Statistical errors of rain rate estimators due to natural variations in raindrop size distribution (DSD) are studied for 3-cm wavelength polarimetric radar. Four types of estimators are examined: A classical estimator R(Z_H), and three types of polarimetric radar estimators R(K_DP), R(Z_H, Z_DR), and R(K_DP, Z_DR), where R is the rain rate, Z_H is the reflectivity factor at horizontal polarization, K_DP is the specific differential phase, and Z_DR is the differential reflectivity. The T-matrix method is employed for the scattering calculations, and a total of 7,664 one-minute raindrop size spectra, measured with a Joss-Waldvogel type disdrometer are used.
According to simulation results, the normalized errors (NEs) of R(Z_H), R(K_DP), R(K_DP,Z_DR), and R(Z_H,Z_DR) for all DSD samples are 25%, 14%, 9%, and 10%, respectively. The NEs of all estimators, except R(Z_H), tend to decrease with increasing rain rate. For rain rates larger than 10 mmh⁻¹, e.g., the average NEs of R(Z_H), R(K_DP), R(K_DP, Z_DR), and R(Z_H,Z_DR) are 25%, 9%, 5%, and 7%, respectively. The simulation results show that the classical estimator R(Z_H) is the most sensitive to variations in DSD and the estimator R(K_DP, Z_DR) is the least sensitive.
The lowest sensitivity of the rain estimator R(K_DP, Z_DR) to variations in DSD can be explained by the following facts. The difference in the forward-scattering amplitudes at horizontal and vertical polarizations, which contributes K_DP, is proportional to the 4.78th power of the drop diameter. On the other hand, the exponent of the backscatter cross section, which contributes to Z_H, is proportional to the 6.38th power of the drop diameter. Because the rain rate R is proportional to the 3.67th power of the drop diameter, K_DP is less sensitive to DSD variations than Z_H. However, DSD spectra with unusually large median volume diameter D₀ can increase the estimation error of R(K_DP). The differential reflectivity Z_DR reduces the effect of unusual D₀ and is useful for further improvement of the estimator R(K_DP). This is due to the fact that Z_DR itself is a good measure of D₀.