## 1. Introduction

[2] Quantitative interpretation of precipitation measurements by radars involves the representation of the radar signal in terms of the reflectivity factor (i.e., *Z* expressed in units of mm^{6} m^{−3}). For vertically pointing radars operating in the VHF band (i.e., meter wavelengths), we have the advantage of measuring also air vertical velocities, in addition to the precipitation signal. For this reason, meter wavelength radars might be more desirable, for the study of precipitation physics, than traditional centimeter wavelength radars. For this to happen, however, we must

[3] 1. Calibrate the measured power density spectra.

[4] 2. Extract the signal originating for precipitation, an additional step compared to centimeter wavelength radars.

[5] 3. Express this received power *P*_{r} in terms of the scatterers cross sections (i.e., radar reflectivity, *η*, expressed in units of m^{−1}).

[6] 4. Express this radar reflectivity in terms of the reflectivity factor *Z*.

[7] For the case of precipitation particles being the scatterers of our VHF radar pulse (i.e., Rayleigh scatter), requirement 4 can be satisfied by using the following expression [e.g., *Rinehart*, 1997, equation (5.13)]:

where ∣K∣^{2} is the dielectric factor, and *λ* is the wavelength of the radar transmitted pulse (in m). *Z* is the reflectivity factor (expressed in mm^{6} m^{−3}). The value of ∣K∣^{2} depends upon the scatterer material, the scatterer temperature, and the radar wavelength.

[8] Unfortunately, estimations for the values of ∣K∣^{2} at VHF band are not readily available in the wind profiler literature. Typically, the ∣K∣^{2} value for S band (∣K∣^{2} ≈ 0.93 for water sampled at 10 cm wavelengths) is used instead [e.g., *Chilson et al.*, 1993].

[9] By convention [e.g., *Smith*, 1984], if ∣K∣^{2} is taken equal to 0.93 (the value corresponding to liquid water at near 20°C, and wavelengths in the S band), then *Z* = *Z*_{e}, the equivalent radar reflectivity factor that is generally plotted on radar displays. This convention is adopted because when radar measurements are made, one is often not certain of the hydrometeor phase or composition. However, it is still necessary to verify if the assumption of ∣K∣^{2} = 0.93 is also valid in the VHF band.

[10] The other three requirements are not attained as directly as with requirement 4, and they represent a challenge that has been met only partially in the current literature [e.g., *Lucas et al.*, 2004, and references therein]. For example, we can accomplish requirement 3 by using the radar equation (i.e., the relationship between *η* and *P*_{r}). Unfortunately, there are several versions of this radar equation that are very often valid only for particular radar configurations [e.g., *Probert-Jones*, 1962; *Gage and Balsley*, 1980; *Hocking*, 1985]. Furthermore, the derivation of such equations is not always presented in detail in the literature. Requirement 2, on the other hand, is accomplished through elaborate algorithms of signal processing [e.g., *Rajopadhyaya et al.*, 1993; *Boyer et al.*, 2001] or by multiwavelength techniques [e.g., *Schafer et al.*, 2002; *Maguire and Avery*, 1994]. Adjustment and refinement of these algorithms require long periods of numerical experimentation with the corresponding radar data sets. Concerning requirement 1, we should recognize that, very often, power density spectra recorded by VHF radars are not expressed in units of W per frequency bin, but only in the arbitrary units of the radar receiver hardware. Requirement 1 then involves a radar calibration, which has already been analyzed by *Campos et al.* [2007].

[11] In this paper, we present our efforts toward the accomplishment of requirements 2, 3, and 4. We will be focusing on the case of precipitation being rain, because it gives us a signal that is easy to separate from the clear-air signal in the power density spectrum, and also because it avoids the inconvenience of not knowing the exact ∣K∣^{2} value for solid precipitation (e.g., snow and graupel). Concerning requirement 4, we verify the assumption that ∣K∣^{2} = 0.93 for most of the rain observations at VHF band. For requirement 3, we derive a general version of the radar equation valid for vertically pointing radars, as well as a particular version of this equation valid for the McGill VHF radar. Then, a numerical algorithm for extracting the rain signal out of the VHF power spectra is presented to achieve the requirement 2. The next section describes the theoretical considerations for these three requirements. We then combine our radar equation and our algorithm for extracting rain signal in section 3, which allow us to retrieve reflectivity factors and air vertical velocities during several rainfall observations at Montreal. As well, we validate our method by comparing our results with the rain signal from raindrop sizes measured at the ground. A discussion of our results is presented in the last section.