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Analogies and correspondences between variograms and covariance functions

Part of: Geophysics (general) Geophysics Harmonic analysis in one variable Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Tilmann Gneiting ,
Zoltán Sasvári  and
Martin Schlather
Tilmann Gneiting*
Affiliation:
University of Washington
Zoltán Sasvári*
Affiliation:
Technische Universität Dresden
Martin Schlather*
Affiliation:
Universität Bayreuth
*
∗ Postal address: University of Washington, Department of Statistics, Box 354322, Seattle, Washington 98195-4322, USA. Email address: [email protected]
∗∗ Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Mommsenstr. 13, 01602 Dresden, Germany.
∗∗∗ Postal address: Universität Bayreuth, Abteilung Bodenphysik, 95440 Bayreuth, Germany.
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Abstract

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

Keywords

Conditionally negative definitegeostatisticsisotropiclocally equivalent stationary covariancepositive definitevariogram

MSC classification

Primary: 86A32: Geostatistics
Secondary: 42A82: Positive definite functions 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 617 - 630
Copyright
Copyright © Applied Probability Trust 2001 

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