2012
Abstract. Although stationary phenomena are characterized by their covariance, the structural tool is the variogram. In the case of nonstationary phenomena, the variogram is not experimentally accessible, so that blind methods are often used for the identification of the ordinary or generalized covariance. However, if regularly spaced data are available (at grid nodes or along profiles), the generalized variogram can be computed. It filters any polynomial trend up to a given degree. Its relation with the covariance is nearly as simple as for the ordinary variogram. The theory of the generalized variogram is presented: definition, relation with the ordinary or generalized covariance, estimation and fluctuations in the Gaussian and non-Gaussian cases.
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Davila Saavedra, Luis A · University of Alberta Library · 2024
The variogram characterizes the spatial variability of a regionalized variable. Variogram calculation and modeling require a significant amount of professional time. The variogram has a significant impact on estimation and simulation, and it has a considerable impact in the mining industry since ordinary kriging is one of the most used tools for resource estimation. This thesis proposes novel tools and techniques related to automatic variogram calculation. The first contribution of this thesis is implementing a GSLIB-like program: autovar. This program calculates the variograms for disseminated and tabular deposits. Autovar prompts the user for some basic parameters only. Autovar implements all necessary steps to get the experimental variogram points and its model directly from the data. The program infers the principal directions of continuity by eigendecomposition of the inertia tensor, followed by an order of the vectors. Then, the program calculates the experimental variogram points and models them. This thesis considers standard guidelines for variogram calculation and modeling. Different search parameters are implemented based on the deposit type. An important functionality of the program is the geolocated option, which allows the identification of variability centred at points (anchor locations) selected by the practitioner. The second contribution is a novel technique to estimate a block model using a mix of ordinary kriging models. This mixed estimation technique considers several block kriging models and mixes them at each node of the block model. Each estimated model has a different variogram as input, reflecting different continuity characteristics. The variograms used in the kriging estimations are outputs from the autovar program: a general variogram and geolocated variograms centred at each selected anchor location. The weights are calculated based on the distance from each node to the anchor points. The postprocessing step standardizes the weights to add up to one, ensuring that kriging is exact. Generally, this technique recognizes local variability better than a regular technique. This thesis presents different examples to show the potential and limitations of the proposed methodologies. This research presents a comparison of the proposed workflows with regular methodologies. Finally, a summary of tests is shown at the end to compare the results of the mixed estimation methodology with a regular estimation workflow.
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Falk Heße · Zenodo (CERN European Organization for Nuclear Research) · 2022
This is a data set that contains world-wide estimates of variogram of covariancve functions for aquifer and soil sites.
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Alison M. Flanagan (4759584), Roger D. Flood (4759587), Michael G. Frisk (419576) et al. · Figshare · 2018
<p>Symbols refer to model parameters in Eqs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189313#pone.0189313.e002" target="_blank">2</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189313#pone.0189313.e006" target="_blank">6</a>.</p>
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Hans Wackernagel · 1995
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Mosobalaje, O. O., Orodu, O. D., Ogbe, David O. · Covenant University Repository (Covenant University) · 2019
Deposits of heavy oil and natural bitumen have been long-discovered in the Dahomey basin south-western Nigeria. However, inconsistency in estimates of volumes of hydrocarbon contained in these deposits has inhibited commercial interest in the deposits. The inconsistency is attributable to the little or no consideration for spatial variability in those studies. This work is therefore motivated by the need for spatially-coherent geomodels leading to reliable volumetric estimates. An existing database of porosity, depth-to-top and thickness attributes of a section of the deposits located at Agbabu is the subject of this work. This work conducted exploratory spatial data analysis (ESDA) as well as empirical variogram estimation, interpretation and modeling of the attributes. Here, the estimation and interpretation of empirical variogram faced a number of challenges with potentials to render the estimates uninterpretable, unstable and inconsistent with geologic information. These include presence of spatial outlier data, clusteredness of variogram clouds, data paucity, and irregular distribution of point-pairs on variogram clouds. Spatial outliers were either removed or correlated with existing geologic information. The clusteredness issues were resolved using a machine-learning – aided variogram estimation technique recently proposed. Variogram cloud binning approach was deployed to handle irregular distribution of point-pairs. In attempting to deploy an automatic fitting algorithm, cases of insufficient empirical points leading to lack of convergence were encountered. Such cases were resolved by adopting a combination of manual and automatic fitting approaches. Ultimately, this work presents a three-dimensional anisotropic (zonal) porosity variogram model and two-dimensional anisotropic (geometric) models for the depth-to-top and thickness variograms. These models are suitable inputs to spatial interpolation algorithms in generating maps of these volumetric attributes.
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Hans Wackernagel · 1995
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Catherine A. Calder, Noel Cressie · Elsevier eBooks · 2009
Cited by 40
Manuel Ribeiro (17889313), Leonardo Azevedo (17889316), André Peralta Santos (5629808) et al. · Figshare · 2024
<p>The exponential variogram model was fit to trace-variogram estimates (blue filled circles) used to weight the dissimilarity matrix in hierarchical clustering (as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0297772#pone.0297772.e009" target="_blank">Eq 7</a>). The x-axis refers to the Euclidean distance between pairs of municipalities, while the y-axis refers to the trace-variogram function values. The nugget-effect is the y-intercept. As distance increases, the values increase asymptotically up to the sill (nugget-effect + partial sill), after which the municipalities are considered to be no longer correlated.</p>
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Genton, Marc · Infoscience (Ecole Polytechnique Fédérale de Lausanne) · 1996
In the context of spatial statistics, the classical variogram estimator proposed by Matheron is not robust against outliers in the data, nor is Cressie and Hawkins' estimator. Therefore, we suggest the use of a variogram estimator based on a highly robust estimator of scale. The robustness properties of these three estimators are analysed and compared. Several simulations are carried out from a spherical underlying variogram, with various amounts of outliers in the data. Results show that the highly robust variogram estimator improves the estimation significantly
Cited by 6Open access
Luca Bonaventura, Stefano Castruccio, Laura M. Sangalli et al. · 2009
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Orlíková, Lucie · DSpace VŠB-TUO (VŠB-TUO)
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Zhen Liu (74646), Ye Guo (224500), Shuheng Du (4033172) et al. · Figshare · 2017
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M. Z. Rashad, Gad El-QADY, Keisuke Ushijima · Medical Entomology and Zoology · 2000
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2010
For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR 52.227-19, Commercial Computer Software-Restricted Rights (June 1987).
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Hans Wackernagel · 2003
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Dean Price, Majdi I. Radaideh, Brendan Kochunas · Deep Blue (University of Michigan) · 2020
Presented at the 2020 University of Michigan Data Science Symposium
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Tatsiana V. Tsekhavaya · SHILAP Revista de lepidopterología · 2018
Properties of the semivariogram of an intrinsically stationary continuous-time random process with finite second moment are investigated. A necessary and sufficient conditions for a continuous function to be semivariogram are found. Confidence intervals for the semivariogram of Gaussian stationary stochastic process are defined. Properties of ꭓ2-distribution are used for constructing confidence intervals for semivariogram. The proposed confidence intervals are more informative compared with point estimates of the semivariogram.
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Nhut Truong, G.B.M. Heuvelink, John Paul Gosling · Socio-Environmental Systems Modeling · 2012
The variogram is the keystone of Kriging. Much research has been devoted to modelling the variogram from structural analysis of observations. However, there are many instances when the variogram is needed and there are no observations to base on. This can be due to budget constraints, physical or temporal obstacles, or a demand for a priori variogram in Bayesian geostatistics and spatial sampling design. Using expert knowledge that is elicited with a formal statistical expert elicitation procedure is suggested. In this study, we designed a protocol for a wellstructured elicitation procedure to elicit the variogram from expert knowledge. The protocol has two main stages: elicitation of the marginal probability distribution and elicitation of the variogram. We built a web-based tool to facilitate the procedure. A case study was carried out to quantify spatial accuracy of a legacy map of the volumetric soil water content at field capacity. Expert elicitation returns a variogram that has a Matérn model shape with nugget = 0.45, partial sill = 54.57, range = 25410 meters and kappa = 0.40. The results show that the online elicitation tool satisfactorily captured expert opinion although, currently, it is only a prototype that needs to be further developed.
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Mary-Huard, Tristan, Jean-Jacques Daudin, Robin, Stéphane et al. · Figshare · 2011
<b>Copyright information:</b>Taken from "Spotting effect in microarray experiments"BMC Bioinformatics 2004;5():63-63.Published online 19 May 2004PMCID:PMC434492.Copyright © 2004 Mary-Huard et al; licensee BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL.e. corrected signal) after reference normalization, for the Arabidopsis slide
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