a. Radar equation with a parallax correction factor

The radar equation for meteorological targets is described in various texts and articles, including Doviak and Zrnić (1984), Smith (1986), and Battan (1973). For the discussion here we note that the received power reflected from clouds or precipitation is proportional to the size of the radar sample volume (V):

View Expanded

where P_r is the average received power referenced to the digitizer input, V is the radar sample volume, Z_e is the equivalent radar reflectivity factor, β is a constant, and r is the range to the center of the sampling volume. For a single-antenna radar with a Gaussian beam the volume (V) can be approximated as a simple function of antenna half-power antenna beamwidth (θ_o) and pulse length (τ), as described in Doviak and Zrnić (1984).

For a dual-antenna radar, Eq. (1) is multiplied with a parallax correction factor [f(r)], which is a complex function of radar geometry and transmit and receive antenna characteristics, and the radar equation becomes

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In Eq. (2), V is the sample volume of one antenna, while f(r) accounts for the range-dependent overlap of the transmit and receive antenna beams.

b. Numerical model of dual-antenna parallax

The sample volume for a single-antenna pulsed radar with a Gaussian beam is given by

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In Eq. (3) the antenna points in the z direction, which coincides with the range dimension (r), and only the main beam is considered since side lobe contributions are negligible for most MMW cloud radar systems. The parameters σ_x(z) and σ_y(z) are obtained from the antenna half-power beamwidth (θ_o). Ferraro (1992) shows that, for a Gaussian antenna pattern,

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Because antenna beamwidth may not be identical in the xz and yz planes, σ_x(z) and σ_y(z) are explicit in Eq. (3).

The sample volume for a two-antenna radar is derived for the configuration shown in Fig. 1, where the beam for one antenna (No. 1) is fixed pointing in the z direction, and the beam direction for the second antenna (No. 2) varies to simulate alignment error. The angles θ_s and ϕ_s quantify alignment errors, or offsets, from the z direction in the yz and xz planes, respectively. The distance Δ represents the physical separation of the two antennas, which lie along the y axis. Using this geometry, we model the sample volume for a dual-antenna radar as the overlap between the transmit and receive antenna beams:

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The parallax correction factor [f(r)] in Eq. (2) is defined as the ratio of the dual-antenna radar sample volume to that for the single-antenna radar:

View Expanded

where z is replaced by r in Eqs. (3) and (4).

Note that Eq. (6) can be simplified by eliminating dependence on pulse length (τ), because f(r) is only sensitive to τ at short distances. This property of f(r) is illustrated in Fig. 2, which shows f(r) for Δ = 1 m and θ_o = 0.2° using several pulse lengths. Beyond a few hundred meters the different curves are indistinguishable from one another. Because single-antenna MMW cloud radars have transmit/receive switches with finite switching speeds that limit their minimum distance to a few hundred meters, eliminating τ from Eq. (6) has little impact on the present set of comparisons. We therefore approximate f(r) as

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where

View Expanded

When Eqs. (8) and (9) are substituted into Eq. (7), cτ/2 is found in both the numerator and denominator, thus eliminating dependence on pulse length. Equation (7) is represented by the solid line in Fig. 2.

Application of Eq. (7) to the Penn State radar system yields the results in Figs. 3 and 4. As Fig. 3 shows, f(r) is symmetric versus ϕ_s and drops off rapidly as the absolute value of ϕ_s increases. The factor f(r) is equally sensitive to θ_s, but the dependence of f(r) on θ_s is not symmetric, as the two families of curves for positive and negative values of θ_s illustrate in Fig. 4. (Note that, in Fig. 4, ϕ_s = 0 and the dotted curves represent the baseline correction that is used when the antenna beams are aligned parallel to one another.) In practice, the beams are aligned by maximizing the echo from a distant target, such as a cirrus cloud, and applying the baseline curve. It is normally assumed that the two antenna beams are exactly parallel with θ_s = 0 and ϕ_s = 0. However, Fig. 4 makes clear that f(r) is extremely sensitive to alignment errors of the 0.2° beamwidth antennas used on the Penn State radar.

Antenna misalignment leads to reflectivity errors and to a general reduction in system sensitivity. At near range the reflectivity error can be larger or smaller than the baseline correction depending upon the sign of θ_s. At far range misalignment leads to underestimation of radar reflectivity. These calculations indicate that angular errors on the order of the half-power beamwidth (θ_o) translate into reflectivity errors and sensitivity reductions on the order of 10 dB. For angular errors of twice the half-power beamwidth, sensitivity is reduced by approximately 50 dB.

The dependence of the baseline f(r) on antenna separation (Δ) is presented in Fig. 5 for several altitudes. As Fig. 5 illustrates, f(r) is sensitive to Δ at low altitudes. For example, at a height of 250 m, a 15-cm (or 15%) increase in Δ translates into a 4-dB decrease in f(r). However, this sensitivity decreases significantly with height.

a. Low-altitude liquid clouds

As illustrated in Fig. 4, the shape of f(r) at low altitudes is sensitive to the sign of the alignment error. Therefore, deep low-level liquid clouds, heavy fog, and precipitation are ideal hydrometeor targets for estimating f(r), θ_s, and ϕ_s, as well as δβ, for a dual-antenna radar system. Such a case of low-level liquid clouds and fog occurred on 11 April 1994 at the U.S. Department of Energy (DOE) Atmospheric Radiadion Measurement (ARM) Program Southern Great Plains (SGP) Cloud and Radiation Testbed (CART) site, when the Penn State and UMass CPRS radar systems were collocated for a cloud observation experiment. The CPRS detected no depolarization on 11 April; hence, we can assume there were few flying insects that could contaminate the cloud data (Sekelsky et al. 1998). Time-averaged vertically pointing radar reflectivity profiles are plotted in Fig. 6, showing that the CPRS Ka-band (33 GHz) and W-band (95 GHz) reflectivity measurements diverge with height. This difference between the two sets of profiles is caused by frequency-dependent attenuation (i.e., scattering and absorption) from cloud droplets and frequency-dependent absorption from water vapor and oxygen (Lhermitte 1990). Because of the possibility of scattering from relatively large drops (i.e., “Mie scattering”) and differences in cloud and water vapor absorption, f(r) is calculated using only W-band radar measurements.

Ratios Z_dual/Z_single of the reflectivity data illustrated in Fig. 6 are presented in Fig. 7, where the measured ratios of Z_dual/Z_single are averaged over 46 min and offset by −0.98 dB, which is the difference δβ in UMass and Penn State radar constants as determined from the offset of the measured from computed ratio curves. A least mean-squares fit of f(r) in Eq. (7) to the measured data leads to angular offsets of θ_s = 0.052° and ϕ_s = ±0.068° for the Penn State radar transmit and receive antennas. Because parallax dependence on ϕ_s is symmetric (Fig. 3), it is not possible to resolve the sign ambiguity of ϕ_s.

The sensitivity to changes in θ_s and ϕ_s of the least root-mean-square difference fit of the model to the data is shown in Fig. 8 for the solutions θ_s = 0.052° and ϕ_s = +0.068°. In the figure, contours of the root-mean-square difference between the measured parallax and model are plotted as a function of ϕ_s and θ_s. For example, the 5% contour indicates solutions with root-mean-square differences within 5% of the minimum value.

Summary histograms of the liquid cloud reflectivity data from 11 April 1994 for altitudes from 200 to 1200 m both before and after applying corrections for parallax, calibration offset, and attenuation are presented in Figs. 9 and 10. Reflectivity data plotted in Fig. 10 have been corrected for gaseous absorption and cloud extinction using models developed by Liebe (1985) and Liebe et al. (1989), together with sounding data from the ARM SGP CART site. Note that these corrections assume that the hydrometeors are small compared to the W-band radar wavelengths.

b. High-altitude cirrus clouds

Unfortunately, the deep low-level liquid clouds and fog that we used to estimate f(r) on 11 April 1994 were observed only once at the beginning of the ARM SGP CART site April 1994 cloud experiment. However, high-altitude cirrus clouds occurred after 11 April 1994, and we used reflectivity measurements from these clouds to monitor the product f(r)δβ throughout the remainder of the experiment. Because f(r) is nearly independent of range at cirrus altitudes, it is not possible to separate variations of f(r) from those of δβ.

A detailed comparison of cirrus measurements for 15 April 1994 is shown in Figs. 11 and 12, which correspond to Figs. 8 and 9 for the liquid cloud case. At cirrus altitudes the baseline parallax correction for perfectly aligned antenna beams is negligible. However, the Penn State reflectivity values are approximately 6.3 dB lower than the UMass data for this day. For the liquid cloud case of 11 April, f(r)δβ is only −0.98 dB at these altitudes. Further comparisons of the product f(r)δβ from cirrus cloud reflectivities made during the experiment are plotted in Fig. 13.

Several possibilities may explain the large variations present in Fig. 13. The Penn State values are consistently lower than the UMass values, suggesting either a parallax problem, lowered values of β for the Penn State system, or elevated values of β for the UMass system. Because component performance problems will not increase β, but rather reduce it, we can conclude that either the antenna alignment or β, or both, for the Penn State radar was not stable over the course of the experimental period.