Download Investigation of Roughness Algorithms Applied to JRC ARMA 15-493

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
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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 i1  x i    yi1  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:
Nh
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

j1
1
m j2
mj
 y
iw 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.
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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.
Eight promising roughness algorithms were chosen from
diverse fields such as geomorphology, rock mechanics
and signal processing. The algorithms were applied to
digitized JRC values at three different sampling
intervals. Results from this study included:
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