Radial Covariance Functions Motivated by Spatial Random Field Models with Local Interactions

We derive explicit expressions for a family of radially symmetric, non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve the modified Bessel function of the second kind. In addition to the characteristic length and the amplitude coefficient, the Spartan covariance parameters include the rigidity coefficient $η_{1}$ which determines the shape of the covariance function. If $η_{1} >> 1$ Spartan covariance functions exhibit multiscaling. We also derive a family of radially symmetric, infinitely differentiable Bessel-Lommel covariance functions valid in $\mathbb{R}^{d}, d\ge 2$. We investigate the parametric dependence of the integral range for Spartan and Bessel-Lommel covariance functions using explicit relations and numerical simulations. Finally, we define a generalized spectrum of correlation scales $λ^{(α)}_{c}$ in terms of the fractional Laplacian of the covariance function; for $0 \le α\le1$ the $λ^{(α)}_{c}$ extend from the smoothness microscale $(α=1)$ to the integral range $(α=0)$. The smoothness scale of mean-square continuous but non-differentiable random fields vanishes; such fields, however, can be discriminated by means of $λ^{(α)}_{c}$ scales obtained for $α<1$.

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