Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities
Use of the scanline mapping technique in geometric surveys of rock discontinuities can often lead to a bias, in that discontinuities are not always observed when they are at small angles to the scanline. Terzaghi introduced the concept of a blind zone to explain this bias, and developed a widely use...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2016
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nasplib_isofts_kiev_ua-123456789-1735602025-02-09T20:57:49Z Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities Распознавание и минимизация погрешностей при оценке несплошностей в негомогенных материалах методом коррекции Терцаги Tang, H.M. Huang, L. Bobet, A. EzEldin, M.A.M. Wang, L.Q. Wu, Y.P. Hu, X.L Научно-технический раздел Use of the scanline mapping technique in geometric surveys of rock discontinuities can often lead to a bias, in that discontinuities are not always observed when they are at small angles to the scanline. Terzaghi introduced the concept of a blind zone to explain this bias, and developed a widely used procedure to correct for it. Unfortunately, little is known about errors that may occur when the Terzaghi procedure is used outside the blind zone. This paper presents a detailed derivation to show that such errors arise with this application of the Terzaghi procedure. This error was evaluated using simulated orientation data and a case study at Wenchuan, Sichuan, China (2008). The results of these tests suggest optimal values of grid size and sample density for reducing the error. Анализ несплошностей в скальных породах с помощью сканирующей методики картирования часто приводит к погрешностям, в результате чего оказываются незамеченными несплошности, находящиеся под малым углом к линии сканирования. Концепция слепой зоны , предложенная Терцаги, позволила объяснить природу этих погрешностей и минимизировать их по предложенной методике. Однако малоизвестно, что применение последней вне слепой зоны также приводит к возникновению погрешностей. Детально проанализированы погрешности при использовании метода Терцаги для модельного материала скальной породы из эпицентра земелетрясения 2008 года в местности Вэньчуань (провинция Сычуань, Китай). Предложены оптимальные параметры расчетной сетки и плотности образца для минимизации погрешностей. Аналіз несуцільностей в скельових породах за допомогою сканувальної методики картування часто призводить до похибки, в результаті чого залишаються непоміченими несуцільності, що знаходяться під малим кутом до лінії сканування. Концепція сліпої зони , запропонована Терцагі, дозволила з'ясувати природу цих похибок і мінімізувати їх за запропонованою методикою. Однак маловідомо, що використання останньої поза сліпою зоною також призводить до виникненя похибок. Детально проаналізовано похибки при використанні методу Терцагі для модельного матеріалу скельової породи з епіцентру землетрусу 2008 року у місцевості Веньчуань (провінція Сичуань, Китай). Запропоновано оптимальні параметри розрахунково'ї сітки і щільності зразка для мінімізації похибок This research was supported by the National Natural Science Foundation of China (Grant Nos. 41230637, 41302231, and 41272309). The authors would like to thank our group for the orientation observations. 2016 Article Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities / H.M. Tang, L. Huang, A. Bobet, M.A.M. EzEldin, L.Q. Wang, Y.P. Wu, X.L. Hu // Проблемы прочности. — 2016. — № 6. — С. 111-121. — Бібліогр.: 12 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173560 539.4 en Проблемы прочности application/pdf Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Tang, H.M. Huang, L. Bobet, A. EzEldin, M.A.M. Wang, L.Q. Wu, Y.P. Hu, X.L Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities Проблемы прочности |
| description |
Use of the scanline mapping technique in geometric surveys of rock discontinuities can often lead to a bias, in that discontinuities are not always observed when they are at small angles to the scanline. Terzaghi introduced the concept of a blind zone to explain this bias, and developed a widely used procedure to correct for it. Unfortunately, little is known about errors that may occur when the Terzaghi procedure is used outside the blind zone. This paper presents a detailed derivation to show that such errors arise with this application of the Terzaghi procedure. This error was evaluated using simulated orientation data and a case study at Wenchuan, Sichuan, China (2008). The results of these tests suggest optimal values of grid size and sample density for reducing the error. |
| format |
Article |
| author |
Tang, H.M. Huang, L. Bobet, A. EzEldin, M.A.M. Wang, L.Q. Wu, Y.P. Hu, X.L |
| author_facet |
Tang, H.M. Huang, L. Bobet, A. EzEldin, M.A.M. Wang, L.Q. Wu, Y.P. Hu, X.L |
| author_sort |
Tang, H.M. |
| title |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities |
| title_short |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities |
| title_full |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities |
| title_fullStr |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities |
| title_full_unstemmed |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities |
| title_sort |
identification and mitigation of error in the terzaghi bias correction for inhomogeneous material discontinuities |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2016 |
| topic_facet |
Научно-технический раздел |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/173560 |
| citation_txt |
Identification and Mitigation of Error in the Terzaghi Bias Correction for Inhomogeneous Material Discontinuities / H.M. Tang, L. Huang, A. Bobet, M.A.M. EzEldin, L.Q. Wang, Y.P. Wu, X.L. Hu // Проблемы прочности. — 2016. — № 6. — С. 111-121. — Бібліогр.: 12 назв. — англ. |
| series |
Проблемы прочности |
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| fulltext |
UDC 539.4
Identification and Mitigation of Error in the Terzaghi Bias Correction for
Inhomogeneous Material Discontinuities
H. M. Tang,
a
L. Huang,
a,1
A. Bobet,
b
M. A. M. EzEldin,
c
L. Q. Wang,
a
Y. P. Wu,
a
and
X. L. Hu
a
a Faculty of Engineering, China University of Geosciences, Wuhan, China
b School of Civil Engineering, Purdue University, West Lafayette, USA
c College of Petroleum Geology & Minerals, University of Bahri, Khartoum, Sudan
1 278908378@qq.com
ÓÄÊ 539.4
Ðàñïîçíàâàíèå è ìèíèìèçàöèÿ ïîãðåøíîñòåé ïðè îöåíêå íåñïëîøíîñòåé
â íåãîìîãåííûõ ìàòåðèàëàõ ìåòîäîì êîððåêöèè Òåðöàãè
Õ. Ì. Òàíã
à
, Ë. Õóàíã
à
, À. Áîáåò
á
, Ì. À. Ì. ÝçÝëäèí
â
, Ë. Ê. Âàíã
à
, Þ. Ï. Âó
à
,
Êñ. Ë. Õó
à
à Êèòàéñêèé óíèâåðñèòåò ãåîôèçèêè, Óõàíü, Êèòàé
á Óíèâåðñèòåò Ïåðäüþ, Âåñò-Ëàôàéåòò, ÑØÀ
â Óíèâåðñèòåò Áàõðè, Õàðòðóì, Ñóäàí
Àíàëèç íåñïëîøíîñòåé â ñêàëüíûõ ïîðîäàõ ñ ïîìîùüþ ñêàíèðóþùåé ìåòîäèêè êàðòèðîâàíèÿ
÷àñòî ïðèâîäèò ê ïîãðåøíîñòÿì, â ðåçóëüòàòå ÷åãî îêàçûâàþòñÿ íåçàìå÷åííûìè íåñïëîø-
íîñòè, íàõîäÿùèåñÿ ïîä ìàëûì óãëîì ê ëèíèè ñêàíèðîâàíèÿ. Êîíöåïöèÿ “ñëåïîé çîíû”, ïðåä-
ëîæåííàÿ Òåðöàãè, ïîçâîëèëà îáúÿñíèòü ïðèðîäó ýòèõ ïîãðåøíîñòåé è ìèíèìèçèðîâàòü èõ ïî
ïðåäëîæåííîé ìåòîäèêå. Îäíàêî ìàëîèçâåñòíî, ÷òî ïðèìåíåíèå ïîñëåäíåé âíå “ñëåïîé çîíû”
òàêæå ïðèâîäèò ê âîçíèêíîâåíèþ ïîãðåøíîñòåé. Äåòàëüíî ïðîàíàëèçèðîâàíû ïîãðåøíîñòè
ïðè èñïîëüçîâàíèè ìåòîäà Òåðöàãè äëÿ ìîäåëüíîãî ìàòåðèàëà – ñêàëüíîé ïîðîäû èç ýïè-
öåíòðà çåìåëåòðÿñåíèÿ 2008 ãîäà â ìåñòíîñòè Âýíü÷óàíü (ïðîâèíöèÿ Ñû÷óàíü, Êèòàé).
Ïðåäëîæåíû îïòèìàëüíûå ïàðàìåòðû ðàñ÷åòíîé ñåòêè è ïëîòíîñòè îáðàçöà äëÿ ìèíèìè-
çàöèè ïîãðåøíîñòåé.
Êëþ÷åâûå ñëîâà: îðèåíòàöèÿ ðàçðûâíîñòè, íàêëîí âûáîðêè, “ñëåïàÿ çîíà”, ðàçìåð
ñåòêè, ïëîòíîñòü âûáîðêè.
Introduction. Rock is a naturally inhomogeneous material due to the presence of
discontinuities, including bedding planes, faults, fissures, fractures, joints, etc. The
orientation of these geological interfaces is a critical factor that governs the stability,
deformation and failure of a rockmass [1–3]. Orientation is typically observed in the field
using the scanline mapping technique [4]. However, this technique can introduce a bias,
i.e., the scanline tends to preferentially intersect discontinuities that make large angles with
the scanline [4, 5].
Terzaghi [6] suggested the following four-step procedure to correct this bias, which
has been widely used by researchers including Fouché and Diebolt [7] and Goodman [8]:
1. Mesh the projection net into grids.
2. Count the poles within each grid.
3. Weight the frequencies according to the Terzaghi equation
© H. M. TANG, L. HUANG, A. BOBET, M. A. M. EZELDIN, L. Q. WANG, Y. P. WU, X. L. HU, 2016
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 111
N
N
90 � �
�sin
, (1)
where � is the intersection angle between the discontinuity and the scanline (Fig. 1), N � is
the number of discontinuities intersected by the scanline at angle �, and N 90 is the number
of discontinuities that would have been intersected at 90�. Note that N 90 also represents
the frequency of discontinuities in a three-dimensional space.
4. Because the frequencies defined in this manner must be integers, the weighted
frequencies are approximated to the nearest integer. Many authors have adopted this step,
e.g., Fouché and Diebold [7], Park and West [9].
The Terzaghi procedure is invalid when applied to orientations that make a shallow
angle (0 to 30°) with respect to a scanline [4, 5]. This angle interval is the so-called blind
zone (shown in Fig. 2). Although several recommendations have been proposed to avoid
the effect of the blind zone (e.g., Park and West [9] and Priest [10]), little attention has been
paid to the accuracy of the Terzaghi procedure outside the blind zone. To the best of the
authors’ knowledge, there have been no previous studies on the level of this accuracy or the
source of low accuracy, and no methods of its improvement are yet available.
In this paper, a detailed derivation of the Terzaghi equation [Eq. (1)] is used to reveal
the source of low accuracy. Next, to improve the accuracy, the optimal values for the grid
size and sample density are determined by comparing the accuracies of correction with
different grid sizes and sample densities. Finally, the derived values are verified by a case
study on the model material – a rock from Wenchuan, China.
1. Derivation of the Terzaghi Equation. Discontinuities that make small angles to a
sampling line (0 to 30�) are not always observed, and Terzaghi [6] introduced the concept
of a blind zone to explain this effect (Fig. 2). Park and West [9] used borehole data to
confirm the 30� value. Noteworthy is that the blind zone does not depend on the particular
data set, but instead is related to the choice of the scanline.
Terzaghi [6] gave a simple derivation of the equation. To help identify the source of
low accuracy, we present here a more detailed derivation using the analytical geometry,
probability theory and integrals.
Let set A denote the discontinuities that exist in the rockmass, and set B denote the
subset of those discontinuities that intersect the scanline. The conditional probability
density pB A| ( , )� � denotes the likelihood that set B occurs, given that set A is certain.
H. M. Tang, L. Huang, A. Bobet, et al.,
112 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6
Fig. 1. Intersection angle � between a scanline and a discontinuity. The discontinuity is assumed to
be disc-shaped. The line segment AC is the projection of the line segment AB on the discontinuity,
while � is the intersection angle between AB and AC.
Here, the case that set A is certain means that both the dip direction and angle of all of the
discontinuities in the rockmass are uniform; i.e., all of the discontinuities in the rockmass
are parallel (Fig. 3). Hence, pB A| ( , )� � represents the probability density of the
discontinuities that intersect the scanline given that all of the discontinuities are parallel.
For these parallel discontinuities, as shown in Fig. 3, the scanline length, l, is
l
L
�
sin
,
�
(2)
then
p
N
l
N
L
B A| ( , )
sin
,� �
�
� � (3)
which is equivalent to
Identification and Mitigation of Error ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 113
Fig. 2. Blind zone, according to Terzaghi [6]. Equal-angle upper-hemisphere projection for a scanline
with a trend/ plunge of 90/45�. The two isogonic lines represent discontinuities that intersect the
scanline at an angle of 30� and define the boundaries of the blind zone. The contours represent the
Terzaghi correction factor sin � in Eq. (1), and the blind zone is the region where � � �30 .
Fig. 3. Intersection between the scanline and parallel discontinuities at angle �. The N
discontinuities are parallel to each other. The distance between discontinuity I and discontinuity N
is L.
p kB A| ( , ) sin� � �� (� � � � � � [ , ], [ , ]1 1N N ), (4)
where k is an undetermined coefficient.
The joint probability density that a particular dip direction and dip angle occur in the
rockmass, p A ( , ),� � is
p
p
p
A
AB
B A
( , )
( , )
( , )
,
|
� �
� �
� �
� (5)
where p AB ( , )� � is the joint probability density of the dip direction and dip angle that is
observed by the scanline.
Any grid D can be subdivided into n sub-grids
1 2 3, , , ... , n (Fig. 4). Thus,
the probability that the given orientation falls within in the interval D can be regarded as
the sum of the probabilities for these sub-grids:
p d d p d dA
D
A
i
n
i
( , ) ( , ) .� � � � � � � �
�� ����
�1
(6)
Substituting Eq. (4) into Eq. (5), and then substituting Eq. (5) into Eq. (6), we obtain
p d d
k
p
d dA
D
AB
i
n
i
( , )
( , )
sin
.� � � �
� �
�
� �
�� ����
�
1
1
(7)
Note that sin � in Eq. (7) is a function of the variables � and �. Suppose � ci is the
intersection angle between the scanline and the discontinuity mapped at the center of the
sub-grid. The variable sin � is equal to the constant sin � ci only if the sub-grids are
infinitesimal (i.e., n
�). In this case only, Eq. (7) can be rewritten as,
p d d
k
p d d
A
D
n
AB
cii
n
i( , ) lim
( , )
sin
� � � �
� � � �
�
��
��
��
�
�
1
1
�
�
�
�
1
1
k
P
n
i
cii
n
lim
sin
,
�
(8)
where Pi is the probability of the observed orientations within the sub-grid
1.
However, a true sub-grid cannot be infinitesimal. In this case, the above substitution
means that all of the nonparallel discontinuities in sub-grid
1 are approximated as
114 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6
Fig. 4. Subdivision of grid D.
H. M. Tang, L. Huang, A. Bobet, et al.,
parallel discontinuities with a uniform orientation mapped at the center of the sub-grid
(Fig. 5). This substitution leads to the error under consideration, which can be expressed as
p d d
k
P
A
D
i
cii
n
( , )
sin
,� � � �
�
�� ��
�
1
1
(9)
Equation (9) is consistent with the Terzaghi equation [6] as shown in Eq. (1).
2. Error Reduction in Taking N 90 as an Integer. Next, we identify the optimal
values of grid size and sample density to improve the accuracy of the Terzaghi procedure.
Grid size is arbitrarily set within the Terzaghi procedure. Sample density is the sample size
per grid size (1 1�� �). In this paper, firstly the accuracy is tested under a series of these two
parameters. The values of these two parameters that yield the highest accuracy are
determined. In the test, the accuracy of correction is characterized in terms of the difference
between the true and corrected distributions.
Firstly, the true distribution of orientations is assumed, along with three other
parameters that are necessary for modeling (size, intensity and aperture). The parameters
used in the model are as follows: length, width, and height of the simulated zone are 20, 20
and 20 m, respectively; number of discontinuity centers per rock volume is 5 m�3; dip
direction follows a uniform distribution with a lower limit of 175� and an upper limit of
185� ; dip angle follows a uniform distribution with a lower limit of 40� and an upper limit
of 50�; radius follows an exponential distribution with a mean of 1 m; in each model, the
trend/plunge values for seven scanlines are 0/45, 10/45, 20/45, 30/45, 40/45, 50/45, and
60/45�, respectively; and the sample density is 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5��2,
respectively.
Secondly, the parameters are entered into the discrete fracture network modeling
software AutoCAD, and seven different models of a discontinuity network are constructed
based on the different scanline directions.
Thirdly, in these models, the orientations observed by scanlines are obtained and then
corrected by the Terzaghi procedure. In each model, orientations are observed under six
different sample densities: 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5��2. Thus, for the seven models,
forty-two series of observed orientations are obtained, all of which fall outside the blind
zone. These orientations are corrected under four different grid sizes: 1 1� , 2 2� , 5 5� , and
10 10�� � . The corrected results are omitted here for brevity sake.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 115
Fig. 5. Interpretation of the approximate substitution in the Terzaghi equation.
Identification and Mitigation of Error ...
Finally, the difference between the true and corrected distributions is tested by a
non-parametric statistical approach (the Kolmogorov–Smirnov two-sample test) with the
SPSS software (version 24.0, IBM, Armonk, NY). This non-parametric hypothesis test
evaluates the difference between cumulative distribution functions of distributions of two
sample data vectors. The test returns an asymptotic significance to quantify the difference.
The significance ranges from 0 to 1; a smaller number reflects a greater difference. More
information about this test can be found in Senger and Çelik [11] and Ozçomak and
co-workers [12].
The relation between this difference and the parameters grid size and sample density
is analyzed, and the values of the two parameters that yield the smallest difference are
determined to be the optimal ones.
2.1. Effect of Grid Size. The test provided two results corresponding to dip direction
and dip angle, respectively. These two results are combined, and their average is shown in
Fig. 6. For most sample densities and intersection angles, significance is an increasing
function of grid size in the size interval between 1 1�� � and 2 2�� �, a decreasing function
between 2 2�� � and 5 5�� �, and a constant function between 5 5�� � and 10 10�� �. This
suggests that a decrease in grid size from 10 10�� � to 5 5�� � cannot improve the accuracy of
correction, a decrease from 5 5�� � to 2 2�� � can improve the accuracy, while a decrease
from 2 2�� � to 1 1�� � will reduce the accuracy. In addition, the greatest significance occurs at
a grid size of 2 2�� � for the most sample densities and intersection angles. To ensure that
this optimal value is independent of the sample density and intersection angle, a partial
correlation test was performed using the SPSS. When the control variable is sample density,
and the test variables are intersection angle and this optimal grid size, the correlation
coefficient is 0.467. Similarly, when the control variable is the intersection angle, and the
test variables are sample density and this optimal grid size, the correlation coefficient is
� 0.301. The test results indicate that this optimal grid size poorly correlates with the
sample density and the intersection angle. Thus, the optimal grid size is assessed as 2 2�� � .
116 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6
a b
dc
H. M. Tang, L. Huang, A. Bobet, et al.,
2.2. Effect of Sample Density. The relation between accuracy and sample density was
evaluated for the grid size of 2 2�� �. Figure 7 shows significance versus sample density
curves at different intersection angles. For intersection angles of 89, 83, 63, 56, and 49�, the
greatest significance occurs at a sample density of 0.05��2. In contrast, the greatest
significance is observed at 0.1��2 and at 0.5��2 for 77� and 70�, respectively. To ensure
that this optimal value of sample density (0.05��2) is independent of the intersection angle,
a Pearson correlation test was performed using the SPSS. The correlation coefficient of
these two test variables was 0.039, which demonstrates that this optimal value poorly
correlates with the intersection angle. Thus, the optimal value of sample density is derived
to be 0.05��2.
3. Case Study: Slope Cut East of Baihua Bridge. The derived optimal values of grid
size and sample density were verified by true data from a study area near the epicenter of
the 2008 Wenchuan Earthquake. Firstly, the observed orientations were obtained by the
scanline mapping technique. Secondly, the observed orientations were corrected by the
Terzaghi procedure. Thirdly, with the use of these corrected orientations the
three-dimensional network of discontinuity was modeled by means of discrete fracture
network modeling. In this model, the orientations that intersected a virtual scanline were
observed. The direction of this virtual scanline was set to be the same as that of the actual
scanline used in the field. To distinguish these orientations from the discontinuity orientations
observed in the field, the discontinuities derived from the model are referred to as model
discontinuities. Finally, the difference between the observed and model orientations was
evaluated by the Kolmogorov–Smirnov two-sample test under different grid sizes.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 117
e f
Fig. 6. Significance versus grid size. The significances of dip direction and angle were tested by the
Kolmogorov–Smirnov two-sample test. The average significance was then calculated from these two
values. Sample density: (a) 0.05��2; (b) 0.1��2; (c) 0.2��2; (d) 0.3��2; (e) 0.4��2; (f) 0.5��2.
Fig. 7. Significance versus sample density, under a grid size of 2 2�� �.
Identification and Mitigation of Error ...
The study area is located near the town of Yingxiu, in Wenchuan, Sichuan Province,
China, about 1800 m east of the epicenter of the 2008 Wenchuan Earthquake. The area is
11 m long, 5 m wide and 6 m high, and consists of exposures of upper Triassic lithic arkose
of the Xujiahe formation along a road cut. The rock has two joint sets, one of which is the
bedding. A scanline with a trend/plunge of 108/15� was fixed on the outcrop to observe the
bedding planes. Their poles are shown in Fig. 8.
The sampling bias of the observed orientations was corrected using the Terzaghi
procedure. For verification of the proposed grid size, the projection net was subdivided into
grids of 1 1� , 2 2� , 5 5� , and 10 10�� �, respectively. For verification of the proposed
sample density, the correction used the first 5, 14, 27, and 55 observed orientations,
corresponding to sample densities of 0.02, 0.05, 0.1, and 0.2��2, respectively. Figure 9
depicts the corrected orientations under a grid size of 5 5�� � and a sample density of 0.2��2.
The volume intensity, radius and aperture were calculated by using the respectrive methods.
The volume intensity was 4 m�3, the radius followed an exponential distribution with a
mean of 0.25 m, while the aperture followed an exponential distribution with a mean of
3.2 mm.
118 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6
Fig. 8. Contour plot of stereographic projections of observed orientations.
Fig. 9. Contour plot of stereographic projections of the corrected orientations under a grid size of
5 5�� � and a sample density of 0.2��2. Because some poles coincide, it appears that there are only 11
poles, while, in fact, there are 74 poles.
H. M. Tang, L. Huang, A. Bobet, et al.,
By multiplying the volume intensity and the volume of the simulated zone (1000 m3),
we calculated that the total number of discontinuities for modeling was 4000. For each of
these discontinuities, pseudo-random numbers were generated for each of five elements,
i.e., the X-coordinate, Y-coordinate, Z-coordinate, diameter and aperture (not shown). After
the pseudo-random numbers and the corrected orientation data were entered into the
modeling software, models were constructed that included a scanline with the same
direction as the field scanline. In the models, the orientations of discontinuities that
intersected the scanline were measured. The number of model discontinuities was equal to
the number observed in the field. Figure 10 shows the model orientations.
The difference between the observed and the model orientations was evaluated using
the Kolmogorov–Smirnov two-sample test. This test produced the asymptotic significances
corresponding to the dip direction and dip angle, as shown in Figs. 11 and 12. Figure 11
shows the average significances under different grid sizes. The curve is an increasing
function of grid size in the size interval between 1 1�� � and 2 2�� �, decreases between 2 2�� �
and 5 5�� �, and is approximately constant between 5 5�� � and 10 10�� � . Moreover, the
highest accuracy is achieved at 2 2�� �. The two results are consistent with the findings
regarding grid size in Section 2.1. Figure 12 presents the findings related to sample density
under a grid size of 2 2�� �. The highest accuracy is obtained at a sample density of 0.05��2.
This result is consistent with the findings regarding sample density in Section 2.2.
Fig. 10. Contour plot of stereographic projections of model orientations under a grid size of 5 5�� �
and a sample density of 0.2��2.
Fig. 11. Significance versus grid size.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 119
Identification and Mitigation of Error ...
4. Discussion. As shown in Figs. 6, 7, 11, and 12, neither the proposed grid size of
2 2�� � nor the proposed sample density of 0.05��2 could completely eliminate the error.
This may be due to the fact that some errors originate from estimating, measuring, or
aprroximating. Furthermore, as mentioned in Section 1, one cannot exclude the possibility
of substitution error. Therefore, application of the proposed values reduces, but cannot fully
eliminate the theoretical error introduced by the Terzaghi procedure.
Conclusions. An approximate substitution in the derivation of the Terzaghi equation
gives rise to a theoretical error when the Terzaghi procedure is applied outside the blind
zone. In this study, we developed a method for reducing this theoretical error. The highest
accuracy was achieved with a grid size of 2 2�� � and a sample density of 0.05��2. The
application of these optimal parameters to scanline mapping via the Terzaghi procedure has
improved the accuracy of the results obtained.
Acknowledgments. This research was supported by the National Natural Science
Foundation of China (Grant Nos. 41230637, 41302231, and 41272309). The authors would
like to thank our group for the orientation observations.
Ð å ç þ ì å
Àíàë³ç íåñóö³ëüíîñòåé â ñêåëüîâèõ ïîðîäàõ çà äîïîìîãîþ ñêàíóâàëüíî¿ ìåòîäèêè
êàðòóâàííÿ ÷àñòî ïðèçâîäèòü äî ïîõèáêè, â ðåçóëüòàò³ ÷îãî çàëèøàþòüñÿ íåïîì³-
÷åíèìè íåñóö³ëüíîñò³, ùî çíàõîäÿòüñÿ ï³ä ìàëèì êóòîì äî ë³í³¿ ñêàíóâàííÿ. Êîí-
öåïö³ÿ “ñë³ïî¿ çîíè”, çàïðîïîíîâàíà Òåðöàã³, äîçâîëèëà ç’ÿñóâàòè ïðèðîäó öèõ ïîõè-
áîê ³ ì³í³ì³çóâàòè ¿õ çà çàïðîïîíîâàíîþ ìåòîäèêîþ. Îäíàê ìàëîâ³äîìî, ùî âèêîðèñ-
òàííÿ îñòàííüî¿ ïîçà “ñë³ïîþ çîíîþ” òàêîæ ïðèçâîäèòü äî âèíèêíåíÿ ïîõèáîê.
Äåòàëüíî ïðîàíàë³çîâàíî ïîõèáêè ïðè âèêîðèñòàíí³ ìåòîäó Òåðöàã³ äëÿ ìîäåëüíîãî
ìàòåð³àëó – ñêåëüîâî¿ ïîðîäè ç åï³öåíòðó çåìëåòðóñó 2008 ðîêó ó ì³ñöåâîñò³ Âåíü-
÷óàíü (ïðîâ³íö³ÿ Ñè÷óàíü, Êèòàé). Çàïðîïîíîâàíî îïòèìàëüí³ ïàðàìåòðè ðîçðàõóí-
êîâî¿ ñ³òêè ³ ù³ëüíîñò³ çðàçêà äëÿ ì³í³ì³çàö³¿ ïîõèáîê.
1. B. H. G. Brady and E. T. Brown, Rock Mechanics for Underground Mining, 3rd edn,
Vol. 2, Springer, Dordrecht (2006), pp. 133–139.
2. E. Hoek, Practical Rock Engineering, 1st edn, Vol. 1, 6, Evert Hoek Consulting
Engineer Inc., North Vancouver (2007).
3. N. Barton, “Shear strength criteria for rock, rock joints, rockfill and rock masses:
Problems and some solutions,” J. Rock Mech. Geotech. Eng., 5, 249–261 (2013).
120 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6
Fig. 12. Significance versus sample density, under a grid size of 2 2�� �.
H. M. Tang, L. Huang, A. Bobet, et al.,
4. C. Xu and P. Dowd, “A new computer code for discrete fracture network modeling,”
Comput. Geosci., 36, 292–301 (2010).
5. A. K. Manda and S. B. Mabee, “Comparison of three fracture sampling methods for
layered rocks,” Int. J. Rock Mech. Mining Sci., 47, 218–226 (2010).
6. R. D. Terzaghi, “Source of error in joint surveys,” Geotechnique, 15, 287–304 (1965).
7. O. Fouché and J. Diebolt, “Describing the geometry of 3D fracture systems by
correcting for linear sampling bias,” Math. Geol., 36, 33–63 (2004).
8. R. Goodman, “Toppling – a fundamental failure mode in discontinuous materials –
description and analysis,” in: C. L. Meehan, D. Pradel, M. A. Pando, and J. F. Labuz
(Eds.), Geo-Congress 2013: Stability and Performance of Slopes and Embankments
III (Proc. of a meeting held 3–7 March 2013, San Diego, CA), Geotechnical Special
Publication No. 231, in 3 volumes, American Society of Civil Engineers (2013),
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9. H. J. Park and T. R. West, “Sampling bias of discontinuity orientation caused by
linear sampling technique,” Eng. Geol., 66, 99–110 (2002).
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Hong Kong (1995).
11. Ö. Senger and A. K. Çelik, “A Monte Carlo simulation study for Kolmogorov–
Smirnov two-sample test under the precondition of heterogeneity: upon the changes
on the probabilities of statistical power and type I error rates with respect to skewness
measure,” J. Statist. Econom. Meth., 2, No. 4, 1–16 (2013).
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Received 20. 10. 2015
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 121
Identification and Mitigation of Error ...
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]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|