Dual-frequency Spaceborne Retrieval Methods and the Role of Non-Uniform Beam Filling

Many of the recent dual-frequency retrieval methods seek to estimate parameters of the particle size distribution (PSD) by relying on the fact that the dual-frequency ratio of the radar reflectivity factors is a function of a characteristic size parameter of the PSD, independent of particle number concentration. An essential part of the retrieval is its account of attenuation where the PSD estimates from prior range gates are used to update the attenuation to the gate of interest. The retrieval can be framed either as a forward (from the radar outward), backward (from the surface or final gate to the radar) or iterative recursion. The primary advantage of the iterative and forward recursion methods is that independent estimates of the path-integrated attenuations (PIA) are not needed; the major weakness appears to be instability in the estimates especially when the attenuation is large and contributions to the attenuation from cloud liquid water and mixed phase precipitation are significant. The primary advantage of the backward recursion is the stability of its estimates and its independence of cloud water and mixed phase precipitation in estimating characteristics of the rain layer, assuming that these constituents appear above the rain. The major disadvantage of the method is the need for independent estimates of PIA, usually using the surface reference technique (SRT). We explore some of the factors that affect the accuracy of the estimates using the backward formulation. One issue is how to account for attenuation at the gate of interest. An inspection of the equations shows that an account of the attenuation at this gate results in coupled equations that can be solved by means of Broyden's method. Comparisons between the more exact and approximate formulations show the differences depend on the strength of the attenuation and the extent of the range gate. Other factors relate to fact that, even when the PIA's are accurate, the SRT-derived attenuations differ from the attenuations needed in the retrieval. In particular, the path-length from the radar to the surface is longer than that from the radar to the last gate above the surface clutter. Moreover, a typical off-nadir application of the SRT provides an estimate of PIA along the central column of the beam and not the beam-averaged PIA needed in the retrieval. Comparisons of retrievals from high resolution antenna fields of view (FOV) versus results from a typical FOV (characteristic of the TRMM-PR and GPM-DPR radars) indicate the error sources and possible methods to mitigate the problem. At higher rain rates, jumps in median mass diameter and number concentration are sometimes seen in the simulation results between the last and next-to-last range gates. A simple smoothing procedure can be used to recompute the estimated PSD parameters at the final gate, along with a recalculation of the PIA, often leading to more accurate PSD and rain rate profiles. Because of strong non-linearities in the dual-frequency equations, multiple runs of the retrieval for changes in PIA, µ and parameters of the cloud water and mixed phase precipitation provide insight into and a quantitative estimate of the errors in the retrieval. It is worth noting that a useful check on the rain rates can be made using the fact that the k-R relationship at Ka-band is nearly independent of the raindrop size distribution so that consistency can be tested between the estimated PIA and profiled rain rates that is relatively insensitive to the raindrop size distributions. Finally, we investigate the relationship between the backward and iterative retrieval methods, noting that iterations can be performed in several ways: iterating on both PIA's, as in the standard approach, or iterating on either the Ka-band or Ku-band attenuation or some linear combination of the two while keeping the ‘orthogonal' component constant. In particular, iteration on PIA(Ku)+PIA(Ka), while keeping the differential attenuation, PIA(Ka)-PIA(Ku), fixed may prove to be an attractive option in view of the higher accuracy in the differential attenuation estimate that can be attained in cases where the surface and incidence angle are such that the normalized surface cross sections at Ku and Ka-band are highly correlated.