Patch-wise Least Squares Collocation is introduced as a localized and optimized form of the conventional Least Squares Collocation method. It aims to mitigate the negative effects of the stationarity assumption and to better recover the high-frequency components of local signals. Conceptually, patching refers to the spatial subdivision of the study area into smaller regions, where local modelling within each patch reduces the error arising from assuming global stationarity. In Patch-wise Least Squares Collocation, inspired by the Remote–Compute–Restore concept, patching is performed by considering the spatial–spectral correspondence of the field. Accordingly, the patch dimensions are selected in proportion to the degree or order used in removing the long-wavelength components of the field.
Patch-wise Least Squares Collocation can also be implemented using anisotropic covariance functions. In this approach, for each patch, an anisotropic covariance is determined based on the observations within that region. Assuming local stationarity and anisotropy, the field is modelled in each patch independently of observations in other patches, and the estimated signal is represented as a vector of patch-wise estimates. In this regard, optimising the patching strategy in Patch-wise Least Squares Collocation is an attempt to enhance the gravity modelling. To do this, available geological and geophysical information could be useful resources to be used. As an initial try in this field, the anisotropic covariance derived from gravity anomalies and density anomalies are computed in a region in north-western of Iran. Results in this case study shows a considerable correlation between the anisotropy direction and elongation of these two covariances.