Precipitation radar measurement depends on drop size distribution (DSD) rather than precipitation rates. Generally, DSD is assumed to follow an exponential or a Gamma distribution with two unknown parameters. Here, DSD can be represented on a 2-dimensional plane. For a single-frequency radar measurement, as two parameters cannot be determined, an empirical power law between radar reflectivity factor Z and rain rate R (Z-R relationship) is used. Assuming Z-R relationship is equivalent to constraining DSD on a 1-dimenisonal curve. In the standard algorithm of PR, an empirical power law between specific attenuation k and Z (k-Z relationship) is given to correct attenuation (Hitschfeld-Bordan method; HB method). Assuming k-Z relationship is also equivalent to constraining DSD on another 1-dimensional curve. Fortunately, space-borne or air-borne radars such as PR can measure surface backscattering cross section and surface reference technique (SRT) is applied to estimate path integrated attenuation (PIA). By referring to PIA estimates, k-Z relationship can be adjusted. However, the adjustment depends on the accuracy of SRT, and is applied simultaneously for all range bins. Therefore, DSD cannot have 2-dimensional degrees of freedom. The accuracy of DSD estimation by PR depends on not only the accuracy of Z but the accuracy of constraints such as k-Z relationship.
By using a dual-frequency radar measurement, without any constraints, two DSD parameters are expected to be retrieved accurately, but it is not true even under ideal conditions where measurement of Z is always perfect (Seto et al. 2011). Two sets of DSD parameters give the same set of Z at the two frequencies. For N range bins, the number of solutions is at most 2 to the Nth. In addition to that, considering the accumulation of numerical errors and measurement errors of Z, it cannot be expected to retrieve DSD parameters from dual-frequency radar measurements without any constraints.
In a developing algorithm for DPR, k-Z relation is assumed and the HB method is applied for each frequency. Once attenuation-corrected radar reflectivity factor Ze is given at both frequencies, the dual-frequency ratio (DFR) of Ze's is calculated, and DSD parameters are retrieved easily from the DFR (DFR method). However, the retrieved DSD generally does not agree with assumed k-Z relations. Then, k-Z relations are adjusted to fit the DSD. The HB method and DFR method can be applied again by using the adjusted k-Z relations. By iterating a combination of the HB method and the DFR method, k-Z relations are improved. This is termed HB-DFR method (Seto et al. 2013). Though k-Z relations are adjusted simultaneously for all range bins using SRT method, this method can adjust k-Z relation at a range bin independently of other range bins. Therefore, in this method, DSD is represented on a 2-dimensinal plane.
HB-DFR method stops iterations when k-Z relations are judged to be converged or when the number of iterations reaches to limitation. In the former case, one solution is selected among multiple candidates of solutions. Here, the selection largely depends on the first guess of k-Z relations. Therefore, k-Z relations are constrains on the HB-DFR method. Generally, errors in rain rates retrieved by the HB-DFR method become larger at lower (farther) range bins. This is common characteristics of forward retrieval methods including the HB-DFR method. SRT is also necessary to mitigate such errors. This method has no big differences from other dual-frequency retrieval methods in terms of the performance for dual-frequency measurement. But, this method can be smoothly applied to single-frequency measurement or the case that some range bins lack either of dual-frequency measurements, just by setting that DFR method and adjustment of k-Z relation are not applied for range bins without dual-frequency measurements. By the DPR, dual-frequency measurements are limited to inner swath of normal scans. In outer swath and in interleaved scans, single-frequency measurement by KuPR or KaPR is available. To have seamless 2-dimensional estimates of precipitation, this method is suitable. In the future, the DPR algorithm is applied to the PR measurements to produce long-term dataset with temporally constant quality.
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