**M.A. Holdaway
National Radio Astronomy Observatory
949 N. Cherry Ave.
Tucson, AZ 85721-0655
email: mholdawa@nrao.edu**

**September 30, 1997**

We have investigated the effects of inhomogeneously
distributed water vapor, as characterized by our 11.2 GHz site testing
interferometer database, upon antenna pointing. This effect, known as
``anomalous refraction'', has been seen at poorer sites with
millimeter wavelength telescopes for years (Altenhoff *et al.*
1987). Because of the structure of atmospheric turbulence, the
pointing error in arcseconds will be smaller for larger antennas, but
the pointing error will be larger in terms of the fraction of the beam
size. The time scale of the pointing error will be nearly the time
required for the atmosphere to cross the dish. To first order, water
vapor is non-dispersive, so the anomalous refraction pointing errors
will be independent of frequency. However, there is mild dispersion
in the submillimeter, resulting in slightly larger pointing errors in
the submillimeter windows. For an 8 m dish on the Chajnantor site,
the atmospheric contribution to the pointing errors will usually be
well under an arcsecond, except during poor weather and while
observing at the lowest elevation angles.

Recently, there has been some concern over the stringent 1 arcsec
pointing error requirement on the MMA antennas. While the pointing
requirement is one of the main drivers for the antennas, it is clear
that we need the good pointing to make high quality mosaic images at
millimeter and submillimeter wavelengths. Some investigators have
questioned if 1 arcsec pointing can even be useful at all, given that
the turbulent atmosphere causes both the synthesized beam and the
primary beam to dance about the sky. If the columns of water vapor
above two antennas are different, a phase error will result,
effectively causing that baseline's contribution to the synthesized
beam to shift on the sky. If there is a gradient in the water vapor
distribution above one antenna, anomalous refraction, or a shift in
the apparent pointing, will result (see Figure 1). Since
both effects are caused by inhomogeneously distributed water vapor, we
can predict the magnitude of these effects from the site testing
interferometer data (Holdaway *et al.*, 1995). Simulations of
mosaicing with phase errors appropriate to the Chajnantor site have
been found to have little effect on imaging, which is dominated by the
physical pointing errors of the antennas (Holdaway, 1997). Here, we
investigate the severity of the pointing errors caused by anomalous
refraction.

Anomalous refraction (Altenhoff *et al.*, 1987) has been seen with
several millimeter wavelength telescopes. If a wedge of water vapor
falls across the antenna's line of site, refraction occurs. There is
nothing special about these wedges, they are just part of the
distribution of turbulent water vapor, and as such go back and forth
very quickly, as opposed to a systematic, persistent wedge. Hence,
anomalous refraction causes pointing errors with time scale
approximately equal to the time it takes the atmosphere to cross the
antenna. Indeed, some of the early investigations of anomalous
refraction found that the pointing error reversed itself before the
dish could scan over a bright quasar, producing an apparent double
beam.

We approximate the instantaneous phase screen as a wedge. Figure 2 shows an example of an actual phase time series which, assuming frozen turbulence, has been converted into a one dimensional slice through a spatial phase screen. The phase bump directly over the antenna is assumed to be a wedge, and smaller scale deviations from the wedge will participate in Ruze scattering, making the beam wider.

In Figure 3 we work through the mathematics of the model
for zenith observations. The geometry of our wedge of water vapor
is set by the angle , which can be related to the
dish diameter *d* and the path length
structure function through:

where *n* is the index of refraction. Snell's law relates the angle
between the incident ray and the line perpendicular to the
wedge to the angle between the refracted ray and the normal:

The pointing error due to anomalous refraction
is given by

Then, using small angle approximations,

The path length structure function is typically approximated as a power
law in the baseline (here *d*), but the amplitude and power law
exponent change with the atmospheric conditions:

On the Chajnantor site, the median structure function power law exponent is
1.2 (or the rms path length, , varies as .
Hence, as the dish size increases, the amount of anomalous refraction
actually decreases as . However, the anomalous refraction does
not decrease as fast as the beam size, so the pointing error as a fraction
of the beam will increase as .

Now, the zenith case is the best case with the least amount of anomalous refraction. We also consider worse cases, with arbitrary observing elevation and the wedge in the plane of the line of sight (see Figure 4). We spare the reader the mathematical details, but point out a few of the complicating factors:

- the effective baseline at which the path length structure function must be evaluated is , where is the observing elevation angle.
- the structure function depends upon elevation as (ie, Holdaway and Ishiguro, 1995).
- small angle approximations cannot be used, except for .

A person adept in trigo-algebraic manipulations might have gotten the
expression in a nicer form, but the computer doesn't mind that the
expression for the anomalous refraction pointing error for non-zenith
observations is given by

with now related to the structure function via

Equation 12 reproduces the results of
Equation 9 at the zenith.

From our site testing efforts on Chajnantor, we have a good statistical knowledge of the path length structure function. Even though we measure the interferometric phase on a 300 m baseline, we sample the phase with 1 s integrations. The temporal fluctuations on 1 s timescales are interpreted to be spatial structure flowing over the interferometer. The signal to noise on 1 s is not always sufficient to characterize the spatial fluctuations on size scales of , but when there is high enough SNR, we find that the temporal structure function is always well fit by a power law. Hence, low SNR conditions still permit a fit to the higher SNR medium to long time scale fluctuations and extrapolation back to the short time scales we are interested in here. A comparison between the temporal structure function and the single point we measure on the spatial structure function at 300 m effectively permits us to solve for the velocity, or to convert the temporal structure function into the spatial structure function, which is the required quantity for this analysis.

We have calculated the rms pointing error due to anomalous refraction on the Chajnantor site for the three quartiles of the rms path length fluctuations, for a range of elevation angles, and for dish diameters 8 m, 12 m, 15 m, and 50 m. The three smaller dish diameters are under discussion for building on the Chajnantor site, and the 50 m calculations are for the benefit of the LMT project. In Table 1 we present the results for the pointing error in arcseconds, along with the fraction of the pointing specification in parentheses. The pointing specification is taken to be at 300 GHz for the three smaller dishes (1 arcs, 0.67 arcs, and 0.53 arcs respectively) and 0.6 arcs over 2 hours for the 50 m dish. If the LMT's site has worse phase stability than the Chajnantor site, then the pointing errors due to anomalous refraction will be larger than the values quoted here.

Also presented in Table 1 are the quartile rms atmospheric path length differences across the dish, in microns. These numbers can get quite large, 50-100 microns for the 50 m dish. As stated earlier, the wedge shape (ie, tilt) will dominate, but there is also structure on the smaller scales deviating from the wedge. These smaller scale deviations will behave as surface errors, effectively broadening the primary beam. Since physical surface errors on the antenna affect the path length twice (pre and post main reflector) and these atmospheric fluctuations only affect the path length once (except for the presumably rare case where the turbulent water vapor is dominated by thin layer between the main reflector and the subreflector!) the residual path length deviations must be divided by two before applying the Ruze formula. We have not investigated this effect further, as we expect it to usually be small for the small interferometric dishes.

d = 8m, (pointing spec = 1.0 arcs) | ||||||

elevation | ||||||

Q | , [] | |||||

25% | 8.8 | 0.22 | 0.31 | 0.52 | 0.81 | 1.73 |

(0.22) | (0.31) | (0.52) | (0.81) | (1.73) | ||

50% | 18.0 | 0.47 | 0.64 | 1.07 | 1.65 | 3.55 |

(0.47) | (0.64) | (1.07) | (1.65) | (3.55) | ||

75% | 39.2 | 1.01 | 1.40 | 2.31 | 3.59 | 7.75 |

(1.01) | (1.40) | (2.31) | (3.59) | (7.75) | ||

d = 12m, (pointing spec = 0.67 arcs) | ||||||

elevation | ||||||

Q | , [] | |||||

25% | 11.2 | 0.20 | 0.26 | 0.44 | 0.68 | 1.47 |

(0.29) | (0.39) | (0.66) | (1.00) | (2.20) | ||

50% | 22.9 | 0.39 | 0.55 | 0.90 | 1.40 | 3.02 |

(0.58) | (0.81) | (1.40) | (2.11) | (4.50) | ||

75% | 50.0 | 0.86 | 1.20 | 1.98 | 3.05 | 6.59 |

(1.30) | (1.79) | (3.00) | (4.60) | (9.91) | ||

d = 15m, (pointing spec = 0.53 arcs) | ||||||

elevation | ||||||

Q | , [] | |||||

25% | 12.8 | 0.17 | 0.25 | 0.40 | 0.62 | 1.34 |

(0.31) | (0.47) | (0.75) | (1.17) | (2.51) | ||

50% | 26.2 | 0.36 | 0.49 | 0.82 | 1.27 | 2.76 |

(0.68) | (0.92) | (1.53) | (2.39) | (5.16) | ||

75% | 57.2 | 0.78 | 1.09 | 1.81 | 2.79 | 6.02 |

(1.46) | (2.05) | (3.39) | (5.24) | (11.3) | ||

d = 50m, (pointing spec = 0.6 arcs) | ||||||

elevation | ||||||

Q | [] | |||||

25% | 26.3 | 0.10 | 0.16 | 0.25 | 0.39 | 0.83 |

(0.17) | (0.26) | (0.41) | (0.65) | (1.20) | ||

50% | 53.9 | 0.22 | 0.31 | 0.51 | 0.79 | 1.70 |

(0.37) | (0.52) | (0.85) | (1.32) | (2.84) | ||

75% | 118. | 0.48 | 0.68 | 1.12 | 1.73 | 3.72 |

(0.80) | (1.12) | (1.86) | (2.88) | (6.20) |

We have assumed that the water vapor is non-dispersive, or that the anomalous refraction is independent of frequency. This is nearly true up to 300 Ghz, but begins to break down above 300 GHz, and the pointing errors due to anomalous refraction will be about 30% larger in the submillimeter windows due to large values of the index of refraction of water vapor.

To first order, the time scale of the anomalous refraction pointing errors will be the time it takes the atmosphere to cross the dish, which is on the order of a second. The details of the structure of the atmosphere will increase the pointing error time scale, especially for small dishes. We can calculate the time scale of the pointing errors using the raw phase monitor data. The pointing errors are proportional to the spatial derivative of the phase screen evaluated on spatial scales of the dish diameter. The pointing error time scale can be estimated from these time series (see Figure 5 and Table 2. These pointing errors will be more or less random over the array. For observations which are long compared to the pointing error time scale, the pointing errors will have a minimal effect, as they are both random in time and with antenna. For observations which are short compared to the pointing error time scale, as for On-The-Fly mosaicing or total power work, the pointing errors will be more problematic. Finally, for a large single dish performing On-The-Fly imaging, the pointing errors are most damaging, as they persist over several beams on the sky, but not long enough to be calibrated, and there are no other antennas to average down spatially random pointing errors with.

Antenna Diameter | Average Pointing |

[m] | Time Scale [s] |

10 | 2.7 |

20 | 3.9 |

30 | 5.0 |

50 | 6.7 |

For arrays of small antennas on a good site, there is little point to correcting the anomalous refraction. However, for a single large antenna, pointing errors caused by anomalous refraction may limit the performance of the antenna some of the time. If one could measure the distribution of water vapor above the antenna on second time scales, one could correct for the pointing in real time or in data post-processing One could do this by mounting four or five 22 GHz or 183 GHz (depending upon the quality of the site) water vapor spectrometers of the sort David Woody is making, at the edge of the large single dish, and possibly one on the back of the subreflector.

**References**

Altenhoff, W.J., *et al.*, 1987, ``Observations of anomalous refraction at radio wavelengths'', A&A **184**, 381.

Holdaway, M.A., and Ishiguro, M., 1995, MMA Memo 127,
``Experimental Determination of the Dependence of Tropospheric
Path Length Variation on Airmass''.

Holdaway, M.A. *et al.*, 1995, MMA Memo 129,
``Data Processing for Site Test Interferometers''.

Holdaway, M.A. *et al.*, 1997, MMA Memo 178,
``Effects of Pointing Errors on Mosaic Images with 8m,
12m, and 15m Dishes''.

**Figure 1:** Water vapor distributions resulting in phase errors
and anomalous refraction.

**Figure 2:** An example water vapor column screen derived from
phase monitor time series data, showing the wedge approximation.

**Figure 3:** Geometry for the anomalous refraction calculation for the zenith.

**Figure 4:** Geometry for the anomalous refraction calculation away from the zenith.

**Figure 5:** A time series of pointing errors as calculated for an 8 m and
a 50 m dish from site testing data during poor conditions.
The larger dish has smaller pointing errors in arcseconds
(larger as a fraction of the beam), but the time scale of the
fluctuations are larger, and therefore more damaging to
mapping work.