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

Variance-Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.Pardo-Igúzquiza E. and Dowd P

IRFK2D: a computer program for simulating intrinsic random functions of order k

Copyright © 2003 Elsevier Science LtdIRFK2D is an ANSI Fortran-77 program that generates realizations of an intrinsic function of order k (with k equal to 0, 1 or 2) with a permissible polynomial generalized covariance model. The realizations may be non-conditional or conditioned to the experimental data. The turning bands method is used to generate realizations in 2D and 3D from simulations of an intrinsic random function of order k along lines that span the 2D or 3D space. The program generates two output files, the first containing the simulated values and the second containing the theoretical generalized variogram for different directions together with the theoretical model. The experimental variogram is calculated from the simulated values while the theoretical variogram is the specified generalized covariance model. The generalized variogram is used to assess the quality of the simulation as measured by the extent to which the generalized covariance is reproduced by the simulation. The examples given in this paper indicate that IRFK2D is an efficient implementation of the methodology.Eulogio Pardo-Igúzquiza and Peter A. Dowdhttp://www.sciencedirect.com/science/journal/0098300