A two-level variational algorithm in the Sobolev–Orlicz space to predict daily surface reflectance at LANDSAT high spatial resolution and MODIS temporal frequency

We propose a new two-level variational model in Sobolev–Orlicz spaces with nonstandard growth conditions of the objective functional and discuss its applications to the spatiotemporal interpolation of multispectral satellite images. At the first level, we deal with the temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection–diffusion equation, whereas at the second level we solve a constrained minimization problem with a nonstandard growth energy functional that lives in variable Sobolev–Orlicz spaces. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with nonstandard growth in spatial interpolation problem, is unknown a priori and it depends on the solution of the first-level optimal control problem. It makes this spatiotemporal interpolation problem rather challenging. In view of this, we discuss the consistency of the proposed model, study the existence of optimal solutions, and derive the corresponding optimality systems. In particular, we apply this approach to the well-known prediction problem of the Daily MODIS Surface Reflectance at the Landsat-Like Resolution.