Date: July 12, 2005 </br>
Subject: Bandpass ripples </br>
From: Jun-Hui Zhao </br>
Updated: July 21, 2005
</br>

<center><b> ABSTRACT </b></center>
Based on the data taken by Dan Marrone on 2005-06-16,
we investigated the origin of the bandpass ripples seen in the SMA spectra. 
Two strong QSOs 3c279 (10Jy) and 3C454.3 (35Jy) 
were observed at 1.3mm during the 12.5 hr track. The antenna-based
bandpass solutions are calculated using either 3c279 or
3C454.3 in each of the UT-hour bins
and are compared to those derived from a reference time bin.
The bandpass solutions derived for antenna 1 shows a significant
variation in time, which appears to be responsible for the ripples
seen in the SMA spectra. A Miriad task <b>SMAtBpass</b> has been
written. It is  shown that the bandpass ripples can be adequately removed
with <b>SMAtBpass</b> along with other tasks in Miriad. 
</br></br>

<b>1. Data Description</b></br>
The data were taken by Dan Marrone on 2005-6-16
during his observation of GC polarization with the dual receiver
(230/690) setup. The upper side band of the lower frequency receiver band
was tuned to 231.900 GHz for observation of
the millimter recombination line H30alpha. 
The spectral data were sampled uniformly across the
24 chunks in each of the sidebands with 16 channels
in each of the chunks. The channel width is 6.5 MHz.
The observation was started at UT 4:30 and ended at UT 19. 
Many sources were observed during the 12.5 hrs including 
two strong QSOs 3c279 in the time range between UT4:30-7:00 and
3C454.3 in the range of UT11:00-19:00.
</br></br> 

<img src="./050616.gif">

<br>
Fig.1: A part raw visibility data observed during the 12.5 Hrs. 
</br></br>
The above diagram is the uv plot to
show part of the visibility (RR) baselines during the entire
track. The visibility data points are color-coded for 
the program sources. The weather condition was great
for 230 GHz. The antenna-based bandpass can be easily
solved from the
 strong ponit sources, either QSOs 3C279 (10Jy at 1mm) 
or 3C454.3 (35Jy at 1mm)
without the confusion/contamination due
to complex structure of a large planet. In addition,
3C 279 was observed from 4:30 to 7:00 and 3C454.3
was observed from 11:00 to 19:00. The data of the two QSOs
can be used to investigate the issue of bandpass
ripples that were hard to be calibrated out in the past.
We only use the RR data throughout this analysis.
</br></br>
<b>2. Antenna-Based Bandpass Solutions</b></br>
The observations were divided into 12 bins of each UT hour.
Either 3C 279 or 3C 454.3 in each of the UT hour bins is used
to calculate the antenna-based bandpass in Miriad. 
The mean value of Tsys averaged from the entire track is about 104 Kelvin.
Assuming JyperK=130 Jy/K and 3 min on the bandpass calibrator
in a UT hour bin, the r.m.s. per antenna per channel is about 0.13 Jy.
Then, we  have an adequate S/N to solve the 
antenna-based bandpass using either 3C279 or 3C 454.3 for 
our analysis. 
Here 
are the typical solutions in the UT-hour bin of UT12-13:
</br></br>
<img src="./3c454_12h13hbpamp.gif">
<br>
Fig.2: Bandpass solution in Amplitude.
</br></br>
<img src="./3c454_12h13hbppha.gif">
<br>
Fig.3: Bandpass solution in Phase.
</br></br>
The above two diagrams are the bandpass in amplitude and phase across the 2 GHz band
(24 spectral chunks color-coded). The dots are the solutions
and the lines are the polynomial (5th order) fits to each of spectral chunks.
Antenna 3 was used as the reference antenna. </br>
</br>

<b>3. The Characters of Bandpass Variations in Time</b>
</br>
We solved the antenna-based bandpass in each
of the UT hour in which an adequate amount of the calibration 
data (on either 3C 279 or 3C 454.3) are present.
The complex bandpass solutions (bp[uth,uth+1]) 
derived from each of UT hour bins
are divided by the solutions calculated in UT 17h-18h (bp[17,18]),
i.e. bp[uth,uth+1]/bp[17,18] and uth from 5 to 18,
in order to investigate the possibility of the bandpass
varying with time. Here are the results of the
bandpass ratio of the solutions in each UT hour bin divided by
the reference  solutions in UT 17h-18h (produced with the task SMAGPPLT
in Miriad):
</br></br>
<a href="./bpratio_amp5:00-6:00.ps">Fig.4.1a-Amp:UT05-UT06 (3c279)</a>,
<a href="./bpratio_pha5:00-6:00.ps">Fig.4.1b-Pha:UT5-UT06 (3c279)</a>
</br> 
<a href="./bpratio_amp6:00-7:00.ps">Fig.4.2a-Amp:UT06-UT07 (3c279)</a>,
<a href="./bpratio_pha6:00-7:00.ps">Fig.4.2b-Pha:UT06-UT07 (3c279)</a>
</br>
<a href="./bpratio_amp11:00-12:00.ps">Fig.3a.5-Amp:UT11-UT12 (3c454.3)</a>,
<a href="./bpratio_pha11:00-12:00.ps">Fig.3b.6-Pha:UT11-UT12 (3c454.3)</a>
</br>
<a href="./bpratio_amp12:00-13:00.ps">Fig.4.4a-Amp:UT12-UT13 (3c454.3)</a>,
<a href="./bpratio_pha12:00-13:00.ps">Fig.4.4b-Pha:UT12-UT13 (3c454.3)</a>
</br>
<a href="./bpratio_amp13:00-14:00.ps">Fig.4.5a-Amp:UT13-UT14 (3c454.3)</a>,
<a href="./bpratio_pha13:00-14:00.ps">Fig.4.5b-Pha:UT13-UT14 (3c454.3)</a>
</br>
<a href="./bpratio_amp14:00-15:00.ps">Fig.4.6a-Amp:UT14-UT15 (3c454.3)</a>,
<a href="./bpratio_pha14:00-15:00.ps">Fig.4.6b-Pha:UT14-UT15 (3c454.3)</a>
</br>
<a href="./bpratio_amp15:00-16:00.ps">Fig.4.7a-Amp:UT15-UT16 (3c454.3)</a>,
<a href="./bpratio_pha15:00-16:00.ps">Fig.4.7b-Pha:UT15-UT16 (3c454.3)</a>
</br>
<a href="./bpratio_amp16:00-17:00.ps">Fig.4.8a-Amp:UT16-UT17 (3c454.3)</a>,
<a href="./bpratio_pha16:00-17:00.ps">Fig.4.8b-Pha:UT16-UT17 (3c454.3)</a>
</br>
<a href="./bpratio_amp17:00-18:00.ps">Fig.4.9a-Amp:UT17-UT18 (3c454.3)</a>,
<a href="./bpratio_pha17:00-18:00.ps">Fig.4.9b-Pha:UT17-UT18 (3c454.3)</a>
</br>
<a href="./bpratio_amp18:00-19:00.ps">Fig.4.10a-Amp:UT18-UT19 (3c454.3)</a>,
<a href="./bpratio_pha18:00-19:00.ps">Fig.4.10b-Pha:UT18-UT19 (3c454.3)</a>
</br></br>

If the antenna-based bandpass had no variations in time,
we would expect the amplitude of the bandpass ratio to be unity and the 
phase to be zero. The solution from antenna 6 appears
to be noisy by a factor of 2 as compared to others, which was also
indicated in the system temperature.<br>
</br>
From antenna 1, a ripple (sinusoid-like) in both ampplitude and phase
across the 2 GHz band  appears to be significant 
(exceeding 20% (max-min)/2mean in amplitude and 30 degree in phase).
<br></br>
The intensity of the variations in both ampltiude and phase
appears to increase as the separation  time between
observations of the targets and calibrators. In other words, if a target
source observed in 12 hrs apart from the bandpass calibrator,
errors at a level of 20% in amplitude and 30 degree in phase
can be generated due to the bandpass variation seen on antenna 1
in the 050616 data. Within a few hrs, the errors stay at a level of
10% in amplitude and 10 degree in phase. 
<br><br>
The change in bandpass appears to be not due to change of
the observing source since both 3c279 and 3c454.3 (UT11-UT12, UT12-UT13)
show a similar pattern.
<br><br>
Other antennas seem to be relatively stable.
<br><br>
Such a ripple might be also present in other data sets (as noticed
by Crystal Brogan and Todd Hunter).
<br><br>
We checked the Miriad programs MFCAL and SMAMFCAL.
The solutions from both programs agree with each other
and are correct.
</br></br>

<b>4. Corrections for Time-Dependent Bandpass Ripples</b>
</br>
Clearly, the variations in bandpass introduce errors,
which need to be calibrate out. In particular,
for weak and broad spectral line emission such as the mm/submm 
hydrogen recombination
lines from  a galactic nucleus, the signals can be easily
washed out.
A new Miriad task <b>SMAtBpass</b> has been implemented.
<b>SMAtBpass</b> removes the antenna-based time-dependent
bandpass ripples by interpolating/extrapolating the bandpass
solutions solved from multiple independent time-intervals
in which a bandpass calibrator observed and applying the fitted
bandpass to the data. Two options for the fitting algorithms
have been used in <b>SMAtBpass</b>: 
</br></br>
1. linear fit to the two nearby time points;
</br>
2. least-square fit to the time variaition in
   bandpasses solved from the multiple independent time-intervals
   with an orthogonal polynomial of degree n. 
</br></br>

A demonstration below is used with the linear fit 
to illustrate the method to remove the bandpass ripples.
</br></br>
<b> 4.1. Normal, Time-independent BP correction</b></br>
With either smamfcal or mfcal, we solved for the overall 
antenna-based bandpass using all the usb data of 3c454.3 
observed on 2005-06-16. The following plot shows the 
spectra of 3c454.3 for the baselines after applying the "averaged"
bandpass corrections.

</br></br>
<img src="./3c454normalbp.gif">
<br>
Fig.5: Spectra of 3C454.c after normal bandpass corrections (time-indpendent).
</br></br>

The bandpass ripples clearly remain on the baselines related to antenna 1.
If the same bandpass corrections are applied to SgrA*, similar ripples
are present on the baselines related to antenna 1 (see the plot below).
</br></br>
<img src="./sgranormalbp.gif">
<br>
Fig.6: Spectra of SgrA* after normal bandpass corrections (time-indpendent).
</br></br>
<b> 4.2. Time-dependent BP correction</b></br>

Now, we correct for the time-dependent bandpass using the bandpass
solutions solved in the UT intervals described in section 3.
A linear fit to two nearby solutions are used in the interpolation/
extrapolation of the bandpass corrections to the data. The following
figure (Fig.7) shows the spectra of 3c454.c with the same scale and 
the same baselines as those in Fig.5 but corrected for time-dependent
bandpasses using the task <b>SMAtBpass</b> in Miriad.  
  
</br></br>
<img src="./3c4542pt1hbp.gif">
<br>
Fig.7: Spectra of 3C454.c after time-dependent bandpass corrections.
</br></br>
The ripples related to the antenna 1 have been effectively removed.
The time-dependent bandpass corrections are applied to the SgrA* data.
The corrected spectra are shown in Fig.8:

</br></br>
<img src="./sgra2pt30mbp.gif">
<br>
Fig.8: Spectra of SgrA* after time-dependent bandpass corrections.
</br></br>
The scale of Fig 8 is the same as that used in Fig. 7.
Again, the bandpass ripples related to the antenna 1 appear to
have been taken out. 
</br></br>
<b> 4.3. Comments </b></br>
We have shown that the antenna-based bandpass ripples seen in the SMA
data are due to
slow variations of the bandpass shape as a function of time. A 5-minute
sampling (on 3c454.3 at 230 GHz) of bandpass data in every a few hrs with linear 
interpolation/extrapolation appear
to be able to adequately remove the bandpass ripples. The residuals
in antenna-based bandpass error remain at a level of a few percent in amplitdue
for  most of the 
antennas
but the residual error for antenna 6 appears to be larger than
a few percent.
</br></br>

Upto 10 percent bandpass errors are still clearly present on a few baselines
(1-6, 3-6) related to chunk 15 and baselines (2-6, 4-6) related
to chunks 9, 10, 11, 12 or block 3. These are likely related to the correlator.