Variogram model selection

2002

Geostatistics is a set of statistical techniques that is increasingly used to characterize spatial dependence in spatially referenced ecological data. A common feature of geostatistics is predicting values at unsampled locations from nearby samples using the kriging algorithm. Modeling spatial dependence in sampled data is necessary before kriging and is usually accomplished with the variogram and its traditional estimator. Other types of estimators, known as non-ergodic estimators, have been used in ecological applications. Non-ergodic estimators were originally suggested as a method of choice when sampled data are preferentially located and exhibit a skewed frequency distribution. Preferentially located samples can occur, for example, when areas with high values are sampled more intensely than other areas. In earlier studies the visual appearance of variograms from traditional and non-ergodic estimators were compared. Here we evaluate the estimators' relative performance in prediction. We also show algebraically that a non-ergodic version of the variogram is equivalent to the traditional variogram estimator. Simulations, designed to investigate the effects of data skewness and preferential sampling on variogram estimation and kriging, showed the traditional variogram estimator outperforms the non-ergodic estimators under these conditions. We also analyzed data on carabid beetle abundance, which exhibited large-scale spatial variability (trend) and a skewed frequency distribution. Detrending data followed by robust estimation of the residual variogram is demonstrated to be a successful alternative to the non-ergodic approach.

Mathematical Geology, 2001

tics 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.

Revista Interdisciplinar de Pesquisa em Engenharia - RIPE, 2017

1997

The variogram plays a central role in the analysis of geostatistical data. A valid variogram model is selected and the parameters of that model are estimated before kriging (spatial prediction) is performed. These inference procedures are generally based upon examination of the empirical variogram, which consists of average squared dierences of data taken at sites lagged the same distance apart in the same direction. The ability of the analyst to estimate variogram parameters eciently is aected signi®cantly by the sampling design, i.e. the spatial con®guration of sites where measurements are taken. In this paper, we propose design criteria that, in contrast to some previously proposed criteria oriented towards kriging with a known variogram, emphasize the accurate estimation of the variogram. These criteria are modi®cations of design criteria that are popular in the context of (nonlinear) regression models. The two main distinguishing features of the present context are that the addition of a single site to the design produces as many new lags as there are existing sites and hence also produces many new squared dierences from which the variogram is estimated. Secondly, those squared dierences are generally correlated, which inhibits the use of many standard design methods that rest upon the assumption of uncorrelated errors. Several approaches to design construction which account for these features are described and illustrated with two examples. We compare their eciency to simple random sampling and regular and space-®lling designs and ®nd considerable improvements.

Environmetrics, 1999

The variogram plays a central role in the analysis of geostatistical data. A valid variogram model is selected and the parameters of that model are estimated before kriging (spatial prediction) is performed. These inference procedures are generally based upon examination of the empirical variogram, which consists of average squared dierences of data taken at sites lagged the same distance apart in the same direction. The ability of the analyst to estimate variogram parameters eciently is aected signi®cantly by the sampling design, i.e. the spatial con®guration of sites where measurements are taken. In this paper, we propose design criteria that, in contrast to some previously proposed criteria oriented towards kriging with a known variogram, emphasize the accurate estimation of the variogram. These criteria are modi®cations of design criteria that are popular in the context of (nonlinear) regression models. The two main distinguishing features of the present context are that the addition of a single site to the design produces as many new lags as there are existing sites and hence also produces many new squared dierences from which the variogram is estimated. Secondly, those squared dierences are generally correlated, which inhibits the use of many standard design methods that rest upon the assumption of uncorrelated errors. Several approaches to design construction which account for these features are described and illustrated with two examples. We compare their eciency to simple random sampling and regular and space-®lling designs and ®nd considerable improvements.

In Proceedings of the 3th International Seminar on Geology for the Mining Industry GEOMIN 2013. Ambrus, J. Beniscelli, J. Brunner, F. Cabello, J. Ibarra, F. (eds.), Gecamin Ltda, Santiago, Chile, 7 p

In geostatistical data analysis, the assessment of spatial variability through sample variograms is a traditional and often difficult problem. In many situations, the scarcity of data, the heavy-tailed data distributions and the presence of preferential and/or irregular sampling designs lead to considerable fluctuations in the sample variogram, making the fitting of a model and the subsequent spatial prediction process complex. To attenuate such fluctuations, an interesting approach that has been proposed in recent literature is to calculate weighted sample variograms, in which the weighting depends on the spatial configuration of the data. This paper compares three approaches to the calculation of weighted sample variograms in the three-dimensional space. The first one corresponds to the traditional sample variogram, for which each data pair receives the same weight. In the second approach, the weight assigned to a data pair is proportional to the product of the declustering weights assigned to each individual data. The last approach applies the declustering method directly to the data pairs, by working in a six-dimensional space. Each approach is tested on two sampling designs -a regular one and a highly irregular oneand applied to multiple sets of simulated data for which the true variogram is known. For each data set and sampling design, the relative estimation error (difference between the true variogram and the calculated sample variogram, relative to the true variogram value) is assessed.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 2016

Abstract—Within geostatistics research, effective estimation of the variogram points has been examined, particularly in developing robust alternatives. The parametric fit of these variogram points which eventually defines the kriging weights, however, has not received the same attention from a robust perspective. This paper proposes the use of the non-linear Wilcoxon norm over weighted non-linear least squares as a robust variogram fitting alternative. First, we introduce the concept of variogram estimation and fitting. Then, as an alternative to non-linear weighted least squares, we discuss the non-linear Wilcoxon estimator. Next, the robustness properties of the non-linear Wilcoxon are demonstrated using a contaminated spatial data set. Finally, under simulated conditions, increasing levels of contaminated spatial processes have their variograms points estimated and fit. In the fitting of these variogram points, both non-linear Weighted Least Squares and non-linear Wilcoxon fits a...

Stochastic Environmental Research and Risk Assessment, 2010

This study adds to our ability to predict the unknown by empirically assessing the performance of a novel geostatistical-nonparametric hybrid technique to provide accurate predictions of the value of an attribute together with locally-relevant measures of prediction confidence, at point locations for a single realisation spatial process. The nonstationary variogram technique employed generalises a moving window kriging (MWK) model where classic variogram (CV) estimators are replaced with information-rich, geographically weighted variogram (GWV) estimators. The GWVs are constructed using kernel smoothing. The resultant and novel MWK-GWV model is compared with a standard MWK model (MWK-CV), a standard nonlinear model (Box-Cox kriging, BCK) and a standard linear model (simple kriging, SK), using four example datasets. Exploratory local analyses suggest that each dataset may benefit from a MWK application. This expectation was broadly confirmed once the models were applied. Model performance results indicate much promise in the MWK-GWV model. Situations where a MWK model is preferred to a BCK model and where a MWK-GWV model is preferred to a MWK-CV model are discussed with respect to model performance, parameterisation and complexity; and with respect to sample scale, information and heterogeneity.

Environmetrics, 1995

Variograms are used to describe the spatial variability of environmental variables. In this study, the parameters that characterize the variogram are obtained from a variogram in a different but comparably polluted area. A procedure is presented for improving the variogram modelling when data become available from the area of interest. Interpolation is carried out by means of a Bayesian form of kriging, where prior distributions of the variogram parameters are used. This procedure differs from current procedures, since commonly applied least squares estimation for the variogram is avoided. The study is illustrated with data from a cadmium pollution in the Netherlands, where this form of extrapolation was compared with ordinary kriging. When sufficient data are available (more than 140), ordinary kriging gave the most precise predictions. When the number of data was small (i.e. less than 60), predictions obtained with Bayesian kriging were more precise as compared to those obtained with ordinary kriging. This leads to a considerable reduction of costs, without loss of information.