  Covariance functions are a fundamental tool for modeling the dependence
structure of spatial processes. This work investigates novel constructions for
covariance functions that enable the integration of anisotropies and hole
effects in complex and versatile ways, having the potential to provide more
accurate representations of dependence structures arising with real-world data.
We show that these constructions extend widely used covariance models,
including the Mat\'ern, Cauchy, compactly-supported hypergeometric and cardinal
sine models. We apply our results to a geophysical data set from a
rock-carbonate aquifer and demonstrate that the proposed models yield more
accurate predictions at unsampled locations compared to basic covariance
models.
