Copula-based geostatistical modeling of continuous and discrete data including covariates

It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.     

Curvelet reconstruction of non-uniformly sampled seismic data using the linearized Bregman method

Seismic data reconstruction, as a preconditioning process, is critical to the performance of subsequent data and imaging processing tasks. Often, seismic data are sparsely and non-uniformly sampled due to limitations of economic costs and field conditions. However, most reconstruction processing algorithms are designed for the ideal case of uniformly sampled data. In this paper, we propose the non-equispaced fast discrete curvelet transform-based three-dimensional reconstruction method that can handle and interpolate non-uniformly sampled data effectively along two spatial coordinates. In the procedure, the three-dimensional seismic data sets are organized in a sequence of two-dimensional time slices along the source–receiver domain. By introducing the two-dimensional non-equispaced fast Fourier transform in the conventional fast discrete curvelet transform, we formulate an L1 sparsity regularized problem to invert for the uniformly sampled curvelet coefficients from the non-uniformly sampled data. In order to improve the inversion algorithm efficiency, we employ the linearized Bregman method to solve the L1-norm minimization problem. Once the uniform curvelet coefficients are obtained, uniformly sampled three-dimensional seismic data can be reconstructed via the conventional inverse curvelet transform. The reconstructed results using both synthetic and real data demonstrate that the proposed method can reconstruct not only non-uniformly sampled and aliased data with missing traces, but also the subset of observed data on a non-uniform grid to a specified uniform grid along two spatial coordinates. Also, the results show that the simple linearized Bregman method is superior to the complex spectral projected gradient for L1 norm method in terms of reconstruction accuracy.     

Characterizing the spatial variability of groundwater quality using the entropy theory: I. Synthetic data

This paper, the first in a series of two, applies the entropy (or information) theory to describe the spatial variability of synthetic data that can represent spatially correlated groundwater quality data. The application involves calculating information measures such as transinformation, the information transfer index and the correlation coefficient. These measures are calculated using discrete and analytical approaches. The discrete approach uses the contingency table and the analytical approach uses the normal probability density function. The discrete and analytical approaches are found to be in reasonable agreement. The analysis shows that transinformation is useful and comparable with correlation to characterize the spatial variability of the synthetic data set, which is correlated with distance. Copyright © 2004 John Wiley & Sons, Ltd.     

多相离散随机介质模型及其探地雷达波场特征研究

沥青混凝土是由骨料、沥青胶浆、空气按照一定的体积百分比混合而成的多相非匀质混合物,其骨料、沥青胶浆和空气的体积不等、形状各异、介电特性不同、空间位置随机分布,具有明显的多相、离散、随机介质特征.本文基于随机介质模型理论,(1)测量与统计了介电常数在典型沥青混凝土芯样空间上的随机分布统计特征;(2)估算了沥青混凝土介质的自相关函数及其特征参数(自相关长度、自相关角度等),确定其随机介质类型;(3)提出了量化约束下的多相离散随机介质建模算法,以混合型椭圆自相关函数为基础,构建了不同粗糙度因子的多相离散随机介质模型;(4)构建了不同空隙率的多相离散随机介质模型,正演模拟与对比分析了探地雷达波在均匀介质、连续型随机介质和多相离散随机介质中的传播特征.结果表明:多相离散随机介质模型不仅描述了沥青混凝土的多相、离散与空间随机分布统计特征,而且进一步描述了其各组成物质体积百分比,能更全面、准确地描述沥青混凝土的介质特征,同时也为描述其他类似材料或介质提供了新的方法和途径;在多相离散随机介质模型中,探地雷达波散射强烈,随机、无序传播的散射波相互叠加干涉,形成了明显的随机扰动和"噪声",致使异常体反射波扭曲变形、不连续,降低了探地雷达回波的信噪比和分辨率.研究探地雷达波的随机扰动特征与多相离散随机介质模型参数之间的关系,将为定量评价多相离散随机介质的属性参数提供参考和帮助.     

A ubiquitous knowledgeable data representation model (UKRM) for three-dimensional geographic information systems (3D GIS)

In the face of complicated, diversified three-dimensional world, the existing 3D GIS data models suffer from certain issues such as data incompatibility, insufficiency in data representation and representation types, among others. It is often hard to meet the requirements of multiple application purposes (users) related to GIS spatial data management and data query and analysis, especially in the case of massive spatial objects. In this study, according to the habits of human thinking and recognition, discrete expressions (such as discrete curved surface (DCS), and discrete body (DB)) were integrated and two novel representation types (including function structure and mapping structure) were put forward. A flexible and extensible ubiquitous knowledgeable data representation model (UKRM) was then constructed, in which structurally heterogeneous multiple expressions (including boundary representation (B-rep), constructive solid geometry (CSG), functional/parameter representation, etc.) were normalized. GIS’s ability in representing the massive, complicated and diversified 3D world was thus greatly enhanced. In addition, data reuse was realized, and the bridge linking static GIS to dynamic GIS was built up. Primary experimental results illustrated that UKRM was overwhelmingly superior to the current data models (e.g. IFC, CityGML) in describing both regular and irregular spatial objects.     

An integrated optimization algorithm for parameter structure identification in groundwater modeling

The inverse problem of parameter structure identification in a distributed parameter system remains challenging. Identifying a more complex parameter structure requires more data. There is also the problem of over-parameterization. In this study, we propose a modified Tabu search for parameter structure identification. We embed an adjoint state procedure in the search process to improve the efficiency of the Tabu search. We use Voronoi tessellation for automatic parameterization to reduce the dimension of the distributed parameter. Additionally, a coarse-fine grid technique is applied to further improve the effectiveness and efficiency of the proposed methodology. To avoid over-parameterization, at each level of parameter complexity we calculate the residual error for parameter fitting, the parameter uncertainty error and a modified Akaike Information Criterion. To demonstrate the proposed methodology, we conduct numerical experiments with synthetic data that simulate both discrete hydraulic conductivity zones and a continuous hydraulic conductivity distribution. Our results indicate that the Tabu search allied with the adjoint state method significantly improves computational efficiency and effectiveness in solving the inverse problem of parameter structure identification.     

Spatial and temporal variations of atmospheric aerosol optical thickness in northwestern Mexico

The purpose of this paper was to study aerosol particles in the Northwestern region of Mexico (NWM) through Aerosol Optical Thickness (AOT) parameter in the atmosphere. This parameter represents one of the extinction coefficients of solar radiation and the rate of suspended particles in the atmosphere. For determination of AOT, we considered the use of remote sensors outside of the atmosphere. In particular, Moderate Resolution Imaging Spectroradiometer (MODIS) which can measure the atmospheric AOT thickness. Data from the MODIS sensor must be validated before they are considered reliable. For this task, we required surface measurements to obtain a correlation with the data acquired with the remote radiometer. The paper describes the validation process performed for data obtained with MODIS through measurements provided by an AErosol RObotic NETwork (AERONET) photometer located in the city of Hermosillo, Sonora, NWM. Additionally, we carried out a temporal analysis based on the behavior of the AOT graphics and spatial analysis supported in maps with sufficient information.     

Linear geophysical inversion via the discrete cosine pseudo‐inverse: application to potential fields

In this paper, we present a methodology to perform geophysical inversion of large‐scale linear systems via a covariance‐free orthogonal transformation: the discrete cosine transform. The methodology consists of compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal transformations to define an approximation of the Moore–Penrose pseudo‐inverse. This methodology is also highly scalable since the model reduction achieved by these techniques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic two‐dimensional gravimetric problem for different dimensionalities and different levels of white Gaussian noise and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correlation. The numerical results show that the discrete cosine transform pseudo‐inverse outperforms the classical least‐squares techniques, mainly in the presence of noise, since the solutions that are obtained are more stable and fit the observed data with the lowest root‐mean‐square error. Besides, we show that model reduction is a very effective way of parameter regularisation when the conditioning of the reduced discrete cosine transform matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) obtaining very successful results in this non‐linear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and non‐linear inverse problems (through linearisation) arising in technology and, particularly, in geophysics, independently of the geophysical model discretisation and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill‐conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations.     

结合序贯指示模拟的贝叶斯高斯混合线性反演

高斯混合模型(Gaussian Mixture Model, GMM)可以用来描述储层性质的多峰分布特性,多峰特性主要是由于它们在不同离散变量内的变化而引起的.在高斯混合模型中,高斯分量的权值代表离散变量的概率.然而,基于高斯混合模型的贝叶斯线性反演可能会对某些点的离散变量错误地分类,进而影响连续变量的反演结果,尤其存在强噪声的时候.在本文中,我们考虑了离散变量的空间变化性,并将高斯混合模型与序贯指示模拟(Sequential Indicator Simulation, SIS)相结合来确定离散变量的后验条件权值,形成了结合序贯指示模拟的贝叶斯高斯混合线性反演方法.该方法能够准确地对离散变量进行归类,且具有良好的抗噪性.通过模型试算,我们证明了这种方法的可行性,并在实际资料中取得了较好的结果.     

用Euler反褶积方法反演台湾海峡磁异常

用Ｅｕｌｅｒ反褶积方法反演台湾海峡磁异常，选取的窗口不能大，而决定取舍反褶积解的误差限又不能小，这势必对解的质量有较大影响．为此，不仅用了不同延深的单体模型，而且进行了多体模型实验．结果表明，复杂分布的磁性体，用Ｅｕｌｅｒ反褶积方法确定磁性体的轮廓可能是困难的，但可确定磁性体的水平位置和深度，从而降低了对资料精度和计算参数选取的要求．在磁异常比较复杂的地区．     

Characterizing the spatial variability of groundwater quality using the entropy theory: II. Case study from Gaza Strip

This paper, the second in the series, uses the entropy theory to describe the spatial variability of groundwater quality data sets. The application of the entropy theory is illustrated using the chloride observations obtained from a network of groundwater quality monitoring wells in the Gaza Strip, Palestine. The application involves calculating information measures, such as transinformation, the information transfer index and the correlation coefficient. These measures are calculated using a discrete approach, in which contingency tables are used. An exponential decay fitting approach was applied to the discrete models. The analysis shows that transinformation, as a function of distance, can be represented by the exponential decay curve. It also indicates that, for the data used in this study, the transinformation model is superior to the correlation model for characterizing the spatial variability. Copyright © 2004 John Wiley & Sons, Ltd.     

多震相走时联合三参数同时反演成像

采用新近研制的分区多步不规则最短路径多震相地震射线追踪正演技术,结合流行的子空间反演算法,提出了一种联合多震相走时资料进行地震三参数 （速度、反射界面和震源位置） 同时反演的方法技术.数值模拟反演实例、以及与双参数 （速度和反射界面或速度和震源位置） 同时反演的对比分析表明：三参数同时反演成像结果大体接近双参数同时反演成像的结果.另外,噪声敏感性试验表明：所提算法对到时数据中可容许的随机误差并不敏感,结果说明多震相走时的联合三参数同时反演成像方法技术不失为一种提高走时成像空间分辨率、进而降低重建模型参数失真度、行之有效的方法技术.     

利用地震活动性参数b值判断张渤地震带中西段地震危险性

利用2000年7月-2017年6月地震资料,计算张渤地震构造带中西段地震活动性参数b值、a值和a/b值。基于沿断裂带的b值空间分布,结合多地震活动参数值组合、历史强震背景分布特点,分析张渤地震构造带中西段不同段落的强震危险性。研究结果表明,河北涿鹿及山西大同一带具有低b值、低a值、较高a/b值的参数组合,反映该区域具有高应力积累,未来具有强震发生的危险。     

Analysis of the Impact of Source Region Structures on Seismological Parameter Scanning

By taking moderate-strong earthquakes in South,North and West China as the research subjects and taking into consideration the fault strikes in these regions,this paper selects 8 kinds of seismology indexes with clear physical significance and strong independence to carry out spatial scanning of the parallel,vertical and oblique slip of fault along the fault strike.Based on the size of correlation coefficients between the scanning curve and source region curve we quantitatively analyze the difference between scan results among different slip modes and study the impact of fault strike in different tectonic divisions on scanning results and variation rules of seismological parameters.The results show that not only does the change of spatial parameters have a great influence on seismological parameter scanning,but so does the fault strike in the source region.This paper presents the optimum condition parameters with least spatial influencing scanning scope for different magnitude seismology indexes and analyzes the possible influence of fault strike on seismological parameter scanning results.     

Inverse distance interpolation for facies modeling

Inverse distance weighted interpolation is a robust and widely used estimation technique. In practical applications, inverse distance interpolation is oftentimes favored over kriging-based techniques when there is a problem of making meaningful estimates of the field spatial structure. Nowadays application of inverse distance interpolation is limited to continuous random variable modeling. There is a need to extend the approach to categorical/discrete random variables. In this paper we propose such an extension using indicator formalism. The applicability of inverse distance interpolation for categorical modeling is then illustrated using Total’s Joslyn Lease facies data.     

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.     

Direct forecasting of subsurface flow response from non-linear dynamic data by linear least-squares in canonical functional principal component space

Inverse modeling is widely used to assist with forecasting problems in the subsurface. However, full inverse modeling can be time-consuming requiring iteration over a high dimensional parameter space with computationally expensive forward models and complex spatial priors. In this paper, we investigate a prediction-focused approach (PFA) that aims at building a statistical relationship between data variables and forecast variables, avoiding the inversion of model parameters altogether. The statistical relationship is built by first applying the forward model related to the data variables and the forward model related to the prediction variables on a limited set of spatial prior models realizations, typically generated through geostatistical methods. The relationship observed between data and prediction is highly non-linear for many forecasting problems in the subsurface. In this paper we propose a Canonical Functional Component Analysis (CFCA) to map the data and forecast variables into a low-dimensional space where, if successful, the relationship is linear. CFCA consists of (1) functional principal component analysis (FPCA) for dimension reduction of time-series data and (2) canonical correlation analysis (CCA); the latter aiming to establish a linear relationship between data and forecast components. If such mapping is successful, then we illustrate with several cases that (1) simple regression techniques with a multi-Gaussian framework can be used to directly quantify uncertainty on the forecast without any model inversion and that (2) such uncertainty is a good approximation of uncertainty obtained from full posterior sampling with rejection sampling.     

A statistical framework of data fusion for spatial prediction of categorical variables

With rapid advances of geospatial technologies, the amount of spatial data has been increasing exponentially over the past few decades. Usually collected by diverse source providers, the available spatial data tend to be fragmented by a large variety of data heterogeneities, which highlights the need of sound methods capable of efficiently fusing the diverse and incompatible spatial information. Within the context of spatial prediction of categorical variables, this paper describes a statistical framework for integrating and drawing inferences from a collection of spatially correlated variables while accounting for data heterogeneities and complex spatial dependencies. In this framework, we discuss the spatial prediction of categorical variables in the paradigm of latent random fields, and represent each spatial variable via spatial covariance functions, which define two-point similarities or dependencies of spatially correlated variables. The representation of spatial covariance functions derived from different spatial variables is independent of heterogeneous characteristics and can be combined in a straightforward fashion. Therefore it provides a unified and flexible representation of heterogeneous spatial variables in spatial analysis while accounting for complex spatial dependencies. We show that in the spatial prediction of categorical variables, the sought-after class occurrence probability at a target location can be formulated as a multinomial logistic function of spatial covariances of spatial variables between the target and sampled locations. Group least absolute shrinkage and selection operator is adopted for parameter estimation, which prevents the model from over-fitting, and simultaneously selects an optimal subset of important information (variables). Synthetic and real case studies are provided to illustrate the introduced concepts, and showcase the advantages of the proposed statistical framework.     

Distributed parameterization of complex terrain

This paper addresses the incorporation of high resolution topography, soils and vegetation information into the simulation of land surface processes in atmospheric circulation models (ACM). Recent work has concentrated on detailed representation of one-dimensional exchange processes, implicitly assuming surface homogeneity over the atmospheric grid cell. Two approaches that could be taken to incorporate heterogeneity are the integration of a surface model over distributed, discrete portions of the landscape, or over a distribution function of the model parameters. However, the computational burden and parameter intensive nature of current land surface models in ACM limits the number of independent model runs and parameterizations that are feasible to accomplish for operational purposes. Therefore, simplications in the representation of the vertical exchange processes may be necessary to incorporate the effects of landscape variability and horizontal divergence of energy and water. The strategy is then to trade off the detail and rigor of point exchange calculations for the ability to repeat those calculations over extensive, complex terrain. It is clear the parameterization process for this approach must be automated such that large spatial databases collected from remotely sensed images, digital terrain models and digital maps can be efficiently summarized and transformed into the appropriate parameter sets. Ideally, the landscape should be partitioned into surface units that maximize between unit variance while minimizing within unit variance, although it is recognized that some level of surface heterogeneity will be retained at all scales. Therefore, the geographic data processing necessary to automate the distributed parameterization should be able to estimate or predict parameter distributional information within each surface unit.