Sufficient conditions for reconstructing 2-d rainfall maps

The ground level rainfall at a given time is modeled as a 2-D spatial random process r(x, y), the rain field. Existing measurement equipment, such as rain gauges, weather stations, or recently proposed microwave links, samples r(x, y) spatially in specific points or along lines. Given these samples, our purpose is to reconstruct r(x, y). In this paper, we study the question: 'under what conditions can a given topology of ground measurements guarantee reconstructability of the rain field?' Based on the assumption that rain fields are sparse, we present a statistical approach to this problem by first characterizing the statistics of the measurements, and then answering the question by applying methods from compressed sensing theory, and in particular Donoho and Tanner's phase transition diagram for sparse recovery. We conclude by suggesting a solution in a form of a simple diagram, allowing one to evaluate the potential reconstruction of r(x, y) in different resolutions without the need for computations.