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.     

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.     

矿床统计预测单元划分的方法与程序

在矿床统计预测中,如何确定最佳的统计预测单元,目前尚无通用的准则和算法,它的划分往往取决于地质学家对研究区控矿条件的认识,矿化的实际分布殂态及统计预测所采用的评价模型等多种因素。系统功能齐全,基本上满足了资源评价的各种需求,实现了预测单元划分的自动化和智能化。     

Estimation of the spatial distribution of ground motion parameters for two recent earthquakes in Japan

A recent development in strong motion instrumentation in Japan provides an opportunity to collect valuable data sets, especially after moderate and large magnitude events. Gathering and modeling these data is a necessity for better understanding of regional ground motion characteristics. Estimations of the spatial distribution of earthquake ground motion plays an important role in early-stage damage assessments for both rescue operations by disaster management agencies as well as damage studies of urban structures. Subsurface geology layers and local soil conditions lead to soil amplification that contributes to the estimated ground motion parameters of the surface. We present a case study of the applicability of the nationally proposed GIS-based soil amplification ratios [J. Soil Dyn. Earthqu. Eng. 19 (2000) 41–53] to the October 6, 2000 Tottori-ken Seibu (western Tottori Prefecture) and the March 24, 2001 Geiyo earthquakes in Japan. First, ground motion values were converted to those at a hypothetical ground base-rock level (outcrop) using an amplification ratio for each 1×1 km area, based on geomorphological and subsurface geology information. Then a Kriging method, assuming an attenuation relationship at the base-rock as a trend component, is applied. Finally, the spatial distribution of ground motion at ground surface is obtained by applying GIS-based amplification factors for the entire region. The correlation between the observed and estimated ground motion values is reasonable for both earthquakes. Thus, the proposed method is applicable in near real-time early-damage assessments and seismic hazard studies in Japan.