Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.arcetri.astro.it/~olmi/LMT/h2735.ps Äàòà èçìåíåíèÿ: Thu Nov 3 02:47:12 2005 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 04:54:24 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: diurnal motion

A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
03.01.2; 03.20.3; 03.20.9
ASTRONOMY
AND
ASTROPHYSICS
9.5.2001
Systematic observations of anomalous refraction at
millimeter wavelengths
L. Olmi 1
LMT/GTM Project, Dept. of Astronomy, 815J Lederle GRT Tower B, University of Massachusetts, 710 N. Pleasant st., Amherst,
MA 01003, U.S.A., olmi@lmtgtm.org
Received date; accepted date
Abstract. It is well known that the water vapor in the
troposphere plays a fundamental role in radio propaga-
tion. The refractivity of water vapor is about 20 times
greater in the radio range than in near-infrared or optical
regimes. As a consequence, phase uctuations at frequen-
cies higher than about 1GHz are predominantly caused
by uctuations in the distribution of water vapor, and
thus radio seeing at these frequencies is predominantly
caused by tropospheric turbulence. Radio seeing shows
up on lled-aperture telescopes as an anomalous refrac-
tion (AR), i.e. an apparent displacement of a radio source
from its nominal position, corrected for large-scale refrac-
tive eects. The magnitude of this eect, as a fraction of
the beam width, is bigger on larger telescopes and thus its
impact on the pointing is likely to become critically im-
portant in the next generation of electrically large lled-
aperture radio telescopes (D= > 10 4 ) and in particular
on the Large Millimeter Telescope. AR eects are expected
to reduce the total eective observing time at the highest
frequencies and will aect on-the- y mapping. Here we
present the results of systematic AR measurements car-
ried out with the 13.7-m telescope of the Five College Ra-
dio Astronomy Observatory. The measured AR pointing
errors range from 1 00 3 00 (winter) to about 20 00 (sum-
mer) and most of the events last less than about 4 sec.
We analysed the structure function, power spectrum and
Allan variance of the data and we have carried out a sta-
tistical analysis to identify correlations of the statistical
functions with selected observing parameters such as pre-
cipitable water vapor, time of day, season and elevation
angle. Our results suggest that uncompensated AR may
be the most important dynamic environmental source of
pointing errors on the new large radio telescopes (ALMA,
GBT, LMT, SRT) and may guide the design of active
AR-compensation devices and help allocating suitable ob-
serving time through dynamic scheduling.
Send oprint requests to: L. Olmi
Key words: atmospheric eects { Methods: observational
{ telescopes
1. Introduction
Radio-seeing eects on centimeter- and millimeter-
wavelength interferometers are a consequence of the inho-
mogeneously distributed atmospheric water vapor which
can cause spatial and temporal variations in the optical
path length of radio waves. Several studies of the problem
of phase uctuations with both centimeter (e.g., Arm-
strong & Sramek 1982) and millimeter (e.g., Bieging et
al. 1984, Olmi & Downes 1992, Wright 1996) interferom-
eters have led to the development of a number of radio-
metric devices to compensate for these uctuations and
restore the uncorrupted phase o-line (e.g., Bremer 1995,
Marvel & Woody 1998).
On the other hand, radio seeing on lled-aperture
telescopes shows up as an anomalous refraction (AR),
i.e., an apparent displacement of a radio source from its
true position, caused by the phase dierence introduced
between the opposite extremities of the receiving aper-
ture because the propagation paths traverse air masses of
varying humidity. AR pointing eects caused by turbu-
lence in the \wet" atmosphere are similar to the \quiver-
ing" of stars observed with visual-wavelength telescopes,
which are also known as angle of arrival uctuations in
the eld of clear- or dry-air propagation eects (see, e.g.,
Fante 1975 and Lawrence & Strohbehn 1970). The mag-
nitude of this eect, as a fraction of the beam width, is
bigger on larger telescopes and thus its impact on the
pointing is likely to become critically important in the
next generation of electrically large lled-aperture radio
telescopes (D= > 10 4 ), and especially in the case of the
Large Millimeter Telescope (or \Gran Telescopio Milimet-
rico", in Spanish, LMT/GTM; see Olmi 1998, Kaercher &

2 L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths
Baars 2000) with a FWHM' 5 00 at  = 1 mm and a re-
quired pointing accuracy at this wavelength < 1 00 .
The rst extensive measurements of AR were carried
out by Altenho et al. (1987), Downes & Altenho (1990),
and also Church & Hills (1990) who found that AR events
are characterized by angular displacements of the sources
from their true positions by a few arc seconds, in both
azimuth and elevation, for a few seconds of time, but oc-
casionally showing much larger events that could last for
tens of seconds. This is similar to what is observed in
near-infrared astronomy, where, for small telescope diam-
eter to Fried parameter ratio, D=r ô () <  6 (the Fried pa-
rameter represents the seeing cell size), the short-exposure
point spread function (PSF) randomly moves in the fo-
cal plane (e.g., Close & McCarthy 1994). On the new
large radio telescopes that are either under construction
(GBT 1 , LMT) or beeing designed (ALMA, SRT 2 ), for
which D=r ô () >  1 at the highest frequencies, phase gradi-
ents across the antenna aperture (i.e., tilt) will dominate,
but there will be also higher order aberrations that can ef-
fectively broaden the primary beam (Olmi 2000a). In more
recent years these projects have also prompted serious in-
vestigations of techniques to compensate AR eects (see
Holdaway 1997, Butler 1997, Holdaway & Woody 1997,
and Olmi 2000a, 2000b).
There were therefore several reasons to carry out an ex-
tensive, systematic study of the AR eects using a single-
dish antenna: (i) AR is the most critical dynamic environ-
mental source of pointing errors on large millimeter and
submillimeter telescopes; (ii) the measurements of phase
uctuations with millimeter interferometers and \seeing
monitors" is sensitive to the relative orientation of the
baseline and the wind direction (Lay 1997), and they are
often carried out over large spatial scales compared to the
diameter of single-dish antennas; (iii) it is important to
determine the potential eects of AR during On-The-Fly
(OTF) mapping; (iv) the next generation of mm-wave tele-
scopes represent big time, eort, and money investments
and thus must meet their design goals and yield a high
observing eôciency; (v) a better knowledge of AR would
also improve the design of active AR-compensation de-
vices and help allocating suitable observing time through
dynamic scheduling.
The main goal of this work is to present the re-
sults of systematic AR observations carried out with the
13.7-m telescope of the Five College Radio Astronomy
Observatory 3 (FCRAO) located in New Salem (U.S.A.)
at an elevation of 314m above sea level. They show that
AR is clearly detectable with the FCRAO 60 00 beam-width
at 86 GHz even when the precipitable water vapor (PWV)
1 \Green Bank Telescope"
2 \Atacama Large Millimeter Array", \Sardinia Radio
Telescope"
3 The Five College Radio Astronomy Observatory is operated
with support from the National Science Foundation and with
permission of the Metropolitan District Commission
is a few mm only. Measured values range from as \little"
as 1 00 2 00 (winter) to as much as 20 00 (summer). The
main purposes of these observations were: (i) detect AR
eects and characterize their magnitude (and time-scales)
as a function of time of the day, season, and elevation;
(ii) detect and measure systematic changes in AR sta-
tistical properties (slopes, turn-overs, etc.). Some results
from an incomplete data sample can be found in Olmi
(2000b, 2001) where we also discuss the basic technical
problems of a tip-tilt compensation device at millimeter
wavelengths for the LMT as well as other related issues.
The outline of the paper is as follows: in Sect. 2 we de-
scribe the measurement technique; in Sect. 3 we analyze
and discuss the AR data using several statistical functions;
nally, we draw our conclusions in Sect. 4.
2. Observations
The AR observations have been carried out using the
FCRAO radome-enclosed 13.7-m telescope located in
western Massachusetts. The telescope site is characterized
by at terrain surrounded by woods, with PWV values
(calculated using the measured ground-level dew point
temperature) ranging from < 1 mm in winter to more
than 10 mm in summer time. The occurrence of AR was
recorded by tracking a strong SiO maser ( = 86 GHz)
pointlike source at the azimuth half-power points of the
response pattern. The source intensity was then com-
pared with the ON-source intensity to determine the ap-
parent angular shift, assuming a given main beam pat-
tern that is well represented by a Gaussian prole (Ladd
& Heyer 1996). There is a tendency for the beam to be
broader at lower elevations, because of (mainly) gravita-
tional eects, but at 86 GHz the maximum FWHM vari-
ation is about 5% for elevations >  30 ô and is 8% for
elevations between 20 ô and 30 ô (Ladd & Heyer 1996). Be-
cause the pointing errors are obtained through a relative
measurement they are not aected by gain variations as a
function of elevation angle.
The pointing and focus of the telescope were checked
at the beginning of a new observing session and about
every 30 min thereafter. The typical absolute pointing ac-
curacy was about 6 00 , although the critical parameter of
interest to AR measurements is the tracking accuracy (see
below). Likewise the ON-source intensity was checked be-
fore and after an AR time series. Using this technique one
measures the modied angular distance, , of the target
source from the beam center, i.e.
 =   ô , where  ô
is the beam FWHM (see Fig. 1 of Olmi 2000b).
 is the
quantity of interest to determine the antenna pointing er-
ror, and the data used in this work are time series of the
observable
(t) = (t)  ô . Further informations about
the observing technique can be found in Olmi (2000b).
Two sets of data were obtained: a large sample where
the source intensity was sampled every  s = 3 s, for as
long as 5.5 min, and a smaller set of data with  s = 1 s

L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths 3
Fig. 1. Histograms showing the distribution of the data as a
function of PWV (top), elevation angle (middle) and local time
(bottom), for combined data sample I and II.
and with durations of up to 10 min. The data sampled
with  s =3 s were obtained under cloud-free conditions in
the period of February 1999 to June 2000 and we will
refer to them as sample I. The data with  s =1 s were
obtained during similar and, very often, during the same
weather conditions as sample I during spring 2000, and we
will refer to them as sample II. The values of the outside
temperature and PWV were recorded during the obser-
vations. However, no data on wind speed and direction
were available. The data are not uniformly distributed in
the observing parameters' space. In particular, most of the
data have been taken during conditions of either low or
high PWV, as shown in Fig. 1, due to the availabilty of
observing time during the regular observing season (when
typically PWV<  5 mm) and during the month of June
before the receivers are shut down.
The range of spatial frequencies analyzed was '
0:003 Hz to a Nyquist frequency of fN = 0:167 Hz for
sample I and ' 0:002 Hz to fN = 0:5 Hz for sample
Fig. 2. Histograms of AR-induced angular displacements (
)
observed on dierent days: 13-APR-2000 with PWV=2.7 mm
(dotted line) corresponding to an optical depth at 225 GHz of
225 = 0:27; 14-APR-2000 with PWV=4.0 mm or 225 = 0:33
(dashed line); and 26-MAY-2000 with PWV' 7:2 mm or
225 = 0:59 (solid line). The bins have a width ' 1 00 .
II. Consequently, we were unable to observe AR events
with time scales shorter than  s = 1 s. Our measure-
ments carried out during very dry conditions in winter
(PWV< 1 mm) indicate that the telescope tracking error
is < 1 00 , which we then consider as the sensitivity limit of
our AR observations. No correlation was found between
the AR pointing error and various recorded telescope pa-
rameters, such as subre ector's motors readings and the
electronic levels of the AZ track.
3. Results and analysis
3.1. Probability density distribution
Fig. 2 shows the distribution of angular displacements ob-
served on three days with dierent PWV of sample II. The
data were taken during a short period of time (between
13:00 and 16:00 local time) in stable weather and thus
during conditions of very similar PWV and temperature.
Because the three histograms in Fig. 2 are not the aver-
age of many days of data, with very dierent conditions,
they represent a good approximation of the AR probabil-
ity density function (PDF) for a given PWV. Moreover,
the air-mass range was approximately the same for each
of the three days considered (1.1 to 1.6) and therefore the
dierent widths of the histograms cannot be explained as
an elevation eect (see Sect. 3.5 for a discussion about
elevation eects).
The widths of the PDFs in Fig. 2 corresponding to
about 75% of the total area underneath the histograms are
' 6 00 and ' 4 00 for the 26-MAY-2000 and the April 2000
data, respectively. Therefore, during typical dry weather
(PWV< 4 mm, or  225 < 0:33 at the FCRAO site) the
magnitude of AR can be of up to several arcseconds, al-

4 L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths
Fig. 3. Histograms of the duration of AR events for the same
days shown in Fig. 2 (line styles are the same). The bins have
a width ' 1 s.
though we cannot exclude that it might be even larger on
time-scales shorter than  s . In summer time at FCRAO,
or during conditions of high PWV, the AR pointing er-
ror can be a considerable fraction of the FCRAO beam-
width. All other observing conditions being approximately
the same, the PWV seems to be a good tracer of AR ac-
tivity (but see the important discussion in Sect. 3.5) and
may thus allow an extrapolation of these results to other
sites. We note, however, that we obtained these values on
a at terrain and there is certainly more turbulence on a
mountain top. Therefore, under similar PWV conditions
we might expect more variations in the AR-induced point-
ing errors measured on the LMT/GTM site, due to its
more complex topography, than on a large open site such
as the FCRAO. Furthermore, because the largest AR-
induced pointing errors will occur at the shortest wave-
lengths, and because it is likely that the LMT/GTM will
operate in this high-frequency regime only during condi-
tions of low ( <  5 10 m/s) wind-speed, we should not
expect that the reduced time-scale of AR events result-
ing during conditions of high wind-speed will contribute
to average the AR eects down. It is clear, however, that
extrapolating these results to sites with dierent charac-
teristics and at higher frequencies is diôcult as well as
uncertain and specic on-site measurements should be ob-
tained.
3.2. Duration of AR events
We dene the duration of an \AR event" as the time inter-
val between two measurements, preceeding and following
a peak (either positive or negative) in the AR time series,
and having a value of
 smaller than 50% of the peak
value. In Fig. 3 we show the distribution of the durations
of the apparent displacements of the source as observed on
the same days as in Fig. 2. Two interesting features can
be seen: rst, the three histograms are remarkably simi-
lar and do not show any specic feature associated with
dierent PWVs. Second, most ( >  75%) of the events last
less than about 3 4 s. To rst order, the typical dura-
tion of an AR event is consistent with the time it takes a
moist element to cross the dish and it is expected to be
longer on larger antennas (see Holdaway 1997). Moreover,
the distribution of the event durations has a tail which
may stretch to times  10 20 s, although these events
are much rarer. The AR events durations obtained using
data from sample I have similar distributions but they fail
to show that most of the events are of very short duration
(see Olmi 2000b). Therefore, observations that are short
compared to the typical duration of the AR events, as is
the case in OTF mapping, will be seriously aected by
the AR pointing errors. Multiple sweeps across the source
may somewhat reduce the average pointing error but will
incur in a ux density loss and primary beam broadening
anyway.
Are the longest AR events also the ones with larger
magnitudes? To answer this question we generated the
three-dimensional plots shown in Fig. 4 where the distri-
bution of the data points is plotted as a function of the
AR-induced pointing error and the corresponding event
duration. The double-peak structure is due to the equal
probability of having, for each duration time, either a pos-
itive (the angular distance of the target source from the
center of the beam increases as an eect of AR) or negative
(the target source approaches the beam center) pointing
error. Clearly, the PDFs of the AR pointing errors are very
similar at any given event duration, except of course for
the total number of occurrences that decreases for longer
durations as already discussed above. Therefore, our data
show that while AR events of short duration are more
likely to occur, the distribution of their magnitudes re-
mains approximately constant. Multiple events that would
show up as long duration ones are possible but they would
be indistinguishable from individual events and we have
not attempted any sophisticated procedure to select them.
3.3. AR Structure function
3.3.1. Observed data
The temporal structure function, D  () where  is the
time lag, for the observable AR-induced pointing error

(t) can be dened as (Olmi 2000b):
D  ()

[
(t + )
(t)] 2

= c 
2   (1)
where the second equality indicates that in most cases the
structure function, or parts of it, can be t using a power
law. In this case the slope of the structure function in a
log-log plot, , is a measure of how rapidly the intensity
of the uctuations of
 increases with increasing  ; the
parameter c  2 is called the structure function coeôcient

L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths 5
Fig. 4. Distribution of the data points as a function of the
event duration and pointing error, for the same days as in
Fig. 2.
and is a direct measure of the strength of the AR eects.
We show two examples of measured structure functions
from sample II in Fig. 5; in particular, the top panel shows
an example of saturation of the AR structure function, i.e.
a attening of the slope at higher lag times and thus a
break in the slope, whereas the bottom panel shows an
example of structure function that can be t with a single
power law.
The distribution of  and of the intercept, log c  2 , are
shown in Fig. 6 for the entire set of data, i.e. samples I and
II. Most of the slopes of D  () are <  0:4 and the weighted
average is hi = 0:2. However, log c  2 can vary over two
orders of magnitude from about 4 to about 2.
Fig. 5. Top. Example of structure function with a break in the
slope and saturation at longer lag times. Bottom. Example of
structure function with no saturation.
3.3.2. Comparison with phase structure function
The AR structure function and the phase structure func-
tion measured with interformeters are dierent, and this
may also explain why in many cases the AR structure
function can saturate as shown in Fig. 5. The spatial phase
structure function at a given time t is dened as:
S (b)

[(r + b; t) (r; t)] 2

(2)
where  is the wavefront phase measured at two posi-
tions separated by the distance b. If we replace the base-
line b with the distance separating two points on the an-
tenna diameter D and assume that the wavefront across
the antenna is tilted with respect to the optical axis but
has no higher-order aberrations otherwise, then the phase
dierence,
(t), across the antenna aperture can be ap-
proximated as:

(t)  (r +D; t) (r; t) '
2

D
(t) (3)
where
(t) is the instantaneous tilt of the wavefront,
equivalent to the pointing error. The procedure for nding
the relationship between S (D) and D  () involves the
use of Taylor's hypothesis of frozen turbulence so that
(Lawrence & Strohbehn 1970):
(r; t + ) = (r v; t) (4)

6 L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths
Fig. 6. Histograms showing the distribution of , the slope
of the structure function (top), and of the intercept, log c 
2
(bottom), for data sample I and II.
Fig. 7. Histogram showing the distribution of  at elevations
< 30 ô (solid line) and at elevations > 50 ô (dashed line).
where v is the average wind velocity component perpen-
dicular to the line-of-sight of the imaging system. We can
now relate S  (D) and the temporal phase structure func-
tion, D  ( ):
D ()

[(r; t + ) (r; t)] 2

= S  (v) (5)
Using equations (1) and (3) one can then write the AR
structure function as:
D  () = 1
kD < ([(r +D; t + ) (r; t +  )] +
[(r +D; t) (r; t)]) 2 > (6)
where k = 2=. We can then use equations (2) to (4) in
Eq. (6) to nally obtain the relationship between phase
and AR structure functions:
D  () = 2
kD fS  (D) + S (v) +
1
2 [S  (jD v j) + S (D + v )]g (7)
where we have dropped the time dependence and we have
also assumed that the turbulence is isotropic. Depending
on the value of v two dierent regimes can be dened in
Eq. (7):
D  () '
8
<
:
2
kD S (D) if v >> D
2
kD S (v) if v << D
(8)
where D  () is independent of  for large values of v ,
corresponding to saturation of the structure function, and
D  () can be t with a power law for small values of
v . The break in the slope of D  () occurs at lag times
  D=v   D when the telescope is pointed to the zenith.
For Kolmogorov turbulence S  (b) = c 2 b 5=3 and thus equa-
tions (8) become:
D  () '
8
> <
> :
2 c 2 D 2=3
k if v >> D
2 v 5=3 c 2
kD  5=3 if v << D
(9)
and comparing with Eq. (1) we obtain:
c 
2 = 2 v 5=3 c 2
kD
: (10)
where c 2 has units of m 5=3 and thus c 
2 has units of
s 5=3 .
A comparison between Eq. (9) and Fig. 6 shows that
the measured value of hi = 0:2 is much smaller than the
Kolmogorov value  ' 1:67. However, the distribution in
Fig. 6 could be explained if most of the data were taken
near or at saturation, i.e.  >   D , or if Taylor's hypothesis
is not entirely correct. If we use a typical value of v w '
5 m/s for the wind speed parallel to the ground and a
zenith angle of z = 45 ô , and if we also assume that the
wind speed vector and the line of sight lie on the same
plane, then:
 D = D
v w
sec z ' 3:9 s (11)
We thus see that using typical observing parameters  D > 
 s . Moreover, steeper slopes of the structure function

L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths 7
Fig. 8. Power spectra of the same time series whose structure
functions are shown in Fig. 5. In the top-panel a break in the
slope of the spectrum can be seen at a frequency of about
0.05 Hz.
should be observed more frequently at low wind speed
and/or small elevation angles.
In Fig. 7 we plot the histograms of  for the data with
EL < 30 ô and for those with EL > 50 ô , and one can see
that the distribution of the data taken at lower elevations
is skewed towards higher  vales, as conrmed by their
weighted averages of 0.23 and 0.08 for the two histograms,
respectively. The data taken at higher elevations have no
values of  >  0:2.
Although Fig. 7 suggests that saturation (i.e., a
smaller slope of the structure function) is more likely to
be observed at higher elevation angles, we could not nd
any clear correlation between  D and sec z, as suggested
by Eq. (11). This may be due to: (i) wind speed vector
not coplanar, on average, with the telescope line of sight,
and (ii) distribution of elevation angles biased towards
intermediate values (see Fig. 1).
3.4. Power spectra and Allan variance
The one-sided power spectrum can also be calculated from
the
(t) time series, and it is dened as:
P  (f) = 2 jF  (f)j 2 / f  (12)
where F  (f) is the Fourier transform of
(t) and f is the
temporal frequency. Most of the power spectra can also
Fig. 9. Allan standard deviations of the same time series whose
structure functions are shown in Fig. 5.
be t using a power law, hence the proportionality sign in
Eq. (12). This is shown in Fig. 8 where we have plotted
the power spectra of the same AR time series from sam-
ple II whose structure functions were presented in Fig. 5.
Despite the noise it is clear that the power spectrum for
the day 13-APR-2000 cannot be t with a single line and
that a break in the slope of the spectrum occurs at a fre-
quency of about 0.05 Hz, consistent with the time lag cor-
responding to the beginning of saturation in the structure
function shown in the top panel of Fig. 5.
It can be shown that if the interferometer phase struc-
ture function S (b) / b  then the one-dimensional tem-
poral phase power spectrum P 1 (f) / f (1+) (Armstrong
& Sramek 1982). Because the slope of the phase structure
function, S , is the same as the slope of the AR struc-
ture function, D  , as shown by equations (9), one should
expect that if D  () /   then P  (f) / f (1+) (see
Sect. 3.5 for a discussion of this point). The weighted av-
erage of the slopes of the AR power spectra is 0:73, and
we can compare this value with the model of the angle
of arrival spectrum obtained by Fante (1975). If we use
Fante's equation (84) with v = 5 m/s we nd that in the
frequency range 0:004 0:16 Hz the slope is ' 0:8, con-
sistent with the AR average value. The intensity values of
the power spectra (e.g., for the examples shown in Fig. 8)
are also consistent with Fante's model assuming the turbu-

8 L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths
lence outer scale is L ô ' 500 m and C 2
n L ' 510 8 m 1=3
(see Sect. 3.5 for a discussion on the parameter C 2
n L).
The Allan variance is another useful method to de-
scribe the atmopsheric phase uctuations (see, e.g., Arm-
strong & Sramek 1982, Thompson et al. 1996, Olmi &
Downes 1992, Wright 1996). The Allan variance of the AR
uctuations removes linear drifts from the data and is de-
ned as:
 2
y = 1
2() 2
* 
(t + 2) + (t)
2 (t + )
 2
+
/
/  2 (13)
where  is the observing frequency. The Allan variance
is related to the one-dimensional power spectrum and it
can be shown that if the structure function can be de-
scribed with a power law of slope  then  y /  where
= ( 2)=2 (Armstrong & Sramek 1982, Thompson
et al. 1996) (see also Sect. 3.5). Two examples of Allan
standard deviations are shown in Fig. 9, where we have
plotted  y for the same AR time series whose structure
functions and power spectra are shown in gures 5 and 8,
respectively. We can see that the plot of  y for the data of
13-APR-2000 also shows a change in the slope, as it was
already the case for the power spectrum shown in Fig. 8,
and as a consequence the t to the whole data is not as
good as in the example of 14-APR-2000.
Fig. 10. Plot of log c 
2 vs. PWV.
3.5. Correlations
We have carried out a statistical analysis of the data to
identify possible correlations of either the power law in-
dices or log c  2 with selected observing parameters such
as PWV, time of day, season and elevation angle.
Because as we mentioned earlier in Sect. 3.1 the PWV
is one of the major environmental factors associated with
the magnitude of the AR errors, we rst present in Fig. 10
Fig. 11. Plot of log c 
2 vs. ground-level temperature.
the correlation of log c  2 with PWV. From the t to our
data we nd the empirical formula:
log c 
2 = 0:13 PWV 3:57 (14)
with c 
2 expressed in SI units and PWV in mm. In Fig. 10
we can see that the winter data, located in the region of
lower PWV, have also lower log c 
2 values whereas the
summer data are characterized by both higher PWV and
log c  2 values. The mid-season data occupy the interme-
diate region of the plot.
We have previously seen in Eq. (10) that the structure
coeôcient of AR is proportional to the phase structure co-
eôcient, which can be written for Kolmogorov turbulence
( = 5=3) as (see, e.g., Roggemann & Welsh 1996):
c 2 = 2:91

2

 2
C 2
n L (15)
where C 2
n is called the refractive index structure param-
eter, assumed constant up to a height L and zero there-
after. It can be shown that at radio wavelengths the real
part of C 2
n , which determines the phase delay in an ra-
dio wave propagating through the atmosphere, is propor-
tional
to
(ôQ) 2

= hQi 2 , i.e. the relative humidity uctu-
ations (Hill et al. 1980, 1988 and Olmi 1994). Thus, the
correlation shown in Fig. 10 might be explained if these
uctuations get stronger under conditions of higher hu-
midity, which are typical of warmer weather when strong
thermal gradients create considerable ground-level turbu-
lence. As discussed by Coulman (1991) the morning and
early afternoon heating disperses the nocturnal \inver-
sions" because of vertical motion of convective elements,
which may also explain the variation of log c  2 with time
shown in Fig. 13. Fig. 11, which shows a clear correlation
of log c  2 with the ground-level temperature, also sug-
gests that increased AR uctuations are in fact associ-
ated with increased convective activity near the ground.

L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths 9
Figures 10 and 11 indicate that an increase in the humid-
ity does not by itself lead to an increase in AR activity,
as also discussed by Wright (1996) in regards of interfer-
ometer phase uctuations, but that both are associated
with
larger
(ôQ) 2

= hQi 2 uctuations as a result of an
increased tropospheric turbulence.
Fig. 12. Plots of log c 
2 (top) and  (bottom) vs. log(sec z).
Only data in the 13:00 to 16:00 time interval have been used
in the top panel. The data have been binned to reduce the
scatter.
In Fig. 12 we show log c  2 and  plotted as a function
of log(sec z). Because log c  2 shows a correlation with
the local time (see Fig. 13) we selected only data in the
13:00 to 16:00 time interval in the top panel of Fig. 12.
However, because  does not seem to correlate with time
(see discussion below) we used all data in the bottom panel
of Fig. 12. Despite some big error bars it is clear from
Fig. 12 that the AR structure coeôcient tends to increase
at larger zenith angles. This trend was expected since for
non-zenith angles Eq. (1) must be rewritten as:
D  () = c 
2   sec z (16)
as shown by Coulman (1991) and Olmi & Downes (1992).
Some decorrelation is expected to occur because the beam-
width changes with elevation angle, causing an error in
retrieving the pointing error from the uctuations in the
source intensity, but because the beam FWHM changes
by 8% at most (see Sect. 2) this eect is negligible. The
correlation of  with sec z is less clear, as shown in the
bottom panel of Fig. 12. If subsequent observations will
conrm that  tends to get bigger at smaller elevations,
this would suggest that on the average the magnitude of
the AR{induced pointing uctuations, for a xed time lag,
becomes larger at smaller elevation angles.
Fig. 13. Plot of log c 
2 vs. local time. The selected data in the
elevation range 30 ô 60 ô have been grouped into 2 hrs bins to
reduce the scatter. No selection based on the PWV or ambient
temperature values has been applied.
We then wanted to nd out whether a correlation ex-
ists between log c 
2 and the local time. We present our
results in Fig. 13 where it can be clearly seen that the
AR structure coeôcient tends to decrease during the af-
ternoon and evening hours. The night hours are less well
sampled, and in particular we have no data between ap-
proximately 05:00 and 09:00, as shown in Fig. 1; however,
the data suggest that a log c  2 peak may be reached some-
time before 10:00. The day-night cycle of log c  2 follows
a similar pattern of both PWV and ambient temperature,
and thus it seems to suggest that this diurnal variation is
associated with increased turbulence near the ground dur-
ing the day-time, as we suggested earlier in this section.
A variability of the structure coeôcient with time was
also found by Olmi & Downes (1992) in the case of the
phase structure function measured with the IRAM inter-
ferometer. On the other hand, we nd no clear correlation
between  and the local time whereas such a correlation
was also observed by Olmi & Downes.
In Fig. 14 we plot the indices of the power laws describ-
ing the power spectrum and the Allan variance, dened in
equations (12) and (13), as a function of the power law
index describing the structure function (sample II only).
Clearly,  and correlate with , and the two best-t
lines are given by the equations  = 1:43  0:55 and
= 0:88  1:29, which are consistent with the predicted
values of  = (1 + ) and = 0:5  1, respectively,
as described in Sect. 3.4. This consistency between the
measured best-t values and the theoretical values of the
various power law slopes, obtained applying the standard

10 L. Olmi: Systematic observations of anomalous refraction at millimeter wavelengths
Fig. 14. Plots of , the power law index of the power spectrum
(top), and of , the power law index of the Allan variance
(bottom), vs. , the power law index of the structure function.
model of turbulence to the phase uctuations of an in-
terferometer, suggests that the same model can also be
applied to the AR uctuations measured with a single-
dish antenna. In fact, in Sect. 3.3.2 we showed how the
structure functions of the phase and AR uctuations can
be related.
4. Conclusions
We have carried out systematic measurements of AR-
induced pointing errors with the radome-enclosed FCRAO
13.7m telescope located in western Masschusetts on a at
terrain, during the period February 1999 to June 2000.
The data are based on the time series of the uctua-
tions of the angular distance of the source from the beam
center, as measured at the 3 dB points. We have de-
tected AR-induced pointing errors with the FCRAO 60 00
beam. The measured values range from ' 2 00 (winter) to
' 20 00 (summer). The probability density distributions
of the AR pointing errors are narrower for low PWV
and wider for high PWV, and during typical dry weather
(PWV< 4 mm) the FWHM of the distributions can be of
several arcseconds. We have also measured the duration of
the individual \AR events" and found that most of them
last less than 3 4 s, with tails in the distribution stretch-
ing to ' 10 20 s. Such short-duration uctuations will
not be averaged out during a typical OTF scan and can
thus aect the reliability of OTF mapping on AR-limited
telescopes.
Several statistical functions have been used to analyse
the data. We found that many structure functions can be
t with a single power law of type D  () = c 2
   , where
usually  < 0:4 and log c 2
  4 to 2. The slope of
the AR structure functions is much lower than that of
the phase structure functions measured with millimeter-
wave interferometers. Power spectra and Allan variance
plots can also be t with single power laws, and we found
that the three dierent power law slopes correlate and are
consistent with the standard model of atmospheric turbu-
lence. The magnitude of the AR uctuations, represented
by the structure coeôcient, c 2
 , correlates well with PWV
and ground-level temperature, decreases with increasing
elevation angle and also varies during the day. These char-
acteristics indicate that stronger AR uctuations are asso-
ciated with increased convective activity near the ground,
which is typical of warmer, and more humid, weather when
strong thermal gradients create considerable ground-level
turbulence. Correlations of  with the observing parame-
ters are still unclear.
Extrapolation of these results to other telescope sites,
such as the LMT/GTM site, is uncertain because of the
dierent latitude, elevation and terrain characteristics.
However, it seems reasonable to expect similar AR eects
during conditions of similar PWV and ambient tempera-
ture. If this is indeed the case then the expected magni-
tude of the AR-induced pointing errors can be compara-
ble with the beam width of the LMT/GTM, under certain
conditions, and all antenna measurements (OTF mapping,
pointing, focusing, beam switching, etc.) would then be se-
riously aected. The LMT/GTM is currently studying the
design of a radiometric wave front sensor to compensate
AR eects.
Acknowledgements. This work was sponsored by the Ad-
vance Research Project Agency, Sensor Technology Oôce
DARPA Order No. C134 Program Code No. 63226E issued
by DARPA/CMO under contract No. MDA972-95-C-0004, and
by the NSF grant AST-9725951. The author thanks M. Brewer
and M. Heyer of FCRAO for help with the observing technique
and reduction of some of the data used in this work.
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