# A New Approach for Estimating Rock Discontinuity Trace Intensity Based on Rectangular Sampling Windows

Authors: Xiaoxue Huo; Qiong Wu; Huiming Tang; Zhen Meng; Di Wang; Yuxin Liu; Shiyu Li
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Trace intensity is defined as mean total trace length of discontinuities per unit area, which is an important geometric parameter to describe fracture networks. The probability of each trace appearing in the sampling surface is different since discontinuity orientation has a scatter and is probabilistically distributed, so this factor should be taken into account in trace intensity estimation. This paper presents an approach to estimate the two-dimensional trace intensity by considering unequal appearing probability for discontinuities sampled by rectangular windows. The estimation method requires the number of discontinuities intersecting the window, the appearing probability of discontinuities with both ends observed, one end observed, and both ends censored, and the mean trace length of discontinuities intersecting the window. The new estimator is validated by using discontinuity data from an outcrop in Wenchuan area in China. Similarly, circular windows are used along with Mauldon’s equation to calculate trace intensity using discontinuity trace data of the same outcrop as a contrast. Results indicate that the proposed new method based on rectangular windows shows close accuracy and less variability than that of the method based on circular windows due to the influence of finite sample size and the variability of location of the window and has advantage in application to sampling surfaces longer in one direction than in the other such as tunnel cross sections and curved sampling surfaces such as outcrops that show some curvature.


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## Body

## 1. Introduction

The discontinuities have a significant influence on the stability, deformability, strength, and percolation characteristics of rock mass [1–5]. Characteristics of discontinuity and discontinuity sets are commonly inferred from discontinuity trace parameters such as trace intensity, length, and density. The aforementioned parameters can be used as typical geometric parameters to describe fracture networks by computer programs [6–9]. Trace intensity defined as mean total trace length of discontinuities per unit area, which is a pattern characteristic that incorporates both fracture density and mean length. Hence, intensity can substitute for dimensionally equivalent density as an input value of fracture networks programs. Intensity also can calculate equivalent mean spacing from a case of two-dimensional subparallel impersistent traces [10]. In addition, the estimation of intensity provides tests for determining whether fracture network is truly representative.There are three types of intensities including linear, areal, and volumetric. The dimension of all types isL−1 (L is length). The linear intensity is the simplest and most common type of intensity used for the whole discontinuities [11]. More recently, researches focus on volumetric intensity (discontinuity area per unit volume of rock mass). Zhang and Einstein [12] proposed an equation to calculate the mean fractures area per unit volume of the rock mass. Grossmann [13] proposed mathematical calculation method on volumetric intensity. Hekmatnejad et al. [14] addressed the problem of predicting the volumetric intensity in space and of quantifying the uncertainty in the true values, using information from observed discontinuities intersecting boreholes. Sanderson and Nixon [15] introduced some simple techniques to characterize the topology of a fracture network, which extended concepts of topological to 3 dimensions. No research has been done on estimation of trace intensity studied in this paper based on rectangular sampling windows.Several methods have been developed to estimate intensity on two-dimensional exposure. Zeeb et al. [16] reclassified the methods to acquire aforementioned geometric parameters into these: (1) scanline sampling (e.g., Priest and Hudson [17]), (2) window sampling (e.g., Paul [18]), and (3) circular estimator method (e.g., Mauldon et al. [19]). A few researchers have performed investigations on the method of estimating trace intensity in two-dimensional exposures. Ferrero and Umili [20] compared the existing fracture intensity estimation method and applied them to the North face of Aiguille du Marbree. Bandpey et al. [21] made a comparison of the methods for calculation geometrical characteristics of discontinuities in a cavern of the Rudbar Lorestan power plant. Kamali et al. [22] focused on fisher constant to research effect of shape and size of sampling window on the determination of intensity of trace discontinuity. Mahe et al. [23] applied mathematical calculations of fracture intensity on one site using measurements of three distinct fracture sets to evaluate and explain variability in collected data. Watkins et al. [24] developed a work flow for fracture data collection including fracture intensity in a region of heterogeneous fractures in a fold and thrust belt. Ortega et al.’s [25] used approach makes use of fracture-size distributions for two carbonate beds in Mexico and illustrates how size-cognizant measurements cast new light on widely accepted interpretation of geologic controls of fracture intensity. It can be seen from previous studies that a scanline survey allows a quick analysis of fracture network characteristics on outcrops, but it has size, truncation, and censoring biases [26]. The window sampling technique provides much data in a relatively short period of time and in the smallest part of the outcrop that is considered as one of the most important advantages of the window sampling technique.This paper contains deduction of trace intensity estimators based on rectangular windows by considering that the probability of each trace appearing in the sampling surface is different. Based on previous studies on mean trace length estimation [27, 28], this paper focuses on the areal trace intensity estimation based on rectangular windows, which has advantage in application to sampling surfaces longer in one direction than in the other such as tunnel cross sections and curved sampling surfaces such as outcrops that show some curvature. The proposed new estimator is then validated by using discontinuity data from an outcrop in Wenchuan area in China. Similarly, circular windows are used along with Mauldon’s equation to calculate trace intensity using discontinuity trace data of the same outcrop as a contrast.


## 2. New Estimation Method of Trace Intensity Based on Rectangular Sampling Windows

### 2.1. Deduction Process

Trace intensity is defined as mean total trace length of fractures per unit area [19]. According to the definition, trace intensity (I) can be expressed as(1)I=L′wh,where L′ is the total observed trace length of discontinuities in rectangular sampling window; w and h are, respectively, width and height of the rectangular window.Depending on the position of the discontinuity, the intersection may occur in three ways: (a) both ends are censored, (b) one end is censored, and (c) both ends are observable in the window [28]. Let N0, N1, and N2 denote the number of discontinuities with both ends censored, one end censored, and both ends observable, respectively. Let L denote the total trace length of discontinuities intersecting the rectangular sampling window. Let K denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window.(2)K=L′L.In order to estimateL′, we first deduct K by considering these three types of discontinuities respectively:


#### 2.1.1. Discontinuities with Both Ends Observable

LetK1 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends observable. It is easy to find that K1 = 1.


#### 2.1.2. Discontinuities with One End Censored

LetK2 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with one end censored. Figure 1 shows a one end censored trace intersecting the lower boundary of the rectangular sampling window. Let xi denote the distance between the midpoint of the ith trace and the lower boundary of the window. xi is positive when the midpoint is within the window, while xi is negative when the midpoint is out of the window. φi denotes the apparent dip angle of the ith trace. li and li’ are, respectively, the full length and observed length in the window of the ith fracture trace. K2 can be expressed as(3)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Figure 1
Intersection relationship between a trace with one end censored and the sampling window.The apparent dip angle of discontinuities in the same set does not vary a lot, so (3) can be simplified as(4)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Assume that the midpoints of trace lengths are uniformly distributed in two-dimensional space; we can get the expression∑i=1N1xi=0. So in this case, K2 = 1/2. Similarly, we can obtain the same conclusion if these discontinuities with one end censored intersect the other boundaries of the sampling window.


#### 2.1.3. Discontinuities with Both Ends Censored

LetK3 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends censored. Consider the problem of a discontinuity with both ends censored with trace length, l, and apparent dip angle, φ. According to the intersection relationships between discontinuity and sampling window, four cases will be discussed in the following. They are①l≥h/sinφ and φ≥tan−1h/w (Figure 2), ② l<h/sinφ and φ≥tan−1h/w (Figure 3), ③ l≥h/sinφ and φ<tan−1h/w (Figure 4), and ④ l<h/sinφ and φ<tan−1h/w (Figure 5).Figure 2
Discontinuities with both ends censored,l≥h/sinφ and φ≥tan−1h/w.Figure 3
Discontinuities with both ends censored,l<h/sinφ and φ≥tan−1h/w.Figure 4
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.Figure 5
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.LetK31, K32, K33, and K34 denote the ratio of total observed trace length and total trace length of discontinuities in the aforementioned four cases, respectively. Use P31, P32, P33, and P34 as the probability of the traces appearing in the window, respectively, in each case. The equations of P31, P32, P33, and P34 are given as follows:(5a)P31=∫tan−1h/w∞∫h/sinφ∞fl,φdldφ,(5b)P32=∫tan−1h/w∞∫0h/sinφfl,φdldφ,(5c)P33=∫0tan−1h/w∫h/sinφ∞fl,φdldφ,(5d)P34=∫0tan−1h/w∫0h/sinφfl,φdldφ,where f(l, φ) is the probability density function of l and φ.It is important to note that it is not necessary to have a befitting theoretical probability density function for the length and apparent dip angle of discontinuity to calculateP31, P32, P33, and P34 by (5a)–(5d). The summation sign can be used instead of the integral sign in the absence of a theoretical probability density function. Let t be the total number of traces that satisfy the condition of each case. Then (5a)–(5d) can be replaced by(6a)P31≈∑i=1tRfli,φi,(6b)P32≈∑i=1tRfli,φi,(6c)P33≈∑i=1tRfli,φi,(6d)P34≈∑i=1tRfli,φi,where φi denotes the apparent dip angle of the ith trace in each case and Rf(li, φi) denotes the relative frequency of the ith trace appearing on the window with length li and the apparent dip angle φi. Rf(li, φi) 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 [29]. According to the stated hypothesis, Rf(li, φi) can be calculated by (7)–(9) as given below.(7)Vi=whdicos2θi+sin2θicos2αr−αi0.5+πdi24wsinθicosαr−αi+hcosθi,where αi and θi denote the dip direction and dip angle of the ith discontinuity, respectively; di is the diameter of the ith discontinuity and αr 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 (7).The weighting function for theith discontinuity (Wi) is given by(8)Wi=1Vi.Then, the corrected relative frequency of theith discontinuity is given by(9)Rfli,φi=Wi∑i=1NWi.The derivations ofK31, K32, K33, and K34 are given in the following:(1)
l≥h/sinφ and φ≥tan−1h/w.The shaded zone in Figure2 shows the probable location of center point of traces in this case. In zone gicd, li′/li=h/lsinφ. And the area of zone gicd Sgicd is(10)Sgicd=w−hcotφlsinφ−h.In the zoneabci, li′/li=zsecφ/l. The meaning of z is shown in Figure 2. And the area of zone abci Sabci can be expressed as(11)Sabci=∫0hcotφl−zsecφsinφdz.On the whole shaded zone in Figure2, the mean ratio (k31) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(12)k31=h/lsinφw−hcotφlsinφ−h+2∫0hcotφzsecφ/ll−zsecφsinφdzw−hcotφlsinφ−h+2∫0hcotφl−zsecφsinφdz=wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K31 is(13)K31=∫tan−1h/w∞∫h/sinφ∞wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−whfl,φdldφ.According to (6a)–(6d), probability density function in (13) can be estimated through replacing the integral sign by the summation sign using the empirical orientation distribution obtained for the discontinuity set. The simplified formula is as follows:(14)K31≈∑i=1twh−wh2/lcscφi+h3/3licotφicscφiwlisinφi+hlicosφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(2)
l<h/sinφ and φ≥tan−1h/w.The shaded zone in Figure3 shows the probable location of center point of traces in this case. The area of abc or a’b’c’ is(15)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l , so the mean ratio (k32) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(16)k32=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.Therefore, for all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K32 = 1/3.(3)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure4 shows the probable location of center point of traces in this case. In zone icdg, li′/li=w/lcosφ. The area of zone icdg Sicdg can be expressed as(17)Sicdg=lcosφ−wh−wtanφ.In zoneabci, li′/li=zsecφ/l. Area abci Sabci is(18)Sabci=∫0wl−zsecφsinφdz.So the mean ratio (k33) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(19)k33=w/lcosφlcosφ−wh−wtanφ+2∫0wzsecφ/ll−zsecφsinφdzlcosφ−wh−wtanφ+2∫0wl−zsecφsinφdz=wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K33 is(20)K33=∫0tan−1h/w∫h/sinφ∞wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−whfl,φdldφ.Similarly, a simplified formula of (20) can be expressed as(21)K33≈∑i=1twh−hw2/lisecφi+w3/3litanφisecφihlicosφi+wlisinφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(4)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure5 shows the probable location of center point of traces in this case. The area of zone abc or a’b’c’ is(22)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l, so the mean ratio (k34) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(23)k34=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K34 = 1/3.By integrating the above four cases,K3 can be obtained by the formula as follows:(24)K3=K31P31+K32P32+K33P33+K34P34.For all three types of discontinuities, the ratio of L′ andL can be expressed as(25a)L′L=K=K1P1+K2P2+K3P3,where(25b)P1=∫φiaφsa∫lialsafl,φdldφ,(25c)P2=∫φibφsb∫liblsbfl,φdldφ,(25d)P3=∫φicφsc∫liclscfl,φdldφ.In (25a)–(25d), P1, P2, and P3 denote the probability of discontinuities of both ends observed (type a), one end observed (type b), and both ends censored (type c) intersecting the sampling window, respectively; φ is apparent dip angle and l is the length of discontinuities; f(l, φ) is the probability density function of discontinuity orientation and length with φi ≤ φ ≤ φs and li ≤ l ≤ ls, where subscripts i and s denote inferior and superior limits, respectively; the subscripts a,b,c indicate that the limits on φ and l are for the traces of type (a), type (b), and type (c), respectively.From the abovementioned method,P1, P2, and P3 can be simplified as follows:(26a)P1≈∑i=1N2Rfli,φi,(26b)P2≈∑i=1N1Rfli,φi,(26c)P3≈∑i=1N0Rfli,φi,whereN2, N1, and N0 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 discontinuity in each type; Rf(l, φ) denotes the corrected relative frequency of discontinuities appearing on the window with length li and the apparent dip angle φi. P1, P2, and P3 can be calculated by (7)–(9).So intensify can be estimated by(27)I=L′wh=μNKwh,whereµ is the mean trace length of discontinuities; w and h are width and height of a rectangular window, respectively; N denotes the total number of traces intersecting the window; K can be estimated by (25a)–(25d).The mean trace length of discontinuitiesµ can be calculated as given below [28]:(28a)μ=wh1+R0−R21−R0+R2wB+hA,where(28b)A=∫αiαs∫θiθscosφfθ,αdθdα≈∑i=1Ncosφi×Rfφi,(28c)B=∫αiαs∫θiθssinφfθ,αdθdα≈∑i=1Nsinφi×Rfφi,(28d)cosφ=11+tan2θcos2δ1/2,(28e)sinφ=11+cot2θsec2δ1/2.In (28a)–(28e), R0 and R2 denote the fractions of discontinuities with both ends censored and both ends observable, respectively; θ and α are dip angle and dip direction; f(θ, α) is the probability density function of discontinuity orientation with θi ≤ θ ≤ θs and αi ≤ α ≤ αs, where subscripts i and s denote inferior and superior limits; δ denotes the acute angle between the dip direction and the vertical sampling plane; φi denotes the apparent dip angle of the ith trace; and Rf(φi) denotes the corrected relative frequency of the ith trace appearing on the window with the apparent dip angle φi resulting from a corresponding dip angle and a dip direction combination (θ, α).


### 2.2. Simplification ofK3 in Practical Work

As deduced in Section2.1, the estimation of K3 needs the length of traces intersecting the window. However, it is hard to acquire the actual whole length for every trace in practical discontinuity sampling by windows with limited dimensions. Therefore, simplification on K3 would be carried out to solve this problem as follows.Consider a discontinuity with both ends censored of trace lengthl≥h/sinφ and apparent dip φ≥tan−1h/w where the probable location of midpoint of trace is shown in Figure 6. Assume the ratio of observed trace length and trace length of discontinuities when the midpoint of the trace falls on the zone icm is the same as that on the zone abci (in the same way for the symmetric part of zone gnd and zone fgde). And assume that the mean ratio (k31) can be equal to average 1/3 when the midpoint is inside the zone nimd. Thus (12) can be simplified as follows:(29)k31≈1/3w−lcosφlsinφ−h+2∫0lcosφzsecφ/ll−zsecφsinφdzw−lcosφlsinφ−h+2∫0lcosφl−zsecφsinφdz=13.Figure 6
Discontinuities with both ends censored,l≥h/sinφ and φ≥tan−1h/w.The simplification process ofk33 is the similar with k31 as shown in Figure 7. Thus (19) can be simplified as given below:(30)k33≈1/3h−lsinφlcosφ−w+2∫0lcosφzsecφ/ll−zsecφsinφdzh−lsinφlcosφ−w+2∫0lcosφl−zsecφsinφdz=13.Figure 7
Traces with both ends censored,l≥h/sinφ and φ≥tan−1h/w.Thus, for all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K31 = K33 = 1/3. So K3 = 1/3.Then,(31)I=μNP1+1/2P2+1/3P3wh,where μ, N, w, and h have the same meaning as those in (25a)–(25d); P1, P2, and P3 can be calculated by (25a)–(25d); μ can be calculated by (28a)–(28e).The simplification is based on the fact that the number of discontinuities with both ends censored (type c) in the window is relatively small compared to the other two types. If this condition is contrary to the fact, it will be indicated that the window dimensions used for surveying may be too small. The unsuitability of measurement range may influence the evaluation of the trace intensity of discontinuities, and the sampling bias would be still unquenchable even if complicated formulas are adopted. In addition, the midpoints of the trace have slim chance of falling in the simplified region. In conclusion, the simplification ofK3 in this section is reasonable.


## 2.1. Deduction Process

Trace intensity is defined as mean total trace length of fractures per unit area [19]. According to the definition, trace intensity (I) can be expressed as(1)I=L′wh,where L′ is the total observed trace length of discontinuities in rectangular sampling window; w and h are, respectively, width and height of the rectangular window.Depending on the position of the discontinuity, the intersection may occur in three ways: (a) both ends are censored, (b) one end is censored, and (c) both ends are observable in the window [28]. Let N0, N1, and N2 denote the number of discontinuities with both ends censored, one end censored, and both ends observable, respectively. Let L denote the total trace length of discontinuities intersecting the rectangular sampling window. Let K denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window.(2)K=L′L.In order to estimateL′, we first deduct K by considering these three types of discontinuities respectively:


### 2.1.1. Discontinuities with Both Ends Observable

LetK1 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends observable. It is easy to find that K1 = 1.


### 2.1.2. Discontinuities with One End Censored

LetK2 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with one end censored. Figure 1 shows a one end censored trace intersecting the lower boundary of the rectangular sampling window. Let xi denote the distance between the midpoint of the ith trace and the lower boundary of the window. xi is positive when the midpoint is within the window, while xi is negative when the midpoint is out of the window. φi denotes the apparent dip angle of the ith trace. li and li’ are, respectively, the full length and observed length in the window of the ith fracture trace. K2 can be expressed as(3)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Figure 1
Intersection relationship between a trace with one end censored and the sampling window.The apparent dip angle of discontinuities in the same set does not vary a lot, so (3) can be simplified as(4)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Assume that the midpoints of trace lengths are uniformly distributed in two-dimensional space; we can get the expression∑i=1N1xi=0. So in this case, K2 = 1/2. Similarly, we can obtain the same conclusion if these discontinuities with one end censored intersect the other boundaries of the sampling window.


### 2.1.3. Discontinuities with Both Ends Censored

LetK3 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends censored. Consider the problem of a discontinuity with both ends censored with trace length, l, and apparent dip angle, φ. According to the intersection relationships between discontinuity and sampling window, four cases will be discussed in the following. They are①l≥h/sinφ and φ≥tan−1h/w (Figure 2), ② l<h/sinφ and φ≥tan−1h/w (Figure 3), ③ l≥h/sinφ and φ<tan−1h/w (Figure 4), and ④ l<h/sinφ and φ<tan−1h/w (Figure 5).Figure 2
Discontinuities with both ends censored,l≥h/sinφ and φ≥tan−1h/w.Figure 3
Discontinuities with both ends censored,l<h/sinφ and φ≥tan−1h/w.Figure 4
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.Figure 5
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.LetK31, K32, K33, and K34 denote the ratio of total observed trace length and total trace length of discontinuities in the aforementioned four cases, respectively. Use P31, P32, P33, and P34 as the probability of the traces appearing in the window, respectively, in each case. The equations of P31, P32, P33, and P34 are given as follows:(5a)P31=∫tan−1h/w∞∫h/sinφ∞fl,φdldφ,(5b)P32=∫tan−1h/w∞∫0h/sinφfl,φdldφ,(5c)P33=∫0tan−1h/w∫h/sinφ∞fl,φdldφ,(5d)P34=∫0tan−1h/w∫0h/sinφfl,φdldφ,where f(l, φ) is the probability density function of l and φ.It is important to note that it is not necessary to have a befitting theoretical probability density function for the length and apparent dip angle of discontinuity to calculateP31, P32, P33, and P34 by (5a)–(5d). The summation sign can be used instead of the integral sign in the absence of a theoretical probability density function. Let t be the total number of traces that satisfy the condition of each case. Then (5a)–(5d) can be replaced by(6a)P31≈∑i=1tRfli,φi,(6b)P32≈∑i=1tRfli,φi,(6c)P33≈∑i=1tRfli,φi,(6d)P34≈∑i=1tRfli,φi,where φi denotes the apparent dip angle of the ith trace in each case and Rf(li, φi) denotes the relative frequency of the ith trace appearing on the window with length li and the apparent dip angle φi. Rf(li, φi) 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 [29]. According to the stated hypothesis, Rf(li, φi) can be calculated by (7)–(9) as given below.(7)Vi=whdicos2θi+sin2θicos2αr−αi0.5+πdi24wsinθicosαr−αi+hcosθi,where αi and θi denote the dip direction and dip angle of the ith discontinuity, respectively; di is the diameter of the ith discontinuity and αr 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 (7).The weighting function for theith discontinuity (Wi) is given by(8)Wi=1Vi.Then, the corrected relative frequency of theith discontinuity is given by(9)Rfli,φi=Wi∑i=1NWi.The derivations ofK31, K32, K33, and K34 are given in the following:(1)
l≥h/sinφ and φ≥tan−1h/w.The shaded zone in Figure2 shows the probable location of center point of traces in this case. In zone gicd, li′/li=h/lsinφ. And the area of zone gicd Sgicd is(10)Sgicd=w−hcotφlsinφ−h.In the zoneabci, li′/li=zsecφ/l. The meaning of z is shown in Figure 2. And the area of zone abci Sabci can be expressed as(11)Sabci=∫0hcotφl−zsecφsinφdz.On the whole shaded zone in Figure2, the mean ratio (k31) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(12)k31=h/lsinφw−hcotφlsinφ−h+2∫0hcotφzsecφ/ll−zsecφsinφdzw−hcotφlsinφ−h+2∫0hcotφl−zsecφsinφdz=wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K31 is(13)K31=∫tan−1h/w∞∫h/sinφ∞wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−whfl,φdldφ.According to (6a)–(6d), probability density function in (13) can be estimated through replacing the integral sign by the summation sign using the empirical orientation distribution obtained for the discontinuity set. The simplified formula is as follows:(14)K31≈∑i=1twh−wh2/lcscφi+h3/3licotφicscφiwlisinφi+hlicosφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(2)
l<h/sinφ and φ≥tan−1h/w.The shaded zone in Figure3 shows the probable location of center point of traces in this case. The area of abc or a’b’c’ is(15)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l , so the mean ratio (k32) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(16)k32=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.Therefore, for all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K32 = 1/3.(3)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure4 shows the probable location of center point of traces in this case. In zone icdg, li′/li=w/lcosφ. The area of zone icdg Sicdg can be expressed as(17)Sicdg=lcosφ−wh−wtanφ.In zoneabci, li′/li=zsecφ/l. Area abci Sabci is(18)Sabci=∫0wl−zsecφsinφdz.So the mean ratio (k33) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(19)k33=w/lcosφlcosφ−wh−wtanφ+2∫0wzsecφ/ll−zsecφsinφdzlcosφ−wh−wtanφ+2∫0wl−zsecφsinφdz=wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K33 is(20)K33=∫0tan−1h/w∫h/sinφ∞wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−whfl,φdldφ.Similarly, a simplified formula of (20) can be expressed as(21)K33≈∑i=1twh−hw2/lisecφi+w3/3litanφisecφihlicosφi+wlisinφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(4)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure5 shows the probable location of center point of traces in this case. The area of zone abc or a’b’c’ is(22)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l, so the mean ratio (k34) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(23)k34=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K34 = 1/3.By integrating the above four cases,K3 can be obtained by the formula as follows:(24)K3=K31P31+K32P32+K33P33+K34P34.For all three types of discontinuities, the ratio of L′ andL can be expressed as(25a)L′L=K=K1P1+K2P2+K3P3,where(25b)P1=∫φiaφsa∫lialsafl,φdldφ,(25c)P2=∫φibφsb∫liblsbfl,φdldφ,(25d)P3=∫φicφsc∫liclscfl,φdldφ.In (25a)–(25d), P1, P2, and P3 denote the probability of discontinuities of both ends observed (type a), one end observed (type b), and both ends censored (type c) intersecting the sampling window, respectively; φ is apparent dip angle and l is the length of discontinuities; f(l, φ) is the probability density function of discontinuity orientation and length with φi ≤ φ ≤ φs and li ≤ l ≤ ls, where subscripts i and s denote inferior and superior limits, respectively; the subscripts a,b,c indicate that the limits on φ and l are for the traces of type (a), type (b), and type (c), respectively.From the abovementioned method,P1, P2, and P3 can be simplified as follows:(26a)P1≈∑i=1N2Rfli,φi,(26b)P2≈∑i=1N1Rfli,φi,(26c)P3≈∑i=1N0Rfli,φi,whereN2, N1, and N0 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 discontinuity in each type; Rf(l, φ) denotes the corrected relative frequency of discontinuities appearing on the window with length li and the apparent dip angle φi. P1, P2, and P3 can be calculated by (7)–(9).So intensify can be estimated by(27)I=L′wh=μNKwh,whereµ is the mean trace length of discontinuities; w and h are width and height of a rectangular window, respectively; N denotes the total number of traces intersecting the window; K can be estimated by (25a)–(25d).The mean trace length of discontinuitiesµ can be calculated as given below [28]:(28a)μ=wh1+R0−R21−R0+R2wB+hA,where(28b)A=∫αiαs∫θiθscosφfθ,αdθdα≈∑i=1Ncosφi×Rfφi,(28c)B=∫αiαs∫θiθssinφfθ,αdθdα≈∑i=1Nsinφi×Rfφi,(28d)cosφ=11+tan2θcos2δ1/2,(28e)sinφ=11+cot2θsec2δ1/2.In (28a)–(28e), R0 and R2 denote the fractions of discontinuities with both ends censored and both ends observable, respectively; θ and α are dip angle and dip direction; f(θ, α) is the probability density function of discontinuity orientation with θi ≤ θ ≤ θs and αi ≤ α ≤ αs, where subscripts i and s denote inferior and superior limits; δ denotes the acute angle between the dip direction and the vertical sampling plane; φi denotes the apparent dip angle of the ith trace; and Rf(φi) denotes the corrected relative frequency of the ith trace appearing on the window with the apparent dip angle φi resulting from a corresponding dip angle and a dip direction combination (θ, α).


## 2.1.1. Discontinuities with Both Ends Observable

LetK1 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends observable. It is easy to find that K1 = 1.


## 2.1.2. Discontinuities with One End Censored

LetK2 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with one end censored. Figure 1 shows a one end censored trace intersecting the lower boundary of the rectangular sampling window. Let xi denote the distance between the midpoint of the ith trace and the lower boundary of the window. xi is positive when the midpoint is within the window, while xi is negative when the midpoint is out of the window. φi denotes the apparent dip angle of the ith trace. li and li’ are, respectively, the full length and observed length in the window of the ith fracture trace. K2 can be expressed as(3)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Figure 1
Intersection relationship between a trace with one end censored and the sampling window.The apparent dip angle of discontinuities in the same set does not vary a lot, so (3) can be simplified as(4)K2=∑i=1N1li′∑i=1N1li=∑i=1N1xi/sinφi+∑i=1N1li/2∑i=1N1li.Assume that the midpoints of trace lengths are uniformly distributed in two-dimensional space; we can get the expression∑i=1N1xi=0. So in this case, K2 = 1/2. Similarly, we can obtain the same conclusion if these discontinuities with one end censored intersect the other boundaries of the sampling window.


## 2.1.3. Discontinuities with Both Ends Censored

LetK3 denote the ratio of total observed trace length and total trace length of discontinuities intersecting the sampling window with both ends censored. Consider the problem of a discontinuity with both ends censored with trace length, l, and apparent dip angle, φ. According to the intersection relationships between discontinuity and sampling window, four cases will be discussed in the following. They are①l≥h/sinφ and φ≥tan−1h/w (Figure 2), ② l<h/sinφ and φ≥tan−1h/w (Figure 3), ③ l≥h/sinφ and φ<tan−1h/w (Figure 4), and ④ l<h/sinφ and φ<tan−1h/w (Figure 5).Figure 2
Discontinuities with both ends censored,l≥h/sinφ and φ≥tan−1h/w.Figure 3
Discontinuities with both ends censored,l<h/sinφ and φ≥tan−1h/w.Figure 4
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.Figure 5
Traces with both ends censored,l≥h/sinφ and φ<tan−1h/w.LetK31, K32, K33, and K34 denote the ratio of total observed trace length and total trace length of discontinuities in the aforementioned four cases, respectively. Use P31, P32, P33, and P34 as the probability of the traces appearing in the window, respectively, in each case. The equations of P31, P32, P33, and P34 are given as follows:(5a)P31=∫tan−1h/w∞∫h/sinφ∞fl,φdldφ,(5b)P32=∫tan−1h/w∞∫0h/sinφfl,φdldφ,(5c)P33=∫0tan−1h/w∫h/sinφ∞fl,φdldφ,(5d)P34=∫0tan−1h/w∫0h/sinφfl,φdldφ,where f(l, φ) is the probability density function of l and φ.It is important to note that it is not necessary to have a befitting theoretical probability density function for the length and apparent dip angle of discontinuity to calculateP31, P32, P33, and P34 by (5a)–(5d). The summation sign can be used instead of the integral sign in the absence of a theoretical probability density function. Let t be the total number of traces that satisfy the condition of each case. Then (5a)–(5d) can be replaced by(6a)P31≈∑i=1tRfli,φi,(6b)P32≈∑i=1tRfli,φi,(6c)P33≈∑i=1tRfli,φi,(6d)P34≈∑i=1tRfli,φi,where φi denotes the apparent dip angle of the ith trace in each case and Rf(li, φi) denotes the relative frequency of the ith trace appearing on the window with length li and the apparent dip angle φi. Rf(li, φi) 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 [29]. According to the stated hypothesis, Rf(li, φi) can be calculated by (7)–(9) as given below.(7)Vi=whdicos2θi+sin2θicos2αr−αi0.5+πdi24wsinθicosαr−αi+hcosθi,where αi and θi denote the dip direction and dip angle of the ith discontinuity, respectively; di is the diameter of the ith discontinuity and αr 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 (7).The weighting function for theith discontinuity (Wi) is given by(8)Wi=1Vi.Then, the corrected relative frequency of theith discontinuity is given by(9)Rfli,φi=Wi∑i=1NWi.The derivations ofK31, K32, K33, and K34 are given in the following:(1)
l≥h/sinφ and φ≥tan−1h/w.The shaded zone in Figure2 shows the probable location of center point of traces in this case. In zone gicd, li′/li=h/lsinφ. And the area of zone gicd Sgicd is(10)Sgicd=w−hcotφlsinφ−h.In the zoneabci, li′/li=zsecφ/l. The meaning of z is shown in Figure 2. And the area of zone abci Sabci can be expressed as(11)Sabci=∫0hcotφl−zsecφsinφdz.On the whole shaded zone in Figure2, the mean ratio (k31) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(12)k31=h/lsinφw−hcotφlsinφ−h+2∫0hcotφzsecφ/ll−zsecφsinφdzw−hcotφlsinφ−h+2∫0hcotφl−zsecφsinφdz=wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K31 is(13)K31=∫tan−1h/w∞∫h/sinφ∞wh−wh2/lcscφ+h3/3lcotφcscφwlsinφ+hlcosφ−whfl,φdldφ.According to (6a)–(6d), probability density function in (13) can be estimated through replacing the integral sign by the summation sign using the empirical orientation distribution obtained for the discontinuity set. The simplified formula is as follows:(14)K31≈∑i=1twh−wh2/lcscφi+h3/3licotφicscφiwlisinφi+hlicosφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(2)
l<h/sinφ and φ≥tan−1h/w.The shaded zone in Figure3 shows the probable location of center point of traces in this case. The area of abc or a’b’c’ is(15)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l , so the mean ratio (k32) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(16)k32=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.Therefore, for all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K32 = 1/3.(3)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure4 shows the probable location of center point of traces in this case. In zone icdg, li′/li=w/lcosφ. The area of zone icdg Sicdg can be expressed as(17)Sicdg=lcosφ−wh−wtanφ.In zoneabci, li′/li=zsecφ/l. Area abci Sabci is(18)Sabci=∫0wl−zsecφsinφdz.So the mean ratio (k33) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(19)k33=w/lcosφlcosφ−wh−wtanφ+2∫0wzsecφ/ll−zsecφsinφdzlcosφ−wh−wtanφ+2∫0wl−zsecφsinφdz=wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−wh.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K33 is(20)K33=∫0tan−1h/w∫h/sinφ∞wh−hw2/lsecφ+w3/3ltanφsecφhlcosφ+wlsinφ−whfl,φdldφ.Similarly, a simplified formula of (20) can be expressed as(21)K33≈∑i=1twh−hw2/lisecφi+w3/3litanφisecφihlicosφi+wlisinφi−whRfli,φi,wheret, f(l, φ), and Rf(li, φi) have the same meaning as those in (6a)–(6d).(4)
l≥h/sinφ and φ<tan−1h/w.The shaded zone in Figure5 shows the probable location of center point of traces in this case. The area of zone abc or a’b’c’ is(22)Sabc=Sa′b′c′=∫0lcosφl−zsecφsinφdz.In these shaded zones,li′/li=zsecφ/l, so the mean ratio (k34) of observed trace length and trace length of discontinuities with trace length l and apparent dip angle φ can be expressed as(23)k34=∫0lcosφzsecφ/ll−zsecφsinφdz∫0lcosφl−zsecφsinφdz=13.For all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K34 = 1/3.By integrating the above four cases,K3 can be obtained by the formula as follows:(24)K3=K31P31+K32P32+K33P33+K34P34.For all three types of discontinuities, the ratio of L′ andL can be expressed as(25a)L′L=K=K1P1+K2P2+K3P3,where(25b)P1=∫φiaφsa∫lialsafl,φdldφ,(25c)P2=∫φibφsb∫liblsbfl,φdldφ,(25d)P3=∫φicφsc∫liclscfl,φdldφ.In (25a)–(25d), P1, P2, and P3 denote the probability of discontinuities of both ends observed (type a), one end observed (type b), and both ends censored (type c) intersecting the sampling window, respectively; φ is apparent dip angle and l is the length of discontinuities; f(l, φ) is the probability density function of discontinuity orientation and length with φi ≤ φ ≤ φs and li ≤ l ≤ ls, where subscripts i and s denote inferior and superior limits, respectively; the subscripts a,b,c indicate that the limits on φ and l are for the traces of type (a), type (b), and type (c), respectively.From the abovementioned method,P1, P2, and P3 can be simplified as follows:(26a)P1≈∑i=1N2Rfli,φi,(26b)P2≈∑i=1N1Rfli,φi,(26c)P3≈∑i=1N0Rfli,φi,whereN2, N1, and N0 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 discontinuity in each type; Rf(l, φ) denotes the corrected relative frequency of discontinuities appearing on the window with length li and the apparent dip angle φi. P1, P2, and P3 can be calculated by (7)–(9).So intensify can be estimated by(27)I=L′wh=μNKwh,whereµ is the mean trace length of discontinuities; w and h are width and height of a rectangular window, respectively; N denotes the total number of traces intersecting the window; K can be estimated by (25a)–(25d).The mean trace length of discontinuitiesµ can be calculated as given below [28]:(28a)μ=wh1+R0−R21−R0+R2wB+hA,where(28b)A=∫αiαs∫θiθscosφfθ,αdθdα≈∑i=1Ncosφi×Rfφi,(28c)B=∫αiαs∫θiθssinφfθ,αdθdα≈∑i=1Nsinφi×Rfφi,(28d)cosφ=11+tan2θcos2δ1/2,(28e)sinφ=11+cot2θsec2δ1/2.In (28a)–(28e), R0 and R2 denote the fractions of discontinuities with both ends censored and both ends observable, respectively; θ and α are dip angle and dip direction; f(θ, α) is the probability density function of discontinuity orientation with θi ≤ θ ≤ θs and αi ≤ α ≤ αs, where subscripts i and s denote inferior and superior limits; δ denotes the acute angle between the dip direction and the vertical sampling plane; φi denotes the apparent dip angle of the ith trace; and Rf(φi) denotes the corrected relative frequency of the ith trace appearing on the window with the apparent dip angle φi resulting from a corresponding dip angle and a dip direction combination (θ, α).


## 2.2. Simplification ofK3 in Practical Work

As deduced in Section2.1, the estimation of K3 needs the length of traces intersecting the window. However, it is hard to acquire the actual whole length for every trace in practical discontinuity sampling by windows with limited dimensions. Therefore, simplification on K3 would be carried out to solve this problem as follows.Consider a discontinuity with both ends censored of trace lengthl≥h/sinφ and apparent dip φ≥tan−1h/w where the probable location of midpoint of trace is shown in Figure 6. Assume the ratio of observed trace length and trace length of discontinuities when the midpoint of the trace falls on the zone icm is the same as that on the zone abci (in the same way for the symmetric part of zone gnd and zone fgde). And assume that the mean ratio (k31) can be equal to average 1/3 when the midpoint is inside the zone nimd. Thus (12) can be simplified as follows:(29)k31≈1/3w−lcosφlsinφ−h+2∫0lcosφzsecφ/ll−zsecφsinφdzw−lcosφlsinφ−h+2∫0lcosφl−zsecφsinφdz=13.Figure 6
Discontinuities with both ends censored,l≥h/sinφ and φ≥tan−1h/w.The simplification process ofk33 is the similar with k31 as shown in Figure 7. Thus (19) can be simplified as given below:(30)k33≈1/3h−lsinφlcosφ−w+2∫0lcosφzsecφ/ll−zsecφsinφdzh−lsinφlcosφ−w+2∫0lcosφl−zsecφsinφdz=13.Figure 7
Traces with both ends censored,l≥h/sinφ and φ≥tan−1h/w.Thus, for all the possiblel and φ in this case, the ratio of total observed trace length and total trace length K31 = K33 = 1/3. So K3 = 1/3.Then,(31)I=μNP1+1/2P2+1/3P3wh,where μ, N, w, and h have the same meaning as those in (25a)–(25d); P1, P2, and P3 can be calculated by (25a)–(25d); μ can be calculated by (28a)–(28e).The simplification is based on the fact that the number of discontinuities with both ends censored (type c) in the window is relatively small compared to the other two types. If this condition is contrary to the fact, it will be indicated that the window dimensions used for surveying may be too small. The unsuitability of measurement range may influence the evaluation of the trace intensity of discontinuities, and the sampling bias would be still unquenchable even if complicated formulas are adopted. In addition, the midpoints of the trace have slim chance of falling in the simplified region. In conclusion, the simplification ofK3 in this section is reasonable.


## 3. Case Study

### 3.1. Application of the Proposed New Method Based on Rectangular Windows

The Wenchuan earthquake triggered enormous landslides in China, 2008. An almost vertical rock slope caused by the earthquake is an excellent exposed outcrop to investigate the discontinuity geometry characteristics of the rock mass, which is significant to better understand the formation mechanism of earthquake landslides. The discontinuity trace network of the outcrop as shown in Figure8 was used to estimate rock discontinuity mean trace length and density in a previous study [27]. The discontinuity data of this outcrop are utilized in this section to validate the proposed new method based on rectangular windows. According to the orientation data, discontinuities on the outcrop can be divided into two sets. Rectangular sampling windows with dimensions of 90 m × 45 m, 110 m × 55 m, and 130 m × 65 m have been placed at four locations on the outcrop. Trace intensity values, respectively, for two sets of discontinuities sampled by all twelve windows are estimated according to the methodology (in (31)) explained in Section 2. The estimation results are presented in Table 1 and Figure 9. The obtained overall mean value and coefﬁcient of variation related to set 1 are equal to 0.151 m−1 and 0.299 and the ones related to set 2 are equal to 0.102 m−1 and 0.205, respectively. True intensity values are calculated by using the data of trace length for discontinuities intersecting sampling windows on the outcrop. The obtained true mean values for all the windows related to set 1 and set 2 are also shown in Table 1 and Figure 9. Both the estimated and the true values indicate that the discontinuity trace intensity varies with the size and the location of the rectangular windows.Figure 8
Sizes and locations of rectangular windows used to estimate mean trace length [27].Table 1
Estimated intensity and its % error based on rectangular sampling windows.
Window size (m)/location numberEstimated value (m−1)True value (m−1)% error ofIeSet 1Set 2Set 1Set 2Set 1Set 290 × 45/location 10.1140.0920.0840.088−36.3−4.890 × 45/location 20.2230.0730.2270.0531.6−38.990 × 45/location 30.1820.1520.1890.1253.5−21.890 × 45/location 40.0930.1110.0810.082−14.9−34.9110 × 55/location 10.1280.0960.0960.091−33.8−6.1110 × 55/location 20.2070.0910.2160.0694.3−32.4110 × 55/location 30.1790.1280.1790.1260.4−1.7110 × 55/location 40.1010.0910.0840.085−20.0−7.1130 × 65/location 10.1420.0970.1060.093−34.7−3.7130 × 65/location 20.1740.0880.2030.07814.0−13.2130 × 65/location 30.1710.1100.1730.1221.29.6130 × 65/location 40.0920.0960.0860.087−6.9−10.0Mean value0.1510.1020.1440.0914.811.6Coefficient of variation0.2990.2050.4090.247Range−36.3 to 14.0−38.9 to 9.6Figure 9
Estimated intensity based on new estimation method using rectangular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)An estimation error can be deﬁned as(32)errorI=It−IeIt×100%,where It and Ie denote the true and estimated trace intensity values, respectively. This error is to evaluate the estimation accuracy of the proposed method. In Table 1 and Figure 10, the estimation errors for set 1 and set 2 related to different sizes of rectangular windows at different locations are presented. The error range caused by finite sample sizes and spatial variability for rectangular window is −36.3% to +14.0% for set 1 and −38.9% to +9.6% for set 2.Figure 10
Prediction error of mean trace length associated with finite sample sizes and spatial variability based on new estimation method using rectangular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)


### 3.2. Application of the Method Based on Circular Windows

The equations (in (33)) to estimate trace intensity using circular windows proposed by Mauldon [6] are also used to estimate the trace intensity for discontinuities on the same outcrop.(33)I=N−NC+NT4R,where N denotes the total number of traces intersecting the circular window, NC denotes the number of traces that are contained in the circular window, and NT denotes the number of traces that transect the circular window.A series of circular windows with radii of 20 m, 30 m, and 40 m at five locations were placed on the discontinuity trace network as shown in Figure11. The calculation results of trace intensity according to (33) are shown in Table 2 and Figure 12. The overall mean value and coefﬁcient of variation related to set 1 are equal to 0.139 m−1 and 0.482 and the ones related to set 2 are equal to 0.095 m−1 and 0.324, respectively. The obtained true values are for all the circular windows related to set 1 and set 2 are also shown in Table 2 and Figure 12. Both the estimated and the true values indicate that the discontinuity trace intensity also varies with the size and the location of the circular windows. The calculating methods of estimation errors were the same as that illustrated in Section 3.1. Table 2 and Figure 13 depict the estimation errors of Mauldon’s method using circular windows. The error range for circular windows is −56.1% to +52.0% for set 1 and −57.2% to +38.4% for set 2.Figure 11
Sizes and locations of the circular windows used to estimate mean trace length [27].Table 2
Estimated intensity and its % error based on circular sampling windows.
Window size (m)/location numberEstimated value (m−1)True value (m−1)% error ofIeSet 1Set 2Set 1Set 2Set 1Set 220/location 10.0630.1250.0710.089−31.7−31.520/location 20.0880.1000.1140.07011.9−0.920/location 30.2750.1000.2810.0696.7−57.220/location 40.2130.1500.2540.143−43.15.320/location 50.0880.0380.0920.05752.08.330/location 10.0750.1250.0630.109−56.19.330/location 20.0750.1080.0870.079−12.517.530/location 30.2330.0670.2610.075−1.6−35.830/location 40.1830.1080.2070.123−28.54.230/location 50.0750.0420.0780.050−31.4−30.040/location 10.1190.1060.0830.110−25.3−4.240/location 20.1310.0630.1090.083−13.638.440/location 30.2060.1000.2320.081−20.0−13.840/location 40.1810.1130.1870.121−2.85.140/location 50.0880.0750.0810.0616.1−14.3Mean value0.1390.0950.1470.088−4.97.4Coefﬁcient of variation0.4820.3240.5290.299Range−56.1 to 52.0−57.2 to 38.4Figure 12
Estimated intensity based on Mauldon’s equation using circular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)Figure 13
Prediction error of intensity associated with finite sample sizes and spatial variability based on Mauldon’s equation using circular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)


### 3.3. Comparison of Results from the Two Intensity Estimation Methods

Errors of overall mean predictions obtained through the rectangular windows method were close to that obtained through the circular windows method. The error ranges obtained for Mauldon’s method based on the circular windows are larger than that resulting from the proposed method based on rectangular windows. The coefficient of variation values depicted in Tables1 and 2 also indicates that the estimation results based on the proposed method show less variability than that of the method based on circular windows due to the variability of window location and size. The proposed method allows one to incorporate the relative frequency of each trace appearing on the exposure. Because this equation uses the angle between the discontinuity plane and the exposure plane, it can also be utilized to estimate trace intensity for traces on curved sampling surfaces such as outcrops that show some curvature. In this case, the curved surface could be divided into several planar surfaces. Then the relative frequencies can be estimated for the traces appearing on each planar surface. Besides, rectangular window has advantage in sampling discontinuities on rock mass surfaces longer in one direction than in the other such as tunnel cross sections.


## 3.1. Application of the Proposed New Method Based on Rectangular Windows

The Wenchuan earthquake triggered enormous landslides in China, 2008. An almost vertical rock slope caused by the earthquake is an excellent exposed outcrop to investigate the discontinuity geometry characteristics of the rock mass, which is significant to better understand the formation mechanism of earthquake landslides. The discontinuity trace network of the outcrop as shown in Figure8 was used to estimate rock discontinuity mean trace length and density in a previous study [27]. The discontinuity data of this outcrop are utilized in this section to validate the proposed new method based on rectangular windows. According to the orientation data, discontinuities on the outcrop can be divided into two sets. Rectangular sampling windows with dimensions of 90 m × 45 m, 110 m × 55 m, and 130 m × 65 m have been placed at four locations on the outcrop. Trace intensity values, respectively, for two sets of discontinuities sampled by all twelve windows are estimated according to the methodology (in (31)) explained in Section 2. The estimation results are presented in Table 1 and Figure 9. The obtained overall mean value and coefﬁcient of variation related to set 1 are equal to 0.151 m−1 and 0.299 and the ones related to set 2 are equal to 0.102 m−1 and 0.205, respectively. True intensity values are calculated by using the data of trace length for discontinuities intersecting sampling windows on the outcrop. The obtained true mean values for all the windows related to set 1 and set 2 are also shown in Table 1 and Figure 9. Both the estimated and the true values indicate that the discontinuity trace intensity varies with the size and the location of the rectangular windows.Figure 8
Sizes and locations of rectangular windows used to estimate mean trace length [27].Table 1
Estimated intensity and its % error based on rectangular sampling windows.
Window size (m)/location numberEstimated value (m−1)True value (m−1)% error ofIeSet 1Set 2Set 1Set 2Set 1Set 290 × 45/location 10.1140.0920.0840.088−36.3−4.890 × 45/location 20.2230.0730.2270.0531.6−38.990 × 45/location 30.1820.1520.1890.1253.5−21.890 × 45/location 40.0930.1110.0810.082−14.9−34.9110 × 55/location 10.1280.0960.0960.091−33.8−6.1110 × 55/location 20.2070.0910.2160.0694.3−32.4110 × 55/location 30.1790.1280.1790.1260.4−1.7110 × 55/location 40.1010.0910.0840.085−20.0−7.1130 × 65/location 10.1420.0970.1060.093−34.7−3.7130 × 65/location 20.1740.0880.2030.07814.0−13.2130 × 65/location 30.1710.1100.1730.1221.29.6130 × 65/location 40.0920.0960.0860.087−6.9−10.0Mean value0.1510.1020.1440.0914.811.6Coefficient of variation0.2990.2050.4090.247Range−36.3 to 14.0−38.9 to 9.6Figure 9
Estimated intensity based on new estimation method using rectangular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)An estimation error can be deﬁned as(32)errorI=It−IeIt×100%,where It and Ie denote the true and estimated trace intensity values, respectively. This error is to evaluate the estimation accuracy of the proposed method. In Table 1 and Figure 10, the estimation errors for set 1 and set 2 related to different sizes of rectangular windows at different locations are presented. The error range caused by finite sample sizes and spatial variability for rectangular window is −36.3% to +14.0% for set 1 and −38.9% to +9.6% for set 2.Figure 10
Prediction error of mean trace length associated with finite sample sizes and spatial variability based on new estimation method using rectangular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)


## 3.2. Application of the Method Based on Circular Windows

The equations (in (33)) to estimate trace intensity using circular windows proposed by Mauldon [6] are also used to estimate the trace intensity for discontinuities on the same outcrop.(33)I=N−NC+NT4R,where N denotes the total number of traces intersecting the circular window, NC denotes the number of traces that are contained in the circular window, and NT denotes the number of traces that transect the circular window.A series of circular windows with radii of 20 m, 30 m, and 40 m at five locations were placed on the discontinuity trace network as shown in Figure11. The calculation results of trace intensity according to (33) are shown in Table 2 and Figure 12. The overall mean value and coefﬁcient of variation related to set 1 are equal to 0.139 m−1 and 0.482 and the ones related to set 2 are equal to 0.095 m−1 and 0.324, respectively. The obtained true values are for all the circular windows related to set 1 and set 2 are also shown in Table 2 and Figure 12. Both the estimated and the true values indicate that the discontinuity trace intensity also varies with the size and the location of the circular windows. The calculating methods of estimation errors were the same as that illustrated in Section 3.1. Table 2 and Figure 13 depict the estimation errors of Mauldon’s method using circular windows. The error range for circular windows is −56.1% to +52.0% for set 1 and −57.2% to +38.4% for set 2.Figure 11
Sizes and locations of the circular windows used to estimate mean trace length [27].Table 2
Estimated intensity and its % error based on circular sampling windows.
Window size (m)/location numberEstimated value (m−1)True value (m−1)% error ofIeSet 1Set 2Set 1Set 2Set 1Set 220/location 10.0630.1250.0710.089−31.7−31.520/location 20.0880.1000.1140.07011.9−0.920/location 30.2750.1000.2810.0696.7−57.220/location 40.2130.1500.2540.143−43.15.320/location 50.0880.0380.0920.05752.08.330/location 10.0750.1250.0630.109−56.19.330/location 20.0750.1080.0870.079−12.517.530/location 30.2330.0670.2610.075−1.6−35.830/location 40.1830.1080.2070.123−28.54.230/location 50.0750.0420.0780.050−31.4−30.040/location 10.1190.1060.0830.110−25.3−4.240/location 20.1310.0630.1090.083−13.638.440/location 30.2060.1000.2320.081−20.0−13.840/location 40.1810.1130.1870.121−2.85.140/location 50.0880.0750.0810.0616.1−14.3Mean value0.1390.0950.1470.088−4.97.4Coefﬁcient of variation0.4820.3240.5290.299Range−56.1 to 52.0−57.2 to 38.4Figure 12
Estimated intensity based on Mauldon’s equation using circular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)Figure 13
Prediction error of intensity associated with finite sample sizes and spatial variability based on Mauldon’s equation using circular windows of different sizes placed at different locations on the outcrop. (a) Set 1. (b) Set 2.
(a)(b)


## 3.3. Comparison of Results from the Two Intensity Estimation Methods

Errors of overall mean predictions obtained through the rectangular windows method were close to that obtained through the circular windows method. The error ranges obtained for Mauldon’s method based on the circular windows are larger than that resulting from the proposed method based on rectangular windows. The coefficient of variation values depicted in Tables1 and 2 also indicates that the estimation results based on the proposed method show less variability than that of the method based on circular windows due to the variability of window location and size. The proposed method allows one to incorporate the relative frequency of each trace appearing on the exposure. Because this equation uses the angle between the discontinuity plane and the exposure plane, it can also be utilized to estimate trace intensity for traces on curved sampling surfaces such as outcrops that show some curvature. In this case, the curved surface could be divided into several planar surfaces. Then the relative frequencies can be estimated for the traces appearing on each planar surface. Besides, rectangular window has advantage in sampling discontinuities on rock mass surfaces longer in one direction than in the other such as tunnel cross sections.


## 4. Conclusions

Trace intensity is one of the most important characteristics of fractures that affect the mechanical properties of the rock mass. In this study, a new estimation method of discontinuity trace intensity based on rectangular windows has been proposed by considering that the probability of each trace appearing on the two-dimensional exposures is different. The discontinuity traces on the outcrop in Wenchuan area were used along with rectangular windows to estimate the trace intensity using the proposed equations. Similarly, circular windows were used along with Mauldon’s equation [19] to calculate the trace intensity for the discontinuity traces appearing in the same outcrop as a comparison. Errors of overall mean predictions obtained through the rectangular window method were close to that obtained through the circular window method. The error ranges obtained for Mauldon’s method based on the circular windows are larger than that resulting from the proposed method based on rectangular windows. The coefficient of variation values depicted in Tables 1 and 2 also indicates that the estimation results based on the proposed method show less variability than that of the method based on circular window due to the variability of window location and size. It should be note that the proposed method based on rectangular sampling windows has advantage in application to sampling surfaces longer in one direction than in the other such as tunnel cross sections and curved sampling surfaces such as outcrops that show some curvature.



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*Source: 8834861-2020-11-26.xml*