In addition to a proper radar calibration, quantitative estimation of precipitation from radars also requires the extraction of the precipitation signal out of the Doppler spectra. It also requires the proper conversion of this precipitation signal into reflectivity factor. This study shows how the measurement of rainfall and vertical air velocities can be performed using only observations from a radar operating at the VHF band (i.e., meter wavelengths). We verify the assumption that the dielectric factor ∣K∣2 = 0.93 is valid for rain observations in the VHF band. We then derive, analytically and numerically, a more general version of the radar equation valid for vertically pointing radars with targets within a few kilometers range but still within the antenna far-field region. Following this, we describe a new algorithm for extraction of rain signal out of VHF Doppler spectra. To validate our methods, we made colocated measurements of VHF Doppler spectra aloft and raindrop sizes at the ground. The analytical version of our radar equation compares well with similar equations available in current literature, and this validates the particular case of our numerically derived radar equation. We combine our numerical version of the radar equation and our algorithm for extracting precipitation signal. This combination allows us to obtain reflectivity factors (from rain signals) and vertical velocities (from air signals), these being simultaneous observations within the same sampling volume. From the data set collected, we found good agreement (linear correlation coefficient around 0.8) between the rain signals derived from VHF observations aloft and from drop sizes at ground level. Hence we are able to measure rainfall amounts and vertical air velocities in a simpler and more efficient way, using only observations from a VHF wind profiler. This represents a promising step toward the analysis of precipitation from large sets of radar data.
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 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 mm6 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
 1. Calibrate the measured power density spectra.
 2. Extract the signal originating for precipitation, an additional step compared to centimeter wavelength radars.
 3. Express this received power Pr in terms of the scatterers cross sections (i.e., radar reflectivity, η, expressed in units of m−1).
 4. Express this radar reflectivity in terms of the reflectivity factor Z.
 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 mm6 m−3). The value of ∣K∣2 depends upon the scatterer material, the scatterer temperature, and the radar wavelength.
 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].
 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 = Ze, 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.
 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 Pr). 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. .
 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.
2.1. Computing the Dielectric Factor at VHF
 In order to accomplish requirement 4, from section 1, we consider equation (1). It is clear that the knowledge of ∣K∣2 at VHF band is required in our analysis (and in any quantitative analysis of precipitation using radars). For Rayleigh scattering, the scatterer dielectric factor, ∣K∣2, is a function of the scatterer's complex refractive index, m, such that [e.g., Battan, 1973, p. 38; Marshall and Gunn, 1952, p. 322]
 At the same time, m varies with scatterer temperature and radar wavelength. Unfortunately, these functional relations are not widely known for the VHF band, and generally it is simply assumed that the ∣K∣2 value is the same as in S band. We chose to try to obtain an expression for the complex refractive index for liquid water as a function of raindrop temperature and VHF wavelength.
where i = , ξ = ξ′ − i ξ″ is the relative dielectric constant (i.e., the ratio between the media dielectric constant and the dielectric constant of empty space), ξ′ is the relative permittivity (energy storage), and ξ″ is the loss factor (energy lost as heat).
 The Debye  model describes well the frequency dependence of the dielectric constant for different temperatures. Although this model is limited to radar frequencies below 100 GHz and to scatterers consisting of pure water particles [Liebe et al., 1991], it is sufficient for our purposes (measuring rainfall with 50 MHz radars). The Debye equations are [e.g., Ulaby et al., 1986, p. 2020]:
where f = 3 × 108ms−1 / λ is the frequency of the electromagnetic radiation, ξ∞ is the high-frequency dielectric constant (when f approaches infinite), ξS is the static dielectric constant, and f0 is the relaxation frequency of pure water (expressed in the same units of f). Then, from Liebe et al. [1991, p. 661, equation (1)] we have
where f0 is expressed in units of Hz. Therefore, by combining equations (2)–(11), we are able to compute the variation of ∣K∣2 with rain temperature and radar frequency, and verify if the assumption of ∣K∣2 = 0.93 is adequate at VHF band.
Figure 1 presents the relative dielectric constants for various raindrop temperatures and radar wavelengths. As validation, the top plot in Figure 1 compares the model by Debye  and the empirical equations by Liebe et al.  with actual measurements in liquid water by Hippel  in Table 1. The bottom plot in Figure 1 plots the relative complex permittivity for pure liquid water at temperatures from −15°C to 35°C. Similarly, the top plot in Figure 2 presents the complex refractive index for liquid water at typical tropospheric temperatures. Here, the curves in the bottom plot of Figure 1 are used as input into equations (4) and (5) to obtain complex refractive indexes (plotted in the top plot of Figure 2).
Table 1. Measurements of Dielectric Properties for Liquid Water at 25°Ca
(ξ″/ξ′) × 104
Measurements are from Hippel . Uncertainties are about ±2% in ξ′ and about ±5 % in ξ′′/ξ′.
1.0 × 108
3.0 × 108
3.0 × 109
1.0 × 1010
 Refractive index factors for various raindrop temperatures and radar wavelengths are plotted in the bottom plot of Figure 2. These were computed by using the top plot curves in Figure 2 as inputs into equations (3) and (2). For these cases, it is clear that ∣K∣2 varies between 0.92 and 0.94 at VHF band. Therefore, in the quantitative analysis of rain using VHF radars, it is also safe (within a 1% or 0.05 dB error) to use the standard weather radar approximation that
However, if the temperature profile above the VHF radar is known (e.g., from radiosonde measurements), then the model described here can provide a more precise value for ∣K∣2.
where Pr is the backscatter power input into the radar antennas (expressed in W), PTx is the power output by the radar transmitter (in W), eT is the antenna efficiency during transmission, Dmax is the maximum directivity of the antenna, λ is the radar wavelength (in m), L is the transmitted pulse length (expressed in units of m), and L/2 is the range resolution. Variables θ, ϕ, and r correspond to the zenith, azimuth, and range (the spherical coordinates), respectively. The range gate is centered at R, and the values R − L/4 and R + L/4 correspond to the radial boundaries of our range gate (near-range and far-range boundaries, respectively). The radar reflectivity η is expressed in m−1. F is the normalized, one-way polar diagram of the radar antenna. We have assumed a square transmitted pulse although in reality this can be untrue (see Appendix A).
 In order to solve equation (13), the main challenge is within the multivariate integral, since the coefficients outside this integral are simply hardware constants (that will be derived in the following sections). Therefore let us focus on this multivariate integral. It is common practice to assume that the spatial variability of η is negligible within a one-gate sampling volume; that is,
if and are in the main lobe of the polar diagram, and if r ∈ [R − , R + ].
 The integral in square brackets in equation (15) is easy to obtain:
 Therefore we only have to deal with the expression
 Let us now focus on solving integral I, and particularly on the antenna pattern F. In the following section, two approaches are presented for solving equation (17).
2.2.1. Analytical Derivation (Gaussian Lobe)
 Assume that F is a Gaussian lobe; that is,
where θ0 is the half beam width at one-way half power. Therefore, by combining equations (17) and (19), we have that
 However, we know that
Equation (22) is verified in Figure 3, where the numerical computation of the right and left sides of equation (22) confirm the agreement within 10−6 units. Given the shape of F in Figure 3 (small dynamic range in θ), it is also safe to assume that the spatial variability of η is negligible within the sampling volume. Therefore we verify equation (14) as well.
 With these assumptions, equation (20) can take the following shape:
 We can now solve the integral in equation (23) by substitution, with
⇒ π2 2 ln2 ≫ θ02 ⇒ θ0 ≪ 3.70 radians = 212°; which is valid all the time.
 Therefore the radar equation will be given by
 Notice that equation (28) is equivalent to other earlier radar equations [e.g., Hocking, 1985, equation (33a); Probert-Jones, 1962, equation (3)]. In general, traditional radar equations do not deal with the power input into the antennas during reception, Pr, but with the power detected at the receiver (i.e., eRPr, where eR is the antenna efficiency during reception). For our analysis, we consider eR during the calibration stage, which is described by Campos et al. . Taking this into account, equation (28) will differ from more traditional expressions only at the factor [R2 − (L/4)2 ]. This factor comes from integral I0 in equation (16). Traditional radar equations generally assume that the radar range resolution is much smaller than the range of the sampling volume, and therefore
Equation (29) is inaccurate for VHF radars when the ranges are comparable to the transmitted pulse lengths. Equation (28) is therefore a more general radar equation than the ones previously published in the literature.
 It should be noted that the assumption in equation (19) is just an approximation that does not consider sidelobes in the antenna pattern nor the pulse shape. However, if somehow we know the antenna polar diagram valid for a particular radar of interest, we then can solve equation (17) numerically. As an example, we present the case of F that is valid for the McGill VHF radar (given in Figure 4). This antenna pattern was provided by Modular Antenna Radar Designs of Canada , the company manufacturing this radar system, and it was obtained from accurate numerical computations of the antenna array response to an input power. Notice that here
The most relevant details in the structure of F can be observed from Figure 5, which indicates that the half beam width at one-way half power for this radar is 2.3 degrees.
 Solving equation (17) by using the antenna pattern in Figure 4 implies dealing with the integrand expression [F(ϕ, θ )]2 sin θ. Figure 6 plots (in solid lines) cross sections for this expression, similar to the ones in Figure 5. For comparison, the corresponding curves for F being a Gaussian lobe (as in section 2.2.1) are also plotted (in dashed lines). The main lobe of (F2 sin θ) lies at zenith angles between zero and five degrees. As well, the main differences between the Gaussian lobe approximation and the computed antenna pattern are located only within the sidelobes (i.e., θ between 5° and 90°). The numerical computation of integral I (a solid angle, in sr) gives as a result
with an uncertainty of 10−8 sr (i.e., 10−8 is the only digit that will vary if the computation resolution is increased). Note that the analytical expression for I, derived from equations (19)–(25) in section 2.2.1, for our case in which θ0 = 2.3°, gives (also in sr)
 Therefore, from equations (18) and (31), the radar equation for our system is given by
 Notice that equation (33) applies only to range gates within the antenna far-field region (also known as the Fraunhofer region [e.g., Ulaby et al., 1981, p. 117–121]). For the McGill VHF radar, the far field would begin at around 1.7 km range. At ranges smaller than the far-field range, the antenna polar diagram in Figure 4 is no longer valid.
 There are other hardware factors that, although they do not invalidate equation (33), can affect our ability to interpret Pr (i.e., the power received at the antennas) from the power output by the radar signal processing, Pout. The most important one is the recovery times of the radar receiver (after being hit by the transmitter pulse). In the McGill VHF radar, this effect manifests as an abrupt decrease in the power intensities as we descend in range. We have noticed this effect at the 2.0 km gate and below. For example, systematic power differences between the 2.0 km and the 2.5 km gates (the second gate not being affected by these hardware factors) are already of the order of 9 dB. We have then performed our precipitation analysis only at range gates above 2 km.
 There are a few other antenna parameters that depend on F and that are worth obtaining. We compute them numerically as follows. The solid angle of the one-way main lobe, which describes the effective width of this main lobe, is given in sr by
Notice that the 5° integration limit (in the zenith angle) comes from Figure 6, which indicates that the main lobe can be located at θ between 0 and 5 degrees. Also note that we would obtain ΩM = 6.446 × 10−3, if we would have used the approximation that the solid angle of a single-lobe radiation pattern is equal to the square of the half-power beam width [Ulaby et al., 1981, p. 102]. The solid angle of the one-way full antenna pattern is given in sr by
Finally, the solid angle of the two-way main lobe is given (in sr) by
which, using equation (31), implies that about 85% of the radar signal is transmitted and received from the two-way main lobe; that is,
2.3. Extracting the Rain Signal From VHF Power Spectra
 Concerning requirement 2, from section 1, we should notice that the automatic separation of the rain signal from the total VHF received power represents an interesting challenge in terms of radar signal processing. On the one hand, Doppler spectra measured by VHF radar during rain events present clearly separated modes. One mode corresponds to the clear air signal (the slowest) and the other to rain signal (the fastest). Since ground clutter has to be previously removed, we use both a signal-processing software for ground clutter filtering [Hocking, 1997] and a radar antenna layout particularly designed for good ground clutter suppression (larger than 100 dB in two-way mode). One spectrum example is presented in Figure 7, which corresponds to observations by the McGill VHF radar at a range gate centered at 2.5 km height (i.e., the gate between 2.25 and 2.75 km above the ground level). This spectrum has a population of scatterers peaking at −3.5 Hz (i.e., a Doppler velocity of about −10 m s−1, typical magnitude for raindrop fall velocities), and a slower population peaking at −0.05 Hz (i.e., a Doppler velocity of −0.14 m s−1, a weak downdraft). We have noticed that, at rain rates of about 4 mm h−1 or higher, it is not rare to observe rain spectral peaks being as strong as (or even stronger than) the clear air peak. On the other hand, part of the clear air signal often overlaps within the rain spectral range.
 To deal with this challenge, we developed a method for extracting the rain signal out of the total Doppler power spectra that is valid for any vertically pointing VHF radar. This method has been developed from an empirical basis, and it is described as follows. Our method starts with the raw spectra measured by the VHF radar (i.e., noncalibrated spectra, expressed in receiver arbitrary units per spectral bin, au Hz−1). For a given range gate, a spectrum is obtained every 35 s, for a spectral range within −10.0 and 10.0 Hz, and a spectral bin resolution of 0.067 Hz. The ground clutter signal has already been removed by a notch filter near 0 Hz (see Hocking  for details on the Doppler power spectra derivation).
 The second step consists in finding the clear-air spectral peak. To do this, we search for the four largest power density values located in the spectral range between −3 and 10 m s−1. Notice that these vertical Doppler velocities correspond (in our radar) to Doppler frequencies between −1.0 and 3.45 Hz. After observing several thousands of power spectra taken by the McGill VHF radar, we have determined that the clear-air peak is generally located within these Doppler velocities. If the four largest power densities are spaced at velocity intervals larger than 1.5 m s−1 (for our radar, frequency intervals larger than 0.5 Hz), then we stop the procedure and conclude that no clear-air signal can be retrieved. Otherwise, we compute the average frequency for these points, and the frequency bin for the clear-air peak, fj, will be the one closer to this average frequency. For the McGill VHF radar, approaching targets will correspond to negative frequencies (and downward, negative Doppler velocities). The vertical line in Figure 7 (near zero Hz) indicates the clear-air peak obtained for this particular case (i.e., −0.1 m s−1).
 During the third step, we subtract the clear air signal from the recorded Doppler power spectrum, and the remaining spectrum will then be the one corresponding to precipitation. We assume that the clear-air spectrum is symmetrically distributed around its peak. Therefore the clear-air signal at n spectral bins to the right of the clear-air peak should be the same (on average) as at n spectral bins to the left of the clear-air peak. In the recorded Doppler spectrum, we will not expect to have precipitation signal to the right of the clear-air peak. Precipitation signal will be present only to the left of the clear-air peak, since (for the vertical beam direction) precipitation Doppler velocities are always more negative than clear-air Doppler velocities. Therefore it is safe to assume that the rain power density is given by
where Sprecip(fn) is the Doppler power density of precipitation at the nth spectral bin (in W per Hz), S(fn) is the Doppler power density of recorded spectrum at the nth spectral bin (in W per Hz), j is the spectral bin corresponding to the clear-air peak, and i is any given spectral bin.
Figure 8 presents the result of applying equation (39) to the Doppler power spectrum in Figure 7. From multiple observations of the performance of this method with real data, we have estimated that the largest Doppler frequency that we can retrieve in the rain spectrum is located at 1.0 Hz to the left of the clear-air peak. Therefore
where fprecip corresponds to all Doppler frequencies in the retrieved precipitation spectrum, and fmin is the smallest Doppler frequency of the retrieved precipitation spectrum. The largest Doppler frequency in equation (40) roughly corresponds to a 0.8 mm raindrop, and it is simply the smallest raindrop that our method can retrieve without being contaminated by the corresponding clear-air spectrum. Since raindrops are easily smaller than 0.8 mm, some underestimation (a few tenths of a dB) by the VHF radar is expected because of this truncation of the largest Doppler frequencies in the rain spectra.
 Notice that fmin corresponds to the Doppler velocity of the largest precipitation particle. We assume that this Doppler velocity matches the terminal velocity of a 5.8 mm raindrop, falling in a standard atmosphere [International Civil Aviation Organization (ICAO), 1993] according to the altitude adjustment by Beard ; that is,
where v is the terminal fall velocity (in m s−1) for a raindrop of diameter De (in mm) at any given height, v0 is the terminal fall velocity (in m s−1) of that drop at sea level, ρ is the air density around the falling raindrop at the given height (in kg m−3), and ρ0 is the air density at sea level (in kg m−3). For the change of air density with height, we use the values of the ICAO standard atmosphere [ICAO, 1993]. The terminal velocity of this hypothetical, largest raindrop is given in Figure 9, and the computation for Figure 9 uses De = 5.8 mm, v0 = 9.17 m s−1 [from Gunn and Kinzer, 1949], and ρ0 = 1.225 kg m−3 [from ICAO, 1993]. Therefore the smallest Doppler frequency of precipitation (fmin) depends on the height of the radar range gate, according to the top x axis in Figure 9. For reference, the spectral regions located within - 10.0 Hz ≤ f < fmin, and within (fj − 1.0 Hz) ≤ f < fj, are plotted as dotted lines in Figure 8. We eliminated these regions from the precipitation spectra since they still contain some remnants of nonprecipitation signal.
 In the last step, we integrate the precipitation power densities Sprecip over the Doppler spectral range in equation (40). As a result, we are finally able to express the VHF integrated precipitation signal. Notice that the input VHF spectrum, S(fn), can be expressed in any signal strength units (e.g., power in arbitrary units or W, reflectivity in m−1, or reflectivity factor in mm6 m−3) per frequency bin (i.e., Hz).
2.4. Calibrating the VHF Spectra
 To deal with the requirement 1, from section 1, we calibrated the VHF power density spectra using the method described by Campos et al. ; that is,
where the subscript out correspond to the radar raw output (expressed in the arbitrary units of the analog-to-digital converter, in the radar receiver), the subscript cal corresponds to the calibrated power (expressed in W), and the subscript sky corresponds to the values derived from a calibration using the sky noise. Therefore the calibration equation of power densities (S) for the ith spectral bin is given by [Campos et al., 2007, equation (24)]
where Scal(fi) is the calibrated spectral density (in W) at the Doppler frequency bin fi, S'out(fi) is the measured spectral density (in arbitrary units) at fi, and Ncoh is the number of coherent averages. Notice that the sampling frequency, fsampling = (PRF/Ncoh), is used here for correcting the fact that not all the Doppler spectral range has been stored during signal processing (only spectral densities within ±10 Hz are being kept). Table 2 provides the values we use for the constant terms in equation (44).
Table 2. Parameters of the McGill VHF Radar
Transmitted wavelength (λ), m
Peak transmitted power (PTx), kW
Antenna efficiency (eT)
Transmitted pulse length (L), km
Pulse repetition frequency (PRF), kHz
Number of coherent averages (Ncoh)
First calibration coefficient (Asky), W
−1.797 × 10−14
Second calibration coefficient (Bsky), W au−1
2.095 × 10−20
2.5. Validating Our Rain Measurements
 In order to measure rainfall reflectivity factors, using only observations from a VHF radar, we first extracted VHF rain signals (expressed as power Pr, in W) applying the method already described in section 2.3. Then we combined equations (1) and (33) in order to express the rain signal as reflectivity factor Z. For this procedure, the values in Table 2 were used. In addition, we required that the VHF radar measurements be taken during an event of widespread precipitation, having a melting level much higher than the lowest range gate of our radar. These requirements provided a sufficiently large data set of rain measurements at least at the very first range gate. We prefer to focus on rain measurements (instead of any other precipitation particles) because this will avoid the inconvenience of not knowing the exact ∣K∣2 value for solid or melting particles. Measured equivalent reflectivity factors Ze are then simply equal to theoretical reflectivity factors Z (from equation (1)). As well, rain signals are easier to separate (from clear-air signals) than snow signals. For our radar data set, the lowest-range gate is between 2.25 and 2.75 km height. It is not often that widespread precipitation over Montreal presents bright bands above these heights. However, we managed to collect VHF data during several precipitation events (more than 23 hours of rainfall) that fulfill these requirements.
 Colocated with the McGill VHF radar, we operated a Precipitation Occurrence Sensor System (POSS, described by Sheppard ) for these precipitation events. POSS is a bistatic, X band (10.5 GHz, 2.85 cm), continuous-wave, Doppler radar. This sensor points upward and detects precipitation particles in its sampling volume, which is located only a few centimeters above the instrument. The POSS allowed us the measurement of raindrop size distributions at the ground, and from these, the radar reflectivity factor was computed by using [e.g., Rogers and Yau, 1989, p.190, equation (11.7)]
where Z is given in mm6 m−3, N(De) is the raindrop size distribution (in mm−1 m−3), and De is the diameter of a sphere that is equivalent to the raindrop (in mm). The Z values obtained from drop sizes at ground were then compared to the VHF reflectivity factors obtained aloft. The comparison is presented in section 3.2.
3.1. Expressing VHF Rain Signal as Reflectivity Factor
 VHF power density spectra (expressed in arbitrary units, au), were selected for a precipitation event occurring on 9 September 2004. This day corresponded to the passage of the remnants of hurricane Frances over the radar site. We selected the range gate located between 2.25 and 2.75 km height. A new power spectrum was obtained every 35 s. Considering the transmitted pulse length and the half beam width at two-way half power of the VHF radar (i.e., 500 m and 1.6°, respectively), these observations are representative of a sampling volume (per unit time) of the order of 2.3 × 105 m3 s−1.
 We first calibrated the raw VHF spectra (in au Hz−1) by using equation (44) and Table 2. Then, for each particular calibrated spectrum (in W Hz−1), we subtracted the noise to the total spectral densities. It is important that we subtract the noise at each calibrated power spectrum, since the next step is to combine spectra taken at different times (for smoothing), and these spectra may not share the same noise. The noise level is estimated here by computing the median spectral power densities in the outer 1 Hz of the spectrum at each end (i.e., near −10 and 10 Hz) and then using the minimum value of these two estimates. Notice that this method for noise estimation is a modification of the method by Hocking , where he uses the mean of the outer spectral densities instead of the median. We prefer to use the median because it is much less affected by extreme power density values, which would be artifacts generated by nonmeteorological targets.
 At this point, the calibrated noise-subtracted VHF spectra (now a signal expressed in W Hz−1) were time smoothed by computing (for each spectral bin) the median value within a 10-min moving window. The reason for applying this smoothing is to homogenize the volume representativeness of the VHF and POSS observations.
 We then obtain average reflectivity densities (in m−1 Hz−1) in the following stage. After calibration, noise subtraction, and smoothing of the power densities, we substituted in equation (33) the input values of Pr by the power density at each spectral bin. We also replaced the output values of (also in equation (33)) with the average reflectivity density (in m−1 Hz−1) at each spectral bin. After this, we multiplied the average reflectivity densities by 2/λ in order to express these spectra in units of reflectivity per Doppler velocity bin (i.e., m−1 (m s−1) −1).
 For each spectral bin, the average reflectivity density (in m−1 (m s−1) −1) was input as a substitute for ηprecip into equation (1). As a result, we obtained VHF Doppler spectra of the reflectivity factor (i.e., the clear-air plus precipitation spectra, valid for the entire spectral range). From these Doppler spectra, we then extracted the precipitation-only spectra, SVHF, following the procedure in section 2.3. An example of these spectra is presented in Figure 10, where the clear-air plus precipitation spectrum is plotted as a continuous line, and the SVHF spectrum is plotted as a dotted line.
 Finally, all SVHF spectra were integrated in order to obtain a time series of VHF reflectivity factors, ZVHF. We eliminate those spurious or very weak observations where ZVHF was less than 10 dBZ. Notice that, from a climatological relation between reflectivity and rain rate valid for Montreal (e.g., Z = 210 R1.47, by Lee and Zawadzki ), Z = 10 dBZ corresponds to a rain rate of R = 0.13 mm h−1. The resulting VHF time series is plotted in Figure 11 as a continuous line, for a precipitation event lasting about 10 hours, on 9 September 2004.
3.2. Comparing VHF and POSS
 In order to validate our VHF measurements of rain reflectivity factor, measurements of raindrop size distributions where taken at ground level by the POSS instrument. The POSS is perhaps the drop size sensor with the largest sampling volume currently available. Its drop size distributions are representative of a sampling volume (per unit time) sized between 0.32 and 190 m3 s−1 (depending on the drop diameters, De [Campos and Zawadzki, 2000]), located at a height of about 2 m above ground, and measured at 1-min resolution.
 From the drop size measurements, we computed the reflectivity factors as a function of drop diameters, Z(De) = N(De) De6. The POSS diameter channels were converted into Doppler velocity channels by first using the following polynomial fit to the observations by Gunn and Kinzer :
where v0 is the terminal fall velocity (in m s−1) at sea level for a raindrop of diameter De (in mm). Each terminal velocity at sea level, v0, where then converted into a terminal velocity at 2.5 km height (the center of our VHF range gate), v, by using equations (41) and (42) in an ICAO standard atmosphere [ICAO, 1993]. Finally, the air vertical velocity (obtained from the clear-air peak in the VHF spectrum, as in section 2.3) was added to the 2.5-km raindrop velocities to obtain the Doppler velocity channels. (An equally valid comparison would remove the clear-air velocity from the VHF rain spectrum, and then would match this spectrum to the POSS observations. In that case, however, the comparison would correspond to the ground conditions, instead of the conditions at 2.5 km height.)
 At this point, we computed the Doppler spectra of reflectivity factors, SPOSS, from the ratio between the reflectivity factor at each diameter channel and the Doppler-velocity width of the corresponding spectral bin. These SPOSS spectra were then smoothed by computing the median value within a 10-min moving window, for each spectral bin. An example of the results is presented in Figure 10, where the smoothed SPOSS spectrum is plotted as a dashed line. The general features in this Figure 10 are typical of those we have observed at other times, and they indicate good agreement between the VHF and POSS spectra. We can now proceed, in the following paragraphs, with a more quantitative comparison.
 For each particular time of observation, we integrated the smoothed SPOSS spectra over the diameter range in order to obtain a POSS reflectivity factor, ZPOSS. We also eliminate observations where ZPOSS was less than 10 dBZ, as we did with ZVHF. Figure 11 compares a 10-hour time series of ZPOSS (plotted in dashed line) and ZVHF (in continuous line). The VHF systematic underestimation is clear during the second half of the period. However, before we quantified this bias, we corrected a small time lag found between the two time series. The magnitude of this time lag was obtained from the cross correlation between ZPOSS and ZVHF. We found that the time when the cross correlation reaches its maximum is 1.8 min for this data set, which corresponds to the time lag between the two time series. This time lag relates to a mismatch between the instruments clocks, as well as the time the raindrops took to fall from 2.5 km (where the VHF radar measured them) to 2 m above the ground (where the POSS measured them). We also found a maximum cross correlation of 0.83 units for this data set, which corresponds to the linear correlation coefficient when the time lag is corrected. This high correlation coefficient is indicative of the good efficiency in our method for extracting the precipitation signal out of the VHF Doppler spectra (i.e., section 2.3).
 We compared simultaneous measurements of ZPOSS and ZVHF during several precipitation events occurring on 24 May and 9 September 2004. In some cases we detected rain only in one of the two instruments. This is expected because of the different volumes that the POSS and the VHF radar represent, for example in cases when the raindrops directly above our VHF radar never fall over our POSS. However, we were able to collect more than 23 hours of rainfall simultaneously measured by our POSS and VHF sensors (a total of 2308 pairs).
Figure 12 presents a scatterplot for these ZPOSS and ZVHF observations, obtained after applying the time lag correction explained in the previous paragraph. For the most intense rates of rainfall, the VHF radar seems to measure lower rates of rainfall than the POSS. However, the situation is the opposite for the numerous observations at moderate and weak rates of rainfall. Given the spread on the Figure 12 scatterplot, we decided to quantify the VHF-POSS comparison simply by providing an average bias. (Any other function of best fit would be statistically equivalent to a linear function of slope one and offset equal to the average bias.) From this data set, the average bias of ZVHF (with respect to ZPOSS) was obtained from the ratio between the total ZPOSS and the total ZVHF (totals integrated over the whole observation period). This ratio has a value of 2.55, which indicates that VHF reflectivity factors are about 4 dB lower than the reference POSS values. This is not a large difference considering the fact that the measurements from these instruments do not represent exactly the same volume in space. Using a radar at X band that points vertically, we verified that the differences between ZVHF and ZXband (when using similar sampling volumes) are of the order of 1 dB [Campos, 2006]. For reference, Figure 12 also plots (as a continuous line) the linear relation corresponding to this 4 dB bias. The correlation coefficient between ZPOSS and ZVHF time series was also computed, which has a value of 0.82 (0.76 for the reflectivity factors expressed in dBZ). This high correlation coefficient, obtained in such complex conditions, validates our analysis methods.
4. Discussion and Conclusions
 This work extends the operational capabilities of the VHF radar to measure precipitation intensity (in units of mm6 m−3) in addition to air velocity. To accomplish this, we presented the mathematical derivation of the VHF radar equation. We have also validated the assumption that ∣K∣2 is 0.93 ± 0.01 for rainfall measured by VHF radars. In addition, we provided an efficient method for extracting the precipitation signal out of VHF Doppler spectra. These aspects have been tested using rain observations taken by the McGill VHF radar and by a POSS distrometer. In particular, we compared VHF reflectivity factors (using equation (33)) with the corresponding reflectivity factors from reference raindrop sizes.
 It is safe to ignore the occurrence of any specular (Fresnel) echoes in our analyses. In fact, the echoes observed by our VHF radar are predominantly due to turbulence (Bragg), from ground up to about 4 km height (including the 2.5 km height of our analysis), during spring, summer and fall conditions. We have verified this by the absence of spikes in the analyzed Doppler spectra, and by the operational observations of the aspect sensitivity factor (i.e., values greater than 5°, as in work by Hocking and Hamza ) over the VHF radar site. In addition, we would eliminate any specular echoes when we apply the 10-min or 3-min median smoothing to the VHF Doppler spectra.
 We acknowledge the fact that the POSS and VHF measurements correspond to different spatial volumes. On one hand, they correspond to different ranges. The VHF measurements correspond to the range at 2.5 km, while the POSS observations correspond to 2 m height (above the ground). On the other hand, the magnitude of the VHF and POSS sampling volumes are very different. As well, the precipitation being measured aloft may not fall directly below, but it can be horizontally advected by the wind. We diminished the problem of representativeness by applying a time smoothing for both the VHF and POSS observations. We also corrected the POSS raindrop velocities for the changes of air density with height. As well, we selected cases of widespread precipitation, where the vertical and horizontal gradients of reflectivity are generally small. In spite of these complex sources of uncertainty, we found a 4-dB bias between drop size measurements and VHF time series, which is mainly due to representativeness errors. Even better, the linear correlation coefficients between ZPOSS and ZVHF observations were of the order of 0.8. These results validate our entire analysis, which includes not only the derived VHF radar equation, but also the method for extracting precipitation signals out of VHF power spectra, and the VHF radar calibration [as in the method by Campos et al. .
 Our retrieval of rain reflectivity factor (from observations of a single VHF radar) has not yet considered the effects of the space-variable reflectivity and antenna sidelobes. We then recognize that our radar equation in (15) is appropriate when dealing with radars that have a narrow transmitted beam and high range resolution. Relationship (15) may not be valid for radars with antenna pattern having significant side lobes (e.g., the McGill VHF radar). The reason is that the radar will receive additional power from scatterers located at the same distance but in a different direction than the range gate of the main beam. Therefore the radar equation (13) is the one to be solved. For the particular case of the McGill VHF radar, we have obtained equation (33) from equation (15), which applies for scatterers in the far-field region and assumes a constant reflectivity within one-gate sampling volume. Future work will include the solution of equation (13) in a space-variable field of reflectivity, as well as the application of our methods in the analysis of precipitation formation.
Appendix A:: Radar Equation
 To begin the derivation of our radar equation, we considered a hypothetical monostatic, vertically pointing, VHF radar. Figure A1 depicts this radar during transmission.
 For an isotropic radar antenna, the transmitted power flux within a small and finite area (perpendicular to the radiation direction) is given by
where Pt is the total power transmitted by the antennas toward the space, r is the range, and dAt is the finite area perpendicular to the radiation direction. Recall that equation (A1) gives the power flux density per unit area, also called intensity of radiation. However, for a real antenna, we have that the power flux is given by
where PTx is the power input into the antennas by the transmitter hardware, and eT is the antenna efficiency. D is the directivity (as a function of azimuth ϕ and zenith θ), and it is given as the ratio between the power flux transmitted by the real antenna and the power flux that an ideal isotropic antenna would transmit; that is [e.g., Ulaby et al., 1981, p.102, equation [(3.22)],
where F is the normalized (i.e., its maximum value is one) one-way polar diagram (or antenna pattern), and Dmax is the maximum directivity (i.e., the D value when the zenith angle is equal to the radar beam direction).
 During backscattering of the radar transmitted pulse (Figure A2), we have that the scattered power from targets contained in a volume V (i.e., the sampling volume) is given by
where dPs is the scattered power and dAs is the average scattering cross section of the targets. This cross section is given (in spherical coordinates) by
where η is the radar reflectivity (expressed in units of m−1). Variable η is also called the scatterer cross section per unit volume, and it assumes that power is scattered isotropically with an intensity equal to that of the backscattered radiation [e.g., Hocking, 1985].
 During reception of the backscattering power into the antenna (Figure A3), the following relation applies for the scattered power flux:
 In this analysis, we do not need to consider the antenna efficiency during reception, eR. Instead, we consider this efficiency during our calibration procedure [Campos et al., 2007], when applying the conversion between the backscatter power input into the antennas, Pr, and the power output by the radar signal processing, Pout.
 In order to obtain the radar equation, let us solve equation (A8) within the limits of a volume confined into a given range gate. This implies that
where L is the transmitted pulse length (expressed in units of m), and L/2 is the range resolution. The range gate is centered at R, and the values R − L/4 and R + L/4 correspond to the radial boundaries of our range gate (near-range and far-range boundaries, respectively). In equation (A8), η is usually convolved with the shape of the transmitted pulse. This is so because the received signal is a convolution between the reflectivity profile and the radar transmitted pulse [e.g., Hocking and Rottger, 1983, section 4]. For simplicity, we approximate here the transmitted pulse as a square pulse, and this implies that the convolution between η and the pulse is simply equal to η.
 When considering the second range gate, strictly speaking, Pt would not be the one given by equation (A2). Instead, it would be only the power that passes the first range gate without being backscattered (i.e., the power incident into the first range gate minus the power backscattered in this same first gate). However, for any given range gate, the power scattered is six or more orders of magnitude smaller than the incident power. Therefore it is safe to assume that the incident power (per unit solid angle) at any given range gate is the same as the power (per unit solid angle) incident into the very first range gate, i.e., the one given by equation (A2). This is known as the Born approximation [e.g., Ulaby et al., 1986, p.1066]. Therefore equation (A9) is still valid for any range gate.
 The authors are indebted to Barry Turner, from the Department of Atmospheric and Oceanic Sciences at McGill University, for proofreading the first version of this manuscript. The McGill VHF radar is operated and funded by the Marshall Radar Observatory.