A fundamental theorem of Earth's surface modelling

Ground observation is able to obtain highly accurate data with high temporal resolution at observation points, but these observation points are too sparse to satisfy some application requirements. Satellite remote sensing and system models can frequently supply spatially continuous information about the Earth's surface, which is impossible from ground-based investigations, but remote sensing description and system model simulation are not able to directly obtain process parameters. The most effective method for Earth surface modelling entails the integration of satellite observations or system models with ground observations. However, the full integration was ignored in most of the methods. For finding a solution for this problem, we suggest an alternative method, high accuracy surface modeling (HASM), which takes global approximate information (e.g., remote sensing images or model simulation results) as its driving field and local accurate information (e.g., ground observation data and/or sampling data) as its optimum control constraints. HASM completes its operation when its output satisfies the iteration stopping criterion which is determined by application requirement for accuracy. A Fundamental Theorem of Earth Surface Modelling (FTESM) is abstracted on the basis of applying HASM to simulating surfaces of elevation, soil properties, changes of ecosystem services, and driving forces of the changes on multi-scales for about 20 years. FTESM is described as an Earth's surface system or a component surface of the Earth's surface environment can be simulated with HASM when its spatial resolution is fine enough, which is uniquely defined by both extrinsic and intrinsic invariants of the surface''. From FTESM, seven corollaries have been deduced, corresponding to interpolation, upsaling, downscaling, data fusion and data assimilation respectively.