standard Deviation (σ) and Signal Energy (Es) The Standard Deviation (σ) and Signal Energy (Es) algorithms exhibit bimodal shape that peaks consistently at JRC 10-12 and JRC 14-16 followed by a sudden drop in roughness (Fig. 3 and 4). Compared to the other roughness indices investigated, the Standard Deviation (σ) and Signal Energy (Es) show little to no variation between the three sampling intervals. In fact, the average deviation from the mean value of data points corresponding to the same JRC value between the three sampling intervals was roughly equal to 1.37x10-4 for Standard Deviation (σ) and to 1.55x10-4 for Signal Energy (Es). Were the correlation coefficients with JRC more satisfactory for these two measures, they might be applicable for JRC estimation. Fig. 6. Plot of Z2 against JRC over large range of sampling intervals. Plots of JRC versus roughness indices exhibit one of two general shapes: either a generally, but sometimes monotonically, increasing curve with occasional small peaks at JRC 6-8 and JRC 12-14 (Z2, Structure Function (SF), Sinuosity (S), Mean Absolute Angle (θA), and Semivariance (γv)) or a bimodal shape that peaks consistently at JRC 1012 and JRC 14-16 followed by a sudden drop in roughness (Standard Deviation (σ) and Signal Energy (Es)). The Fractal Dimension (D) algorithm does not fall into either category. Z2 and Mean Absolute Angle (θA) show the highest linearity of the eight promising algorithms. Z2 has been used numerous times to assess JRC profiles but Mean Absolute Angle (θA) is a new and promising roughness algorithm. Standard Deviation (σ) and Signal Energy (Es) show little to no variation between the three sampling intervals but their relatively low linearity with JRC make them less attractive for quantifying weathering. At this point, Fractal Dimension (D) does not seem like a viable methodology for assessing weathering. A new method for estimating JRC is presented. The method uses the familiar Z2 and the c, which is the absolute value of the intercept of the log-log plot of Z2 versus sampling interval. 5. PROPOSED METHOD It is disappointing, though not surprising given prior research, that the fractal dimension alone does not adequately quantify JRC roughness, as it should theoretically describe a roughness that is independent of scale. As an alternative that remains in the spirit of fractal-based methods, a method that utilizes the results from Z2 calculation as sampling interval increases to generate a cross-scale assessment of roughness is proposed. This index is described by the ratio of Z2 calculated at the original sampling interval over the absolute value of the intercept of the log-log plot of Z2 versus sampling interval. Unlike a, this intercept, which will be referred to as c, does not require the regression to be performed on a specific range, as the log-log plot is roughly linear over all ranges investigated (depicted in Figure 6). Applying this index to the standard JRC profiles and fitting a logarithmic regression yields R2 of 0.988. The resultant function for JRC estimation is: Z JRC 12.457 ln 2 21.953 c (8) This correlation is slightly stronger than the one presented by Tse and Cruden using only Z2 [3]. However, the correlation of the Tse and Cruden method has been found both here and in other works [4, 6, 8, and 9] to be highly specific to the original sampling interval used. The standard error of this new estimation using Equation 8 is roughly 0.659 as compared to 3.56 using the Tse & Cruden [3] method at the smallest sampling interval available (0.0142 cm). This method accounts for the tendency of Z2 to increase as sampling interval decreases by reducing Z2 proportionally to its rate of increase. REFERENCES 1. Patton, F.D., 1966. Multiple modes of shear failure in rocks. In Proceedings of the 1st ISRM Congress, Lisbon, Portugal, 25 September – 1 October, 1966, International Society for Rock Mechanics. 2. Barton, N. and Choubey, V., 1977. The shear strength of rock joints in theory and practice. Rock Mechanics 10: 12, 1-54. 3. Tse, R., and Cruden, D.M., 1979. Estimating joint roughness coefficients. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 16: 5, 303-307. 4. Yu, X., and Vayssade, B., 1991. Joint profiles and their roughness parameters. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 28: 4,333-336. 5. McCarroll D. and Nesje, A., 1996. Rock surface roughness as an indicator of degree of rock surface weathering. Earth Surface Processes and Landforms 21:10, 963-977. 6. Jang, B.A., Jang, H.S., and Park, H.J., 2006. A new method for determination of joint roughness coefficient. In Proceedings of the 10th IAEG International Congress, 6. SUMMARY Joint roughness coefficient profiles, which are most often used to assess the shear strength of rock joints and incorporated into rock mass classification schemes, were used as a baseline to investigate the applicability of eight roughness algorithms to assess weathering state. Weathering state, which is often assessed qualitatively, is subject to a number of biases. The use of a quantitative measure to assess weathering should alleviate any ambiguities. 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Deriving roughness characteristics of rock mass discontinuities from terrestrial laser scan data. In Proceedings of the 10th IAEG International Congress, Nottingham, UK, 6-10 September, 2006, eds. J.S. Griffiths et al., The Geological Society of London. 21. Yang, Z.Y., Lo, S.C., and Di, C.C., 2001. Reassessing the joint roughness coefficient (JRC) estimation using Z2. Rock Mechanics and Rock Engineering, 34:3, 243-251. 22. Shear Strength of Discontinuities, https://www.rocscience.com/hoek/corner/4_Shear_strengt h_of_discontinuities.pdf