Comparison of inference methods for estimating semivariogram model parameters and their uncertainty: The case of small data sets

The semivariogram model is the fundamental component in all geostatistical applications and its inference is an issue of significant practical interest. The semivariogram model is defined by a mathematical function, the parameters of which are usually estimated from the experimental data. There are important application areas in which small data sets are the norm; rainfall estimation from rain gauge data and transmissivity estimation from pumping test data are two examples from, respectively, surface and subsurface hydrology. Thus a benchmark problem in geostatistics is deciding on the most appropriate method for the inference of the semivariogram model.The various methods for semivariogram inference can be classified as indirect methods, in which there is an intermediate step of calculating the experimental semivariogram, and direct approaches that obtain the model parameter values directly as the values that minimize some objective function.To avoid subjectivity in fitting models to experimental semivariograms, ordinary least squares (OLS), weighted least squares (WLS) and generalized least squares (GLS) are often used. Uncertainty evaluation in indirect methods is done using computationally intensive resampling procedures such as the bootstrap method.Direct methods include parametric methods, such as maximum likelihood (ML) and maximum likelihood cross-validation (MLCV), and non-parametric methods, such as minimization of cross-validation statistics (CV).The bases for comparing the previous methods are the sampling distribution of the various parameters and the "goodness" of the uncertainty evaluation in a sense that we define. The final questions to be answered are (1) which is the best method for estimating each of the semivariogram parameters? (2) Which is the best method for assessing the uncertainty of each of the parameters? (3) Which method best selects the functional form of the semivariogram from among a set of options? and (4) which is the best method that jointly addresses all the previous questions? © 2012 Elsevier Ltd.Eulogio Pardo-Igúzquiza, Peter A.Dow

Geostatistical modelling of a coal seam for resource risk assessment

The evaluation of a coal seam for profitable extraction requires the estimation of its thickness and quality characteristics together with the spatial variability of these variables. In many cases the only data available for the estimation are from a limited number of exploration and feasibility drill holes. Spatial variability can be quantified by geostatistical modelling, which provides the basis for estimation (kriging). In cases where the spatial variability of the seam thickness and quality characteristics has a significant impact on how the coal is extracted and stored, geostatistical simulation may be preferable to geostatistical kriging methods. The aim of this paper is to present an improved approach to resource risk assessment by propagating the uncertainty in semi-variogram model parameters into the spatial variability of coal variables. We show that a more realistic assessment of risk is obtained when the uncertainty of semi-variogram model parameters is taken into account. The methodology is illustrated with a coal seam from North-western Spain. © 2012 Elsevier B.V.E. Pardo-Igúzquiza, P.A. Dowd, J.M. Baltuille, M. Chica-Olm

The imprint of global climate cycles in the Fuentillejo maar-lake record during the last 50 ka cal BP (central Spain)

We have analysed the geochemical (element analysis), mineralogical and sedimentary facies to characterize the sedimentary record in Fuentillejo maar-lake in the central Spanish volcanic field of Campo de Calatrava and thus be able to reconstruct the cyclicity of the sedimentary and paleoclimatic processes involved. The upper 20 m of core FUENT-1 show variations in clastic input and water chemistry in the lake throughout the last 50 ka cal BP. Being a closed system, the water level in this maar-lake depends primarily on the balance between precipitation and evaporatio

The spatial structure of lithic landscapes : the late holocene record of east-central Argentina as a case study

Fil: Barrientos, Gustavo. División Antropología. Facultad de Ciencias Naturales y Museo. Universidad Nacional de La Plata; ArgentinaFil: Catella, Luciana. División Arqueología. Facultad de Ciencias Naturales y Museo. Universidad Nacional de La Plata; ArgentinaFil: Oliva, Fernando. Centro Estudios Arqueológicos Regionales. Facultad de Humanidades y Artes. Universidad Nacional de Rosario; Argentin

EMLK2D: a computer program for spatial estimation using empirical maximum likelihood kriging

Copyright © 2004 Elsevier LtdThe authors describe a Fortran-90 program for empirical maximum likelihood kriging. More efficient estimates are obtained by solving the estimation problem in the ‘Gaussian domain’ (i.e., using the normal scores of the experimental data), where the simple kriging estimate is equivalent to the maximum likelihood estimate and to the conditional expectation. The transform to normality is done using the empirical cumulative probability distribution function. A Bayesian approach is adopted to ensure a conditionally unbiased estimate, which is obtained as the mean of the posterior distribution. The posterior distribution also provides a complete specification of the probability of the variable and thus provides the basis for a more realistic evaluation of uncertainty by various methods: inverting Gaussian confidence intervals, confidence intervals measured from the posterior distribution, variance measured from the posterior distribution or intervals obtained using the likelihood ratio statistic. A detailed case study is used to demonstrate the use of the program.Eulogio Pardo-Igúzquiza and Peter A. Dowdhttp://www.sciencedirect.com/science/journal/0098300

Estimating the boundary surface between geologic formations from 3D seismic data using neural networks and geostatistics

©2005 Society of Exploration GeophysicistsThe exact locations of horizons that separate geologic sequences are known only at physically sampled locations (e.g., borehole intersections), which, in general, are very sparse. 3D seismic data, on the other hand, provide complete coverage of a volume of interest with the possibility of detecting the boundaries between formations with, for example, contrasted acoustic impedance. Detection of boundaries is hampered, however, by coarse spatial resolution of the seismic data, together with local variability of acoustic impedance within formations. The authors propose a two-part approach to the problem, using neural networks and geostatistics. First, an artificial neural network is used for boundary detection. The training set for the neural net comprises seismic traces that are collocated with the borehole locations. Once the net is trained, it is applied to the entire seismic grid. Second, output from the neural network is processed geostatistically to filter noise and to assess the uncertainty of the boundary locations. A physical counterpart is interpreted for each structure inferred from the spatial semivariogram. Factorial kriging is used for filtering, and uncertainty in the shape of the boundaries is assessed by geostatistical simulation. In this approach, the boundary locations are interpreted as random functions that can be simulated to incorporate their uncertainty in applications. A case study of boundary detection between sandstone and breccia formations in a highly faulted zone is used to illustrate the methodologies.Peter A. Dowd and Eulogio Pardo-Igúzquiz

Assessment of the uncertainty of spatial covariance parameters of soil properties and its use in applications

Copyright © 2007, Lippincott Williams & WilkinsThe spatial variability of soil variables is a critical component of modelling, estimation, prediction and risk assessment in soil science. On one hand, spatial variability must be taken into account for optimal spatial interpolation (e.g., kriging) and risk assessment (e.g., evaluating the probability that the value of a given property is higher than an established threshold); on the other, spatial variability influences the output of physically based models (e.g., rainfall-runoff). As soil variables are usually known at only a small number of experimental locations, their spatial variability must be evaluated using statistical tools such as the spatial covariance (or the semivariogram), which, in turn, are modelled by a few parameters (e.g., nugget variance, sill, range). These covariance parameters, however, can only be inferred with an associated statistical uncertainty. The main objective of this paper is to show that this uncertainty can be assessed by a maximum likelihood approach to inference. Three different methods for obtaining joint confidence intervals of spatial covariance parameters are considered: (i) the Fisher information matrix, (ii) likelihood intervals, and (iii) the likelihood ratio statistic. The authors show that, when expressed as likelihood intervals, the confidence regions provided by the likelihood ratio statistic are especially suitable for applications because of its straightforward calculation. A second objective of the paper is to demonstrate how the uncertainty of the spatial covariance parameters can be included in applications. The interpolation uncertainty is included by Bayesian kriging whereas it is included in risk assessment and physically based models by geostatistical simulation. A case study of the zinc content of soils of the Swiss Jura is used to illustrate the methodology.Pardo-Iguzquiza, Eulogio; Dowd, Peter A