| This methodological contribution aims to present some new covariance models with
applications in the stochastic analysis of hydrological processes. More specifically, we
present explicit expressions for radially symmetric, non-differentiable, Spartan covariance
functions in one, two, and three dimensions.
The Spartan covariance parameters include a characteristic length, an amplitude
coefficient, and a rigidity coefficient which determines the shape of the covariance function.
Different expressions are obtained depending on the value of the rigidity coefficient and the
dimensionality. If the value of the rigidity coefficient is much larger than one, the Spartan
covariance function exhibits multiscaling. Spartan covariance models are more
flexible than the classical geostatatistical models (e.g., spherical, exponential). Their
non-differentiability makes them suitable for modelling the properties of geological
media.
We also present a family of radially symmetric, infinitely differentiable Bessel-Lommel
covariance functions which are valid in any dimension. These models involve combinations
of Bessel and Lommel functions. They provide a generalization of the J-Bessel covariance
function, and they can be used to model smooth processes with an oscillatory decay of
correlations.
We discuss the dependence of the integral range of the Spartan and Bessel-Lommel
covariance functions on the parameters. We point out that the dependence is not
uniquely specified by the characteristic length, unlike the classical geostatistical
models.
Finally, we define and discuss the use of the generalized spectrum for characterizing
different correlation length scales; the spectrum is defined in terms of an exponent α. We
show that the spectrum values obtained for exponent values less than one can be used
to discriminate between mean-square continuous but non-differentiable random
fields.
References
[1] D. T. Hristopulos and S. Elogne, 2007. Analytic properties and covariance functions
of a new class of generalized Gibbs random fields, IEEE Transactions on Information Theory,
53(12), 4667 – 4679.
[2] D. T. Hristopulos and M. Zukovic, 2011. Relationships between correlation lengths
and integral scales for covariance models with more than two parameters, Stochastic
Environmental Research and Risk Assessment, 25(1), 11–19.
[3] D. T. Hristopulos, 2014. Radial Covariance Functions Motivated by Spatial Random
Field Models with Local Interactions, arXiv:1401.2823 [math.ST]
. |