变差函数球状模型的自动拟合与实现

根据变差函数的随机性和空间结构性,综合利用变差函数计算方法和加权线性规划拟合方法,分别拟合出各主要方向上的模型参数,再根据各向异性情况进行结构套合,实现了变差函数的计算及球状模型的自动拟合.针对样本中存在特异值的情况,算法中还提供了相对变差函数方法,有效地抑制了特异值对变差函数的影响,保证了球状模型拟合的精度.本算法在VC++6.0中实现,利用拟合出的模型,对样本区域进行插值得到网格文件,调用Surfer8.0绘制了等值线图.通过交叉验证和综合法验证,表明拟合度较高.     

二级套合结构实验变差函数的几何拟合

叠加地球化学场表现为各地球化学元素的变差函数具有双重套合结构。拟合实验交差函数是分解叠加地球化学场的关键。本文依据地球化学场自相关与自相似的内在联系,提出用多标度分形谐方法拟合具有二级套合结构的实验交差函数。     

基于改进群搜索优化算法的变差函数拟合

群搜索优化算法是一种群智能优化算法,通过研究群搜索优化算法的优劣以及其改进的方法,并将改进的群搜索优化算法应用于变差函数的高斯、指数和一阶及多阶球状模型的最优拟合。实例表明,群搜索优化算法能够有效地应用于变差函数拟合。     

最佳关联度的计算方法及其应用

着重了关联度理论计算公式中分辨系数的取值问题，提出了最佳关联度的一种计算方法－变差函数最优拟合法，即用关联度曲线与变差函数球状模型达最优拟合时的变程作为最佳分辨系数，从而求得最佳关联度。笔者将此方法用于大兴安岭地槽区砂金矿床被给因素的定量评价中，取得较好效果。     

理论变异函数球状模型的加权线性规划法似合

对理论变异函数球状模型及其套合结构拟合这一问题作了探讨，提出了加权线性规划拟合法。同于该法在目标函数中既可考虑到对不同滞后h下所得实验变异函数值进行加权，又可保证拟合成功，同时还可进行人工干预，因此，方法综合了现有加权多项式拟合法及线性规划拟合法的各自优点，且在计算上较目标规划拟合法更为简单。     

淮北平原年降水量空间插值模型的比选

为对比不同变差模型在降水量空间插值中的优劣,对淮北平原185个雨量站的2009年年降水观测数据,分别用不同理论变差模型与实验变差值拟合,然后选择普通克里格法进行降水场的变异分析及插值。经交叉验证法等多指标对插值结果检验后,证实了区域降水量具有明显的空间相关性,且三种拟合后变差模型的差异主要在短距离(h<10km)内。同时在现有站网布设方式下,三种模型插值结果并无显著差异,但以球状函数拟合插值结果整体效果最佳。     

球状模型的最优参数估计

在地质统计学中,变差函数理论模型的拟合一直没有满意的算法。本文结合加权回归多项式法和线性规划法的优点,提出用目标规划法进行球状模型的参数估计,为地质统计学计算全过程自动化提供了重要的方法。     

序贯高斯模拟在矿石品位估计中的应用研究

随机模拟是地质统计方法的重要内容。在矿石品位估计方法中克里格方法作为一种无偏估计方法,常被用于矿石品位的估计。但克里格法估值存在平滑效应。作者在分析序贯高斯模拟和普通克里格法基本原理的基础上,运用序贯高斯模拟方法和普通克里格方法对某铁矿体内全铁(TFe)品位进行估计,给出了品位估计结果模型。研究从勘探线方向、垂直勘探线方向和竖直方向分别计算变差函数,对球状模型、指数模型、高斯模型的变差函数拟合效果进行了优选,结果表明球型模型拟合效果最好。针对序贯模拟和克里格品位估值效果进行了分析,结果显示:序贯高斯模拟结果在品位分布形态上更接近样品品位分布形态,其平滑效应更小;克里格方法估计与序贯高斯模拟方法相比仅在品位均值方面更接近样本品位均值。因此,认为序贯高斯模拟方法可以更好地刻画矿体内品位分布状态。     

变差函数的参数模拟

本文给出利用线性方程组非负解理论，进行地质统计学变差函数理论模型参数最优拟合，并给出了计算实例，从而为实现地质统计学计算过程的自动化提供了重要方法。     

地质统计学变差函数人机对话拟合

变差函数的研究在地质统计学中具有十分重要作用，本文运用界面图形图像处理强的Ｃ￣（＋＋）语言实现了界面友好汉化人机对话变差函数的拟合，主要包括管理菜单的生成，实验变差值的求解，变差图的图形显示，标准函数模型的计算及变差函数人机对话求解等部分。最后对比一下回归分析与人机对话拟合结果。     

半变异函数及取样间距对克里金法在海洋地层分析中的影响研究

以浅剖数据为源数据,钻孔实测数据为验证数据,利用普通克里金法对海底地层厚度进行空间插值得到地层分布特征,采用3种半变异函数模型和不同取样间距对某井场3组地层厚度进行普通克里金插值并验证其插值效果。结果表明:普通克里金是一种有效的海底地层厚度预测方法;结构分析最佳的模型不一定是误差最小的模型,应对不同模型下的插值结果进行综合分析来选择最合适的模型,并提出球状模型在该井场厚度估计中最优,高斯模型次之;对于球状模型,增大取样间距对地层厚度变化剧烈的地层回归效果影响较小,对地层厚度变化不大的地层回归效果影响较大;同时,SE预测值变化率分析表明对于地层厚度变化剧烈的地层,减小取样间距可以大幅度地减少插值误差,而对于地层厚度变化不大的地层,减小取样间距对插值精度提高的意义不大。     

Indicator Variogram Models: Do We Have Much Choice?

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.     

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.     

重磁场的变差函数特征及应用

用变差函数研究重磁场的区域变化特征.变差函数的变程反映重磁场的相干范围, 块金效应反映随机干扰, 基台值反映变异程度.重磁场的理论模拟说明: 重力场的相干范围大于磁场, 重磁场变程主要取决于场源深度, 浅源重磁场变差函数近似为球状模型或指数模型, 深源重磁场近似为连续性更好的高斯模型.磁场场源深度近似等于变程的一半, 重力场场源深度近似等于变程的四分之一.湖北大冶铁矿垂直分量磁异常具有几何各向异性, 北西-南东走向, 变差函数推测磁铁矿平均深度为250m.磁异常小波多尺度分解细节和逼近部分磁场具有协调几何各向异性, 变差函数的各阶场源深度估计结果与功率谱估计结果吻合.      

Teacher''s Aide: Modeling Hole-Effect Variograms of Lithology-Indicator Variables

Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations.     

Teacher's Aide: Modeling Hole-Effect Variograms of Lithology-Indicator Variables

Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations.     

Automatic Variogram Modeling by Iterative Least Squares: Univariate and Multivariate Cases

In this paper, we propose a new methodology to automatically find a model that fits on an experimental variogram. Starting with a linear combination of some basic authorized structures (for instance, spherical and exponential), a numerical algorithm is used to compute the parameters, which minimize a distance between the model and the experimental variogram. The initial values are automatically chosen and the algorithm is iterative. After this first step, parameters with a negligible influence are discarded from the model and the more parsimonious model is estimated by using the numerical algorithm again. This process is iterated until no more parameters can be discarded. A procedure based on a profiled cost function is also developed in order to use the numerical algorithm for multivariate data sets (possibly with a lot of variables) modeled in the scope of a linear model of coregionalization. The efficiency of the method is illustrated on several examples (including variogram maps) and on two multivariate cases.     

Scale Effect on Principal Component Analysis for Vector Random Functions

Principal component analysis (PCA) is commonly applied without looking at the spatial support (size and shape, of the samples and the field), and the cross-covariance structure of the explored attributes. This paper shows that PCA can depend on such spatial features. If the spatial random functions for attributes correspond to largely dissimilar variograms and cross-variograms, the scale effect will increase as well. On the other hand, under conditions of proportional shape of the variograms and cross-variograms (i.e., intrinsic coregionalization), no scale effect may occur. The theoretical analysis leads to eigenvalue and eigenvector functions of the size of the domain and sample supports. We termed this analysis growing scale PCA, where spatial (or time) scale refers to the size and shape of the domain and samples. An example of silt, sand, and clay attributes for a second-order stationary vector random function shows the correlation matrix asymptotically approaches constants at two or three times the largest range of the spherical variogram used in the nested model. This is contrary to the common belief that the correlation structure between attributes become constant at the range value. Results of growing scale PCA illustrate the rotation of the orthogonal space of the eigenvectors as the size of the domain grows. PCA results are strongly controlled by the multivariate matrix variogram model. This approach is useful for exploratory data analysis of spatially autocorrelated vector random functions.     

Variogram Model Selection via Nonparametric Derivative Estimation

Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented.