Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances

A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features that could not be generated with traditional kriging.     

Conditional curvilinear stochastic simulation using pixel-based algorithms

In geology, structures displaying differing local directions of continuity are widespread, a typical example being a flusial depositional system. Conventional pixel-based geostatistical algorithms, may fail to reproduce such curvilinear structures. Conversely, object-based algorithms can reproduce curvilinear shapes but are difficult to condition to dense local data. Local depositional directions as obtained from dipmeter data. 3D seismic data, and geological interpretation represent critical information. An improved pixel-based geostatistical algorithm is proposed to account for such directional information. Case studies demonstrate the potential and limitations of the algorithm.     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

When do we need a trend model in kriging?

Under usual estimation practice with local search windows for data and for interpolation situations, universal kriging and ordinary kriging yield the same estimates, using a data set with apparent trend, for both the unknown attribute and its trend component. Modeling the trend matters only in extrapolation situations. Because conditions of the case study presented arise most frequently in practice, the simpler ordinary kriging is the preferred option.     

Stochastic Distance Based Geological Boundary Modeling with Curvilinear Features

Obtaining accurate geological boundaries and assessing the uncertainty in these limits are critical for effective ore resource and reserve estimation. The uncertainty in the extent of an ore body can be the largest source of uncertainty in ore resource estimation when drilling is sparse. These limits are traditionally interpreted deterministically and it can be difficult to quantify uncertainty in the boundary and its impact on ore tonnage. The proposed methodology is to consider stochastic modeling of the ore boundary with a distance function recoding of the available data. This technique is modified to incorporate non-stationarities in the form of a locally varying anisotropy field used in kriging and sequential Gaussian simulation. Implementing locally varying anisotropy kriging retains the geologically realistic features of a deterministic model while allowing for a stochastic assessment of uncertainty. A case study of a gold deposit in Northern Canada is used to demonstrate the methodology. The proposed technique generates realistic, curvilinear geological boundary models and allows for an assessment of the uncertainty in the model.     

Geostatistical Mapping with Continuous Moving Neighborhood

An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models.     

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.     

专家克里金插值法在空间插值中的应用

克里金插值是一种进行局部估值的方法,但它并不能对局部的地质特征进行智能分析.因此,在很多时候其插值结果不能体现数据的一种局部异构特征.针对这一问题,提出了一种专家克里金插值法,这种方法是在普通克里金插值法的基础上加入专家地质知识,使得插值结果更能符合实际的地质情况.对该方法的原理、模型进行了详细的论述,并通过实例验证该方法的有效性.     

Creating an advanced backpropagation neural network toolbox within GIS software

An artificial neural network (ANN) toolbox is created within GIS software for spatial interpolation, which will help GIS users to train and test ANNs, perform spatial analysis, and display results as a single process. The performance is compared to that of the open source Fast Artificial Neural Network library and conventional interpolation methods by creating digital elevation models (DEMs) given that nearly exact solutions exist. Simulation results show that the advanced backpropagations such as iRprop speed up the learning, while they can get stuck in a local minimum depending on initial weight sets. Besides, the division of input–output examples into training and test data affects the accuracy, particularly when the distribution of the examples is skewed and peaked, and the number of data is small. ANNs, however, show the similar performance to inversed distance weighted or kriging and outperform polynomial interpolations as a global interpolation method in high-dimensional data. In addition, the neural network residual kriging (NNRK) model, which combines the ANN toolbox and kriging within GIS software, is performed. The NNRK outperforms conventional methods and well captures global trends and local variations. A key outcome of this work is that the ANN toolbox created within the de facto standard GIS software is applicable to various spatial analysis including hazard risk assessment over a large area, in particular when there are multiple potential causes, the relationship between risk factors and hazard events is not clear, and the number of available data is small given its performance for DEM generation.     

Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics

Traditional simulation methods that are based on some form of kriging are not sensitive to the presence of strings of connectivity of low or high values. They are particularly inappropriate in many earth sciences applications, where the geological structures to be simulated are curvilinear. In such cases, techniques allowing the reproduction of multiple-point statistics are required. The aim of this paper is to point out the advantages of integrating such multiple-statistics in a model in order to allow shape reproduction, as well as heterogeneity structures, of complex geological patterns to emerge. A comparison between a traditional variogram-based simulation algorithm, such as the sequential indicator simulation, and a multiple-point statistics algorithm (e.g., the single normal equation simulation) is presented. In particular, it is shown that the spatial distribution of limestone with meandering channels in Lecce, Italy is better reproduced by using the latter algorithm. The strengths of this study are, first, the use of a training image that is not a fluvial system and, more importantly, the quantitative comparison between the two algorithms. The paper focuses on different metrics that facilitate the comparison of the methods used for limestone spatial distribution simulation: both objective measures of similarity of facies realizations and high-order spatial cumulants based on different third- and fourth-order spatial templates are considered.     

Spatial interpolation of severely skewed data with several peak values by the approach integrating kriging and triangular irregular network interpolation

It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values.     

The influence of conceptual model of sedimentary formation hydraulic heterogeneity on contaminant transport simulation

Development of heterogeneity model of layered sandy-clay formation and impact of this model on transport is considered. The lithological data of more than 250 wells that captured 300 meters formation at the investigated area of 40 km² are used for model of heterogeneity construction. Two models of heterogeneity were developed with using these well data: TP/MC model based on 3D Markov chain simulation for four hydrofacies and 2D kriging interpolation of thicknesses of elementary lithological layers. Simulation of conservative transport by particle tracking algorithm shows that horizontal transport along layers is similar for both models. The main difference is in vertical transport cross formation bedding. The kriging interpolation model gives more conservative results than TP/MC model due to larger characteristic horizontal length of layers in the kriging model. As the result vertical effective hydraulic conductivity of formation is in two times larger and the first particle arriving time is in four times faster in TP/MC model.     

基于DEM的青海贵德地区地形起伏度的研究

基于贵德地区1：5万地形图等高线数据内插生成的DEM数据,在ArcMap软件空间分析支持及Spss软件拟合曲线统计下,得到该地区地形起伏度提取的最佳窗口为400m×400m;基于400×400m2的分析窗口,提取贵德地区的地形起伏度,完成地形起伏度的专题图,统计分析高程与起伏度之间的相互关系,其结果与实际情况基本吻合。     

Optimal Interpolation of Gravity Maps Using a Modified Neural Network

This paper proposes an interpolation method based on a modified Kohonen artificial neural network, and is used to interpolate marine gravity data on a regular grid. This method combines accuracy comparable to that of kriging with a much shorter computing time than kriging. It is particularly efficient when both the size of the grid and the quantity of available data are large. Under some hypotheses similar to those of kriging with a trend, the unbiasedness and optimality of the method can be demonstrated. Comparison with kriging with a trend using marine gravity data shows similar results. Although neural interpolation is slightly less efficient, it is more robust outside of the marine data area.     

Hydrographic data modeling methods for determining precise seafloor topography

Acoustic and light detection and ranging are the recent methods used in hydrographical surveying. Depth values depending on X and Y horizontal coordinates are measured in both methods. While processing the hydrographical data, all data with different density are interpolated and modeled for determining the seafloor model using different interpolation techniques. In this study, effects of different surface modeling methods are investigated. Data obtained from single-beam echo sounder (SBES) are modeled using inverse distance, kriging, local polynomial, minimum curvature, moving average, nearest-neighbor, and Delaunay interpolation methods. Interpolation results are compared with the multibeam echo sounder data which were collected on the same area for determining the accuracy of modeling methods. Depending on the maximum total vertical uncertainty values in hydrographic survey standards, the best results were determined by using the kriging method. The Delaunay, minimum curvature, and inverse distance methods can be used for modeling the SBES data in shallow waters.     

Evaluation of Interpolation Accuracy of Neural Kriging with Application to Temperature-Distribution Analysis

An interpolation method based on a multilayer neural network (MNN), has been examined and tested for the data of irregular sample locations. The main advantage of MNN is in that it can deal with geoscience data with nonlinear behavior and extract characteristics from complex and noisy images. The training of MNN is used to modify connection weights between nodes located in different layers by a simulated annealing algorithm (one of the optimization algorithms of the network). In this process, three types of errors are considered: differences in values, semivariograms, and gradients between sample data and outputs from the trained network. The training is continued until the summation of these errors converges to an acceptably small value. Because the MNN trained by this learning criterion can estimate a value at an arbitrary location, this method is a form of kriging and termed Neural Kriging (NK). In order to evaluate the effectiveness of NK, a problem on restoration ability of a defined reference surface from randomly chosen discrete data was prepared. Two types of surfaces, whose semivariograms are expressed by isotropic spherical and geometric anisotropic gaussian models, were examined in this problem. Though the interpolation accuracy depended on the arrangement pattern of the sample locations for the same number of data, the interpolation errors of NK were shown to be smaller than both those of ordinary MNN and ordinal kriging. NK can also produce a contour map in consideration of gradient constraints. Furthermore, NK was applied to distribution analysis of subsurface temperatures using geothermal investigation loggings of the Hohi area in southwest Japan. In spite of the restricted quantity of sample data, the interpolation results revealed high temperature zones and convection patterns of hydrothermal fluids. NK is regarded as an interpolation method with high accuracy that can be used for regionalized variables with any structure of spatial correlation.     

尕斯油田岩性建模算法优选

比较岩性模型建立方法。首先,在高分辨率层序地层学的指导下,最大限度地应用地质、露头、三维地震、测井等静态资料,发挥井点资料垂向分辨率高,地震资料横向信息丰富的优势,在地质规律约束下建立不同时间的高精度等时地层格架模型。然后,在精细格架模型的基础上,以测井解释得到的岩相数据作为条件数据,分别采用指示克里格、截断高斯模拟、Object-modeling算法、贯指示模拟建立砂体展布模型。最后,通过抽稀检验评价不同算法对模拟结果的影响,实现算法及其参数的优选,从而指导整个区块不同开发阶段,不同井网密度时全区三维精细地质模型的建立,也可为具有相似地质环境的油田建立三维地质模型提供参考。通过比较,优选出指示克里格、序贯指示模拟两种算法都能较好表征本研究区地质情况。     

划分重磁区域异常与局部异常的一种方法泛克立格法

在重（磁）资料的分析处理中,经常需要划分区域异常与局部异常。本文对用于矿床品位预测的方法—泛克立格法加以改进、完善,使其适合于求重（磁）资料的区域异常与局部异常。根据某点周围若干个信息点上的重（磁）观测数据估计出该点处的区域异常值,而且这种估计是在满足线性、无偏、最小估计方差条件下求得的。从重（磁）观测数据中减去区域异常就可得到局部异常,这就是泛克立格法。它比经常用来划分区域异常与局部异常的方法—趋势分析方法有许多优点。趋势分析方法只是泛克立格法的一个特例。在本文中,作者还提出在小范围内,重力区域异常适合于用一次多项式拟合;而磁法区域异常适合于用二次多项式拟合的观点     

A method for assessing local variation among scattered measurements

Automatic triangulation of scattered locations permits analysis of local variation in a dependent variable through calculation of a roughness index. This is approached by treating triangles of the triangulation (including the dependent variable) as vectorial structures, and accumulating at each data point the vector sum of the cluster of triangles surrounding it. The roughness index is defined as the complement of the ratio of the area of a triangle cluster to the area of component triangles as projected onto a gradient plane defined by their vector sum. The roughness index provides a measure of consistency of data values relative to surrounding observations and can be interpreted as a local index of reliability of interpolation.     

http://www.sciencedirect.com/science/article/pii/S1674987111001411

正Three-dimensional geological modeling(3DGM) assists geologists to quantitatively study in three-dimensional(3D) space structures that define temporal and spatial relationships between geological objects.The 3D property model can also be used to infer or deduce causes of geological objects.3DGM technology provides technical support for extraction of diverse geoscience information,3D modeling,and quantitative calculation of mineral resources.Based on metallogenic concepts and an ore deposit model, 3DGM technology is applied to analyze geological characteristics of the Tongshan Cu deposit in order to define a metallogenic model and develop a virtual borehole technology;a BP neural network and a 3D interpolation technique were combined to integrate multiple geoscience information in a 3D environment. The results indicate:(1) on basis of the concept of magmatic-hydrothermal Cu polymetallic mineralization and a porphyry Cu deposit model,a spatial relational database of multiple geoscience information for mineralization in the study area(geology,geophysics,geochemistry,borehole,and cross-section data) was established,and 3D metallogenic geological objects including mineralization stratum,granodiorite, alteration rock,and magnetic anomaly were constructed;(2) on basis of the 3D ore deposit model,23,800 effective surveys from 94 boreholes and 21 sections were applied to establish 3D orebody models with a kriging interpolation method;(3) combined 23,800 surveys involving 21 sections,using VC++ and OpenGL platform,virtual borehole and virtual section with BP network,and an improved inverse distance interpolation(IDW) method were used to predict and delineate mineralization potential targets (Cu-grade of cell not less than 0.1%);(4) comparison of 3D ore bodies,metallogenic geological objects of mineralization,and potential targets of mineralization models in the study area,delineated the 3D spatial and temporal relationship and causal processes among the ore bodies,alteration rock,metallogenic stratum,intrusive rock,and the Tongshan Fault.This study provides important technical support and a scientific basis for assessment of the Tongshan Cu deposit and surrounding exploration and mineral resources.