On the Existence,Uniqueness and Correctness of the Fracture Diameter Distribution Given the Fracture Trace Length Distribution

Based on the fracture trace length distribution, conditions for the existence, uniqueness, and correctness of the fracture diameter distribution are given using Warburton’s fracture model. In particular, a solution for the fracture diameter distribution exists and is unique for all trace length probability density functions, h _A (y), such that \(h_{A}(y)/\sqrt{y^{2}-x^{2}}\) is Lebesgue integrable on [x,∞). This condition is met by the uniform, exponential, gamma, lognormal, and power-law trace length distributions as well as by the trace length distributions that arise from a deterministic fracture diameter or from a discontinuous fracture diameter length distribution. Exponential, gamma, lognormal, and power-law trace length distributions satisfy necessary conditions for the diameter distribution to be non-negative, and necessary and sufficient conditions for the distribution to have unit integral over the real line. Negative values of the fracture diameter distribution arise when the trace has a uniform distribution and the lower bound of the fracture trace is greater than zero.     

Structure modelling and cation partitioning of spinel solid solutions at high <Emphasis Type="Italic">T</Emphasis>,<Emphasis Type="Italic">P</Emphasis> conditions

This paper presents evaluation of cation distributions from diffraction data collected at high T, P, and is an extension of the spinel structure modelling procedure by Lavina et al. (2002). Optimised cation-to-oxygen distances are modified for thermal expansion and compressibility at T and P of interest following Hazen and Prewitt (1977) and Hazen and Yang (1999). The procedure is applied to literature data concerning hercynite, spinel s.s., Zn aluminate, Zn ferrite, magnetite and the (Fe₃O₄)_{1– x} (MgAl₂O₄) _x join. Calculated cation distribution is strongly affected by standard deviations in cell parameters and oxygen coordinates. The underestimated values often reported in the literature for powder profile refinements may strongly affect the cation distribution; however, if standard deviations are increased to physically realistic values, consistent results are obtained. For P up to 10 GPa, reasonable evaluations of cation distribution are obtained for spinel s.s., Zn aluminate and magnetite, whereas for Zn ferrite they are limited to 1.8 GPa. For P beyond 10 GPa, compressibility cannot be assumed to be linear; the relationship between cell parameter and pressure is well-defined, but the inaccuracy of oxygen coordinate prevents simple modelling of bond distances with pressure.     

Unified scaling theory for distributions of temporal and spatial characteristics in seismology

The paper describes the unified scaling theory for distribution functions of temporal and spatial characteristics in seismology. It is based on the scaling of seismological characteristics calculated for various energy–spatial–temporal intervals. The common mathematical methods for the scaling of distribution functions are developed. The means to test possibility of such scaling are found as well. The relationship between the unified scaling theory and other present scaling approaches is determined. The theory is applied to two characteristics of different seismoactive regions. The first characteristic is the waiting time between earthquakes ΔT, the second one is a new space parameter ΔD_min, which is the minimum distance of a current seismic event to the nearest (in space) neighbor in an energy–spatial–temporal interval. The distribution of the characteristics ΔT and ΔD_min allows estimating the time interval to the next earthquake and the distance of the following earthquake from previous earthquakes. Thus, these characteristics are very important for seismic hazard estimations. Scaling of distributions functions is proven to be successful for ΔD_min in all energy–spatial–temporal intervals and for ΔT with variations of energy/magnitude range. The distribution function of ΔT for various time domains was stable in only 60% of the cases, and near to unstable for spatial variations.     

Normal Alkane Distributions in Soil Samples along a Lhasa-Bharatpur Transect

We present n-alkane distributions in the soil samples along a transect from Lhasa on the Tibetan Plateau to Bharatpur in Nepal, which covers a large geographical area and a wide range of climatic conditions. These data allow us to assess the significance of n-alkane distributions in different vegetation types and their relationships to temperature and precipitation. In the tropical rainforest and broadleaved forest zones, n-alkanes exhibit a bimodal distribution pattern with dominant homologues around n-C_(31) and n-C_(23). The bimodal distribution of tropical rainforest n-alkanes may be a general pattern because of the presence of many lianas, epiphytic plants, algae and the strong microbial activity and degradation involved in the postdepositional process. In the warm-temperate mixed forest and needle-leaved forest zones, the long-chain alkanes have a pattern of n-C_(31) n-C_(33) n-C_(27). In the alpine shrub and grassland zone, although the most abundant homologue is n-C_(31) , relatively high n-C_(23) concentrations have also been observed in some samples. The statistical results show a good correlation between n-alkane proxies and climatic factors. The average chain length(ACL) values are positively correlated with precipitation and temperature across the transect. The carbon preference index(CPI) values show a negative correlation with temperature and precipitation along a Lhasa–Bharatpur transect. Although there is a positive relationship between ACL and temperature, it is difficult to separate the two climatic variables(temperature and precipitation) because they are well coupled in the monsoon region.     

Grain sorting in gravel-bed streams and the choice of particle sizes for flow-competence evaluations

Flow-competence assessments of floods have been based on the largest particle sizes transported, and yield either the mean flow stress, mean velocity, or discharge per unit flow width. The use of extreme particle sizes has potential problems in that they may have been transported by debris flows rather than by the flood, it may be difficult to locate the largest particles within the flood deposits, and there are questions concerning how representative one or a few large particles might be of the transported sediments and therefore of the flood hydraulics. Such problems would be eliminated for the most part if competence evaluations are based on median grain sizes of transported sediments, or perhaps on some coarse percentile that is established by a reasonable number of grains. In order to examine such issues, the gravel-transport data of Milhous from Oak Creek, Oregon, and of Carling from Great Eggleshope Beck, England, have been analysed in terms of changing grain-size percentiles with varying flow stresses. A comparison between these two data sets is of added interest because the bed material in Oak Creek is segregated into well-developed pavement and subpavement layers, while such a layering of bed materials is largely absent in Great Eggleshope Beck. The analyses show that the trend of increasing sizes of the largest particles in the bedload samples (diameter D_m) with increasing flow stresses is consistent with similar dependencies based on sieve percentiles ranging from the medians (D₅₀) to the 95th percentiles (D₉₅). This indicates that the largest particles are an integral part of the overall distributions of bedload grain sizes, and respond to changing flow hydraulics along with the rest of the size distribution. In Oak Creek, the median grain size shows the largest change with increasing flow stresses, followed by D₆₀, and so on to D₉₅ which shows the smallest change. The variations in D_m continue this trend, and are similar to those for D₉₅. This systematic variation of grain-size percentiles in Oak Creek is consistent with changes in the overall distributions which tend to be symmetrical and Gaussian for low discharges, but become skewed Rosin distributions for high discharges. In contrast, in Great Eggleshope Beck the several percentiles and D_m show the same rate of shift to coarser sizes as flow stresses increase. This results in part from differences in sampling techniques wherein the bedload samples from Great Eggleshope Beck represent a complete flood event, while shorterterm samples at a specific flow stage were obtained in Oak Creek. As a result of the integrated sampling in Great Eggleshope Beck, the bedload grain-size distributions are more complex, commonly with a bimodal pattern. However, after accounting for differences in sampling schemes in the two streams, contrasting patterns in changing grain-size distributions remain, and these are concluded to reflect grain sorting differences as the bedload grain-size distributions approach the distributions of the bed materials. It is surprising that if criteria commonly employed to demonstrate the equal mobility of different grain sizes are used in the comparison, then Great Eggleshope Beck is far closer to this condition in spite of its minimal development of a pavement. It is concluded that the respective shapes of the bed-material grain-size distributions, in particular their degrees of skewness, are more important to the observed sorting patterns than are the effects of a pavement layer regulating grain entrapment to produce an equal mobility of different grain sizes. Therefore, the comparison has established that flow-competence relationships will differ from one stream to another, depending on the pattern of grain sorting which is a function of the bedmaterial grain-size distribution.     

A comparison of seismic and structural measurements of scaling exponents during tensile subcritical crack growth

The observed fractal nature of both fault length distributions and earthquake magnitude-frequency distributions suggests that there may be a relationship between the structure of active fault systems and the resulting seismicity. In previous theoretical work, a positive correlation between the exponent D from the fracture length distribution, and the seismic or acoustic emission (AE) b-value has been inferred from a simple dislocation model of the seismic source. Here, we present the first experimental evidence for a correlation between D and b from a series of tensile fracture mechanics tests on crystalline rock, carried out in different environmental conditions, both air-dry and water-saturated, and at ambient temperature and pressure. The microseismic acoustic emissions were monitored during subcritical crack growth under controlled conditions of constant stress intensity, K_I, and quantitative analyses of the resulting fracture patterns were carried out on the same specimens. It is found that AE b-values, ranging from 1.0 to 2.3, correlate negatively with the normalized stress intensity K_I/KIC, where K_IC is the fracture toughness of the specimen. The microcrack length distribution exponent D, ranges from 1.0 to 1.7. Fluid presence has a first-order influence on both the AE and structure produced in these experiments. For experiments at low stress intensity or high fluid content, the activation of the stress corrosion mechanism for K_I < K_IC leads to a greater relative proportion both of small cracks and of low amplitude acoustic emissions, reflected in higher values of D and b. The exponent D is found to correlate positively with the AE b-value.     

Estimating and mapping snow hazard based on at-site analysis and regional approaches

The estimation of snow hazard and load faces the small sample size effect because of the short snow depth record at a station. To reduce such an effect, we propose to estimate the return period value of the annual maximum ground snow depth S, s_T, for Canada sites by applying the regional frequency analysis (RFA) and the region of influence approach (ROIA). The use of RFA and ROIA to map Canadian snow hazard is new. The comparison of their performance for snow hazard mapping has not been explored in the literature. We also consider the at-site analysis approach (ASA) for estimating s_T by using three often used probability distributions for S. A comparison of the estimated s_T by using the three approaches (ASA, RFA, ROIA) indicates that there is considerable scatter between the estimated s_T value although the identified overall spatial trends of s_T are similar. It is shown that the two-parameter lognormal distribution for S at most Canadian sites, based on the at-site analysis, is preferred; this differs from the Gumbel distribution used to develop the design snow load in Canadian structural design code. The new findings indicate that it is valuable to consider the lognormal distribution for developing design snow load for Canadian sites.

     

Spatial declustering weights

Because of autocorrelation and spatial clustering, all data within a given dataset have not the same statistical weight for estimation of global statistics such mean, variance, or quantiles of the population distribution. A measure of redundancy (or nonredundancy) of any given regionalized random variable Z(u_α)within any given set (of size N) of random variables is proposed. It is defined as the ratio of the determinant of the N X Ncorrelation matrix to the determinant of the (N - 1) X (N - 1)correlation matrix excluding random variable Z(u_α).This ratio measures the increase in redundancy when adding the random variable Z(u_α)to the (N - 1 )remainder. It can be used as declustering weight for any outcome (datum) z(u_α). When the redundancy matrix is a kriging covariance matrix, the proposed ratio is the crossvalidation simple kriging variance. The covariance of the uniform scores of the clustered data is proposed as a redundancy measure robust with respect to data clustering.     

Reliability analysis for bearing capacity of surface strip footing using fuzzy finite element method

ABSTRACT

This paper presents the reliability analysis on the basis of the foundation failure against bearing capacity using the concept of fuzzy set theory. A surface strip footing is considered for the analysis and the bearing capacity is estimated using the conventional Finite Element Method (FEM). The spatial variability of the variables is taken into consideration to capture the physical randomness of the soil parameters for an isotropic field. A variation of the probability of failure (P_f) against a varying limiting applied pressure (q) is presented for different Coefficient of Variation (COV) of the variables and different scale of fluctuation (θ). The results reveal that the friction angle of soil (?) is the most influencing parameter among the other variables. Further, the influence of the scale of fluctuation (θ) on the probability of failure (P_f) is also examined. It is observed that for a particular COV of ?, higher value of θ predicts higher P_f whereas, P_f increases as COV of ? increases for a particular θ value. Later, a comparison study is accomplished to verify the viability of the present method and it can be noticed that the present method compares well with the other reliability method (First Order Reliability Method) to a reasonably good extent.     

The sampling distribution of theW statistic of disjunction for the arbitrary division of a random rectangular distribution

The sampling distribution of the Wstatistic of disjunction has been estimated by Monte Carlo simulation for the case where the underlying distribution is a random rectangular (Poisson) variable that is divided into two groups at an arbitrary position. A transformation to sinh ^–1 log _e Wgave a variable that was acceptably normal, and from this a simple approximation for the distribution is given, together with a diagram of confidence limits of Wfor this case.     

Rapid estimation of maximum and minimum void ratios of granular soils

Maximum and minimum void ratios (e_max and e_min) of granular soils are commonly used as indicators of many engineering properties. However, few methods, apart from laboratory tests, are available to provide a rapid estimation of both e_max and e_min. In this study, we present a theoretical model to map the densest and the loosest packing configurations of granular soils onto the void space. A corresponding numerical procedure that can predict both e_max and e_min of granular soils with arbitrary grain size distributions is proposed. The capacity of the proposed method is evaluated by predicting the maximum and minimum void ratios of medium to fine mixed graded sands with different contents of fines. The influence of the grain size distribution, characterized quantitatively by uniformity parameter and the fractal dimension, on e_max and e_min is discussed using the proposed method. Moreover, application of this method in understanding the controlling mechanism for the void ratio change during grain crushing is presented.

     

Surface tension of natural silicate melts from 1,200°–1,500° C and implications for melt structure

The surface tension between silicate liquid and gas has been measured for four lava compositions (limburgite to andesite) from 1,200° to 1,500° C. The magnitude of surface tension () is in the range 350–370 dynes/cm. Variations found in as a function of liquid composition were small and had no obvious relation to liquid composition. was also found to vary little with furnace atmosphere — air, Ar, CO₂, H₂, CO and their mixtures. A relaxation time of hours to days, depending on temperature, is required before reproducible results can be obtained from originally crystalline starting material.The reproducible temperature dependence of for complex silicate liquid solutions was found to be small, positive, and a relatively simple function of liquid composition. To a first approximation, the dependence of d/dT on composition found by King (1953) for simple silicate binaries appears to extend to the complex solutions measured here. The extent of progressive dissociation of liquid constituents implied by the increase of with T appears to be principally determined by the average field strength (Z/R) of the network-modifying cations in the liquid and abundance of network-forming constituents . The actual amount of structural dissociation implied by these results reduces the molar volume of the average structural unit by at most 5% for a 100° C increase in temperature. Evidently the polymerization of coordination polyhedra is fairly stable over this temperature and composition range.     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Modeling Bedload Transport of Juvenile Bivalves: Predicted Changes in Distribution and Scale of Postlarval Dispersal

The potential consequences of bedload transport of postlarvae for patterns of distribution of marine invertebrates were explored by developing a bedload transport model for juvenile bivalves in a small estuary in New Jersey, USA. A simple numerical model of tidal current hydrodynamics was developed based on field measurements of shear stresses near the bottom. Burrowing behavior of bivalves was incorporated into the model of bedload transport by using estimates of entrainment rates of Gemma gemma and Mya arenaria in a laboratory flume, and jump lengths of the bivalves were estimated by methods previously developed for noncohesive particles. Based on the flood domination and strong gradient of shear stresses in the Navesink estuary, our model predicted that juvenile bivalves would accumulate in the center of the estuary, traveling up to several kilometers over 30 days. Field distributions of juvenile bivalves were consistent with the model predictions for other species of bivalves but not for G. gemma, for which field distributions of both <500- and >500-μm individuals were concentrated in the eastern end of the estuary. Differences between the bedload model and G. gemma distributions suggest that spatial variation in burrowing behavior or biological interactions are playing an important role in maintaining distribution patterns of this species in spite of high levels of bedload transport. This modeling approach is applicable to other juvenile benthic invertebrates that disperse as bedload and is a useful model against which to compare field observations of rates of transport and patterns of distribution and abundance.     

Statistical treatment of experimental errors in the fission track dating method

The major stochastic elements in the fission track dating method are (i) the number of spontaneous fission tracks (N _s ) in a sample, and (ii) the number of induced tracks (N _i ) observed when the sample is irradiated with neutrons. The foundations for the statistical uncertainty in these measures are of two kinds: (i) there exists a definite probability of uranium fission by means of natural decay and by neutron activation, and (ii) within a crystal the distribution of uranium is not uniform and perhaps follows something like a Poisson law. In any event, the natural logarithm of the ratio (N _s /N _i ) is proportional to age. A plausible statistical fission track dating model should, therefore, start by considering the joint distribution of N _s and N _i . In this paper a joint bivariate normal model is described which allows the rigorous definition of the probability distributions of N_s, N _i , the ratio N _s /N _i , and age itself. A general computer program (FISSION) has been developed to perform all the necessary computations. By accounting for the correlation between N _s and N _i , the statistical model here ascribes smaller standard errors to N _s /N _i (and therefore age) than do previous methods. In addition, the error associated with neutron flux is a significant factor in the age relationships and has been incorporated into the model.     

A geophysical, geochemical, petrological model of the sub-marine lithosphere

A schematic but quantitative geochemical, petrological, model of the sub-marine lithosphere and its genesis is given. With this model we calculate numerically, a priori, the geophysical characteristics of the lithosphere, its acoustic properties, density, oceanic heat flow and ocean bottom topography. Comparison with observational data for these characteristics shows good agreement. Particular attention is given to anomalous upper mantle in the vicinity of spreading centres. Compressional and shear wave velocity distributions are given in tabular form for the submarine lithosphere as a function of age. Comparison between observations for V_p, V_s and the calculated acoustic properties suggests that the lower marine lithosphere is anisotropic. Possible thickening of layer 3 with age is discussed. Melt distribution in the ridge axial region has been evaluated. All calculations were done for a plate velocity of 1 cm y⁻¹.     

Statistical treatment of outlier data in epithermal gold deposit reserve estimation

A common characteristic of gold deposits is highly skewed frequency distributions. Lognormal and three-parameter lognormal distributions have worked well for Witwatersrand-type deposits. Epithermal gold deposits show evidence of multiple pulses of mineralization, which make fitting simple distribution models difficult. A new approach is proposed which consists of the following steps: (1) ordering the data in descending order. (2) Finding the cumulative coefficient of variation for each datum. Look for the quantile where there is a sudden acceleration of the cumulative C.V. Typically, the quantile will be above 0.85. (3) Fitting a lognormal model to the data above that quantile. Establish the mean above the quantile, Z _H ^* . This is done by fitting a single or double truncated lognormal model. (4) Use variograms to establish the spatial continuity of below-quantile data (Z_L) and indicator variable (1 if below quantile, 0 if above). (5) Estimate grade of blocks by (1*) (Z _L ^* )+(1 – 1*) (Z _H ^* ), where 1* is the kriged estimate of the indicator, and Z _L ^* is the kriged estimate of the below quantile portion of the distribution. The method is illustrated for caldera, Carlin-type, and hot springs-type deposits. For the latter two types, slight variants of the above steps are developed.     

Prediction of the tunnel displacement induced by laterally adjacent excavations using multivariate adaptive regression splines

Excavations may cause excessive ground movements, resulting in potential damage to laterally adjacent tunnels. The aim of this investigation is to present a simple assessment technique using a multivariate adaptive regression splines (MARS) model, which can map the nonlinear interactions between the influencing factors and the maximum horizontal deformation of tunnels. A high-quality case history in Tianjin, China, is presented to illustrate the effect of excavation on the tunnel deformation and to validate the FEM. The hypothetical data produced by the FEM provide a basis for developing the proposed MARS model. Based on the proposed model, the independent and coupled effects of the input variables (i.e. the normalized buried depth of tunnels H_t/H_e, the normalized horizontal distance between tunnels and retaining structures L_t/H_e, and the maximum horizontal displacement of retaining structures, δ_Rmax) on the tunnel response are analysed. The prediction precision and accuracy of the MARS model are validated via the artificial data and the collected case histories.

     

Occurrences and distributions of branched alkylbenzenes in the Dongsheng sedimentary uranium ore deposits,China

A series of branched alkylbenzene ranging from C₁₅ to C₁₉ with several isomers (2–5) at each carbon number were identified in sediments from the Dongsheng sedimentary uranium ore deposits, Ordos Basin, China. The distribution patterns of the branched alkylbenzenes show significant differences in the sample extracts. The branched alkylbenzenes from organic-rich argillites and coals range from C₁₅ to C₁₉ homologues, in which the C₁₇ or C₁₈ dominated. On the other hand, the C₁₉ branched alkylbenzenes dominated in the sandstone/siltstone extracts. The obvious differences of the branched alkylbenzene distributions between the uranium-host sandstones/siltstones and the interbedded barren organic-rich mudstones/coals probably indicate their potential use as biological markers associated with particular depositional environments and/or maturity diagenetic processes. Possible origins for these branched alkylbenzenes include interaction of simple aromatic compounds with, or cyclization and aromatization reactions of, these linear lipid precursors such as fatty acids, methyl alkanoates, wax esters or alkanes/alkenes that occur naturally in carbonaceous sediments. The possible simple aromatic compounds may include substituted benzenes, functionalized compounds such as phenols that are bound to kerogen at the benzyl position, and phenols that are decomposition products derived from aquatic and terrestrial sources. The distributions of methyl alkanoates and n-alkanes were found to be different between organic-rich mudstone/coal and sandstone/siltstone. From this result, it can be concluded that such differences of the alkylbenzene distributions were mainly resulting from the differences of organic precursors, although maturity effect and radiolytic alteration cannot be completely excluded.