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Estimation of joint length distribution using window sampling
International Journal of Rock Mechanics & Mining Sciences 38 (2001) 519–528
Estimation of joint length distribution using window sampling Jae-Joon Song, Chung-In Lee* School of Civil, Urban and Geosystem Engineering, Seoul National University, Seoul, South Korea Accepted 18 March 2001
Abstract One of the most difficult procedures in statistical joint modeling is the determination of the joint length distribution. In this study, a window sampling method was adopted to estimate the trace length distribution for the Poisson disc joint model. Four kinds of equations were derived for estimating the trace length distribution from contained or dissecting trace length distributions in a rectangular or circular window. The equations were tested for accuracy through Monte Carlo simulation and their efficiencies for estimating the trace length distribution were compared. This new technique using window sampling was compared with the method using a semi-trace or complete trace length distribution from scanline sampling. A numerical technique for determining the diameter distribution from the trace length distribution was also suggested. To check the validity of this numerical method, it was applied to solving four example cases and their results were compared with theoretical solutions. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The statistical joint modeling technique is frequently applied to analyses of hydraulic conductivity or of the stability of rock blocks, which can be strongly affected by the geometrical characteristics of the joints. One of the least reliable procedures in joint modeling is the estimation of the joint length distribution. When the joints are assumed to be Poisson discs, there are two steps in determining the joint length distribution: the estimation of the trace length distribution on an infinite plane and the estimation of the joint diameter distribution in an infinite space. The joint length distribution has been inferred mainly from complete trace length or semi-trace length distributions obtained in scanline surveys [1]. For whatever sampling policy is used, however, it is an ill-posed problem to build the joint diameter distribution using a trace length distribution estimated from field survey data. This is due to the fact that trace length distributions are not sensitive to major changes in joint diameter distributions [2]. An alternative solution to this problem is to assume the distribution of the trace length or joint diameter in advance and then to find the proper *Corresponding author. Tel.: +82-2-880-8708; fax: +82-2-8732717. E-mail address: [email protected] (C.-I. Lee).
trace length or joint diameter distribution that best reflects the experimental data [2,3]. This method is useful, both for obtaining a well-defined continuous distribution of the trace length or joint diameter, and especially for overcoming the limit of a monotonically decreasing function of the semi-trace length distribution. With this method, however, it is still an unresolved problem that the final trace length or joint diameter distribution is extremely sensitive even to a small sampling error in the experimental data. Moreover, this is a distribution-dependent method. To improve the accuracy of the estimated joint diameter and trace length distributions, an effective technique is desired by which the trace length distribution can be more precisely estimated using the experimental data. From this, the diameter distribution may be obtained without any assumption regarding its form. To find a more reliable and distribution-independent method of estimating the trace length distribution for the Poisson disc joint model, trace lengths observed in sampling windows were used instead of scanline sampling in this study. Several studies for estimating a mean trace length [4–7] or an areal frequency [7] using window sampling have been reported. Areal frequency has the same meaning as areal density, that is, the average number of trace centers per unit area [3]. Hereafter, areal frequency will be used consistent with linear frequency, which is defined as the average
1365-1609/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 1 8 - 1
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number of traces intersected by a unit length of a sampling line. Traces can be classified into three categories according to the number of their end points observed in a window. A contained trace has two observed ends, a dissecting trace has only one and a transecting trace has zero. Here, four kinds of equations were derived for estimating the trace length distribution from contained or dissecting trace length distributions in a rectangular or circular window. Transecting traces were indirectly involved in developing the four kinds of equations in process of obtaining an areal frequency, a commonly used factor in each equation. The trace length distribution was not estimated using the transecting trace lengths because when estimated from partial lengths of traces, the trace length distribution inevitably becomes more erroneous than when it is estimated from the same number of full trace lengths. This is intuitive, and it has also been confirmed by comparing the errors on trace length distributions estimated from contained and dissecting trace length distributions. Biases in sampling trace lengths can be divided into three categories: size (length) bias, truncation bias and censoring bias [5]. The size bias means that longer joints are more likely to intersect a sampling plane and make traces. The truncation bias occurs when trace lengths below some cutoff length are not recorded. The censoring bias refers to traces whose end points cannot be seen, so only lower bound estimates of their lengths may be determined. In this study, the size bias problem is resolved by considering geometric characteristics of contained or dissecting traces in the process of deriving each equation. The truncation bias is not considered because in practice, its effects can be made negligible by choosing a cutoff length that is small compared to the average trace length [5]. It is impossible to determine probability density values for trace lengths longer than the maximum observable length unless the sampled length or the trace length distribution is predefined. To avoid this problem, the sampling window here is assumed to be large enough to observe the maximum joint diameter and its trace length. In a practical environment where the censoring bias can seriously affect confidence in the estimated trace length distribution, one possible solution has been given in this paper. However, the further effort for the censoring bias remains as future work. The efficiency of using a dissecting trace length distribution from a rectangular window in the estimation of the trace length distribution was compared with the use of a semi-trace length distribution. Then, the same comparison was performed using the contained and dissecting trace length distributions from rectangular windows, and using the complete trace length distribution from the scanline, that is, the distribution of whole lengths of traces intersecting a scanline.
We also suggested a numerical technique to determine directly the diameter distribution from the trace length distribution. This approach was verified by comparing the numerical solutions of example cases with their theoretical predictions.
2. Estimating trace length distributions from rectangular windows 2.1. Contained trace length distribution All the joints are assumed to have the same orientation but not to be parallel with a rectangular window plane of length W and height H. Let the complete trace length of a joint in the window be l and the angle between the trace and the horizontal boundary of the window be y (Fig. 1). Orientations of joints are usually recorded as strike/dip or dip direction/dip in field measurements. The first step in the calculation of y is to determine the normal vectors of the joint and the sampling window. Then, the direction vector of a trace on a sampling exposure can be determined by the cross product of the two normal vectors of the joint and sampling planes. Finally, y can be obtained from the dot product of the unit direction vectors of the trace and the horizontal boundary. If the midpoint of any trace is in the shaded rectangular area of the window, the trace becomes a contained trace. The area of the shaded rectangular Acl can be obtained by the following equation: Acl ¼ ðW l cos yÞðH l sin yÞ ¼ cos y sin yl 2 þ ðW sin y H cos yÞl þ WH ¼ al 2 þ bl þ c
ðl5LX Þ;
ð1Þ
where a ¼ cos y sin y; b ¼ W sin y H cos y; c ¼ WH, and LX is the maximum length of the contained trace, which is the smaller of W=cos y and H=sin y. In Eq. (1), Acl is expressed by a second order function of l for which
Fig. 1. Area that includes the midpoints of contained discontinuities.
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the coefficients are determined by the geometry of the window and the traces. When l is greater than LX , Acl becomes zero. The expected number of contained traces Nlc can be obtained by the product of the areal frequency of traces having lengths of l l þ dl and the area of midpoints shown by the shaded rectangular area in the figure as Nlc ¼ rA Acl f ðlÞ dl;
ð2Þ
where rA is the areal frequency of whole traces, and f ðlÞ is the probability density function of the trace length distribution. Both are defined in an infinite sampling plane. Because the probability that the contained trace has a c length of l l þ dl is f c ðlÞ dl and equals Nlc =Nall , the trace length density f ðlÞ can be obtained using Eqs. (1) and (2) as follows: f ðlÞ ¼
rA
ðal 2
c Nall f c ðlÞ; þ bl þ cÞ
ð3Þ
where f c ðlÞ is the probability density function of the c contained trace length distribution, and Nall is the number of all contained traces observed in the window. Mauldon [7] suggests the following equation for the areal frequency of the joint traces independent of the trace length distribution: t c Nall Nall þ Nall ; ð4Þ 2WH t where Nall is the number of all the transecting traces in the rectangular window. Because Nall indicates the number of all the traces intersecting the window, the c d numerator in Eq. (4) can be also written as ‘2Nall þ Nall ’, d where Nall is the number of all dissecting traces in the rectangular window. The trace length distribution f ðlÞ can be determined c d from f c ðlÞ, Nall and Nall , which are obtained from the site investigation.
rA ¼
Fig. 2. Center lines that include the midpoints of the dissecting discontinuities showing length of l 0 in a rectangular window.
and their expected number Nld0 can be expressed using Eq. (5) as follows: Z SX d Nl 0 ¼ rA Adl 0 f ðlÞ dl l0
¼ 2rA ð2al 0 þ bÞ dl 0
Let the complete length of a dissecting trace be l and its partial length in the window be l 0 when the geometric conditions of the window and traces are as shown in Fig. 2. Then the midpoints of the dissecting traces are restricted to a region of width dx or dy as illustrated in the figure, with area Adl0 given by Adl 0 ¼ 2ðdyðW l 0 cos yÞ þ dxðH l 0 sin yÞÞ ¼ 2 dl 0 ðsin yðW l 0 cos yÞ þ cos yðH l 0 sin yÞÞ ¼ 2 dl 0 ð2al 0 þ bÞ;
ð5Þ
where a ¼ cos y sin y and b ¼ W sin y H cos y. If the maximum diameter of the joints is SX , the complete lengths of dissecting traces that have lengths of l 0 in the sampling window are in the range of l 0 SX
SX
f ðlÞ dl:
ð6Þ
l0
Because the proportion of dissecting traces having length l 0 among all the dissecting traces in the window d can be expressed as Nld0 =Nall , the probability density function of the dissecting traces f d ðl 0 Þ can be obtained from Eq. (6) as Z SX 2rA d 0 0 f ðl Þ ¼ ð2al þ bÞ f ðlÞ dl: ð7Þ d Nall l0 The integration on the right side of Eq. (7) can be expressed as 1 Fðl 0 Þ where Fðl 0 Þ is the cumulative probability value of f ðlÞ in the range of 0 l 0 . Eq. (7) can be rewritten for the cumulative probability function replacing l 0 by l as FðlÞ ¼ 1 þ
2.2. Dissecting trace length distribution
Z
d Nall f d ðlÞ: 2rA ð2al þ bÞ
ð8Þ
Thus FðlÞ can be obtained by determining the parameters on the right side of this equation from a field investigation. It should be noted that on the right side of Eq. (8), the variable l is the partial length of dissecting traces observed in the window while on the left side it represents the entire length of traces from an infinite plane exposure. A noteworthy expression about censoring bias is obtained if Eq. (8) is rewritten replacing l by LX as follows: 1 FðLX Þ ¼
d Nall f d ðLX Þ: 2rA ð2aLX þ bÞ
This equation indicates that the area of the trace length distribution, beyond the maximum observable length where the censoring takes place, can be determined
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using the probability density value f d ðLX Þ. This relationship is useful when there are some traces that are longer than the maximum observable length of the sampling window. However, the probability density f d ð LX Þ is a discrete function, so in practical cases, it is affected by the column width of its histogram. As the column width gets smaller, the frequency in the column goes to zero. It remains as future work to choose a proper column length for correctly determining the censored area of the trace length distribution.
3. Estimating trace length distributions from circular windows A sampling exposure that is circular in shape can also be conveniently used as a sampling window [6,8]. The relationship between the trace length distribution and the contained or dissecting trace length distribution in a circular window can be obtained by a procedure similar to that we used in the case of the rectangular window described above. The terminology for the area of midpoints, the number of traces and the probability density functions is the same as that for the rectangular window, except that values of the circular window are represented by dots above the symbol letters in italics.
Fig. 3. The three areas that include the midpoints of the different types of discontinuities in the circular sampling window.
3.1. Contained trace length distribution There are three kinds of regions for the midpoints of traces that have a complete length of l in a circular window having a radius of R (Fig. 3). The rugby ballshaped area filled with horizontal lines is the region for the midpoints of the contained traces and the area with lattice marks is for the transecting traces. The empty regions in the two dashed circles are for the dissecting traces. The region for the midpoints of the contained traces exists when l is less than the diameter of the window (2R) (Fig. 3) but it disappears when l is greater than 2R (Fig. 4). In the former case, the area of the midpoint c region of the contained traces A’ l and the number of c contained traces N’ l can be calculated as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 l l c þ 2R2 sin1 @ R2 =RA; A’ l ¼ l R2 2 2 ð9Þ c c N’ l ¼ rA A’ l f ðlÞ dl:
ð10Þ
The trace length distribution can be determined in the same way as for Eq. (3) as follows: c N’ all ’ c f ðlÞ ¼ c f ðlÞ; rA A’ l
ð11Þ
Fig. 4. The narrow bands include the midpoints of the dissecting discontinuities that have a partial length of l 0 in the circular sampling window.
where f’ ðlÞ is the probability density function of the c contained trace length distribution and N’ all is the number of all contained traces in the circular window. c
3.2. Dissecting trace length distribution If the complete length of a dissecting trace is l and its partial length in the window is l 0 , its midpoint will exist in the area of one of two arc bands (Fig. 4). Let the width of the band parallel to such traces be dl 0 and let the bands be divided equally into such small units that each of them can be thought of as a parallelogram. Then the area of an arc band can be considered to be the same as that of a linear band with a width of dl 0 and a length equal to the distance between the two ends of the arc band. Therefore, the area of the two arc bands is determined by d A’ l0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 l 2 dl 0 : ¼4 R 2
ð12Þ
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The number of dissecting traces that have a partial length of l 0 in the window can be calculated as follows: Z SX d d ’ N l0 ¼ rA A’ l0 f ðlÞ dl l0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 Z SX l ¼ 4rA R2 dl 0 f ðlÞ dl 2 l0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 l 2 ð1 Fðl 0 ÞÞ dl 0 : ¼ 4rA R ð13Þ 2 The trace length distribution can be obtained in the same way as the rectangular window (Eqs. (6)–(8)): FðlÞ ¼ 1
d d N’ all f’ ðlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ; 4rA R2 l=2
ð14Þ
where f’ ðlÞ is the probability density function of the d dissecting trace length distribution and N’ all is the number of all the dissecting traces in the window. d
Fig. 5. Decrease in sampling error due to increase in sample size.
4. Checking the validity of the derived equations To make sure that Eqs. (3), (8), (11) and (14) were correctly derived, we applied the Monte Carlo simulation technique as follows. The distribution of the joint diameters was assumed to be uniform, then a theoretical trace length distribution was calculated from an equation suggested by Warburton [9]. Next, the contained and dissecting trace length distributions were obtained using the four equations above. The theoretical distributions were then compared with those obtained from Monte Carlo simulation in which the joint discs were generated in three-dimensional space and their intersections (i.e., traces) with a rectangular or circular window were recorded. The maximum diameter of the joint discs was 10 m. To avoid the censoring bias, the width and height of the rectangular window were set to 20 and 10 m, respectively, and the diameter of the circular window was taken to be 10 m. Considering the end effect in the sampling windows, the center points of the joint discs were located as far as 10 m from the windows in every direction. Of course, all joints were parallel with each other and the angle between the traces and the horizontal boundaries of the rectangular window was selected as one of 08, 308, 608 or 908. Incidentally, joint normals were not required to be parallel to the sampling plane at this time because the presence of an angle between the joint normals and the sampling plane makes no difference in estimating the joint trace length distribution. Only the sampling frequency is affected; as 908 is approached, joint traces are rarely sampled on the plane. The error between the theoretical contained or dissecting trace distribution and the simulated distribu-
tion was examined. The error was calculated by summing the absolute differences between the theoretical and the simulated distributions at each trace length level. The error decreased as the number of sampled traces increased regardless of the kind of distributions or the angle between the traces and the sampling window (Fig. 5). The various symbols in this figure, represent different distributions or different trace angles, but since all cases converged into one fitted curve, the distinction among the symbols can be considered meaningless. From these results, we conclude that Eqs. (3), (8), (11) and (14) were correctly derived. Sampling error is known to be inversely proportional to the square root of the number of samples when there is no other factor causing error. The fact that the exponent of the number of samples in the equation of the fitting curve was very close to 0.5 also supports the validity of the four equations.
5. Comparison of efficiencies in estimating trace length distributions We compared the efficiencies of the contained and the dissecting trace length distributions in a rectangular or circular window with each other in terms of the error in the estimation of the trace length distribution. Then we considered the similarity between the dissecting trace and the semi-trace length distribution and compared the complete trace length distribution from the scanline sampling with the contained and dissecting trace length distributions obtained through Monte Carlo simulation.
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FðlÞ is expressed as [1] hðlÞ ¼
1 FðlÞ ; mL
ð15Þ
where mL is the mean length of the joint traces. As for joint traces normal to the scanline, their areal frequency rA and linear frequency rS have the following relationship [10]: rS ¼ mL rA :
ð16Þ
If the length of the scanline is LS and the number of h traces intersecting the scanline is Nall , the linear h frequency can be expressed by Nall =LS . Using this expression and then substituting Eq. (16) into Eq. (15), the following result is obtained: hðlÞ ¼
Fig. 6. Errors in the trace length distributions estimated from contained and dissecting trace distributions as a function of sample size.
5.1. Contained and dissecting trace length distributions The errors in the trace length distributions that were estimated from the contained and the dissecting trace distributions from Monte Carlo simulation were compared with each other (Fig. 6). The experimental conditions, such as the dimensions of the sampling windows and the distribution of the joint diameters, were the same as in the simulation described previously for checking the validity of the equations. The errors in the estimation obtained from the contained trace length distribution in a rectangular or circular window were the smallest and the error from the dissecting trace length distribution in a circular window was the largest at every number of sampled traces. The estimation method using the dissecting trace distribution is worse than the method using the contained one because the former uses only partial lengths of traces for the estimation of the complete length distribution while the latter does not. For the same reason, we concluded that transecting traces are surely inferior to contained traces for an economical estimation of the trace length distribution.
5.2. Relationship between dissecting trace and semi-trace length distributions The relationship between the semi-trace length distribution from the scanline survey hðlÞ and the cumulative function of the trace length distribution
rA LS ð1 FðlÞÞ: h Nall
ð17Þ
After a and b in Eq. (8) are substituted by cos y sin y and W sin y H cos y, respectively, and y by 08 or 908, Eq. (8) can be rewritten as 2rA H ð1 FðlÞÞ at y ¼ 08; d Nall 2r W f d ðlÞ ¼ Ad ð1 FðlÞÞ at y ¼ 908; Nall f d ðlÞ ¼
ð18Þ
Comparing Eqs. (17) and (18), note that the semitrace length distribution is identical to the dissecting trace length distribution when the length of the horizontal or vertical boundary of the rectangular window is half that of the scanline. This is true only when the traces intersect perpendicularly the scanline and boundary lines of the window. Even though the directions of the traces are random, the difference between these two distributions is caused only by the traces near the corners of the rectangular window. Therefore, we conclude that the dissecting trace and the semi-trace length distributions have basically the same characteristics (or efficiency) for estimating the trace length distribution. 5.3. Comparison of contained, dissecting and complete trace length distributions Although the number of complete trace lengths sampled was generally less than the number of semitrace lengths on a scanline in the limited extent of exposure, unlike the semi-trace length, the complete trace length is not a monotonically decreasing function insensitive to the change of a trace length distribution. Here, the complete trace length means the whole length of a trace that intersects a scanline. This whole length may not be observed in the finite sampling exposure. The term ‘complete trace length’ is in contrast to ‘semitrace length’. The trace length distribution can be
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obtained by the complete trace length distribution from scanline sampling [1] as f ðlÞ ¼
mL gðlÞ; l
ð19Þ
where mL is the mean length of the joint traces and gðlÞ is the probability density function of the complete trace length distribution. Note that the trace length distribution f ðlÞ can be directly acquired from the complete trace length distribution gðlÞ, while the cumulative function of f ðlÞ is obtained using the semi-trace length distribution hðlÞ as in Eq. (15). If sampling errors in gðlÞ and hðlÞ observed in a finite exposure are at the same level, f ðlÞ estimated from the latter has a larger variance than that estimated from the former [10]. This is because the variance of the sampling error in hðlÞ is expanded into f ðlÞ in the process of calculating f ðlÞ by f ðlÞDl ¼ Fðl þ DlÞ FðlÞ ¼ mL ðhðlÞ hðl þ DlÞÞ. Here, the trace length distribution f ðlÞ estimated from the observed gðlÞ was compared with those estimated from contained or dissecting trace length distributions through Monte Carlo simulation. In this simulation, joints were assumed to be Poisson discs with a diameter following a uniform distribution with a mean value of 5 m. All joint discs had the same normal vector and their intersections with a rectangular window made an angle of y with the horizontal boundaries of the window (Fig. 7). y was fixed as one of 308, 608 or 908 for each simulation case. The width and height of the window were 20 and 10 m, respectively. For complete trace sampling, two kinds of scanline installations were used: 20 scanlines or one scanline parallel to the horizontal boundaries of the sampling window. In the former case, complete traces could be sampled repeatedly as they intersected several scanlines at the same time. The figure shows only three scanlines as a matter of convenience. Six traces are sampled by scanline 1: a, b, c, d, e and f; and six by scanline 2: a, b, e, g, h and i. In this case, traces a, b and e are sampled by both scanlines. The results of the simulation are presented in Table 1. The average number in Table 1 refers to the average number of traces sampled in a rectangular window by each of the three methods: contained and dissecting traces by window sampling, and complete trace lengths by scanline sampling. One trace length distribution was estimated by each method using sample traces assembled from 100 sampling windows, and the trace length distribution was obtained repeatedly up to 20 times. The error in Table 1 indicates the average error of the 20 distributions from a theoretical trace length distribution. Although the average number of complete trace lengths by 20 scanlines is four times greater than that of the contained traces, the average error of the trace length distribution estimated from it is three times or
Fig. 7. Joint traces and horizontal scanlines.
Table 1 Error of trace length distributions estimated from a rectangular window and scanline sampling Angle/number of scanlines
308 608 908
20 1 20 1 20 1
Average number/error Contained
Dissecting
Scanline
12.2/0.097 12.3/0.097 10.1/0.12 10.2/0.118 9.9/0.133 9.8/0.141
17.4/0.62 17.3/0.642 21.4/0.636 21.4/0.622 21.7/0.637 21.8/0.633
54.9/0.325 4.0/0.25 72.9/0.508 5.8/0.446 78.4/0.621 6.0/0.638
more than that of the contained traces at any direction of traces. As for the one-scanline method, error from complete traces is slightly lower than that for dissecting traces but three times higher than that for the contained traces, while the average number of samples is about half of the others. This result indicates that with the same sampling window, we can estimate the trace length distribution more precisely through the contained trace length distribution than through dissecting traces or the complete traces of a scanline survey. Table 1 also shows that repeated sampling of complete trace lengths in a scanline survey is not helpful in reducing the estimation error.
6. Numerical solution of the diameter distribution We developed and applied a numerical technique to obtain the joint diameter distribution from the trace length distribution to several cases to test its validity. Once a trace length distribution is determined, the joint diameter distribution can be obtained using this numerical technique irrespective of what sampling method, such as window or scanline sampling, has been used. 6.1. Approach Warburton [9] suggests the following equation for determining the trace length distribution from the joint
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diameter distribution: Z Z s 1 SX l dl pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðlÞ dl ¼ cðsÞ ds 1 FðlÞ ¼ m l l s l s2 l 2 Z SX pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ s2 l 2 cðsÞ ds; ð20Þ ms l Z
SX
where cðsÞ is the probability density function of the joint diameter distribution, and ms is the mean diameter. Eq. (20) shows that the trace length distribution can be obtained after the diameter distribution has been determined. The goal of this section, however, is to get the diameter distribution from the trace length distribution without any assumption regarding the nature of its distribution. For the first step, in order to express the diameter distribution as a function of a cumulative distribution of the trace length, Eq. (20) is approximated with a discrete form as 1 FðlÞ
SX pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ds X s2 l 2 cðsÞ: ms s¼l
1 FðSX DsÞ ¼
Ds ms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2X ðsX DsÞ2 cðsX Þ;
Ds ms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsX DsÞ2 ðsX 2DsÞ2 cðsX DsÞ
.. . cðN iÞ ¼ 2 6 4
Pi1
ms ð1 FðN ði þ 1ÞÞ Ds qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
2
3 7 5
j¼0 ðSX jDsÞ ðSX ði þ 1ÞDsÞ cðN jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX iDsÞ2 ðSX ði þ 1ÞDsÞ2 : ð23Þ
6.2. Verification of the numerical solution
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðs2X ðsX 2DsÞ2 cðsX Þ; 1 FðSX 3DsÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðsX 2DsÞ2 ðsX 3DsÞ2 cðsX 2DsÞ 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 Ds6 2 2 6 ¼ 6 þ ðsX DsÞ ðsX 3DsÞ cðsX DsÞ 7 7 ms 4 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ ðsX ðsX 3DsÞ cðsX Þ .. .
cðN 2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ms ð1 FðN 3ÞÞ SX2 ðSX 3DsÞ2 cðNÞ 6 Ds 7 4 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðSX DsÞ2 ðSX 3DsÞ2 cðN 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX 2DsÞ2 ðSX 3DsÞ2
When the diameter distribution is calculated using Eq. (23), the mean diameter ms is initially assigned a non-zero random value, then after all values of cðN iÞ are determined, ms can be obtained by scaling the area of the diameter distribution to one [10].
1 FðSX 2DsÞ
¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX DsÞ2 ðSX 2DsÞ2 ;
ð21Þ
If the whole range of the diameter (0 SX ) is divided into N small portions of DS, l can be replaced by SX ; iDS, in which i is 0, 1, 2. . ., and Eq. (21) can be expanded as follows: 1 FðSX Þ ¼ 0;
diameter distribution c as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m cðNÞ ¼ s ð1 FðN 1ÞÞ= SX2 ðSX DsÞ2 ; Ds " m cðN 1Þ ¼ s ð1 FðN 2ÞÞ Ds # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 SX ðSX 2DsÞ cðNÞ
ð22Þ
If SX in FðSX Þ or cðSX Þ is replaced by N, and SX iDS by N i, Eq. (22) can be rewritten in terms of the
To verify the numerical solution (Eq. (23)), four kinds of theoretical diameter distributions were chosen: a uniform distribution, two kinds of right triangles and a negative exponential. The trace length distribution of each was determined using Eq. (20). Each trace length distribution was applied to Eq. (23) to obtain the diameter distribution, which was compared with the theoretical distribution. Among the four distributions, the numerically calculated uniform distribution corresponds well to the theoretical one except in the vicinity of zero and especially at a diameter of 10 m (Fig. 8). For all cases, the error near to 10 m is generally bigger than the error near to zero. This tendency is most clearly shown for the ascending right triangle distribution. The error is relatively greater near the maximum diameter because frequency values in the diameter distribution can be calculated by reference to other frequencies at larger diameters (Eq. (23)).
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Fig. 8. Comparison of four theoretical distributions of diameter and their estimated distributions.
From the results revealed in the figure, we concluded that the numerical technique for determining the diameter distribution from the trace length distribution has been properly developed and is valid. An estimated trace length distribution and diameter distribution were compared with theoretical ones (Fig. 9). The trace length distribution was estimated from a contained trace length distribution and the diameter distribution was obtained numerically from the estimated trace length distribution. In this test, the theoretical diameter distribution was set to be a negative exponential with a mean of 2 m. Then, contained traces were sampled among the traces that occurred by the intersection of simulated joint discs and the sampling window. The estimated distributions of trace length and diameter show good agreement with the theoretical ones. The number of contained traces is 998 in a rectangular window of 20 10 m. In real cases, however, the contained traces are not usually sampled as many times as in this test. Sometimes, the number of sampled traces is so small that only a rough histogram of the contained trace length distribution can be obtained. In such cases, it can be helpful to estimate the trace length distribution from a continuous distribution fitted over the histogram of contained traces. An example is presented (Fig. 10) in which only 42 contained traces were sampled and their histogram was fitted with a lognormal function. These data are from one of the joint sets surveyed at an underground storage cavern for liquefied petroleum gas in Korea. In the figure, the trace length distribution almost coincides with the fitted contained trace distribution, which means the sampling window was sufficiently large for the joint size. The numerically calculated diameter distribution seems
Fig. 9. Estimation of a diameter distribution in Monte Carlo simulation.
Fig. 10. Estimation of a diameter distribution in a real case.
somewhat narrower than the trace length distribution and increases much higher, at about 0.25 m in length. 7. Conclusions In this study, four kinds of equations were derived for estimating the trace length distribution using the window sampling method. The equations were tested through Monte Carlo simulation, which showed that they had been properly derived. The contained trace length distribution in a rectangular or circular window turned out to be better than the dissecting one for estimating the trace length distribution.
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The dissecting trace length distribution from a rectangular window and the semi-trace length distribution were similar in efficiency to each other and the contained trace length distribution was more efficient than the complete trace length distribution from the scanline. The complete trace length distribution proved to be more efficient than the dissecting trace length distribution. A numerical technique for determining the diameter distribution from the trace length distribution was suggested and verified. By sampling the contained trace length distribution in a window and using the numerical technique proposed here, more reliable and direct estimation of the trace length and the joint diameter distribution can be performed.
Acknowledgements This work has been supported by R&D Project: 960042, Korea Institute of Construction Technology. This support is gratefully acknowledged.
References [1] Priest SD, Hudson JA. Estimation of discontinuity spacing and trace length using scanline surveys. Int J Rock Mech Sci & Geomech Abstr 1981;18:183–97. [2] Villaescusa E, Brown ET. Maximum likelihood estimation of joint size from trace length measurements. Rock Mech Rock Eng 1992;25:67–87. [3] Priest SD. Discontinuity analysis for rock engineering. London: Chapman & Hall, 1993. [4] Pahl PJ. Estimating the mean length of discontinuity traces. Int J Rock Mech Min Sci & Geomech Abstr 1981;18:221–8. [5] Kulatilake PHSW, Wu TH. Estimation of mean trace length of discontinuities. Rock Mech Rock Eng 1984;17:215–32. [6] Zhang L, Einstein HH. Estimating the mean trace length of rock discontinuities. Rock Mech Rock Eng 1998;31(4):217–35. [7] Mauldon M. Estimating mean fracture trace length and density from observations in convex windows. Rock Mech Rock Eng 1998;31(4):201–16. [8] Zhang L, Einstein HH. Estimating the intensity of rock discontinuities. Int J Rock Mech Sci & Geomech Abstr 2000;37:819–37. [9] Warburton PM. A stereological interpretation of joint trace data. Int J Rock Mech Sci & Geomech Abstr 1980;17:181–90. [10] Song JJ. Statistical modeling of joint length distribution and removability analysis of rock blocks around a tunnel. Ph.D. thesis, Seoul National University, Seoul, Korea, 2000.
Estimation of joint length distribution using window sampling Jae-Joon Song, Chung-In Lee* School of Civil, Urban and Geosystem Engineering, Seoul National University, Seoul, South Korea Accepted 18 March 2001
Abstract One of the most difficult procedures in statistical joint modeling is the determination of the joint length distribution. In this study, a window sampling method was adopted to estimate the trace length distribution for the Poisson disc joint model. Four kinds of equations were derived for estimating the trace length distribution from contained or dissecting trace length distributions in a rectangular or circular window. The equations were tested for accuracy through Monte Carlo simulation and their efficiencies for estimating the trace length distribution were compared. This new technique using window sampling was compared with the method using a semi-trace or complete trace length distribution from scanline sampling. A numerical technique for determining the diameter distribution from the trace length distribution was also suggested. To check the validity of this numerical method, it was applied to solving four example cases and their results were compared with theoretical solutions. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The statistical joint modeling technique is frequently applied to analyses of hydraulic conductivity or of the stability of rock blocks, which can be strongly affected by the geometrical characteristics of the joints. One of the least reliable procedures in joint modeling is the estimation of the joint length distribution. When the joints are assumed to be Poisson discs, there are two steps in determining the joint length distribution: the estimation of the trace length distribution on an infinite plane and the estimation of the joint diameter distribution in an infinite space. The joint length distribution has been inferred mainly from complete trace length or semi-trace length distributions obtained in scanline surveys [1]. For whatever sampling policy is used, however, it is an ill-posed problem to build the joint diameter distribution using a trace length distribution estimated from field survey data. This is due to the fact that trace length distributions are not sensitive to major changes in joint diameter distributions [2]. An alternative solution to this problem is to assume the distribution of the trace length or joint diameter in advance and then to find the proper *Corresponding author. Tel.: +82-2-880-8708; fax: +82-2-8732717. E-mail address: [email protected] (C.-I. Lee).
trace length or joint diameter distribution that best reflects the experimental data [2,3]. This method is useful, both for obtaining a well-defined continuous distribution of the trace length or joint diameter, and especially for overcoming the limit of a monotonically decreasing function of the semi-trace length distribution. With this method, however, it is still an unresolved problem that the final trace length or joint diameter distribution is extremely sensitive even to a small sampling error in the experimental data. Moreover, this is a distribution-dependent method. To improve the accuracy of the estimated joint diameter and trace length distributions, an effective technique is desired by which the trace length distribution can be more precisely estimated using the experimental data. From this, the diameter distribution may be obtained without any assumption regarding its form. To find a more reliable and distribution-independent method of estimating the trace length distribution for the Poisson disc joint model, trace lengths observed in sampling windows were used instead of scanline sampling in this study. Several studies for estimating a mean trace length [4–7] or an areal frequency [7] using window sampling have been reported. Areal frequency has the same meaning as areal density, that is, the average number of trace centers per unit area [3]. Hereafter, areal frequency will be used consistent with linear frequency, which is defined as the average
1365-1609/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 1 8 - 1
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number of traces intersected by a unit length of a sampling line. Traces can be classified into three categories according to the number of their end points observed in a window. A contained trace has two observed ends, a dissecting trace has only one and a transecting trace has zero. Here, four kinds of equations were derived for estimating the trace length distribution from contained or dissecting trace length distributions in a rectangular or circular window. Transecting traces were indirectly involved in developing the four kinds of equations in process of obtaining an areal frequency, a commonly used factor in each equation. The trace length distribution was not estimated using the transecting trace lengths because when estimated from partial lengths of traces, the trace length distribution inevitably becomes more erroneous than when it is estimated from the same number of full trace lengths. This is intuitive, and it has also been confirmed by comparing the errors on trace length distributions estimated from contained and dissecting trace length distributions. Biases in sampling trace lengths can be divided into three categories: size (length) bias, truncation bias and censoring bias [5]. The size bias means that longer joints are more likely to intersect a sampling plane and make traces. The truncation bias occurs when trace lengths below some cutoff length are not recorded. The censoring bias refers to traces whose end points cannot be seen, so only lower bound estimates of their lengths may be determined. In this study, the size bias problem is resolved by considering geometric characteristics of contained or dissecting traces in the process of deriving each equation. The truncation bias is not considered because in practice, its effects can be made negligible by choosing a cutoff length that is small compared to the average trace length [5]. It is impossible to determine probability density values for trace lengths longer than the maximum observable length unless the sampled length or the trace length distribution is predefined. To avoid this problem, the sampling window here is assumed to be large enough to observe the maximum joint diameter and its trace length. In a practical environment where the censoring bias can seriously affect confidence in the estimated trace length distribution, one possible solution has been given in this paper. However, the further effort for the censoring bias remains as future work. The efficiency of using a dissecting trace length distribution from a rectangular window in the estimation of the trace length distribution was compared with the use of a semi-trace length distribution. Then, the same comparison was performed using the contained and dissecting trace length distributions from rectangular windows, and using the complete trace length distribution from the scanline, that is, the distribution of whole lengths of traces intersecting a scanline.
We also suggested a numerical technique to determine directly the diameter distribution from the trace length distribution. This approach was verified by comparing the numerical solutions of example cases with their theoretical predictions.
2. Estimating trace length distributions from rectangular windows 2.1. Contained trace length distribution All the joints are assumed to have the same orientation but not to be parallel with a rectangular window plane of length W and height H. Let the complete trace length of a joint in the window be l and the angle between the trace and the horizontal boundary of the window be y (Fig. 1). Orientations of joints are usually recorded as strike/dip or dip direction/dip in field measurements. The first step in the calculation of y is to determine the normal vectors of the joint and the sampling window. Then, the direction vector of a trace on a sampling exposure can be determined by the cross product of the two normal vectors of the joint and sampling planes. Finally, y can be obtained from the dot product of the unit direction vectors of the trace and the horizontal boundary. If the midpoint of any trace is in the shaded rectangular area of the window, the trace becomes a contained trace. The area of the shaded rectangular Acl can be obtained by the following equation: Acl ¼ ðW l cos yÞðH l sin yÞ ¼ cos y sin yl 2 þ ðW sin y H cos yÞl þ WH ¼ al 2 þ bl þ c
ðl5LX Þ;
ð1Þ
where a ¼ cos y sin y; b ¼ W sin y H cos y; c ¼ WH, and LX is the maximum length of the contained trace, which is the smaller of W=cos y and H=sin y. In Eq. (1), Acl is expressed by a second order function of l for which
Fig. 1. Area that includes the midpoints of contained discontinuities.
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the coefficients are determined by the geometry of the window and the traces. When l is greater than LX , Acl becomes zero. The expected number of contained traces Nlc can be obtained by the product of the areal frequency of traces having lengths of l l þ dl and the area of midpoints shown by the shaded rectangular area in the figure as Nlc ¼ rA Acl f ðlÞ dl;
ð2Þ
where rA is the areal frequency of whole traces, and f ðlÞ is the probability density function of the trace length distribution. Both are defined in an infinite sampling plane. Because the probability that the contained trace has a c length of l l þ dl is f c ðlÞ dl and equals Nlc =Nall , the trace length density f ðlÞ can be obtained using Eqs. (1) and (2) as follows: f ðlÞ ¼
rA
ðal 2
c Nall f c ðlÞ; þ bl þ cÞ
ð3Þ
where f c ðlÞ is the probability density function of the c contained trace length distribution, and Nall is the number of all contained traces observed in the window. Mauldon [7] suggests the following equation for the areal frequency of the joint traces independent of the trace length distribution: t c Nall Nall þ Nall ; ð4Þ 2WH t where Nall is the number of all the transecting traces in the rectangular window. Because Nall indicates the number of all the traces intersecting the window, the c d numerator in Eq. (4) can be also written as ‘2Nall þ Nall ’, d where Nall is the number of all dissecting traces in the rectangular window. The trace length distribution f ðlÞ can be determined c d from f c ðlÞ, Nall and Nall , which are obtained from the site investigation.
rA ¼
Fig. 2. Center lines that include the midpoints of the dissecting discontinuities showing length of l 0 in a rectangular window.
and their expected number Nld0 can be expressed using Eq. (5) as follows: Z SX d Nl 0 ¼ rA Adl 0 f ðlÞ dl l0
¼ 2rA ð2al 0 þ bÞ dl 0
Let the complete length of a dissecting trace be l and its partial length in the window be l 0 when the geometric conditions of the window and traces are as shown in Fig. 2. Then the midpoints of the dissecting traces are restricted to a region of width dx or dy as illustrated in the figure, with area Adl0 given by Adl 0 ¼ 2ðdyðW l 0 cos yÞ þ dxðH l 0 sin yÞÞ ¼ 2 dl 0 ðsin yðW l 0 cos yÞ þ cos yðH l 0 sin yÞÞ ¼ 2 dl 0 ð2al 0 þ bÞ;
ð5Þ
where a ¼ cos y sin y and b ¼ W sin y H cos y. If the maximum diameter of the joints is SX , the complete lengths of dissecting traces that have lengths of l 0 in the sampling window are in the range of l 0 SX
SX
f ðlÞ dl:
ð6Þ
l0
Because the proportion of dissecting traces having length l 0 among all the dissecting traces in the window d can be expressed as Nld0 =Nall , the probability density function of the dissecting traces f d ðl 0 Þ can be obtained from Eq. (6) as Z SX 2rA d 0 0 f ðl Þ ¼ ð2al þ bÞ f ðlÞ dl: ð7Þ d Nall l0 The integration on the right side of Eq. (7) can be expressed as 1 Fðl 0 Þ where Fðl 0 Þ is the cumulative probability value of f ðlÞ in the range of 0 l 0 . Eq. (7) can be rewritten for the cumulative probability function replacing l 0 by l as FðlÞ ¼ 1 þ
2.2. Dissecting trace length distribution
Z
d Nall f d ðlÞ: 2rA ð2al þ bÞ
ð8Þ
Thus FðlÞ can be obtained by determining the parameters on the right side of this equation from a field investigation. It should be noted that on the right side of Eq. (8), the variable l is the partial length of dissecting traces observed in the window while on the left side it represents the entire length of traces from an infinite plane exposure. A noteworthy expression about censoring bias is obtained if Eq. (8) is rewritten replacing l by LX as follows: 1 FðLX Þ ¼
d Nall f d ðLX Þ: 2rA ð2aLX þ bÞ
This equation indicates that the area of the trace length distribution, beyond the maximum observable length where the censoring takes place, can be determined
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using the probability density value f d ðLX Þ. This relationship is useful when there are some traces that are longer than the maximum observable length of the sampling window. However, the probability density f d ð LX Þ is a discrete function, so in practical cases, it is affected by the column width of its histogram. As the column width gets smaller, the frequency in the column goes to zero. It remains as future work to choose a proper column length for correctly determining the censored area of the trace length distribution.
3. Estimating trace length distributions from circular windows A sampling exposure that is circular in shape can also be conveniently used as a sampling window [6,8]. The relationship between the trace length distribution and the contained or dissecting trace length distribution in a circular window can be obtained by a procedure similar to that we used in the case of the rectangular window described above. The terminology for the area of midpoints, the number of traces and the probability density functions is the same as that for the rectangular window, except that values of the circular window are represented by dots above the symbol letters in italics.
Fig. 3. The three areas that include the midpoints of the different types of discontinuities in the circular sampling window.
3.1. Contained trace length distribution There are three kinds of regions for the midpoints of traces that have a complete length of l in a circular window having a radius of R (Fig. 3). The rugby ballshaped area filled with horizontal lines is the region for the midpoints of the contained traces and the area with lattice marks is for the transecting traces. The empty regions in the two dashed circles are for the dissecting traces. The region for the midpoints of the contained traces exists when l is less than the diameter of the window (2R) (Fig. 3) but it disappears when l is greater than 2R (Fig. 4). In the former case, the area of the midpoint c region of the contained traces A’ l and the number of c contained traces N’ l can be calculated as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 l l c þ 2R2 sin1 @ R2 =RA; A’ l ¼ l R2 2 2 ð9Þ c c N’ l ¼ rA A’ l f ðlÞ dl:
ð10Þ
The trace length distribution can be determined in the same way as for Eq. (3) as follows: c N’ all ’ c f ðlÞ ¼ c f ðlÞ; rA A’ l
ð11Þ
Fig. 4. The narrow bands include the midpoints of the dissecting discontinuities that have a partial length of l 0 in the circular sampling window.
where f’ ðlÞ is the probability density function of the c contained trace length distribution and N’ all is the number of all contained traces in the circular window. c
3.2. Dissecting trace length distribution If the complete length of a dissecting trace is l and its partial length in the window is l 0 , its midpoint will exist in the area of one of two arc bands (Fig. 4). Let the width of the band parallel to such traces be dl 0 and let the bands be divided equally into such small units that each of them can be thought of as a parallelogram. Then the area of an arc band can be considered to be the same as that of a linear band with a width of dl 0 and a length equal to the distance between the two ends of the arc band. Therefore, the area of the two arc bands is determined by d A’ l0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 l 2 dl 0 : ¼4 R 2
ð12Þ
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The number of dissecting traces that have a partial length of l 0 in the window can be calculated as follows: Z SX d d ’ N l0 ¼ rA A’ l0 f ðlÞ dl l0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 Z SX l ¼ 4rA R2 dl 0 f ðlÞ dl 2 l0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 l 2 ð1 Fðl 0 ÞÞ dl 0 : ¼ 4rA R ð13Þ 2 The trace length distribution can be obtained in the same way as the rectangular window (Eqs. (6)–(8)): FðlÞ ¼ 1
d d N’ all f’ ðlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ; 4rA R2 l=2
ð14Þ
where f’ ðlÞ is the probability density function of the d dissecting trace length distribution and N’ all is the number of all the dissecting traces in the window. d
Fig. 5. Decrease in sampling error due to increase in sample size.
4. Checking the validity of the derived equations To make sure that Eqs. (3), (8), (11) and (14) were correctly derived, we applied the Monte Carlo simulation technique as follows. The distribution of the joint diameters was assumed to be uniform, then a theoretical trace length distribution was calculated from an equation suggested by Warburton [9]. Next, the contained and dissecting trace length distributions were obtained using the four equations above. The theoretical distributions were then compared with those obtained from Monte Carlo simulation in which the joint discs were generated in three-dimensional space and their intersections (i.e., traces) with a rectangular or circular window were recorded. The maximum diameter of the joint discs was 10 m. To avoid the censoring bias, the width and height of the rectangular window were set to 20 and 10 m, respectively, and the diameter of the circular window was taken to be 10 m. Considering the end effect in the sampling windows, the center points of the joint discs were located as far as 10 m from the windows in every direction. Of course, all joints were parallel with each other and the angle between the traces and the horizontal boundaries of the rectangular window was selected as one of 08, 308, 608 or 908. Incidentally, joint normals were not required to be parallel to the sampling plane at this time because the presence of an angle between the joint normals and the sampling plane makes no difference in estimating the joint trace length distribution. Only the sampling frequency is affected; as 908 is approached, joint traces are rarely sampled on the plane. The error between the theoretical contained or dissecting trace distribution and the simulated distribu-
tion was examined. The error was calculated by summing the absolute differences between the theoretical and the simulated distributions at each trace length level. The error decreased as the number of sampled traces increased regardless of the kind of distributions or the angle between the traces and the sampling window (Fig. 5). The various symbols in this figure, represent different distributions or different trace angles, but since all cases converged into one fitted curve, the distinction among the symbols can be considered meaningless. From these results, we conclude that Eqs. (3), (8), (11) and (14) were correctly derived. Sampling error is known to be inversely proportional to the square root of the number of samples when there is no other factor causing error. The fact that the exponent of the number of samples in the equation of the fitting curve was very close to 0.5 also supports the validity of the four equations.
5. Comparison of efficiencies in estimating trace length distributions We compared the efficiencies of the contained and the dissecting trace length distributions in a rectangular or circular window with each other in terms of the error in the estimation of the trace length distribution. Then we considered the similarity between the dissecting trace and the semi-trace length distribution and compared the complete trace length distribution from the scanline sampling with the contained and dissecting trace length distributions obtained through Monte Carlo simulation.
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FðlÞ is expressed as [1] hðlÞ ¼
1 FðlÞ ; mL
ð15Þ
where mL is the mean length of the joint traces. As for joint traces normal to the scanline, their areal frequency rA and linear frequency rS have the following relationship [10]: rS ¼ mL rA :
ð16Þ
If the length of the scanline is LS and the number of h traces intersecting the scanline is Nall , the linear h frequency can be expressed by Nall =LS . Using this expression and then substituting Eq. (16) into Eq. (15), the following result is obtained: hðlÞ ¼
Fig. 6. Errors in the trace length distributions estimated from contained and dissecting trace distributions as a function of sample size.
5.1. Contained and dissecting trace length distributions The errors in the trace length distributions that were estimated from the contained and the dissecting trace distributions from Monte Carlo simulation were compared with each other (Fig. 6). The experimental conditions, such as the dimensions of the sampling windows and the distribution of the joint diameters, were the same as in the simulation described previously for checking the validity of the equations. The errors in the estimation obtained from the contained trace length distribution in a rectangular or circular window were the smallest and the error from the dissecting trace length distribution in a circular window was the largest at every number of sampled traces. The estimation method using the dissecting trace distribution is worse than the method using the contained one because the former uses only partial lengths of traces for the estimation of the complete length distribution while the latter does not. For the same reason, we concluded that transecting traces are surely inferior to contained traces for an economical estimation of the trace length distribution.
5.2. Relationship between dissecting trace and semi-trace length distributions The relationship between the semi-trace length distribution from the scanline survey hðlÞ and the cumulative function of the trace length distribution
rA LS ð1 FðlÞÞ: h Nall
ð17Þ
After a and b in Eq. (8) are substituted by cos y sin y and W sin y H cos y, respectively, and y by 08 or 908, Eq. (8) can be rewritten as 2rA H ð1 FðlÞÞ at y ¼ 08; d Nall 2r W f d ðlÞ ¼ Ad ð1 FðlÞÞ at y ¼ 908; Nall f d ðlÞ ¼
ð18Þ
Comparing Eqs. (17) and (18), note that the semitrace length distribution is identical to the dissecting trace length distribution when the length of the horizontal or vertical boundary of the rectangular window is half that of the scanline. This is true only when the traces intersect perpendicularly the scanline and boundary lines of the window. Even though the directions of the traces are random, the difference between these two distributions is caused only by the traces near the corners of the rectangular window. Therefore, we conclude that the dissecting trace and the semi-trace length distributions have basically the same characteristics (or efficiency) for estimating the trace length distribution. 5.3. Comparison of contained, dissecting and complete trace length distributions Although the number of complete trace lengths sampled was generally less than the number of semitrace lengths on a scanline in the limited extent of exposure, unlike the semi-trace length, the complete trace length is not a monotonically decreasing function insensitive to the change of a trace length distribution. Here, the complete trace length means the whole length of a trace that intersects a scanline. This whole length may not be observed in the finite sampling exposure. The term ‘complete trace length’ is in contrast to ‘semitrace length’. The trace length distribution can be
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obtained by the complete trace length distribution from scanline sampling [1] as f ðlÞ ¼
mL gðlÞ; l
ð19Þ
where mL is the mean length of the joint traces and gðlÞ is the probability density function of the complete trace length distribution. Note that the trace length distribution f ðlÞ can be directly acquired from the complete trace length distribution gðlÞ, while the cumulative function of f ðlÞ is obtained using the semi-trace length distribution hðlÞ as in Eq. (15). If sampling errors in gðlÞ and hðlÞ observed in a finite exposure are at the same level, f ðlÞ estimated from the latter has a larger variance than that estimated from the former [10]. This is because the variance of the sampling error in hðlÞ is expanded into f ðlÞ in the process of calculating f ðlÞ by f ðlÞDl ¼ Fðl þ DlÞ FðlÞ ¼ mL ðhðlÞ hðl þ DlÞÞ. Here, the trace length distribution f ðlÞ estimated from the observed gðlÞ was compared with those estimated from contained or dissecting trace length distributions through Monte Carlo simulation. In this simulation, joints were assumed to be Poisson discs with a diameter following a uniform distribution with a mean value of 5 m. All joint discs had the same normal vector and their intersections with a rectangular window made an angle of y with the horizontal boundaries of the window (Fig. 7). y was fixed as one of 308, 608 or 908 for each simulation case. The width and height of the window were 20 and 10 m, respectively. For complete trace sampling, two kinds of scanline installations were used: 20 scanlines or one scanline parallel to the horizontal boundaries of the sampling window. In the former case, complete traces could be sampled repeatedly as they intersected several scanlines at the same time. The figure shows only three scanlines as a matter of convenience. Six traces are sampled by scanline 1: a, b, c, d, e and f; and six by scanline 2: a, b, e, g, h and i. In this case, traces a, b and e are sampled by both scanlines. The results of the simulation are presented in Table 1. The average number in Table 1 refers to the average number of traces sampled in a rectangular window by each of the three methods: contained and dissecting traces by window sampling, and complete trace lengths by scanline sampling. One trace length distribution was estimated by each method using sample traces assembled from 100 sampling windows, and the trace length distribution was obtained repeatedly up to 20 times. The error in Table 1 indicates the average error of the 20 distributions from a theoretical trace length distribution. Although the average number of complete trace lengths by 20 scanlines is four times greater than that of the contained traces, the average error of the trace length distribution estimated from it is three times or
Fig. 7. Joint traces and horizontal scanlines.
Table 1 Error of trace length distributions estimated from a rectangular window and scanline sampling Angle/number of scanlines
308 608 908
20 1 20 1 20 1
Average number/error Contained
Dissecting
Scanline
12.2/0.097 12.3/0.097 10.1/0.12 10.2/0.118 9.9/0.133 9.8/0.141
17.4/0.62 17.3/0.642 21.4/0.636 21.4/0.622 21.7/0.637 21.8/0.633
54.9/0.325 4.0/0.25 72.9/0.508 5.8/0.446 78.4/0.621 6.0/0.638
more than that of the contained traces at any direction of traces. As for the one-scanline method, error from complete traces is slightly lower than that for dissecting traces but three times higher than that for the contained traces, while the average number of samples is about half of the others. This result indicates that with the same sampling window, we can estimate the trace length distribution more precisely through the contained trace length distribution than through dissecting traces or the complete traces of a scanline survey. Table 1 also shows that repeated sampling of complete trace lengths in a scanline survey is not helpful in reducing the estimation error.
6. Numerical solution of the diameter distribution We developed and applied a numerical technique to obtain the joint diameter distribution from the trace length distribution to several cases to test its validity. Once a trace length distribution is determined, the joint diameter distribution can be obtained using this numerical technique irrespective of what sampling method, such as window or scanline sampling, has been used. 6.1. Approach Warburton [9] suggests the following equation for determining the trace length distribution from the joint
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diameter distribution: Z Z s 1 SX l dl pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðlÞ dl ¼ cðsÞ ds 1 FðlÞ ¼ m l l s l s2 l 2 Z SX pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ s2 l 2 cðsÞ ds; ð20Þ ms l Z
SX
where cðsÞ is the probability density function of the joint diameter distribution, and ms is the mean diameter. Eq. (20) shows that the trace length distribution can be obtained after the diameter distribution has been determined. The goal of this section, however, is to get the diameter distribution from the trace length distribution without any assumption regarding the nature of its distribution. For the first step, in order to express the diameter distribution as a function of a cumulative distribution of the trace length, Eq. (20) is approximated with a discrete form as 1 FðlÞ
SX pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ds X s2 l 2 cðsÞ: ms s¼l
1 FðSX DsÞ ¼
Ds ms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2X ðsX DsÞ2 cðsX Þ;
Ds ms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsX DsÞ2 ðsX 2DsÞ2 cðsX DsÞ
.. . cðN iÞ ¼ 2 6 4
Pi1
ms ð1 FðN ði þ 1ÞÞ Ds qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
2
3 7 5
j¼0 ðSX jDsÞ ðSX ði þ 1ÞDsÞ cðN jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX iDsÞ2 ðSX ði þ 1ÞDsÞ2 : ð23Þ
6.2. Verification of the numerical solution
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðs2X ðsX 2DsÞ2 cðsX Þ; 1 FðSX 3DsÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðsX 2DsÞ2 ðsX 3DsÞ2 cðsX 2DsÞ 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 Ds6 2 2 6 ¼ 6 þ ðsX DsÞ ðsX 3DsÞ cðsX DsÞ 7 7 ms 4 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ ðsX ðsX 3DsÞ cðsX Þ .. .
cðN 2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ms ð1 FðN 3ÞÞ SX2 ðSX 3DsÞ2 cðNÞ 6 Ds 7 4 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðSX DsÞ2 ðSX 3DsÞ2 cðN 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX 2DsÞ2 ðSX 3DsÞ2
When the diameter distribution is calculated using Eq. (23), the mean diameter ms is initially assigned a non-zero random value, then after all values of cðN iÞ are determined, ms can be obtained by scaling the area of the diameter distribution to one [10].
1 FðSX 2DsÞ
¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ðSX DsÞ2 ðSX 2DsÞ2 ;
ð21Þ
If the whole range of the diameter (0 SX ) is divided into N small portions of DS, l can be replaced by SX ; iDS, in which i is 0, 1, 2. . ., and Eq. (21) can be expanded as follows: 1 FðSX Þ ¼ 0;
diameter distribution c as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m cðNÞ ¼ s ð1 FðN 1ÞÞ= SX2 ðSX DsÞ2 ; Ds " m cðN 1Þ ¼ s ð1 FðN 2ÞÞ Ds # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 SX ðSX 2DsÞ cðNÞ
ð22Þ
If SX in FðSX Þ or cðSX Þ is replaced by N, and SX iDS by N i, Eq. (22) can be rewritten in terms of the
To verify the numerical solution (Eq. (23)), four kinds of theoretical diameter distributions were chosen: a uniform distribution, two kinds of right triangles and a negative exponential. The trace length distribution of each was determined using Eq. (20). Each trace length distribution was applied to Eq. (23) to obtain the diameter distribution, which was compared with the theoretical distribution. Among the four distributions, the numerically calculated uniform distribution corresponds well to the theoretical one except in the vicinity of zero and especially at a diameter of 10 m (Fig. 8). For all cases, the error near to 10 m is generally bigger than the error near to zero. This tendency is most clearly shown for the ascending right triangle distribution. The error is relatively greater near the maximum diameter because frequency values in the diameter distribution can be calculated by reference to other frequencies at larger diameters (Eq. (23)).
J.-J. Song, C.-I. Lee / International Journal of Rock Mechanics & Mining Sciences 38 (2001) 519–528
527
Fig. 8. Comparison of four theoretical distributions of diameter and their estimated distributions.
From the results revealed in the figure, we concluded that the numerical technique for determining the diameter distribution from the trace length distribution has been properly developed and is valid. An estimated trace length distribution and diameter distribution were compared with theoretical ones (Fig. 9). The trace length distribution was estimated from a contained trace length distribution and the diameter distribution was obtained numerically from the estimated trace length distribution. In this test, the theoretical diameter distribution was set to be a negative exponential with a mean of 2 m. Then, contained traces were sampled among the traces that occurred by the intersection of simulated joint discs and the sampling window. The estimated distributions of trace length and diameter show good agreement with the theoretical ones. The number of contained traces is 998 in a rectangular window of 20 10 m. In real cases, however, the contained traces are not usually sampled as many times as in this test. Sometimes, the number of sampled traces is so small that only a rough histogram of the contained trace length distribution can be obtained. In such cases, it can be helpful to estimate the trace length distribution from a continuous distribution fitted over the histogram of contained traces. An example is presented (Fig. 10) in which only 42 contained traces were sampled and their histogram was fitted with a lognormal function. These data are from one of the joint sets surveyed at an underground storage cavern for liquefied petroleum gas in Korea. In the figure, the trace length distribution almost coincides with the fitted contained trace distribution, which means the sampling window was sufficiently large for the joint size. The numerically calculated diameter distribution seems
Fig. 9. Estimation of a diameter distribution in Monte Carlo simulation.
Fig. 10. Estimation of a diameter distribution in a real case.
somewhat narrower than the trace length distribution and increases much higher, at about 0.25 m in length. 7. Conclusions In this study, four kinds of equations were derived for estimating the trace length distribution using the window sampling method. The equations were tested through Monte Carlo simulation, which showed that they had been properly derived. The contained trace length distribution in a rectangular or circular window turned out to be better than the dissecting one for estimating the trace length distribution.
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The dissecting trace length distribution from a rectangular window and the semi-trace length distribution were similar in efficiency to each other and the contained trace length distribution was more efficient than the complete trace length distribution from the scanline. The complete trace length distribution proved to be more efficient than the dissecting trace length distribution. A numerical technique for determining the diameter distribution from the trace length distribution was suggested and verified. By sampling the contained trace length distribution in a window and using the numerical technique proposed here, more reliable and direct estimation of the trace length and the joint diameter distribution can be performed.
Acknowledgements This work has been supported by R&D Project: 960042, Korea Institute of Construction Technology. This support is gratefully acknowledged.
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