A methodology for merging multisensor precipitation estimates based on expectation-maximization and scale recursive estimation

Scale recursive estimation (SRE) is a Kalman filter-based methodology, which can be used to produce optimal (in terms of bias and minimum variance) estimates of a field at any desired scale given uncertain and sparse observations at different scales. SRE requires the specification of the state equation, which describes the variability of the precipitation process across scales, and the observation equation, which relates the observations to the state. Typical models for describing the multiscale rainfall variability are the multiplicative cascade models. However, in order to convert them into the additive form needed by SRE, one needs to work in the log space, creating thus a problem in handling zero-intermittency in a satisfactory way. In this paper, we propose an alternative approach, based on a data-driven identification methodology, which operates directly on the data and does not require a pre-specified multiscale model structure. Rather, system identification and estimation is performed simultaneously via a likelihood-based Expectation-Maximization (EM) procedure. The merits of the proposed approach versus approaches based on multiplicative cascade models are explored via several examples of synthetic and real precipitation fields. For practical application, the proposed approach will need to be extended to include the temporal evolution of storms. This extension presents theoretical challenges and until these are addressed, a simple alternative is explored of coupling the EM-SRE approach with a spatial downscaling methodology to merge precipitation observations available at different spatial and temporal scales. An example application is presented motivated by its relevance to the Global Precipitation Measuring (GPM) mission.