Many quantitative analyses of radar signal require a radar calibration. Established calibration methods for VHF radar provide only partial information about antenna or receiver parameters. We propose that a more complete approach to calibrate VHF radar can be obtained by combining multiple calibration methods. To test this, we developed a calibration technique by combining a first calibration method that compares the recorded VHF signal to power coming from a noise generator and a second calibration method that compares recorded VHF signal to cosmic radiation. We derive four equations that allow us to retrieve antenna and receiver-chain parameters (such as noises, efficiency, and gain), and four other equations for the corresponding errors. In addition, we develop an equation for calibrating Doppler spectra. To test our calibration technique, we collected an extensive data set from the McGill VHF radar. For validation, we performed a third calibration using measurements of voltage and impedance to compute power losses in the antenna transmission lines. On the basis of our equations, we have found the values for the antenna and receiver-chain parameters in the McGill VHF radar, and their corresponding uncertainties, and we have compared these to the energy losses obtained by the third calibration method. The antenna efficiencies derived by our technique and by the third calibration method agreed within 0.5 dB. Furthermore, analyses of our calibrated Doppler spectra in rain demonstrate the potential of this calibration technique for absolute measurement of precipitation by wind-profiler radar.
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 Numerous meteorological applications have been made possible through VHF radar techniques (see for example the reviews by Rottger and Larsen , Gage , and Gage and Gossard ). Measurements of absolute backscatter power by VHF (Very High Frequency) radars are one important aspect of atmospheric studies of clear-air turbulence [e.g., Hocking, 1985] and precipitation [e.g., Wakasugi et al., 1986]. There is, however, a central issue that must be dealt with before attempting any quantitative study of precipitating weather systems with VHF radars: the radar calibration.
 In a typical setting for VHF radars (Figure 1), a pulse of known power PTx is sent from the transmitter hardware toward the antennas. Actual antennas also have power losses, in particular due to impedance mismatches and thermal-energy dissipation in the antenna structure and cables. Therefore the power radiated to space Pt is actually smaller than the power available at the antenna input PTx. The ratio of these quantities is the antenna transmission efficiency (or radiation loss factor, eT = Pt / PTx). Further power losses are also experienced during the antenna reception, i.e., between the point at which the backscattered power is input into the antenna (Pr) and the point at which the power is output from the antenna toward the receiver hardware (PRx). The reception efficiency is then given by eR = PRx / Pr. In addition, the transmitter can leak small amounts of power into the receiver, cables, and antenna structure, generating electromagnetic noise at the radar VHF frequency. These leaked powers (expressed here as antenna noise Na and receiver noise NRx) can be particularly significant during the radar reception period. We then can write:
 Of further relevance, however, is the fact that the power output after signal processing (Pout) has been usually converted, by an analog-to-digital-converter (ADC), into numbers with arbitrary units (au). In the linear region of a receiver with linear amplifiers, the conversion from W to au is mathematically expressed by the gain of the receiver chain, gRx, such that
Notice that gRx is not an efficiency because efficiency denotes some loss (of power). Variables gRx and NRx (the noise of the receiver chain) combine several factors: amplification in the receiver, the conversion factor from Watts into arbitrary units (made by the ADC), and the signal processing (made by the computer). Therefore gRx and NRx are global converting factors representative of a whole receiving chain.
 Measurement or retrieval of several meteorological variables (such as turbulence and precipitation intensity) requires that Pout must be given in Watts (W) instead of arbitrary units (au). A calibration procedure is thus required. The standard calibration procedure involves a noise-generator hardware that is connected directly to the receiver. However, this calibration does not take into account the antenna parameters (antenna noise and efficiencies), nor transmitter characteristics (PTx).
 On the other hand, using known sources of cosmic radiation is a common method for calibrating radio telescopes [e.g., Léna et al., 1998, section 3.5]. We can also apply this method to calibrate VHF radars, given the fact that at VHF band the power from cosmic sources is large, and that this cosmic power can in principle be computed from the Rayleigh-Jeans approximation to the Plank’s law [e.g., Ulaby et al., 1981, section 4-3.3]. Unfortunately, attempts at VHF radar-calibration using cosmic radiation have been reported in the literature only a few times [e.g., Hocking et al., 1983; Green et al., 1983; Campistron et al., 2001]. A calibration from known sources of cosmic radiation is more complex than the calibration from a noise generator. There is also the inconvenience that, when computing the receiver power from the radar equation, we need to know the antenna parameters (for example, Na and eR) independently from the receiver-chain parameters (for example, NRx and gRx); and this cannot be calculated from cosmic-noise calibrations only.
 In this paper, we overcome these radar calibration difficulties by both improving the model for cosmic radio sources and also by combining this method with the known noise-generator method. We present here our new VHF calibration approach that provides estimates of both the antenna parameters and the receiver-chain parameters. We also perform an independent check through a third calibration method, which uses measurements of voltage and impedance to compute power losses at different points along the antenna transmission lines.
 Any radar power calibration involves a comparison between a known power source and the radar power measurement. In the first part of our calibration technique, the known power source corresponds to the input (in Watts) from a noise-generator. In the second part, the known power source is the cosmic radio emissions (in Watts). We then combine both calibrations results to retrieve particular antenna and receiver-chain parameters. We will now explain each part in more detail.
2.1. Noise-Generator Calibration
 For the first part of the calibration, the noise-generator calibration, the radar hardware was configured as in Figure 2. In this case, the power PNG from the noise-generator hardware (N-G) was input into the receiver hardware Rx. This power was digitized by the analog-to-digital converter (ADC) and then sent to a computer, where the signal processing took place. This gave as a result the output power Pout (in arbitrary units, au). The objective here was to obtain a linear relation between the power input by the noise-generator (PNG, in Watts) and the radar power output after all signal processing (Pout, in au); i.e.,
where ANG is the power (noise) generated within the receiver hardware, measured in Watts. BNG corresponds to the conversion factor between the input and output powers, measured in W/au. It should be noted that this calibration cannot be used to obtain any antenna parameters (for example, efficiency and noise). However, we will relate the coefficients ANG and BNG to usual radar parameters later in section 2.3.
2.2. Sky-Noise Calibration
Figure 3 illustrates the second part of our method, the sky-noise calibration. This figure presents a radar hardware configuration where the power received by the antennas comes exclusively from cosmic sources. Under these conditions, a linear relation can be obtained between the VHF cosmic radio emissions (sky power: Psky = Pr, in Watts) and the radar output power (Pout, in au); i.e.,
where Asky corresponds to the power (noise) generated within the radar hardware, measured in Watts, and Bsky (measured in W/au) is the conversion factor between the power received by the antennas and the power output after the signal processing. (We will relate the coefficients Asky and Bsky to usual radar parameters later in section 2.3.).
where the brightness temperatures are both given in Kelvins, and β is the so-called spectral index. Although β varies according to the position in the sky as well as the ratio f2 / f1 (e.g., Roger et al.  present a sky survey for f1 = 22 MHz and f2 = 408 MHz, with β in the range 2.40 to 2.55, and its average is 2.5), it is generally assumed that β ≈ 2.5. This assumption leads to a relative error smaller than 3% in the retrieved temperature at VHF band [Campistron et al., 2001].
where kBoltzmann = 1.381 × 10−23 J/K is the Boltzmann constant [e.g., Mohr and Taylor, 2003; recall also that J/K = W/(Hz K)], and BPFwidth is the band-pass filter width of the radar receiver, in Hertz. The derivation of equation (6) takes into account the facts that the cosmic radiation is unpolarized, and that our linearly polarized antenna will then collect half of the incident (unpolarized) cosmic power [Ulaby et al., 1981].
 This sky-noise calibration only provided information about the antenna and receiver parameters in a general sense. Particular values such as antenna efficiency or receiver-chain noise could not be retrieved in this manner. However, we were able to retrieve these antenna and receiver-chain parameters by combining both the sky-noise calibration and noise-generator calibration methods. The next section explains the procedure.
 In addition, expressions for the uncertainties in the antenna and receiver estimates (eR, Na, gRx, and NRx) were derived from the following expression [e.g., Press et al., 1986, page 505]:
where σ2(f) is the variance uncertainty of the function f, which is a function that depends on variables x1, x2,…, xn−1, xn. As well, σ2(xi) is the variance uncertainty for the i-th variable. Equation (12) was then applied to equations (7), (8), (10), and (11) in order to obtain the following one-standard-deviation uncertainties:
where the values for σ2(Asky), σ2(Bsky), σ2(ANG), and σ2(BNG) were obtained from the square of uncertainties in the coefficients of equations (3) and (4). Notice that the uncertainties in Pout are already considered when we compute the uncertainties in coefficients ANG, BNG, Asky, and Bsky.
 Once we have the calibration of the radar measured power (Pout), we proceed with a calibration of the power densities. To do this we produce Doppler spectra from a recorded time series and then proceed as follows. We assume that the spectra are recorded at steps Δf. Then, from equation (4) we know that
where Δf is the spectral bin resolution (in Hz). The variables Sout(fi) and Ssky(fi) correspond to the Doppler power densities at the i-th spectral bin (a total of n spectral bins), given in au/Hz for the variables with the subscript “out” and in W/Hz for the variables with the subscript “sky”. The previous equation can also be expressed as
Therefore the power-densities calibration equation for the i-th spectral bin is given by
Note that, for the derivation of equation (17), the linear relation in equation (4) must be applied for the powers in linear units.
 It is not rare to have radar signal processing performing coherent averaging (for example, in the McGill VHF radar). Under these conditions, the full spectral range is defined by the radar sampling rate as follows:
where PRF is the radar pulse repetition frequency and Ncoh is the number of samples used for the coherent averages (given in Table 1).
Table 1. McGill VHF Radar Parameters
Transmitted Wavelength (Frequency)
5.77 m (52.0 MHz)
Peak Transmitted Power
One-Way Half-Power Half-Beam Width
Pulse Repetition Frequency (PRF)
Band Pass Rx Filter Width (BPFwidth)
Number of Coherent Averages (Ncoh)
Doppler Spectral Range (DSR, After Signal Processing)
Approx. 35 s/profile
Spectral Bin Resolution (Δf)
Location (Lat., Long.)
45.409° N, 73.937° W
 Another common signal processing practice (also performed by the McGill VHF radar) is to store only Doppler power spectra within a range of interesting frequencies [i.e., Sout'(fi)]. If the full spectral range [i.e., Sout(fi)] corresponds to Doppler frequencies within ± 0.5 fsampling, then the quantity Pout, corresponding to the full spectral range, is given by
and Pout′ is the total power integrated within the stored Doppler spectral range (DSR).
Equation (17) has then to be modified according to equations (18) and (19). For this, we use the fact that the power density is conserved for a white-noise spectrum. As well, we recognize that the application of coherent averaging (of in-phase and quadrature time series, as it is done in our signal processing) reduces both the full spectral range [reduction already included in the definition of fsampling; i.e., equation (18)] and the measured Psky (since the spectral density magnitude is preserved at all frequency bins; e.g., Lyons [1997, p. 321]). Therefore the calibrated power-density spectra, Scal, at the frequency bin fi, must be such that
where Ncoh is the number of samples used for the coherent averages, Δf is the spectral bin resolution, and Psky is given in Watts. From here, n is the number of stored spectral bins (not longer the total number of bins). In addition, equation (4) provides us with a conversion between Watts and arbitrary units. Therefore from equations (4) and (20) we obtain that
where Sout'(fi) is the measured spectral density (in au/Hz) at the Doppler frequency bin fi. Then, by combining equations (21) and (22), we have that
Since Δf = DSR / n, we thus get,
2.5. Operational Background
 To show the application of our method we used data from the McGill VHF radar [described by Campos and Hocking, 2003] working under the configuration described in Table 1. For the sky-noise calibration, we could use any oblique direction as well, but the vertical direction already provides enough observations to successfully calibrate the radar. Our radar system carefully monitors, on a regular basis, the offset and quadrature of the real and imaginary channels. The nonorthogonality of these channels introduces small errors on instantaneous power estimates, which are always less than about 10%. As well, the analog-to-digital converter uses a 16-bit digitizer, and typical recorded amplitudes are at least two orders of magnitude greater than the digitalization resolution. Therefore the errors (on instantaneous power estimates) at the ADC level (induced by the data quantization) are in the order of 1% or less. Notice that all these errors do not introduce biases in our VHF precipitation measurements.
 The signal processing used here was the same as in the work of Hocking [1997, section 4]. Every 35 s, a profile of 45 Doppler power spectra (300-point discrete-spectrum within a spectral range of ± 10.0 Hz, for 45 range gates between 0.5 and 23.0 km) was produced. We integrated each of these spectra in order to obtain corresponding Pout' values; i.e., the integrated powers (in au) within the Doppler spectral range (DSR, see Table 1).
 As described in section 2.1, during the noise-generator calibration, a small modification was made in the reception hardware. The noise-generator output was connected to the receiver, instead of the line from the transmitter-receiver switch. Then, different noise sources were obtained by changing the factor F in the noise-generator hardware. One unit increment in F was equivalent to a 290-Kelvin increase in brightness temperature. At F = 0, the noise generator still introduces a small amount of power into the receiver. This amount depends on the noise generator temperature (approximately 290 K) in a manner similar to equation (6). Therefore power input by the noise generator into the radar receiver was given by:
where PNG is the noise-generator power (in Watts, measured in the radar receiver just after the band-pass filter). As before, kBoltzmann is the Boltzmann constant and BPFwidth is the band-pass filter width of the radar receiver (given in Table 1), in Hertz.
 For the second part of our calibration, there was no need to disconnect the transmitter, or to alter the normal operation of the radar in any way. We kept the radar hardware and software working as usual (i.e., with line from transmitter-receiver switch connected to receiver, as in Figure 1). The known power sources from cosmic radio emissions (in Watts) were then compared with the corresponding radar integrated power (in au) measured only at very high range gates (between 17.5 and 22.5 km). At these ranges, backscattering of the transmitted power and other terrestrial VHF radio sources is negligible. Thus the radar received powers, at these high ranges only, were considered as coming exclusively from cosmic sources.
3.1. Noise-Generator Calibration
 The noise-generator calibration was performed using observations made on 21 October 2004, a day without precipitation. The results are presented in Figure 4. For a given PNG value, there are 45 Pout' values plotted in the x axis. These Pout' values correspond to the 45 radar range gates (between 0.5 and 23.0 km) available at each profile. Considering the 300 spectral points and the 45-range gates used for computing Pout', we estimate [from Petitdidier et al., 1997, equation (10)] that the expected error in these computations is 100% 1 %. We then computed the Pout values plotted in the abscissa (x axis) of Figure 4 by using equation (19). The range of F values, from 0 to 30 units, was sampled twice (the two data sets are represented in Figure 4 as small crosses). A Chi-square linear fit [Press et al., 1986, section 14.2] was then used to obtain the relation
where the units are given in square brackets, and the uncertainties correspond to one standard-deviation errors in the coefficients estimates. The relationship (26) is presented as a line in Figure 4.
3.2. Cosmic-Noise Calibration
3.2.1. Sky Map
 The cosmic noise power Psky at the radar operating frequency (52 MHz) was obtained from a sky brightness temperatures map at 45 MHz (the closest available frequency). We used data published by Campistron et al. , which corresponds to epoch-J1999 equatorial-coordinates. These coordinates, right ascension and declination, are continuously changing in time, primarily as a result of the precession of the equinoxes. We then had to convert the figure coordinates from the epoch J1999 to the epoch J2004 (the epoch of the radar observations). For this, we used the standard procedure given in section B42 of The Astronomical Almanac [Nautical Almanac Offices et al., 2003]. The resulting sky map is presented in Figure 5, which has a resolution of 1.5 min in right-ascension hour and one degree in declination angle.
 To test the reliability of this 45 MHz map, we compared its brightness temperatures at a particular declination angle (matching our VHF radar observations) with the corresponding values from the maps by Milogradov-Turin and Smith  and by Roger et al. . The first map corresponds to a 38-MHz frequency and epoch J1967, and the second map corresponds to 22-MHz frequency and epoch B1950. For this comparison, we then had to convert the 45, 38, and 22 MHz temperatures to 52 MHz by using equation (5) with β = 2.5. As well, we had to precess the coordinates to a common astronomical epoch, in this case J2004 (the epoch of our VHF radar observations). Finally, we use linear interpolation in order to obtain the temperature value at the declination of 45.409° and with a right ascension resolution of 0.25 hours. Figure 6 presents the comparison of sky brightness temperatures for the three maps. The only significant disagreement is with the 22 MHz map, at right ascension between 19 and 22 hours, probably due to contamination by the strong signal from Cygnus A. However, there is general agreement between the three sky maps, which indicates the reliability of the 45-MHz map.
 Considering our radar antenna pattern and time resolution, the radar observations and the sky map did not match in resolution (the data sets representativeness are not the same). Therefore the sky brightness temperatures, at the radar declination, were smoothed in order to resemble our VHF radar resolution. We did this by convolving the 45-MHz map (Figure 5) with a direct numerical simulation of the one-way antenna pattern (i.e., the antenna one-way polar-diagram). This antenna pattern was provided by the radar manufacturer (Mardoc Inc., of London, Ontario, Canada) and it is presented in Figure 7. Notice that, as the kernel of the convolution operation, we used only a section of the full antenna pattern (zenith angles smaller than 13°, having the same resolution as the sky map, i.e., 1.5 min per 1°). Zenith angles greater than 13° were not used since they imply a kernel outside the sky map. In any case, the sidelobes of the antenna pattern located outside 12° zenith angles are not significant (their magnitudes are generally smaller than −15 dB). For all right ascension hours (at a resolution of 1.5 min), the convolution was performed with the kernel centered at the declination of our radar observations (i.e., 45.409° declination angle, at the dashed line in Figure 5). The result for this convolution, between the sky brightness temperatures and the antenna pattern, was used as input for equation (5). The resulting 52-MHz brightness-temperatures are plotted in Figure 9 as the red line.
3.2.2. Sky Noise
 Between 14 and 17 October 2004, the McGill VHF radar was operated according to the specifications given in Table 1. We selected the period in Figure 8, where the sky noise could be assumed to be due only to cosmic sources. From the measured Doppler power spectra, we computed the total integrated power (for spectral Doppler frequencies between −10.0 Hz and +10.0 Hz) at ranges between 17.5 and 22.5 km. At these high ranges, the Doppler power spectra received by VHF radars are basically formed by white noise, and when we integrate these spectra we obtain the so-called sky noise. We then used equation (19) to correct the total integrated power for not storing the full Doppler spectra. Considering the 300 spectral points and the 11 range gates used for computing the total power of the sky noise, we estimate [from Petitdidier et al., 1997, equation (10)] that the expected error in these computations is 100% 2 %. Notice in Figure 8 that the temporal evolution of the sky noise power has a 23-hours-56-min cycle (i.e., a sidereal day). This confirms the dominant cosmic origin of the noise observed by our VHF radar.
 In some cases, a few extreme, spurious power observations can be measured by VHF radars, and these observations correspond to signals from noncosmic sources (for example, human interference or broadcasting). These signals must be eliminated before proceeding with our calibration. In Figure 8, we have already filtered out most of these spurious data by eliminating sky noise values that were six or more median-absolute-deviations away from the median sky noise (the median for the whole observation period).
 By knowing the direction in the sky at which our radar is pointing at a given time, we can compute the equatorial coordinates (right ascension and declination) of this direction. We computed the radar pointing directions (for the cosmic sky-noise periods in Figure 8) by using standard astronomical procedures valid for the epoch J2004 [e.g., Lang, 1999]. Since our radar was located at a fixed longitude and elevation angle (vertical direction), our cosmic sky noises correspond to a fixed declination with varying right ascension. This is shown in Figure 9, where the VHF cosmic sky-noises (black and blue points, in 105 au) are plotted as a function of right ascension. Since our radar measurements correspond to a declination of 45.409°, we can compare our integrated powers with the corresponding 52 MHz sky brightness temperatures computed in section 3.2.1. These temperatures are over-plotted in Figure 9 as red points (in kiloKelvins).
 The black observations in Figure 9, which correspond to nighttime radar-measurements taken between 23.1 UTC (7:06 pm local time) and 11.1 UTC (7:06 am local time), match well with the corresponding sky temperatures (red points). However, the observations in blue, which corresponds to daytime measurements taken between 11.1 UTC and 23.1 UTC, tend to be above the corresponding sky brightness temperatures. For radio waves, daytime sky-noise is very challenging to analyze. On one hand, we have the power contribution from the Sun, which for our VHF band corresponds to a brightness temperature in the order of 105 K [Subramanian, 2004]. This temperature corresponds to about 10−13 Watts [from equations (5) and (6)]. On the other hand, there is the ionospheric absorption of radio waves, which affects all cosmic radiation when passing through the D and E ionospheric layers (at 60- to 100-km altitude). Ionospheric absorption is a well-known phenomenon, which is controlled in part by solar activity (i.e., sunspot number). Observations taken during the night are practically free from these inconveniences. We therefore filtered out all the measurements taken between 7:06 am and 7:06 pm (i.e., approximately between sunrise and sunset).
3.2.3. From Arbitrary Units to Watts
 In order to obtain the sky-noise powers, the brightness temperatures (red points) in Figure 9 were multiplied by the Boltzman constant and the radar Band-Pass-Filter width [i.e., equation (6)]. However, the radar measurements (black points) in Figure 9 still had a large amount of scatter, which could complicate the empirical derivation of the coefficients in equation (4). We reduced this scatter in the following manner: for each Psky observation (in Watts), we selected all radar observations (black points in Figure 9, in arbitrary units) that were within 45 s around the Psky hour angle. (Recall that the resolution of the Psky observations is 1.5 min.) The median of these radar observations was then the radar output power, Pout, to be matched to the Psky observation. The matched pairs are shown in Figure 10 as right ascension time series, where the line corresponds to the sky-noise powers (Psky, in Watts), and the points correspond to the Pout values (in arbitrary units).
 To eliminate the unlikely possibility of having a lag between the two time series in Figure 10, we computed the cross correlation between the two series. The maximum cross correlation was found at lag time equal zero (not shown). This means that no time lag can be found between the two time series, and if there is one, it will be less than the interval between two consecutive observations (i.e., 1.5 min). Thus no lag-time correction was applied.
Figures 9 and 10 indicate minor departures between observed and expected sky noise patterns (Pout and Psky, respectively). We have minimized these departures by filtering out spurious observations, using the methods described in section 3.2.2. Small remaining departures, however, can be due to small, spurious power observations from noncosmic sources, to small departures of the cosmic sources from the corresponding sky-map values, to local departures in the spectral index (β) from its average 2.5 value, and to changes in the ionospheric composition [e.g., Campistron el al., 2001].
 We can also visualize the data in Figure 10 by plotting Psky as a function of Pout. This leads to the scatterplot in Figure 11 and the linear relation for power in Watts as a function of power in arbitrary units (the line in the figure). As described in section 2.2, we expect a linear relation, but the uncertainties about the variation of β in space [see equation (5)] could deviate the expected linear relation slightly. Fortunately, Figure 11 indicates that this small effect can be neglected in our case. Therefore a linear relation between power in Watts and power in arbitrary units was derived by minimizing the Chi-square error statistic, as in Press et al. , and it is given as follows:
As before, the units are given in square brackets, and the uncertainties correspond to one standard-deviation errors in the coefficients estimates.
3.3. Radar Hardware Coefficients
 In order to calculate the values of antenna and receiver-chain parameters, we need to compare our two sets of calibration equations [i.e., the sky-noise calibration in equation (27) and the noise-generator calibration in equation (26)]. The comparison is shown in Figure 12, where the sky-noise calibration is plotted as a dashed line and the noise-generator calibration is given as a continuous line. Of course, the slope of the noise-generator calibration is smaller than the slope of the sky-noise calibration, and we expect this difference from equation (10).
 The hardware parameters can now be computed from equations (7), (8), (10), and (11) simply by noticing the correspondence between equations (4) and (27), and between (3) and (26). As well, their corresponding uncertainties are estimated from equations (13) to (16). These values are given in Table 2. Notice that the antenna efficiency in Table 2, eRx = 44%, refers only to reception. The antenna system was originally designed to maximize transmitted power, and the overall power losses on transmission are estimated to be less than 2 dB; i.e., eT = 63%.
Table 2. Hardware Parameters
1.14 × 10-14 W
4 × 10−16 W
1.081 × 1020 au/W
3 × 1017 au/W
3.70 × 105 au
7 × 103 au
 To compute the noise temperature of the receiver chain, TRx, we use an equation similar to equation (6); i.e., NRx(W) = kBoltzmannTRxBPFwidth, where NRx(W) is the receiver-chain noise expressed in units of Watts. This receiver-chain noise can be computed from the NRx value in Table 2 and the slope in equation (26), or from the offset in equation (26). In both cases, we obtain that the noise temperature of the receiver chain (for the McGill radar) is about 619 ± 12 K.
4. Antenna Matching Unit
 To validate our results, we will now study the various subcomponents of the radar antenna that are most likely leading to power losses. This analysis leads to a third calibration method, which will provide an independent estimation of the antenna efficiency.
 In this regard the antenna transmission lines are the most important. In order to minimize energy losses, the impedance in the antenna aerials is matched to the transmitter impedance through an arrangement of coaxial cables. These assemblies of cables are then called the antenna matching units. For the McGill VHF radar, we use a matching arrangement like the one shown in Figure 13. This includes matching cables made from RG213 coaxial cable, with lengths as indicated in the figure, and beam-pointing boxes that are used to introduce phase delays to the antennas in order to implement beam pointing. The internal details of the beam-pointing boxes are not shown on the figure, but the efficiency of these units will be considered separately in due course. The matching boxes at the transmitter end hold inductors of approximately 75 nH and capacitors to earth of about 60 pF, which are tunable in order to provide final accurate matching.
 The arrangement in Figure 13 includes matching boxes and switching boxes. In order to assess the performance of this arrangement, we have built a slightly simpler system which contains no switching boxes, and used it for performing the third calibration method. This arrangement is shown in Figure 14. We will first discuss the operation of this unit, and consider theoretical efficiencies. Following this, we will report the results of a series of measurements on the system, and compare with theory. Finally, we will return to the original matching arrangement (Figure 13) and make further measurements, which can then be interpreted in terms of our results using Figure 14.
 In order to properly understand the efficiency of an impedance matching system, like that shown in Figure 14, it is necessary to consider both its forward and backward transmission characteristics. The cable impedance is assumed to be 50 Ω. The simplified transmission lines shown in Figure 14 had cables lengths of one half of a wavelength between H and I, one quarter of a wavelength between G and F, one half of a wavelength between E and D, and one quarter of a wavelength between C and B. We assume that the antennas are all tuned to 50 Ω. Where two cables come together as at G/H, the point G (looking out toward the antenna aerials) sees an impedance of 25 Ω. The quarter wave section F-G transforms this to 100 Ω. The point E, looking out toward the antenna aerials, sees an effective impedance of 2 × 100 Ω impedances in parallel, or 50 Ω. The point D, looking out toward the antenna aerials, also sees 50 Ω. Point C sees 2 × 50 Ω impedances in series, and so sees 25 Ω. This maps to 100 Ω at B (looking toward the antenna aerials). Finally, point A sees 2 × 100 Ω impedances in parallel, or 50 Ω. These results are summarized in the fourth column of Table 3.
Table 3. Impedance at Different Points of the Antenna Transmission Lines, for the McGill VHF Radar
Looking Toward Transmitter (Transmitter Terminated in 50 Ω)
Looking Toward Aerials (All Aerials Terminated in 50 Ω)
Measurement ± [0.5 Ω, 1.0°]
Measurement ± [0.5 Ω, 1.0°]
[50.0 Ω, 0°]
[50.0 Ω, 0°]
[50.0 Ω, 0°]
[48.0 Ω, 13.0°]
[33.3 Ω, 0°]
[100.0 Ω, 0°]
[75.0 Ω, 0°]
[75.5 Ω, −1.4°]
[25.0 Ω, 0°]
[26.8 Ω, 5.9°]
[30.0 Ω, 0°]
[31.5 Ω, 5.6°]
[50.0 Ω, 0°]
[83.3 Ω, 0°]
[80.9 Ω, 3.5°]
[50.0 Ω, 0°]
[49.0 Ω, 8.8°]
[45.45 Ω, 0°]
[44.4 Ω, 3.5°]
[100.0 Ω, 0°]
[55.0 Ω, 0°]
[57.5 Ω, −4.0°]
[25.0 Ω, 0°]
[26.5 Ω, 11.8°]
[27.25 Ω, 0°]
[50.0 Ω, 0°]
[27.25 Ω, 0°]
[29.0 Ω, 7.5°]
[50.0 Ω, 0°]
 The fifth column in Table 3 shows actual measurements of the impedances, expressed as magnitudes and angles, as measured by a Hewlett-Packard Vector-Impedance Meter. Agreement with theoretical expectations is good, and differences are due to the facts that (1) the characteristic impedance of the cable was actually close to 51 Ω and (2) slight errors in cutting the lengths of the cables to exact multiples of a quarter of a wavelength. (Optimal cable lengths were determined using a vector-impedance meter, with the cables being open circuit. The quarter-wavelength cables were cut until impedance was zero, and the half-wavelength cables were cut until impedance was maximum.)
 In the previous paragraph we examined impedance transformations in the matching unit, comparing theoretical and experimental values. It is also necessary to examine power transmission, which is best done by looking at voltages at various points along the antenna transmission lines.
 A continuous-wave 52.00-MHz signal, of peak-to-peak voltage equal to 1.16 V (as measured into a Cathode Ray Oscilloscope loaded with 50 Ω), was fed into the point A in Figure 14. In addition, all terminations except that at “I” were given 50 Ω loads. The voltage measured into a 50 Ω load at “I” was then equal to 28-mV peak to peak. It would be expected that the applied power should be equally distributed across all loads, so that if the input power is , then the output voltage should be 29 mV. The total cable length from input to output is 1.5 wavelengths, or 5.71 m, since the velocity propagation factor for RG213 cable is 0.66. This RG213 coaxial cable has a loss factor of 1.3 dB per 30 m at 52 MHz, so losses of 0.25 dB are expected. This should reduce the received signal to a peak voltage of 28.2 mV, consistent with our measured value. Hence the losses on transmission through such a matching unit are about 0.2 dB, mainly due to cable attenuation.
 In considering the system efficiency of a transmit-receive system like this, it is also necessary to consider the return path of the signal. It is well known that with a well-designed antenna array, the sky noise received by the radar is independent of the number of antenna aerials, provided that the sky noise is isotropic in origin. Suppose that a single antenna aerial is used, and fed directly into point A in Figure 14, from where it passes through a transmit-receive switch to a receiver. Let the signal power received be P. Now suppose that 16 antenna aerials are now used, and are fed by the matching arrangement in Figure 14. Each antenna aerial receives power P, but as the signal passes back through the stages of the matching unit, more and more is lost by reflections. Some of it ends up being reradiated by other antenna aerials in the array. In fact, the power expected at the point A due to the signal received at one antenna is only P/16. The accumulated power from all antenna aerials is 16 times this, or P. In terms of polar diagrams, this result can be determined by recognizing that the collection of 16 antenna aerials has a narrower polar diagram than a single antenna aerial, but in terms of the actual matching used, the result arises because of power losses due to reflections on the return path. The received signal strength (and therefore the sky-noise temperature) is thus independent of the number of antenna aerials used. This is a well-known result that is employed in calibrating many radio systems, and it has been used in the early sections of this paper as well.
 To see this more clearly, it is a simple matter to determine the impedances seen at various stages of the matching path looking back toward the transmitter (as opposed to the previous cases, which were determined looking out toward the antenna aerials). To begin, consider the point B (in Figure 14) looking back toward the transmitter. It sees a 50-Ω load in the form of the transmitter, and a 100-Ω load coming in from the other arm of the V-section closest to the transmitter. Hence point B sees 33.3 Ω. This maps to 75 Ω at point C due to the quarter-wavelength section. By working along the matching unit from A to I, the impedances seen in the second column of Table 3 can easily be deduced. In Table 3, column 3 shows experimental values of the impedances, and again agreement between theory and experiment is good.
 It is now necessary to determine the power expected to be received at the point A, assuming that this point is terminated in 50 Ω, and all other cables above the point I in Figure 14 are also terminated in 50 Ω. This can be calculated by looking at transmission efficiencies at each point. For example, a 50-Ω input applied at point I sees an impedance of 27.25 Ω, so a voltage reflection coefficient of (50 − 27.25)/(50 + 27.25) = 0.2945 applies. Hence the reflected power is 8.7% of the original. The transmitted power is therefore 91.3% of the original. This transmitted signal progresses to the junction between G and H, where some signal passes through to G, some is reflected back, and yet more of the signal passes into the adjoining cable and is transmitted into the next termination (or, in a real radar, is transmitted into the next antenna aerial). The signal that is reflected back to I is partly retransmitted from I into the adjoining load, and partly rereflected back to H, and so forth. The signal that passes through G suffers further reflection and splitting at E/F, and so forth. Eventually, only one sixteenth of the original signal arrives at the point A.
 Experimental testing of this pathway was carried out. An input signal of 1.16 V peak-to-peak fed in at “I” produced a signal of 0.26V peak-to-peak at point A. Inserting inputs at other locations similar in location to point “I” gave outputs at the point A in the range 0.24 to 0.27 V peak-to-peak. These results are entirely consistent with the above expectations, and indicate that even on the return path the losses of this matching unit are very modest, and certainly less than 10%, even including losses due to cable attenuation.
 We now return to Figure 13. Having performed the above tests, a subunit of Figure 13 (shaded in the figure) was extracted for further tests. As for the circuit in Figure 14, forward propagation (from the transmitter out to the antenna aerials) was very efficient. For the reverse direction, Figure 15 shows a series of measurements. In this case the input signal was 90 mV peak-to-peak. The beam-pointing units were removed from the circuit.
Figure 15 shows more clearly the distribution of power around the circuit. Notice that the input power is proportional to (90)2 V2, but as before, only about 91% of the input power enters the matching unit, and the rest is reflected back into the signal generator, due to the mismatch at the input. Since all voltages were measured into 50 Ω, we will dispense with converting powers to Watts, and express them in terms of Voltage squared. The transmitted power is therefore proportional to 7200 V2. It should be noted that if all of the power produced at all the other remaining ports are summed, (512 + 252 + … + 10.82 + 272), the result is 5300 V2: less than, but comparable to, the total input power. Some of the signal travels over relatively long paths, up to 3 wavelengths (for example, signal that travels form the input, to I3, and back to one of the antenna aerials), so losses of the order of 0.5 dB are possible due to cable losses and connector losses. If such losses are considered, the total available power is proportional to 7200 × 10−0.05 = 6400 V2, very similar to the 5300 V2 outputted. Of most importance is the fact the signal strength returned to the receiver at I3 is very close to the ideal value of 90/ = 31.8 mV, and some of this lost is due to cable losses. Thus the efficiency of this matching unit for reception is of the order of (27/31.8)2, or in other words the losses are of the order of 1.4 dB. Measurements along other arms gave slightly less losses, and overall the system loss due to this matching unit for the returned signal should be less than 1 dB.
 The above test was repeated, but this time we included the beam-pointing boxes. These units added a further 0.5 to 1 dB to the system losses, varying slightly from one unit to the next. In addition, the cable to the antennas is Andrews ” Heliax, which has a loss of 0.5 dB per 30 meters. Each output port feeds to a separate quartet of four antenna aerials, and distances to the inner antenna aerials are typically 35 meters, and the outer ones are 76 meters. All cables are carefully cut to integral numbers of wavelengths in length. Therefore cable losses are of the order of 0.5 to 1 dB. Some small losses can be expected at the final antenna-matching unit, but they should not be large. Hence due to antenna-matching issues, we anticipate that the overall system efficiency should be of the order of −2 to −3 dB, being comprised of about 1 dB in the matching unit, 0.5 to 1 dB in the beam-pointing boxes, and 0.5 to 1 dB in the cables that carries the signals to and from the antenna aerials. Checks of interpath coupling between different paths in the beam-pointing units showed that such coupling is generally of the order of −30 dB, and this is not likely to introduce further inefficiencies.
 Therefore these antenna-matching calculations give an extreme value of −3 dB for antenna losses in the McGill VHF radar. This laborious estimation have not considered other, less significant, power losses (for example, at the final antenna-matching unit, where the cable feeds four antenna aerials). Consequently, the estimation is in general agreement with the antenna efficiency value obtained in section 3 (from Table 2, eR = 0.442 = −3.5 dB).
5. Precipitation Applications
 When we applied equation (24) to the noncalibrated power spectrum output by our radar signal processing (i.e., power densities in au/Hz), we obtain the calibrated power spectrum of Figure 16 (i.e., power densities in W/Hz). This figure corresponds to a 2.5-km range gate, for a rain event on 9 September 2004, at 14:05:45 UTC. The bimodality is due to the simultaneous detection of clear-air signal (peak near 0.2 Hz) and rain signal (peak near 3.5 Hz). As a reference, we measured at ground level (for the same time) a 1-min rainrate of 13 mm/h. Similarly to our observations in Figure 16, Gage [1990, and references therein] discusses an example of Doppler spectrum showing clear-air and precipitation echoes during light rain. The advantage in our case is that we express our ordinate (calibrated by our technique) in W/Hz, while the power densities in Gage’s Figure 3.7 are in arbitrary units.
 Furthermore, we can convert our calibrated power spectra into reflectivity spectra (expressed in units of m−1/Hz) by using a proper radar equation, e.g. [Campos et al., 2007, equation (28)]
where is the reflectivity averaged over the sampling volume, R is the range (in meters), L is the transmitter pulse length (in meters), Dmax is the antenna maximum directivity, λ is the radar wavelength (in meters), and θ0 is the one-way half-power half-beam width. By integrating the part of the reflectivity spectra that corresponds to clear-air signal (frequencies larger than −1.25 Hz for the observations in Figure 16), we could compute an estimate of air turbulence in precipitation conditions; i.e., energy dissipation rates as in the method by Hocking [1985, Appendix A]. We can also obtain an estimate of the precipitation intensity from the spectra in Figure 16.
 As an example, Figure 17 shows reflectivity-factor spectra obtained simultaneously by the McGill VHF radar and by ground measurements of raindrop-size distributions. The abscissa (x axis) has been changed from Doppler frequency shift, f, into Doppler velocity, V, by using the relation f = 2V / λ. The VHF precipitation spectra are wider and a bit shifted toward the negative velocities. This is because the beam width is larger in the VHF than in the raindrop-size sensor, because air velocities are different in the sampling volume of each sensor, and because the change of air density with height implies a 10% increase in raindrop fall velocity at 2.5-km height [e.g., Beard, 1985]. However, in general there is good agreement between both spectra, which demonstrates the potential of using power spectra, calibrated by our technique, for retrieving meteorological information such as precipitation bulk quantities (for example, reflectivity factor and rain rates). These meteorological variables are typical in radar meteorology, where microwaves are most often used instead of longer-wavelength radio waves. The advantage is that, with the use of VHF radio waves, we can also retrieve information about the air motion independently and simultaneously to the precipitation. We will discuss this application more in detail in our companion paper [Campos et al., 2007].
6. Discussion and Conclusions
 When dealing with the power measured by VHF radars, it is often necessary to convert power units (from the arbitrary units of the analog-to-digital converter) into Watts. A radar calibration is then required. This paper discussed an integrated, multiple-method approach for obtaining this calibration, using noise-generator calibration and sky-noise calibration methods, and intelligent integration of the methods. There are important inconveniences associated with using exclusively one or the other. The noise-generator method requires hardware (the noise generator) that is not always available at the radar site, and the normal operation of the radar has to be interrupted to connect this hardware. Furthermore, the calibration equation that results does not take into account the antenna losses, and is therefore not accurate. On the other hand, attempts to calibrate VHF radars using the sky-noise method have only been reported a few times in the literature. This is most probably related to difficulties in obtaining reference sources of cosmic radiation at VHF band. Although this limitation has now been overcome, sky-noise calibration-methods do not provide independent information on the receiver or antenna parameters. This information on radar parameters is fundamental when applying the radar equation to derive meteorological variables such as turbulence and precipitation.
 We overcame these calibration difficulties by combining the sky-noise and the noise-generator methods. We present here a more complete approach to radar calibration for operations in the VHF band. In addition, our technique allows derivation of several antenna and receiver-chain parameters and their corresponding uncertainties. We give these parameters for the McGill VHF radar in Table 2. The application of our calibration technique to the McGill VHF radar measurements generates calibrated power spectra like the one in Figure 16.
 Another advantage of our calibration technique is that, once the noise-generator part has been applied, the rest of the calibration can be performed during routine observations (without the need for additional hardware or modification of the radar operation). Furthermore, a change in the radar hardware does not require a new noise-generator calibration. We simply perform a new sky-noise calibration [i.e., we obtain Asky(new) and Bsky(new) for equation (4)]. For a change in the radar antenna, the noise-generator coefficients in equation (3) will remain the same. We will then apply our calibration technique using the old noise-generator coefficients and the new sky-noise coefficients. For a change in the radar receiver chain, the antenna efficiency and antenna noise would remain the same. Then, we obtain from equation (10) that
At this point, the new coefficients for equations (3) and (4) are available and our calibration technique can be applied.
 For best implementation of our calibration technique, it is very important to select night observation periods where unknown variations of cosmic power (for example, solar emissions and ionosphere attenuation of the cosmic power) are minimal. It is also important to minimize any noncosmic radio sources (for example, broadcasting signals) from the calibration data. The amount of noncosmic radio sources depends on the radar location (an urban site will probably have much more noncosmic radio sources than a remote site), and the removal of affected periods can be done as in section 3.2.2.
 Our calibration technique does not consider the power losses in the radar transmitter or between the transmitter and the transmitter-receiver switch. In general, these omissions are not very relevant, since the length of the cables between the transmitter and the transmitter-receiver switch are not very long (i.e., very high transmitter efficiencies). As well, radar manufacturers usually provide a calibrated transmitter.
 In order to validate the results from our calibration technique, we applied a third calibration method. The third method corresponded to antenna-matching calculations, which provided an independent estimate of the antenna power lost. We found this estimate to agree with the antenna efficiency derived by our calibration technique.
 This work has concentrated on the correct measurement (in units of Watts) of power by VHF radars. However, we have also demonstrated the potential of using the Doppler spectra calibrated by our technique, in combination with the values of radar hardware parameters derived by our technique, for retrieving meteorological information such as precipitation bulk quantities (for example, reflectivity factor and rain rates). Nevertheless, the derivation of precipitation quantities requires relating the spectra and hardware parameters to a proper radar equation (i.e., the relationship between power and targets backscattering cross sections). As well, a method for separating the precipitation mode from the air mode has to be implemented. We will elaborate more on this application in our companion paper [Campos et al., 2007].
 The authors are indebted to Dr. Bernard Campistron, from the Aeronomy Laboratory at the Midi-Pyrénées Observatory, Université Paul Sabatier (Lannemezan, France), for providing the sky survey data sets by Campistron et al.  and by Milogradov-Turin and Smith . Dr. Tom Landecker, from the Dominion Radio Astrophysical Observatory (Penticton, British Columbia, Canada), kindly provided us the sky survey data set by Roger et al. . The collaboration given by Dr. Trevor Carey-Smith, from the Air Quality Research Branch at the Meteorological Service of Canada (Downsview, Ontario, Canada), when taking the antenna-matching measurements in Figures 16 and 17, was very appreciated. We are also grateful to Dr. Barry Turner, from the Department of Atmospheric and Oceanic Sciences at McGill University, for proofreading the first version of this manuscript. Prof. Linda Cooper and her Science Writing and Publishing class, at McGill University, provided helpful editing for some parts of this manuscript.