#  Platform Deformation Phase Correction for the AMiBA-13 Co-planar
Interferometer

Yu-Wei Liao11affiliation: Institute of Astronomy and Astrophysics, Academia
Sinica, P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>,
Kai-Yang Lin11affiliation: Institute of Astronomy and Astrophysics, Academia
Sinica, P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>,
Yau-De Huang11affiliation: Institute of Astronomy and Astrophysics, Academia
Sinica, P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>,
Jiun-Huei Proty Wu22affiliation: Department of Physics, Institute of
Astrophysics, & Center for Theoretical Sciences, National Taiwan University,
Taipei 10617, Taiwan<EMAIL_ADDRESS>, Paul T. P. Ho11affiliation:
Institute of Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141,
Taipei 10617, Taiwan<EMAIL_ADDRESS>33affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138,
USA , Ming-Tang Chen11affiliation: Institute of Astronomy and Astrophysics,
Academia Sinica, P. O. Box 23-141, Taipei 10617, Taiwan;
<EMAIL_ADDRESS>Chih-Wei Locutus Huang11affiliation: Institute of
Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141, Taipei 10617,
Taiwan<EMAIL_ADDRESS>22affiliation: Department of Physics,
Institute of Astrophysics, & Center for Theoretical Sciences, National Taiwan
University, Taipei 10617, Taiwan<EMAIL_ADDRESS>, Patrick M.
Koch11affiliation: Institute of Astronomy and Astrophysics, Academia Sinica,
P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>, Hiroaki
Nishioka11affiliation: Institute of Astronomy and Astrophysics, Academia
Sinica, P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>,
Tai-An Cheng22affiliation: Department of Physics, Institute of Astrophysics, &
Center for Theoretical Sciences, National Taiwan University, Taipei 10617,
Taiwan<EMAIL_ADDRESS>, Szu-Yuan Fu22affiliation: Department of
Physics, Institute of Astrophysics, & Center for Theoretical Sciences,
National Taiwan University, Taipei 10617, Taiwan<EMAIL_ADDRESS>, Guo-
Chin Liu44affiliation: Department of Physics, Tamkang University, 251-37
Tamsui, New Taipei City, Taiwan , Sandor M. Molnar55affiliation: Leung Center
for Cosmology and Particle Astrophysics, National Taiwan University, Taipei
10617, Taiwan , Keiichi Umetsu11affiliation: Institute of Astronomy and
Astrophysics, Academia Sinica, P. O. Box 23-141, Taipei 10617, Taiwan;
<EMAIL_ADDRESS>, Fu-Cheng Wang22affiliation: Department of
Physics, Institute of Astrophysics, & Center for Theoretical Sciences,
National Taiwan University, Taipei 10617, Taiwan<EMAIL_ADDRESS>, Yu-
Yen Chang66affiliation: Max-Planck-Institut für Astronomie Königstuhl 17.
D-69117 Heidelberg, Germany , Chih-Chiang Han11affiliation: Institute of
Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141, Taipei 10617,
Taiwan<EMAIL_ADDRESS>, Chao-Te Li11affiliation: Institute of
Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141, Taipei 10617,
Taiwan<EMAIL_ADDRESS>, Pierre Martin-Cocher11affiliation:
Institute of Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141,
Taipei 10617, Taiwan<EMAIL_ADDRESS>, and Peter
Oshiro11affiliation: Institute of Astronomy and Astrophysics, Academia Sinica,
P. O. Box 23-141, Taipei 10617, Taiwan<EMAIL_ADDRESS>
###### Abstract

We present a new way to solve the platform deformation problem of co-planar
interferometers. The platform of a co-planar interferometer can be deformed
due to driving forces and gravity. A deformed platform will induce extra
components into the geometric delay of each baseline, and change the phases of
observed visibilities. The reconstructed images will also be diluted due to
the errors of the phases. The platform deformations of The Yuan-Tseh Lee Array
for Microwave Background Anisotropy (AMiBA) were modelled based on
photogrammetry data with about 20 mount pointing positions. We then used the
differential optical pointing error between two optical telescopes to fit the
model parameters in the entire horizontal coordinate space. With the platform
deformation model, we can predict the errors of the geometric phase delays due
to platform deformation with given azimuth and elevation of the targets and
calibrators. After correcting the phases of the radio point sources in the
AMiBA interferometric data, we recover $50\%-70\%$ flux loss due to phase
errors. This allows us to restore more than $90\%$ of a source flux. The
method outlined in this work is not only applicable to the correction of
deformation for other co-planar telescopes but also to single dish telescopes
with deformation problems. This work also forms the basis of the upcoming
science results of AMiBA-13.

###### Subject headings:

Astronomical Instrumentation

##  1\. Introduction

Co-planar interferometers have been used for cosmological observations for
years. Several known co-planar interferometers – including Degree Angular
Scale Interferometer (DASI) (Leitch et al., 2008), Cosmic Background Imager
(CBI) (Padin et al., 2001), and AMiBA (Ho et al., 2009) – are dedicated to a
variety of topics in cosmology. A co-planar interferometer array benefits from
several advantages over an array of individually steerable antennas. Most
notably, the co-planar design is good for close-packing a large number of
small antennas without the worry of shadowing each other. The servo design of
a co-planar array is also simpler because there is no need to control
individual antennas simultaneously.

The Yuan-Tseh Lee Array for Microwave Background Anisotropy (AMiBA) is a co-
planar radio interferometer for research in cosmology. With an operating
frequency band between 86 and 102 GHz, which corresponds to a wavelength
$\approx 3$mm, the main goal of AMiBA is to map the Sunyaev-Zeldovich effect
(SZE) (Sunyaev & Zel’dovich, 1972) in galaxy clusters. The AMiBA antennas and
receivers are mounted on a 6-meter carbon fiber platform. The platform itself
is mounted on a hexapod mount. The mount allows the platform to be pointed to
all the positions in the sky above elevation $30^{\circ}$. The mount can also
rotate the platform within a $\pm 30^{\circ}$ range around the normal
direction. The polarization angle rotated around the normal direction is
called hexpol below.

With 7 platform-mounted 0.6-meter antennas, the former AMiBA-7 observed six
galaxy clusters in 2007 and 2008. High quality science data were obtained for
these six clusters with AMiBA-7. An overview of AMiBA-7 is given in Ho et al.
(2009). Koch et al. (2009) detailed the mount and the platform of AMiBA. The
stiffness of the AMiBA platform is studied in Huang, Raffin, & Chen (2011).
The introductions of correlator and receiver can be found in Li et al. (2006)
and Chen et al. (2009). The details of AMiBA-7 observation, calibration, data
analysis, foreground contamination estimation, and quality checking are given
in Lin et al. (2009), Wu et al. (2009), Liu et al. (2010), and Nishioka et al.
(2009). The science results of AMiBA-7 are shown in Huang et al. (2010), Liao
et al. (2010), Liu et al. (2010), Wu et al. (2009), and Umetsu et al. (2009).

In 2008 and 2009, AMiBA was upgraded to AMiBA-13, which has 13 1.2-meter
dishes (Koch et al., 2011) mounted on the original AMiBA-7 platform. The
collecting area of AMiBA-13 is 7.5 times of AMiBA-7. The angular resolution of
AMiBA was also improved from previously $7^{\prime}$ to $2^{\prime}$. With the
new AMiBA-13, we can observe fainter objects with a shorter integration time.
The ability of AMiBA-13 to constrain intracluster gas models is predicted by
Molnar et al. (2010). Figure 1 shows pictures of AMiBA-7 and AMiBA-13.

Figure 1.— The pictures of AMiBA-7 (left) and AMiBA-13 (right). The antennas
(0.6m for AMiBA-7 and 1.2m for AMiBA-13) and receivers are mounted in a
compact configuration on the 6-meter platform.

Driven by the six legs of the hexapod mount, the platform of AMiBA is under
complicated pushing and pulling forces from the actuators. The reaction from
the actuators, combined with the gravity loading of the interferometer array,
cause deformations of the platform. Using photogrammetry, the deformation of
the AMiBA platform was measured to be up to several hundred $\mu$m (see Huang
et al. (2008) and Koch et al. (2008)). The saddle-shaped platform deformation,
which changes its amplitude and direction with different platform pointing and
rotation angles, were found to be repeatable in the photogrammetry data. Since
we use platform-mounted optical telescopes to calibrate the pointing of the
mount, the change of the platform deformations, also induces a change in the
relative tilt between the optical telescopes and the platform itself. We found
differences in the optical pointing measurement to be $\approx 1^{\prime}$.
Koch et al. (2008) described how the pointing errors have been dealt with in
the stage of AMiBA-7.

The other important effect of platform deformations is phase delay. The
deformations of the platform will induce extra geometric delays between
different antenna elements. In other words, these elements are no longer in
the same plane. Those geometric delays will cause artificial phase delays in
the observed visibilities from each baseline. For AMiBA-7, all the antennas
were mounted at the radius $r\leq 1.2$m. Within this range, the platform
generally deforms by less than $0.3$mm, or $0.1$ times of the observing
wavelength of AMiBA. Therefore, the phase delays are negligible in the AMiBA-7
data. On the other hand, antennas are mounted on the whole 6-m platform for
AMiBA-13. The geometric delays between antennas can easily rise to $0.5$ times
of the observing wavelength. With this magnitude of geometric delays, the
signals are essentially decorrelated, thereby significantly distorting the
reconstructed images. Therefore, it is important to find a way to correct the
phase delays. The same problem can also be encountered by other co-planer
interferometers. Hence our experience of correcting the phase delays might be
useful for other telescopes. Furthermore, the way we model and determine the
platform deformations can also be applied to study the pointing dependent
deformations of single dish telescopes.

In this paper, we detail the way we model the platform deformations and
correct the phase delays (see Figure 2 for the flowchart of our method). In
section 2, we describe a new way to model the deformations of the AMiBA
platform. We fit the photogrammetry data with a new two-rotating-saddle model.
We describe how we correlate the pointing errors of the two platform-mounted
optical telescopes to fit the model parameters for the entire pointing
parameter space of the mount. In section 3, we show how we correct the
visibility data base using the platform deformation model. In order to verify
the efficiency of our phase correction method, we apply it to the data of
planets and radio sources at different pointings on the sky. The results are
also shown here. In section 4, we briefly discuss the improvement of data
analysis contributed from the phase correction method, and the limits of such
corrections.

Figure 2.— The flow chart of our method to correct the platform deformation
for AMiBA. The red arrows show the real situations we met.

##  2\. Platform Deformation Modeling

Before correcting the phase delays contributed by platform deformations, we
need to understand more about the phenomenon. The platform deformations of
AMiBA are known to be a function of the pointing and the polarization angle of
the mount with a repeatability within $50\mu$m (Huang et al. (2008), Koch et
al. (2008)), which is much smaller than the wavelength of $3$mm considered for
its application. Therefore, the platform deformations are considered to be
modelable and correctable. We describe the photogrammetry-measured platform
deformation with an empirical model. However, the photogrammetry surveys were
very time-consuming and were, therefore, only done for a sample of selected
limited positions. Using two optical telescopes mounted on the platform, it
was verified that the relative pointing errors between the two follow the
relative tilts of them predicted by the platform deformation patterns. Then we
used the relative pointing errors between the two optical telescopes over the
whole sky to determine an empirical model of how the deformation pattern
changes with pointing.

Figure 3.— Platform deformation. Mount pointing positions (az, el, and hexpol)
are indicated in the titles of each panel. The unit of $x$ and $y$ is m, and
the unit of $dz$ in color is mm. Obvious saddle features are apparent for most
positions.

Figure 4.— Saddle pattern amplitude and phase of platform deformation at
$el=60^{\circ}$ (left two panels) and $40^{\circ}$ (right two panels). The
hexpol is $0^{\circ}$. The blue dots and green curve show the photogrametric
measurements and model prediction, respectively. Figure 5.— The left and
middle panels show the residual platform deformation after averaging out all
the information dependent on pointing azimuth. The elevation of the left and
middle panel are $40^{\circ}$ and $60^{\circ}$, respectively. The unit of $x$
and $y$ is m, and the unit of $dz$ is mm. All data were taken at
$hexpol=0^{\circ}$. The right panel shows the azimuthally averaged platform
deformation after removing the contribution of pointing azimuth.

###  2.1. Photogrammetry data and deformation modeling

Since the loading of the AMiBA platform changed after the upgrade to 13
elements, we did a new photogrammetry test in Nov 2009 (see Huang et al.
(2008) for the details of photogrammetry test). The test included 26 sets of
data. Each set of data contains several hundreds of photogrammetry
measurements taken at different positions on the platform with specified
platform pointing and polarization angles. These 26 sets can be split into
three groups. The 11 data sets in the first group share the same platform
$el=60^{\circ}$ and polarization angle $hexpol=0^{\circ}$. Their platform
pointings were distributed uniformly in azimuth between $0^{\circ}$ and
$360^{\circ}$. The platform pointings of the 9 data sets in the second group
were distributed in the same way in azimuth with $el=40^{\circ}$ and
$hexpol=0^{\circ}$. The other 6 data sets were taken at $el=30^{\circ}$,
$az=0^{\circ}$, and 6 different polarization positions. Part of the results
are shown in Figure 3.

The platform deformation can be described as a ”saddle pattern”, which can be
parametrized as following (Koch et al., 2008):

$dz=A[(x^{2}-y^{2})\cos 2\theta+2xy\sin 2\theta],$ (1)

where $dz$ is the platform deformation in the normal direction of the
platform, $x$ and $y$ indicate the position on the platform. Here we set east
direction as positive $x$, north as positive $y$, and positive $z$ above the
platform. The saddle pattern is described by two parameters: $A$, the
amplitude of the saddle pattern, and $\theta$, the phase of it.

For each set of photogrammetry data, we can find the best fit $A$ and
$\theta$. The best fit parameters of the first two groups of photogrammetry
data sets are shown in Figure 4 as blue dots. As we can see, the amplitude of
the best fit saddle pattern has a 3-fold symmetry. The phase of the saddle
pattern changes as $\theta=0.5az+\theta_{0}+\epsilon$, where $\theta_{0}$ is a
constant, and $\epsilon$ looks like a periodic perturbation. (Note: because of
our definition of $x$ and $y$, the directions of $\theta$ and $az$ are
opposite to each other.)

It can be shown that if there are two saddle patterns,
$dz_{1}(x,y)=A_{1}[(x^{2}-y^{2})\cos 2\theta_{1}+2xy\sin 2\theta_{1}]$ and
$dz_{2}(x,y)=A_{2}[(x^{2}-y^{2})\cos 2\theta_{2}+2xy\sin 2\theta_{2}]$, the
combination of these two saddle patterns
$dz_{1}+dz_{2}=A_{3}[(x^{2}-y^{2})\cos 2\theta_{3}+2xy\sin 2\theta_{3}]$ will
be another saddle pattern. Furthermore, we can derive that $A_{3}\cos
2\theta_{3}=A_{1}\cos 2\theta_{1}+A_{2}\cos 2\theta_{2}$ and $A_{3}\sin
2\theta_{3}=A_{1}\sin 2\theta_{1}+A_{2}\sin 2\theta_{2}$. In other words, a
saddle pattern can be described as a vector $\vec{p}$ with absolute value $A$
and phase angle $2\theta$, and the combinations of saddle patterns can be
considered as summations of vectors.

We can consider the best fit saddle pattern parameters shown in Figure 4 as
the absolute value and half of the phase angle of $\vec{p}$. The 3-fold
symmetry of $|\vec{p}|$ and the periodic perturbation of $angle(\vec{p})$ can
be explained as the summation of two vectors
$\vec{p}_{1}=(A_{1}\cos{\alpha_{1}},A_{1}\sin{\alpha_{1}})$ and
$\vec{p}_{2}=(A_{2}\cos{\alpha_{2}},A_{2}\sin{\alpha_{2}})$ rotating along the
opposite directions as $az$ grows. If $\alpha_{1}$ rotates with angular
velocity $w_{1}=1$ as the azimuth grows and $\alpha_{2}$ rotates with angular
velocity $w_{2}=-2$, $\vec{p}_{1}$ and $\vec{p}_{2}$ will be parallel to each
other while $az=0,2/3\pi,4/3\pi$, and anti-parallel while
$az=1/3\pi,\pi,5/3\pi$. That can explain the 3-fold symmetry of $A$, which
corresponds to the absolute value of the summation of $\vec{p}_{1}$ and
$\vec{p}_{2}$. If we assume $A_{1}>A_{2}$, the phase angle $\alpha$ of the
combined vector $\vec{p}$ will be swinging around $\alpha_{1}$ with a period
of $2/3\pi$. That can also explain what we see in Figure 4.

The two-saddle model described above can be summarized as following:

$\displaystyle dz$ $\displaystyle=$ $\displaystyle A_{1}[(x^{2}-y^{2})\cos
2\theta_{1}+2xy\sin 2\theta_{1}]$ (2) $\displaystyle+A_{2}[(x^{2}-y^{2})\cos
2\theta_{2}+2xy\sin 2\theta_{2}],$

where

$\displaystyle\theta_{1}$ $\displaystyle=$
$\displaystyle\frac{az}{2}+\phi_{1}$ $\displaystyle\theta_{2}$
$\displaystyle=$ $\displaystyle-az+\phi_{2}.$ (3)

Equation (2) remains a saddle pattern.

There are four parameters: $A_{1}$, $A_{2}$, $\phi_{1}$, and $\phi_{2}$, in
the model described by Equation (2). The green curves in Figure 4 show $A$ and
$\theta$ as predicted by our model with the parameters fitted to the
photogrammetry data. We found that our model fits the blue dots well. We also
noticed that both $A_{1}$ and $A_{2}$ grow significantly as the platform
elevation is lowered from $60^{\circ}$ to $40^{\circ}$. But $\phi_{1}$ and
$\phi_{2}$ do not change much ($<10^{\circ}$) between elevation $40^{\circ}$
and $60^{\circ}$.

There can be additional secondary features of the platform deformation other
than the saddle pattern model described above. To investigate this, we
subtract the best fit saddle patterns from each set of photogrammetry data,
then average the residuals over pointing azimuth. By averaging over pointing
azimuth, we can extract the deformation features independent of it. The left
and middle panels of Figure 5 show the averaged residual data at elevation
$40^{\circ}$ and $60^{\circ}$, respectively. We notice that the averaged
residual has isotropic features. The residual $dz$ tends to be higher at the
center and lower at the edge of the platform. In the right panel of Figure 5
we average the residual feature along concentric circles, which share the same
center with the platform itself, on the platform. The difference of $dz$ at
the center and the edge is about $0.15$mm, or $0.05\lambda$. The azimuthal
average does not change much ($<0.03$)mm as the pointing elevation changes
between $40^{\circ}$ and $60^{\circ}$.

We are now addressing the question whether the secondary features are
important to our phase correction work. Considering our calibration method, if
the cluster scans and the calibration scans share the constant secondary-
feature deformation, the phase delays induced by it will be subtracted during
the calibration scheme. Therefore, the key concern is how the platform
deformation changes with platform position. This will then determine how to
calibrate the data in the absence of a nearby strong calibrator. Based on this
consideration, the saddle pattern model described in Equation (2) plays the
main role in our platform deformation corrections. Since the secondary
deformation pattern only changes within $0.01\lambda$ as pointing changes from
$el=40^{\circ}$ to $el=60^{\circ}$ at $\simeq 95\%$ of positions on the
platform (i.e.: comparing the left and middle panels of Figure 5), it is
considered to be negligible in this work. Consequently, secondary features are
not further taken into account in our correction scheme.

###  2.2. Correlation of 2 Optical Telescopes

In principle we can directly measure the platform deformation with
photogrammetry measurements. However, it takes time to sample the entire
3-dimensional pointing parameter space ($az$, $el$, and $hexpol$) with
acceptable resolution. Fortunately, there is another way using the optical
pointing measurement to estimate the platform deformation model for the whole
sky.

Figure 6.— The upper left panel is the X-direction pointing error measured by
OT1, the middle left panel is the X-direction pointing error measured by OT2,
and the lower left panel is the difference between OT1 and OT2 measurement on
X-direction pointing error. The right column panels show the Y-direction
pointing error in the same order. The unit of $az$ and $el$ are both degree,
and the unit of pointing error is arc-second. All the pointing errors in this
figure are obtained at $hexpol=0^{\circ}$. The interpolation table built based
on OT1 pointing test was applied to correct the pointing error. Figure 7.—
Amplitude of Fourier modes obtained by Fourier transforming the difference of
pointing errors measured by two OTs (as shown in Figure 6) along iso-elevation
bands. The left panel shows the results obtained with X-direction pointing
error, and the right panel shows the Y-direction results.

There are two optical telescopes (OTs) mounted on the AMiBA platform to test
the pointing of the mount. Because the pointing of a single optical telescope
can be significantly affected by the platform deformation and other local
effects, we use two of them to do cross checking. The first optical telescope
(OT1) is mounted at the distance of about 0.6 meters away from the center
point of the platform. The other optical telescope (OT2) is mounted at the
same radius, but $120^{\circ}$ apart from OT1.

The main idea of the optical pointing test is that the difference of pointing
errors taken by two OTs provides information of the relative tilt between two
OTs. We can then use the information of this relative tilt to fit the platform
deformation model described above. Comparing with the several hundreds of
samples provided by the several hundreds of targets on the platform in the
photogrammetry test for a single pointing position, the sampling for a single
pointing position provided by optical pointing is really poor (only 2 samples
for a pointing position). However, if the platform deformation itself can be
described well by a simple model like our two-saddle pattern model, the poor
sampling on the platform will not cause any severe problem. On the other hand,
the optical pointing test can provide $\sim 500$ samples per night in the
parameter space of platform pointing and polarization angle. With
photogrammetry measurements it would take 2 months to achieve the same
sampling level. Since we need to build all-sky phase corrections, optical
pointing tests are more efficient and suitable for this goal.

Our optical pointing tests for AMiBA-13 were conducted in 2010. We followed a
similar procedure as described in Koch et al. (2008). The sky was partitioned
into 500 zones with equal solid angle above elevation $=30^{\circ}$. One
bright star per zone was chosen. We took images at
$hexpol=-24^{\circ},-12^{\circ},0^{\circ},12^{\circ},$ and $24^{\circ}$ for
each star with 2 OTs. Therefore, we have 10 images for each star. It took 5
nights to finish the optical pointing observations. Then we grouped those 2500
images according to OTs and $hexpol$. With each image we derived the pointing
error at a specified pointing coordinate. Therefore, we have 500 data points
of pointing error with each combination of specified OT and polarization.
After removing the constant tilt of the OTs (Koch et al., 2008), we
interpolated a pointing error map (e.g.: Figure 6) at different hexpol for
each OT.

First, we use OT1 to construct an interpolation table (IT), which is used to
correct the mount pointing error. The rms of OT1 pointing error is about
$0.3^{\prime}$ with IT correction. However, the difference of pointing between
OT1 and OT2 is at the level of $1^{\prime}-2^{\prime}$. Figure 6 shows the
pointing errors of both OTs and the differences between the errors. In Figure
6 we have transformed the pointing errors into platform coordinates.

The differences of pointing errors in $X$ and $Y$ directions can be expressed
as $perr_{diff,x}(az,el)$ and $perr_{diff,y}(az,el)$, respectively. We fix the
elevation to extract

$\displaystyle g_{x,el_{0}}(az)=perr_{diff,x}(az,el=el_{0})$ $\displaystyle
g_{y,el_{0}}(az)=perr_{diff,y}(az,el=el_{0}).$ (4)

We then perform Fourier transformations on $g_{x}$ and $g_{y}$ with respect to
$az$. Finally, we get Fourier modes as functions of elevation.

Figure 7 shows the Fourier transformation results of the relative pointing
errors between 2 OTs. As shown in Figure 7, the dominating Fourier modes of
the pointing differences are the modes with $n=0,1,2$. The higher order modes
are negligible.

Ideally, the relative pointing errors between OTs can be explained by the
platform deformation. Here, we assume that the pointing errors measured by the
optical telescopes are the combination of the mount pointing error and the
relative tilt between OT and platform. We also assume that the relative tilt
between OT and platform is contributed by two origins: generic tilting of the
OT box (assumed to be constant) and the platform deformation (assumed to be
variable). We adopted the way described in Koch et al. (2008) to remove the
constant tilt. In this section we further assume that the remaining tilt can
be fully explained by the local normal vector of the platform at the positions
of the OTs. Because of the platform deformation, the local normal vector will
not be exactly parallel to the true pointing of the platform. That will cause
a relative tilt of the OTs.

With Equation (1), the normal vector change contributed by the saddle-shape
platform deformation can be written as:

$\displaystyle\delta n_{x,sadd}=-\frac{\partial dz}{\partial x}$
$\displaystyle=$ $\displaystyle-2A(y\sin 2\theta+x\cos 2\theta)$
$\displaystyle=$ $\displaystyle-2Ar\cos(\alpha-2\theta)$ $\displaystyle\delta
n_{y,sadd}=-\frac{\partial dz}{\partial y}$ $\displaystyle=$ $\displaystyle
2A(y\cos 2\theta-x\sin 2\theta)$ (5) $\displaystyle=$ $\displaystyle
2Ar\sin(\alpha-2\theta),$

where $r$ and $\alpha$ are the polar coordinate coefficients of the OT
location.

With Equation (2), $\delta n_{x,sadd}$ and $\delta n_{y,sadd}$ can be written
as the combination of two saddle pattern contributions:

$\displaystyle\delta n_{x,sadd}$ $\displaystyle=$
$\displaystyle-2A_{1}r\cos(\alpha-2\theta_{1})-2A_{2}r\cos(\alpha-2\theta_{2})$
$\displaystyle\delta n_{y,sadd}$ $\displaystyle=$ $\displaystyle
2A_{1}r\sin(\alpha-2\theta_{1})+2A_{2}r\sin(\alpha-2\theta_{2}).$ (6)

$A_{1}$, $A_{2}$, $\phi_{1}$, and $\phi_{2}$ are assumed to be functions of
elevation.

Considering the common $r=0.6$m of the two optical telescopes, we can derive
the relative tilt $\Delta n_{x,sadd}$ and $\Delta n_{y,sadd}$, which can be
attributed to platform deformation, between two optical telescopes as
following.

$\displaystyle\Delta n_{x,sadd}$ $\displaystyle=$ $\displaystyle
2A_{1}r[(\cos\Delta\alpha-1)\cos\beta_{1}-\sin\Delta\alpha\sin\beta_{1}]$
$\displaystyle+2A_{2}r[(\cos\Delta\alpha-1)\cos\beta_{2}-\sin\Delta\alpha\sin\beta_{2}]$
$\displaystyle\Delta n_{y,sadd}$ $\displaystyle=$ $\displaystyle
2A_{1}r[(1-\cos\Delta\alpha)\sin\beta_{1}-\sin\Delta\alpha\cos\beta_{1}]$ (7)
$\displaystyle+2A_{2}r[(1-\cos\Delta\alpha)\sin\beta_{2}-\sin\Delta\alpha\cos\beta_{2}],$

where $\beta_{1}=\alpha_{1}-az-2\phi_{1}$,
$\beta_{2}=\alpha_{1}+2az-2\phi_{2}$, and
$\Delta\alpha=\alpha_{2}-\alpha_{1}$. $\alpha_{1}$ and $\alpha_{2}$ are the
angular positions of OT1 and OT2, respectively.

We can further put $\Delta\alpha=-2\pi/3$ into Equation (7) and get:

$\displaystyle\Delta n_{x,sadd}$ $\displaystyle=$ $\displaystyle
2\sqrt{3}A_{1}r\cos(\beta_{1}-\frac{5\pi}{6})+2\sqrt{3}A_{2}r\cos(\beta_{2}-\frac{5\pi}{6})$
(8) $\displaystyle\Delta n_{y,sadd}$ $\displaystyle=$ $\displaystyle
2\sqrt{3}A_{1}r\cos(\beta_{1}-\frac{\pi}{3})+2\sqrt{3}A_{2}r\cos(\beta_{2}-\frac{\pi}{3}).$
(9)

Considering the definition of $\beta_{1}$ and $\beta_{2}$, we can see that the
two terms in the right hand side of Equation (8) and (9) correspond to the
$n=1$ mode and $n=2$ mode in Figure 7. (Note: The measured pointing error
differs from the tilt of the OT by a minus sign.) Therefore, we can use the
amplitudes and phases of $n=1$ and $n=2$ modes to determine deformation model
parameters $A_{1}$, $A_{2}$, $\phi_{1}$, and $\phi_{2}$ as functions of
elevation. Furthermore, we can also derive these parameters at different
$hexpols$ with different sets of pointing error data. In other words, we can
predict the platform deformation using the model (Equation (2)) with given
pointing coordinate and polarization angle of the platform. With the knowledge
of the location of each antenna on the platform, we can also predict the
vertical displacements of all of the 13 antennas. That will lead us to the
crucial geometric delays and phase delays.

However, the $n=0$ mode in Figure 7 is still not explained by our saddle
model. Because this $n=0$ mode is independent of pointing azimuth, we should
be able to see significant changes of the residual as shown in Figure 5 at
different elevations if this $n=0$ mode is caused by the global pattern of the
platform deformations. In Figure 5 we see small differences between the
residual deformation at $el=40^{\circ}$ and $el=60^{\circ}$. This difference
is not large enough to explain the large $n=0$ mode shown in Figure 7.
Consequently, we think this $n=0$ mode might be mainly due to some local
effect near the locations where the OTs were mounted. In this case, we do not
need to further consider the $n=0$ mode while determining the global platform
deformation pattern. It is shown in section 3.1 that the platform deformation
corrections fit the observed phases of visibilities very well without taking
into account the $n=0$ mode.

One can also combine more than two rotating saddle patterns by rewriting
Equation 2 as linear combination of rotating saddle pattern terms such as
$A_{m}[(x^{2}-y^{2})\cos 2\theta_{m}+2xy\sin 2\theta_{m}]$, where
$\theta_{m}=(m/2)\times Az+\phi_{m}$, and $m$ is a integer. Following the
similar way from Equation 6 to Equation 9, one can see that the saddle pattern
rotating with $\theta_{m}$ will corresponds to Fourier mode with $n=\mid
m\mid$. Therefore, for a platform with verified saddle pattern deformation,
one can use Fourier modes of relative pointing errors of different OTs to
determine the parameters of each of the rotating saddle patterns. In the case
of AMiBA-13, the Fourier modes with $n>2$ are negligible. This fact implies
that the saddle patterns with larger $m$ are also negligible.

##  3\. Correction for Platform Deformation

Using the model parameters derived above at different elevations, one can
reconstruct the platform deformation with given platform pointing using
Equation (2). By feeding the positions of the 13 antennas on the platform into
the platform deformation model, the vertical displacement of each antenna
$d_{i}$ can be obtained. In this section, we will describe how we verify the
model-predicted $d_{i}$ with real radio observations, and the way we correct
the phase errors induced by platform deformations using the model above. We
will also present how much the performance of AMiBA-13 has been improved with
this correction.

Figure 8.— Comparison between SVD solutions of vertical offsets (black dots)
and predicted vertical offsets of the model (green dots). The SVD solutions
were solved from the data of seven 6-hour long tracks on Jupiter. The x axis
is hour angle in unit of degree, and the y axis is vertical offset in unit of
mm. The vertical offsets of antenna 1 are set to be 0 and not shown here. The
antenna 9 was temporarily off-line while doing the long tracks on Jupiter.

###  3.1. Solving Deformations from Visibilities

One important aspect of verification of the parametrized deformation model is
to check if the modeled geometric delays are consistent with those measured
from the visibility phases. For this purpose, we observed Jupiter, which is
considered a point source for AMiBA, with the two-patch subtraction method (Wu
et al., 2009) over several nights, covering as much as possible in the range
of hour angles. The uncertainty in the observed visibility phases were
dominated by systematic effects, while thermal noise could be ignored. The
major systematic effect was attributed to the band smearing effect when
transforming the limited 4-lag correlations to visibilities in two frequency
channels. Given a fixed transformation kernel, this effect sensitively depends
on the bandpass function of the receiver, intermediate frequency (IF)
electronics, and the individual analogue correlators (Lin et al., 2009). The
systematic uncertainty on the visibility phase thus has an antenna-based
contribution (from receivers and IFs) and a baseline-based contribution (from
correlators). One thing to remember is that the band smearing effect depends
on the total geometric delay (i.e. source offset and platform deformation),
and that it varies with telescope positions.

The measured visibility phase between any antenna pair is related to their
geometric delays, instrumental delays, phase difference of local oscillator
(LO) signals, and the lag-to-visibility systematics. With a strong point
source, its pointing offset can easily be determined from the coherent phase
distortion and be corrected. Furthermore, Jupiter data taken each night were
self-calibrated with the visibility taken closest to its transit (highest
elevation), leaving only effects that vary with telescope positions:

$\phi_{ij}=k(\bar{d}_{i}-\bar{d}_{j})+(\varphi_{i}-\varphi_{j})+\psi_{ij},$
(10)

where $\phi_{ij}$ is the calibrated phase between antenna $i$ and $j$, $k$ is
the wave number, $\bar{d}_{i}$ is the vertical displacement for antenna $i$
relative to when it is at near transit, $\varphi_{i}$ is the antenna-based
relative systematic uncertainty, and $\psi_{ij}$ is the baseline-based
relative uncertainty.

Since the baseline-based systematics do not correlate with antennas, as an
approximation $\psi_{ij}$ can be dropped when solving Equation (10). Equation
(10) can be rewritten as

$\displaystyle\phi_{\alpha}$ $\displaystyle=$ $\displaystyle(\delta_{\beta
i}-\delta_{\beta j})_{\alpha}\tilde{d}_{\beta}$ (11) $\displaystyle=$
$\displaystyle M_{\alpha\beta}\tilde{d}_{\beta},$

where the index $\alpha$ refers to distinct pairs of antenna $i$ and $j$,
$\beta$ refers to distinct antennas, $\delta_{\beta i}$ denotes the Kronecker
delta, and $\tilde{d}_{\beta}$ is the relative vertical displacement of
antenna $\beta$ including the antenna based systematics. In the case of
AMiBA-13, we have index $i$, $j$, and $\beta$ ranging from $1$ to $13$,
referring to 13 antennas, and index $\alpha$ ranging from $1$ to $78$,
referring to 78 distinct antenna pairs.

The sparse matrix is clearly not square and is singular. We used the LAPACK
routine _sgesvd_ to perform the singular value decomposition (SVD) of the
sparse matrix $M_{\alpha\beta}$, zeroing singular values smaller than
$10^{-6}$, and then constructed the pseudo-inverse matrix
$M^{-1}_{\beta\alpha}$ that was applied to the measurement
$\tilde{d}_{\alpha}$. However, since the equations are based on differences of
the variables, an arbitrary constant can be added to the solutions. Here we
define the vertical displacements of the central antenna 1 to be zero (Figure
8).

Once the solutions of the vertical displacements of each antenna from real
radio observations have been obtained, one can compare the solutions with the
predictions of the model above. Figure 8 shows the comparison between model-
predicted vertical displacements, and those solved from radio observation, for
each antenna. As shown in Figure 8, the model predictions match the solved
vertical displacements within a $0.2$mm range for most of the antennas and
pointing coordinates. The good match between solved $\tilde{d}_{\beta}$ and
the predictions of deformation model is a good evidence showing that other
antenna-based systematics are secondary compared with the platform
deformation.

Figure 9.— The upper panels show the comparison between the peak values
constructed from phase corrected visibilities (blue stars), uncorrected
visibilities (black dots), and the simulated perfect phase correction (red
crosses), of Saturn (left) and Jupiter (right) data. The middle panels show
the corrected peaks and uncorrected peaks divided by perfect peaks. The lower
panels show the pointing elevations of Saturn and Jupiter at each frame. The
declination of Saturn and Jupiter was about $-2^{\circ}$ and $+11^{\circ}$,
respectively. All the Saturn data were calibrated by the Saturn two-patches
scans at the highest elevation every night. All the Jupiter data were
calibrated by the Jupiter two-patches scans at $az\sim 119^{\circ}$ and
$el\sim 72^{\circ}$ every night. The expected flux of Saturn and Jupiter was
238 Jy and 1600 Jy, respectively.

###  3.2. Phase Correction of Visibilities

With the platform deformation model and pointing coordinate, we can predict
the $d_{i}$ and $d_{j}$ required in Equation (10) to derive the geometric
phase delay $\phi_{ij}$, which is induced by the platform deformation, for
each pair of antennas. We simply multiply the calibrated visibilities
$V_{ij,calibrated}$ by the exponential term
$exp[i(\phi_{ij,target}-\phi_{ij,calibrator})]$ to correct the platform
deformation induced phase error. $\phi_{ij,target}$ and $\phi_{ij,calibrator}$
are $\phi_{ij}$ derived with the coordinates of the observed target and
calibration events, respectively.

We used Jupiter and Saturn data to quantify the improvement of the phase
correction. Because the Jupiter and Saturn data are self-calibrated, we
exactly know how strong the central flux should be with perfect correction. In
order to do that, we take several 2-patches scans across the transit point on
Jupiter and Saturn. Then we use one of the planet scans for each night as a
calibrator to calibrate the data of the other scans, and construct the images
with and without the platform deformation phase correction. For Saturn, we
choose the scans closest to the transit point as the calibrators. For Jupiter,
we choose scans with hour angles of about $-1$h to control the pointing error.
Each two-patches scan was used to reconstruct one image with phase correction
and one without correction. Figure 9 shows the comparison of the reconstructed
central fluxes of Saturn with and without the phase corrections. We can see
that the phase correction can recover about $30\%$ of flux lost of Saturn at
$el\sim 45^{\circ}$. We can also recover about $50\%$ of flux lost of Jupiter
at $el\sim 55^{\circ}$ (see Figure 9).

We also constructed Saturn and Jupiter images using only amplitudes of the
visibilities to simulate perfect phase correction. The peak values of these
images are shown as red crosses in the upper panels of Figure 9. By looking at
these red crosses, we can find out that there will be still some flux lost
even with perfect phase correction. One possible reason is likely to be the
band smearing effect. Since AMiBA operates in a wide frequency band with two
frequency channels, the incoming signal within a wide frequency range is
integrated together. However, the phase delay due to the same geometrical
delay changes at different frequency. Therefore, the signal integrated over a
wide band is smeared by non-constant phase changes within the band. This
smearing effect reduces the amplitude of observed visibilities. In order to
evaluate the efficiency of the phase correction itself, we divided the peak
flux values reconstructed with and without phase correction by the peak values
derived with complete phase removal. The results are shown in the middle
panels of Figure 9. The phase correction can recover more than $50\%$ of flux
lost due to phase error, restoring more than $90\%$ of the flux of the
’perfect phase correction’, in most of the planet two-patches scans.

If the flux loss is due to band smearing, given that the amplitude of the
platform deformation increases as the pointing elevation decreases, one can
expect that simulated flux should decrease while the elevation becomes lower.
The red crosses in the upper panels of Figure 9 show the expected trend. By
simulating the perfect phase correction with Saturn and Jupiter data, we see
that the remaining flux lost above elevation $40^{\circ}$ is smaller than
$10\%$. Since the response across the entire bandwidth has not been measured
in detail, it is difficult to estimate and correct the band smearing effect.

###  3.3. Verification of Phase Correction with Radio Sources

In the sections above, we corrected the phase delays and verified the
efficiency of our correction with Jupiter and Saturn data. However, we can
only verify the method within part of the pointing range with small changes of
the declination of the planets. So we selected several radio sources with
expected fluxes higher than 2 Jy at 90GHz from the ATNF catalogue
777http://www.narrabri.atnf.csiro.au/calibrators/. We then did continuous two-
patches scans on these radio sources. With one two-patches scan containing two
4-minutes patches, we can achieve S/N ratios higher than 30 for these strong
radio sources. Other than the sources from ATNF catalogue, Neptune is also
included in the samples of radio sources. The declinations of these targets
are evenly spread between $+50^{\circ}$ and $-30^{\circ}$. The calibrator for
this test is Jupiter. Most of the calibration scans were taken near
$az=115^{\circ}$ and $el=73^{\circ}$, to control the pointing error induced by
the calibration process. Figure 10 shows an example of the impact of phase
correction on the image making. The peak fluxes of the images constructed with
uncorrected and corrected visibilities are shown in Figure 11.

Figure 10.— An illustration of the improvement in image reconstruction
contributed by phase correction. The target here is QSO B0355+508. The upper
panels show the dirty maps reconstructed with visibilities after phase
correction, while the lower panels show the results without phase correction.
The angular unit here is arc-second, and the intensity unit is mJy. The
azimuth and elevation of the platform pointings are shown in the image titles.

Assuming the phase correction is valid for these radio sources, we expect to
see two features. The first one is an overall flux rise. The variances of the
observed fluxes are also expected to be reduced after correction because phase
correction is supposed to be able to remove the flux variances caused by
deformation. As we can see in Figure 11, the phase correction increases the
observed flux in almost all the 2-patch scans. The impacts of phase correction
are more significant at low elevations as we expected. The variances of the
flux between different scans on the same target are also reduced after phase
correction.

The phase corrections are expected to be smaller while the pointing of the
observations are close to the calibration events. So one can also expect to
see smaller flux changes after correction at pointing coordinates closer to
the calibration events. Since most of the calibration events are in the
eastern part of the sky, we can expect a stronger impact of phase correction
in the western part of the sky as what we see in real data.

As proven to be valid on point radio sources, the phase correction method is
applied to galaxy cluster data of AMiBA-13 in Lin et al. (2013, in
preparation).

Figure 11.— The impact of phase correction on observed flux of radio sources
and Neptune. The upper panels for each target show the observed fluxes. The
black dots are uncorrected fluxes and the red stars are corrected ones. Each
dot or star shows the result from one 2-patch scan which takes 8 minutes. The
middle panels show the ratios between corrected fluxes and the uncorrected
ones. The lower panels show the elevations of each 2-patch scan.

##  4\. Discussions and Conclusion

The deformation of the AMiBA platform is negligible in the AMiBA-7
configuration, but becomes a critical component of image reconstruction in the
AMiBA-13 configuration. We present a new way to cope with the phase delays
induced by the platform deformation. We first use photogrammetry data taken at
a few platform pointings to derive a generic platform deformation model. Then
we use the difference of the optical pointing errors measured by two platform-
mounted optical telescopes to estimate the deformation parameters within the
entire pointing parameter space of the AMiBA platform. With the platform
deformation modeling and phase correction method, we can control the flux loss
to be less than $10\%$ for most of the platform pointings.

The major part of the remaining flux loss is likely to be the result of a band
smearing effect which is difficult to quantify exactly due to the low
frequency resolution of the current AMiBA. With known response functions of
each receiver and each baseline, it would be possible to further correct the
band smearing effect with the platform deformation model. However, this part
is currently not yet developed.

The saddle pattern deformation of AMiBA platform has been confirmed with both
AMiBA-7 and AMiBA-13 configuration. Photogrammetry tests also show that
changing the loading on the platform does not alter the saddle pattern
deformation of it. Therefore, if the AMiBA antenna configuration changes in
the future, one can perform multi-OT pointing test and use Equation (5) to
check the saddle pattern. If saddle pattern is verified with multi-OT test,
Fourier transformation of relative pointing errors between OT pairs can be
done in the same way as described in Section 2.2. The azimuthal dependence of
phases of saddle patterns can be determined by different Fourier modes of
$g_{x}$ and $g_{y}$, as described in Section 2.2. If the multi-OT test results
are not consistent to Equation (5), more photogrammetry might be needed to
determine the new platform deformation pattern. The comparison between three
or more OTs can also help to determine the origin of the Fourier mode with
$n=0$. The $n=0$ mode is likely to be related to some local effects on or near
a specified OT if $n=0$ mode is only seen in the OT pairs contain it.

The deformations of other instruments are not guaranteed to be like saddle
pattern. Therefore, while applying the idea of this work onto another
instrument, photogrammetry data should be taken at several well chosen mount
position to understand the deformation model for that specified instrument.
However, using optical telescopes to probe the model parameters for all of the
telescope positions would be still valuable because photogrammetry tests for
all of the positions could be avoid. That saves a lot of time.

Platform deformation can also be a critical issue for observation and image
reconstruction as outlined here with other co-planar interferometers while the
platform is not stiff enough. The same approach can be applied to other co-
planar interferometers while coping with the platform deformation problem. It
is also possible to apply this approach to investigate the deformation of
single dish telescopes with several optical telescopes mounted on different
locations on the primary mirror. The work of this paper also forms the basis
for the AMiBA-13 data analysis and science results. Now we are using the
method described in this paper to correct the galaxy cluster data of AMiBA-13.
The corrections and analysis in the cluster data will be presented in Lin et
al. (2013, in preparation).

We thank the Ministry of Education, the National Science Council (NSC), and
the Academia Sinica, Taiwan, for their funding and supporting of the AMiBA
project. YWL thanks M. Birkinshaw for helpful suggestions. The guiding,
supporting, hard working, and helpful discussions of the entire AMiBA team are
also acknowledged.

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