Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions

Isotropic reproducing kernels for a sphere or spherical shell are derived as weighted product sums o fL ₂ orthonormal base functions. For the sphere these functions are products of the surface spherical harmonics and the Jacobi polynomials of degree (0, 2). Reproducing kernels for a sphere are consistent with the covariance function of the outer anomalous gravity potential of the Earth. These reproducing kernels may be used for gravity field modeling which include density (anomaly) data as observations or which aims at predicting such quantities using optimal estimation methods, that is for solving the inverse gravimetric problem.     

A new cluster algorithm for orientation data

An algorithm to classify data points on the sphere in distinct cluster groups is defined. The characteristics of the cluster groups and the rule for assigning data to the groups are related to a continuous differentiable density estimation. The modes of the estimated density are assumed to be representative of the groups; data points are then assigned to the mode reached by the steepest ascent. The major advantage of this procedure is its sensitivity in detecting cluster groups independently of their geometry and configuration. As a consequence, the procedure is capable of handling orientation data that may be arranged in girdles.     

Grain size distribution expressed as tanh-functions

The grain size distribution within a unimodal sediment can be described as a lognormal distribution when the distribution is formed by only one process. However, most sediments are formed by more than one process giving polymodal sediments. Polymodal sediments have to be described as the sum of several normal distributions, one for each process involved within the formation. Grain size distributions are usually interpreted with the help of graphical methods. Interpretations of polymodal sediments require mathematical methods. In mathematical terms a unimodal sediment can be described as a tangential hyperbolic function (tanh) and a polymodal sediment can generally be described by the sum of two or three tanh-functions. The tanh-method is a tool for identifying and estimating the number of modes within a grain size distribution and helps interpret the processes involved within the formation of a deposit. The mathematical method can also be used to computerize sediment data, allowing storage with just a few numbers. Different samples can easily be compared and classified. Also, this method could be a valuable tool for calculations of various sediment parameters both in geotechnology and hydrogeology.     

Conditional estimation variances: An empirical approach

One of the tasks routinely carried out by geostatisticians is the evaluation of global mining reserves corresponding to a given cutoff grade and size of selective mining units. A long with these recovery figures, the geostatistician generally provides an assessment of the global estimation variance, which represents the precision of the overall average grade estimate, when no cutoff is applied. Such a global estimation variance is of limited interest for evaluating mining projects; what is required is the reliability of the estimate of recovered reserves or, in other words, the conditional estimation variance. Unfortunately, classical linear geostatistical methods fail to provide an easy way to estimate this variance. Through the use of simulated deposits (representing various types of regionalization)the present paper reviews and discusses the effects of changes in cutoff grade and selective mining unit size on the conditional estimation variance. It is shown that, when the cutoff grade is applied to a pointsupport (sample-size)distribution, the conditional estimation variance appears to be readily accessible by classical formulas, once the conditional semivariogram is known. However, the evaluation of the conditional estimation variance seems to be less straightforward for the general case when a cutoff is applied to the average grade distribution of selective mining units. Empirical approximation formulas for the conditional estimation variance are tentatively proposed, and their performance in the case of the simulated deposits is shown. The limitations of these approximations are discussed, and possible ways of formalizing the problem suggested.     

Fast Summation of Functions on the Rotation Group

Computing with functions on the rotation group is a task carried out in various areas of application. When it comes to approximation, kernel based methods are a suitable tool to handle these functions. In this paper, we present an algorithm which allows us to evaluate linear combinations of functions on the rotation group as well as a truly fast algorithm to sum up radial functions on the rotation group. These approaches based on nonequispaced FFTs on SO(3) take O(M+N)\mathcal{O}(M+N) arithmetic operations for M and N arbitrarily distributed source and target nodes, respectively. In this paper, we investigate a selection of radial functions and give explicit theoretical error bounds, as well as numerical examples of approximation errors. Moreover, we provide an application of our method, namely the kernel density estimation from electron back scattering diffraction (EBSD) data, a problem relevant in texture analysis.     

Block size selection with the objective of minimizing the discrepancy in real and estimated block grade

The valuation of a mining project depends upon the accuracy of geological block model. Sampling density, estimation method, and proper block size mainly affect the accuracy of estimated block. This paper aims to answer three questions: (1) which estimation method is more accurate, (2) what is the relation between sampling density and block size, and (3) what the optimum block size is. Conditional Gaussian simulation (CGS) was used to generate a hypothetical deposit, considered as a real block model. A range of different block dimensions were estimated by ordinary kriging, inverse squared distance, and nearest neighbor methods based on tow-simulated drilling grids database. The comparison of estimated and real block grades reveals that increasing the sampling density results the similar outcomes of geostatistics and deterministic interpolation methods. Furthermore, it was deduced that sampling density could not be a viable alternative in choosing appropriate block dimension and the variogram rang a was suggested as an affective parameter in block size selection. Then a geometrical formula was developed to obtain the block size based on the variogram range. The increment in project value that a mine planner can expected from the additional information of the dense drilling grid was also calculated and it was concluded that the block size obtained based on the suggested formula results acceptable information value. Finally, the database of Chador Malu iron ore mine which is located in 180 km northeast of Yazd city in the central part of Iran were used to validate the suggested formula.     

球体任意堆积的统计几何分析

为了对球体任意堆积的密度作出理论的解析,作者提出,以核心球中心为中心,2R为半径,作一个包裹球的数学模型。在包裹球中被包裹的有核心球、与核心球相切的球和次近邻球。后两种球被包裹的部分是双凸透镜体。这两种球数之和是最大配位数。最大配位数可以从包裹球面积与相切球在包裹球面上投影面积的比值得到。在此基础上,通过统计几何分析方法,可以计算出等大球和非等大球各种任意堆积时的密度常数,计算值与实验值符合得很好。并且证明二元非等大球任意堆积在球径比上有一阈值0.70,大于此值可视为等大球,小于此值堆积密度会逐渐增加,计算的变化曲线与实验曲线完全重合。从而得出结论,包裹球的数学模型是非常合理的。     

CO_2 Density-Raman Shift Relation Derived from Synthetic Inclusions in Fused Silica Capillaries and Its Application

Abstract: The densities of CO2 inclusions in minerals are commonly used to determine the crystallizing conditions of the host minerals. However, conventional microthermometry is difficult to apply for inclusions of small size (< 5–10 μm) or low density. Raman analysis is an alternative method for determining CO2 density, provided that the CO2 density–Raman shift relation is known. This study aims to establish this CO2 density–Raman shift relation by using CO2 inclusions synthesized in fused silica capillaries. By using this newly-developed synthetic technique, we formed pure CO2 inclusions, and their densities were determined by microthermometry. The Raman analysis showed that the relation between CO2 density (D in g/cm3) and the separations (Δ in cm?1) between the two main bands (i.e. Fermi diad bands) in CO2 Raman spectra can be represented by a cubic equation: D (g/cm3)=0.74203(?0.019Δ3+5.90332Δ2?610.79472Δ+21050.30165)?3.54278 (r2=0.99920). Our calculated D value for a given Δ is between those obtained from two previously-reported equations, which were derived from different experimental methods. An example was given in this study to demonstrate that the densities of natural CO2 inclusions that could not be derived from microthermometry could be determined by using our method.     

Inter‐instrument comparison of particle‐size analysers

This paper presents a methodological framework for inter‐instrument comparison of different particle‐size analysers. The framework consists of: (i) quantifying the difference between complete particle‐size distributions; (ii) identifying the best regression model for homogenizing data sets of particle‐size distributions measured by different instruments; (iii) quantifying the precision of a range of particle‐size analysers; and (iv) identifying the most appropriate instrument for analysing a given set of samples. The log‐ratio transform is applied to particle‐size distributions throughout this study to avoid the pitfalls of analysing percentage‐frequency data in ‘closed‐space’. A Normalized Distance statistic is used to quantify the difference between particle‐size distributions and assess the performance of log‐ratio regression models. Forty‐six different regression models are applied to sediment samples measured by both sieve‐pipette and laser analysis. Interactive quadratic regression models offer the best means of homogenizing data sets of particle‐size distributions measured by different instruments into a comparable format. However, quadratic interactive log‐ratio regression models require a large number of training samples (n > 80) to achieve optimal performance compared to linear regression models (n = 50). The precision of ten particle‐size analysis instruments was assessed using a data set of ten replicate measurements made of four previously published silty sediment samples. Instrument precision is quantified as the median Normalized Difference measured between the ten replicate measurements made for each sediment sample. The Differentiation Power statistic is introduced to assess the ability of each instrument to detect differences between the four sediment samples. Differentiation Power scores show that instruments based on laser diffraction principles are able to differentiate most effectively between the samples of silty sediment at a 95% confidence level. Instruments applying the principles of sedimentation offer the next most precise approach.     

Relation between dislocation density and subgrain size of naturally deformed olivine in peridotites

The subgrain size and the dislocation density of subgrain interiors were measured by the oxidation-decoration method under the optical microscope on naturally deformed olivine from peridotite xenoliths and alpine-type peridotites. Relation of the subgrain size, d and the dislocation density, , within subgrains is represented by the equation, d=15/. Combining with relations of the differential stress and the dislocation density proposed by Kohlstedt and Goetze (1974), relation between the stress () and the subgrain size becomes d=45 Gb/, where G and b are the rigidity and the magnitude of the Burgers vector of olivine. This relation is in good agreement with those in a simple oxide (MgO), and alkali halides (NaCl, LiF) given by Hüther and Reppich (1973), Poirier (1972), and Streb and Reppich (1973), respectively.     

Spectral Estimation of Irregularly Sampled Nonstationary Multidimensional Processes by Time-Varying Periodogram

A multidimensional version of the time varying periodogram has been developed. The estimation method based on the multidimensional time-varying periodogram has been applied to a nonstationary multidimensional storm model. This work proposes that the multidimensional time varying periodogram is capable of estimating nonstationary spectral density functions in space and time.     

Estimating the crush zone size under a cutting tool in coal

Summary As part of an effort to understand the mechanics of fine fragment formation in coal, which is important in studies of respirable dust due to mining, fracture toughness measurements and the strain energy density (SED) theory were applied to calculate the crush zone size under a cutting tool in coal. This zone is the major source of fine fragments in the 1 to 10 µm size range. The model used in these calculations is a boundary element program containing a failure criterion based on the SED theory. The boundary element program calculates linear elastic stresses at numerous points in the coal material ahead of a cutting bit. These stresses are then input to a subroutine called critical flaw length and orientation (CFLO) which uses the SED theory to determine the CFLO for a small crack at the boundary element stress computation point. The extent of crushing is based on earlier postulates about the role of inherent flaws in a fragmentation process. To form 1 to 10 µm fragments requires firstly a local stress strong enough to activate flaws with a characteristic length less than 1 to 10 µm and secondly, a flaw density sufficient to provide an average spacing between flaws also on the order of 1 to 10 µm. The locus of active 10 µm flaws represents the maximum possible extent of fine fragmentation in the 10 µm or less size range assuming that a sufficient inherent flaw density exists. The approach offers a first order approximation to the extent of crushing under a tool tip. The size and shape of the crush zone volume is affected by the attack angle and geometry of the tool.     

The effect of sample size on geological strain estimation from passively deformed clastic sedimentary rocks

The role of sample size in the estimation of geological strain, both finite strain (R_s) and that of the orientation of the finite strain ellipse (φ_s), is investigated for clastic sedimentary rocks. This study looks at four strain methods, the Robin method, the linearization method, the Mulchrone and Meere method and the mean radial length method that are initially tested using simulated strained data sets and subsequently by applying the methods to real data. It is found that the optimum strain analysis sample size for a clastic sedimentary rock is primarily dependant on the intensity of strain suffered by that rock because of the error behavior associated with R_s estimates. An iterative process is therefore recommended starting with a minimum sample size of 150, which can be maintained or reduced based on the initial R_s estimates.     

Variance as a function of sample support size

The effect of sample support size on variance is examined and evaluated. Results based on variograms and geostatistics are compared to the classical relationship developed by H. F. Smith in 1938; that is, that the variance is reduced fromV ₁ toV ₁ /n ^b as the support area increases from I ton plots for uniformity trials. The exponentb is between zero and one. Theoretical results are based on use of auxiliary functions and account for the size and shape of the sample support and the overall field geometry. Results are given in terms of approximations by rational functions for ease of calculation. Experimental results for uniformity trials, infiltration measurements, and spectral data from satellites are compared to theoretical and empirical results. Applications include not only uniformity trials, but also measurement theory.     

Percolation-theory and fuzzy rule-based probability estimation of fault leakage at geologic carbon sequestration sites

Leakage of CO₂ and displaced brine from geologic carbon sequestration (GCS) sites into potable groundwater or to the near-surface environment is a primary concern for safety and effectiveness of GCS. The focus of this study is on the estimation of the probability of CO₂ leakage along conduits such as faults and fractures. This probability is controlled by (1) the probability that the CO₂ plume encounters a conductive fault that could serve as a conduit for CO₂ to leak through the sealing formation, and (2) the probability that the conductive fault(s) intersected by the CO₂ plume are connected to other conductive faults in such a way that a connected flow path is formed to allow CO₂ to leak to environmental resources that may be impacted by leakage. This work is designed to fit into the certification framework for geological CO₂ storage, which represents vulnerable resources such as potable groundwater, health and safety, and the near-surface environment as discrete “compartments.” The method we propose for calculating the probability of the network of conduits intersecting the CO₂ plume and one or more compartments includes four steps: (1) assuming that a random network of conduits follows a power-law distribution, a critical conduit density is calculated based on percolation theory; for densities sufficiently smaller than this critical density, the leakage probability is zero; (2) for systems with a conduit density around or above the critical density, we perform a Monte Carlo simulation, generating realizations of conduit networks to determine the leakage probability of the CO₂ plume (P _leak) for different conduit length distributions, densities and CO₂ plume sizes; (3) from the results of Step 2, we construct fuzzy rules to relate P _leak to system characteristics such as system size, CO₂ plume size, and parameters describing conduit length distribution and uncertainty; (4) finally, we determine the CO₂ leakage probability for a given system using fuzzy rules. The method can be extended to apply to brine leakage risk by using the size of the pressure perturbation above some cut-off value as the effective plume size. The proposed method provides a quick way of estimating the probability of CO₂ or brine leaking into a compartment for evaluation of GCS leakage risk. In addition, the proposed method incorporates the uncertainty in the system parameters and provides the uncertainty range of the estimated probability.     

Behaviour of model footing on pond ash

This paper focuses on the effective utilization of pond ash, as foundation medium. A series of laboratory model tests have been carried out using square, rectangular and strip footings on pond ash. The effects of dry density, degree of saturation of pond ash, size and shape of footing on ultimate bearing capacity of shallow foundations are presented in this paper. Local shear failure of a square footing on pond ash at 37% moisture content (optimum moisture content) is observed up to the values of dry density 11.20 kN/m³ and general shear failure takes place at the values of dry density 11.48 kN/m³ and 11.70 kN/m³. Effects of degree of saturation on ultimate bearing capacity were studied. Experimental results show that degree of saturation significantly affects the ultimate bearing capacity of strip footing. The effect of footing length to width ratio (L/B), on increase in ultimate bearing capacity of pond ash, is insignificant for L/B ≥ 10 in case of rectangular footings. The effects of size of footing on ultimate bearing capacity for all shapes of footings viz., square, rectangular and strip footings are highlighted.     

Examination of the Chayes-Kruskal procedure for testing correlations between proportions

The Chayes-Kruskal procedure for testing correlations between proportions uses a linear approximation to the actual closure transformation to provide a null value,p _ij , against which an observed closed correlation coefficient,r _ij , can be tested. It has been suggested that a significant difference betweenp _ij andr _ij would indicate a nonzero covariance relationship between theith andjth open variables. In this paper, the linear approximation to the closure transformation is described in terms of a matrix equation. Examination of the solution set of this equation shows that estimation of, or even the identification of, significant nonzero open correlations is essentially impossible even if the number of variables and the sample size are large. The method of solving the matrix equation is described in the appendix.     

Particle size characteristics of aeolian ripple crests and troughs

Ripples are prevalent in aeolian landscapes. Many researchers have focused on the shape and formation of sand ripples, but few have studied the differences in the particle size of sand on crests and in troughs along bed, especially the variations caused by changes in friction velocity and the wind‐blowing duration. A particle size of 158 μm (d ) was used to create aeolian ripples in a wind tunnel under four friction velocities (u *) with different wind duration times (t ). Samples were collected from the surfaces of ripple crests and troughs, respectively, at seven sites, and particle sizes were measured using a Malvern Mastersizer 2000. The main results were: (i) The particle size distributions of sand in troughs are unimodal with slight variations of particle size parameters, including mean particle size, standard deviation, skewness and kurtosis, etc., under different conditions, while these particle size parameters of sand on crests change with friction velocity and deflation time. Moreover, some of the particle distribution curves for the sand on crests do not follow typical unimodal curves. (ii) With increasing friction velocity or deflation duration, the sand on the crests shows a coarsening process relative to those on the bed surface. The particle size of sand on crests at a 1 m bed increases linearly with friction velocity (d  =  344·27 + 34·54 u *) at a given wind‐blowing duration. The particle sizes of sand on crests at 1 m, 2 m and 4 m beds increase with a power‐law relationship (d  = a + t ^b, where a and b are fitting parameters) with deflation time at a given friction velocity. (iii) The probability cumulative curves of sand showed a three‐section pattern in troughs and on most of the crests but a four‐section pattern at crest locations due to increased influence by friction velocity and deflation time. The proportions of the sediment moved by suspension, saltation and creep in the three‐section pattern were within the ranges of 0·2% to 2·0%, 97·0% to 98·9%, and 0·8% to 3·0%, respectively. For the four‐section pattern, suspension accounted for 0·3% and 3·0%, and the proportion of creep increased with friction velocity and deflation time, while saltation decreased accordingly. Although these results require additional validation, they help to advance current understanding of the grain‐size characteristics of aeolian ripples.     

On the Optimal Estimation of the Cumulative Distribution Function in Presence of Spatial Dependence

For spatial analysis of data, the optimal prediction of the cumulative distribution function (c.d.f.) is made difficult by the presence of spatial dependence, particularly when data are clustered. Several methods have been proposed in the literature to correct for the redundancy which is present when the observed values are correlated. Most of them are based on an unequal weighting of the values for the computation of the c.d.f. The methodology is extended by defining weights which depend both on the spatial correlation and on the threshold value. If a parametric multivariate distribution is assumed, optimal weights can be obtained conditionally to the choice of this distribution, and a nonparametric indicator estimate of the c.d.f. can be defined a posteriori. The method can be extended in order to provide kernel smoothed estimates of the c.d.f., by replacing the indicator functions by other functions taken here as complementary differentiable c.d.f., whose parameters are defined in order to preserve the unbiasedness and minimum variance property of the estimate, conditionally to the choice of the a priori distribution. A simulation study is conducted in order to emphasize the properties of the various methods. It is found that, even for bounded and positively skewed distributions, a simplified version of the method seems to perform quite well without significant loss of efficiency, and provides smooth estimates of the c.d.f. which can be differentiated in order to obtain density estimates. These estimates do not depend highly on the choice of the parameters for the a priori distribution, and yet they respect major characteristics of the a priori c.d.f., such as lower and upper bounds, which are mainly dictated by the nature of the variable under study. A practical case study is conducted on cobalt concentrations for various rock types in the Swiss Jura, illustrating the applicability of the methodology for real datasets.     

An efficient approach to generating DEM samples having coincident void ratios with actual sand

The ranges of initial void ratios that can be achieved for numerical samples with the same grading as actual sand under gravity are explored using three-dimensional discrete element method (DEM). A corrected compressible accumulation model is proposed to predict the packing density of sand. Compared with the measured results, the errors of the prediction results are small. For three-dimensional DEM samples, when the ratio of sample size to particle size L/R is equal to or larger than 30, the maximum and the minimum void ratios are minimally affected by the sample size L. The maximum void ratios and the minimum void ratios of numerical samples with spherical particles are much smaller than those of actual sands. The generation method for numerical samples with non-spherical particles is proposed based on the theory of CPM. The variations of void ratios of numerical samples with non-spherical particles are wider than actual sands. So it can meet all the need of DEM simulation on the void ratio of actual sand.