Quadratic covariance estimation and equivalence of predictions

This article illustrates the use of linear and nonlinear regression models to obtain quadratic estimates of covariance parameters. These models lead to new insights into the motivation behind estimation methods, the relationships between different methods, and the relationship of covariance estimation to prediction. In particular, we derive the standard estimating equations for minimum norm quadratic unbiased translation invariant estimates (MINQUEs) from an appropriate linear model. Connections between the linear model, minimum variance quadratic unbiased translation invariant estimates (MIVQUEs), and MINQUEs are examined and we provide a minimum norm justification for the use of one-step normal theory maximum likelihood estimates. A nonlinear regression model is used to define MINQUEs for nonlinear covariance structures and obtain REML estimates. Finally, the equivalence of predictions under various models is examined when covariance parameters are estimated. In particular, we establish that when using MINQUE, iterative MINQUE, or restricted maximum likelihood (REML) estimates, the choice between a stationary covariance function and an intrinsically stationary semivariogram is irrelevant to predictions and estimated prediction variances.     

On the estimation of the generalized covariance function

The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.     

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.     

Generalized covariance functions in estimation

I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods.     

A modification of minimum norm quadratic estimation of a generalized covariance function for use with large data sets

Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency.     

Global estimation variance: Formulas and calculation

This paper discusses the combination of kriging variances, which have been considered heretofor unfeasible since linearity of the problem and considerable simplifications which follow were overlooked. A simplified expression for global estimation variance is presented and an algorithm discussed with respect to precision and computer cost. A case study is presented, and, finally, an optimum calculation method is recommended.     

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.     

Bernstein preprocessing for nonlinear least squares inversion of gravity and magnetic data

Nonlinear, nonlocal and adaptive optimization algorithms, now readily available, as applied to parameter estimation problems, require that the data to be inverted should not be very noisy. If they are so, the algorithm tends to fit them, rather than smoothening the noise component out. Here, use of Bernstein polynomials is proposed to prefilter noise out, before inversion with the help of a sophisticated optimization algorithm. Their properties are described. Inversion of gravity and magnetic data for basement depth estimation, singly and jointly, and without and after Bernstein-preprocessing is conducted to illustrate that the inversion of Bernstein-preprocessed gravity data alone may be slightly superior to the joint inversion of gravity and magnetic data.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

GPS satellite and receiver instrumental biases estimation using least squares method for accurate ionosphere modelling

The positional accuracy of the Global Positioning System (GPS) is limited due to several error sources. The major error is ionosphere. By augmenting the GPS, the Category I (CAT I) Precision Approach (PA) requirements can be achieved. The Space-Based Augmentation System (SBAS) in India is known as GPS Aided Geo Augmented Navigation (GAGAN). One of the prominent errors in GAGAN that limits the positional accuracy is instrumental biases. Calibration of these biases is particularly important in achieving the CAT I PA landings. In this paper, a new algorithm is proposed to estimate the instrumental biases by modelling the TEC using 4th order polynomial. The algorithm uses values corresponding to a single station for one month period and the results confirm the validity of the algorithm. The experimental results indicate that the estimation precision of the satellite-plus-receiver instrumental bias is of the order of ±0.17 nsec. The observed mean bias error is of the order −3.638 nsec and −4.71 nsec for satellite 1 and 31 respectively. It is found that results are consistent over the period.     

Nonparametric estimation of generalized covariances by modeling on-line data

Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods (i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions (the system is indeterminate). The method is presented in a general context of a k order trend in R^d, although the full solution is given only fork = I in Rⁱ. In Rⁱ, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric and nonparametric method.     

On maximum likelihood estimation of parameters in incorrectly specified models of covariance for spatial data

The analysis of, and from, models of spatial data usually proceeds under the assumption, often implicit, that the correct model has been specified. However, any model identification procedures based on sample data are subject to error, and consequences of such errors then permeate subsequent analysis. Thus, an attempt to quantify some of these consequences is of interest. A standard framework for analysis is extended here, by introduction of information theory, to permit the study of effects of model misspecification on maximum likelihood estimators of parameters of model covariance. Asymptotically valid theoretical results are presented, and the relevance of these results to samples of finite sizes met in practice is assessed in a series of simulation experiments. The effect of model misspecification, and use of estimators of parameters of misspecified covariance models, on the practical problem of prediction at a previously unsampled location is considered briefly, and further areas for possible investigation are outlined.     

Minimum norm quadratic estimation of components of spatial covariance

Quadratic estimators of components of a nested spatial covariance function are presented. Estimators are unbiased and possess a minimum norm property. Inversion of a covariance matrix is required but, by assuming that spatial correlation is absent, a priori, matrix inversion can be avoided. The loss of efficiency that results from this assumption is discussed. Methods can be generalized to include estimation of components of a generalized polynomial covariance assuming the underlying process to be an intrinsic random function. Particular attention is given to the special case where just two components of spatial covariance exist, one of which represents a nugget effect.     

化爆作用下岩石介质自由场应力波传播方程拟合方法研究

为得到不同工程条件下岩石化爆自由场分布特征,采用工程类比的方法,在试验数据和经验公式的基础上,采用加权最小二乘法对改进自由场应力波公式参数进行拟合。在分析参照工程与参照试验岩石及炸药参数的基础上,根据设计工程与参照工程和参照试验中岩石介质的波阻抗、弹性模量、炸药当量等参数的相似程度定义权重,将等值平均改为加权平均,对各经验公式进行加权最小二乘拟合。在确保类似工程和试验对设计工程具有更大参考价值的前提下,确定自由场应力波计算的改进公式,使爆炸荷载计算公式对设计工程具有更好的适用性。     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

拉东域基于最小二乘反演的斜缆数据鬼波压制方法

斜缆采集资料中鬼波的陷频特征限制了地震记录的频带宽度和分辨率,给地震资料反演、解释带来困难。通过压制鬼波,可获得高分辨率的宽频数据。基于拉东域的水平拖缆鬼波压制方法,结合斜缆鬼波随偏移距变化的特点,推导出一次波和鬼波在频率-拉东域的逆变换算子,建立检波器处总波场和海表面处上行波场之间新的关系式,利用最小二乘反演精确求解获得海表面处上行波场,并延拓得到拖缆处鬼波压制后的记录。通过考虑鬼波延迟时间受偏移距和出射角的影响,弥补鬼波延迟时间估计中存在的误差,无需进行反演迭代求取最优鬼波延迟时间,提高了计算效率。合成数据及海上实际斜缆数据测试结果表明,研究方法能较好地压制鬼波,达到拓宽地震记录频带的目的。     

Spectral simulation of multivariable stationary random functions using covariance fourier transforms

A coregionalization simulation consists of the generation of realizations of a group of spatially related random variables. The Fourier integral method is presented, modified to carry out such a multivariable simulation. This method allows the simulation of realizations with any specified symmetrical covariance matrix and it is not limited to the classic linear model of coregionalization. The results of gaussian nonconditinal simulations from a case study modeling the spatial characteristics of a layer of coal are given.     

高密度电阻率法比值参数基于阻尼最小二乘反演

三电位电极系中装置的探测精度和数据处理方法的研究一直是地球物理工作者研究的一个热点,同时也存在很大的争议.利用正演模拟结果讨论了岩溶地区几种可能存在的地质条件下三种装置的探测精度,发现β装置和γ装置的探测效果明显优于α装置;然后利用正演计算得到的数据合成比值参数(T),对合成数据T进行最小二乘反演,发现T值反演结果和视电阻率反演结果一致,并在噪声影响较大的区域,T值反演结果优于T值等值线图,可作为判断异常体特征的一个依据,也可验证视电阻率的反演结果,弥补由于噪声对某种单一装置探测效果的影响.以义马某地的水文地质勘察为例,T值最小二乘反演结果表明,在含水低阻区域T值也呈现小值异常,且显示的异常体边界准确,结构特征明显.利用T值反演对数据处理具有重要的意义,应予以重视.     

Moving least squares method for reliability assessment of rock tunnel excavation considering ground-support interaction

A practical approach is proposed in this paper for the reliability assessment of rock tunnel excavations using the moving least squares method (MLSM) and the uniform design. The failure probability is computed by the first-order and the second-order reliability method (FORM/SORM), which is based on the generated MLSM response surface (MLSM-RS) via an iterative algorithm. The proposed approach is first implemented in the analysis of a circular tunnel that consists of three limit state functions to illustrate the efficiency and accuracy of the approach. Then, the method is applied to a non-circular tunnel to demonstrate the feasibility and validity of the method for practical problems, in which numerical procedures are commonly employed to solve the implicit limit state functions.     

Estimation of covariance parameters in kriging via restricted maximum likelihood

In kriging, parametric approaches to covariance (or variogram) estimation require that unknown parameters be inferred from a single realization of the underlying random field. An approach to such an estimation problem is to assume the field to be Gaussian and iteratively minimize a (restricted) negative loglikelihood over the parameter space. In doing so, the associated computational burden can be considerable. Also, it is usually not easy to check whether or not the minimum achieved is global. In this note, we show that in many practical cases, the structure of the covariance (or variogram) function can be exploited so that iterative minimizing algorithms may be advantageously replaced by a procedure that requires the computation of the roots of a simple rational function and the search for the minimum of a function depending on one variable only. As a consequence, our approach allows one to observe in a straightforward fashion the presence of local minima. Furthermore, it is shown that insensitivity of the likelihood function to changes in parameter value can be easily detected. The note concludes with numerical simulations that illustrate some key features of our estimation procedure.