Geostatistics Valencia 2016

Inhaltsverzeichnis

1. Frontmatter

2. In Honor of Professor Danie Krige

   1. Frontmatter

   2. Professor Danie Krige’s First Memorial Lecture: The Basic Tenets of Evaluating the Mineral Resource Assets of Mining Companies, as Observed in Professor Danie Krige’s Pioneering Work Over Half a Century

      W. Assibey-Bonsu

      Abstract

      This paper provides a write-up of the first Professor Danie Krige memorial lecture in 2014, which was organised by the University of the Witwatersrand in collaboration with the Southern African Institute of Mining and Metallurgy (SAIMM) and the Geostatistical Association of Southern Africa, where his wife Mrs Ansie Krige, the SAIMM and Professor R.C.A. Minnitt also spoke. The memorial lecture was presented by his previous PhD graduate student, Dr Winfred Assibey-Bonsu.

      During that inaugural memorial lecture, the SAIMM highlighted three activities that the institute would hold going forward, so as to remember this great South African mining pioneer:

      • The publication of a Danie Krige Commemorative Volume of the SAIMM Journal
      • An annual Danie Krige Memorial Lecture to be facilitated by the School of Mining Engineering of the University of the Witwatersrand
      • The annual award of a Danie Krige medal

      What follows is both a tribute to his work and a testimony to the great man’s deep personal integrity, belief in family, humility and faith in Christ, all of which led him to become not only a giant in the South African mining industry but indeed worldwide.

3. Theory

   1. Frontmatter

   2. Functional Decomposition Kriging for Embedding Stochastic Anisotropy Simulations

      J. A. Vargas-Guzmán, B. Vargas-Murillo

      Abstract

      Functional analysis of the kriging algorithm is accomplished with consecutive projections of vectors in Hilbert space. The innovation unveils “functional decomposition kriging” (FDK), which can forecast fields on spatially continuous domains without using blocks, cells, or elements. FDK assembles the random field as a summation of field analytic functions, which are sample pivoted and nonstationary. Furthermore, spatially variable uncertain anisotropy is represented as a continuous tensor random field, which is formed from non-orthogonal members. FDK predicts tensor members using physical data collected at sparse sample locations. Particular interest is on structural anisotropy tensor fields representing curvilinear and folded patterns of structural uncertainty. Therefore, spatially variable eigenvector and eigenvalue tensor fields give continuously varying orientation and range of principal stochastic anisotropy of covariances that are used as input to stochastic functionals. FDK enables simulation of anisotropic properties (e.g., permeability, rock stiffness, or structural anisotropy), with stochastic covariance parameter fields. Integration of field analytic functions delivers upscaled multiresolution moments. Since FDK can be stopped, optimized, and updated without repeating computations, it is suitable for inverse, adaptive, and real-time modeling.

   3. Can Measurement Errors Be Characterized from Replicates?

      Chantal de Fouquet

      Abstract

      Sample measurements (of grade, depth, etc.) are almost inevitably affected by errors. Several error models were studied in the literature. But the interest of replicates for selecting the error model received limited attention. If measurement errors are supposed to be additive, homoscedastic, without correlation between them, and spatially not correlated with the exact values, the variances of the measurement errors are computable from the sample, simple, and cross-variograms of replicate data sets, even if the variogram of the exact value is pepitic (Aldworth W, Spatial prediction, spatial sampling, and measurement error. Retrospective Theses and Dissertations. Paper 11842. Iowa State University Digital Repository @ Iowa State University, 1998; Faucheux et al. Characterisation of a hydrocarbon polluted soil by an intensive multi-scale sampling. Geostats 2008, proceedings of the 8th international geostatistics congress, 1–5 Dec. 2008, Santiago, Chile. Ortiz J-M, Emery X (eds) for an example, 2008). But what about the other cases? When the error is additive, its correlation with the exact value can remain undetectable. The variance of the measurement errors is thus not always computable. It’s the same for an error of multiplicative type. Except in some special cases, keeping the different measurement values rather than their average improves the precision of the estimation.

   4. Modelling Asymmetrical Facies Successions Using Pluri-Gaussian Simulations

      Thomas Le Blévec, Olivier Dubrule, Cédric M. John, Gary J. Hampson

      Abstract

      An approach to model spatial asymmetrical relations between indicators is presented in a pluri-Gaussian framework. The underlying gaussian random functions are modelled using the linear model of co-regionalization, and a spatial shift is applied to them. Analytical relationships between the two underlying gaussian variograms and the indicator covariances are developed for a truncation rule with three facies and cut-off at 0. The application of this truncation rule demonstrates that the spatial shift on the underlying gaussian functions produces asymmetries in the modelled 1D facies sequences. For a general truncation rule, the indicator covariances can be computed numerically, and a sensitivity study shows that the spatial shift and the correlation coefficient between the gaussian functions provide flexibility to model the asymmetry between facies. Finally, a case study is presented of a Triassic vertical facies succession in the Latemar carbonate platform (Dolomites, Northern Italy) composed of shallowing-upward cycles. The model is flexible enough to capture the different transition probabilities between the environments of deposition and to generate realistic facies successions.

   5. Considerations for the Use of Sequential Sampling Techniques

      J. Leguijt

      Abstract

      Sequential sampling is a well-known and efficient method to generate probabilistic realisations of models that are constrained by two-point statistics.

      These two-point statistics consist of second-order moments that are defined by a variogram. The statistics describe the lateral continuity behaviour of the models. It can be shown that the sequential sampling method correctly generates samples from a probability density function (pdf), when this pdf honours only the statistics that define the lateral continuity constraints. In Bayesian statistics, this is named a prior pdf. The sequential sampling method is also used to generate models from a probability density function that is constrained by observations, similar to those that are derived from seismic data. This is known as a posterior pdf. To justify this approach, some assumptions have to be made that are not strictly valid and the result is often a significant error. The errors will be investigated using a realistic synthetic example. The probabilistic seismic inversion programme that has been developed by Shell contains a module that is able to account for lateral continuity. In this module, an alternative approach has been used to mitigate the problems with the sequential sampling method. To realise this, each location needs to be visited repeatedly.

   6. A New High-Order, Nonstationary, and Transformation Invariant Spatial Simulation Approach

      Amir Abbas Haji Abolhassani, Roussos Dimitrakopoulos, Frank P. Ferrie

      Abstract

      This paper presents a new high-order, nonstationary sequential simulation approach, aiming to deal with the typically complex, curvilinear structures and high-order spatial connectivity of the attributes of natural phenomena. Similar to multipoint methods, the proposed approach employs spatial templates and a group of training images (TI). A coarse template with a fixed number of data points and a missing value in the middle is used, where the missing value is simulated conditional to a data event found in the neighborhood of the middle point of the template, under a Markovian assumption. Sliding the template over the TI, a pattern database is extracted. The parameters of the conditional distributions needed for the sequential simulation are inferred from the pattern database considering a set of weights of contribution given for the patterns in the database. Weights are calculated based on the similarity of the high-order statistics of the data event of the hard data compared to those of the training image. The high-order similarity measure introduced herein is effectively invariant under all linear spatial transformations.

      Following the sequential simulation paradigm, the template chosen is sequentially moved on a raster path until all missing points/nodes are simulated. The high-order similarity measure allows the approach to be fast as well as robust to all possible linear transformations of a training image. The approach respects the hard data and its spatial statistics, because it only considers TI replicate data events with similar high-order statistics. Results are promising.

   7. A Truly Multivariate Normal Score Transform Based on Lagrangian Flow

      Ute Mueller, K. Gerald van den Boogaart, Raimon Tolosana-Delgado

      Abstract

      In many geostatistical applications, a transformation to standard normality is a first step in order to apply standard algorithms in two-point geostatistics. However, in the case of a set of collocated variables, marginal normality of each variable does not imply multivariate normality of the set, and a joint transformation is required. In addition, current methods are not affine equivariant, as should be required for multivariate regionalized data sets without a unique, canonical representation (e.g., vector-valued random fields, compositional random fields, layer cake models). This contribution presents an affine equivariant method of Gaussian anamorphosis based on a flow deformation of the joint sample space of the variables. The method numerically solves the differential equation of a continuous flow deformation that would transform a kernel density estimate of the actual multivariate density of the data into a standard multivariate normal distribution. Properties of the flow anamorphosis are discussed for a synthetic application, and the implementation is illustrated via two data sets derived from Western Australian mining contexts.