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We derive explicit expressions for a family of radially symmetric,non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involvethe modified Bessel function of the second kind. In addition to thecharacteristic length and the amplitude coefficient, the Spartan covarianceparameters include the rigidity coefficient $\eta_{1}$ which determines theshape of the covariance function. If $ \eta_{1} >> 1$ Spartan covariancefunctions 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 theintegral range for Spartan and Bessel-Lommel covariance functions usingexplicit relations and numerical simulations. Finally, we define a generalizedspectrum of correlation scales $\lambda^{(\alpha)}_{c}$ in terms of thefractional Laplacian of the covariance function; for $0 \le \alpha \le1$ the$\lambda^{(\alpha)}_{c}$ extend from the smoothness microscale $(\alpha=1)$ tothe integral range $(\alpha=0)$. The smoothness scale of mean-square continuousbut non-differentiable random fields vanishes; such fields, however, can bediscriminated by means of $\lambda^{(\alpha)}_{c}$ scales obtained for $\alpha<1$.

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