An integral equation method for dual-wavelength radar data using a “soft” constraint

A set of integral equations for dual-wavelength radar data provides a means of estimating liquid water content and parameters of the particle size distribution (PSD) and constitutes a potential algorithm for the proposed DPR (Dual-wavelength Precipitation Radar) on the Global Precipitation Mission (GPM) satellite. One version of the method, the forward-recursion, starts at the storm top. Although this implementation requires no constraint, like the Hitschfeld-Bordan solution, it tends to be unstable when the path attenuation becomes significant. As such, the method can be applied with confidence only in regions of frozen hydrometeors or light rain. For moderate rain or mixed-phase precipitation, the backward recursion is preferable because it is more robust and accurate. The backward recursion, however, requires constraints of path-integrated attenuation (PIA) down to the surface at both wavelengths, estimates of which have typically come from the surface reference technique (SRT).

There has been work on improving the stability of the method by constraining the behavior of the PSD parameters along the range direction. In addition, an iteration and consistency approach has been investigated with the purpose of deriving estimates that are independent of the SRT. In this study, we first assess the performance of the backward recursion/SRT by using a simple rain simulation where measured radar reflectivities at Ku and Ka-band are generated using an assumed set of PSD data. In particular, we examine the behavior of the method with respect to the number of independent samples and the variability of the PIA estimate. The behavior of the solutions with respect to variations in the PIA suggests an alternative to the SRT using the difference in the measured radar reflectivity factors near the surface. This difference, however, is a “soft” constraint in the sense that it is itself a function of one of the unknowns - the characteristic size parameter (at the range gate in question) of the distribution. This implies, in turn, that there are multiple solutions consistent with the constraint. Three factors, however, tend to reduce the ambiguities. The first is that, when the attenuation is moderate or strong, the solutions often converge quickly as the solution progresses toward the storm top. The second mitigating factor is that the solution that assumes the correct or nearly correct size parameter typically exhibits the smallest variability in the rain rate and PSD parameters as the solution approaches the surface. Thirdly, out-of-range values of the size parameter can be eliminated because they are inconsistent with the integral equations – i.e., inconsistent with the measured reflectivity data. Thus, if we assume that the PSD parameters are smoothly varying as the surface is approached, a candidate solution, close to the true profile, often can be chosen, particularly if the number of independent samples is large and the µ value is close to the true value.

Some of the drawbacks and error sources of this approach, as well as comparisons to the SRT constraint, are discussed and illustrated.