Comment on “Indicator principal component kriging” by V. Suro-Pérez and A. G. Journel

Conclusions The indicators covariance is equivalent to the bivariate distribution of the class index. The factor decomposition of the indicator covariance proposed by SPJ is an isofactorial model of the corresponding bivariate distribution. However, the choices of cumulative indicators and of principal component analysis produce unacceptable inconsistencies. In LL, a correspondence analysis of the bivariate distribution was used to produce more satisfactory empirical factors. These were used in the procedure of identification of discrete isofactorial models, with improved consistency, and the benefit of change of support models.     

Change of support for diffusion-type random functions

Diffusion-type random functions are a first attempt at a non-Markovian and multidimensional generalization of the Ito stochastic integral theory. Within this framework, the variation f of the pdf can be evaluated for a small change of support. These results are compared with the provisions of the usual approximate models. The conclusions are: the affine correction is false for the first order approximation, unless there is a linear factor. The isofactorial model is true for the first order, and in the multi-Gaussian case, almost correct for the second-order approximation. Counterexamples are given for the discontinuous case, and a more general model is suggested.     

Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.     

Isofactorial models for granulodensimetric data

Existing isofactorial models developed for disjunctive kriging using a cutoff grade on one variable are extended to the bivariate case which arises when dealing with granulo-densimetric data, such as are obtained from coal washing or mineral processing.     

Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions

This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

On Some Consistency Conditions for Geostatistical Change-of-Support Models

This work deals with a family of geostatistical models used in application fields concerned with a change of support, such as mineral resources evaluation and polluted soil management. Three models are examined: the discrete Gaussian, Hermitian and Laguerrian models, which rely on a transformation of the variable of interest (mineral grade or pollutant concentration) defined on point and block supports into variables with Gaussian or gamma univariate distributions and isofactorial bivariate distributions. The focus is given to the relationships between the transformed variables at both supports, and to the conditions that these relationships imply on the model parameters. Additionally, guidelines are given for improving the variogram analysis of the transformed variables and for validating the change-of-support model.     

Application of disjunctive kriging for estimating economic grade distribution in an iron ore deposit: A case study of the Choghart North Anomaly,Iran

Most significant iron ore deposits in Iran are located in Central Iran Zone. These deposits belong to the Bafq mining district. The Bafq mining district is located in the Early Cambrian Kashmar-Kerman volcanic arc of Central Iran. Linear estimation of regionalized variables (for example by inverse distance weighting or ordinary Kriging) results in relatively high estimation variances, i.e. the estimates have very low precision. Assessment of project economics (or other critical decision making) based on linear estimation is therefore risky. Non-linear estimation methods like disjunctive kriging perform better and the lower estimation variance allows less risky economic decision-making. Another advantage of disjunctive kriging is that it allows estimation of functions of the primary variable, which here is the grade (Fe %) of the ore. In particular it permits estimation of indicator functions defined using thresholds on the primary variable. This paper is devoted to application of disjunctive kriging method in Choghart North Anomaly iron ore deposit in Central Iran, Yazd province, Iran. In this study, the Fe concentration of Choghart North Anomaly iron ore deposit was modelled and estimated. The exploration data consists of borehole samples measuring the Fe concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to estimate the regionalized variable (Fe %) at unsampled locations and to assess the probabilities that the actual concentrations exceed a threshold value at a given location. Consequently a three dimensional model of probability of exceeding a threshold value and the estimated value are provided by disjunctive kriging to divide the ore into an economic and uneconomic part on the basis of estimation of indicator functions using thresholds grades defined on point support. The tools and concepts are complemented by a set of computer programs that are applied to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade of an ore deposit. Results showed that the correlation between the estimated value and real value at locations close to each other is 81.9%.     

Inverse methods and kinetic models of hydrocarbon generation. II. Case histories for residual kerogen analysis

Using the results of the first paper in this series, which dealt with an inverse procedure for hydrocarbon generation, this paper focuses on case histories and numerical implementation of the model. Experimental data on residual kerogen analysis have been used with the inverse procedure to examine the applicability of the model to hydrocarbon generation directly from the degradation of kerogen, and to derive the kinetic parameters for the construction of a prediction equation. Energy-dependent channeling probability distributions are used to exhibit the stability of the model in describing the processes of kerogen degradation. The numerical methods for the solution of the inverse procedure and a sensitivity analysis are also discussed.     

在1:5万成矿预测中建立综合信息预测模型

本文研究了一种用于1:5万成矿预测的专属性模型——矿田综合信息预测模型。对建模基本思路、建模工作流程和建模中的几个关键技术问题(地质概念模型的建立、信息的提取与综合、成矿规律的分析、模型图件的编制)进行了总结,并以实例说明在1:5万成矿预测中建立和应用这种模型可大大提高预测的可靠性。     

Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix

The geostatistical analysis of multivariate data involves choosing and fitting theoretical models to the empirical matrix. This paper considers the specific case of the model of linear coregionalization, and describes an automated procedure for fitting models, that are adequate in the mathematical sense, using a least-squares like technique. It also describes how to decide whether the number of parameters of the cross-variogram matrix model should be reduced to improve stability of fit. The procedure is illustrated with an analysis of the spatial relations among the physical properties of an alluvial soil. The results show the main influence of the scale and the shape of the basic models on the goodness of fit. The choice of the number of basic models appears of secondary importance, though it greatly influences the resulting interpretation of the coregionalization analysis.     

表碛下冰面消融模型的改进

针对已有的表碛下冰面消融模型的缺点,通过对模型算法、模型假设和模型结构等方面进行改进,在模型的易用性和模拟精度上有了较大的提高.科其喀尔冰川的实例研究表明,改进模型对于表碛区不同试验点上地温的模拟是较好的,变化趋势基本一致,变化幅度也相当.但也存在模拟值与实测值之间变化相位差异的突出问题,实际操作可以通过适当的校正程序得以较好的解决.模型中对表碛热参数的常数化处理不会产生较大的模拟误差,此外计算开始时均一表碛温度的假设不会对表碛温度的整体模拟效果产生较大的影响.通过地温模拟得到的冰面消融速率与利用实测表碛热通量计算的冰面消融速率的比较说明,模型思路合理且算法可行.     

高压旋喷桩复合土钉墙Plaxis有限元分析

根据高压旋喷桩复合土钉墙研究现状,提出了有限元数值分析软件Plaxis8．2选择理由,论述了土体本构模型选择原则,支护结构材料模型选择原则,参数取值原则,分步计算过程原则。重点给出了高压旋喷桩、土钉、砼面层相关参数的计算公式。通过代表性的示例研究了高压旋喷桩贡献作用规律。最后通过基坑支护实例,介绍了该程序在高压旋喷桩复合土钉墙设计方面的应用和验证情况。     

Micro-scale flood risk analysis based on detailed 2D hydraulic modelling and high resolution geographic data

The paper presents a consistent micro-scale flood risk analysis procedure, relying on detailed 2D inundation modelling as well as on high resolution topographic and land use database. The flow model is based on the shallow-water equations, solved by means of a finite volume scheme on multi-block structured grids. Using highly accurate laser altimetry, the simulations are performed with a typical grid spacing of 2 m, which is fine enough to represent the flow at the scale of individual buildings. Consequently, the outcomes of hydraulic modelling constitute suitable inputs for the subsequent exposure analysis, performed at a micro-scale using detailed land use maps and geographic database. Eventually, the procedure incorporates social flood impact analysis and evaluation of direct economic damage to residential buildings. Besides detailing the characteristics and performance of the hydraulic model, the paper describes the flow of data within the overall flood risk analysis procedure and demonstrates its applicability by means of a case study, for which two different flood protection measures were evaluated.     

The role of observations in the inverse analysis of landslide propagation

Model calibration is usually based on trial-and-error procedures that, in turn, rely on expert judgment or previously acquired experiences for similar phenomena. Efficient and reliable procedures for model calibration of the propagation stage of landslides are still needed. This paper addresses this issue by proposing an inverse analysis procedure and applying it to the case history of a short run-out landslide triggered by a rising perched water table after a heavy rainfall. It focuses on the key role played by the field observations used to set up the inverse analysis, and evaluating the reliability of the numerical simulations. It also investigates the effect of different types of optimization parameters on the inverse analysis results, referring to a mixed-phase model or to a two-phase model for the propagating soil. Several sets of observations are used; all of them refer to the soil deposit thickness at the end of propagation, but differ in both location and number of the adopted values. The numerical analysis of the case history is performed through the academic “GeoFlow_SPH” model, and model calibration by inverse analysis is conducted using the “UCODE” software. The results obtained are discussed with the aim to provide practical criteria to identify the minimum amount of information required for a satisfactory model calibration.     

A new data assimilation procedure to develop a debris flow run-out model

Parameter calibration is one of the most problematic phases of numerical modeling since the choice of parameters affects the model’s reliability as far as the physical problems being studied are concerned. In some cases, laboratory tests or physical models evaluating model parameters cannot be completed and other strategies must be adopted; numerical models reproducing debris flow propagation are one of these. Since scale problems affect the reproduction of real debris flows in the laboratory or specific tests used to determine rheological parameters, calibration is usually carried out by comparing in a subjective way only a few parameters, such as the heights of soil deposits calculated for some sections of the debris flows or the distance traveled by the debris flows using the values detected in situ after an event has occurred. Since no automatic or objective procedure has as yet been produced, this paper presents a numerical procedure based on the application of a statistical algorithm, which makes it possible to define, without ambiguities, the best parameter set. The procedure has been applied to a study case for which digital elevation models of both before and after an important event exist, implicating that a good database for applying the method was available. Its application has uncovered insights to better understand debris flows and related phenomena.     

An efficient probabilistic back-analysis method for braced excavations using wall deflection data at multiple points

This paper presents an efficient Bayesian back-analysis procedure for braced excavations using wall deflection data at multiple points. Response surfaces obtained from finite element analyses are adopted to efficiently evaluate the wall responses. Deflection data for 49 wall sections from 11 case histories are collected to characterize the model error of the finite element method for evaluating the deflections at various points. A braced excavation project in Hang Zhou, China is chosen to illustrate the effectiveness of the proposed procedure. The results indicate that the soil parameters could be updated more significantly for the updating that uses the deflection data at multiple points than that only uses the maximum deflection data. The predicted deflections from the updated parameters agree fairly well with the field observations. The main significance of the proposed procedure is that it improves the updating efficiency of the soil parameters without adding monitoring effort compared with the traditional method that uses the maximum deflection data.     

Shear wave propagation along infinite slopes: A theoretically based numerical study

In this paper, the seismic response of ‘infinitely’ long slopes is numerically analysed via the formulation of a 1D analytical/numerical model, in which the soil mechanical behaviour is assumed to be elasto‐perfectly viscoplastic and simple shear (SS) kinematical constraints are imposed. In order to simplify the problem, a theoretically based procedure to set up a fully 1D shear constitutive model is defined, within which the mechanical response of a multiaxial relationship is condensed. The use of a 1D shear constitutive model is aimed at reducing the number of unknowns and, therefore, the computational costs. In particular, the case of the Mohr–Coulomb yield criterion is considered, while an enhanced Taylor–Galerkin finite element algorithm is employed to simulate the seismic wave propagation within the soil stratum. The proposed ‘condensation’/calibration procedure captures both the ‘pseudo’‐hardening pre‐failure behaviour and the influence of dilation on the occurrence of strain‐localization, which characterize, under SS conditions, the static response of virgin perfectly plastic soils. The effectiveness of the conceived method is shown with reference to freshly deposited deposits, while, in the case of highly overconsolidated strata, some difficulties arise because of the brittle behaviour induced both by unloading and non‐associativeness. Copyright © 2011 John Wiley & Sons, Ltd.