Estimating Mean Fracture Trace Length and Density from Observations in Convex Windows

Summary Although window samples of fracture traces are widely recognized to be length-biased and censored, they are often the best source of data for inferring statistical parameters of fracture sets. This paper presents new estimators for mean fracture trace length and density that correct for the effects of bias and censoring. A stereological estimator of mean trace length is derived for parallel traces in a rectangular sampling window, an end-point estimator of mean trace length is derived for windows with arbitrary convex boundaries and for arbitrary trace length distributions, and an end-point estimator of trace density is derived for windows with arbitrary convex boundaries and for arbitrary trace length and trace orientation distributions. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. When applied to circular windows, the end-point estimator of mean trace length is, in addition, independent of the trace orientation distribution. The estimators are easily determined from field data. The end-point estimator of trace density requires knowing only the area of the window and the number of end-points inside the window. The end-point estimator of mean trace length, when applied to circular windows, requires, in addition, the number of end-points outside the window (of those traces that intersect the window) and the stereological estimator of mean trace length requires only the apparent mean trace length and the height of the window.     

Circular scanlines and circular windows: new tools for characterizing the geometry of fracture traces

We introduce new estimators for fracture trace intensity, trace density and mean trace length that exploit the use of circles as efficient sampling tools. A fracture trace is the commonly observed surface expression of a fracture, i.e. the intersection of a fracture with an exposed surface such as a rock pavement or a mine drive wall. Trace intensity, trace density and mean trace length estimators are derived and shown to form a self-consistent set of two-dimensional fracture abundance measures. The intensity estimator n/4r uses the number, n, of intersections between fracture traces and a circular scanline of radius r. The density estimator m/2πr² uses the number, m, of trace endpoints inside a circular window. The mean trace length estimator (n/m)πr/2 uses the ratio of the number of trace intersections on the circle to the number of endpoints in the circle.The circular sampling tools and estimators described here eliminate most sampling biases due to orientation and also correct many errors due to censoring and length bias that plague established scanline and areal measurement techniques. Performance of the estimators is demonstrated by comparison with areal samples of a synthetic fracture trace population with known intensity, density and mean trace length. The estimators are also applied successfully to a natural rock pavement with two orthogonal fracture sets, one of which is severely censored. Because the new circle-based estimators only require counts of trace–circle intersections and/or trace endpoints, they are more time-efficient than current methods for estimating geometric characteristics of fracture traces.     

不连续面平均速长概率估值法及其应用

在进行不连续面3-D网络模拟过程中, 不连续面的大小是一个重要的参数, 该数据主要来源是在现场露头上测量不连续面出露的迹长。由于取样中不可避免的偏差而使结果有误差, 在进行平均迹长估计时取样的误差必须予以校正, 本文介绍一种用概率的方法考虑取样误差通过有限2-D露头上测量的不连续面数据求算无限2-D空间平均迹长的技术。该方法可以借助计算机进行处理。     

不连续面平均速长概率估值法及其应用

在进行不连续面３－Ｄ网络模拟过程中，不连续面的大小是一个重要的参数，该数据主要来源是在现场露头上测量不连续面出露的迹长。由于取样中不可避免的偏差而使结果有误差，在进行平均迹长估计时取样的误差必须予以校正，本文介绍一种用概率的方法考虑取样误差通过有限２－Ｄ露头上测量的不连续面数据求算无限２－Ｄ空间平均迹长的技术。该方法可以借助计算机进行处理。     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据，有效的减少了野外工作量     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据, 有效的减少了野外工作量。     

取样窗口迹长均值估计方法的讨论

采用窗口取样法测量岩体中随机不连续面的迹长, 相对于无限二维空间的岩体而言存在着不可避免的偏差。因此, 在实际岩体工程中要对实地测量迹长的偏差进行校正, 以获得校正后的平均迹长。本文着重讨论了Kulatilake教授于 1984年提出的不连续面迹长均值估算法, 并对算法作了必要的修正, 使算法更客观, 更完善, 编制的计算机代码可以很方便地求取无限二维岩体不连续面的校正迹长。     

Appraisal of fracture sampling methods and a new workflow to characterise heterogeneous fracture networks at outcrop

Characterising fractures at outcrop for use as analogues to fractured reservoirs can use several methods. Four important fracture data collection methods are linear scanline sampling, areal sampling, window sampling and circular scanline sampling. In regions of homogeneous fracture networks these methods are adequate to characterise fracture patterns for use as outcrop analogues, however where fractures are heterogeneous, it is more difficult to characterise fracture networks and a different approach is needed.We develop a workflow for fracture data collection in a region of heterogeneous fractures in a fold and thrust belt, which we believe has applicability to a wide variety of fracture networks in different tectonic settings. We use an augmented circular scanline method, along with areal sampling to collect a range of fracture attribute data, including orientation, length, aperture, spatial distribution and intensity. This augmented circular scanline method more than halves the time taken for data collection, provides accurate, unbiased data that is representative of local fracture network attributes and involves data collection of a wider range of fracture attributes than other sampling techniques alone.     

Comparison of rock discontinuity mean trace length and density estimation methods using discontinuity data from an outcrop in Wenchuan area,China

The equations that exist in the literature to estimate corrected mean trace length and corrected two-dimensional density of a rock discontinuity set using area sampling technique are critically reviewed. The discontinuity traces appearing in an outcrop in Yingxiu area in China are used along with rectangular windows to calculate the corrected mean trace length and two-dimensional density using Kulatilake and Wu’s equations. Similarly, circular windows are used along with Mauldon’s and Zhang and Einstein’s equation to calculate the mean trace length and Mauldon’s equation to calculate the two-dimensional density for the same discontinuity sets using the same outcrop discontinuity trace data. For both parameters, the predictions based on the rectangular window methods were found to be more accurate than that based on the circular window methods.     

An application of the deterministic variogram to design-based variance estimation

When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.     

Laslett算法和广义H-H算法在不同迹长层次下的适用性讨论

结构面平均迹长估算的Laslett算法和广义H-H算法虽然在理论上具有严密的推导过程,但实际应用时,由于迹长层次不同,两种算法估值精度不同。在利用开挖边坡所揭露的全迹长资料对不同迹长层次下两种算法的估值精度进行检验的基础上,讨论了两种算法的适用性,并针对工程中所采用的测量窗口高度通常较小这一实际情况,对广义H-H算法在小窗口长迹长层次下的迹长估计值进行适当修正。综合研究结果,建议:(1)小窗口短小迹长层次情况下采用Laslett算法进行迹长估计; (2)小窗口长大迹长层次情况下采用乘以修正系数x=2的广义H-H算法。     

Estimation of mean trace length of discontinuities

Summary Trace lengths of discontinuities observed on finite exposures are biased due to sampling errors. These errors should be corrected in estimating mean trace length. A technique, which takes into account the sampling errors, is proposed for estimating the mean trace length on infinite, vertical sections from the observations made on finite, rectangular, vertical exposures. The method is applicable to discontinuities whose orientation is described by a probability distribution function. The method requires that the numbers of discontinuities with both ends observed, one end observed, and both ends censored be known. The lengths of observed traces and the density function of trace length are not required. The derivation assumes that midpoints of traces are uniformly distributed in the vertical plane. Also independence between trace length and orientation is assumed. Data on a Pennsylvania shale in Ohio, U. S. A., were used as an example.Notations dip direction - direction of sampling plane - acute angle between dip direction and sampling plane - dip angle - _A apparent dip angle - mean density of trace mid-points per unit area - mean trace length - D diameter of discontinuity - f (.),g (.) probability density function - h height of rectangular window - estimator of mean trace length - m sample size, number of discontinuities intersecting window - m ₀ number of discontinuities intersecting window with both ends censored - m ₂ number of discontinuities intersecting window with both ends observed - n, N expected number of discontinuities intersecting the window - n ₀,N ₀ expected number of discontinuities intersecting the window with both ends censored - n ₂,N ₂ expected number of discontinuities intersecting the window with both ends observed - Pr (.) probability - w width of rectangular window - x trace length     

Stereological and Other Methods Applied to Rock Joint Size Estimation—Does Crofton''s Theorem Apply?

A number of authors concerned with the analysis of rock jointing have used the idea that the joint areal or diametral distribution can be linked to the trace length distribution through a theorem attributed to Crofton. This brief paper seeks to demonstrate why Crofton's theorem need not be used to link moments of the trace length distribution captured by scan line or areal mapping to the moments of the diametral distribution of joints represented as disks and that it is incorrect to do so. The valid relationships for areal or scan line mapping between all the moments of the trace length distribution and those of the joint size distribution for joints modeled as disks are recalled and compared with those that might be applied were Crofton's theorem assumed to apply. For areal mapping, the relationship is fortuitously correct but incorrect for scan line mapping.     

Estimating the Mean Trace Length of Rock Discontinuities

Summary Rock discontinuities appear as traces on exposures such as natural outcrops or tunnel walls. Discontinuity size which has important effects on rock mass behavior is related to trace length.  This paper presents a technique for estimating mean trace length from the observations made on finite, circular sampling windows. The method takes sampling errors into account and it requires, like existing methods using rectangular sampling windows, that the numbers of discontinuities with both ends censored, with one end observed and one end censored, and with both ends observed be known. Knowledge of the lengths of the observed traces and the distribution of trace lengths is not required. A major advantage of the proposed method over the existing methods is that it does not need sampling data about the orientation of discontinuities, i.e. the proposed method is applicable to traces with arbitrary orientation distributions.  To check the validity of the proposed method, theoretical relations between the mean trace length and the mean diameter of circular discontinuities, respectively for lognormal and negative exponential distributions of the diameter of discontinuities, are derived. The proposed method is then applied to analyze data simulated with the FracMan code, and the predicted results are compared to the corresponding theoretical solutions. The results show that the proposed method is satisfactory. Comparisons of the predicted mean trace length with the mean of observed trace lengths show that it is important to consider sampling biases when estimating mean trace length.     

Stereological and Other Methods Applied to Rock Joint Size Estimation—Does Crofton's Theorem Apply?

A number of authors concerned with the analysis of rock jointing have used the idea that the joint areal or diametral distribution can be linked to the trace length distribution through a theorem attributed to Crofton. This brief paper seeks to demonstrate why Crofton's theorem need not be used to link moments of the trace length distribution captured by scan line or areal mapping to the moments of the diametral distribution of joints represented as disks and that it is incorrect to do so. The valid relationships for areal or scan line mapping between all the moments of the trace length distribution and those of the joint size distribution for joints modeled as disks are recalled and compared with those that might be applied were Crofton's theorem assumed to apply. For areal mapping, the relationship is fortuitously correct but incorrect for scan line mapping.     

基于交切条件的不连续面平均迹长估算法

从不连续面与窗口交切的充要条件出发,推导了窗口法不连续面剖面测量的平均迹长估算公式,实例分析表明用雇在算的平均迹长与实测平均迹长吻合较好。     

拱坝坝肩侧裂面的面连通率研究

采用窗口法对拱坝坝肩不连续面进行描绘测量,对获取的不连续面的产状和迹长进行取样偏差校正和蒙特卡罗模拟,从而形咸三维空间的不连续面网络模型。据工程要求对网络模型按一定带宽进行切割、投影、计算,由此可得到侧裂结构面的面连通率。在此基础上,可以进行坝肩岩体质量评价,为工程设计提供一定的理论依据。      

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

Demarcation of homogeneous structural domains within a rock mass based on joint orientation and trace length

This paper presents a new method for determining the structural domain boundaries within the rock mass. This new method is based on a statistical comparison of data from pairs of sample regions. The stereonet is divided into 100 windows with approximately equal areas. The poles of joints occurring in each corresponding window on the two projection plots of the regions being compared are then merged and arranged in ascending order with respect to their trace lengths. Finally, the Wald–Wolfowitz runs test is used to identify the homogeneity of structural populations by analyzing the joint sequence. Based on a significance level of 0.01, the homogeneity of structural populations collected from four adjacent adits at the Songta dam site is determined using the proposed method. The results show that the boundaries of structural domain change with the sizes of the sampling domains being compared. The initial sampling domains should be selected according to the engineering geological conditions of the studied area. In addition, the clear advantage of the proposed method is that both joint orientation and trace length are considered.     

用窗口法估计不连续面的连通率

基于不连续面与窗口交切的必需而充分的条件,推导了不连续面迹长、间断长和不连续面连通率的估算公式,通过实例验证了该公式的正确性。