The function computes the distance between locations, with geometric anisotropy. Consider real parameters \(\theta_1\) and \(\theta_2\), and the transformation \(\psi=\arctan(\theta_1/\theta_2)/2\) and \(r=1 +\theta_1^2 + \theta_2^2\). The dilation and rotation matrix is $$\left(\begin{matrix} \sqrt{r}\cos(\rho) & -\sqrt{r}\sin(\rho) \\ \sin(\rho)/\sqrt{r} & \cos(\rho)/\sqrt{r} \end{matrix} \right).$$ The parametrization is convenient for optimization purposes, as the parameter vector is unconstrained and the transformation has unit Jacobian.
References
Rai, K. and Brown, P.E. (2025), A parameter transformation of the anisotropic Matérn covariance function. Canadian Journal of Statistics e11839. doi:10.1002/cjs.11839