An approximation to the effective beam weighting function for scanning meteorological radars with an axisymmetric antenna pattern

To obtain statistically stable reflectivity measurements by meteorological radars, it is common practice to average over several consecutive pulses during which the antenna rotates at a certain angular velocity. Taking into account the antenna's continuous motion, the measured reflectivity is determined by an effective beam weighting function, which is different from a single-pulse weighting function-a fact that is widely ignored in applications involving beam weighting. In this paper, the effective beam weighting function is investigated in detail. The theoretical derivation shows that the effective weighting function is essentially a simple moving sum of single-beam weighting functions. Assuming a Gaussian shape of a single pulse, a simple and easy-to-use parameterization of the effective beam weighting function is arrived at, which depends only on the single beamwidth and the ratio of the single beamwidth to the rotational angular averaging interval. The derived relation is formulated in the "radar system" (i.e., the spherical coordinate system consisting of azimuth and elevation angles) that is often applied in practice. Formulas for the "beam system" (two orthogonal angles relative to the beam axis) are also presented. The final parameterization should be applicable to almost all meteorological radars and might be used (i) in specialized radar data analyses (with ground-based or satellite radars) and (ii) for radar forward operators to calculate simulated radar parameters from the results of NWP models.