Estimation of the drop size distribution parameter in the rain profiling algorithm for the TRMM Precipitation Radar

The standard rain profiling algorithm for the TRMM Precipitation Radar, which is known as 2A25, uses the Hitschfeld-Bordan method for the rain attenuation correction. When the path integrated attenuation (PIA) exceeds a certain threshold, the attenuation estimate from the H-B method is compared with the PIA estimated given by the surface reference technique (SRT). By taking into consideration of measurement errors, the specific attenuation assumed in the H-B method is adjusted so that the modified PIA from the H-B method with the new specific attenuation is consistent with the PIA estimate from the SRT. In this process, the maximum likelihood method is used in version 5 of the algorithm. However, a more probabilistic idea was introduced when version 6 was released. In version 6, the adjustment parameter is regarded as a random variable and its posterior probability distribution is derived for given data of rain and surface echoes. The a posteriori probability density function is calculated in a Bayesian way. All physical variables to be estimated are expressed in terms of the adjustment parameter and their expected values are calculated with the derived posterior pdf.

In stead of calculating the expected values of the output variables, if we first calculate the expected value of the adjustment parameter and if we use it to calculate all other variables deterministically, the average of the final estimates of surface rain rates decreases by a few percent. If we use the maximum likelihood estimate of the adjustment parameter as in version 5 of the algorithm, the average increases by a few percent. These differences occur because the posterior pdf is skewed and the output variables non-linearly depend on the adjustment parameter.

This paper examines the pros and cons of the maximum likelihood estimate and various Bayesian estimates of the adjustment parameter. The structure of the algorithm and the assumptions used in the estimation of the default drop size distribution parameters are reviewed in order to find which estimation method is more appropriate for the TRMM mission among the possible methods examined in the paper.