Calibrating short-timescale tropospheric phase fluctuations seen by a radio telescope: Limits from subreflector and Cassegrain feed ring radiometer placement

Authors

  • Roger Linfield

    1. Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, California, USA
    2. Now at Ball Aerospace Corporation, Boulder, Colorado, USA.
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Abstract

[1] Water vapor radiometers (WVRs) measure tropospheric brightness temperatures and use those measurements to infer path delay. Calibration of short-timescale phase fluctuations at a radio telescope requires that the WVR and radio telescope sample a similar volume of the troposphere. Using a statistical (Kolmogorov frozen flow) model of tropospheric fluctuations, the short-timescale calibration capability of two WVR configurations has been quantified. The first configuration is a WVR mounted, with its own antenna, on the back side of the main radio telescope subreflector, giving a conical beam that is coaxial with the main cylindrical near-field beam of the large telescope. The second configuration uses a Cassegrain feed ring, with the WVR and radio astronomy feeds at different positions on the ring. This second configuration gives a cylindrical calibration near-field beam, offset in angle to the main cylindrical beam. An important application of short-timescale phase calibration is improving the coherence of high-frequency interferometric observations. For two cases of current/near future interest (86 GHz very long baseline interferometry with the Very Long Baseline Array; 350 GHz observations with the Atacama Large Millimeter Array, ALMA), useful calibration could be achieved with either geometry (coaxial conical beam or offset cylindrical beam). For a coaxial conical beam, a 2° WVR beam width would allow significant coherence improvement, but a beam width ≤1° (full width at half maximum) is needed for optimum performance. For an offset cylindrical beam, the desired angular offset (on the sky) is ≤1° for 43 GHz Very Large Array observations, or ≤0.3° for 350 GHz ALMA observations.

1. Introduction

1.1. Effects of Tropospheric Fluctuations

[2] Radio signals propagating through the Earth's atmosphere have their phase corrupted by irregularities in the index of refraction. The contribution of charged particles in the ionosphere and interplanetary medium to the phase of the signal scales as ν−1, compared to ν+1 for the neutral troposphere (ν is the radio frequency). At frequencies above approximately 3–8 GHz, fluctuations of the density and the humidity in the neutral troposphere produce the dominant contributions to the total phase fluctuations.

[3] Two classes of precision phase measurements of radio signals propagating through the Earth's atmosphere will be limited in accuracy by these fluctuations. The first involves round trip propagation between a radio antenna on the ground and a spacecraft (round trip propagation avoids the phase impurity of a frequency standard on board the spacecraft). Low-frequency gravitational wave searches can be conducted in this manner [Tinto and Armstrong, 1998], and high phase precision is needed for a sensitive search.

[4] The second class of measurements is interferometry, involving simultaneous observations of a celestial radio source with two or more radio telescopes. In connected element interferometry (e.g., the Very Large Array in New Mexico), a single frequency standard is distributed to all the telescopes. In very long baseline interferometry (VLBI), separate high-stability frequency standards (e.g., hydrogen masers) are used at each telescope. In either case, the relative phase stability of oscillators at the different radio telescopes is sufficiently good that the coherence times of observations are limited by phase fluctuations in the local medium (the Earth's troposphere at high radio frequencies). The coherence time decreases rapidly toward higher observing frequency, and is often only 10–30 s at 86 GHz [Rogers et al., 1984]. Interferometric observations require that a source (either the target object or an angularly nearby calibrator) be detectable within one coherence time. The short coherence times at high frequencies result in poor sensitivity. The time variability of the atmospheric coherence time leads to poor calibration of visibility amplitudes, and poor imaging capability.

1.2. Calibration With Water Vapor Radiometers

[5] At radio frequencies, the refractivity of water vapor is roughly 20 times larger than for dry air [Hill et al., 1982; Owens, 1967]. Because of its high refractivity and inhomogeneous distribution in the atmosphere, water vapor is responsible for most of the tropospheric refractivity fluctuations.

[6] Water vapor radiometers (WVR) [Elgered, 1993] measure the thermal emission from tropospheric water vapor along a specific line of sight on the sky. The 22 GHz spectral line of water vapor has generally been used. The atmosphere is optically thin in this line under most conditions, so that brightness temperatures are approximately proportional to line-of-sight column densities. At least two spectral channels are used. The first is near 20.8 or 23.8 GHz, on a shoulder of the line. At these two frequencies, the emissivity is nearly independent of the effect of pressure broadening, minimizing the sensitivity of the brightness temperature to the height distribution of the water vapor. A second sensing channel is near 31 GHz, beyond the spectral line. The emission at this frequency is only weakly sensitive to water vapor, but is strongly sensitive to the presence of liquid water in the atmosphere. This high-frequency channel is thus useful for detection of clouds, which corrupt WVR determinations of water vapor.

[7] The refractivity of water vapor is nearly proportional to ρv/T, where ρv is the vapor density and T is the absolute temperature. Therefore, WVR brightness temperatures can be used to infer the time-variable delay in the same direction, potentially improving the accuracy of radio science measurements [Resch et al., 1984] and the coherence of high-frequency interferometry [Welch, 1999]. Knowledge of the temperature profile in the atmosphere improves the accuracy of the brightness temperature to path delay conversion.

[8] A high-performance WVR-based troposphere calibration system has been designed and built at the Jet Propulsion Laboratory (JPL), in support of radio science measurements on the link between NASA's Deep Space Network (DSN) and the Cassini spacecraft [Tanner, 1998; Keihm and Marsh, 1996]. As a test of the accuracy of this system, the level of phase fluctuations measured with radio interferometry over a 21 km baseline at Goldstone, California, was reduced after application of atmosphere calibration measurements [Naudet et al., 2000].

[9] The Cassini Troposphere Calibration System was designed to optimize calibration of fluctuations on timescales of 100–10,000 s, with performance at 1000–10,000 s given the highest priority. Because of the timescales of interest, the WVR in this system will be located on the ground, approximately 50 m from the axis of the radio telescope used for the link between Earth and Cassini.

[10] The offset location of the WVR causes its sampled troposphere volume to be different (little or no overlap) from the tropospheric volume traversed between the spacecraft and the radio telescope. As a result, accurate calibration of fluctuations on short timescales (<100 s) is not possible [Linfield and Wilcox, 1993]. A different WVR location is needed for calibration of short-timescale radio science measurements or millimeter-wavelength interferometry.

[11] The volume mismatch can be eliminated by integrating a WVR into a beam waveguide (BWG) antenna. Tests of such a configuration have been encouraging [Tanner, 2000]. However, there will be significant challenges in designing such a system, especially the difficulty of splitting the three WVR frequencies from the signals used for the spacecraft uplink and downlink. Such a splitting must be extremely stable, in terms of amplitude loss and added noise. In addition, a calibration capability is desired for non-BWG antennas. It is worthwhile considering other WVR configurations.

[12] Two WVR configurations were analyzed in this study. The first consists of a radiometer mounted on the back of a radio telescope subreflector, as illustrated schematically in Figure 1. The WVR would have a conical beam that is coaxial with the cylindrical near-field beam of the radio telescope. Because of the substantial overlap between the cylindrical and conical troposphere volumes sampled by the radio telescope and WVR, the volume mismatch can be much smaller than for an offset beam. As with an offset location, a radiometer on the back of a radio telescope subreflector can use a clear aperture reflector to minimize sidelobes. This location avoids two of the problems with an integrated BWG location: scattering off the radio telescope feed legs (causing time-variable ground pickup) and the complication of splitting the signal between the radio science and WVR frequencies.

Figure 1.

Geometry of a WVR, mounted on the back of a radio telescope secondary. The radio telescope samples a cylindrical volume. The WVR samples a truncated conical volume. The light gray indicates volumes that are sampled only by the radio telescope or by the WVR. The dark gray indicates the volume that is sampled by both.

[13] The drawbacks of this subreflector location are as follows: (1) It does not completely eliminate the volume mismatch problem, (2) the WVR must have low weight to avoid causing excess flexure of the feed legs, and (3) access to the WVR for maintenance is not as convenient as for an offset location. For reference, the weight of the radiometer in the Cassini Troposphere Calibration System is 150 lb. and the weight of its clear aperture, off-axis parabolic antenna is 60 lb. Minimizing the weight was not a primary consideration in the design of the system.

[14] The second configuration has an off-axis feed in the focal plane of the radio telescope. This feed samples a cylindrical tropospheric volume which is offset in angle from the source direction. The geometry is illustrated in Figure 2. Like the first configuration, this does not completely eliminate the volume mismatch problem. It has the additional problem of scattering (and time-dependent ground pickup) off the feed legs of the radio telescope. It has the advantage of not adding any weight to be supported by the feed legs. Furthermore, it is compatible with telescopes that have Cassegrain feed rings, such as the Very Large Array (VLA), Very Long Baseline Array (VLBA), and the proposed Atacama Large Millimeter Array (ALMA). This configuration has been proposed for the VLA (using a 22 GHz receiver) [Butler, 1999] and ALMA (using a 183 GHz receiver) [Hills and Richer, 2000; Gibb and Harris, 2000], as a means of improving coherence in high-frequency interferometry.

Figure 2.

Geometry of radiometry using an off-axis Cassegrain feed. The radio telescope samples a cylindrical tropospheric volume. The radiometer samples a cylindrical volume that is offset in angle from the source direction. The dark gray indicates the volume that is sampled by both the source feed and the calibration feed. The light gray indicates the volumes that are sampled by only one of the two feeds.

2. Fluctuation Model and Calculations

2.1. Basic Turbulence Model

[15] The refractivity fluctuations were assumed to have a Kolmogorov spectrum, with uniform turbulence strength from the surface to a height of 2 km. The fluctuation level was assumed to be zero above 2 km, which is the mean scale height of the wet troposphere. A refractivity structure constant (Cn) of 1.1 × 10−7 m−1/3 was used; this and a turbulent slab height of 2 km reproduces the mean conditions at the three DSN sites. These values were derived from the magnitude of variations in the zenith delays during multiple years of intercontinental astrometric VLBI observations. In addition, WVR monitoring at Goldstone (the driest of the three DSN sites) yielded data on variability there [Keihm, 1995]. The scaling to other levels of turbulence is discussed in section III (for Allan deviation, the scaling is linear in Cn).

[16] A frozen flow model (the Taylor hypothesis) was used to relate spatial and temporal fluctuations, using a constant wind velocity equation image. Calculations were performed as given by Treuhaft and Lanyi [1987].

2.2. Conical-Cylindrical Mismatch

[17] The volume of wet troposphere sampled by the radio telescope, for incoming plane waves from a compact radio source, was modeled as a cylinder of radius r and length h/sinθsource, where h is the thickness of the turbulent layer (2 km) and θsource is the elevation angle of the radio telescope (i.e., the source direction). The center of the base of the cylinder was defined as the coordinate system origin, and the pointing direction was in the xz plane (The region of tropospheric fluctuations was slightly expanded to include the “corners” of the cylinder that were below z = 0 or above z = 2 km). The wet tropospheric delay seen by the radio telescope was then:

equation image

N is the refractivity.

equation image

The volume of wet troposphere sampled by the WVR was modeled as a truncated cone, whose axis was coincident with the radio telescope axis. The truncation (with a diameter of 2 m) gave a slightly more realistic treatment of the beam of a WVR like the one used for the Cassini troposphere calibration system. More importantly, removing a small region at the apex of the cone significantly speeded up convergence of the numerical integrations. All derivatives of the refractivity structure function become infinite at zero separations, so terms with small physical separations in the numerical integration introduce relatively large numerical errors. The length of the axis of the truncated cone was h/sinθsource. A Gaussian beam profile was used, truncated (for numerical convenience) at a diameter 3.4 times the full width at half maximum (FWHM) of the Gaussian profile. The center of the WVR antenna was at the coordinate system origin. In practice, there would be an extra path length ∼2r in the volume sampled by the radio telescope, but this extra volume is small compared to the total tropospheric path length. The wet tropospheric delay seen by the conical WVR beam is:

equation image
equation image

Here Ωcon is the solid angle of the Gaussian conical WVR beam, which is truncated at an angular radius ψmax, W(u) is the amplitude (weight) of the beam profile an angle u from the axis, and θ(u, ω) and ϕ(u, ω) are the elevation angle and azimuth for rays originating at location (u, ω) in the beam. dWVR is the length (along its axis) of the truncated section of the cone, N is the refractivity, and equation image is the wind velocity.

[18] The error (uncalibrated residual) after subtracting the WVR delay from the radio telescope delay will be:

equation image

In order to derive the expected Allan deviation and coherence, we first must calculate the delay structure function:

equation image

Here the angle brackets (〈〉) represent ensemble averages, or expectation value.

[19] Substitution of equations (1) and (2) into (4) leads to an expression for Dτt) (see Treuhaft and Lanyi [1987] and Linfield [1998] for details), consisting of six-dimensional integrals of the refractivity structure function Dnequation image)

equation image

These integrals were evaluated numerically for all integer seconds up to 200 s. The majority of the cpu time was used for the shorter time intervals.

[20] The actual mismatch will be somewhat different than is described by the above equations. The wet troposphere will be entirely in the near field of the radio telescope beam, but mostly in the far field of the WVR beam. Therefore, the weighting across each beam will be different. In the far field, the weighting will be smooth, and the characterization as a Gaussian profile is appropriate. In the near field of the radio telescope beam, the weighting will have numerous small scale ‘ripples,’ and the characterization (equation (1)) as a uniform cylinder will break down. However, the volume mismatch between a cone and a cylinder occurs on a large scale, and will average over multiple ripples. In light of the typical ∼50% variations in the level of the structure constant, the effect of these ripples should be minor.

2.3. Cylindrical-Cylindrical Mismatch

[21] For the case of a cylindrical calibration beam, offset in angle from the source direction, τerr was the difference between two expressions for τcyl, with different pointing directions. In order to simplify the calculations, the length of the calibration cylinder was set equal to the length of the cylinder in the source direction, even when their elevation angles were different.

2.4. Allan Deviation

[22] The measure of the power in τerr(t) at different timescales is Allan deviation [Allan, 1966]. The Allan Variance σy2t) of a delay process τ(t) is:

equation image

A little algebra leads to:

equation image

The factor of c (velocity of light) is needed when Dτt) is expressed in length2 units.

2.5. Coherence

[23] The coherence Coh(T) of a delay process τ(t) at an observing frequency ν is [Thompson et al., 1986]:

equation image

(Dτ(t) is expressed in length2 units, as in the rest of this paper). With interferometry, both telescopes of a baseline contribute to the coherence loss, with independent fluctuations on the timescales over which coherence loss usually occurs. We can account for the two telescopes in equation (8) by doubling the value of Dτt) from a single telescope.

2.6. Thermal WVR Noise

[24] On short timescales, thermal noise from a WVR is important. With a Dicke-switched, gain-stabilized radiometer, the noise N(tint) in an integration time tint is:

equation image

For the main vapor-sensing frequency channel of 21.5 or 23.8 GHz, 1 K brightness temperature corresponds to ≈6 mm of path delay. Expressing N(tint) in units of path delay, we get:

equation image

Here T100 is the system temperature in units of 100 K, BW100 is the detected bandwidth in units of 100 MHz, and tints is the integration time in seconds.

[25] An optimistic lower bound to the WVR thermal noise contributions to the Allan deviation comes from setting Δt = tint (we lose all information on timescales shorter than Δt), BW = 400 MHz, and system temperatures of 300 K (uncooled amplifier) or 60 K (cryogenic amplifier). (For comparison, the total system temperature for the Cassini troposphere calibration system WVR is ≈600 K). The contribution of thermal noise to the Allan deviation is σyt) = N(tint)equation image/(cΔt), or

equation image
equation image

WVRs operating at the 22 GHz water vapor line cannot do better than these limits. (At sites with very low water vapor column density, the 183 GHz water vapor line is optically thin; measurement at this frequency could give lower thermal noise in path delay units.)

[26] The contribution of thermal WVR noise to the structure function for τerr for each radio telescope is independent of Δt:

equation image

For the WVR parameters given above, and including an extra factor of 2 in the exponent in equation (8) for the effect of two WVRs, we get:

equation image

With an integration time ≥5 s, the coherence loss from WVR thermal noise will be minor at 86 GHz, even for an uncooled WVR. For interferometric observations at higher frequencies, cooled receivers or longer integration times would be needed to minimize coherence loss due to WVR noise. In addition, there will be atmospheric coherence loss during the WVR integration time. As presented in the next section, typical integration times for 95% coherence are: 7 s (86 GHz, low elevation angle VLBI at average sites) and 5 s (350 GHz observations with ALMA). WVR integration times shorter than these values are desired.

3. Results

3.1. Allan Deviation

[27] Figures 35 show the Allan deviation for WVR beam sizes (FWHM) of 1°, 2°, and 4°, for coaxial radio telescope and WVR beams. A 1° FWHM is approximately the smallest beam width that could be achieved with an antenna that (1) could be mounted on the back of a radio telescope subreflector and (2) is underilluminated in order to achieve a very low sidelobe level. (for comparison, the Cassini Troposphere Calibration System WVR has a FWHM of approximately 1°). The elevation angle is 30° and the antenna diameter is 34 m (the size of most DSN antennas) for the results shown in Figures 35. Both the total Allan deviation seen by the large antenna and the Allan deviation of the calibrated signal (τerr) are shown. The thermal noise limits for uncooled (equation 11) and cryogenic (equation 12) WVRs are also plotted in each figure.

Figure 3.

Allan deviation of troposphere fluctuations seen with a 34 m diameter radio telescope. The upper (thick black) curve shows the total fluctuations. The lower (thick gray) curve shows the residual, after calibration with an ideal WVR having a 1° FWHM beam. The thin straight black and gray lines show the limits resulting from thermal WVR noise. Read 1.00E-13 as 1.00 × 10−13.

Figure 4.

Allan deviation of troposphere fluctuations seen with a 34 m diameter radio telescope. The upper (thick black) curve shows the total fluctuations. The lower (thick gray) curve shows the residual, after calibration with an ideal WVR having a 2° FWHM beam. The thin straight black and gray lines show the limits resulting from thermal WVR noise.

Figure 5.

Allan deviation of troposphere fluctuations seen with a 34 m diameter radio telescope. The upper (thick black) curve shows the total fluctuations. The lower (thick gray) curve shows the residual, after calibration with an ideal WVR having a 4° FWHM beam. The thin straight black and gray lines show the limits resulting from thermal WVR noise.

[28] Figure 6 shows the dependence on elevation angle: θsource = 60°, 30°, and 20°, for a 2° FWHM WVR beam. The mismatch error is smallest at high elevation angles, where the WVR beam has less chance to spread out before it reaches the top of the wet troposphere. At low elevation angles (≤30°), there is a large benefit to reducing the beam width below FWHM = 2°.

Figure 6.

Elevation angle dependence of beam mismatch error. All three plots show the total and calibrated (residual) fluctuations for a WVR with 2° FWHM, mounted on a 34 m diameter radio telescope.

[29] Figure 7 shows the Allan deviation for calibration with an offset cylindrical beam. The three panels of the plot are for offset angles of 0.1°, 0.5°, and 2°, for a 25 m diameter radio telescope at 30° elevation angle. The orientation of the offset is 45° from horizontal; tests showed that the dependence on the orientation is weak.

Figure 7.

Mismatch error for a cylindrical calibration beam that is offset in angle from the beam of a radio telescope, as would occur with a radiometer on a Cassegrain feed ring. The telescope diameter is 25 m.

[30] Figure 8 shows the case of an offset WVR location for comparison: 50 m between the WVR and the radio telescope axis, 1° FWHM, and 30° elevation angle.

Figure 8.

This plot shows the consequences of locating the WVR off the radio telescope. The black curve shows the total tropospheric fluctuations. The gray curve shows the residual fluctuations, after application of WVR calibration, with a 1° FWHM WVR beam, offset 50 m from the radio telescope axis.

[31] All these calculations used a wind velocity of 8 m/s for the turbulent layer, with an orientation 45° to the source direction.

[32] All the Allan deviation curves (total and calibrated) scale linearly with Cn; the ratio of the calibrated to total value will not change. If the wind velocity of the turbulent wet troposphere is different from 8 m/s, the amplitude of the total and calibrated Allan deviation will not be affected. However, the timescale for both curves will change, inversely proportional to wind velocity (i.e., for a larger wind velocity, a given fluctuation level will occur on a shorter timescale). The thermal noise limits do not depend on Cn or wind velocity.

3.2. Coherence

3.2.1. On-Axis Conical WVR Beams

[33] Figure 9 shows the expected coherence for interferometric observations at 86 GHz, with 25 m radio telescopes, with and without calibration by an on-axis WVR. For this plot, the elevation angle was 20°, the wind was 8 m/s at azimuth 45°, and the structure constant was 1.5 times the average value at the three DSN sites (these parameters are appropriate for VLBI observations, where at least one site is likely to have turbulent tropospheric conditions, and the elevation angle is usually low at some telescopes). The three panels in the figure are for WVR beam sizes (FWHM) of 1°, 2°, and 4°. For comparison, Figure 10 shows the coherence for a 50 m offset WVR location, a beam size of 1°, and an elevation angle of 20°, also for 25 m diameter radio telescopes, with the same level of tropospheric fluctuations as in Figure 9.

Figure 9.

Coherence improvements for 86 GHz interferometry, using ideal WVRs mounted on the subreflectors of 25 m diameter radio telescopes. All three plots are for observations at 20° elevation angle.

Figure 10.

Coherence improvement for 86 GHz interferometry, using ideal WVRs located 50 m from the axes of 25 m diameter radio telescopes. The elevation angle is 20°, and the WVR beam size is 1° FWHM.

[34] Figure 11 shows a similar set of plots, using parameters appropriate for ALMA. The antenna diameter is 12 m, the elevation angle is 30°, and the structure constant is 0.5 times the average value at the three DSN sites [Holdaway, 1997; Holdaway et al., 1997]. The observing frequency is 350 GHz. The three panels in the figure are for WVR beam sizes (FWHM) of 0.5°, 1°, and 2°. The subreflector of a 12 m ALMA antenna will not allow a WVR antenna as large as could be mounted on the subreflector of a 25 m VLBA or VLA antenna. However, the very low atmospheric opacity at the ALMA site should allow the use of the 183 GHz line for WVR use, compared to 22 GHz at other sites. This much higher WVR frequency will enable WVR beams as narrow as 0.5° FWHM.

Figure 11.

Coherence improvements for 350 GHz interferometry, using ideal WVRs mounted on the subreflectors of 12 m diameter radio telescopes. A low level of atmospheric turbulence has been assumed, appropriate for the ALMA site. The elevation angle is 30°.

3.2.2. Cylindrical WVR Beams With Angle Offset

[35] Figures 12 and 13 show coherence plots for offset cylindrical calibration beams. The plots in Figure 12 are for 25 m diameter radio telescopes and 43 GHz observing frequency, modeling a proposed VLA configuration. The structure constant is 1.5 times the average DSN value (e.g., fair to poor conditions at the VLA site), and the elevation angle is 30°. The angular offset between the source and calibration beam is 0.1°, 0.5°, and 2° for the three panels. This is the same configuration as for Figure 7.

Figure 12.

Coherence improvements for 43 GHz interferometry, using ideal WVRs mounted on the Cassegrain feed rings of 25 m diameter radio telescopes. A fairly high level of atmospheric turbulence has been assumed, appropriate for the VLA site under fair to poor conditions. The elevation angle is 30°.

Figure 13.

Coherence improvements for 350 GHz interferometry, using ideal WVRs mounted on the Cassegrain feed rings of 12 m diameter radio telescopes. A low level of atmospheric turbulence has been assumed, appropriate for typical conditions at ALMA. The elevation angle is 30°.

[36] Figure 13 uses parameters corresponding to ALMA: 12 m diameter antennas, 350 GHz observing frequency, and a structure constant 0.5 times the average DSN value. The elevation angle is 30°, and the angular offsets are 0.1°, 0.2°, and 0.5°.

[37] For small coherence loss, 1-Coh is proportional to ν2Cn2, for both the calibrated and uncalibrated cases in Figures 913, where ν is the observing frequency. The timescale in both figures is inversely proportional to vw, as for the Allan deviation.

3.3. Consequences of a Thinner Turbulence Layer

[38] The dependence of the results on the turbulence structure constant (Cn) and the wind speed (vw) have been described above. The other key parameter of the tropospheric model is the distribution of the turbulence with height above the Earth's surface. The volume mismatch described here occurs on scales comparable to the radio telescope diameter. Therefore, the effects of the mismatch will be insensitive to the turbulence thickness, down to a thickness of a few telescope diameters, as long as the total delay/phase fluctuation level at short timescales is held constant.

[39] The mismatch will depend strongly on the average height of the turbulence, since that height determines (to first order) how far two beams diverge, or how much a conical beam expands. If the average turbulence height is a factor N smaller than 1 km, the effect of the mismatch is the same as for a 1 km average height with an offset (or WVR beam width) N times smaller. For example, with a uniform turbulence layer in the lower 400 m (average height of 200 m), the consequence of an 0.5° beam offset with ALMA is the same as shown in Figure 13 for an 0.1° offset.

4. Discussion

4.1. Spacecraft Tracking Application

[40] A WVR mounted on the back side of a radio telescope subreflector could greatly reduce the beam mismatch problem of an off-axis WVR location. A 4° FWHM on-axis WVR gives slightly better performance than a narrow beam WVR located 50 m off axis (see Figures 5 and 8). The performance improves quickly as the WVR beam width is reduced. Calibration with an on-axis 1° FWHM WVR could reduce the Allan deviation of troposphere fluctuations by a factor of 3 at 10 s, and a factor of 20 at 100 s. Future spacecraft tracking measurements could benefit from this performance if they use two-way tracking to eliminate dependence on onboard oscillator stability, or if they fly an advanced onboard oscillator that is more stable than the troposphere.

[41] With an uncooled WVR, the total error (thermal noise plus beam mismatch) at Δt ≥ 10 s continues to decrease as the WVR beam size decreases, down to 1° FWHM (see Figures 35). Further reduction in WVR beam size, or integration into a BWG antenna, would not give better total performance with an uncooled WVR.

[42] Conversely, changing from an uncooled to a cryogenic WVR would yield an improvement in total performance only for an on-axis WVR with FWHM < 2°, or for a WVR integrated into a BWG antenna. For an integrated location (zero mismatch error) with a cryogenic WVR, thermal WVR noise sets a lower limit for useful calibration (improvement by a factor ≥3) of Δ t ≈ 5 s.

[43] A radio telescope with a Cassegrain feed ring could achieve similar performance by using an off-axis feed for the radiometer. In order to achieve comparable calibration accuracy to a 1° FWHM on-axis WVR beam, the angular offset between the source and calibration beams would need to be smaller than ≈0.2° (see Figure 7). We can compare the volume mismatches of these two configurations. A 1° FWHM conical beam spreads out to a diameter of 70 m at the top of a 2 km thick wet troposphere, at 30° elevation angle. A 0.2° offset cylinder diverges by 14 m (separation of axes) at the same height. The volume difference is somewhat larger for the conical beam, but the non-overlapping volume is symmetric about the source cylindrical volume, compared to the one-sided mismatch for the offset feed case. These two effects approximately cancel.

4.2. Coherence

[44] Use of a WVR in either of these two locations on a radio telescope could greatly improve the coherence of high-frequency interferometric observations, at least during clear weather, when WVR measurements are not corrupted by clouds. The higher coherence would have two large benefits. First, the signal-to-noise ratio (SNR) is linearly proportional to the coherence, so calibration with a WVR would increase the SNR substantially, and allow detection of weaker sources. Second, when the atmospheric coherence is low, it will also be highly variable (because atmospheric fluctuations vary in magnitude from one integration time to the next). This highly variable coherence leads to a large amplitude calibration uncertainty, and large errors in the images obtained from interferometry. The coherence resulting from calibration by a WVR with suitably small beam mismatch can be close to 1.0 Therefore, the calibrated coherence would exhibit only mild variability, allowing better amplitude calibration and more accurate images of radio sources.

[45] For an on-axis subreflector location, a WVR with 2° FWHM would allow a large improvement in coherence for several cases of interest (86 GHz VLBI using the VLBA, Figure 9; 350 GHz interferometry with ALMA, Figure 11). A beam width near 1° would be preferred, as it would allow a calibrated coherence near 1.0 under a wide range of atmospheric conditions.

[46] For a radiometer connected to an offset feed, excellent performance under a range of conditions could be achieved with an angular offset ≤1° for 43 GHz VLA observations (Figure 12) or ≤0.3° for 350 GHz ALMA observations (Figure 13).

4.3. Cautions/Other Error Sources

[47] Beam mismatch and WVR thermal noise are not the only error sources in WVR calibration. At the short timescales (<100 s) of interest in this analysis, retrieval errors [Keihm and Marsh, 1996] may be important. Of greater concern is the magnitude of dry delay fluctuations, which we have no current idea how to calibrate. Fluctuations in the temperature and density of dry air on spatial scales of a few cm limit the angular resolution of optical telescopes. Extrapolating to the much larger spatial scales (tens to hundreds of meters) that correspond to timescales of 10–100 s is quite uncertain. Dry fluctuations could be as large as ∼30% of the total fluctuation level. However, measurements in the near infrared [Linfield et al., 2001] suggest that these fluctuations grow more slowly with spatial scale than simple Kolmogorov turbulence theory predicts, so the actual magnitude may be much less than this 30% level.

Acknowledgments

[48] I thank A. Tanner for providing information on current and future radiometer capabilities. C. Naudet made a number of useful suggestions after reading an early draft of this manuscript and provided crucial help with file conversion for the figures. This work was performed at the Infrared Processing and Analysis Center, California Institute of Technology.

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