Kriging in a finite domain

Adopting a random function model {Z(u),u ε study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed.     

Kriging with strings of data

The concept of a random function and, consequently, the application of kriging cells for the implicit assumption that the data locations are embedded within an infinite domain. An implication of this assumption is that, all else being equal, outlying data locations will receive greater weight because they are seen as less redundant, hence, more informative of the infinite domain. A two- step kriging procedure is proposed for correcting this siring effect. The first step is to establish the total kriging weight attributable to each string. The distribution of that total weight to the samples in the string is accomplished by a second stage of kriging. In the second stage, a spatial redundancy measure r_(n) is used in place of the covariance measure in the data-data kriging matrix. This measure is constructed such that each datum has the same redundancy with the (n)data of the string to which it belongs. This paper documents the problem of kriging with strings of data, develops the redundancy measure r_(n),and presents a number of examples.     

Kriging with strings of data

The concept of a random function and, consequently, the application of kriging cells for the implicit assumption that the data locations are embedded within an infinite domain. An implication of this assumption is that, all else being equal, outlying data locations will receive greater weight because they are seen as less redundant, hence, more informative of the infinite domain. A two- step kriging procedure is proposed for correcting this siring effect. The first step is to establish the total kriging weight attributable to each string. The distribution of that total weight to the samples in the string is accomplished by a second stage of kriging. In the second stage, a spatial redundancy measure r_(n) is used in place of the covariance measure in the data-data kriging matrix. This measure is constructed such that each datum has the same redundancy with the (n)data of the string to which it belongs. This paper documents the problem of kriging with strings of data, develops the redundancy measure r_(n),and presents a number of examples.     

The Variance-Based Cross-Variogram: You Can Add Apples and Oranges

The variance-based cross-variogram between two spatial processes, Z₁ (·) and Z₂ (·), is var (Z₁ ( u ) – Z₂ ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z₁ (·) and Z₂ (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z₁ ( u ) – Z₁ ( v ), Z₂ ( u ) – Z₂ ( v )). One concern with the variance-based cross-variogram has been that Z₁ (·) and Z₂ (·) might be measured in different units (apples and oranges). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z₁ (·) and Z₂ (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.     

Conditional Simulation of Random Fields by Successive Residuals

This paper presents a new approach to the LU decomposition method for the simulation of stationary and ergodic random fields. The approach overcomes the size limitations of LU and is suitable for any size simulation. The proposed approach can facilitate fast updating of generated realizations with new data, when appropriate, without repeating the full simulation process. Based on a novel column partitioning of the L matrix, expressed in terms of successive conditional covariance matrices, the approach presented here demonstrates that LU simulation is equivalent to the successive solution of kriging residual estimates plus random terms. Consequently, it can be used for the LU decomposition of matrices of any size. The simulation approach is termed conditional simulation by successive residuals as at each step, a small set (group) of random variables is simulated with a LU decomposition of a matrix of updated conditional covariance of residuals. The simulated group is then used to estimate residuals without the need to solve large systems of equations.     

The construction of extreme compositions

If a geochemical compositional dataset X (n×p)is a realization of a physical mixing process, then each of its sample (row) vectors will approximately be a convex combination (mixture) of a fixed set of (l×p)extreme compositions termed endmembers. The kpoints in p-space corresponding to a specified set of k (klinearly independent endmember estimates associated with a p-variate (n×p)compositional dataset X,define the vertices of a (k–1)dimensional simplex H.The nestimated mixtures X (n×p)which together account for the systematic variation in the dataset X,should each be convex combinations of the kfixed endmember estimates. Accordingly,the npoints in p-space which represent these mixtures should be interior points of the simplex H.Otherwise, for each sample point which lies outside H,at least one of the mixture coefficients (endmember contributions) will be negative. The purpose of this paper is to describe procedures for expanding H in the situation that its vertices are not a set of extreme points for the set which represents the mixtures.     

Development of Analytical Models to Estimate Pile Setup in Cohesive Soils Based on FE Numerical Analyses

This paper presents the numerical simulation of pile installation and the subsequent increase in the pile capacity over time (or setup) after installation that was performed using the finite element software Abaqus. In the first part, pile installation and the following load tests were simulated numerically using the volumetric cavity expansion concept. The anisotropic modified Cam-Clay and Dracker–Prager models were adopted in the FE model to describe the behavior of the clayey and sandy soils, respectively. The proposed FE model proposed was successfully validated through simulating two full-scale instrumented driven pile case studies. In the second part, over 100 different actual properties of individual soil layers distracted from literature were used in the finite element analysis to conduct parametric study and to evaluate the effect of different soil properties on the pile setup behavior. The setup factor A was targeted here to describe the pile setup as a function of time after the end of driving. The selected soil properties in this study to evaluate the setup factor A include: soil plasticity index (PI), undrained shear strength (S _u ), vertical coefficient of consolidation (C _v ), sensitivity ratio (S _r ), and over-consolidation ratio (OCR). The predicted setup factor showed direct proportion with the PI and S _r and inverse relation with S _u , C _v and OCR. These soil properties were selected as independent variables, and nonlinear multivariable regression analysis was performed using Gauss–Newton algorithm to develop appropriate regression models for A. Best models were selected among all based on level of errors of prediction, which were validated with additional nineteen different site information available in the literature. The results indicated that the developed model is able to predict the setup behavior for individual cohesive soil layers, especially for values of setup factor greater than 0.10, which is the most expectable case in nature.     

On solution of the inverse problem for confined aquifer flow via maximum likelihood

Joint estimation of transmissivity (T) and storativity (S) in a confined aquifer is done via maximum likelihood (ML). The differential equation of groundwater flow is discretized by the finite-element method, leading to equation _t+x _t=u _t. Elements of matrices and , as well as estimated covariance matrix of noise termu _t, are functions of T and S. By minimizing the negative loglikelihood function corresponding to discretized groundwater flow equation with respect to T and S, ML estimators are obtained. The ML approach is found to yield accurate estimates of T and S (within 9 and 10% of their actual values, respectively) and showed quadratic convergence in Newton's search technique. Prediction of aquifer response, using ML estimators, results in estimated piezometric heads accurate to ±0.5 m from their actual, exact values. Statistical properties of ML estimators are derived and some basic results for statistical inference are given.     

Kriging in terms of projections

In the last few years, an increasing number of practical studies using so-called kriging estimation procedures have been published. Various terms, such as universal kriging, lognormal kriging, ordinary kriging, etc., are used to define different estimation procedures, leaving a certain confusion about what kriging really is. The object of this paper is to show what is the common backbone of all these estimation procedures, thus justifying the common name of kriging procedures. The word kriging (in French krigeage) is a concise and convenient term to designate the classical procedure of selecting, within agiven class of possible estimators, the estimator with a minimum estimation variance (i.e., the estimator which leads to a minimum variance of the resulting estimation error). This estimation variance can be seen as a squared distance between the unknown value and its estimator; the process of minimization of this distance can then be seen as the projection of the unknown value onto the space within which the search for an estimator is carried out.     

Interval-valued random functions and the kriging of intervals

Estimation procedures using data that include some values known to lie within certain intervals are usually regarded as problems of constrained optimization. A different approach is used here. Intervals are treated as elements of a positive cone, obeying the arithmetic of interval analysis, and positive interval-valued random functions are discussed. A kriging formalism for interval-valued data is developed. It provides estimates that are themselves intervals. In this context, the condition that kriging weights be positive is seen to arise in a natural way. A numerical example is given, and the extension to universal kriging is sketched.     

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     

Improvement of Fourier-Based Unconditional and Conditional Simulations for Band Limited Fractal (von Kármán) Statistical Models

We evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation. The method is applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization (the von Kármán model). The simple Fourier unconditional simulation is conducted by Fourier transform of the amplitude spectrum model, sampled on a discrete grid, multiplied by a random phase spectrum. Although computationally efficient, this method failed to adequately match the intended statistical model at small scales because of sinc-function convolution. Attempts to alleviate this problem through the covariance method (computing the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function) created artifacts and spurious high wavenumber content. A modified Fourier method, consisting of pre-aliasing the wavenumber spectrum, satisfactorily remedies sinc smoothing. Conditional simulations using Fourier-based methods require several processing stages, including a smooth interpolation of the differential between conditioning data and an unconditional simulation. Although kriging is the ideal method for this step, it can take prohibitively long where the number of conditions is large. Here we develop a fast, approximate kriging methodology, consisting of coarse kriging followed by faster methods of interpolation. Though less accurate than full kriging, this fast kriging does not produce visually evident artifacts or adversely affect the a posteriori statistics of the Fourier conditional simulation.     

The Second-Order Stationary Universal Kriging Model Revisited

Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.     

Capillary Pressure Curves from Centrifuge Data: A Semi-Iterative Approach

The problem of determining capillary pressure functions from centrifuge data leads to an integral equation of the form _a ^x K(x,t)f(t)dt=g(x),x[a,b],(1)where the kernel K is known exactly and given by the underlying mathematical model. g is only known with a limited degree of accuracy in a finite and discrete set of points x ₁,...,x _M . However, the sought function f(t) is continuous. By the nature of the right-hand side, g(x), equation (1) is a discrete inverse problem which is ill-posed in the sense of Hadamard [9]. By a parameterization of the sought function, equation (1) reduces to a system of linear equations of the form Ac=b+ ,where b is the observation vector and A arises from discretization of the forward problem. is the error vector associated with b, and c contains the model parameters. The matrix A is usually ill-conditioned. The ill-conditioning is closely connected to the parameterization of the problem [23].In this paper a semi-iterative regularization method for solving the Volterra integral equation in the ₂-norm, namely, Brakhage's -method [2], is investigated. The iterative method is tested on synthetically generated, and on experimental data.     

Some asymptotic properties of kriging when the covariance function is misspecified

The impact of using an incorrect covariance function on kriging predictors is investigated. Results of Stein (1988) show that the impact on the kriging predictor from not using the correct covariance function is asymptotically negligible as the number of observations increases if the covariance function used is compatible with the actual covariance function on the region of interestR. The definition and some properties of compatibility of covariance functions are given. The compatibility of generalized covariances also is defined. Compatibility supports the intuitively sensible concept that usually only the behavior near the origin of the covariance function is critical for purposes of kriging. However, the commonly used spherical covariance function is an exception: observations at a distance near the range of a spherical covariance function can have a nonnegligible effect on kriging predictors for three-dimensional processes. Finally, a comparison is made with the perturbation approach of Diamond and Armstrong (1984) and some observations of Warnes (1986) are clarified.     

Mononodal indicator variography—Part I: Theory

For any distribution of grades, a particular cutoff grade is shown here to exist at which the indicator covariance is proportional to the grade covariance to a very high degree of accuracy. The name “mononodal cutoff” is chosen to denote this grade. Its importance for robust grade variography in the presence of a large coefficient of variation—typical of precious metals—derives from the fact that the mononodal indicator variogram is then linearly related to the grade variogram yet is immune to outlier data and is found to be particularly robust under data information reduction. Thus, it is an excellent substitute to model in lieu of a difficult grade variogram. A theoretical expression for the indicator covariance is given as a double series of orthogonal polynomials that have the grade density function as weight function. Leading terms of this series suggest that indicator and grade covariances are first-order proportional, with cutoff grade dependence being carried by the proportionality factor. Kriging equations associated with this indicator covariance lead to cutoff-free kriging weights that are identical to grade kriging weights. This circumstance simplifies indicator kriging used to estimate local point-grade histograms, while at the same time obviating order relations problems.     

Application of fuzzy clustering and fuzzy classification to evaluate the provenance of glacial till

To evaluate the provenance of glacial till, the trace element content of magnetite was used. Magnetite was present in all known rock types and all till samples in the area investigated. By using fuzzy-set theory it was possible to group samples of magnetite taken from bedrock into relatively homogeneous and geologically meaningful groups and also, by fuzzy classification, to relate the till samples to the rocks in such a way that the relative contribution of each rock type to the till is estimated. Each rock and till sample is assigned a membership value between 0 and 1 for each rock type. The membership values, for a certain rock type in the till, are then interpolated by kriging onto maps. Magnetites from skarns associated with sulfide ores especially are rather distinct, and so a map of such membership values for till unveils all known ore deposits some 1–5 km downstreamin the general direction of the ice flow. Other anomalies show up which cannot be related to hitherto known ores or mineralizations.     

Inner product matrices, kriging, and nonparametric estimation of variogram

Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice.     

Bearing capacity of a strip foundation on a sand layer overlying a smooth rigid stratum

The effect of a smooth rigid stratum, located beneath a dense sand layer, on the bearing capacity and settlement of surface and shallow strip footings is investigated using an advanced experimental model. A theoretical analysis is presented for the bearing capacity of surface footings. The results indicate that the bearing capacity reaches a minimum value at a specific sand-layer thickness. Any increase in the layer thickness above this value causes an increase in the bearing capacity up to that corresponding to a continuous media.Notation H= thickness of the sand layer - B= foundation width - N _q and N = bearing capacity factors for a semi-infinite layer - N _qs and N _s= bearing capacity factors for a finite layer - H _o /B= limiting depth - D _r= relative density - = angle of soil internal friction - M= model width - D= depth of surcharge - q= bearing stress, pressure applied on the footing - q _u= bearing capacity - = unit weight of sand