Trend surface analysis as a special case of IRF-k kriging

This short note establishes the equivalence between trend surface analysis with polynomials of orderk and IRF-k (intrinsic random function of orderk) kriging with a nugget effect covariance model.     

Cokriging nonstationary data

Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science.     

Conditional simulation of intrinsic random functions of orderk

Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rⁿ; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada.     

Cokriging nonstationary data

Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science.     

Intrinsic random function of order<Emphasis Type="Italic"> k</Emphasis> kriging of electrical resistivity data for estimating the extent of saltwater intrusion in a coastal aquifer system

A saltwater intrusion study using electrical resistivity distribution was conducted in a coastal aquifer system in the southeastern part of Busan, Korea. This aquifer system is divided into four layers according to the hydrogeologic characteristics, and the horizontal extent of intruded saltwater is determined at each layer through the geostatistical interpretation of electrical resistivity data. Intrinsic random function of order k (IRF-k) kriging is performed with covariance models to produce the plane of spatial mean resistivities. The kriged estimates are evaluated by cross validation, and they showed a good agreement with the true values. The statistics of cross validation represented low errors for the estimates. In the resistivity contour maps more than 5 m below the surface, a dominant direction of saltwater intrusion was observed beginning from the east side. These results led the authors to conclude that the application of geostatistical technique to electrical resistivity data is useful for assessing saltwater intrusion in a coastal aquifer system.     

The equivalence of predictions from universal kriging and intrinsic random-function kriging

A proof is provided that the predictions obtained from kriging based on intrinsic random functions of orderk are identical to those obtained from anappropriate universal kriging model. This is a theoretical result based on known variability measures. It does not imply that people performing traditional universal kriging will get the same predictions as those using intrinsic random functions, because traditionally these methods differ in how variability is modeled. For intrinsic random functions, the same proof shows that predictions do not depend on the specific choice of the generalized covariance function. It is argued that the choice between these methods is really one of modeling and estimating the variability in the data.     

Endmember graphics

Thek points inp-space corresponding to a specified set ofk (k<p) linearly independent endmember estimates associated with ap-variate (n×p) compositional datasetX, define the vertices of a (k?1) dimensional simplexH. Then estimated mixtures which together account for the systematic variation in the datasetX, are each convex combinations of thek fixed endmember estimates. Accordingly, then-points inp-space which represent these mixtures must be interior points of the simplexH. The purpose of this paper is to describe a simple graphical procedure for examining the positions of the estimated mixtures relative tok=3 or 4 putative endmembers, and for assessing their compliance or otherwise with the convexity constraints. Since the mixture coefficients must be estimated first in order to obtain the mixture estimates, the paper begins with a review of the least-squares partitioning procedure.     

Conditional simulation of intrinsic random functions of order k

Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rⁿ; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada.     

Vorticity and non-coaxiality in progressive deformations

A measure of the non-coaxiality involved in progressive deformation histories is proposed in the form of the kinematical vorticity number, W_k. This number is a measure of the relative effects of rotation of material lines (relative to the instantaneous stretching axes) and of stretching of these material lines. As such, W_k, is a measure of the instantaneous degree of non-coaxiality. A detailed example is first presented in the form of a progressive simple shearing in which the shear plane rotates relative to an external coodinate system. This is followed by examples of more complicated deformation histories. Three specific types of progressive, isochoric (constant volume) deformation histories are recognized. Those for which 0 ≤ W_k < 1 correspond to deformation histories where no line that has been extended is shortened in future increments; W_k > 0 is a special case of these corresponding to a coaxial history. Histories with W_k > 0 are non-coaxial. Those histories with W_k = 1 correspond to progressive simple shearing. Those histories with 1 < W_k < ∞ are pulsating and lines that have been extended may be shortened in future increments.     

Comparative study procedure for the safety evaluation of high arch dams

A set of factors for the safety evaluation of the dam heel crack and ultimate bearing capacity of high arch dams is presented: the curtain safety factor k₁, the structural mutation safety factor k₂, and the ultimate safety factor k₃. The factor k₁ reflects the curtain safety at the dam heel, k₂ describes the structural mutation stage for the adjustment of the multiple arch–cantilever system, and k₃ reflects the ultimate bearing capacity of the entire dam structure. A comparative study on the performance and ultimate bearing capacity of dams is performed to give a more reasonable evaluation criterion of safety factors.     

Parameters of the turbulent magnetic field in the solar photosphere: Power spectrum of the line-of-sight field

Ground-based (Big Bear Solar Observatory) and extra-atmospheric (SOHO/MDI) measurements of the photospheric line-of-sight magnetic field of one active and two quiet regions are used to calculate power spectra of the field, taking into account the characteristic function for the diffraction limit of the telescope resolution. At high frequencies, the physically meaningful linear interval in the spectrum extends to a wave number of k=4.6 Mm^?1 (spatial scale l=1.4 Mm) for the quiet regions and k=3.35 Mm^?1 (l=1.9 Mm) for the active region. A high-frequency spectral break at k≥3 Mm^?1 is associated with the characteristic telescope function; the position of the break and the spectral slope beyond the break do not reflect the turbulent state of the field. As the field recording improves, the break shifts toward higher frequencies. The spectral indices in the physically meaningful linear interval are substantially different for the active and quiet regions: in the active region (NOAA 8375), the spectrum behaves as E(k)≈k ^?1.7 (very close to the Kolmogorov index, ?5/3) in the interval 0.78≤k≤3.35 Mm^?1, while in the quiet regions E(k)≈k ^?1.3 for 0.77≤k≤4.57 Mm^?1. This difference can be explained by the additional effect of a small-scale turbulent dynamo in the unperturbed photosphere. In this case, this mechanism can generate at least 6% of the magnetic energy of the photospheric line-of-sight field in quiet regions.     

Approximation of the Velocity by Coupling Discontinuous Galerkin and Mixed Finite Element Methods for Flow Problems

In this paper, we show how to couple the local discontinuous Galerkin method and the Raviart–Thomas mixed finite element method for elliptic equations modeling flow problems. We then show that the approximation of the velocity converges with the optimal order of k when we take the local discontinuous Galerkin that uses polynomials of degree k and the Raviart–Thomas space of polynomials of degree k?1.     

Relationship between hydrogeological parameters for data-scarce regions: the case of the Araripe sedimentary basin, Brazil

This paper applies and validates a method for generating spatially distributed hydraulic conductivity (k) based on the specific capacity (Q _s) for data-scarce regions. This method has been applied to the Araripe sedimentary basin, Brazil, and consists of four steps: (1) selection of (32) wells for which both k and Q _s data are available; (2) estimation of k as a function of Q _s for the (128) wells for which only specific capacity data are available; (3) spatial distribution of k using the kriging geostatistical tool; (4) validation of the method, using (17) representative wells with k measured data. The equation relating k and Q _s showed a statistically significant linear relationship (R = 0.93), from which a database has been generated using kriging with the spherical model. The results showed a calibration coefficient of Nash and Sutcliffe (NS) of 0.54 and moderate spatial dependence ratio of 69 %. The validation process provided only a moderate efficiency (NS = 0.22), possibly due to the geological complexity of the focus system. Despite its limitations, the method indicates the possibility of application of ordinary kriging to generate reliable data from auxiliary variables, especially for the water management of data-scarce areas.     

Estimation of velocity function parameters in unconsolidated sands using semblance velocity analysis

To properly understand seismic wave propagation in unconsolidated sand layers, it is important to estimate the parameters of their continuous velocity–depth functions. This study proposes a procedure to estimate the V ₀ and k parameters of a specific velocity function, where V ₀ is the direct P-wave velocity at the ground surface and k is the velocity gradient. The V ₀ and k parameters are generally independent of each other. However, it is possible to relate them numerically because both depend strongly on the porosity (?) and water saturation (S _w). The proposed procedure starts by tabulating V ₀ and k for 0.05?≤???≤?0.5 sampled at Δ??=?0.05 and S _w?=?0.6, so that only V ₀ is needed for fitting. Then, time–distance (T-X) type curves of the direct arrival are calculated for the corresponding values of V ₀ and k parameters values. The type curves are fitted then to the observed shot gather through a modification of the classic semblance velocity analysis method. Once the best-fit V ₀ value is found, the corresponding k, ?, and S _w values are picked from a V ₀–k–? lookup table. The procedure is applied on synthetic shot gathers with various amounts of additive Gaussian random noise. Results show that the method is robust and tolerant to low to moderate amounts of noise.     

The at-rest earth pressure coefficient prediction using simple elasto-plastic constitutive models

The paper deals with the theoretical prediction of the at-rest earth pressure coefficient for normally consolidated soils (k_0(NC)) using simple elasto-plastic constitutive models. In the first part of the work, the k_0(NC) expressions derived from critical state soil mechanics models are critically discussed. It is shown that, adopting typical values of the models parameters, the experimental k_0(NC) values are usually over-predicted. In the second part, a possible modification of the critical state model formulation, which allows to predict lower k_0(NC) values for a fixed angle of shearing resistance, is considered. Finally, a discussion of the results and some brief conclusions are proposed.     

Evaluation of the hydraulic conductivity of aquifers with piles

Distributed piles are often installed in the foundation pit in sandy soil before dewatering. Hydraulic conductivity should be estimated considering the block effect of piles on groundwater seepage. Research shows that the effective medium theory (EMT) could be used to calculate the equivalent hydraulic conductivity (k _eq) of a heterogeneous medium with other material inclusions. In order to verify the applicability of EMT in an aquifer with piles, an experimental investigation is conducted. In this experiment, a sandy soil is considered as an aquifer and polyvinylchloride pipes are adopted as piles. Piles are distributed in rectangular and triangular layouts. The relationship between k _eq of the aquifer and the volume replacement ratio is plotted for these two layouts. The results indicate that EMT could well predict k _eq for the cases with the rectangular layout; however, for triangular layouts, the prediction becomes poor. To provide a better prediction of k _eq of the aquifer with piles distributed in a triangular layout, a modified EMT (MEMT) is proposed, in which a correction factor is adopted considering different layouts of piles. The MEMT can provide better k _eq predictions for an aquifer with piles in the triangular layout pattern than those obtained using the EMT.     

Time-Variant Reliability Analysis of Unsaturated Soil Slopes Under Rainfall

Time-variant reliability analysis for a typical unsaturated soil slope is performed. Eight rainfall conditions are considered, and three slope models are set up for studying the influence of shear strength parameters, hydraulic conductivity parameters, rainfall intensity and duration on the reliability of the soil slope. Sensitivity analysis shows that when the saturated hydraulic conductivity (k _s) is very small, the variation of hydraulic conductivity has little effect on the reliability index (β). For saving the computation effort, only the shear strength parameters are needed in performing the reliability analysis in this condition. With the increase of k _s, the importance of hydraulic conductivity becomes large. The reliability index of the soil slope is changing with time (t), and the shape of β–t curves for different slope model is quite different for they depend on the value of k _s. When k _s is very small, β keeps decreasing for all the 18 simulation days. With the increase of k _s, β decreases to its minimum value at about the cessation day of rainfall events, and it then increases gradually due to the redistribution of suction in the soil slope.     

Hydraulic Conductivity of Fly Ash-Amended Mine Tailings

The objective of this study was to evaluate the effect of fly ash addition on hydraulic conductivity (k) of mine tailings. Mine tailings used in this study were categorized as synthetic tailings and natural tailings; two synthetic tailings were developed via blending commercially-available soils and natural tailings were collected from a garnet mine located in the U.S. Two fly ashes were used that had sufficient calcium oxide (CaO) content (17 and 18.9 %) to generate pozzolanic activity. Hydraulic conductivity was measured on pure tailings and fly ash-amended tailings in flexible-wall permeameters. Fly ash was added to mine tailings to constitute 10 % dry mass of the mixture, and specimens were cured for 7 and 28 days. The influence of fly ash-amendment on k of mine tailings was attributed to (1) molding water content and (2) plasticity of the mine tailings. Tailings that classified as low-plasticity silts with clay contents less than 15 % exhibited a decrease in k when amended with fly ash and prepared wet of optimum water content (w _opt ). Tailings that classified as low-plasticity clay exhibited a one-order magnitude increase in k with addition of fly ash for materials prepared dry or near w _opt . The decrease in k for silty tailings was attributed to formation of cementitious bonds that obstructed flow paths, whereas the increase in k for clayey tailings was attributed to agglomeration of clay particles and an overall increase in average pore size. The results also indicated that the effect of curing time on k is more pronounced during the early stages of curing (≤7 days), as there was negligible difference between k for 7 and 28-days cured specimens.     

An efficient regularized inversion approach for self-potential data interpretation of ore exploration using a mix of logarithmic and non-logarithmic model parameters

A very fast and efficient approach to self-potential (SP) data inversion for ore exploration has been developed. This approach is based on Tikhonov regularization and the conjugate gradient method, and simultaneously inverts for the depth (z), electric dipole moment (k), and angle of polarization (θ) of a buried anomalous body from SP data measured along a profile. This inversion algorithm works iteratively, and solves for z and k in the logarithmic-space (log(z) and log(k)), and solves for θ in the linear-space (non-logarithmic). It is found that the original inversion formulation that uses the model parameters themselves (z, k and θ) is unstable and divergent. It is also found that the inversion formulation that uses the logarithm of the model parameters (log(z), log(k) and log(θ)) is unstable and divergent. Rather, the new inversion scheme that is based on the aforementioned mixed log-linear combination of the model parameters (log(z), log(k), and θ) overcomes and eliminates the mentioned instability and divergence problems. The sensitivity analysis and numerical experiments investigated have indicated that the new approach has a far better and far more optimized minimization search direction. This proposed technique fits the observed data by some geometrically simple body in the restricted class of vertical cylinder, horizontal cylinder, and sphere models. The applicability of the algorithm has been demonstrated on various reliable synthetic data sets with and without noise. The algorithm has been carefully and successfully applied to six real data examples, with ore bodies buried in different complex geologic settings and at various depths in the subsurface. The method is shown to be highly applicable for mineral exploration, and is of particular value in cases where the SP observed data is due to ore body embedded in the subsurface. On average, it took about 40 s of computation (not CPU) time on a 1 GHz PC.     

Stability of a long unsupported circular tunnel in soils with seismic forces

By using the lower-bound finite element limit analysis, the stability of a long unsupported circular tunnel has been examined with an inclusion of seismic body forces. The numerical results have been presented in terms of a non-dimensional stability number (γH/c) which is plotted as a function of horizontal seismic earth pressure coefficient (k _h) for different combinations of H/D and ?; where (1) H is the depth of the crest of the tunnel from ground surface, (2) D is the diameter of the tunnel, (3) k _h is the earthquake acceleration coefficient and (4) γ, c and ? define unit weight, cohesion and internal friction angle of soil mass, respectively. The stability numbers have been found to decrease continuously with an increase in k _h. With an inclusion of k _h, the plastic zone around the periphery of the tunnel becomes asymmetric. As compared to the results reported in the literature, the present analysis provides a little lower estimate of the stability numbers. The numerical results obtained would be useful for examining the stability of unsupported tunnel under seismic forces.