Surface reference technique required for a primitive retrieval algorithm of GPM/DPR

The Dual-frequency Precipitation Radar (DPR), which is planned to be carried on the core satellite of the Global Precipitation Measurement (GPM) mission, observes vertical profile of radar reflectivity factors (Z) at Ku (13.6 GHz) and Ka (35.5GHz) bands.

Many previous studies have investigated algorithms to retrieve drop size distribution (DSD) from observed Zs. Under some ideal conditions, a retrieval algorithm can be simplified into a set of 2n non-linear equations with 2n unknown DSD parameters (called as primitive retrieval algorithm), where n is the number of vertical ranges. There are three kinds of solving method for the primitive retrieval algorithm, forward method, backward method, and recursive backward method. In every solving method, the primitive retrieval algorithm can be decomposed into lemmas to solve a set of 2 non-linear equations with 2 unknown DSD parameters. A type of lemmas for the backward method and the recursive backward method has unique solution usually, while a type of lemmas for the forward method has dual solutions. It suggests that the forward method can not uniquely determine the solution of the primitive retrieval algorithm. The recursive backward method is equivalent with the forward method as the two methods accept an upper boundary condition but ignore a lower boundary condition in terms of attenuation, while the backward method accept the lower boundary condition and ignore the upper boundary condition. Therefore, the recursive backward method can not determine the unique solution objectively. However, as the recursive backward method tends to select the solution with the smallest PIA among multiple candidates of solutions, the method can normally give the right solution when the rain rate is weak and moderate (up to 20-50 mm/h). This upper limit shall be lower when n is larger. For heavier rainfall, as the recursive backward method does not give the right solution, the backward method with good estimates of the lower boundary condition or path integrated attenuation (PIA) by surface reference technique (SRT) is necessary. When SRT is available in Ku (Ka) band, to retrieve surface rain rate within a relative error of 50%, the accuracy of PIA for Ku (Ka) band should be 1 dB (4 dB) or better.