PLATFORM DEFORMATION PHASE CORRECTION FOR THE AMiBA-13 COPLANAR INTERFEROMETER

Since the loading of the AMiBA platform changed after the upgrade to 13 elements, we did a new photogrammetry test in 2009 November (see Huang et al. 2008 for the details of the 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° and polarization angle hexpol = 0°. Their platform pointings were distributed uniformly in azimuth between 0° and 360°. The platform pointings of the nine data sets in the second group were distributed in the same way in azimuth with el = 40° and hexpol = 0°. The other six data sets were taken at el = 30°, az = 0°, and six different polarization positions. Part of the results are shown in Figure 3.

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The platform deformation can be described as a "saddle pattern," which can be parameterized as follows (Koch et al. 2008):

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

where dz is the platform deformation in the normal direction of the platform, and x and y indicate the position on the platform. Here, we set the 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 θ, the phase of it.

For each set of photogrammetry data, we can find the best fit A and θ. 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 three-fold symmetry. The phase of the saddle pattern changes as θ = 0.5az + θ₀ + , where θ₀ is a constant, and looks like a periodic perturbation. (Note: because of our definition of x and y, the directions of θ and az are opposite from each other.)

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It can be shown that if there are two saddle patterns, dz₁(x, y) = A₁[(x² − y²)cos 2θ₁ + 2xysin 2θ₁] and dz₂(x, y) = A₂[(x² − y²)cos 2θ₂ + 2xysin 2θ₂], the combination of these two saddle patterns dz₁ + dz₂ = A₃[(x² − y²)cos 2θ₃ + 2xysin 2θ₃] will be another saddle pattern. Furthermore, we can derive that A₃cos 2θ₃ = A₁cos 2θ₁ + A₂cos 2θ₂ and A₃sin 2θ₃ = A₁sin 2θ₁ + A₂sin 2θ₂. In other words, a saddle pattern can be described as a vector $\boldsymbol {p}$ with absolute value A and phase angle 2θ, and the combinations of saddle patterns can be considered 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 $\boldsymbol {p}$. The three-fold symmetry of $\vert \boldsymbol {p}\vert$ and the periodic perturbation of $angle(\boldsymbol {p})$ can be explained as the summation of two vectors $\boldsymbol {p}_{1}=(A_{1}\cos {\alpha _{1}},A_{1}\sin {\alpha _{1}})$ and $\boldsymbol {p}_{2}=(A_{2}\cos {\alpha _{2}},A_{2}\sin {\alpha _{2}})$ rotating along the opposite directions as az grows. If α₁ rotates with angular velocity w₁ = 1 as the azimuth grows and α₂ rotates with angular velocity w₂ = −2, $\boldsymbol {p}_1$ and $\boldsymbol {p}_2$ will be parallel to each other while az = 0, 2/3π, 4/3π, and anti-parallel while az = 1/3π, π, 5/3π. That can explain the three-fold symmetry of A, which corresponds to the absolute value of the summation of $\boldsymbol {p}_{1}$ and $\boldsymbol {p}_2$. If we assume A₁ > A₂, the phase angle α of the combined vector $\boldsymbol {p}$ will be swinging around α₁ with a period of 2/3π. That can also explain what we see in Figure 4.

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

$$\begin{eqnarray} dz&=&A_{1}[(x^{2}-y^{2})\cos 2\theta _1 +2xy\sin 2\theta _1] \nonumber \\ & &+A_{2}[(x^{2}-y^{2})\cos 2\theta _2 +2xy\sin 2\theta _2], \end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray} \theta _1 &=& \frac{az}{2}+\phi _1 \nonumber \\ \theta _2 &=&-az+\phi _2. \end{eqnarray} \tag{ 3 }$$

Equation (2) remains a saddle pattern.

There are four parameters: A₁, A₂, ϕ₁, and ϕ₂, in the model described by Equation (2). The green curves in Figure 4 show A and θ 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 A₁ and A₂ grow significantly as the platform elevation is lowered from 60° to 40°. But ϕ₁ and ϕ₂ do not change much (<10°) between elevation 40° and 60°.

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° and 60°, respectively. We note 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 0.05λ. The azimuthal average does not change much (<0.03 mm) as the pointing elevation changes between 40° and 60°.

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We are now addressing the question of 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 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 0.01λ as pointing changes from el = 40° to el = 60° at ≃ 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.

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

There are two OTs mounted on the AMiBA platform to test the pointing of the mount. Because the pointing of a single OT 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 m away from the center point of the platform. The other optical telescope (OT2) is mounted at the same radius, but 120° 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. Compared 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 two 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 problems. On the other hand, the optical pointing test can provide ∼500 samples per night in the parameter space of platform pointing and polarization angle. With photogrammetry measurements it would take two 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°. One bright star per zone was chosen. We took images at hexpol = −24°, −12°, 0°, 12°, and 24° for each star with two OTs. Therefore, we have 10 images for each star. It took five 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.

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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 03 with IT correction. However, the difference of pointing between OT1 and OT2 is at the level of 1'–2'. 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 in pointing errors in the 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

$$\begin{eqnarray} &&g_{x,el_0}(az)=perr_{{\rm diff},x}(az,el=el_0) \nonumber \\ &&g_{y,el_0}(az)=perr_{{\rm diff},y}(az,el=el_0). \end{eqnarray} \tag{ 4 }$$

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 two 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.

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Ideally, the relative pointing errors between OTs can be explained by the platform deformation. Here, we assume that the pointing errors measured by the OTs 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 method 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-shaped platform deformation can be written as

$$\begin{eqnarray} \delta n_{x,{\rm sadd}}=-\frac{\partial dz}{\partial x}&=&-2A(y\sin 2\theta +x\cos 2\theta)\nonumber \\ &=&-2Ar\cos (\alpha -2\theta) \nonumber \\ \delta n_{y,{\rm sadd}}=-\frac{\partial dz}{\partial y}&=&2A(y\cos 2\theta -x\sin 2\theta)\nonumber \\ &=&2Ar\sin (\alpha -2\theta), \end{eqnarray} \tag{ 5 }$$

where r and α are the polar coordinate coefficients of the OT location.

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

$$\begin{eqnarray} \delta n_{x,{\rm sadd}}&=&-2A_{1}r\cos (\alpha -2\theta _1)-2A_{2}r\cos (\alpha -2\theta _2)\nonumber \\ \delta n_{y,{\rm sadd}}&=&2A_{1}r\sin (\alpha -2\theta _1)+2A_{2}r\sin (\alpha -2\theta _2), \end{eqnarray} \tag{ 6 }$$

where A₁, A₂, ϕ₁, and ϕ₂ are assumed to be functions of elevation.

Considering the common r = 0.6 m of the two OTs, we can derive the relative tilt Δn_{x, sadd} and Δn_{y, sadd}, which can be attributed to platform deformation, between two OTs as follows:

$$\begin{eqnarray} \Delta n_{x,{\rm sadd}}&=&2A_{1}r[(\cos \Delta \alpha -1)\cos \beta _{1}-\sin \Delta \alpha \sin \beta _{1}] \nonumber \\ & &+2A_{2}r[(\cos \Delta \alpha -1)\cos \beta _{2}-\sin \Delta \alpha \sin \beta _{2}] \nonumber \\ \Delta n_{y,{\rm sadd}}&=&2A_{1}r[(1-\cos \Delta \alpha)\sin \beta _{1}-\sin \Delta \alpha \cos \beta _{1}] \nonumber \\ & &+2A_{2}r[(1-\cos \Delta \alpha)\sin \beta _{2}-\sin \Delta \alpha \cos \beta _{2}], \nonumber \\ \end{eqnarray} \tag{ 7 }$$

where β₁ = α₁ − az − 2ϕ₁, β₂ = α₁ + 2az − 2ϕ₂, and Δα = α₂ − α₁. α₁ and α₂ are the angular positions of OT1 and OT2, respectively.

We can further put Δα = −2π/3 into Equation (7) and obtain

$$\begin{eqnarray} \Delta n_{x,{\rm sadd}}&=&2\sqrt{3}A_{1}r\cos \left(\!\beta _{1}-\frac{5\pi }{6}\!\right)+2\sqrt{3}A_{2}r\cos \left(\!\beta _{2}-\frac{5\pi }{6}\!\right)\nonumber \\&&\quad \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \Delta n_{y,{\rm sadd}}&=&2\sqrt{3}A_{1}r\cos \left(\beta _{1}-\frac{\pi }{3}\right)+2\sqrt{3}A_{2}r\cos \left(\beta _{2}-\frac{\pi }{3}\right).\nonumber \\ \end{eqnarray} \tag{ 9 }$$

Considering the definition of β₁ and β₂, we can see that the two terms on the right-hand side of Equations (8) and (9) correspond to the n = 1 mode and n = 2 mode in Figure 7. (Note that 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 A₁, A₂, ϕ₁, and ϕ₂ 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 coordinates and a 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, which 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 the pointing azimuth, we should be able to see significant changes in 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° and el = 60°. 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 a linear combination of rotating saddle pattern terms such as A_m[(x² − y²)cos 2θ_m + 2xysin 2θ_m], where θ_m = (m/2) × Az + ϕ_m, and m is a integer. Following the similar method from Equations (6)–(9), one can see that the saddle pattern rotating with θ_m corresponds to a Fourier mode with n = ∣m∣. Therefore, for a platform with a 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.

One important aspect of verification of the parameterized 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 was 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 four-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 analog 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 the geometric delays, instrumental delays, phase difference of local oscillator signals, and the lag-to-visibility systematics. With a strong point source, the 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:

$$\begin{equation} \phi _{ij}=k(\bar{d}_i-\bar{d}_j)+(\varphi _i-\varphi _j)+\psi _{ij}, \end{equation} \tag{ 10 }$$

where ϕ_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 near transit, φ_i is the antenna-based relative systematic uncertainty, and ψ_ij is the baseline-based relative uncertainty.

Since the baseline-based systematics do not correlate with antennas, as an approximation, ψ_ij can be dropped when solving Equation (10). Equation (10) can be rewritten as

$$\begin{eqnarray} \phi _{\alpha } &=& (\delta _{\beta i} - \delta _{\beta j})_{\alpha } \tilde{d}_{\beta }\nonumber \\ &=& M_{\alpha \beta } \tilde{d}_{\beta }, \end{eqnarray} \tag{ 11 }$$

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

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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.

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 ϕ_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(ϕ_{ij, target} − ϕ_{ij, calibrator})] to correct the platform deformation induced phase error. ϕ_{ij, target} and ϕ_{ij, calibrator} are ϕ_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 know exactly how strong the central flux should be with a perfect correction. In order to do that, we take several two-patch 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 hr to control the pointing error. Each two-patch 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 ∼ 45°. We can also recover about 50% of flux lost of Jupiter at el ∼ 55° (see Figure 9).

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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 still be some flux lost even with a perfect phase correction. One possible reason for this 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 frequencies. 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-patch 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° 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.

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 90 GHz from the ATNF catalog.⁷ We then did continuous two-patch scans on these radio sources. With one two-patch scan containing two four-minute patches, we can achieve signal-to-noise ratios higher than 30 for these strong radio sources. Other than the sources from the ATNF catalog, Neptune is also included in the samples of radio sources. The declinations of these targets are evenly spread between +50° and −30°. The calibrator for this test is Jupiter. Most of the calibration scans were taken near az = 115° and el = 73°, 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.

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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 two-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 is 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 from AMiBA-13 in K.-Y. Lin et al. (2013, in preparation).