A Variational Merging Approach to the Spatial Description of Environmental Variables

High‐resolution images of environmental variables are highly valuable sources of information in research and environmental management. Obtaining spatially continuous information from ground observations is challenging due to the wide variety of factors that affect classical interpolation methods. While geostatistical methods have produced noteworthy results, they generally rely on important assumptions and strongly depend on the availability of observed data. In turn, satellite‐based or model‐based gridded images generally represent the global spatial structure of environmental variables but are subject to bias. With the objective of exploiting the benefits of both sources of information, we propose a new mathematical formulation to merge observed data with gridded images of environmental variables using partial differential equations in a variational setting. With a convenient functional, formed as the sum of two competing terms, two simultaneous goals are achieved: to improve the description of the spatial structure in maps generated by simple deterministic interpolation methods and to increase the reliability of satellite‐based or model‐based images. Either satellite‐based or model‐based information is included in a regularity term to provide the global spatial structure, while in situ data are included in a fidelity term. The resulting maps can be considered a merging of interpolated in situ data with satellite or model imagery. The method is first evaluated using simulated images generated by geostatistical simulation and then applied to actual temperature and precipitation data in selected regions in the Tropical Andes. The results indicate that the method is capable of generating realistic maps while performing well in terms of validation.