Corrections to and Considerations of the Spectrum Width Equation

The Fourier transform of the autocorrelation coefficient of weather radar signal is the weighted normalized expected Doppler spectrum. Without other spectral broadening mechanisms, the weighted normalized expected Doppler spectrum equals the probability density function of the turbulent velocity weighted by the beam pattern, range weighting and reflectivity. With non-zero, uniform, and steady radial flow, the weighted normalized expected Doppler spectrum is identical to probability function of total velocity weighted by the beam pattern, range weighting and reflectivity. The probability density of total velocity at a point equals the probability density of its turbulent component at same location, and is centered around the speed of the steady flow at that point.

For a stationary antenna, the square of the total spectrum width is a weighted sum of squared spectrum width associated with each spectral broadening mechanism (e.g. shear, dispersion of the hydrometer's terminal velocity, etc.) and an cross term depending on the cross product of the gradient of mean terminal velocity of hydrometers and wind. If radar is vertically pointed and steady flow is horizontal, or the radar is horizontally pointed and the earth's curvature is negligible so that the radial component of terminal velocity is zero, the cross term will disappear. For a scanning radar, the square of the total spectrum width is still a weighted sum of squared spectrum widths associated with each mechanism and a cross term depending on the cross product of the gradient of mean terminal velocity of hydrometers and wind, but an effective beam pattern must be applied with shear in the direction of rotation.

The weighted expression(s) developed in this study are applicable to both scanning and stationary antenna, and for any elevation angle. But, for scanning radars with the existence of significant azimuth shear, or the radar beams elevated above horizon, the classical spectrum width equation is invalid.

Without hydrometer's terminal velocity, only steady flow contributes to the first moment of the weighted expected normalized Doppler spectrum; turbulent component contributes nothing. Furthermore, if turbulence is homogeneous and there are no hydrometer's oscillation and antenna rotation, the squared spectrum width due to turbulence associated with the weighted normalized expected Doppler spectrum equals the variance of the turbulent velocity at a point and contains the contributions from all scales of eddies without any attenuation. The energy partition theory cannot be derived from the weighted expected normalized Doppler spectrum.