In this study, the combined application of geodesic kernel and Gaussian
process regression was investigated to estimate nonstationary hydraulic
conductivity fields in two-dimensional hydrogeological systems.
Particularly, a semi-analytical form of the geodesic distance based on
the intrinsic geometry of the manifold was derived and used to define
positive definite geodesic covariance matrices that are employed for
Gaussian process regression. Furthermore, the proposed approach was
applied to a series of synthetic hydraulic conductivity estimation
problems and the results show that the incorporation of secondary
information, such as geophysical or geological interpretations, can
considerably improve the estimation accuracy, especially in
nonstationary fields. Moreover, groundwater flow and solute transport
simulations based on the estimated hydraulic conductivity fields
revealed that the accuracy of the simulations was strongly affected by
the inclusion of secondary information. These results suggest that
incorporating secondary information into manifold geometry can
remarkably improve the estimation accuracy and provide new insights on
the underlying structure of geological data. This proposed approach has
crucial implications for hydrogeological applications, such as
groundwater resource management, safety assessments, and risk management
strategies related to groundwater contamination.