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Comparison of rock discontinuity mean trace length and density estimation methods using discontinuity data from an outcrop in Wenchuan area, China
Computers and Geotechnics 38 (2011) 258–268
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Comparison of rock discontinuity mean trace length and density estimation methods using discontinuity data from an outcrop in Wenchuan area, China Qiong Wu a,b, P.H.S.W. Kulatilake c,⇑, Hui-ming Tang a a
Engineering Faculty, China University of Geosciences, Wuhan 430074, China Visiting Research Scholar, Geological Engineering Program, Department of Materials Science & Engineering, University of Arizona, Tucson, AZ 85721, USA c Geological Engineering Program, Department of Materials Science & Engineering, University of Arizona, Tucson, AZ 85721, USA b
a r t i c l e
i n f o
Article history: Received 23 October 2010 Accepted 13 December 2010 Available online 15 January 2011 Keywords: Rock mass Discontinuity geometry Sampling window Mean trace length Density
a b s t r a c t The equations that exist in the literature to estimate corrected mean trace length and corrected twodimensional density of a rock discontinuity set using area sampling technique are critically reviewed. The discontinuity traces appearing in an outcrop in Yingxiu area in China are used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations. Similarly, circular windows are used along with Mauldon’s and Zhang and Einstein’s equation to calculate the mean trace length and Mauldon’s equation to calculate the two-dimensional density for the same discontinuity sets using the same outcrop discontinuity trace data. For both parameters, the predictions based on the rectangular window methods were found to be more accurate than that based on the circular window methods. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction On May 12, 2008, the Wenchuan earthquake of Richter scale 8.0 occurred in the Longmenshan fault zone in Sichuan province, China. This disastrous earthquake triggered enormous landslides. Investigations indicated that Wenchuan earthquake directly caused more than 15,000 geohazards in the form of landslides, rockfalls, and debri flows [1]. In addition, it resulted in approximately 20,000 deaths [1]. Several research studies were initiated to investigate deformation and failure mechanisms of earthquake induced landslides in Wenchuan area [2–4]. Proper research on such topics cannot be carried out without a sound knowledge of the discontinuity geometry system that exists in the rock mass. Yingxiu Town, located in Wenchuan County, is the heaviest disaster area of Wenchuan Earthquake. Fig. 1 shows an almost vertical rock slope in Yingxiu Town. It is an excellent exposed outcrop for the study on rock mass discontinuity structure in this area. In this paper, discontinuity mean trace length and density are estimated using different methods available in the literature. Such studies are valuable to perform subsequent rock mass stability analysis in Wenchuan area under earthquake loading. A high definition photograph of the discontinuities that exist on the outcrop was taken in the field. Comprehensively considering the field sampling data and processing the results of the photograph, the discontinuity trace network of the outcrop was obtained ⇑ Corresponding author. Tel.: +1 520 621 6064; fax: +1 520 621 8059. E-mail address: [email protected] (P.H.S.W. Kulatilake). 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.12.003
(Fig. 2). In addition, discontinuity orientations were collected from the outcrop by running several scanline surveys. These data are used to estimate discontinuity mean trace length and density using different methods available in the literature. Accuracy of the methods is evaluated. 2. Literature review It is generally recognized that the strength, deformability, stability and percolation characteristics of a rock mass are strongly influenced by the discontinuities that are present in the rock mass. Quantitative descriptions of discontinuities have been proposed by the International Society for Rock Mechanics [5]. Discontinuity geometry for a rock mass is characterized by the number of discontinuity sets, mean density, and the distributions for location, orientation, size and spacing for each discontinuity set [6–8]. Estimation of the aforementioned discontinuity geometry parameters are commonly made by means of measurements of the discontinuity geometry on one or two-dimensional exposures. The trace length observed on two-dimensional exposures is related to discontinuity size in three-dimensions. It is important to note that the trace length, produced by a discontinuity on a two-dimensional exposure, depends on the shape and size of the discontinuity, the orientation of the discontinuity, the angle between the discontinuity plane and the plane of exposure, and the dimensions of the exposure [9,10]. The last three variables can be measured in exposures. However, determination of discontinuity shape and size is extremely difficult. The number of trace mid points per unit area can
Q. Wu et al. / Computers and Geotechnics 38 (2011) 258–268
259
Fig. 1. Pictorial view of the exposed outcrop in Wenchuan area, China.
0
50
100
150
Average cut strike:180¡ ã
200 (m) Qel+dl 4
Fig. 2. Discontinuity trace network of the exposed outcrop in Wenchuan area, China.
be used as a measure of discontinuity density in two dimensions [11]. Several sampling biases have been identified in estimating mean trace length and two-dimensional density of discontinuities [9–19]. Estimation of mean trace length and density has received much concern in the literature in the areas of rock mechanics as well as geological and applied statistics. Many investigators have conducted research on the topic of estimation of mean trace length of discontinuities [6–8,12–26]. A few researchers [6,11,21] have performed investigations on density of discontinuities in twodimensional exposures. Sampling plans used for trace length measurements can basically be divided into two types. They are: (1) sampling the traces that intersect a line drawn on the exposure, known as scanline survey and (2) sampling the traces that appear on a finite-size window made on a two-dimensional exposure, known as area sampling survey. According to the selected sampling method, the estimation method of the mean trace length and density varies. Estimation of the corrected mean trace length (corrected for sampling biases) from measured trace lengths using scanline survey has been described by Cruden [12], Priest and Hudson [16], Zhang and Liao [20] and Kulatilake et al. [6,8]. Estimation of corrected mean trace length and two-dimensional density using window sampling is dealt with in detail in this paper. Two main equations have been suggested in the literature to estimate mean trace length based on window sampling. A summary on that is given below. Pahl [17] derived an estimator of corrected mean trace length (l) for discontinuities observed in mine drive walls by assuming that trace midpoints are randomly and uniformly distributed. The method corrects for censoring and size biases. The method is independent of the distribution of trace length and restricted to a discontinuity set whose orientation has a single value. Kulatilake and Wu [19] advanced Pahl’s method to discontinuities whose orientation is described by a probability density function. In deriving the equation, they divided the discontinuities into
three groups: (a) those with both ends censored, (b) those with one end censored, and (c) those with both ends observable. The method provides corrections for the size and censoring biases for traces appearing on a rectangular two-dimensional exposure. The lengths of observed traces and the probability density function of trace length are not required to apply the corrections. The proposed equation is given below:
l¼
whð1 þ R0 R2 Þ ð1 R0 þ R2 ÞðwB þ hAÞ
ð1aÞ
where
A¼
Z
au
Z
al
B¼
Z
au
al
cos u ¼
sin u ¼
hu
j cos ujf ðh; aÞdh da
ð1bÞ
sin uf ðh; aÞdh da
ð1cÞ
hl
Z
hu hl
1 ð1 þ tan2 h cos2 dÞ1=2 1 ð1 þ cot2 h sec2 dÞ1=2
ð1dÞ
ð1eÞ
In Eqs. (1a)–(1e), w and h are width and height of a rectangular window, respectively; R0 and R2 denote the fractions of discontinuities with both ends censored and both ends observable, respectively; h and a are dip angle and dip direction; f(h, a) is the probability density function of discontinuity orientation with hl 6 h 6 hu and al 6 a 6 au , where subscripts l and u denote lower and upper limits, respectively; u is apparent dip angle and d denote the acute angle between the dip direction and the vertical sampling plane. Note that to use the aforementioned equation, it is not necessary to have a fitted theoretical probability density function for the discontinuity set orientation. In the absence of a theoretical
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probability density function, the equation can be used by replacing the integral sign by the summation sign and using the empirical orientation distribution obtained for the discontinuity set. The method allows one to incorporate the corrected relative frequency of each trace appearing on the two-dimensional exposure, due to the orientation bias, in the presence of a probabilistically distributed discontinuity set orientation. In addition, because this equation incorporates the angle between the discontinuity plane and the two-dimensional exposure plane, it can also be used to estimate mean trace length when traces appear on curved sampling surfaces such as tunnels having curved cross sections and outcrops that show some curvature. In such a case, the curved surface should be divided into a series of planar surfaces. Then for the traces appearing on each plane, the corrected relative frequencies can be calculated. Mauldon [21] derived estimators of mean trace length for windows with arbitrary convex boundaries. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. The estimator of corrected mean trace length he obtained for a rectangular window is the same as the one presented by Kulatilake and Wu [19]. The equation given for a circular window is as follows:
l¼
pr N þ NT NC 2
N NT þ NC
k¼
Pnb
Pc þ nj¼1 ½P 0 ðCÞj Nt na þ nb þ nc
i¼1 ½P 1 ðCÞi
ð3bÞ
R1 a R 2a f ðxÞdx f ðxÞdx P0 ðCÞ ¼ 2aR 1xa þ Ra1 f ðxÞdx f ðxÞdx a a
ð3cÞ
where a is the length of the intersected trace appearing within the window and f(x) is the probability density function of trace length (X) on the infinite two-dimensional plane. Note that to use the aforementioned equation, even though it is preferred, it is not necessary to have a fitted theoretical probability density function for the discontinuity set trace length. In the absence of a theoretical probability density function, the equation can be used by replacing the integral sign by the summation sign and using the empirical trace length distribution obtained for the discontinuity set. This equation has been used in practice to first estimate the two-dimensional density and then to estimate three-dimensional density in modeling discontinuity networks in three-dimensions for Stripa Mine, Sweden [6]. Mauldon [21] have suggested the following equation to estimate number of trace mid points per unit area using circular windows and based on the concepts given in the paper by Parker and Cowen [27].
ð2Þ k¼
where r is the radius of the circular sampling window; N, NT and NC are the number of traces intersecting, transecting and contained in the window, respectively. Zhang and Einstein [22] obtained the same equation as Mauldon [21] for a circular window using the general approach given by Kulatilake and Wu [19]. It is important to note that the equation given for a circular window is based on the implicit assumption that each trace has the same chance of appearing on the sampling window. This assumption is applicable only for the deterministic orientation (i.e. parallel traces on the two-dimensional exposure) and it is not applicable in the usual case where the discontinuity orientation has a scatter and is probabilistically distributed. It is important to note that the corrected relative frequency of a discontinuity trace is linked to the orientation bias. Because the aforementioned equation given for a circular window does not have the capability of calculating corrected relative frequencies, it cannot be used to estimate mean trace length when traces appear on non-planar two-dimensional exposures. Song [24] made an attempt to extend Mauldon’s [21] equation for a circular window for non-planar surfaces using the orientation bias concept. Even though he correctly pointed out the need for the extension, the equation given in Song’s [24] paper for the extension is incorrect. Two main equations have been suggested in the literature to estimate two-dimensional density of discontinuities (number of discontinuity trace midpoints per unit area) based on window sampling. A summary on that is given below. Kulatilake and Wu [11] proposed the following equation to estimate number of trace mid points per unit area, k, starting from number of traces per unit area, Nt, counted on a rectangular sampling domain.
na þ
R 2a f ðxÞdx P1 ðCÞ ¼ Ra1 f ðxÞdx a
ð3aÞ
where na, nb, nc denote the number of discontinuities of both ends observed (type a), one end observed (type b) and both ends censored (type c), respectively; i denotes the ith type (b) discontinuity; j denotes the jth type (c) discontinuity; P1(C) and P0(C) respectively denote the probability that the trace midpoint of the type (b) and type (c) discontinuities are within the window. P1(C) and P0(C) can be expressed as
ðN NT þ NC Þ 2p r 2
ð4Þ
where r, N, NT and NC have the same meaning as those in Eq. (2). A trace length distribution is not required to use this method. It is important to note that Parker and Cowen have assumed that the line segment orientation on a two-dimensional plane follows a uniform distribution. It is impossible to satisfy this assumption in dealing with rock discontinuity traces appearing from a distinct discontinuity set orientation. Eq. (4) does not show a contribution from both ends censored traces to the parameter number of mid points per unit area. In addition, the equation implies the number of mid points resulting from one end observed traces to be half the number of such trace type irrespective of the type of the trace length distribution and size of the sampling domain. The latter two concepts are also definitely not acceptable. In summary, the correctness or validity of this equation is highly questionable. The estimation techniques summarized above can be classified into two types: Type 1 uses a rectangular sampling window and type 2 uses a circular sampling window. Discontinuity traces appearing in the outcrop in Yingxiu area are used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations [19,11]. Similarly, circular windows are used along with Mauldon’s [21] and Zhang and Einstein’s equation [22] to calculate the mean trace length and Mauldon’s [21] equation to calculate the twodimensional density for the discontinuity traces that exist on the same outcrop. The obtained results are compared. 3. Mean trace length estimations 3.1. Corrected mean trace length based on rectangular windows Eq. (1) given earlier is used to estimate corrected mean trace length using rectangular windows. The expectation values for sin u and cos u appearing in Eqs. (1b) and (1c) can be computed approximately by
Eðsin uÞ
N X ½sin ui Rf ðui Þ
ð5aÞ
i¼1
Eðj cos ujÞ
N X ½j cos ui j Rf ðui Þ i¼1
ð5bÞ
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where ui denotes the apparent dip angle of the ith trace and Rf(ui) denotes the corrected relative frequency of the ith trace appearing on the window with the apparent dip angle ui resulting from a corresponding dip angle and a dip direction combination (h, a). The Rf(ui) can be calculated as given below. The probability of intersection between a discontinuity and a sampling domain is proportional to the volume (Vi) within which the center of the discontinuity should lie in order to intersect the sampling domain [10]. According to the stated hypothesis, the Rf(ui) can be calculated by Eqs. (6)–(8) as given below [10]. 2
V i ¼ whdi ½cos2 hi þ sin hi cos2 ðar ai Þ0:5 þ
p
2 di
4
½w sin hi j cosðar ai Þj þ h cos hi
ð6Þ
In Eq. (6), ai and hi denote the dip direction and dip angle of the ith discontinuity, respectively; di is the diameter of the ith discontinuity and ar is the strike of the sampling window. In the absence of diameter values for discontinuities, a diameter value equal to 10% higher than the maximum trace length may be used for all the discontinuities in using Eq. (6). The weighting function for the ith discontinuity (Wi) is given by
Wi ¼
1 Vi
ð7Þ
Then the corrected relative frequency of the ith discontinuity is given by
Wi Rf ðui Þ ¼ PN i¼1 W i
ð8Þ
3.2. Application of the proposed mean trace length equation for rectangular windows Orientation data of the discontinuities on this outcrop were measured by five scanlines. The pole (normal vector) concentration plot of these data is shown in Fig. 3. According to the plot, discontinuities on this outcrop can be divided into two sets: set 1 features dip direction from 105° to 200° and dip angle from 20° to 66°, set 2 features dip direction from 301° to 331° and dip angle from 33° to 76°. The strike of the outcrop changes slightly from 165° to 190° with a mean value of 180°. Therefore, two sets of discontinuities respectively produce traces on the outcrop, one dipping approximately to the south (set 1) and the other dipping approximately to the north (set 2). A series of rectangular windows were placed
on the outcrop to estimate the corrected mean trace length of the discontinuities (see Fig. 4). There are four window locations (1–4), and at each location, there are three windows with the following sizes: 90 m 45 m, 110 m 55 m and 130 m 65 m. Apparent dip angles of all the discontinuities appearing on these windows were measured. In order to calculate the corrected relative frequency for each discontinuity trace, the orientation values corresponding to each discontinuity trace were obtained through the following steps: (1) calculate the apparent dip angle for each discontinuity orientation measured by the scanline by using the strike value of the exposure; (2) compare between the measured apparent dip angle and calculated apparent dip angles and choose the orientation values corresponding to each discontinuity trace needed to calculate the corrected mean trace length. Then the corrected mean trace length values for discontinuities of the two sets belonging to all the 12 windows were calculated by the method explained in Section 3.1. The results are shown in Table 1 and Fig. 5. The obtained overall mean value and coefficient of variation for set 1 are respectively 39.8 m and 0.223, and that for set 2 are 33.4 m and 0.118, respectively. These overall estimated mean values are also shown in Fig. 5. The total trace length of each discontinuity trace intersected with each sampling window on the outcrop is known. Using this data, for each window a true mean trace length value was calculated (Table 1). The overall true mean values resulting for set 1 and set 2 from all these true mean trace lengths (Table 1) are also shown in Fig. 5. For set 2, the overall estimated mean value is almost the same as the overall true mean value. For set 1, the difference between the two parameters is only 7.66%. Fig. 5 shows that the individual estimated mean trace length values vary with the location and the size of the rectangular windows. This is expected when estimations are made using finite size samples. An estimation error for the prediction may be defined as
errorl ¼
lt le 100% lt
ð9Þ
where lt and le denote the true and estimated mean trace length values, respectively. This error can be used to evaluate the accuracy of the proposed prediction method. Table 1 and Fig. 6 show the errors resulted by applying Kulatilake and Wu’s [19] method by placing different size rectangular windows at different locations on the outcrop. The range of error resulting from finite sample sizes and spatial variability is 16.0% to +18.0% for set 1 and 19.3% to +21.3% for set 2.
N Fisher Concentrations % of total per 1.0 % area
1m 2m
W
E
2m
0.00 ~ 2.00 % 2.00 ~ 4.00 % 4.00 ~ 6.00 % 6.00 ~ 8.00 % 8.00 ~ 10.00 % 10.00 ~ 12.00 % 12.00 ~ 14.00 % 14.00 ~ 16.00 % 16.00 ~ 18.00 % 18.00 ~ 20.00 %
No Bias Correction Max. Conc. = 18.7784%
1m
Equal Area Lower Hemisphere 107 Poles 107 Entries
S Fig. 3. Pole (normal vector) concentrations of discontinuity orientations on a lower hemispherical equal area plot.
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0
100
50
150
200 (m)
Location 2 Location 1
Location 3
Location 4
Fig. 4. Sizes and locations of rectangular windows used to estimate mean trace length.
Table 1 Estimated corrected mean trace length and its% error based on rectangular sampling windows. Window size (m)/location number
Estimated value (m)
90 45/location 1 90 45/location 2 90 45/location 3 90 45/location 4 110 55/location 1 110 55/location 2 110 55/location 3 110 55/location 4 130 65/location 1 130 65/location 2 130 65/location 3 130 65/location 4 Mean value Coefficient of variation Range
% Error of le
True value (m)
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
23.7 50.0 34.3 46.8 29.4 47.6 33.6 50.6 36.9 39.7 36.0 48.4 39.8 0.223
27.3 33.9 36.8 39.7 29.2 35.9 33.4 33.5 30.9 31.1 29.0 39.9 33.4 0.118
28.9 54.4 40.7 47.8 31.5 50.1 40.7 51.4 31.8 45.0 40.4 54.7 43.1 0.208
34.7 33.3 36.9 35.3 32.2 30.1 35.8 36.5 32.7 31.3 33.9 34.7 34.0 0.062
18.0 8.1 15.7 2.1 6.7 5.0 17.4 1.6 16.0 11.8 10.9 11.5
21.3 1.8 0.3 12.5 9.3 19.3 6.7 8.2 5.5 0.6 14.5 15.0
16.0 to +18.0
19.3 to +21.3
Mean trace length (m)
80
90×45
110×55
130×65
Overall estimated mean
60
Overall true mean
40 20 0 1
2
3
4
Location number
(a) Set
Mean trace length (m)
80
90×45 60
110×55
130×65
Overall estimated mean
Overall true mean
40 20
0 1
2
3
4
Location number
(b) Set 2 Fig. 5. Estimated mean trace length based on Kulatilake and Wu [19] method using rectangular windows of different sizes placed at different locations on the outcrop.
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100
% Error
50
Location 1
Location 2
Location 4
Zero line
Location 3
0 -50 -100 90×45
110×55
130×65
Window size (m)
(a) Set 1 100
% Error
50
Location 1
Location 2
Location 4
Zero line
Location 3
0 -50
-100 90×45
110×55
130×65
Window size (m)
(b) Set 2 Fig. 6. Prediction error of mean trace length associated with finite sample sizes and spatial variability based on Kulatilake and Wu’s [19] method using rectangular windows of different sizes placed at different locations on the outcrop.
3.3. Mean trace length estimation using circular sampling windows As stated in Section 2, Mauldon [21] and Zhang and Einstein [22] proposed Eq. (2) to estimate corrected mean trace length using circular sampling windows. To estimate the corrected mean trace length of discontinuities using Eq. (2), in the outcrop a series of circular windows were placed on the discontinuity trace network as shown in Fig. 7. Five window locations were chosen and at each location three windows of radii 20 m, 30 m and 40 m were placed. Predicted results are shown in Table 2 and Fig. 8. The overall predicted means, overall true means and estimation errors were calculated similar to the way they were done for the rectangular windows. Table 2 and Fig. 9 show the errors resulted by applying Mauldon’s [21] and Zhang and Einstein’s [22] method by placing different size circular windows at different locations on the outcrop. The range of error resulting from finite sample sizes and spatial variability for circular windows is 14.6% to +41.5.0% for set 1 and 66.1% to +36.5% for set 2. 3.4. Comparison of results from the two mean trace length estimation methods Overall mean predictions obtained through the rectangular window method were closer to the overall true values than that
0
50
100
obtained through the circular window method. The estimation error ranges obtained for the circular window method are considerably larger than that resulted from the rectangular window method. The coefficient of variation values given in Tables 1 and 2 also indicate that predictions based on the rectangular window method show less variability than that of the circular window method due to the variability of window location and size. The aforementioned findings indicate that the predictions based on the rectangular window method are more accurate than that of the circular window method. 4. Density estimation 4.1. Density estimation of discontinuity traces using rectangular sampling windows As stated in Section 2, Kulatilake and Wu [11] proposed Eq. (3) to estimate number of discontinuity trace mid points per unit area. To use Eq. (3a) to predict k, three rectangular windows of size (120 m 40 m) were placed on the outcrop at different locations (Fig. 10). To calculate P1(C) and P0(C) given in Eqs. (3b) and (3c), the integration sign was replaced by the summation sign and the probability density function was replaced by relative frequencies as given below:
150
Location 1Location 2 Location 3 Location 4
200 (m)
Location 5
Fig. 7. Sizes and locations of the circular windows used to estimate mean trace length.
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Table 2 Estimated corrected mean trace length and its% error based on circular sampling windows. Window size (m)/location number
20/location 1 20/location 2 20/location 3 20/location 4 20/location 5 30/location 1 30/location 2 30/location 3 30/location 4 30/location 5 40/location 1 40/location 2 40/location 3 40/location 4 40/location 5 Mean value Coefficient of variation Range
Mean trace length (m)
80
% Error of le
Estimated value (m)
True value (m)
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
10.5 56.5 57.6 26.4 44.0 17.0 41.9 47.1 30.0 32.6 29.1 49.3 36.2 36.1 55.0 38.0 0.3718
39.3 41.9 31.4 49.3 18.8 47.1 43.5 20.9 29.4 23.6 30.5 29.9 33.6 34.9 43.0 34.5 0.2722
15.6 64.8 55.0 45.2 41.1 22.2 54.9 53.6 43.5 44.8 25.4 46.4 48.8 38.2 48.1 43.2 0.3078
36.7 29.5 33.9 44.4 26.3 37.1 26.2 32.0 32.1 37.1 33.0 24.5 31.1 35.7 39.8 33.3 0.1628
32.8 12.7 4.63 41.5 6.86 23.7 23.7 12.2 31.1 27.2 14.6 6.41 25.7 5.58 14.3
7.00 41.9 7.42 11.0 28.4 26.9 66.1 34.7 8.42 36.5 7.60 22.2 8.15 2.30 8.06
14.6 to +41.5
66.1 to +36.5
r=20
r=30
r=40
Overall estimated mean
Overall true mean
60 40 20 0 1
2
3
4
5
4
5
Location number
(a) Set 1
Mean trace length (m)
80
r=20 r=30 Overall estimated mean
60
r=40 Overall true mean
40 20
0 1
2
3
Location number
(b) Set 2 Fig. 8. Estimated mean trace length based on Mauldon’s [21] and Zhang and Einstein’s [22] equation using circular windows of different sizes placed at different locations on the outcrop.
P1 ðCÞ ¼
P2a x¼a Rft P 1 ax¼0 Rft
ð10aÞ
Pxmax P0 ðCÞ ¼
1
a x¼2a xi a Rft Pa x¼0 Rft
þ
P2a x¼a Rft P 1 ax¼0 Rft
ð10bÞ
where Rft denotes the relative frequency of trace length in a certain length interval; xi denotes the mean trace length in the length interval and xmax is the maximum value of the trace length. Relative frequencies can be calculated using the histogram obtained for the trace length. A typical histogram obtained for trace length is given in Fig. 11.
Let ke and kt denote the estimated and the true density value, respectively. Because the distributions of all the traces in the three windows are known, kt can be calculated. Table 3 and Fig. 12 show the results of density estimations. Error in the prediction was calculated using the following equation.
errork ¼
kt ke 100% kt
ð11Þ
Values given in Table 3 and Fig. 12 show that predicted densities are quite close to the true values. Fig. 12 also shows that significant differences exist between the values obtained for number of traces per unit area and number of trace mid points per unit area.
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100
Location 1 Location 4
% Error
50
Location 2 Location 5
Location 3 Zero line
0 -50
-100
20
30
40
Radius (m)
(a) Set 1 100
% Error
50
Location 1
Location 2
Location 3
Location 4
Location 5
Zero line
0 -50
-100 20
30
40
Radius (m)
(b) Set 2 Fig. 9. Prediction error of mean trace length associated with finite sample sizes and spatial variability based on Mauldon’s [21] and Zhang and Einstein’s [22] equation using circular windows of different sizes placed at different locations on the outcrop.
0
100
50
Location 1
Location 2
150
200 (m)
Location 3
Fig. 10. Selected size and locations of rectangular windows for density estimation through Kulatilake and Wu’s [11] method.
Relative frequency density (m-1)
0.045 0.04 Relative frequency density=Relative frequency / Interval length
0.035
That indicates importance of the sampling bias correction in estimating two-dimensional density of discontinuity traces. Prediction errors range between 13.6% and +16.0% for set 1 and between 0% and +14.8% for set 2 to reflect the influence of finite sample size and spatial variability.
0.03 0.025
4.2. Density estimation of discontinuity traces using circular windows
0.02 0.015 0.01 0.005 0
0
20
40
60
80
100
120
Trace length (m) Fig. 11. Histogram for trace length of discontinuities of set 1 at location 1.
As stated in Section 2, Mauldon [21] proposed Eq. (4) to estimate number of discontinuity trace mid points per unit area based on circular sampling windows. To apply this method, four circular windows of size r = 25 m were placed on the outcrop as shown in Fig. 13. True density values for circular windows were calculated as for the rectangular windows. Table 3 and Fig. 14 show the obtained results. Values given in Table 3 and Fig. 14 show that predicted densities deviate significantly from the true values. Prediction errors range between 0% and +50.0% for set 1 and between 72.2% and +24.2% for set 2.
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Table 3 Estimated two-dimensional densities of discontinuities and their% errors based on rectangular and circular sampling windows. Nt (m2)
Method/location number
ke (m2)
kt (m2)
% Error of ke
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
Set 1
Rectangular window Location 1 Location 2 Location 3 Mean value Coefficient of variation Range
0.0054 0.0075 0.0050 0.0060 0.2251
0.0054 0.0048 0.0042 0.0048 0.1250
0.0036 0.0042 0.0026 0.0035 0.2332
0.0035 0.0024 0.0023 0.0027 0.2436
0.0035 0.0050 0.0023 0.0036 0.3758
0.0035 0.0027 0.0027 0.0030 0.1557
2.9 16.0 13.0
Circular window Location 1 Location 2 Location 3 Location 4 Mean value Coefficient of variation Range
0.0061 0.0061 0.0112 0.0046 0.0070 0.4126
Set 2 0.0 11.1 14.8
13.0 to +16.0 0.0061 0.0036 0.0051 0.0061 0.0052 0.2261
0.0046 0.0018 0.0038 0.0025 0.0032 0.3970
0.0036 0.0031 0.0046 0.0025 0.0034 0.2576
0.0046 0.0036 0.0061 0.0036 0.0045 0.2640
0.0028 0.0018 0.0033 0.0033 0.0028 0.2525
0.0 50.0 37.7 30.6
0.0 to +14.8 28.6 72.2 39.4 24.2
0.0 to +50.0
72.2 to +24.2
0.01
Density (m-2)
Nt
t
e
0.005
0
1
2
3
Location number
(a) Set 1 0.01
Density (m-2)
Nt
t
e
0.005
0
1
2
3
Location number
(b) Set 2 Fig. 12. Predicted results of two-dimensional discontinuity density by Kulatilake and Wu’s [11] method using rectangular windows.
4.3. Comparison of results from the two density estimation methods
5. Conclusions
Predicted densities obtained through the rectangular window method are quite close to the true values. Prediction error ranges obtained for the circular window method are considerably larger than that resulted for the rectangular window method. The coefficient of variation values given in Table 3 also indicate that predictions based on the rectangular window method show less variability than that of the circular window method due to the influence of finite sample size and the variability of location of the window. The aforementioned findings indicate that predictions based on the rectangular window method are far more accurate than that of the circular window method.
The discontinuity traces appearing in the outcrop in Yingxiu area were used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations [19,11]. Similarly, circular windows were used along with Mauldon’s [21] and Zhang and Einstein’s equation [22] to calculate the mean trace length and Mauldon’s [21] equation to calculate the two-dimensional density for the discontinuity traces that exist on the same outcrop. For both the corrected mean trace length and the two-dimensional density, the predictions based on the rectangular window methods were found to be more accurate than that based on the circular window
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0
50
Location 2 Location 1
100
150
200 (m)
Location 3 Location 4
Fig. 13. Selected sizes and locations of circular windows for density estimation through Mauldon’s [21] method.
0.012
Nt
Density (m-2)
0.01
t
e
0.008 0.006 0.004 0.002 0
1
2
3
4
3
4
Location number
(a) Set 1 0.01
Nt
t
e
Density (m-2)
0.008 0.006 0.004 0.002 0
1
2
Location number
(b) Set 2 Fig. 14. Predicted results of two-dimensional discontinuity density based on Mauldon’s method [21] using circular windows.
methods. A shortcoming of the equation proposed to calculate the corrected mean trace length of a discontinuity set based on a circular sampling window is pointed out. The authors have pointed out the major shortcomings that exist with Mauldon’s equation [21] proposed for the estimation of two-dimensional density of a discontinuity set using circular windows. Due to these shortcomings, the correctness or validity of this equation seems highly questionable.
Acknowledgments This research was funded by The Chinese Geological Survey Project Number 1212010914036, The National Basic Research Program of China (973 Program) Project Number 2011CB710600, the Special Fund for Basic Scientific Research of Central Colleges, China University of Geosciences, Wuhan (CUG090104) and The National Natural Science Youth Foundation Project Number 40702050. The
authors are grateful to the organizations that provided the aforementioned financial support.
References [1] Yin YP, Wang FW, Sun P. Landslide hazards triggered by the 2008 Wenchuan earthquake, Sichuan, China. Recent Landslides 2009;6:139–51. [2] Wang YS, Xu HB, Luo YH, Wu JF. Study of formation conditions and toss motion program of high landslides induced by earthquake. Chin J Rock Mech Eng 2009;28(11):2360–8. [3] Hu XW, Huang RQ, Zhu HY, Lu XP, Zhang X, Shi YB. Earthquake reactivation effects and stability study of Malingyan landslide in Tangjiashan Dammed Lake. Chin J Rock Mech Eng 2009;28(6):1270–8. [4] Zheng Y, Han G, Zhao QH. Basic characteristic and formation mechanism of clastic flow landslide at Hejiagou occurred during the Wenchuan earthquake. J Geol Hazards Environ Preserv 2009;20(4):86–90. [5] International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests. Suggested methods for the quantitative description of discontinuities in rock masses. Int J Rock Mech Min Soc Geomech Abstr 1978; 15: 319–368.
268
Q. Wu et al. / Computers and Geotechnics 38 (2011) 258–268
[6] Kulatilake PHSW, Wathugala DN, Stephansson O. Joint network modeling including a validation to an area in Stripa Mine, Sweden. Int J Rock Mech Min Sci 1993;30(5):503–26. [7] Kulatilake PHSW, Chen J, Teng J, Shufang X, Pan G. Discontinuity geometry characterization for the rock mass around a tunnel close to the permanent shiplock area of the Three Gorges Dam Site in China. Int J Rock Mech Min Sci 1996;33:255–77. [8] Kulatilake PHSW, Um J, Wang M, Escandon RF, Narvaiz J. Stochastic fracture geometry modeling in 3-D including validations for a part of Arrowhead East Tunnel site, California, USA. Int J Eng Geol 2003;70(1–2):131–55. [9] Kulatilake PHSW, Wu TH. Sampling bias on orientation of discontinuities. Rock Mech Rock Eng 1984;17:243–53. [10] Wathugala DN, Kulatilake PHSW. A general procedure to correct sampling bias on joint orientation using a vector approach. Comput Geotech 1990;10:1–31. [11] Kulatilake PHSW, Wu TH. The density of discontinuity traces in sampling windows. Int J Rock Mech Min Sci Geomech Abstr 1984;21(6):345–7. [12] Cruden DM. Describing the size of discontinuities. Int J Rock Mech Min Sci Geomech Abstr 1977;14:133–7. [13] Baecher GB, Lanney NA. Trace length biases in joint surveys. In: Proc 19th US symp rock mech, Mackay School of Mines; 1978. p. 56–65. [14] Baecher GB. Progressively censored sampling of rock joints traces. Math Geol 1980;12(1):33–40. [15] Warburton PM. A stereological interpretation of joint trace data. Int J Rock Mech Min Sci Geomech Abstr 1980;17:181–90. [16] Priest SD, Hudson JA. Estimation of discontinuity spacing and trace length using scanline surveys. Int J Rock Mech Min Sci Geomech Abstr 1981;18:183–97.
[17] Pahl PJ. Estimating the mean length of discontinuity traces. Int J Rock Mech Min Sci Geomech Abstr 1981;18:221–8. [18] Laslett GM. Censoring and edge effects in areal and line transect sampling of rock joint traces. Math Geol 1982;14:125–40. [19] Kulatilake PHSW, Wu TH. Estimation of mean trace length of discontinuities. Rock Mech Rock Eng 1984;17:215–32. [20] Zhang X, Liao GH. Estimation of confidence bounds for mean trace length of discontinuities using scanline surveys. Int J Rock Mech Min Sci Geomech Abstr 1990;27(3):207–12. [21] Mauldon M. Estimating mean fracture trace length and density from observations in convex windows. Rock Mech Rock Eng 1998;31(4):201–16. [22] Zhang L, Einstein HH. Estimating the mean trace length of rock discontinuities. Rock Mech Rock Eng 1998;31(4):217–35. [23] Song JJ, Lee CI. Estimation of a joint length distribution using window sampling. Int J Rock Mech Min Sci 2001;38:519–28. [24] Song JJ. Estimation of areal frequency and mean trace length of discontinuities observed in non-planar surfaces. Rock Mech Rock Eng 2006;39(2):131–46. [25] Jimenez-Rodriguez R, Sitar N. Inference of discontinuity trace length distributions using statistical graphical models. Int J Rock Mech Min Sci 2006;43:877–93. [26] Wu Q, Tang HM, Wang LQ, Zhang GC. Three-dimensional connectivity rate and shear strength parameters calculation based on a new modified method of mean trace length. Chin J Rock Mech Eng 2008;27(Suppl. 2):3941–9. [27] Parker P, Cowan R. Some properties of line segment processes. J Appl Probab 1976;13:96–107.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Comparison of rock discontinuity mean trace length and density estimation methods using discontinuity data from an outcrop in Wenchuan area, China Qiong Wu a,b, P.H.S.W. Kulatilake c,⇑, Hui-ming Tang a a
Engineering Faculty, China University of Geosciences, Wuhan 430074, China Visiting Research Scholar, Geological Engineering Program, Department of Materials Science & Engineering, University of Arizona, Tucson, AZ 85721, USA c Geological Engineering Program, Department of Materials Science & Engineering, University of Arizona, Tucson, AZ 85721, USA b
a r t i c l e
i n f o
Article history: Received 23 October 2010 Accepted 13 December 2010 Available online 15 January 2011 Keywords: Rock mass Discontinuity geometry Sampling window Mean trace length Density
a b s t r a c t The equations that exist in the literature to estimate corrected mean trace length and corrected twodimensional density of a rock discontinuity set using area sampling technique are critically reviewed. The discontinuity traces appearing in an outcrop in Yingxiu area in China are used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations. Similarly, circular windows are used along with Mauldon’s and Zhang and Einstein’s equation to calculate the mean trace length and Mauldon’s equation to calculate the two-dimensional density for the same discontinuity sets using the same outcrop discontinuity trace data. For both parameters, the predictions based on the rectangular window methods were found to be more accurate than that based on the circular window methods. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction On May 12, 2008, the Wenchuan earthquake of Richter scale 8.0 occurred in the Longmenshan fault zone in Sichuan province, China. This disastrous earthquake triggered enormous landslides. Investigations indicated that Wenchuan earthquake directly caused more than 15,000 geohazards in the form of landslides, rockfalls, and debri flows [1]. In addition, it resulted in approximately 20,000 deaths [1]. Several research studies were initiated to investigate deformation and failure mechanisms of earthquake induced landslides in Wenchuan area [2–4]. Proper research on such topics cannot be carried out without a sound knowledge of the discontinuity geometry system that exists in the rock mass. Yingxiu Town, located in Wenchuan County, is the heaviest disaster area of Wenchuan Earthquake. Fig. 1 shows an almost vertical rock slope in Yingxiu Town. It is an excellent exposed outcrop for the study on rock mass discontinuity structure in this area. In this paper, discontinuity mean trace length and density are estimated using different methods available in the literature. Such studies are valuable to perform subsequent rock mass stability analysis in Wenchuan area under earthquake loading. A high definition photograph of the discontinuities that exist on the outcrop was taken in the field. Comprehensively considering the field sampling data and processing the results of the photograph, the discontinuity trace network of the outcrop was obtained ⇑ Corresponding author. Tel.: +1 520 621 6064; fax: +1 520 621 8059. E-mail address: [email protected] (P.H.S.W. Kulatilake). 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.12.003
(Fig. 2). In addition, discontinuity orientations were collected from the outcrop by running several scanline surveys. These data are used to estimate discontinuity mean trace length and density using different methods available in the literature. Accuracy of the methods is evaluated. 2. Literature review It is generally recognized that the strength, deformability, stability and percolation characteristics of a rock mass are strongly influenced by the discontinuities that are present in the rock mass. Quantitative descriptions of discontinuities have been proposed by the International Society for Rock Mechanics [5]. Discontinuity geometry for a rock mass is characterized by the number of discontinuity sets, mean density, and the distributions for location, orientation, size and spacing for each discontinuity set [6–8]. Estimation of the aforementioned discontinuity geometry parameters are commonly made by means of measurements of the discontinuity geometry on one or two-dimensional exposures. The trace length observed on two-dimensional exposures is related to discontinuity size in three-dimensions. It is important to note that the trace length, produced by a discontinuity on a two-dimensional exposure, depends on the shape and size of the discontinuity, the orientation of the discontinuity, the angle between the discontinuity plane and the plane of exposure, and the dimensions of the exposure [9,10]. The last three variables can be measured in exposures. However, determination of discontinuity shape and size is extremely difficult. The number of trace mid points per unit area can
Q. Wu et al. / Computers and Geotechnics 38 (2011) 258–268
259
Fig. 1. Pictorial view of the exposed outcrop in Wenchuan area, China.
0
50
100
150
Average cut strike:180¡ ã
200 (m) Qel+dl 4
Fig. 2. Discontinuity trace network of the exposed outcrop in Wenchuan area, China.
be used as a measure of discontinuity density in two dimensions [11]. Several sampling biases have been identified in estimating mean trace length and two-dimensional density of discontinuities [9–19]. Estimation of mean trace length and density has received much concern in the literature in the areas of rock mechanics as well as geological and applied statistics. Many investigators have conducted research on the topic of estimation of mean trace length of discontinuities [6–8,12–26]. A few researchers [6,11,21] have performed investigations on density of discontinuities in twodimensional exposures. Sampling plans used for trace length measurements can basically be divided into two types. They are: (1) sampling the traces that intersect a line drawn on the exposure, known as scanline survey and (2) sampling the traces that appear on a finite-size window made on a two-dimensional exposure, known as area sampling survey. According to the selected sampling method, the estimation method of the mean trace length and density varies. Estimation of the corrected mean trace length (corrected for sampling biases) from measured trace lengths using scanline survey has been described by Cruden [12], Priest and Hudson [16], Zhang and Liao [20] and Kulatilake et al. [6,8]. Estimation of corrected mean trace length and two-dimensional density using window sampling is dealt with in detail in this paper. Two main equations have been suggested in the literature to estimate mean trace length based on window sampling. A summary on that is given below. Pahl [17] derived an estimator of corrected mean trace length (l) for discontinuities observed in mine drive walls by assuming that trace midpoints are randomly and uniformly distributed. The method corrects for censoring and size biases. The method is independent of the distribution of trace length and restricted to a discontinuity set whose orientation has a single value. Kulatilake and Wu [19] advanced Pahl’s method to discontinuities whose orientation is described by a probability density function. In deriving the equation, they divided the discontinuities into
three groups: (a) those with both ends censored, (b) those with one end censored, and (c) those with both ends observable. The method provides corrections for the size and censoring biases for traces appearing on a rectangular two-dimensional exposure. The lengths of observed traces and the probability density function of trace length are not required to apply the corrections. The proposed equation is given below:
l¼
whð1 þ R0 R2 Þ ð1 R0 þ R2 ÞðwB þ hAÞ
ð1aÞ
where
A¼
Z
au
Z
al
B¼
Z
au
al
cos u ¼
sin u ¼
hu
j cos ujf ðh; aÞdh da
ð1bÞ
sin uf ðh; aÞdh da
ð1cÞ
hl
Z
hu hl
1 ð1 þ tan2 h cos2 dÞ1=2 1 ð1 þ cot2 h sec2 dÞ1=2
ð1dÞ
ð1eÞ
In Eqs. (1a)–(1e), w and h are width and height of a rectangular window, respectively; R0 and R2 denote the fractions of discontinuities with both ends censored and both ends observable, respectively; h and a are dip angle and dip direction; f(h, a) is the probability density function of discontinuity orientation with hl 6 h 6 hu and al 6 a 6 au , where subscripts l and u denote lower and upper limits, respectively; u is apparent dip angle and d denote the acute angle between the dip direction and the vertical sampling plane. Note that to use the aforementioned equation, it is not necessary to have a fitted theoretical probability density function for the discontinuity set orientation. In the absence of a theoretical
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probability density function, the equation can be used by replacing the integral sign by the summation sign and using the empirical orientation distribution obtained for the discontinuity set. The method allows one to incorporate the corrected relative frequency of each trace appearing on the two-dimensional exposure, due to the orientation bias, in the presence of a probabilistically distributed discontinuity set orientation. In addition, because this equation incorporates the angle between the discontinuity plane and the two-dimensional exposure plane, it can also be used to estimate mean trace length when traces appear on curved sampling surfaces such as tunnels having curved cross sections and outcrops that show some curvature. In such a case, the curved surface should be divided into a series of planar surfaces. Then for the traces appearing on each plane, the corrected relative frequencies can be calculated. Mauldon [21] derived estimators of mean trace length for windows with arbitrary convex boundaries. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. The estimator of corrected mean trace length he obtained for a rectangular window is the same as the one presented by Kulatilake and Wu [19]. The equation given for a circular window is as follows:
l¼
pr N þ NT NC 2
N NT þ NC
k¼
Pnb
Pc þ nj¼1 ½P 0 ðCÞj Nt na þ nb þ nc
i¼1 ½P 1 ðCÞi
ð3bÞ
R1 a R 2a f ðxÞdx f ðxÞdx P0 ðCÞ ¼ 2aR 1xa þ Ra1 f ðxÞdx f ðxÞdx a a
ð3cÞ
where a is the length of the intersected trace appearing within the window and f(x) is the probability density function of trace length (X) on the infinite two-dimensional plane. Note that to use the aforementioned equation, even though it is preferred, it is not necessary to have a fitted theoretical probability density function for the discontinuity set trace length. In the absence of a theoretical probability density function, the equation can be used by replacing the integral sign by the summation sign and using the empirical trace length distribution obtained for the discontinuity set. This equation has been used in practice to first estimate the two-dimensional density and then to estimate three-dimensional density in modeling discontinuity networks in three-dimensions for Stripa Mine, Sweden [6]. Mauldon [21] have suggested the following equation to estimate number of trace mid points per unit area using circular windows and based on the concepts given in the paper by Parker and Cowen [27].
ð2Þ k¼
where r is the radius of the circular sampling window; N, NT and NC are the number of traces intersecting, transecting and contained in the window, respectively. Zhang and Einstein [22] obtained the same equation as Mauldon [21] for a circular window using the general approach given by Kulatilake and Wu [19]. It is important to note that the equation given for a circular window is based on the implicit assumption that each trace has the same chance of appearing on the sampling window. This assumption is applicable only for the deterministic orientation (i.e. parallel traces on the two-dimensional exposure) and it is not applicable in the usual case where the discontinuity orientation has a scatter and is probabilistically distributed. It is important to note that the corrected relative frequency of a discontinuity trace is linked to the orientation bias. Because the aforementioned equation given for a circular window does not have the capability of calculating corrected relative frequencies, it cannot be used to estimate mean trace length when traces appear on non-planar two-dimensional exposures. Song [24] made an attempt to extend Mauldon’s [21] equation for a circular window for non-planar surfaces using the orientation bias concept. Even though he correctly pointed out the need for the extension, the equation given in Song’s [24] paper for the extension is incorrect. Two main equations have been suggested in the literature to estimate two-dimensional density of discontinuities (number of discontinuity trace midpoints per unit area) based on window sampling. A summary on that is given below. Kulatilake and Wu [11] proposed the following equation to estimate number of trace mid points per unit area, k, starting from number of traces per unit area, Nt, counted on a rectangular sampling domain.
na þ
R 2a f ðxÞdx P1 ðCÞ ¼ Ra1 f ðxÞdx a
ð3aÞ
where na, nb, nc denote the number of discontinuities of both ends observed (type a), one end observed (type b) and both ends censored (type c), respectively; i denotes the ith type (b) discontinuity; j denotes the jth type (c) discontinuity; P1(C) and P0(C) respectively denote the probability that the trace midpoint of the type (b) and type (c) discontinuities are within the window. P1(C) and P0(C) can be expressed as
ðN NT þ NC Þ 2p r 2
ð4Þ
where r, N, NT and NC have the same meaning as those in Eq. (2). A trace length distribution is not required to use this method. It is important to note that Parker and Cowen have assumed that the line segment orientation on a two-dimensional plane follows a uniform distribution. It is impossible to satisfy this assumption in dealing with rock discontinuity traces appearing from a distinct discontinuity set orientation. Eq. (4) does not show a contribution from both ends censored traces to the parameter number of mid points per unit area. In addition, the equation implies the number of mid points resulting from one end observed traces to be half the number of such trace type irrespective of the type of the trace length distribution and size of the sampling domain. The latter two concepts are also definitely not acceptable. In summary, the correctness or validity of this equation is highly questionable. The estimation techniques summarized above can be classified into two types: Type 1 uses a rectangular sampling window and type 2 uses a circular sampling window. Discontinuity traces appearing in the outcrop in Yingxiu area are used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations [19,11]. Similarly, circular windows are used along with Mauldon’s [21] and Zhang and Einstein’s equation [22] to calculate the mean trace length and Mauldon’s [21] equation to calculate the twodimensional density for the discontinuity traces that exist on the same outcrop. The obtained results are compared. 3. Mean trace length estimations 3.1. Corrected mean trace length based on rectangular windows Eq. (1) given earlier is used to estimate corrected mean trace length using rectangular windows. The expectation values for sin u and cos u appearing in Eqs. (1b) and (1c) can be computed approximately by
Eðsin uÞ
N X ½sin ui Rf ðui Þ
ð5aÞ
i¼1
Eðj cos ujÞ
N X ½j cos ui j Rf ðui Þ i¼1
ð5bÞ
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where ui denotes the apparent dip angle of the ith trace and Rf(ui) denotes the corrected relative frequency of the ith trace appearing on the window with the apparent dip angle ui resulting from a corresponding dip angle and a dip direction combination (h, a). The Rf(ui) can be calculated as given below. The probability of intersection between a discontinuity and a sampling domain is proportional to the volume (Vi) within which the center of the discontinuity should lie in order to intersect the sampling domain [10]. According to the stated hypothesis, the Rf(ui) can be calculated by Eqs. (6)–(8) as given below [10]. 2
V i ¼ whdi ½cos2 hi þ sin hi cos2 ðar ai Þ0:5 þ
p
2 di
4
½w sin hi j cosðar ai Þj þ h cos hi
ð6Þ
In Eq. (6), ai and hi denote the dip direction and dip angle of the ith discontinuity, respectively; di is the diameter of the ith discontinuity and ar is the strike of the sampling window. In the absence of diameter values for discontinuities, a diameter value equal to 10% higher than the maximum trace length may be used for all the discontinuities in using Eq. (6). The weighting function for the ith discontinuity (Wi) is given by
Wi ¼
1 Vi
ð7Þ
Then the corrected relative frequency of the ith discontinuity is given by
Wi Rf ðui Þ ¼ PN i¼1 W i
ð8Þ
3.2. Application of the proposed mean trace length equation for rectangular windows Orientation data of the discontinuities on this outcrop were measured by five scanlines. The pole (normal vector) concentration plot of these data is shown in Fig. 3. According to the plot, discontinuities on this outcrop can be divided into two sets: set 1 features dip direction from 105° to 200° and dip angle from 20° to 66°, set 2 features dip direction from 301° to 331° and dip angle from 33° to 76°. The strike of the outcrop changes slightly from 165° to 190° with a mean value of 180°. Therefore, two sets of discontinuities respectively produce traces on the outcrop, one dipping approximately to the south (set 1) and the other dipping approximately to the north (set 2). A series of rectangular windows were placed
on the outcrop to estimate the corrected mean trace length of the discontinuities (see Fig. 4). There are four window locations (1–4), and at each location, there are three windows with the following sizes: 90 m 45 m, 110 m 55 m and 130 m 65 m. Apparent dip angles of all the discontinuities appearing on these windows were measured. In order to calculate the corrected relative frequency for each discontinuity trace, the orientation values corresponding to each discontinuity trace were obtained through the following steps: (1) calculate the apparent dip angle for each discontinuity orientation measured by the scanline by using the strike value of the exposure; (2) compare between the measured apparent dip angle and calculated apparent dip angles and choose the orientation values corresponding to each discontinuity trace needed to calculate the corrected mean trace length. Then the corrected mean trace length values for discontinuities of the two sets belonging to all the 12 windows were calculated by the method explained in Section 3.1. The results are shown in Table 1 and Fig. 5. The obtained overall mean value and coefficient of variation for set 1 are respectively 39.8 m and 0.223, and that for set 2 are 33.4 m and 0.118, respectively. These overall estimated mean values are also shown in Fig. 5. The total trace length of each discontinuity trace intersected with each sampling window on the outcrop is known. Using this data, for each window a true mean trace length value was calculated (Table 1). The overall true mean values resulting for set 1 and set 2 from all these true mean trace lengths (Table 1) are also shown in Fig. 5. For set 2, the overall estimated mean value is almost the same as the overall true mean value. For set 1, the difference between the two parameters is only 7.66%. Fig. 5 shows that the individual estimated mean trace length values vary with the location and the size of the rectangular windows. This is expected when estimations are made using finite size samples. An estimation error for the prediction may be defined as
errorl ¼
lt le 100% lt
ð9Þ
where lt and le denote the true and estimated mean trace length values, respectively. This error can be used to evaluate the accuracy of the proposed prediction method. Table 1 and Fig. 6 show the errors resulted by applying Kulatilake and Wu’s [19] method by placing different size rectangular windows at different locations on the outcrop. The range of error resulting from finite sample sizes and spatial variability is 16.0% to +18.0% for set 1 and 19.3% to +21.3% for set 2.
N Fisher Concentrations % of total per 1.0 % area
1m 2m
W
E
2m
0.00 ~ 2.00 % 2.00 ~ 4.00 % 4.00 ~ 6.00 % 6.00 ~ 8.00 % 8.00 ~ 10.00 % 10.00 ~ 12.00 % 12.00 ~ 14.00 % 14.00 ~ 16.00 % 16.00 ~ 18.00 % 18.00 ~ 20.00 %
No Bias Correction Max. Conc. = 18.7784%
1m
Equal Area Lower Hemisphere 107 Poles 107 Entries
S Fig. 3. Pole (normal vector) concentrations of discontinuity orientations on a lower hemispherical equal area plot.
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0
100
50
150
200 (m)
Location 2 Location 1
Location 3
Location 4
Fig. 4. Sizes and locations of rectangular windows used to estimate mean trace length.
Table 1 Estimated corrected mean trace length and its% error based on rectangular sampling windows. Window size (m)/location number
Estimated value (m)
90 45/location 1 90 45/location 2 90 45/location 3 90 45/location 4 110 55/location 1 110 55/location 2 110 55/location 3 110 55/location 4 130 65/location 1 130 65/location 2 130 65/location 3 130 65/location 4 Mean value Coefficient of variation Range
% Error of le
True value (m)
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
23.7 50.0 34.3 46.8 29.4 47.6 33.6 50.6 36.9 39.7 36.0 48.4 39.8 0.223
27.3 33.9 36.8 39.7 29.2 35.9 33.4 33.5 30.9 31.1 29.0 39.9 33.4 0.118
28.9 54.4 40.7 47.8 31.5 50.1 40.7 51.4 31.8 45.0 40.4 54.7 43.1 0.208
34.7 33.3 36.9 35.3 32.2 30.1 35.8 36.5 32.7 31.3 33.9 34.7 34.0 0.062
18.0 8.1 15.7 2.1 6.7 5.0 17.4 1.6 16.0 11.8 10.9 11.5
21.3 1.8 0.3 12.5 9.3 19.3 6.7 8.2 5.5 0.6 14.5 15.0
16.0 to +18.0
19.3 to +21.3
Mean trace length (m)
80
90×45
110×55
130×65
Overall estimated mean
60
Overall true mean
40 20 0 1
2
3
4
Location number
(a) Set
Mean trace length (m)
80
90×45 60
110×55
130×65
Overall estimated mean
Overall true mean
40 20
0 1
2
3
4
Location number
(b) Set 2 Fig. 5. Estimated mean trace length based on Kulatilake and Wu [19] method using rectangular windows of different sizes placed at different locations on the outcrop.
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100
% Error
50
Location 1
Location 2
Location 4
Zero line
Location 3
0 -50 -100 90×45
110×55
130×65
Window size (m)
(a) Set 1 100
% Error
50
Location 1
Location 2
Location 4
Zero line
Location 3
0 -50
-100 90×45
110×55
130×65
Window size (m)
(b) Set 2 Fig. 6. Prediction error of mean trace length associated with finite sample sizes and spatial variability based on Kulatilake and Wu’s [19] method using rectangular windows of different sizes placed at different locations on the outcrop.
3.3. Mean trace length estimation using circular sampling windows As stated in Section 2, Mauldon [21] and Zhang and Einstein [22] proposed Eq. (2) to estimate corrected mean trace length using circular sampling windows. To estimate the corrected mean trace length of discontinuities using Eq. (2), in the outcrop a series of circular windows were placed on the discontinuity trace network as shown in Fig. 7. Five window locations were chosen and at each location three windows of radii 20 m, 30 m and 40 m were placed. Predicted results are shown in Table 2 and Fig. 8. The overall predicted means, overall true means and estimation errors were calculated similar to the way they were done for the rectangular windows. Table 2 and Fig. 9 show the errors resulted by applying Mauldon’s [21] and Zhang and Einstein’s [22] method by placing different size circular windows at different locations on the outcrop. The range of error resulting from finite sample sizes and spatial variability for circular windows is 14.6% to +41.5.0% for set 1 and 66.1% to +36.5% for set 2. 3.4. Comparison of results from the two mean trace length estimation methods Overall mean predictions obtained through the rectangular window method were closer to the overall true values than that
0
50
100
obtained through the circular window method. The estimation error ranges obtained for the circular window method are considerably larger than that resulted from the rectangular window method. The coefficient of variation values given in Tables 1 and 2 also indicate that predictions based on the rectangular window method show less variability than that of the circular window method due to the variability of window location and size. The aforementioned findings indicate that the predictions based on the rectangular window method are more accurate than that of the circular window method. 4. Density estimation 4.1. Density estimation of discontinuity traces using rectangular sampling windows As stated in Section 2, Kulatilake and Wu [11] proposed Eq. (3) to estimate number of discontinuity trace mid points per unit area. To use Eq. (3a) to predict k, three rectangular windows of size (120 m 40 m) were placed on the outcrop at different locations (Fig. 10). To calculate P1(C) and P0(C) given in Eqs. (3b) and (3c), the integration sign was replaced by the summation sign and the probability density function was replaced by relative frequencies as given below:
150
Location 1Location 2 Location 3 Location 4
200 (m)
Location 5
Fig. 7. Sizes and locations of the circular windows used to estimate mean trace length.
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Table 2 Estimated corrected mean trace length and its% error based on circular sampling windows. Window size (m)/location number
20/location 1 20/location 2 20/location 3 20/location 4 20/location 5 30/location 1 30/location 2 30/location 3 30/location 4 30/location 5 40/location 1 40/location 2 40/location 3 40/location 4 40/location 5 Mean value Coefficient of variation Range
Mean trace length (m)
80
% Error of le
Estimated value (m)
True value (m)
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
10.5 56.5 57.6 26.4 44.0 17.0 41.9 47.1 30.0 32.6 29.1 49.3 36.2 36.1 55.0 38.0 0.3718
39.3 41.9 31.4 49.3 18.8 47.1 43.5 20.9 29.4 23.6 30.5 29.9 33.6 34.9 43.0 34.5 0.2722
15.6 64.8 55.0 45.2 41.1 22.2 54.9 53.6 43.5 44.8 25.4 46.4 48.8 38.2 48.1 43.2 0.3078
36.7 29.5 33.9 44.4 26.3 37.1 26.2 32.0 32.1 37.1 33.0 24.5 31.1 35.7 39.8 33.3 0.1628
32.8 12.7 4.63 41.5 6.86 23.7 23.7 12.2 31.1 27.2 14.6 6.41 25.7 5.58 14.3
7.00 41.9 7.42 11.0 28.4 26.9 66.1 34.7 8.42 36.5 7.60 22.2 8.15 2.30 8.06
14.6 to +41.5
66.1 to +36.5
r=20
r=30
r=40
Overall estimated mean
Overall true mean
60 40 20 0 1
2
3
4
5
4
5
Location number
(a) Set 1
Mean trace length (m)
80
r=20 r=30 Overall estimated mean
60
r=40 Overall true mean
40 20
0 1
2
3
Location number
(b) Set 2 Fig. 8. Estimated mean trace length based on Mauldon’s [21] and Zhang and Einstein’s [22] equation using circular windows of different sizes placed at different locations on the outcrop.
P1 ðCÞ ¼
P2a x¼a Rft P 1 ax¼0 Rft
ð10aÞ
Pxmax P0 ðCÞ ¼
1
a x¼2a xi a Rft Pa x¼0 Rft
þ
P2a x¼a Rft P 1 ax¼0 Rft
ð10bÞ
where Rft denotes the relative frequency of trace length in a certain length interval; xi denotes the mean trace length in the length interval and xmax is the maximum value of the trace length. Relative frequencies can be calculated using the histogram obtained for the trace length. A typical histogram obtained for trace length is given in Fig. 11.
Let ke and kt denote the estimated and the true density value, respectively. Because the distributions of all the traces in the three windows are known, kt can be calculated. Table 3 and Fig. 12 show the results of density estimations. Error in the prediction was calculated using the following equation.
errork ¼
kt ke 100% kt
ð11Þ
Values given in Table 3 and Fig. 12 show that predicted densities are quite close to the true values. Fig. 12 also shows that significant differences exist between the values obtained for number of traces per unit area and number of trace mid points per unit area.
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100
Location 1 Location 4
% Error
50
Location 2 Location 5
Location 3 Zero line
0 -50
-100
20
30
40
Radius (m)
(a) Set 1 100
% Error
50
Location 1
Location 2
Location 3
Location 4
Location 5
Zero line
0 -50
-100 20
30
40
Radius (m)
(b) Set 2 Fig. 9. Prediction error of mean trace length associated with finite sample sizes and spatial variability based on Mauldon’s [21] and Zhang and Einstein’s [22] equation using circular windows of different sizes placed at different locations on the outcrop.
0
100
50
Location 1
Location 2
150
200 (m)
Location 3
Fig. 10. Selected size and locations of rectangular windows for density estimation through Kulatilake and Wu’s [11] method.
Relative frequency density (m-1)
0.045 0.04 Relative frequency density=Relative frequency / Interval length
0.035
That indicates importance of the sampling bias correction in estimating two-dimensional density of discontinuity traces. Prediction errors range between 13.6% and +16.0% for set 1 and between 0% and +14.8% for set 2 to reflect the influence of finite sample size and spatial variability.
0.03 0.025
4.2. Density estimation of discontinuity traces using circular windows
0.02 0.015 0.01 0.005 0
0
20
40
60
80
100
120
Trace length (m) Fig. 11. Histogram for trace length of discontinuities of set 1 at location 1.
As stated in Section 2, Mauldon [21] proposed Eq. (4) to estimate number of discontinuity trace mid points per unit area based on circular sampling windows. To apply this method, four circular windows of size r = 25 m were placed on the outcrop as shown in Fig. 13. True density values for circular windows were calculated as for the rectangular windows. Table 3 and Fig. 14 show the obtained results. Values given in Table 3 and Fig. 14 show that predicted densities deviate significantly from the true values. Prediction errors range between 0% and +50.0% for set 1 and between 72.2% and +24.2% for set 2.
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Table 3 Estimated two-dimensional densities of discontinuities and their% errors based on rectangular and circular sampling windows. Nt (m2)
Method/location number
ke (m2)
kt (m2)
% Error of ke
Set 1
Set 2
Set 1
Set 2
Set 1
Set 2
Set 1
Rectangular window Location 1 Location 2 Location 3 Mean value Coefficient of variation Range
0.0054 0.0075 0.0050 0.0060 0.2251
0.0054 0.0048 0.0042 0.0048 0.1250
0.0036 0.0042 0.0026 0.0035 0.2332
0.0035 0.0024 0.0023 0.0027 0.2436
0.0035 0.0050 0.0023 0.0036 0.3758
0.0035 0.0027 0.0027 0.0030 0.1557
2.9 16.0 13.0
Circular window Location 1 Location 2 Location 3 Location 4 Mean value Coefficient of variation Range
0.0061 0.0061 0.0112 0.0046 0.0070 0.4126
Set 2 0.0 11.1 14.8
13.0 to +16.0 0.0061 0.0036 0.0051 0.0061 0.0052 0.2261
0.0046 0.0018 0.0038 0.0025 0.0032 0.3970
0.0036 0.0031 0.0046 0.0025 0.0034 0.2576
0.0046 0.0036 0.0061 0.0036 0.0045 0.2640
0.0028 0.0018 0.0033 0.0033 0.0028 0.2525
0.0 50.0 37.7 30.6
0.0 to +14.8 28.6 72.2 39.4 24.2
0.0 to +50.0
72.2 to +24.2
0.01
Density (m-2)
Nt
t
e
0.005
0
1
2
3
Location number
(a) Set 1 0.01
Density (m-2)
Nt
t
e
0.005
0
1
2
3
Location number
(b) Set 2 Fig. 12. Predicted results of two-dimensional discontinuity density by Kulatilake and Wu’s [11] method using rectangular windows.
4.3. Comparison of results from the two density estimation methods
5. Conclusions
Predicted densities obtained through the rectangular window method are quite close to the true values. Prediction error ranges obtained for the circular window method are considerably larger than that resulted for the rectangular window method. The coefficient of variation values given in Table 3 also indicate that predictions based on the rectangular window method show less variability than that of the circular window method due to the influence of finite sample size and the variability of location of the window. The aforementioned findings indicate that predictions based on the rectangular window method are far more accurate than that of the circular window method.
The discontinuity traces appearing in the outcrop in Yingxiu area were used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations [19,11]. Similarly, circular windows were used along with Mauldon’s [21] and Zhang and Einstein’s equation [22] to calculate the mean trace length and Mauldon’s [21] equation to calculate the two-dimensional density for the discontinuity traces that exist on the same outcrop. For both the corrected mean trace length and the two-dimensional density, the predictions based on the rectangular window methods were found to be more accurate than that based on the circular window
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0
50
Location 2 Location 1
100
150
200 (m)
Location 3 Location 4
Fig. 13. Selected sizes and locations of circular windows for density estimation through Mauldon’s [21] method.
0.012
Nt
Density (m-2)
0.01
t
e
0.008 0.006 0.004 0.002 0
1
2
3
4
3
4
Location number
(a) Set 1 0.01
Nt
t
e
Density (m-2)
0.008 0.006 0.004 0.002 0
1
2
Location number
(b) Set 2 Fig. 14. Predicted results of two-dimensional discontinuity density based on Mauldon’s method [21] using circular windows.
methods. A shortcoming of the equation proposed to calculate the corrected mean trace length of a discontinuity set based on a circular sampling window is pointed out. The authors have pointed out the major shortcomings that exist with Mauldon’s equation [21] proposed for the estimation of two-dimensional density of a discontinuity set using circular windows. Due to these shortcomings, the correctness or validity of this equation seems highly questionable.
Acknowledgments This research was funded by The Chinese Geological Survey Project Number 1212010914036, The National Basic Research Program of China (973 Program) Project Number 2011CB710600, the Special Fund for Basic Scientific Research of Central Colleges, China University of Geosciences, Wuhan (CUG090104) and The National Natural Science Youth Foundation Project Number 40702050. The
authors are grateful to the organizations that provided the aforementioned financial support.
References [1] Yin YP, Wang FW, Sun P. Landslide hazards triggered by the 2008 Wenchuan earthquake, Sichuan, China. Recent Landslides 2009;6:139–51. [2] Wang YS, Xu HB, Luo YH, Wu JF. Study of formation conditions and toss motion program of high landslides induced by earthquake. Chin J Rock Mech Eng 2009;28(11):2360–8. [3] Hu XW, Huang RQ, Zhu HY, Lu XP, Zhang X, Shi YB. Earthquake reactivation effects and stability study of Malingyan landslide in Tangjiashan Dammed Lake. Chin J Rock Mech Eng 2009;28(6):1270–8. [4] Zheng Y, Han G, Zhao QH. Basic characteristic and formation mechanism of clastic flow landslide at Hejiagou occurred during the Wenchuan earthquake. J Geol Hazards Environ Preserv 2009;20(4):86–90. [5] International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests. Suggested methods for the quantitative description of discontinuities in rock masses. Int J Rock Mech Min Soc Geomech Abstr 1978; 15: 319–368.
268
Q. Wu et al. / Computers and Geotechnics 38 (2011) 258–268
[6] Kulatilake PHSW, Wathugala DN, Stephansson O. Joint network modeling including a validation to an area in Stripa Mine, Sweden. Int J Rock Mech Min Sci 1993;30(5):503–26. [7] Kulatilake PHSW, Chen J, Teng J, Shufang X, Pan G. Discontinuity geometry characterization for the rock mass around a tunnel close to the permanent shiplock area of the Three Gorges Dam Site in China. Int J Rock Mech Min Sci 1996;33:255–77. [8] Kulatilake PHSW, Um J, Wang M, Escandon RF, Narvaiz J. Stochastic fracture geometry modeling in 3-D including validations for a part of Arrowhead East Tunnel site, California, USA. Int J Eng Geol 2003;70(1–2):131–55. [9] Kulatilake PHSW, Wu TH. Sampling bias on orientation of discontinuities. Rock Mech Rock Eng 1984;17:243–53. [10] Wathugala DN, Kulatilake PHSW. A general procedure to correct sampling bias on joint orientation using a vector approach. Comput Geotech 1990;10:1–31. [11] Kulatilake PHSW, Wu TH. The density of discontinuity traces in sampling windows. Int J Rock Mech Min Sci Geomech Abstr 1984;21(6):345–7. [12] Cruden DM. Describing the size of discontinuities. Int J Rock Mech Min Sci Geomech Abstr 1977;14:133–7. [13] Baecher GB, Lanney NA. Trace length biases in joint surveys. In: Proc 19th US symp rock mech, Mackay School of Mines; 1978. p. 56–65. [14] Baecher GB. Progressively censored sampling of rock joints traces. Math Geol 1980;12(1):33–40. [15] Warburton PM. A stereological interpretation of joint trace data. Int J Rock Mech Min Sci Geomech Abstr 1980;17:181–90. [16] Priest SD, Hudson JA. Estimation of discontinuity spacing and trace length using scanline surveys. Int J Rock Mech Min Sci Geomech Abstr 1981;18:183–97.
[17] Pahl PJ. Estimating the mean length of discontinuity traces. Int J Rock Mech Min Sci Geomech Abstr 1981;18:221–8. [18] Laslett GM. Censoring and edge effects in areal and line transect sampling of rock joint traces. Math Geol 1982;14:125–40. [19] Kulatilake PHSW, Wu TH. Estimation of mean trace length of discontinuities. Rock Mech Rock Eng 1984;17:215–32. [20] Zhang X, Liao GH. Estimation of confidence bounds for mean trace length of discontinuities using scanline surveys. Int J Rock Mech Min Sci Geomech Abstr 1990;27(3):207–12. [21] Mauldon M. Estimating mean fracture trace length and density from observations in convex windows. Rock Mech Rock Eng 1998;31(4):201–16. [22] Zhang L, Einstein HH. Estimating the mean trace length of rock discontinuities. Rock Mech Rock Eng 1998;31(4):217–35. [23] Song JJ, Lee CI. Estimation of a joint length distribution using window sampling. Int J Rock Mech Min Sci 2001;38:519–28. [24] Song JJ. Estimation of areal frequency and mean trace length of discontinuities observed in non-planar surfaces. Rock Mech Rock Eng 2006;39(2):131–46. [25] Jimenez-Rodriguez R, Sitar N. Inference of discontinuity trace length distributions using statistical graphical models. Int J Rock Mech Min Sci 2006;43:877–93. [26] Wu Q, Tang HM, Wang LQ, Zhang GC. Three-dimensional connectivity rate and shear strength parameters calculation based on a new modified method of mean trace length. Chin J Rock Mech Eng 2008;27(Suppl. 2):3941–9. [27] Parker P, Cowan R. Some properties of line segment processes. J Appl Probab 1976;13:96–107.