Formalism for Comparing Rain Estimation Designs

Space-time averages of rain rates are needed in several applications. Nevertheless, they are difficult to estimate because the methods invariably leave gaps in the measurements in space or time. A formalism is developed which makes use of the frequency-wavenumber spectrum of the rain field. The mean square error of the estimate is expressed as an integral over frequency and two-dimensional wavenumber of an integrand consisting of two factors, a design-dependent-filter multiplied by the space-time spectrum of the rain rate field. Such a formalism helps to separate the design issues from the peculiarities of rain rate random fields. Two cases are worked out in detail: a low orbiting satellite which takes cell-wide snapshots at discrete intervals and a network of raingages which are gappy in space but continuous in time.