Effects of search parameters on kriged reserve estimates

Summary Reliable ore reserve estimates for deposits with highly skewed grade distributions are difficult tasks to perform. Although some recent geostatistical techniques are available to handle problems with these estimations, ordinary kriging or conventional interpolation methods are still widely used to estimate the ore reserves for such deposits. The estimation results can be very sensitive to the search parameters used during the interpolation of grades with these methods.This paper compares the ore reserve estimates from ordinary kriging using several cases in which certain search parameters are varied. The comparisons are extended to different mineralizations to show the changing effects of these parameters.     

A Combined Three-Dimensional Geological-Geostatistical-Numerical Model of Underground Excavations in Rock

Summary. This paper exploits geological and borehole geotechnical data obtained in the exploratory phase of a tunneling project to investigate in a first place if the kriging interpolation scheme may effectively reproduce the spatial variability of rock mass quality (Rock Mass Rating, RMR) in the vicinity of tunnels. For this purpose a quick solver in Fortran has been developed that performs variography analysis of 3D spatial data, fast kriging estimations of RMR between borehole sampling locations at the centroids of the elements of the numerical model, and model validation. For the purpose of an integrated underground excavation design, a step further is made by incorporating into the 3D mechanical numerical model of the rock mass, the three-dimensional (3D) solid geological model, thus coupling the geology with the ground (geotechnical) model (i.e. each element of the numerical model is assigned a geological material). The mechanical properties of each finite difference cell (or Representative Elementary Volume) of the ground model were then prescribed according to its geological type, the spatial heterogeneity of the rock mass expressed quantitatively with the kriging model, and the upscaling calculations of the mechanical properties of the intact rocks determined in the laboratory, based on the size-effect (strength dependence on size) and Damage Theory. Furthermore, a preliminary numerical simulation of the advance of unsupported tunnels in the model of the heterogeneous rock mass was performed for illustration purposes.     

Large to intermediate-scale aquifer heterogeneity in fine-grain dominated alluvial fans (Cenozoic As Pontes Basin,northwestern Spain): insight based on three-dimensional geostatistical reconstruction

Facies reconstructions are used in hydrogeology to improve the interpretation of aquifer permeability distribution. In the absence of sufficient data to define the heterogeneity due to geological processes, uncertainties in the distribution of aquifer hydrofacies and characteristics may appear. Geometric and geostatistical methods are used to understand and model aquifer hydrofacies distribution, providing models to improve comprehension and development of aquifers. However, these models require some input statistical parameters that can be difficult to infer from the study site. A three-dimensional reconstruction of a kilometer scale fine-grain dominated Cenozoic alluvial fan derived from more than 200 continuously cored, closely spaced, and regularly distributed wells is presented. The facies distributions were reconstructed using a genetic stratigraphic subdivision and a deterministic geostatistical algorithm. The reconstruction is only slightly affected by variations in the geostatistical input parameters because of the high-density data set. Analysis of the reconstruction allowed identification in the proximal to medial alluvial fan zones of several laterally extensive sand bodies with relatively higher permeability; these sand bodies were quantified in terms of volume, mean thickness, maximum area, and maximum equivalent diameter. These quantifications provide trends and geological scenarios for input statistical parameters to model aquifer systems in similar alluvial fan depositional settings.     

地统计学分析技术及其在气象中的适用性

首先论述了地统计学的一些特点，然后从变异函数和Kriging插值两个方面介绍了地统计学的分析技术，最后讨论了地统计学在气象研究中的适用性。     

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

A six-step practical approach to semivariogram modeling

Geostatistical prediction and simulation are being increasingly used in the earth sciences and engineering to address the imperfect knowledge of attributes that fluctuate over large areas or volumes—pollutant concentration, electromagnetic fields, porosity, thickness of a geological formation. Central to the application of such techniques is the need to know the spatial continuity, knowledge that is commonly condensed in the form of covariance or semivariogram models. Their preparation is subdivided here into the following steps: (1) Data editing, (2) Exploratory data analysis, (3) Semivariogram estimation, (4) Directional investigation, (5) Simple modeling, (6) Nested modeling. I illustrate these stages practically with a real data set from a geophysical survey from Elk County, Kansas, USA. The applicability of the approach is not limited by the physical nature of the attribute of interest.     

辽宁弓长岭一矿区铁矿石品位分布的地质统计学研究

弓长岭铁矿地质条件复杂,使矿山生产长期受到制约。运用地质统计学的方法对该矿区的ＴＦｅ,ＦｅＯ品位分布规律进行了系统研究,确定了品位分布的结构模型,对今后矿山开发和生产提出了建议。     

The use of 137Cs as a tracer in an erosion study in south limburg (the Netherlands) and the influence of chernobyl fallout

The ¹³⁷Cs radioactivity of soils was used as a tracer of soil erosion in a catchment in the Netherlands: 143 samples were analysed to map the ¹³⁷Cs redistribution using geostatistical interpolation methods. Caesium-137 activities on grassland are significantly higher than on arable land. Also, ¹³⁷Cs activities on waning slopes are higher and activities on steep slopes are lower. The soil erosion estimates, derived from the ¹³⁷Cs data, are used to validate the USLE erosion model. The recent Chernobyl nuclear accident also contributed to the ¹³⁷Cs activity. However, the Chernobyl input of ¹³⁷Cs, with a constant ratio of 1.765:1 to ¹³⁴Cs, cannot be used as a tracer of soil erosion. Because of the rapid decay of ¹³⁴Cs, we will not be possible to separate the sources of ¹³⁷Cs in the near future in areas significantly influenced by Chernobyl fallout and in these areas ¹³⁷Cs can no longer be used as a soil erosion tracer.     

Statistical properties of accumulated values and their application in mining estimations

This paper presents results of some investigations on the statistical properties of accumulation of discrete as well as continuous values when a two-dimensional stationary random function is reduced to a one-dimensional function. Based on the results of these investigations, an indirect technique of estimating covariances or variograms of point support grade values is proposed. The efficiency of the technique, particularly in relation to the conventional procedure of covariance estimation, has been demonstrated by an actual case study of a pre-Cambrian high-grade iron ore deposit in eastern India.