# Anisotropy Models for Spatial Data

## Abstract

This work addresses the question of building useful and valid models of anisotropic variograms for spatial data that go beyond classical anisotropy models, such as the geometric and zonal ones. Using the concept of principal irregular term, variograms are considered, in a quite general setting, having regularity and scale parameters that can potentially vary with the direction. It is shown that if the regularity parameter is a continuous function of the direction, it must necessarily be constant. Instead, the scale parameter can vary in a continuous or discontinuous fashion with the direction. A directional mixture representation for anisotropies is derived, in order to build a very large class of models that allow to go beyond classical anisotropies. A turning band algorithm for the simulation of Gaussian anisotropic processes, obtained from the mixture representation, is then presented and illustrated.

## Keywords

Anisotropy Covariance Isotropy Spatial statistics Turning band method Variogram## References

- Bevilacqua M, Gaetan C, Mateu J, Porcu E (2012) Estimating space and space-time covariance functions for large data sets: a weighted composite likelihood approach. J Am Stat Assoc 107:268–280CrossRefGoogle Scholar
- Biermé H, Moisan L, Richard F (2014) A turning-band method for the simulation of anisotropic fractional Brownian fields. J Comput Graph Stat. doi: 10.1080/10618600.2014.946603 Google Scholar
- Chiles JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. John Wiley & Sons, New-YorkCrossRefGoogle Scholar
- Daley DJ, Porcu E (2014) Dimension walks through Schoenberg spectral measures. Proc Am Math Soc 142:1813–1824CrossRefGoogle Scholar
- Davies S, Hall P (1999) Fractal analysis of surface roughness by using spatial data. J R Stat Soc B 61:3–37CrossRefGoogle Scholar
- Dowd A, Igúzquiza E (2012) Geostatistical analysis of rainfall in the West African Sahel. In: Gómez-Hernández J (ed) IX Conference on Geostatistics for Environmental Applications, geoENV2012, Editorial Universitat Politècnica de València, pp 95–108Google Scholar
- Eriksson M, Siska P (2000) Understanding anisotropy computations. Math Geol 32:683–700CrossRefGoogle Scholar
- Gneiting T (1998) Closed form solutions of the two-dimensional turning bands equation. Math Geol 30: 379–390CrossRefGoogle Scholar
- Gneiting T, Sasvari Z, Schlather M (2001) Analogies and correspondences between variograms and covariance functions. Adv Appl Probab 33:617–630CrossRefGoogle Scholar
- Journel A, Froidevaux R (1982) Anisotropic hole-effect modeling. Math Geol 14:217–239CrossRefGoogle Scholar
- Kazianka H (2013) Objective bayesian analysis of geometrically anisotropic spatial data. J Agric Biol Environ Stat 18:514–537CrossRefGoogle Scholar
- Ma C (2007) Why is isotropy so prevalent in spatial statistics? Trans Am Math Soc 135:865–871Google Scholar
- Lantuéjoul C (2002) Geostatistical simulation. Models and algorithms. Springer, BerlinCrossRefGoogle Scholar
- Matheron G (1970) La théorie des variables régionalisées et ses applications. Cahiers du Centre de Morphologie Mathématique de Fontainebleau, Fasc. 5, Ecole des Mines de Paris. Translation (1971): The Theory of Regionalized Variables and Its ApplicationsGoogle Scholar
- Matheron G (1975) Random sets and integral geometry. John Wiley & Sons, New YorkGoogle Scholar
- Porcu E, Schilling R (2011) From Schoenberg to Pick-Nevanlinna: toward a complete picture of the variogram class. Bernoulli 17:441–455CrossRefGoogle Scholar
- Porcu E, Gregori P, Mateu J (2006) Nonseparable stationary anisotropic space-time covariance functions. Stoch Environ Res Risk Assess 21:113–122CrossRefGoogle Scholar
- Schlather M et al (2015) Package ’RandomFields’. http://cran.r-project.org/web/packages/RandomFields/
- Stein M (1999) Interpolation of spatial data: some theory for kriging. Springer, New-YorkCrossRefGoogle Scholar
- Stein M (2013) On a class of space-time intrinsic random functions. Bernoulli 19:387–408CrossRefGoogle Scholar
- Schoenberg IJ (1938) Metric spaces and positive definite functions. Trans Am Math Soc 44:522–536CrossRefGoogle Scholar