Multivariable variogram and its application to the linear model of coregionalization

In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.     

Gradual Deformation and Iterative Calibration of Gaussian-Related Stochastic Models

This paper describes a new method for gradually deforming realizations of Gaussian-related stochastic models while preserving their spatial variability. This method consists in building a stochastic process whose state space is the ensemble of the realizations of a spatial stochastic model. In particular, a stochastic process, built by combining independent Gaussian random functions, is proposed to perform the gradual deformation of realizations. Then, the gradual deformation algorithm is coupled with an optimization algorithm to calibrate realizations of stochastic models to nonlinear data. The method is applied to calibrate a continuous and a discrete synthetic permeability fields to well-test pressure data. The examples illustrate the efficiency of the proposed method. Furthermore, we present some extensions of this method (multidimensional gradual deformation, gradual deformation with respect to structural parameters, and local gradual deformation) that are useful in practice. Although the method described in this paper is operational only in the Gaussian framework (e.g., lognormal model, truncated Gaussian model, etc.), the idea of gradually deforming realizations through a stochastic process remains general and therefore promising even for calibrating non-Gaussian models.     

岩土力学参数空间变异性的集合卡尔曼滤波估值

岩土参数具有结构性和随机性的空间变异特征,该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础,将分布于研究区的岩土参数视为区域化变量,变异函数既描述了岩土参数整体的空间结构性变化,又描述了其局部的随机性变化,用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法,利用时空分布的观测数据,对岩土参数空间变异性进行估值。数值算例表明,EnKF能够有效地融合观测数据,较好地提供岩土参数空间变异性的估值。     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

岩土力学参数空间变异性的集合卡尔曼滤波佶值

岩土参数具有结构性和随机性的空间变异特征，该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础，将分布于研究区的岩土参数视为区域化变量，变异函数既描述了岩土参数整体的空间结构性变化，又描述了其局部的随机性变化，用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法，利用时空分布的观测数据，对岩土参数空间变异性进行估值。数值算例表明，EnKF能够有效地融合观测数据，较好地提供岩土参数空间变异性的估值。     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Bayesian Modelling of Spatial Data Using Markov Random Fields, With Application to Elemental Composition of Forest Soil

Spatial datasets are common in the environmental sciences. In this study we suggest a hierarchical model for a spatial stochastic field. The main focus of this article is to approximate a stochastic field with a Gaussian Markov Random Field (GMRF) to exploit computational advantages of the Markov field, concerning predictions, etc. The variation of the stochastic field is modelled as a linear trend plus microvariation in the form of a GMRF defined on a lattice. To estimate model parameters we adopt a Bayesian perspective, and use Monte Carlo integration with samples from Markov Chain simulations. Our methods does not demand lattice, or near-lattice data, but are developed for a general spatial data-set, leaving the lattice to be specified by the modeller. The model selection problem that comes with the artificial grid is in this article addressed with cross-validation, but we also suggest other alternatives. From the application of the methods to a data set of elemental composition of forest soil, we obtained predictive distributions at arbitrary locations as well as estimates of model parameters.     

克里金参数估值法及其在参数估计分析中的应用

为考虑岩土介质参数的空间分布的结构性和随机性等不确定因素,引入了克里金参数估值法。采用变异函数描述参数在空间结构上的变化,建立其空间变异规律的数学模型,从而实现对岩土参数的估值。通过实例分析揭示了克里金估值法具有反映“过滤效应”和“集团效应”的优点,对不同位置的数据赋予不同的权重系数,能够有效地反映参数空间变异结构,有利于对参数的合理化分析     

Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure

In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.     

Comment: A study of “probabilistic” and “deterministic” geostatistics

Stochastic process theory involves integrals of measurable functions over probability measure spaces. One of these is the ensemble space, Ω, whose members are sample functions on Euclidean spaceR ^k and the other isR ^k itself. What geostatisticians call the “theory of regionalized variables” is said to based on stochastic theory. A recent paper inMathematical Geology proclaims a distinction between “probabilistic” and “deterministic” geostatistics. The former is said to rely on “ensemble integrals” over Ω and the latter on “spatial integrals” overR ^k. This study shows that the proposed distinction rests on an arbitrary choice between two estimators for the covariance of a stochastic process; neither is an ensemble integral, both are spatial integrals, and both are Kolmogorov inconsistent. The “deterministic” estimator is identical with that of classical bivariate least-squares regression in which “spatial structure” is of no consequence. This study shows that both stochastic models are suboptimal approximations to the unique nonstationary classical statistical multivariate regression model generated by each sample pattern. The stochastic process model and its “spatial continuity measures,” thus, appear as questionable mathematical embellishments on suboptimal estimates, correspondence with geomorphic reality is tenuous, and estimates are biased and distorted. Various related misconceptions in the paper are also discussed.     

Trend-surface analysis

This paper is largely an exposition of the work of the French geostatistian G. Matheron and his school in English and at a simpler mathematical level. The probability theory on which it is based is essentially all contained in the references cited, most of which will be unfamiliar to mathematical geologists. The important method of ‘universal kriging” is explained intuitively. The genuine statistical problems, which have yet to be overcome, are pointed out.     

水文随机模拟进展

综述了近20年来水文随机模拟的新进展,包括三方面:①随机水文模型改进和创新;②水文随机模拟应用研究新进展;③水文随机模拟认识新进展。并指出了今后的研究重点:①对水文过程的重要物理特性和统计特性作深入的分析;②加强非参数模型和非线性模型的研究;③加强流域系统随机模型的研究;④加强建立模型时如何综合利用多种信息的研究;⑤加强模型的各种检验和合理分析。     

Study on Migration and Transformation Rule of Organic Pollutants （COD） in Aerated Zone

Collecting waste water with a certain concentration of organic pollutants COD (chemical oxygen demand), static adsorption, static biodegradation and dynamic soil column experiments were made in laboratory,we researched migration and transformation of COD in aerated zone. and put for-ward a mathematical model showing the process.The results show that adsorption of organism in aerated zone is linear,which is erprsented by henry‘s law s=Kdc sd, adsorption coefficient Kd=0.0693; biodegradation diagram accord basically with first-order kinetics equation c=coe^-k1t, biodegradation co-efficient K1=0.0499d^-1; dispersion coefficient D=0.0042m^2/d in experiments. The migration and transformation of organic pollutants (COD) in aerated zone jointly result from many factors such as dispersion, adsorption and biodegradation etc..     

Correcting the Smoothing Effect of Estimators: A Spectral Postprocessor

The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.     

初论地质信息有序系列

信息有序系列（ＩＯＳ）是笔者提出的一个新概念，是指具有非随机性定义的信息、非周期性、存在于大量无序中的有序部分特征的有序性。ＩＯＳ的理论是应用基于较少假设的、较为简单的数字或几何模式去研究地质体客观存在的复杂性，减少类似概率分布、周期性、平稳性、遍历理论等假设要求。ＩＯＳ中的一组现象，表面上看类似于周期，但被广义地定义，它们不能被谱分析和统计分析发现。“系列”一词用以与“模型”相区别，其特点是仅在有限范围和短时间内存在，是不平稳和遍历的。“有序”一词是强调在混沌中存在的有序部分，可应用于预测目的。所以，ＩＯＳ是有序性中一部分。大地震、热点、超大型矿床、节理等时空分布表明了ＩＯＳ的客观存在。     

A structured approach to enhance flood hazard assessment in mountain streams

An evidence-based flood hazard analysis in mountain streams requires the identification and the quantitative characterisation of multiple possible processes. These processes result from specific triggering mechanisms on the hillslopes (i.e. landslides, debris flows), in-channel morphodynamic processes associated with sudden bed changes and stochastic processes taking place at critical stream configurations (e.g. occlusion of bridges, failure of levees). From a hazard assessment perspective, such possible processes are related to considerable uncertainties underlying the hydrological cause-effect chains. Overcoming these uncertainties still remains a major challenge in hazard and risk assessment and represents a necessary condition for a reliable spatial representation of process intensities and the associated probabilities. As a result of an accurate analysis of the conceptual flaws present in the procedures currently employed for hazard mapping in South Tyrol (Italy) and Carinthia (Austria), we propose a structured approach as a means to enhance the integration of hillslope, morphodynamic and stochastic processes into conventional flood hazard prediction for mountain basins. To this aim, a functional distinction is introduced between prevailing one-dimensional and two-dimensional process propagation domains, i.e., between confined and semi- to unconfined stream segments. The former domains are mostly responsible for the generation of water, sediment and wood fluxes, and the latter are where flooding of inactive channel areas (i.e. alluvial fans, floodplains) can occur. For the 1D process propagation domain, we discuss how to carry out a process routing along the stream system and how to integrate numerical models output with expert judgement in order to derive consistent event scenarios, thus providing a consistent quantification of the input variables needed for the associated 2D domains. Within these latter domains, two main types of spatial sub-domains can be identified based on the predictability of their dynamics, i.e., stochastic and quasi-deterministic. Advantages and limitations offered by this methodology are finally discussed with respect to hazard and risk assessment in mountain basins.     

Integrating spatial, temporal, and magnitude probabilities for medium-scale landslide risk analysis in Darjeeling Himalayas, India

Landslide risk assessment is based on spatially integrating landslide hazard with exposed elements-at-risk to determine their vulnerability and to express the expected direct and indirect losses. There are three components that are relevant for expressing landslide hazard: spatial, temporal, and magnitude probabilities. At a medium-scale analysis, this is often done by first deriving a landslide susceptibility map, and to determine the three types of probabilities on the basis of landslide inventories linked to particular triggering events. The determination of spatial, temporal, and magnitude probabilities depend mainly on the availability of sufficiently complete historical records of past landslides, which in general are rare in most countries (e.g., India, etc.). In this paper, we presented an approach to use available historical information on landslide inventories for landslide hazard and risk analysis on a medium scale (1:25,000) in a perennially typical data-scarce environment in Darjeeling Himalayas (India). We demonstrate how the incompleteness in the resulting landslide database influences the various components in the calculation of specific risk of elements-at-risk (e.g., buildings, population, roads, etc.). We incorporate the uncertainties involved in the risk estimation and illustrate the range of expected losses in the form of maximum and minimum loss curves. The study demonstrates that even in data-scarce environments, quantitative landslide risk assessment is a viable option, as long as the uncertainties involved are expressed.     

Geologic Stochastic Modeling and Connectivity Assessment of Fracture Systems in the Boston Area

Summary The MIT stochastic model for representing fracture patterns has been applied to the Boston area. The model is a geometric-mechanical model, which means that underlying mechanisms are geometrically represented. In this paper the model is calibrated by comparing the fracture spacing and trace length simulated by the model with data collected by Billings (1976) in a number of tunnels in the Boston area. Following this, geometric fracture connectivity is investigated. This is done by recording in the model simulations, the number of interconnected fractures in so called sub-networks and by recording the geometric extents of these sub-networks. The results indicate limited horizontal but large vertical connectivity.     

On the possibilities of premonitory swarms for three sequences of earthquakes of the Burma—Szechwan region

Short term spatial and temporal variations in seismicity prior to the three sequences of earthquakes of m_b 5.8 of the Burma—Szechwan region are studied. Six years (1971–1976) of ISC seismicity data, as reported in the Regional Catalogue of Earthquakes, are considered. During the period, six earthquakes of body wave magnitude m_b 5.8 occurred in four sequences. Of these, three sequences are preceded by swarm activity in the epicentral regions. Evison (1977b) suggested that the swarm before the sequences of large shocks is a possible long-term precursor. He derived the conclusion by analyzing earthquakes in New Zealand and California. The analysis of the seismicity data for the region under investigation supports Evison's view and suggests that a relation between swarms and sequences of large events exists. The precursory time period (i.e. the time from beginning of the swarm to the main shock) for the Szechwan earthquakes of m_b = 5.9 (Feb. 6, 1973) and m_b = 5.8 (May 10, 1974) and the Burma earthquake of m_b = 6.2 (Aug. 12, 1976) are 305, 317 and 440 days, respectively.