Measurement of Attenuation by a Dual-Radar Method: Concept and Error Analysis

Abstract

The authors describe a method for the measurement of specific attenuation using two radars operating at virtually the same wavelength at which the attenuation is desired. Use of the same wavelength avoids confounding between differences in received powers due to attenuation and wavelength dependence of reflectivity factors. A governing equation for the specific attenuation and an analytical solution of the equation are derived. The analytical solution gives an explicit expression for the specific attenuation in terms of the specific attenuation at an initial point and the second derivative of estimates of the difference between ensemble average received powers for the two radars along and near the characteristics of the governing equation. An error analysis shows that error in the computed attenuation is due to error in the attenuation coefficient at the initial point and errors in the estimates of the ensemble average received powers. Unless the initial point is poorly chosen, the error due to the initial condition is approximately of the same magnitude as the error in the initial value of the specific attenuation. The error due to errors in estimates of the ensemble average received powers accumulate along the characteristics; the error increases rapidly near the radars and is smaller away from the radars. The error decreases with distance between the radars, the number of independent pulses averaged to find estimates of ensemble average power, the number of independent radar pulse volumes used in smoothing the observed reflectivities, and the angular intervals used in performing the numerical differentiation for the calculation of the above-mentioned second derivative. Using typical radar parameters, it is found that the standard error of the two-way specific attenuation can be less than a tenth or a few tenths of a decibel per kilometer if the number of independent pulses exceeds about 10, smoothing is performed over hundreds of independent radar pulse volumes, and the angular intervals for the differentiation are about 3° or greater.