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.
 Consider the complex refractive index given by [e.g., Ulaby et al., 1986, p. 2018]
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
and T is the temperature in degrees Celsius. Notice that equation (8) is valid for a wide span of temperatures; that is, − 20°C ≤ T ≤ 60°C. (For a more general relationship of ξS as a function of T, see Fernández et al. ). From Liebe et al. [1991, p. 667, equation (2a)], we also have that
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).
Figure 1. Relative dielectric constant for pure liquid water, from the analytical equations by Debye  and the empirical equations by Liebe et al. , i.e., equations (6)–(11). The dashed lines represent the real component, and the solid lines correspond to the imaginary part. The top plot corresponds to a temperature of 25°C, and the corresponding measurements from Table 1 are plotted in red. The bottom plot corresponds to the relative complex permittivity at various temperatures.
Download figure to PowerPoint
Figure 2. (top) Complex refractive indexes and (bottom) scatterer dielectric factors for pure liquid water at various temperatures and wavelengths. In the top plot, the dashed lines represent the real component, and the solid lines correspond to the imaginary part. Values are computed from the data in Figure 1 and equations (2)–(5).
Download figure to PowerPoint
Table 1. Measurements of Dielectric Properties for Liquid Water at 25°Ca
|f, Hz||ξ′||(ξ″/ξ′) × 104|
|1.0 × 108||78.0||50|
|3.0 × 108||77.5||160|
|3.0 × 109||76.7||1570|
|1.0 × 1010||55||5400|
 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.
2.2. Deriving a VHF Radar Equation
 In order to accomplish requirement 3, from section 1, we start from a general form of the radar equation (i.e., equation (A9), derived in Appendix A):
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).
 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.
Figure 3. Gaussian approximation of F, for θ0 = 2.3°. The solid line corresponds to the expression inside the right-side integral in equation (22). The dashed line (on top of the solid line) corresponds to the expression inside the left-side integral in equation (22).
Download figure to PowerPoint
 With these assumptions, equation (20) can take the following shape:
 We can now solve the integral in equation (23) by substitution, with
 From equations (18), (23), and (25) we obtain
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.
2.2.2. Numerical Derivation (Antenna Polar Diagram)
 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.
Figure 4. One-way antenna pattern (also known as polar diagram, F) for the McGill VHF radar. The concentric circles correspond to the zenith angles in the x and y axes. The azimuth angles start clockwise from the positive y axis. Geographic north is located at 48.7° azimuth.
Download figure to PowerPoint
Figure 5. Cross section of the one-way antenna pattern. (top) Transect in Figure 4 along the x axis, at the y axis equal to zero. (bottom) Transect in Figure 4 along the diagonal, where the x axis is equal to the y axis. The one-way half-power half beam width (at 2.3° zenith angle) is indicated by dashed lines.
Download figure to PowerPoint
 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)
Figure 6. Cross section of the integrand expression F2 sinθ. The continuous lines use the one-way antenna patterns in Figure 5, and the dashed lines correspond to the Gaussian antenna pattern in Figure 3.
Download figure to PowerPoint
 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
The maximum directivity is given by [e.g., Ulaby et al., 1981, p.102, equation (3.21)]
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.
Figure 7. Doppler power spectrum observed in rain by the McGill VHF radar (over Montreal). For this example, on 9 September 2004, at 1529:51 UTC, the beam points vertically, and the range gate centers at 2.5 km. The vertical line (near 0 Hz) represents the frequency bin where our method has found the peak in the clear-air spectrum. This spectral peak corresponds to a downward vertical velocity of 0.1 m s−1.
Download figure to PowerPoint
 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.
Figure 8. Precipitation spectrum (solid line) extracted from the Doppler power spectrum in Figure 7 and equation (39). The dotted line to the right of the solid line corresponds to the spectral region located within (fj − 1 Hz) and fj, where fj is the frequency bin for the clear-air peak. The dotted line to the left of the solid line corresponds to the spectral region where Doppler frequencies are smaller than a threshold value fmin. The value of fmin is defined by Figure 9.
Download figure to PowerPoint
 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.
Figure 9. Raindrop of 5.8 mm diameter falling at terminal velocity in an ICAO standard atmosphere. The fall velocity at 0 km height corresponds to observations by Gunn and Kinzer . The top x axis defines the value of fmin to be used in equation (40). For example, the range gate at 2.5 km height corresponds to fmin = −3.61 Hz.
Download figure to PowerPoint
 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||5.77|
|Peak transmitted power (PTx), kW||40|
|Antenna efficiency (eT)||0.631|
|Transmitted pulse length (L), km||1|
|Pulse repetition frequency (PRF), kHz||6|
|Number of coherent averages (Ncoh)||16|
|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.