Implementing an anisotropic and spatially varying Matérn model covariance with smoothing filters

While known to be an important aspect of geostatistical simulations and inverse problems, an a priori model covariance can be difficult to specify and implement, especially where that model covariance is both anisotropic and spatially varying. The popular Matérn covariance function is extended to handle such complications, and is implemented as a cascade of numerical solutions to partial differential equations. In effect, each solution is equivalent to application of an anisotropic and spatially varying smoothing filter. Suitable filter coefficients can be obtained from auxiliary data, such as seismic images. An example with simulated porosities demonstrates the effective use of a Matérn model covariance implemented in this way.