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 $\eta_{1}$ which determines the shape of the covariance function. If $\eta_{1} >> 1$ 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 $\lambda^{(\alpha)}_{c}$ in terms of the fractional Laplacian of the covariance function; for $0 \le \alpha \le1$ the $\lambda^{(\alpha)}_{c}$ extend from the smoothness microscale $(\alpha=1)$ to the integral range $(\alpha=0)$. The smoothness scale of mean-square continuous but non-differentiable random fields vanishes; such fields, however, can be discriminated by means of $\lambda^{(\alpha)}_{c}$ scales obtained for $\alpha <1$.