expectations, but the boxes for the low aspect values (2
and 3) were completely filled! Thus, the curves were
loaded if this selection was made (table 7).

Tal

7.—Computation of fractal dimension using box
method (expected value = 1.3)

Aspect ratio _ Number of boxes
Input values consistent
with prior analysis! , . 10 45
15 73
20 103
22 114
25 132
7 145
30 166
35 198
40 221
45 264
Input values varied? . . 2 4
3 9
4 13
5 18
6 22
18 89
19 92
20 103
First point removed? . . 3 9
4 13
5 18
6 22
18 89
19 92
20 103

1Fractal dimension = 1.166; correlation coefficient = 0.9997.
2Fractal dimension = 1.335; correlation coefficient = 0.9965.
3Fractal dimension = 1.269; correlation coefficient = 0.9987.

The point (2,4) log values of (0.3010, 0.6021) were
exercising undue influence on the slope of the fitted line
and were a candidate for dismissal as an outlier. This de-
cision could be justified on other grounds because 100 pet
of the boxes were filled when only two subdivisions were
made on the coordinate axes. If the point (log 2, log 4)
were left out, but (log 3, log 9), which represents another
set of totally filled boxes, were allowed to remain, then
the point (log 3, log 9) lies on a line consistent with the
remaining set of values (figure 7 and table 7).

By removing one data point, the fractal dimension was
changed from 1.335 to 1.269. This parameter variation
could be continued through many possible combinations
of aspect values and would generate changes in the fractal
dimension. This exercise illustrates how the selection of
input values would change the fractal dimension, in this
instance, from 1.666 to 1.335. (Again, recall that the
expected fractal dimension is 1.3.) These changes were
not a consequence of using the aspect ratio method.
Feder (20, pp. 186) changes the rectangular dimensions of

the covering of boxes, and in so doing, varies the fractal
dimension from 1.03 to 1.51. Thus, the user’s selection
of input values can directly affect the resulting fractal
dimension, certainly not a desirable situation when one
is seeking invariance or stability of answers from a
procedure.

Modified Divider Algorithm and Scaling
Factor Variations

Earlier, the modified divider technique was used in
conjunction with a z-coordinate (i.e., trace height) scaling
factor to alleviate the crossover length problem. A set of
profile traces with given fractal dimension values of 1.05,
1.2, 1.3, 1.4, and 1.5 and a sample size of 1,200 was gen-
erated by spectral synthesis (5). Table 8 shows the pro-
gression of the fractal dimension values as a function of
the scaling factor.

Table 8.—Effect of scaling factor on fractal dimension
computations using modified divider algorithm

Generated fractal Z-coordinate scale factor

dimension 1 5 10 100 1,000
1.05 .. 1.012 1.080 1.110 1.132-1.132
1.20 1.038 1.174 1.213 1.240 1.241
1.30 1.079 1.261 1.301 1.327 1.927
1.40 1.147 1.962 1.308 1.418 1.418
1.50 1.246 1.467 1.497 1.513 1.513

Several points become apparent. First, a scaling factor
of 1.0 underestimated the expected (generated) fractal
dimension values in all cases. Second, the expected fractal
dimension of 1.05 was poorly estimated throughout. Third,
a scaling factor of about 10 seemed to be a best choice for
this set of data in that the expected fractal dimensions
were reasonably close to the derived values. Fourth, the
fractal dimension values asymptote to a constant as the
scale factor increased.

As before, a potential problem arises in that user input
of best scale factor is implied. With experimental data,
there are no guidelines as to what the best choices might
be, leaving the user with difficult decisions as to what
scale factor to choose. In previous work, the asymptotic
approach was chosen, using a scale factor of 1,000.

GNEISS y-DIRECTION ANALYSIS

For the same sample used in the section "Analysis of
Roughness Measures,” 10 profiles oriented along the y-axis
were also digitized using photogrammetry. Four fractal
dimension algorithms and two statistical algorithms were
applied to these profiles. Descriptive statistics were com-
puted relative to these derived measures of the 10 traces,


equation JRC = 32.2 - 32.47 log (Z,) was used to predict
a JRC value (table B-6). With the exception of profiles 5
and 6, the stereo digitizer and the profilometer provided
nearly identical JRC values. Also, the stereo-digitizer-
derived JRC values generally were slightly less than those
derived from the profilometer.

A deterministically derived, nonergodic covariance
estimator (C) was used to compute the geostatistical
covariance of elevations along each of the profiles. This
estimator [(adapted from Isaaks and Srivastava) (27)] is
defined as:

and 2(x) = data value at location x.

The original definition of this estimator relied on spatial
integrals and was an extension of earlier time-series work
by Anderson (2); further explanations are beyond the
scope of the current discussion. Suffice it to say that this
covariance model has several mathematical advantages
over traditional, probabilistic models of covariance (27),
one of which is nonreliance on a stationary mean.

Elevation covariance was computed for each of the
roughness profiles using lag cells 0.1 mm wide. An
example covariance plot is given in figure B-5, which

N(h) displays the results as complement covariance (ic.,
€@) = 1] 2) 2 +h) variance minus covariance) and resembles a traditional
N(h) |i variogram. The covariance estimates obtained from the
detrended elevation data provided paired values that could
FN (h) N(h) be analyzed with the matched-pairs t-test discussed carlier.
-]) ze] | zm) +h Results of the t-tests (table B-6) indicated that roughness
it im data collected by the profilometer and the stereo digitizer
. had similar covariances. Differences in the matched co-
where h = separation distance, or lag, variance values were caused primarily by differences in
variances,
N(h) = number of sample pairs separated by
lag h,
Table B-5.—Sample variances and calculated two-sample F-statistics
for roughness data obtained by mechanical profilometer (MP)
and stereo digitizer (SD), square millimeters
Profile Raw data Detrended data
MP var SDvar_ F-value MPvar_ SDvar _ Fevalue
X-DIRECTION
1. 0.5967 0.4821 1.113 0.1912 0.1871 —*1.022
2. 1.9468 1.8032 1.080 4687-4430 1.058
3 1.6388 1.4550 1.126 ©2444. .22468 1.088
4. 36053239 1.141 25532342 —*1.090
y-DIRECTION
5. 0.5696 1.072 0.1642 0.1579 1.040
6. 5505 1.057 09290803 1.157
7. 2699 1.109 06370597 ‘1.067
8. 2.9481 1.150 1267 «1375 ‘1.085
var
Table B-6.—Comparative roughness measures for rock fracture profiles obtained
by mechanical profilometer (MP) and stereo digitizer (SD)
Estimated Covariance matched pairs C(O) profile Fractal
Profile JRC Calculated Probability length dimension
MP SD _ t-statistic MP. $0 MP sD
1. 10 11 “1.39 0.172 0.0082 —«0.0082—=S«i.262—~—~=C«*« 2.
2. 1918 217 866 0079 0075 1.145 1.074
3. 1812 +33 745 0041 0038 1.194 1.071
4. 20011 -95 348 0043 0040 1.140 1,058
5. 7 4 116 253 0025 0024 1.275 1.091
6. 6 2 06 950 0013 0011 1,269 1.114
7. 2 1 -39 696 0009 0008 1.223 1.094
8. 10 9 72 478 0022 0024 1.229 1.123
C(O) Covariance of estimator.
JRC Estimated joint roughness coetticient.


15.0
7 I I I I I
125 4
10.0
€
E I
East Ht +
iO} I
PA I
=
5.0 ! +
H KEY
———— Photogrammetry
251— 1. Metroplate 4
! Mechanical transducer
1
n ! | | |
° 20 40 80 100

60
DISTANCE, mm

Figure B-2.—Three different measuring methods on trace 1 (middle trace) of accuracy bar.

Photographic Enlargements

In conducting geostatistical analyses of the profile data
sets, unexplainable perturbations in the results were found.
A possible reason was the size of the dot (40 um) on the
analytic plotter. The operator uses this dot to pick the
points for digitizing. Because the sampling density was
0.05 mm, only 0.01-mm spaces were left between points,
which could result in the same point being read for two
different points. Thus, the data would be smoothed out,
there would be less variability, and the roughness of the
profile would be less.

To see if the dots were overlapping, a copy of a
slide was enlarged 2.44 times from the original 2-1/4- by
2-1/4-in slide. This enlarged slide was then processed
using the same profiles as on the original slide.

The enlarged slide was more convenient for the plotter
operator to use, but did not necessarily generate more
accurate measurements because of distortions introduced
by the photographic process. The data were probably
improved because of the casier pace of data sampling,
which may minimize the perturbations seen in original
data sets,

The results showed that indeed there was some
smoothing of the data when the smaller slide was used

HEIGHT, mm

|e ed on oe ee
15 20 25 30 35 40 45
HORIZONTAL DISTANCE, mm _

2

5 10 55

Figure B-3.—Comparison of profiles using original and
enlarged slides. A, Profile of GHJ from original 2-1/4- by 2-1/4-
in slide; B, profile of GEHJ from slide enlarged 2.44 times,

(figure B-3 and table B-2), This smoothing, however, was
not enough to affect the calculated values of the rough-
ness measures in any significant way. For all practical pur-
poses, then, the smaller slide was appropriate to use even
at a point density of 0.05 mm.


partial divider length required to reach the end of the
profile be divided by the selected divider span and then
added to the number of divider spans already accumulated
(eg,, the final number of spans might be 80.25, where the
remainder is one-fourth of the selected divider span).

To alleviate the remainder problem when using the
modified divider method, it is suggested that only those
x-increments that terminate at an x-coordinate within
0.005L, of the end of the profile (L, = horizontal length of
profile) be used. In addition to using this criterion, when
computing fractal dimensions for the 60 profiles, investi-
gators made sure that the largest x-increments were those
that required at least 20 repetitions to cover the profile
being studied.

When the box method was evaluated, the greatest
success came when box sizes were used that allowed 20 to
120 boxes to cover the profile. Again, this criterion was
based on profiles having about 1,000 digitized points. With
the spectral method, a 2 or 5 pet cosine-bell tapering win-
dow provided the most consistent results.

Thus, using the above guidelines for the various meth-
ods, the fractal dimensions of all 60 profiles were com-
puted and compared.

ANALYSIS OF ROUGHNESS MEASURES

Besides fractal dimension, there are several other meas-
ures that can be used to quantify roughness along rock
fractures. The two we investigated were the Z, measure
proposed by Myers (38) and the standard deviation of
heights,

Calculation of the Z, measure for profiles is facilitated
by using a slightly modified version of the original Myers
definition (38), as proposed by Tse and Cruden (44):

1 Not 2
Mont >> Oier- yi,

where N

number of digitized values,
M = number of intervals on x-axis = N - 1,
and dx = width of digitized x-increment.

The Z, measure is related strongly to the variance of
slopes along the profile. To eliminate the effect of trends

in the roughness profiles (i.c., regional slope) on the
calculated standard deviation of heights, a linear trend
was removed from each profile before calculating its
standard deviation. It is realized that this detrended
standard deviation does not serve as an adequate measure
of roughness in itself, but it was included as a standard
reference value for comparing with other roughness
measures.

Comparison of Adjacent Profiles

To investigate surface roughness variability, for each
set of 10 parallel profiles, the autocorrelations of the Z,
measure and of the fractal dimension derived from the
modified divider method were calculated. One of the ap-
pealing aspects of fractal analysis is that the fractal di-
mension of a topographic surface can be obtained by
adding 1.0 to the fractal dimension of a single profile from
that surface (32). However, the question arises as to
which profile to use when rock fractures are being studied;
that is, are such profiles relatively similar along a fracture
surface? The autocorrelation results shown in table 1 (an
autocorrelation of 1.0 implies perfect correlation) indicate
that roughness profiles can be quite different, even when
taken parallel to each other. In this case, the parallel
profiles in each set were spaced 50 mm apart.

Table 1.—Autocorrelations of Z, and fractal dimension D
for adjacent profiles on fracture surfaces

Meas- Basalt Basalt Gnelss Gneiss Quartzite Quartzite
ure x y x y x y
Z, ... 0.351 -0,030  -0.084 0.349 0.072 0.228
D.... 566-209 -.056 101 056-185

Correlation of Roughness Measures

Linear correlation coefficients were computed for the
six roughness measures for each of the six profile series.
Four of the measures were fractal dimensions calculated
by the different methods: D, by box method, D, by
divider method, D,, by modified divider method, and D,
by spectral method. Typical results are displayed in ta-
ble 2, The negative correlation values indicate that the
nonfractal roughness measures were inversely related to
D estimates. Values of D computed by the two divider
methods were the most strongly correlated, followed by
those computed by the spectral method, The nonfractal
measures of roughness were correlated strongly.


correlation between D, and D,, (r = 0.450). Z, correlates
rather well with the fractal measures. The standard devia-
tion does go its own way, being negatively correlated with
all the fractal methods. Unfortunately, it is also nega-
tively correlated with the other statistical measure, Z,.
Thus, when using the original space data, some cor-
relations improve, while other correlations do not.

For the visual ranking of the gneiss profiles, the use of
Kendall’s rank correlation analysis (42) generated table 13.

Table 13.—Rank correlation coefficients
using original scale data, roughness
‘measures versus visual ranking,
gneiss specimens

Da Da
x-direction . 0.750 0.841
y-direction . 315 225

Using the original scaled data gave much better re-
sults for the divider algorithms in the x-direction, but this
comparison was rather inconclusive for algorithms in the
y-direction. Augmenting this result with the preceding
correlation matrices, the conclusion was that using the
original scaled data represented no particular advantage
over using the aspect ratio approach. In particular, the
ambiguities between the fractal measures remained an
important problem.

FRACTAL ALGORITHMS APPLIED
TO NORANDA DATA SET

A reasonable question that must be addressed is: Are
the fractal algorithm anomalies particular to the Bureau’s
Spokane Research Center (SRC) photogrammetric data
set? To provide a partial answer, SRC researchers were
fortunate to obtain a set of rock profile data collected by
Noranda engineers. Noranda investigators hoped that a
fractal dimension analysis would provide additional sci-
entific insights, while SRC researchers appreciated the
opportunity to verify fractal work using an independent set
of profile information.

Profile traces were collected with a mechanical pro-
filometer from a variety of sites. Typical profile lengths
(fig. 9) were 100 cm, with over 800 usable data points per
trace. (Profile lengths were over 10 times those of the
SRC data, although the number of data points per profile
trace was very similar.) Nine profile traces were collected
per joint surface; three traces were oriented in a horizontal
direction, three traces at 45°, and three traces in a vertical
direction. Eighteen parameters were measured on each

profile, including asperity angle, high and low peak values,
amplitude parameters, and the Z, measure. The Z, meas-
ure provided a means of verifying that the Noranda data
were properly transmitted for use at SRC.

Three sets of joints were selected from the Noranda
data file, and three fractal algorithms, D,, D,, and D,,,
were derived from the nine profile traces contained on
a single joint surface. The aspect ratio was not applied
when computing D,. A representative set of Noranda re-
sults is shown in table 14. Then, to verify that the results
were not unique, two other joint sets were analyzed; the
results are shown in table 15.

Table 14.—Fractal dimensions and Z, for Noranda
Joint set NS2U1

Profile Dd Dy Do,
Hat. 1.154 1,0048 1.330 0.238
H212 . 1,082 1.0050 1.268 323
H213 . 1.099 1,0055 1.274 318
Vat1 . 1.096 1.0067 1.139 346
Vai2 . 1.136 1.0052 1.194 241
V213 . 11 1.0023 1.166 176
A2it . 1.122 1,0034 1.287 .202
A212 1.168 1.0034 1.312 184
A213. 1.099 1.0043 1.330 Eihl

Table 15.—Correlation matrix for roughness
measures, Noranda Joint sets

0, 0,
NORANDA JOINT SET NS2J1

Dd - 7.000

Dg - ~335 1,000

Dm 415-223 1,000

Zu. +713 838-169 1,000
NORANDA JOINT SET NS1J1

D, 7.000

Dg - 076 1,000

Dy + 255 158 1.000

Zievees 027 993-197 1.000
NORANDA JOINT SET NS1J5

dD, - 7,000

Dg - 386 1.000

Da - 552 «.272—*1.000

Zz. 201-932 ——.284_——1.000

The expectation of high correlations was tempered by
the fact that the Noranda traces were not collected as
uniformly as were the SRC data. Here, nine traces per
joint surface are included, but the traces are oriented into
three triplets going in three different directions, None-
theless, the joint surface provided a basis of commonality,


°

yp Rh OD

COMPLEMENT
COVARIANCE (h)

°

15

20

2 30 35

LAG DISTANCE (h), mm

Figure B-5.—Complement covariance plot for elevations measured along profile 4 using m
; width of lag cells = 0.1 mm; sample mean = 0.5 mm;

(Number of data points = 1,16;
= 0.3695 mm’). (h = separation distance or lag.)

Another means of comparing the results of the two
profiling methods was based on the Yxx(0) parameter
proposed by Wu and Ali (47). In this scheme, Yxx(0) is
equal to the covariance of estimation [C(O)] divided by
profile length; smooth fractures are those with Yxx(0) less
than 0.2, As shown in table B-6, there was excellent
agreement between the two methods. The small values
seemingly indicated a fairly smooth fracture surface, but
based on experience, the investigators do not agree with
Wu and Allis criterion.

The final comparison was the fractal dimension, which
is a measure of how much a surface profile fills its topo-
logical space (10, 32). Most rock fracture profiles have
fractal dimensions between 1.0 and 1.5, where a higher
number indicates a rougher surface. Using the criteria
suggested by Brown (12), the fractal dimension of cach
profile was calculated by the divider method, taking care
to magnify the elevation values sufficiently to avoid under-
estimating the fractal dimension. These computed fractal
dimensions are presented in table 1. The fractal dimen-
sions calculated using the stereo digitizer were approxi-
mately 10 pct less than those obtained from the profilom-
eter, which indicated that (1) some smoothing was invoked
on the roughness data during the photogrammetric and
stereo-digitizing procedures and/or (2) the profilometer
method had some random noise induced by variability in
electrical current during voltage readings or by differences
in rock hardness.

The differences in paired fractal dimensions also sug-
gested that the fractal dimensions were quite dependent on
the finest available scales of measurement. This notion
was verified by computing the fractal dimensions of pro-
filometer profiles smoothed by a three-point, moving-
window averaging scheme; this scheme typically reduced

nical profilometer.
ind sample variance

the calculated fractal dimension by 1 to 5 pet. Further
evidence for the importance of the digitizing increment
was shown by the relatively small fractal dimensions of
profiles 2, 3, and 4. Earlier studies, where the first four
traces (x-direction) were rougher than the last four traces
(y-direction), indicated anistropy in surface roughness.

Therefore, in this study, the evidence suggests that the
fractal dimension may not be the same type of roughness
measure as that used by other investigators. It appears
that fractal dimensions may be influenced heavily by short
wavelength components of a fracture surface, whereas
variance-based and Z,-type parameters may depend more
on longer wavelength components. More research is
planned to investigate such relationships.

Conclusions on Comparing Photogrammetry
and Profilometers

Close-range photogrammetry and stereo digitizing pro-
vided a means to measure rock fracture roughness. With
the capabilities of current equipment, stereo digitizing had
slightly poorer resolution in the horizontal plane and
slightly less accurate repeatability in the vertical plane than
mechanical profilometers when laboratory-scale rock
specimens were examined.

Five statistical comparisons and four derived rough-
ness measures (visual, JRC, Yxx(0), and fractal dimen-
sion) were used to compare close-range photogrammetry
and stereo digitizing as methods of measuring rock frac-
ture roughness. Of the nine comparison criteria, seven
indicated no significant difference between the tech-
niques. Thus, reasonable compatibility between profilom-
eter methods and stereo-digitizing methods has been
established.



10.

11.
A-1.
B-1.
B-2,
B-3,
B-4,
B-5.

ave

. Table of differences
. Summary statistics comparing an ori
-3. Summary statistics for absolute deviations in measured elevations

-5. Sample variances and calculated two-sample F-statistics for roughness data .
-6. Comparative roughness measures for rock fracture profiles

ILLUSTRATIONS—Continued

Examples of direct-shear test results . Pence ene cece e eee e een e een en eens

Generated synthetic versus actual peak shear-strength envelopes for two natural rock fractures
Photogrammetric setup
Accuracy bar in aluminum reference frame
Three different measuring methods on trace 1 of accuracy bar
Comparison of profiles using original and enlarged slides.
Roughness profiles for rock fracture surfaces . wee
Complement covariance plot for elevations measured along profile 4 using mechanical profilometer .

TABLES

Autocorrelations of Z, and fractal dimension D for adjacent profiles on fracture surfaces
Roughness measures, gneiss x-direction series
Correlation matrix for roughness measures and shape parameters for nonlinear, shear-strength
envelope
Correlation matrix for roughness measures using aspect ratio method, gneiss x-
Roughness measure correlations between regular and enlarged data set bees
Rank correlation coefficients of roughness measures versus visual rank using aspect ratio method, gneiss
x-direction series, y-coordinate ........... 0.0005 .
Computation of fractal dimension using box method ... eee
Effect of scaling factor on fractal dimension computations using modified divider algorithm .
Correlation matrix for roughness measures using aspect ratio method, gneiss y-direction series .
Rank correlation coefficients of roughness measures versus visual rank, gneiss x- and y-direction series . .
Comparison of six roughness measures using aspect ratio method on gneiss specimen, x versus
y-direction oe. eee cece cece c ee cee ete vsseecuaeeceuanteceneteteeunneeeens
Correlation matrix for roughness measures using original data space input, gneiss x- and y-direction
series .
Rank correlation coefficients using ‘original scale data, roughness measures versus visual ranking, gneiss
SPECIMENS ©... eee eee cece teen eee eee
Fractal dimensions and Z, for Noranda joint set NS2J1 .
Correlation matrix for roughness measures, Noranda joint sets
Summary of averaged roughness measures and results of direct-shear tests for six fractures.

nal profile to its enlargement

Matched-j

irs comparisons of local means for roughness data . . .



£22 or f16 at 1/15th of a second. Several combinations
were used on cach specimen, but cach pair of pictures was
always taken with the same combination.

PHOTOGRAPH PROCESSING BY ANALYTIC
PLOTTER

Use of an analytic plotter made the photogrammetric
approach possible. Without photogrammetry, it would
have been too time consuming and expensive to measure
the required number of points on a rock surface (over
80,000 sets on a 9- by 9-cm surface). Such massive
amounts of data could not have been generated using
older mono and stereo comparators, which plot just one
profile at a time, The analytic plotter allows the x, y
coordinates of a profile and the density of points along the
profile to be set. The readings are recorded automatically,
requiring an operator simply to set the initial conditions
and monitor the process.

The profiles were plotted with 40-um (0.04-mm) dots.
This size is almost as large as the space between adjacent
points, which makes it difficult to understand how points
closer than one-half of 40 ym can be measured. An
experiment was conducted in which the original 70-mm

negatives were enlarged 2.44 times to see if the enlarging
process made a difference in the robustness of the data.
This test is discussed in the section "Photographic En-
largements" in appendix B.

When the project was initiated, there were only a few
firms in the continental United States with analytic plotters
and only one firm in the Pacific Northwest, Spencer B.
Gross, Inc., of Portland, OR. A procedure evolved in
which color slides (positives) of the specimen in the con-
trol frame were sent to the firm along with measurements
of the height (distance) of the cameras from the sample,
the angle of convergence of the cameras, and the separa-
tion distance. The company ran the profiles using a Wild
stereo analytic plotter, model Avilyt BC1, and returned
the x, y, z coordinates of the points on 5-1/4-in floppy
discs using the American Standard Code for Informa-
tion Interchange (ASCII) format. The data were then
processed.

In reading the profile data from the photographs,
investigators tried to obtain the most precise measure-
ments the equipment was capable of making, which turned
out to be 0.05 mm between points and between heights.
Points closer than 0.05 mm or differing in height by less
than 0.05 mm were treated as the same point.


24

T T q T

20 4
e Fractal dimension = 1.27 without (log 2, fog 4).
5 \
fm)
Bd
Gt6eb 4
a
Ww
o)
ira
Ww
s
S12b 4
<= Fractal dimension all points = 1.34
8
—

8 4
@ (Log 2, log 4)
4 1 ! L ! L
0.2 0.4 0.6 08 1.0 12 1.4

LOG (NUMBER OF BOX DIVISIONS)

Figure 7.~Effect of one data point on calculated fractal dimension.

giving values for mean, median, standard deviation, stand-
ard error of the mean, etc. Correlations were established
between the six algorithms being considered (table 9).

Table 9.—Correlation matrix for roughness measures using
aspect ratio method, gneiss y-direction series

Dy Ds On Dy Z, Standard
deviation
D. 7.000
Dg 513 1.000
Da 587.924 1,000
0, 433-054 -.007 1.000
Zvvees. 561 167.412.4932 1,000
Standard
deviation -572_ 793-813 -.139_—-.323 ‘1.000

Table 9 is to be compared with table 2. In table 2,
there is a nice dichotomy between the four fractal
measures—D,, D,, D,, and D,—and the statistical
measures—Z, and standard deviation. The fractal meas-
ures are positively correlated with each other and nega-
tively correlated to the statistical measures. The two
statistical measures are positively correlated. In table 9,
three of the fractal measures—D,, D,, and D,,—are posi-
tively correlated with each other, but D, does not con-
form to family behavior as before. Furthermore, Z, is now
positively correlated with the fractal measures; only the
standard deviation stands alone.

Four investigators were again asked to rank the
roughness measure of the 10 gneiss y-profiles subjectively
(fig. 8). Their visual rankings were then compared with


recommend that future researchers should carefully weigh
the pros and cons of the fractal geometry controversy
before making their research commitments.

CONTINUING RESEARCH

Three research efforts using fractal dimensions are
continuing at SRC. In photogrammetry, it is imperative to
measure its effectiveness relative to larger scale situations.
For this reason, a field study program will begin soon,
using photogrammetry first on rock outcrops, then on rock
surfaces underground.

Three-dimensional fractal analysis is a second area of
interest. While there are several algorithms available
for two-dimensional profile work, three-dimensional algo-
rithms are scarce. One such algorithm requires even spac-
ing in the x-y plane; the data are not amenable to this
condition,

Estimated semivariogram functions have been used to
obtain fractal dimensions for rock fracture profiles (39),
where it is assumed that the roughness profiles are
modeled effectively with a self-affine fractal model. In
such cases, a log-log plot of the semivariogram yields
data amenable to linear regression. The slope b of the re-
gression line is then used to estimate D using the formula
D=2-b/2.

Because semivariograms also can be used for surfaces,
a topological D, rather than a profile D, is available, For
the topological D, the formula is

D =3-b/2.

Work is progressing along these lines, with the goal
being to obtain an estimate of the fractal dimension of the
entire fractal surface while also investigating anisotropic
properties of surface roughness.

Finally, another two-dimensional profile of a fractal
dimension algorithm is being investigated. This method,
called the variation method, is attributed to Dubuc,
Roques-Carmes, Tricot, and Zucker (19). These authors
claim that the variation method is superior to the algo-
rithms described in the present RI. The variation meth-
od involves dividing the x-axis of the data into sets of
bins, choosing a set of intervals about the bin centers,
computing maximum and minimum z-values in the in-
tervals, summing the differences between corresponding
maximum-minimum values, and obtaining a fractal dimen-
sion from an ensuing log-log plot. The initial computer
programming has been completed, but reliable reportable
results are not available at this time.

SHEAR STRENGTH ESTIMATION

INTRODUCTION

Using the results obtained from analyzing the gneiss
x-direction series of profiles, it was decided to attempt
to model shear strength as a function of three parameters:
fractal dimension, fractal intercept obtained by the modi-
fied divider algorithm, and Myers’ Z, measure. These pa-
rameters were selected because of their high correlations
with the visual rankings of the profiles (table 10).

Roughness profiles from six clean fractures were
ing close-range photogrammetry and automated
izing. Two of the fractures were contained in
basalt specimens, two in gneiss specimens, and two in
quartzite specimens. The rock specimens were trimmed
to be approximately 8 cm on a side, making the roughness
profiles 6 to 7 cm in length. This size was appropriate
for subsequent casting of the specimens with quick-setting
cement to produce testable direct-shear specimens. Ten
parallel roughness profiles were digitized in each of two
directions, identified as the x- and y-directions. Thus,
20 profiles were obtained from each specimen, making a
total of 120 profiles, To avoid generating excessively large
amounts of data, the regular digitizing interval was approx-
imately 0.2 mm, which produced about 300 observations

along each profile. Figures 5 and 6 show the roughness
profiles for the samples.

The estimated roughness measures were averaged for
each set of 10 parallel profiles (table 16). Autocorrelation
calculations for each of the roughness measures indicated
that adjacent profiles (spaced about 5 mm apart) were
independent of each other with regard to estimated
roughness. This confirmed carlier results regarding the
independence, variability, and uniqueness of adjacent
profiles, implying that it would be difficult to select one
single profile as representative of the entire fracture
surface.

DIRECT-SHEAR TESTING

The six specimens containing the natural fractures were
sheared with a direct-shear apparatus in SRC laboratories
at a rate of 0.3 mm/min, Applied normal stresses were
purposely kept low (j.e., less than 25 mt/m?) to avoid
extensive damage to fracture surfaces. Each specimen was
sheared at six normal loads in the x-direction and six in
the y-direction, Then one of the directions was selected
for shearing at slightly higher normal stresses (ie., up to
about 80 mt/m?).


the ranking obtained by the six roughness measures, after
which Kendall's nonparametric rank correlation test (42)
was applied. Results are tabulated in table 10.

Table 10.—Rank correlation coefficients of roughness
measures versus visual rank, gnelss
x- and y-direction series

D, 0, OD, Dd, A,  & Standard
deviation
x-direction -0.289 -0.511 -0.556 -0.911 0.911 -0.867 0.600

y-direction -244 -556 -.368 135 822 252378

These results were not as encouraging as those shown
for the x-direction profiles, and only A,, was highly cor-
related with the visual assessment.

Another way to compare the answers in the x- and y-
directions is to consider the 10 x-traces and the 10 y-traces
as populations and to use a statistical t-test to test for
equality of means, Table 11 summarizes the results and
demonstrates the anisotropic nature of the gneiss rock
sample,

Table 11.—Comparison of six roughness
measures using aspect ratio method
on gnelss specimen, x- versus
y-direction

tvalue!

Method

1.86
2.80
3.63
2.03

48
3.31

‘Degree of freedom are greater than 11 in
all cases.

Similar analyses were applied to the x- and y-directions
of a basalt sample and a quartzite sample. The anisotropy
shown in the gneiss sample was also apparent in both the
basalt and the quartzite samples.

In summary, the y-direction analysis of the gneiss
sample did not generate results that were consistent with
those obtained for the x-direction analysis. Furthermore,
the y-direction results would not lead to the same ap-
proach to shear strength modeling as used in the section
"Shear Strength Estimation." However, the data for the x-
and y-directions for the gneiss, basalt, and quartzite sam-
ples were fit reasonably well by the model developed in
that section. With regard to the most important conclu-
sion of this RI—the ambiguities of the fractal dimension

measures and their contrast with the statistic measures—
work with the gneiss y-profiles and the basalt and quartz-
ite x- and y-profiles further substantiated the ambiguity
premise.

ORIGINAL DATA SPACE ANALYSIS,
GNEISS x- AND y-DIRECTIONS

Brown’s aspect ratio method (11) was used in the
algorithms for the box, regular divider, and modified
divider computations. As stated earlier, not all researchers
are in favor of the aspect ratio technique. In this work,
the z-coordinates (trace heights) of the profile traces were
multiplied by 1,000 in computing the fractal dimensions for
the divider and modified divider methods. The aspect
ratio artifice was used in determining the number of
divisions of both axes when applying the box-counting
technique. The work that follows does not apply the
scaling factor of 1,000 to the divider methods. The
remaining fractal dimension algorithms (for D, and D,)
and statistical algorithms (Z, and standard deviation) were
computed as before. The goal was to find out whether
better results were obtained using original data space
information than were obtained using the scaling factor.
The ensuing correlation matrices are shown in table 12.

Table 12.—Correlation matrix for roughness measures using
original data space Input, gnelss x- and y-direction series

Dd, D, =D, 0, 2 Standard
deviation
GNEISS x-DIRECTION SERIES
7.000
034 1.000
+050 966 1,000
522-659-720 1,000
Z + 7288 933 925-828 1,000
Standard
deviation -719 554-534-832 _—=-792__—*1.000
GNEISS y-DIRECTION SERIES
7.000
.071 1.000
397 450 1,000
443-100 .270 1.000
561.367.860.432 1.000
deviation -72-.056-.551_-.139 —-.323 ‘1.000

The results in table 12 are to be compared with those
in tables 1 and 8, There is no marked improvement in
the correlations among the four fractal dimension algo-
rithms, although in table 1, D, and D,, are highly cor-
related (r = 0.966). But the data in table 12 show a low


APPENDIX B.—EVALUATION OF CLOSE-RANGE PHOTOGRAMMETRY

ACCURACY ANALYSIS
Accuracy Bar

The purpose of an accuracy bar (fig. B-1) is to test
photogrammetric results against a known standard. Know-
ing how photogrammetric results compare to a known
standard enables better characterization of the accuracy
of the complete system. The standard was a milled alu-
minum bar 10.2 cm long, 4 cm wide, and 2 to 2.5 cm high.
Four tapered holes were drilled to various depths in the
top. In addition, seven steps 2 to 4 mm high and 2 to
4 mm wide were cut into the surface to test how accurately
depth was measured when photogrammetry was used. The
test bar was photographed, and the photographs were dig-
itally processed by Spencer B. Gross, Inc., Portland, OR.
Each trace covered all the steps and one or more of the
holes. Between the traces, the traces covered all the holes.

The same traces were sent to the University of Idaho,
Moscow, ID, where they were measured with a mechan-
ical transducer and with a metroplate and measuring dial
with a long probe that penetrated to the bottom of the
tapered holes and steps. Plots of one of the three traces
depicting the three different measuring schemes are shown
in figure B-2.

It was necessary to use an appropriate statistical test
to determine if the three measuring processes were or

Figure B-1.—Accuracy bar in aluminum reference frame.
Trace 1 runs through middle of bar encompassing two large
holes and seven steps.

were not equivalent. As the traces in figure B-1 indicate,
it was not possible to match exactly the three reading
positions on the horizontal axis. Therefore, to make rea-
sonable comparisons of the three traces, the differences
between the heights (z-valucs) at designated abscissas were
compared (table B-1).

Table B-1.—Table of differences, millmeters

Photogrammetry Mechanical Metroplate
transducer
TRAGE 1
5.94 5.85 6.2
5.97 5.82 6.12
7.49 777 7.67
7.54 7.74 7.63
TRACE 2
0.02 0.02 0.005
1.30 1.41 1.4122
1.33 1.41 1.4046
Ot 03 0025
TRACE 3
1.88 2.03 2.027
1.85 1.95 1.973
1.87 2.0 4.991
7.54 7.74 7.63
1.83 1.99 1.987
17 1.99 2.006
1.05 1.08 1.046
10.79 11.19 11.153
2.87 2.73 2718
2.66 2.73 2.713

This process was possible even though the abscissa
points did not exactly match. This was because the accu-
racy bar is uniformly flat, except for the steps and the
drilled holes, and so the abscissa coordinates on a flat
surface did not matter when determining the differences
between measured heights of two differing-in-height flat
surfaces.

A randomized block design analysis-of-variance model
(14) was the statistical method used. Either by inspect-
ing the data or using the ensuing F-statistic, it can be seen
that the traces vary significantly from each other. The
point of interest is to compare the three measuring
procedures—photogrammetry, metroplate, and mechan-
ical transducer, The resulting sample F-statistic of com-
parison is F(2,39) = 3.04. At the 0.95 significance level,
the tabular F-statistic is 3.21. Because the sample statis-
tic is less than the critical level statistic, the conclusion
is that the three methods are not statistically different.
Thus, photogrammetry is an acceptable method of data
collection.


would be anticipated, and in fact, in three of the five cases,
high correlations were obtained.

Rank correlation coefficients (42) were computed com-
paring the visual assessments with the roughness measures.
Results are summarized in table 6,

Rank correlation coefficients of roughness
versus visual rank using aspect ratio
method, gneiss x-direction serles, y-coordinate

DB O&O, &
0.951 0.042 0.733 -0.697 0.867
335, 564 -.685 -418 822

Regular...
Enlarged ..

Both photographic methods generated similar results;
only the Z, measure correlated positively with the visual
rankings, while the four fractal methods correlated
negatively (in seven of the eight cases) with the visual
assessments.

Original Data Space Analysis

Only two measures, D,, and Z,, were used. For the
regular photographs, the correlation between D,, and Z,
was 0.945; for the enlarged photographs, the correlation
between D,, and Z, was 0.954. This is quite significant in
that the fractal measure was strongly positively correlated
with the Z, measure in contrast with results obtained by
comparing aspect ratios. The correlation for D,, (regular
versus enlarged data set) was 0.969, which was also a
better answer than the 0.777 obtained for the aspect ratio.
Correlations between D,, and the visual rankings were
0.764 (regular data) and 0.733 (enlarged data). Two major
conclusions were reached:

1. The regular and enlarged data sets behaved similarly
when using a scale factor of 1, and

2. A scale factor of 1 promoted better agreement
between D,, and Z, than did previous analyses of the
gneiss x-direction data. This point will be expanded upon
in the section "Gneiss y-Direction Analysis."

Thus, it was felt that the enlarged photographs gener-
ated results similar to those obtained from the regular-
sized photographs, and that using enlargements will not
produce more satisfying results.

FRACTAL DIMENSION AND ASPECT
RATIO CONCEPT

A perplexing problem facing users of the fractal dimen-
sion as a measure of profile roughness is the small magni-
tudes of the generated fractal dimension values. Using the
divider method, the authors generated a fractal dimension
of 1.3 for the coast of Britain and 1.52 for the coast of

Norway (20). Both coastlines represent highly perturbated
profiles, yet these fractal dimensions are rather modest.
In studying profiles taken from rock surfaces adjacent to
Libby Dam, MT, Carr and Warner (15) attained fractal
dimension values of 1.0031, 1.0032, 1.0220, etc. Applying
the divider technique to nonscaled profiles for the rock
surfaces studied here generated fractal dimensions similar
to those found by Carr and Warner. Furthermore, when
the divider technique was applied to fractal-generated
profiles of known dimensions (¢.g., 1.2, 1.3, 1.5), the fractal
dimensions gave results much less than these known
dimensions. The divider method is not unique with regard
to this shortcoming; the box-counting method presents
similar problems.

Two authors, Wong (46) and Brown (11), have recog-
nized and dealt with this problem, which is referred to as
the crossover length problem and is best defined in terms
of the basic divider method. There exists a critical divider
length (crossover length) such that if the chosen divider
radii are larger than the crossover length, the fractal
dimension will be very close to unity, Brown suggests
solutions for the crossover problem as follows:

1, For the divider method, multiply the z-coordinate
(only) by increasingly large values, ¢.g., 10, 100, 1,000...
If one does this, the fractal dimension will indeed asymp-
tote to a value that is not necessarily near unity, but this
value should be more representative of the true fractal
dimension.’ This transformation of the z-coordinate only
may result in an extremely exaggerated portrayal of the
original trace. Effectively, all variability has been trans-
ferred to the z-variable alone.

2. For the box-counting method, choose an equal num-
ber of intervals (for division into box size) along both the
x- and z-axes, Brown (11) develops the mathematics for
this option? Using Brown’s aspect ratio concept provides
excellent verification for generated profiles of known frac-
tal dimension, thus giving credence to the consistency of
this approach.

Several scientists question transforming a profile so
drastically in order to attain a consistent fractal dimension.
The authors of this RI used the aspect ratio concept
exclusively for two symposium papers (36-37). In these
Papers, the authors endeavored to look at the data with
and without using aspect ratio algorithms. At this junc-
ture, it seems unclear as to whether one should commit
unequivocally to either school of thought on the subject of
aspect ratio.

‘Brown's approach (17) circumvents one’s actually finding the critical
crossover length, which is certainly a useful technique,

"This approach provides a very pragmatic artifice for users of the box
‘method because determination of the box’s physical dimensions is critical
to generation of a "correct" fractal dimension. Feder (20, pp. 186-188)
demonstrates varying the fractal dimension from 1.03 to 1.51 based on
the dimensions of the boxes chosen.


HEIGHT, mm

Ic i] I | i] ! !

i | | | | |

10 15 20 25 30 35 40

45 50 55 60 65 70 75

HORIZONTAL DISTANCE, mm

Figure 1.—Examples of fracture roughness profiles. A, Basalt; B, gneiss; and C, quartzite.

approximately 0.052 mm, producing about 1,000 obser-
vations in the x-direction and 1,300 observations in the
y-direction.

Examples of the roughness profiles are shown in fig-
ure 1, In general, a visual assessment of the entire set of
profiles indicated that the quartzite fracture was the
smoothest of the three, followed by the gneiss fracture,
then the basalt fracture. Furthermore, some anisotropy
in roughness was observed, especially for the gneiss frac-
ture, where most of the x-profiles appeared smoother than
the y-profiles,

COMPUTATIONAL METHODS FOR ESTIMATING
FRACTAL DIMENSIONS

Of the four methods used to calculate fractal dimen-
sions of the roughness profiles, the divider and box meth-
ods rely on a deterministic approach, while the spectral
method relies on a stochastic approach. The deterministic
methods are illustrated in figure 2 and are described in
more detail below.

Divider Method

The divider, or yardstick, method is best visualized by
considering a pair of dividers set to a particular span and
then "walked" along the roughness profile. The number of
divider spans (yardsticks) required to cover the entire
profile is counted and then multiplied by the length of the
divider span to give an estimate of the profile length. The
divider span is set to another value, and the process is
repeated several times to produce a discrete relationship

between divider span and profile length. ‘The two are re-
lated linearly in log-log space according to the expression

log(L) = A + (1 - D) log(r),

where L = estimated profile length,
A = y-intercept,
and r = divider span [adapted from Feder (20)].

Thus, slope b of the log-log plot = 1 - D or D =1-b.
Modified Divider Method

A modified divider method proposed by Brown (12)
consists of using horizontal divider spans (x-increments)
rather than walking the dividers along the profile. Thus,
the incremental lengths or segments along the profile from
x, to x, + 1 are summed to obtain the total estimated
profile length for each given x-increment, which is denoted
as r for this case. The above log-log relationship holds,
where r = x-increment and D = 1 - b = slope of the
log-log plot.

Box Method

Rather than use divider spans as the counting in-
strument, the box method relies on small rectangular
boxes. A box size is selected, which establishes a regular
grid, and then the number of such boxes needed to cover


APPENDIX A.-PROCEDURES FOR CLOSE-RANGE PHOTOGRAMMETRY

SPECIMEN DESCRIPTION

For the photogrammetric work in the laboratory, three
different rock types were used: basalt, gneiss, and quartz-
ite. These types were chosen primarily because of conven-
ience; they are the most readily available rock types in the
area. Sedimentary rocks were excluded solely because
there are no sedimentary formations in northern Idaho
and eastern Washington.

Two specimens of each rock type were collected, each
containing a natural fracture. The specimens were then
trimmed to 8- by 9- by 3-cm blocks having 1.5 cm of intact
rock on either side of the fracture. This yielded 12 sam-
ples (6 pairs of mated halves). The 12 trimmed and frac-
tured specimens were then cast in quick-setting cement in
a 10- by 10-cm mold to produce samples for direct-shear
tests,

EQUIPMENT FOR CLOSE-RANGE
PHOTOGRAMMETRY

The use of close-range photogrammetry was suggested
by an article describing the successful use of this technique
to measure wear on asphalt roads in California (1, pp. 530-
551). Although the highway project involved the use of
two 35-mm cameras to take the photographs, only special-
ized photogrammetric cameras or cameras that had been
specially calibrated for photogrammetric work were con-
sidered because the Bureau project required more precise
measurements,

Two Rollei? single-lens reflex cameras with 80-mm Zeis
lenses and two 34-mm-long extension tubes (fig. A-1) were
selected. An 80-mm lens can focus to 1 m, which covers
a viewing area 50 by 50 cm. By using an extension tube,
the distance between the camera and the specimen could
be narrowed to 38.5 cm, covering an area of 10 by 10 cm
and enabling the image of the specimen to fill the frame.
The extension tube and the lens had to be calibrated
together at this distance.

To provide support for the two cameras, a 3-ft by 2- by
1/4-in aluminum bar was milled with a 1/4-in slot cut part
way down the middle; the cameras were then bolted to the
bar by 1/4-in screws passing through the slot in the bar to
a tripod-mounting hole on the cameras. The slot allowed
the distances between the cameras to be adjusted as
desired. The bar was then bolted to the top of the stur-
diest tripod available. The importance of the sturdiness of

{talic numbers in parentheses refer to items in the list of references
preceding this appendix.

?Reference to specific products does not imply endorsement by the
US. Bureau of Mines.

the tripod cannot be overemphasized. Any shaking in the
cameras when taking the pictures will result in blurry
photographs, which will compromise the accuracy of the
gathered data. A cable release is also recommended to
help minimize camera shake.

Sometimes small mounts containing rack-and-pinion
gears allowing movement in both the x- and y-directions
are helpful for fine tuning camera positions. The cameras
are first attached to these mounts and then bolted to the
aluminum bar. For photographing in the laboratory, a
second tripod is handy for holding the specimen and its
reference frame. The tripods should have a rack-and-
pinion shaft to enable fine adjustments. This permits
raising or lowering either the cameras or the specimen to
obtain the desired framing.

Figure A-1.Photogrammetric setup showing two Rollei
cameras mounted on support bar and rock specimen mounted in
aluminum reference frame.


nN [ay S
3 to) roy

SHEAR STRESS, mt/m2
3

i) 10 20 30 40
NORMAL STRESS, mt/m2

Figure 11.—Generated synthetic (solid line) versus actual
(dashed line) peak shear-strength envelopes for two natural rock
fractures, (A = basalt and B = gnelss.)

fractal dimension, fractal intercept, and the Z, measure.
‘Two important observations were made in the early stages
of the study:

1. Parallel profiles of fracture roughness are inde-
pendent of each other when spaced as closely as 5 mm
apart (implying the difficulty of selecting one particular
profile to represent the entire topography of a surface).
Even averaging the results from several parallel profiles
may not provide representative roughness measures for a
fracture surface.

2. Shear strength along natural fractures is often an
anisotropic property (ie., shear strength can depend on
direction of shearing).

Use of the average (ic., arithmetic mean), the median,
and the maximum roughness measures for a set of parallel
profiles was also investigated, but all yielded comparable
correlations with strength envelope parameters. In addi-
tion, none of these two-dimensional measures was deemed
entirely satisfactory as a representative roughness value for
a given fracture surface.

CONCLUSIONS

The photogrammetric method of collecting and digit-
izing information from rock surfaces proved to be quite
successful. Although not as precise as a mechanical pro-
filometer, photogrammetry is accurate enough for many
situations. Additional studies should involve large-scale
surfaces underground.

Photogrammetry equipment is field portable. Photo-
grammetry and subsequent stereo digitizing can produce
reasonable estimates of fracture roughness measures, as
well as permanent records of fracture surfaces that can
be readily profiled in any direction or topographically
mapped to generate detailed digital elevation models.
These data characteristics make it possible to analyze
three-dimensional properties of rock fracture roughness.
This analysis includes descriptions of anisotropy based on
two-dimensional covariance and/or spectral analysis or
on surface fractals rather than on profile fractals. Such
studies can eliminate the uncertainty and possible errors
in using only a few profiles (which often may not be rep-
resentative) to characterize the roughness of an entire
fracture surface.

A shear strength model was developed using the x-
direction data from the surface of a gneiss sample. It is
quite possible that a different model would have evolved
if both x- and y-direction input had been processed. In
spite of using only one direction, however, the resulting

shear strength model did a good job of curve fitting the
available shear strength data from 12 samples—4 gneiss,
4 basalt, and 4 quartzite. It was unfortunate that the
hardness parameter proved to be insignificant when the
available rock specimens were described; it was hypothe-
sized that both parameters—roughness and hardness—
would have generated a better model for shear strength
computations than the model in which only roughness was
used,

The major focus of this work was to find a single
parameter as a measure of the roughness of a rock frac-
ture surface. This effort led to detailed analyses of four
fractal dimension algorithms—box, divider, modified divid-
er, and spectral. Of these algorithms, the modified divider
was the easiest to implement and provided the best agree-
ment with generated profiles of known fractal dimension.
The spectral method proved to be least dependable. Both
the divider and box methods required parameter adjust-
ments for proper implementation. When applied to a
family of profiles, the intercorrelations between the four
fractal measures proved to be disappointing. Furthermore,
when compared to visual perceptions as to degree of
roughness for a family of profiles, there was again a lack
of consistency. The Z, measure was also computed
throughout the analysis, but Z, alone was not sufficient to
describe roughness. Although falling short of some of the


ESTIMATION OF SHEAR STRENGTH USING FRACTALS
AS A MEASURE OF ROCK FRACTURE ROUGHNESS

By P. C. McWilliams," J. C. Kerkering,? and S. M. Miller?

ABSTRACT

Researchers at the U.S, Bureau of Mines investigated the use of close-range photogrammetry and
subsequent stereo digitizing to obtain data from rock fracture roughness profiles. The photogrammetric
process yielded results that were acceptable but somewhat inferior to those obtained by a mechanical
profilometer. On the basis of this study, further pursuit of photogrammetry as a data collection method
in mining is proposed. Fractal geometry was investigated as a means of measuring the roughness of
rock fracture profiles. Four fractal algorithms were used: divider method, modified divider method,
box method, and spectral method. A comparison of the methods gave ambiguous results. Brown’s
modified divider method provided the best means of obtaining the fractal dimension. Shear strength
estimates were obtained using the parameters of the modified divider method and Myers’ Z, measure.
Because of differences in results when comparing the different ways of obtaining the fractal dimension,
future users of fractals in studies of rock fractures are advised to cross-check their results carefully.

1Mathematical statistician (retired), Spokane Research Center, U.S. Bureau of Mines, Spokane, WA.
?Mathematician, Spokane Research Center.
3Mining engineer, Spokane Research Center, and professor of geological engineering, University of Idaho, Moscow, ID.


cm

c/mm

mm

UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT

centimeter

cycle per millimeter
foot

inch

kilonewton

meter

millimeter

square millimeter
micrometer

millimeter per minute
metric ton per square meter
percent

second



EFFECTS OF PARAMETER INPUT
ON FRACTAL DIMENSION

The selection of parameter values has a marked effect
on the calculated fractal dimension. When using fractal
algorithms, the user must be aware that one cannot just
operate on automatic pilot and generate significant an-
swers. Two illustrations are provided, one relative to the
box algorithm, the other to the modified divider algorithm.

Parameter Effects—Box Algorithm
An artificially generated profile trace with 1,200 data

points and a known fractal dimension of 1.3 was used as
the example (fig. 6). The dependent variable had a mean

10

of 7.212 and a standard deviation of 1.295. The generation
routine was a spectral analysis synthesis method proposed
by Saupe (5, pp. 93-94). The advantage of this generated
profile was that the property of self-affinity was guar-
anteed. Visually, one would feel comfortable in assuming
that the trace was of a fractal nature.

The set of aspect ratio values that proved most suc-
cessful relative to the bulk of the experimental data was
used. The computer output is listed in table 7.

The fractal dimension 1.166 obtained earlier was cer-
tainly unsatisfactory relative to the expected value of 1.3.
However, if the user were to choose aspect ratio input
values running consecutively from 2 to 20, the generated
fractal dimension would be 1.335, with a correlation coef-
ficient value of 0.9965. This seems more in line with

HEIGHT, mm
~
T

40 60 80
HORIZONTAL DISTANCE, mm

Figure 6.—Generated fractal profile with fractal dimension of 1.3.


from several centimeters to many meters. Such technology
could provide a workable new approach to measuring rock
fracture roughness, particularly with regard to field situ-
ations where a wide range of fracture sizes are present.
Because this technique relies on photographic stereo pairs,
equipment is portable; a permanent record of the entire
area of interest on the fracture surface (not just selected
traces) is obtained; and the typically cumbersome tasks
of rock sample collection and/or instrument setup are
avoided. An additional benefit is that digitized elevations
can be obtained from a pair of photographs by using a
stereo analytic plotter in conjunction with a data acqui-
sition system and a microcomputer.

Thus, the preferred means of data collecting was close-
range photogrammetry. First, however, it was necessary to
determine that results from this method were as good or
better than those obtainable through the use of a mechan-
ical profilometer. To this end, a comparative trace study
and an accuracy and precision analysis were conducted.
Photogrammetry is more thoroughly discussed in ap-
pendixes A and B, Appendix A is a detailed description
of the procedures used in characterizing rock surfaces
using close-range photogrammetry. Appendix B is a com-
parative evaluation of the use of close-range photogram-
metry and a mechanical profilometer for data gathering,

FRACTAL GEOMETRY AND ROCK FRACTURE ROUGHNESS

INTRODUCTION

In deriving his Q-statistic for rock mass classification,
Barton (7) identified rock fracture roughness as an influ-
ential variable. A numerical description of the roughness
of a rock fracture surface is essential to the estimation
of shear strength, dilatancy, and stiffness of the fracture.
In this context, the term "fracture" refers to any semiplanar
discontinuity in a rock mass (e.g., joint, bedding plane,
fault), In engineering practice, the most commonly used
measure of roughness is the joint roughness coefficient
(IRC) proposed by Barton (7) and adopted by the Inter-
national Society for Rock Mechanics (26). This coefficient
ranges in value from 0 to 20 and is estimated either by
visual matching of surface profiles with "standard" profiles
(fig. 1) or by back calculation using peak shear strength
and basic friction angle (obtained from direct-shear tests)
in conjunction with joint-wall compressive strength. The
first approach is highly subjective, while the second has
little practical merit because roughness should preferably
be used to predict shear strength, not vice versa.

Since Barton’s early work (7), other scientists have
addressed the problem of quantifying his subjective rough-
ness profiles. Ferrero and Giani (27) used a geostatistical
approach in which variograms were employed as a means
of expressing Barton’s profiles. Lee, Carr, Barr, and Haas
(31) used the divider method to establish corresponding
fractal dimensions for Barton’s 10 profiles. The JRC was
then expressed as a second-degree polynomial with the
fractal dimension as the independent variable.

At the time this project was started, fractal geometry
(32) was considered to be a most promising way of de-
scribing geometric configurations. In recent years, inves-
tigators have attempted to apply fractal geometry to objec-
tive descriptions of rock fracture surfaces (3, 10, 15). Such
work relies on the assumption that natural rock fracture
surfaces can be represented by self-affine fractal models,

which seem to be more generally applicable to geologic
phenomena than self-similar fractal models. In essence, a
self-similar fractal is a geometric feature that retains its
statistical properties (statistical moments, to be more
precise) through various magnifications. That is, the visual
and statistical appearances of the feature are similar at
all scales of magnification. In contrast, self-affine fractals
remain statistically similar only if they are scaled differ-
ently in different directions. Examples include the x- or
y-coordinate of two-dimensional Brownian motion plotted
as a function of time, and surface roughness profiles in
which elevation is plotted as a function of horizontal
distance (12), The fractal dimension D of such self-affine
fractals has a value between the topological dimensions of
1 and 2; values of 1.0 to 1.5 are commonly obtained for
rock fracture profiles. Thus, D is potentially an appealing
measure of surface roughness because it is a single value
that is independent of scale.

A literature search revealed four common methods
for calculating fractal dimensions of roughness profiles:
divider, or yardstick, method; modified divider method;
box method; and spectral method. To evaluate these
methods, roughness profiles were obtained from three
natural fractures, each in a different rock type, using
close-range photogrammetry and stereo digitizing. Fractal
dimensions and other roughness measures were computed
for the profiles and then evaluated and compared.

ROUGHNESS PROFILES

The three rock types studied were basalt, gneiss, and
quartzite. The rock specimens were approximately 8 cm
on a side, making the roughness profiles 6 to 7 cm long.
Ten parallel roughness profiles were digitized in cach of
two directions, identified as the x- and y-directions. Thus,
20 profiles were obtained from each specimen, making a
total of 60 profiles, The regular digitizing interval was


HEIGHT, cm
a

oo

a

o°0

T T T T T T T T T
—
! ! ! L i ! ! ! L
T T T T T T T T T
~ =
1 i ! i ! i ! | !
T T T T T T T T T
=U _|
i it l i l 1
T I I T T T
! \ ! \ N ! L L L
T T T T T T T T T
_S |
Nl ! ! ! L L ! L !
T T T T T T T T T
eee SA
| I i | L 1 1 | !
T T T T T T T T T
L | L L | ! L | i
T T T T I I T T TI
L | | L L l I l |
! Nl ! ! \ \ ! ! !
10 20 30 40 50 60 70 80 90 100
HORIZONTAL DISTANCE, cm

Figure 9.—Noranda profiles used In analysis. (Profiles were unidentified.)


REPORT OF INVESTIGATIONS/1993

Estimation of Shear Strength Using Fractals
as a Measure of Rock Fracture Roughness

By P. C. McWilliams, J. C. Kerkering, and S. M. Miller



the box method, an equal number of box divisions were
used in the x- and y-directions to ensure an aspect ratio
equal to that of the profile. Analyses were also done with
divider methods using no exaggeration of the y-coordinate
values.

Spectral Method

The fractal dimension of a surface roughness profile
can be related to the power spectral density S(F) of the
profile, if one assumes a spectral density with power-law
form, where S(F)«F* and F = frequency in units of cycles
per millimeter. In this case, D is related to slope b of the
spectral density (13) according to b = 2D - 5.

Caution must be exercised when interpreting fractal
results obtained from the spectrum, however, because
assumptions of stationarity, ergodicity, and random phase
may not be met by the profiles being studied (25).

The spectral density of each roughness profile was
estimated using the fast Fourier transform method as dis-
cussed by Bendat and Piersol (9). This approach requires
the investigator to select a tapering window and a spectral
smoothing algorithm.

APPLICATIONS OF COMPUTER
ALGORITHMS TO DATA

In applying the four computer algorithms—box, divider,
modified divider, and spectral—it became apparent that

there were difficulties specific to each method. For ex-
ample, when using Brown’s aspect ratio concept (11) with
the box method, an equal number of divisions are made
along both axes of the original profile data. The fractal
dimension is computed from a derived log-log relationship,
where the x-axis is the log of the aspect ratio and the
y-axis is the log of the number of boxes intersected by the
profile. The slope of a line fit to this log-log plot is the
fractal dimension value.

Inherent in this method is the hypothesis that a straight
line is a good model for the log-log data (fig. 3). How-
ever, there are at least three plausible but very different
answers for the fractal dimension, varying from 0.94 to
1.19, depending on which set of box sizes the user chooses.

When using both divider methods, the procedures are
similar. The x-axis of the derived plot equals the log of
the divider span; the y-axis is the log of the corresponding
profile length; and the fractal dimension equals 1 - slope
of the log-log plot. In both methods, the log-log plots are
similar in ambiguity, and the fractal dimensions can vary
significantly, depending on the set of divider spans spec-
ified for measuring the profiles,

Even if a roughness profile reasonably meets station-
arity and random-phase assumptions, fractal dimensions
estimated using the spectral method can depend signif-
icantly on user specifications for obtaining the spectral
density of the profile. The two most apparent specifi-
cations are the tapering window used to mitigate leakage
and the algorithm used to smooth the spectral estimates

v
io
T

LOG (NUMBER OF BOXES HIT)
g ie
T T

b

24 T 1 T
D (points 1-20) = 1.11
e D (points 14-20) = 0.94
D (points 1-7) = 1.19
| i 1 i
0.8 1.0 1.2 1.4 1.6 18 2.0

___ LOG (NUMBER OF BOX DIVISIONS)

Figure 3.—Log-log determination of fractal dimension, box method,


TRACE HEIGHT, mm
a
T

Modified divider method
T=Du 4

(4
|
| I
| Lo!
2
I iy ts
4k I ji 1 | 4
| po Pop tras!
ptr ty
Box method | Divider method ] 1d |
(count number of boxes I B= (divider length) x I I Eaualy spores | |
covering trace) | __ (number of intersections) | toy |
3 \ Lt \ too ty!
5 15 25 35 45 55 65

HORIZONTAL DISTANCE, mm

Figure 2.—Box and divider methods for estimating fractal dimenstons. (L = estimated profile length.)

the entire profile is counted. Two versions of the box
method were used in this study, one taken from Brown
(11) and one from Feder (20). Brown’s approach requires
that the box grid be formed by taking an equal number of
divisions in the x- and y-directions, which maintains an
aspect ratio consistent with that of the profile, and then
determining the relationship between the number of
divisions and the number of boxes. In Feder’s approach,
the relationship between a box-size multiplier and the
number of boxes required to cover the profile is deter-
mined. It was found that both methods provide similar
estimates of D if the specified box sizes are comparable.
For computational convenience, Brown’s method was
preferred, where D is estimated by

log(N) = A + D-log(g),

where N = number of boxes required to cover the
profile,
and g = number of box grid divisions in the x- and

y-directions.

In this case, D = slope of the log-log plot.

In using the divider and box methods described above,
appropriate adjustments were made to mitigate the cross-
over length problem associated with roughness profiles, as
discussed by Brown (12). For the divider methods, the
y-coordinate of the profiles was multiplied by 1,000. For


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Paper in Proceedings



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INTRODUCTION

As part of its continuing efforts to characterize rock
masses for enhanced design of mine openings, researchers
at the U.S. Bureau of Mines studied rock fracture rough-
ness and its influence on structural behavior. The original
goal of this endeavor was to predict the shear strength
of natural rock fractures as a function of two variables—
fracture roughness and fracture hardness. As is often the
case in research, two topics were of special significance to
the Bureau researchers.

First, close-range photogrammetry and subsequent
stereo digitizing were used as the data-collecting method.
This method was contrasted with the more conventional
approach of using a mechanical profilometer to collect in-
formation about a rock surface profile. The primary moti-
vation for using photogrammetry was that large amounts
of information are required for the work. In addition,
photogrammetry has potential as a data-gathering proce-
dure in the field, both above and below ground.

Second, fractal geometry [see Mandelbrot report (33)
was proposed as a means of determining the roughness
of rock surface profiles. It was hoped that by using frac-
tal geometry, investigators could move away from the
qualitative and subjective techniques presently used to
obtain rock surface profiles, which originated with Barton
(7-8) in 1973. However, the fractal methods did not prove
to be a panacea for the problems encountered in quan-
tifying roughness; rather, the ambiguities of the fractal
algorithms became a topic of considerable research in-
terest. Furthermore, since the inception of this work, the
scientific community's attitude toward applicability of
fractal methods has varied considerably and currently tilts
in a more skeptical direction.

This Report of Investigations (RI) endeavors to address
three directly related topics: the photogrammetric data
collection process, use of fractal geometry to describe sur-
face roughness, and a procedure to estimate rock shear
strength using roughness and hardness as input variables.

ACKNOWLEDGMENTS

The authors wish to thank the following people for
their contributions to this RI: William Durham of Law-
rence Livermore Laboratories, Livermore, CA, for con-
ducting precise profilometer measurements that were
used in evaluating photogrammetry as a data-collecting
technique; Paul Germain of the Centre de Technologie

Noranda, Pointe Claire, QU, for providing additional rock
roughness data that were used in evaluating fractal com-
putation methods; and Steven Brown of Sandia Lab-
oratories, Sandia, NM, for sharing his insights into the
nature of fractals and methods of computing the fractal
dimension.

PHOTOGRAMMETRY

This project required a large number of x, y, z meas-
urements* on the natural fracture surfaces of rocks so that
a quantitative roughness measurement for a given surface
could be formulated. Traditionally, the way to generate
data for these measurements has been to use a mechanical
profilometer. This instrument has a stylus that runs on x-y
axes and measures the height of the surface along a given
xy profile, Height readings along the profile are taken at
prescribed intervals (usually every 0.05 mm) by a probe
that registers differences in voltage. Such an instrument
can be quite accurate and reliable.

The mechanical profilometer, however, does have limi-
tations. First, it is rather large, and therefore it is not eas-
ily portable. Second, it requires a reliable power source.
Third, it can usually measure only small specimens (up to
a maximum of a foot square). Fourth, the rock surface
must be small enough and of such a shape to fit on the
profilometer bed.

These requirements make a typical mechanical pro-
filometer cumbersome for field work, where the fracture

surfaces must be measured in situ. Because some faces
are up to 4 m across and can be either vertical or inclined,
it is difficult to attach a mechanical profilometer. Con-
sequently, an alternative data-recording device was sought
that would be portable and yet would be as accurate as a
mechanical profilometer. For these reasons, photogram-
metry was investigated as being more usable than a pro-
filometer for gathering data on rock fracture surfaces.
Advances in computers and photogrammetry during the
last decade now make it possible to collect, reduce, and
display digital elevation data efficiently at distances ranging

“Italic numbers in parentheses refer to items in the list of references
preceding the appendixes at the end of this report.
roughout this report, the z-axis represents the height of a data
point with a base value in the x-y plane. Thus, all two-dimensional
profiles have the z-coordinate as the dependent variable and cither the
x-coordinate or the y-coordinate as the independent variable. All
profiles are read perpendicular to either the x-axis or the y-axis, In
three dimensions, z = f(xy).


HEIGHT, mm

ORAM WATAN WATAN WATOAN WATOAN WROD WAHNOAN WHRUD WERUD WATON
Tet
7

40
DISTANCE, mm

Figure 8.—Ten gneiss profiles (traces A-J) oriented in y-direction.

aa]
1 1 1 1 L 1 a
iid T T T T T
_ __—e ee cw a
[G 1 1 1 i I 1 =
a T T T T T
a —“— a
H 1 1 1 1 l 1
20 30 50 60 70 80


HEIGHT, mm

OANO© AN WAN WAN AN AHN -HH4NO wan

34 49 64

79 4

19 34 49 64 79

X-DIRECTION, mm

Figure B-4.—Roughness profiles for rock fracture surfaces at 2.5 times vertical exaggeration. Profiles
1-4 are In x-direction and profiles 5-8 are in y-direction. A, Profilometer results; B, photogrammetry results,

repeatability of the mechanical profilometer was approx-
imately twice as good as that of the stereo digitizer. The
quartile values and the relatively large standard deviations
are representative of probability distributions skewed to
the right.

Table B-3.—Summary statistics for absolute deviations
in measured elevations obtained by repeating
digitization of profile 1, millimeters

Method Mean Standard Lower Median Upper

deviation quartile quartile
Profilometer.. 0.012 0.013 0.0042 0.0082 0.0182
Stereo digitizer 028.033 .0087_—=0157 0387,

The upper quartiles indicate that 75 pet of the absolute
deviations from the mechanical profilometer were less than
0.0132 mm, whereas 75 pct of the absolute deviations from
the stereo digitizer were less than 0.0357 mm. Thus, con-
servative estimates of measurement errors for the eleva-
tion data were approximately +0.02 mm for the profilom-
eter and +0.04 mm for the stereo digitizer.

Comparisons of Roughness Data

One way of comparing the profile data obtained by the
two different methods was to use a matched-pairs t-test on
elevation data recorded along the same profile. However,
because of the slightly different digitization interval used


in the two methods (0.002 in for the profilometer versus
0.05 mm for the stereo digitizer), it was impossible to
match up any two corresponding points precisely, Further-
more, an inspection of the original profile data files
showed that for any given profile, the horizontal increment
was not a constant, but deviated slightly from the specified
spacing. Thus, for comparative purposes, local means
were computed in nonoverlapping neighborhoods (cells)
approximately 0.6 mm wide, each of which contained about
12 original observations. Although this procedure pro-
vided smoothed elevation values, the new set of roughness
observations was more compatible with the basic statistical
assumptions of independence and normality upon which
matched-pairs t-tests are based. In contrast, neither as-
sumption was very well satisfied by the original roughness
measurements; only three of the eight profiles had eleva-
tion data that approximated a normal distribution, and all
profiles contained spatially dependent elevation data.

The matched-pairs t-test was conducted according to
guidelines from Dixon and Massey (18, pp. 119-121). The
value of the t-statistic is computed as follows:

t-M-9

ol

where d = arithmetic mean of paired differences
A
- 4 > I@ - %)il,
s = standard deviation of paired differences,
and n = number of paired differences (number of

local means).

The null hypothesis, that the compared profiles were
the same, would have been rejected if the calculated
t-value exceeded 1.99, which is the critical value for typ-
ical n values at a significance level of 0.05. For each
t-calculation, the corresponding probability of obtaining
at least the absolute value of the t-statistic was also cal-
culated. A high-probability value implied that the paired
profiles were very similar.

Results from the matched-pairs analyses are given in
table B-4 and indicate that for all eight profiles, there is
negligible difference between the use of the profilometer
or the stereo digitizer with regard to roughness data ex-
pressed as local means. This finding was further con-
firmed by linear regressions of the paired profile data,
which yielded correlation coefficients greater than 0.98 in
all cases. The repeatability trials shown in the last two
rows of table B-4 imply that the stereo digitizer was better

than the profilometer at reproducing consistent values of
local means of the elevations along a profile.

Table B-4.—Matched-pairs comparisons of local means
for roughness data obtained by mechanical
profilometer (MP) and stereo digitizer (SD)

Matched Calculated Probability Linear correlation

Profiles t-statistic value coefficient
MP1-SD1 0.02 0.9839, 0.9928
MP2-SD2 =04 9657 9810
MP3-SD3. 04 9710 9980
MP4-SD4 04 9680 9840
MPS5-SD5 03 9800 9929
MP6-SD6 05 9639 9946
MP7-SD7 04 9644 9910
MP8-SD8 ° 9998 9926
MP1A-MP1B 13 8955 9999
SD1A-SD1B 04 9719 9985

To evaluate the influence of the size of the averaging
window, the interval width was varied from about 0.4 mm
(8 observations per cell) to 1.2 mm (24 observations per
cell). The resulting t-values were small and did not sig-
nificantly differ from those presented in table B-4.

A two-sample variance test (0, versus o,) based on an
F-statistic (14, pp. 285-286) also was applied to the
roughness profiles. The null hypothesis would have been
rejected if the calculated value (F = 0,/a,) exceeded the
critical F-value, which equaled 1.12 for n values typical
of these profiles at a significance level of 0.05.

The variances and calculated F-value for the detrended
data (table B-5) indicated that (1) variances from paired
profiles were similar, but not strongly so; (2) roughness
data generated by the stereo digitizer generally had
smaller variances than data from the profilometer; (3) ani-
sotropy was apparent in the fracture roughness, and the
x-direction profiles were consistently rougher (i.c., higher
variance) than the y-direction profiles; and (4) detrended
roughness data provided a more reasonable basis for com-
paring profiles than did the raw data because variances in
the raw data were influenced by trends.

Comparisons Based on Derived
Roughness Measures

An empirical equation developed by Tse and Cruden
(44) was applied to the profile data to estimate values of
joint roughness coefficient (JRC). This procedure is based
on the Z, measure (root mean square of the first deriva-
tive of the profile) (38) and relies on a digitization interval
of 1.27 mm. To meet the interval requirement a cubic
spline interpolation scheme was used to generate new
elevation values at a constant spacing of 1.27 mm, which
produced profiles with approximately 50 observations each.
After computing the Z, for cach profile, the empirical


le 16.—Summary of averaged roughness measures and results

of direct-shear tests for six fractures
‘Averaged roughness Direct-shear tests,

Fracture Direction measures peak strength envelope!
D _ Intercept A A B e

BASALT
x 1.228 3.917 0.338 «2.1060 0.9119 58.3
y 1.286 3.953278 1,7661 «965658.
x 1.276 3.753451 1.5086 8814. 47.1
y 1.168 3.926.257 __—*'1.7591__—«8590 48.0

GNEISS
1 x 1.185 «3.632 (0.153.«1.9481 0.7139 40.4
1 y 1.211 3.686143 -2.1583 7393 45.1
2 x 1.208 3.728 = 258 1.6723 7827 43.1
2. y 1.168 3.709 —.143—-2.0555 7842 48.9

QUARTZITE

1 x 1212 3.582 0.149 2.2478 0.7277 44.9
1 y 1146 3.728.153 2.2544 7606 47.9
2 x 1.254 3.859327 1.9678 8073 48.0
2 y 1.154 3.854.196 —-2.2224 8083 51.8

‘A,B Curve parameters.
D Fractal dimension,

1Power model is r = Ao3; friction angle ¢° is based on linear envelope.

Output from a shear test can be displayed as a graph
with shear load plotted against shear displacement
(fig. 104). Using the peak shear strength observed for
each trace, the laboratory data were reduced to provide
estimates of shear-strength t as a function of normal stress
2, (fig. 10B). In most situations, a nonlinear power curve
provided a better model for describing the shear-strength
envelope than did a linear model. This result concurs with
observations previously published by Jaeger (28) and
Miller and Borgman (35). Shearing direction had a signif-
icant influence on shear strength in the gneiss and quartz-
ite samples. Results of the laboratory testing and model
fitting are summarized in table 16.

As part of the laboratory work, a Schmidt L-hammer
was used to take several measurements of hardness (i.c.,
joint wall compressive strength) on the surface of each of
the fractures. These measurements then were averaged
for each fracture and compared across the three rock
types. Because there was so little variability among the
results (values ranged from 36 to 44 on the hardness
index), it was concluded that hardness could be considered
uniform for all these specimens and could be ignored in
subsequent analyses aimed at predicting shear strength.
Thus, fracture roughness in conjunction with applied nor-
mal stress seemed to be the principal input needed to esti-
mate peak shear-strength envelopes for such fractures.

SHEAR STRENGTH ESTIMATES
FROM ROUGHNESS MEASURES

The initial goal of this project was to predict shear-
strength envelopes as a function of roughness, hardness,
and applied normal stress. As discussed above, fracture
hardness was dismissed as having any significant influence
on the specimens being studied here. This left three
roughness measures to be considered: the fractal dimen-
sion (obtained from the slope of the log-log fractal plot),
the y-intercept from the log-log fractal plot, and the Z,
measure.

To investigate basic relationships between each rough-
ness measure and each of the nonlinear shear-strength
envelopes (estimated from the direct-shear tests), cor-
relation coefficients were computed and then correlation
matrix (table 3) was constructed based on the 12 direct-
shear tests of the 6 fractures (tables 10 and 16). These
correlation values indicated that the roughness measures
significantly influenced the shape parameters of the
strength envelopes (fig. 10) and that D and the Z, measure
were significantly negatively correlated. (Recall that a
correlation coefficient of 1.0 indicates perfect positive
correlation, while -1.0 indicates perfect negative correlation
ina linear sense.) If the median or the maximum value
of the roughness measures were used for each set of


CONTENTS

Abstract ....
Introduction
Acknowledgments
Photogrammetry . .
Fractal geometry and rock fracture roughness.
Introduction tee
Roughness profiles . .
Computational methods for estimating fractal dimensions
Divider method ....... :
Modified divider method .
Box method .
Spectral method .
Applications of computer algorithms to data
Analysis of roughness measures .........
Comparison of adjacent profiles ...
Correlation of roughness measures
Evaluation of roughness measures using visual assessments of roughness .
Additional topics on fractals as a measure of rock fracture roughness .
Introduction
Effect of enlarged photography .
Aspect ratio analysis .
Original data space analysis .
Fractal dimension and aspect ratio concept
Effects of parameter input on fractal dimension
Parameter effects—box algorithm ........
Modified divider algorithm and scaling factor variations .
Gneiss y-direction analysis 2.2.0.0... 00. eceeeeeeeee bee
Original data space analysis, gneiss x- and y-directions .
Fractal algorithms applied to Noranda data set
Fractal controversy . .
Continuing research .
Shear strength estimation .
Introduction .......
Direct-shear testing .
Shear strength estimates from roughness measures
Conclusions os
References ........ see : :
Appendix A.—Procedures for close-range photogrammetry
Appendix B.—Evaluation of close-range photogrammetry . ..

ILLUSTRATIONS

Examples of fracture roughness profiles ............0.006
Box and divider methods for estimating fractal dimensions ...
Log-log determination of fractal dimension, box method .
Examples of log-log spectral plots
Ten gneiss profiles oriented in x-direction
Generated fractal profile with fractal dimension of 13 .
Effect of one data point on calculated fractal dimension .
. Ten gneiss profiles oriented in y-direction
Noranda profiles used in analysis

PEIAWEY NE

Page

BERBER ER commana aa avwunnne


Surprisingly, the D,-standard deviation and the D,-Z,
correlations were negative. This finding contradicts theo-
retical relationships where variance of heights is the Oth
moment of the spectrum and variance of slopes is the
2d moment of the spectrum (13). Because computer-
generated synthetic profiles produced correlations that
agreed with the stated theory, the above contradictions
may be caused by the scale dependence of Z, and standard
deviation and/or by the failure of the roughness profiles to
be modeled adequately by random processes.

Evaluation of Roughness Measures Using
Visual Assessments of Roughness

Because the two different sets of roughness measures
appeared to be so different when applied to these rock

fractures, the measures were evaluated using visual obser-
vations of the roughness profiles. The 10 profiles for the
gneiss x-direction series are shown in figure 5.

To evaluate the performance of each roughness meas-
ure, four investigators were asked to independently rank
the roughness of profiles in each series on a scale of 1
to 10 (1 was smoothest and 10 was roughest) according
to visual observations. These roughness rankings were
tabulated with rankings obtained using the six roughness
measures, plus another parameter, A,,, which is equal to
the y-intercept of the log-log plot of the modified divider
algorithm. These numbers were used to compute values
of Kendall’s nonparametric rank correlation coefficient
(42). Typical rank correlation results are summarized in
table 2 and illustrate the general conclusion that calculated
fractal dimensions can be, and often are, negatively cor-
related with visual rankings of roughness profiles.

Table 2.Roughness measures, gneiss x-direction series

DB Dy dD, O0, AQ Z, Standard Visual
deviation _rank
CORRELATION MATRIX
7.000
551 1,000
580 857 1.000
522.793 746 1,000
Zo 288-848-677 -.828 1.000
Standard deviation =719  -859 -788 832 792 1,000
‘SUMMARY OF PROFILES
Profile:
A. 1,099 1.161 1.129 (1.258 «3.569 (0.171 0.201 6
B. 1.158 1.192 1.123 1.246 3.546.169 199 5
c. 1.118 1.193 1.157 1.202 3.592.186 268 7
D. 1.159 1.094 1.103 1.191 3813 282 385 9
E. 1.143 1.261 1.199 1.383 3.377.131 118 1
F. 1.122 1155 1.128 1,309 3.495.149 202 2
G. 1.196 1.182 1.200 1.263 3.573.179 238 4
H. 1.146 1.183 1.147 1.274 3.536.161 2 3
1 1.076 14105 1.103 1.225 3731 213, 479 8
J 1.041 1.084 1.058 1.124 3816 245 618 10
Rank correlation coefficients
versus visual rank ...... 1289-511 -556 911911867 600



ADDITIONAL TOPICS ON FRACTALS AS A MEASURE
OF ROCK FRACTURE ROUGHNESS

INTRODUCTION

The work on shear strength used the gneiss specimen.
in the x-direction only (34). The work described in the
following sections applies the fractal geometry approach to
the gneiss y-direction and the quartzite and basalt x- and
y-directions also. There were several topics investigated,
including the effects of using enlarged photographs, the
aspect ratio concept (12), the separate data set supplied
by the Centre de Technologie Noranda, the use of pro-
files versus three-dimensional modeling, and the fractal
controversy.

EFFECT OF ENLARGED PHOTOGRAPHY

One possible source of error in the photogrammetric
studies was the physical size of the photographs being
read and digitized. Since the scanning dot size (0.05 mm)
was quite large relative to the interval between data points,
it was suggested that enlarging the photographs of the
samples might provide more representative results. Thus,
the photographs were enlarged 2,44 times. Care was taken
to align traces exactly. For this work, gneiss x-direction
readings were used, So that comparisons could be made,
it was necessary to truncate the original teadings to a
smaller set of information (80 pet of original trace length);
thus correlation values differed from those discussed in
the section "Analysis of Roughness Measures" (compare
table 3 with table 4). As before, 10 profile traces were
analyzed to generate the set of derived fractal dimensions
and Z, measures. The y-coordinates were scaled two
ways: with Brown’s aspect ratio (71) and with a scale fac-
tor equaling 1. The latter case is denoted as the original
data space analysis throughout the rest of this RI. "Reg-
ular" refers to the standard-sized photographs while “en-
larged" refers to the enlarged photographs.

Aspect Ratio Analysis

As explained earlier, linear correlation coefficients were
computed for the four roughness measures and for the Z,
measure (table 4). It can be seen that the same basic
trends hold: that the fractal measures were not strongly
correlated with each other and were, with one exception,
negatively correlated with the Z, measure. The fractal
measures were slightly better self-correlated for the data
set from the enlarged photographs than for the data set
from the regular photographs, while the set of enlarged
profile traces generated more robust fractal dimension
values (ie., higher mean value and greater variation). This

was true when using the box, divider, and modified divider
methods. However, the spectral method did not display
this robust quality; here, the fractal dimension values from
the regular photographs had a higher mean and greater
variation.

Table 3.—Correlation matrix for roughness
shape parameters for
nonlinear, shear-strength envelope

A B D t z
7.000

-489 1,000

637-578 1,000

-467 785 -.267 1.000
775.678 704 534 1,000

Table 4.—Correlation matrix for roughness
measures using aspect ratio method,
gneiss x-direction series

Bo 0, 0%

REGULAR PHOTOGRAPHS:

D,.... 1.000

Dg 273 1,000

Da 471.827 1.000

D, 237° 846588 1,000

Zz 051-882-608 -.822 1.000
ENLARGED PHOTOGRAPHS

D,.-.- 1.000

Dg 665 1,000

Da, 695 817 1.000

D, 672 822.590 1.000

ZB +522 742-390-919 1.000

Another comparison of interest was the correlation
between the regular and enlarged results for the 10 profile
traces analyzed by the five measures of roughness. The
results are shown in table 5.

Table 5—Roughness measure correlations
between regular and enlarged data set

Method Correlation coefficient
o,. 0.584
Dy 980
Da 77
D, 935
Z 976

If one were to hypothesize similarities between the reg-
ular and enlarged photographs, high correlation values


Library of Congress Cataloging in Publication Data:

McWilliams, P. C. (Paul C.)
Estimation of shear strength using fractals as a measure of rock fracture rough-
ness / by P.C. McWilliams, J.C. Kerkering, and S.M. Miller.
Pp. cm. — (Report of investigations; 9447)
Includes bibliographical references (p. 25).
Supt. of Docs. no.: I 28.23:9447.
1. Rocks—Fracture. 2. Rock mechanics. I. Kerkering, J. C. (John C).

IL Miller, S. M. (Stanley M.). III. Title. IV. Series: Report of investigations
(United States. Bureau of Mines); 9447,

TN23.U43 [TA706] 622 s—de20 [622°.28]  92-23708 CIP



Table B-2.—Summary statistics comparing an original
profile (GHu) to its enlargement (GEXJ)

GHI GEXJ
Data points in sample 873 896
1 75146 76746
564692 588993
75103 767031
Variance number? 564045 588336
Skewness .... 134612 129583

lEstimators for population using (n - 1).
Estimators for sample using n.

COMPARISON OF PROFILOMETER
AND PHOTOGRAMMETRY

For the initial investigation of photogrammetry, it was
necessary to evaluate its usefulness and effectiveness
as compared to more traditional ways of collecting data.
To this end, a test procedure was set up and arrangements
were made for Lawrence Livermore Laboratories of Liv-
ermore, CA, to measure profiles of a given sample. The
profiles were then remeasured using the photogrammetric
technique and the results from the two methods were
compared.

Data Collection

For the comparative study, an irregularly shaped, silty
quartzite specimen containing a natural fracture and
trimmed to 8 by 9 by 3 cm was chosen. To facilitate the
profilometric and photogrammetric studies (as well as
future direct-shear tests), each half of the specimen was
cast in quick-setting cement in a 10- by 10-cm mold,
leaving an exposed fracture surface that could be meas-
ured for surface roughness. Only one of these fracture
surfaces was analyzed during the study.

Profilometric Measurements

William Durham, a rock mechanics researcher who has
been studying rock fracture topography as part of a larger
investigation in fluid flow mechanics in fractured rock,
supervised the profilometric measurements at Lawrence
Livermore Laboratories. Technicians profiled four par-
allel traces spaced 2 cm apart in a defined x-direction
(width) and four in a corresponding y-direction (length)
using a profilometer. The tip of the profilometer stylus is
semirounded and has a nominal width of approximately
0.0015 in. The tip leaves a faint "scribe" line on the frac-
ture surface as each trace is profiled. These marks were
used to align and register the stereo plotter so that the
same cight traces could be digitized with photogrammetric
methods.

Elevation measurements were taken at increments of
0.002 in (0.0508 mm) along each of the eight traces. The
horizontal distance was recorded in inches, while the
elevations were recorded as voltage generated by the
stylus. Both types of measurements were subsequently
converted to millimeters. Traces 1 to 4 in the x-direction
were approximately 60 mm long, while traces 5 to 8 in the
y-direction were approximately 70 mm long. The resulting
profiles are presented in figure B-4.

Photogrammetry and Stereo Digitizing

The rock specimen was photographed at the Bureau’s
Spokane Research Center (SRC) under strict photogram-
metric guidelines. The two cameras were aimed at the
fracture surface at a convergence angle of 9.5° for each
camera. The base distance between the midpoints of the
two film planes was 128 mm. The perpendicular distance
from each film plane to the center of the fracture surface
was 385 mm, while the perpendicular distance from the
baseline to the fracture surface was 380 mm. The 34-mm
extension tubes made it possible to photograph the rock
fracture surface at close range for maximum image size
and to focus the lenses at infinity for concurrence with
photogrammetric calibration factors. ‘The exposure time
was 1/125 s at f16.

The two sets of profiles (fig. B-4) had different starting
coordinates arbitrarily located along a horizontal reference.
To align, or register, a given pair of profiles, the horizontal
coordinate corresponding to the minimum elevation on the
stereo-digitized profile was matched to the same location
on the profilometric profile. A new arbitrary horizontal
axis was then defined. Using the new axis, the profiles
were truncated, if necessary, so that each set of matched
profiles had approximately the same beginning and ending
points in the horizontal plane. Profiles were registered in
the vertical plane by subtracting the mean elevation of a
given profile from each of its recorded elevations and
adding 5 mm. This resulted in all profiles being centered
about an arbitrarily defined mean elevation of 5 mm.

Measurement Repeatability

During the profilometer exercise, profile 1 was digitized
twice so that a replicate would be available to estimate
measurement repeatability of the profilometer. All eight
Profiles were digitized twice with the stereo digitizer,
although only profile 1 was used in the comparisons, Sum-
mary statistics for the absolute values of deviations (i.e.,
|Z, - Z,|, where Z, and Z, = the corresponding elevations
from the original trace and the replicate trace, respec-
tively) are given in table B-3 and indicate that the


and if the measures of roughness are uniform or consist-
ent, it is to be expected that the resulting correlations
should have reasonably high values as before.

The matrices displayed in table 15 seem to verify the
SRC results. The intercorrelations between the fractal
measures are again disappointing; in fact, the only high
correlations in table 15 are between D, and Z,. Unfortu-
nately, the correlation between D, and Z, reverses alge-
braic sign for joint set NS1JS, thus negating positive
speculation about the interrelationship of these two vari-
ables. Therefore, no new research directions or conclu-
sions relative to fractal dimension measures evolved after
considering the Noranda data.

FRACTAL CONTROVERSY

Much of the work discussed in this report was predi-
cated on using the fractal dimension as a potential meas-
ure of the roughness of a rock joint surface. However,
the field of fractal geometry evolved considerably during
the period this research was being undertaken (1987-91).
Many reputable scientists from many disciplines have
written scholarly papers in which fractal geometry served
as a major research tool (6, 29, 45, 48), particularly in the
earth sciences. In 1985, Barton and Larsen published such
a paper, "Fractal Geometry of Two-Dimensional Fracture
Networks at Yucca Mountain, Southwestern Nevada’ (6).
The work of Brown (11-13) has been previously cited.
The article by Aviles, Scholz, and Boatwright, "Fractal
Analysis Applied to Characteristic Segments of the San
Andreas Fault" (3), was published in 1987, Papers have
been published relating the fractal dimension with earth-
quakes (43). In a recent paper by Coughlin and Kranz
(17), the earthquake concept was successfully applied to
relating the fractal dimension with rock-burst-associated
seismicity, Feder (20) is an excellent practical source that
contains many scientific applications.

Certainly the visual geometry generated by fractal tech-
niques has captured the imagination of scientists, scholars,
and others. As a generator of surfaces, fractal geometry
has proven to be most successful, Beautiful two- and
three-dimensional pictures have been produced by fractal
generators, some of which were attractively displayed in
avolume entitled "The Science of Fractal Images,” by
Barnsley, Devaney, Mandelbrot, Peitgen, Saupe, and Voss
@).

However, in spite of the positive applications of frac-
tals, there is increasing criticism of the field of fractal
geometry. Many practitioners have found that fractals are
not necessarily the solution to their particular scientific
problem. Fox’s article (22) pointed out discrepancies

between theory and empirical results with regard to using
spectra to compute the fractal dimension. This is very
significant, for most of the theoretical basis for the
fractal dimension is derived from spectral considerations.
Gilbert’s article (24) also referred to untidy problems
when using the spectral approach. In a very provocative
set of articles and answering correspondence (4, 30, 33),
two noteworthy opinions were put forth. Kranz (30) crit-
icized the field of fractal geometry, stating that "Fractal
geometry has not solved any problems. It is not even clear
that it has created any new ones." A significant letter to
the editor in the journal "The Mathematical Intelligencer"
putting the field of fractals in a somewhat unfavorable
light was then contributed by Brooks (10), who was a co-
worker with Mandelbrot on many early fractal endeavors.
An article entitled "Fractal Fracas" appeared in "Science"
and summarized (as of 1990) the controversy over fractal
geometry (40). Among the topics discussed was the con-
sensus of the mathematical community over the "legit-
imacy’ of fractals. Axler (4), publisher of the "The Math-
ematical Intelligencer," stated that a majority of the
mathematicians line up against the proponents of fractal
geometry.

The authors’ involvement with fractals has focused on
computing the fractal dimension of profile traces and
trying to use that dimension as a measure of roughness.
Results have been disappointing, particularly with regard
to the anomalies between the various fractal algorithms.
These efforts could be easily dismissed by stating that
the profiles were neither self-similar nor self-affine, thus
implying that the fractal dimension was a redundant exer-
cise. However, establishing the properties of self-similarity
or self-affinity is difficult, particularly for empirical data.

In his Ph. D. thesis, Piggott (39, p. 115) used statisti-
cal, geostatistical, and fractal methods as measures of
the roughness of a number of different surface profiles.
The author concluded "that statistical and geostatistical
analyses are the most appropriate means of quantifying
fracture surface topography under the experimental con-
ditions described in this chapter. Used in conjunction, the
procedures describe both the statistical and spatial dis-
tribution of surface elevation in a manner amenable to
analytical modelling of fracture properties. Fractal and
spectral methods may yield superior descriptions of surface
topography under different experimental conditions.”

Some other authors who have enjoyed success using the
fractal dimension postulate that the profiles are self-affine,
rather than attempting to prove that this condition exists.
Other users of fractal geometry seem to implement the
technique without consideration of self-similarity or self-
affinity. Based on the preceding discussion, the authors


Report of Investigations 9447

Estimation of Shear Strength Using Fractals
as a Measure of Rock Fracture Roughness

By P. C. McWilliams, J. C. Kerkering, and S. M. Miller

UNITED STATES DEPARTMENT OF THE INTERIOR
Bruce Babbitt, Secretary

BUREAU OF MINES


Photogrammetry requires both horizontal and vertical
control in the picture area, This control allows calibra-
tion of the digitizing and plotting equipment. For the
laboratory samples, a 12- by 12-cm aluminum reference
frame was fabricated. This frame had a 10-cm-square
opening to allow insertion of the sample. Inscribed on the
frame every few centimeters were crosshairs. These pro-
vided horizontal control. The corners of the frame were
elevated by approximately 1 cm to help provide vertical
control. In addition, a 1-cm cube was attached to the
frame for further vertical control.

Color slide film (Ektachrome ASA 100) was used in-
stead of negative film. The information was read directly
from the positive slide images, and it was felt that this
gave better resolution than converting negative film into
prints and then analyzing the prints. The colored image
also helped to obtain more accurate readings.

EQUIPMENT CALIBRATION

Usable cameras for photogrammetry are either metric
or nonmetric, (Here, "metric' means measured.) The
metric camera is constructed with either fiduciary marks
(marks inside the camera on the film plane that show on
the negative) or a Reseau plate. The Reseau plate sits in
the light path immediately in front of the film plane and
has a grid on it that shows on the exposed film. Both
fiduciary marks and the Reseau plate serve the purpose of
helping to determine the amount of distortion in the film
caused by buckling.

A metric camera also comes with a scale that shows
lens distortion at various focusing distances; it may also
be equipped with a vacuum film back to hold the film
perfectly flat against the platen. Such cameras as the
Hasselblad MK70 are quite expensive, costing in excess of
$20,000 for just the body and lens.

Nonmetric cameras are not equipped with specialized
equipment or scales of measurement. Examples are
Hasselblad or Rollei cameras in the 70-mm? format or
Nikon, Pentax, or Leica cameras in the 35-mm format.
These cameras need to be calibrated to be suitable for
precise photogrammetry, which, if contracted out, can cost
up to $1,000.

Nonmetric cameras were chosen because they cost
much less than metric cameras, even with the necessity of
calibrating them. Nonmetric cameras can also be used for
ordinary photography, thus saving the cost of additional
photographic equipment for another research project.

The calibration procedure requires first engraving
fiduciary marks on the four corners of the film plane.
These marks serve in place of the Reseau plate.

Lens distortion must be measured at each planned
picture-taking distance. At cach position, the radial and
tangential distortions are measured, and this information

is later entered into the computer program used to run the
analytic plotter. This enables the plotter to compensate
automatically for lens distortion and unevenness of the
film,

An analytic self-calibration computer program that
enables individuals to calibrate their own camera systems
is included in reference 23, This reference also includes
sample input and output.

PHOTOGRAPHIC PROCEDURES

The following procedure was used for taking photo-
grammetric pictures in the laboratory. The cement part of
the specimen block was clearly marked as to top, bottom,
left, and right and then placed in the aluminum reference
frame. The block and frame were mounted vertically on
a small platform attached to the top of a tripod. The two
Rollei cameras were mounted on the rack-and-pinion
gears attached to the aluminum bar and this bar was
attached to the top of another tripod (see figure A-1).
The cameras were moved back and forth on the bar until
each camera had the same image and image size. The
angle of inclination was also checked to ensure that it was
the same for both cameras.

This setup procedure was greatly facilitated by using a
measuring tape to determine the distance from the top
center front of the reference frame to the film plane of
each camera. These distances should be equal. The
height (horizontal distance) from the top center front of
the reference plane to an imaginary line connecting the
film planes of the two cameras was also measured. This
distance was a common side of two right triangles and so
was used to calculate the actual measured distances.

Two lights with daylight bulbs were positioned hori-
zontally and vertically at 45° angles on each side of the
specimen. The lights were placed at equal distances from
the specimen to ensure uniform lighting. Alll other lights
in the room were turned out and the windows were dark-
ened. This ensured the correct color balance on the speci-
men and helped maintain surface relief, which would have
been smoothed by light from extraneous sources.

The cameras were tripped successively by means of
cable releases to minimize vibrations. The mirrors on
each camera were locked upright to further reduce picture
fuzziness caused by camera vibrations.

A spot meter and gray card were used to determine
light levels and exposure information. Light meter read-
ings were taken from the gray card held immediately in
front of the specimen. This combination was also used to
check the evenness of the light intensity on different parts
of the sample, These readings were compared with spot
readings taken directly off the face of the sample.

The shutter f-stop combination was chosen to give the
maximum depth of field. Generally, the combination was


ind
ES

nN
°

SHEAR LOAD, kN
S3

&

ES

Pj | fj ft | |
1 2 3 4 6 6 7 8 9
SHEAR DISPLACEMENT, mm

&

ny

8
T
i

3S
T
|

SHEAR STRESS, mt/m2
8
T
L

0 10 20 30 40
__NORMAL STRESS, mim?

Figure 10.—Examples of direct-shear test results, Each trace
represents application of a specified normal load. A, Plot of
shear load as a function of shear displacement; B, peak shear-
strength envelope fitted to test data, wi ra hoe (r=

A and B = curve parameters; and o, = normal

10 parallel profiles instead of mean value, results were
similar (table 3).

Some correlation was shown between the curve param-
eters A and B, but not enough to discourage the use of
these parameters as independent terms in subsequent
least squares regression modeling. The following linear
regression models were applied to the data set:

A=P)+P,D B=P)+P,D
A=P)+Pil B= P)+PiI
A = Py + PiZ, B= Py + PiZy
A=P)+P,D+PJI B=P)+P,D+P,I

0
A=Po+P,D +P +P3;Z, B=P)+P,D + Py +P3Z,
where P= ith regression coefficient

and I = fractal intercept.

The latter two models represent general "hyperplanes"
in multidimensional space. The merit of each model was
judged using an index of determination R,, which is anal-
ogous to the coefficient of determination (i.e., squared cor-
relation coefficient) for simple linear regression. This
index can be defined as follows:

~9.
R,-1- =i HY

3?

LY Mi - yi
where y; = measured value of dependent variable,
9 = value of dependent variable estimated

from regression model,

and J = mean value of dependent variable.

None of the models showed an index greater than 0.6
for both A and B predictors except the model in which all
three roughness measures were used. This model’s indices
were 0.625 for the A model and 0.765 for the B model.

Thus, based on the 12 direct-shear tests of clean frac-
tures at low normal stresses, the curve parameters A and
B for the shear-strength envelope + = Ao® can be esti-
mated from the roughness measures, where A = 4.295
- 1,022 (D) - 0.201 (1) - 1352 (Z,), and B = -1.449
+ 0.568 (D) + 0.412 (1) + 0.076 (Z,).

In developing such models for the experimental data
set, it was observed that the fractal intercept is at least as
important as the fractal dimension for describing surface
roughness and its subsequent influence on the shear
strength of clean fractures.

To evaluate the effectiveness of these models, they
were used to generate synthetic shear-strength envelopes
and then to compare these envelopes with the actual
envelopes produced by laboratory direct-shear tests. Ex-
amples are presented in figure 11 for two of the fractures.
Both generated envelopes seemed reasonably compatible
with actual envelopes, although the generated envelopes
tended toward a hypothetical "mean" envelope for all tests
considered here.

In testing for the influence of roughness on shear

strength, equations of the form + = P,D (2) were also
considered. The successful estimation of the coefficients
P, and P, by nonlinear least squares regression demon-
strated that fractal measures can be used to predict shear-
strength envelopes. However, the sparseness of data by
rock type in this particular study prevented pursuing these
particular models.

Thus, it can be seen that nonlinear, power-curve-type,
shear-strength envelopes for natural rock fractures can
be predicted using three objective roughness measures:


(9, 41). Repeated investigations of computed spectra of
the roughness profiles showed that aliasing was not a
problem (i.e., the spectra fell off to near zero at fre-
quencies of 3 to 4c/mm, much less than the Nyquist
frequency of about 9.7 c/mm). Thus, leakage was the
major concern, and even if a standard Tukey-Hanning
smoothing function was consistently used for the spectral
estimates and only the type of tapering window was
varied,’ quite different fractal dimensions were obtained
for a given roughness profile (fig. 4).

Microcomputer programs written to calculate D were
based on the four methods. Three of these programs were

The following tapering windows were used: cosine-bell with 2, 5,
and 10 pet and full applications; Parzen; and Welch, Estimated fractal
dimensions varied from 1.34 (for 2 pet cosine-bell window) to 1.49 (for
Welch window).

cross-checked with programs written by other computer
programmers. The answers generated by the comparisons
were quite consistent, giving the authors assurance that
the computer programs were not sources of error.
Experience in analyzing the 60 roughness profiles
indicated that the divider method was the most difficult
of the four methods to use consistently and still obtain
reasonable estimates of fractal dimension. The divider
method does not provide unique results, and estimates of
D may vary by 10 to 20 pet for a given profile. However,
some of this ambiguity can be mitigated by selecting those
divider spans that require 30 to 200 intersections to cover
the profiles, which have approximately 1,000 digitized
points. In addition, there are several ways to deal with
remainders left when the divider increments approach the
end of the profile trace (3). It is recommended that the

od Wr T T T

LOG ( G (FREQUENCY) )
$6 & &b
T T T T

4
x
T

BI T T T

-2 0 2

LOG (FREQUENCY), 1/mm

4-6 4 -2 ° 2 4

Figure 4.—Examples of log-log spectral plots showing the sensitivity of calculated fractal dimension to tapering.

A, 2 pet cosine-bell window; B, Welch window.


HEIGHT, mm

6 T T T T
el 7
afe 4
3 i 1 L 1

s T T T —————
5 4
e_—_—o___oOCOoOoOoOC, 4
3 n 1 i 1

6 T T I =

i 7
4 =
3 iT 1 1 iT

6 T I T

5 =
ae —]
3 1 1 J L

6 T T T T

sf 4
4b 7]
3 1 1 j 1

6 T T T T

5

4 —
3 i 1 l 1

6 T T T T

a ~~ =
4e J
3 i i 1 lL

6 T T T T

sk 4
4 ——
3 i n 1 n

6 T T T

5 =
4— 4
3 1 L 1 1

7 T T T T

6 +
5re a
4r- =
3 f i | l

10 20 30 40 50 60

DISTANCE,

Figure 5.—Ten gnelss profiles (traces A-J) oriented in x-direction.

mm