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Non-isoplanatic Imaging

At low frequencies, the imaging problems related to the ionosphere are aggravated. The field of view is large and the ionospheric phases cannot be considered to be a function of time alone - they become a function of direction in the sky as well. The phase delays goes as $\lambda^{2}_{}$ at low frequencies and that makes it essential to correct the raw phases as a function of direction in the sky to get moderate dynamic range maps. Also the 2D approximation in inverting the visibilities to get the image breaks down. A 2D Fourier Transform is no longer valid.

The problem of 3D inversion is well understood and implemented (in SDE etc.). Here we attempt to describe the proposed algorithm for doing SelfCal for the non-isoplanatic case. Implementing these algorithms in AIPS++ will be a good test of the flexibility of the system and also a requirement.

The problem

In ordinary SelfCal, the gain corrections applied to the visibilities, are assumed to be antenna based and constant for the entire field of view of the individual antennas for the given time range (the SelfCal integration time interval). This means that a single gain is associated with a given antenna, at a given time. In other words, the properties of the ionosphere do not vary appreciably over each antenna's PB. However, when the field of view is large, the gain associated with the individual antennas is a function of the direction in the sky. Therefore, the gain solution needs to be a function of time as well as the direction in the sky.

One of the solution suggested (Schwab), divides the PB of each antenna into smaller patches, each of which can be assumed to be isoplanatic and then solve for the gains associated with each of these patches, for each antenna. Thus for an array with N antennas, and each PB divided into M patches, the number of unknowns to be solved for will be (M*N) - 1 (phases for all the patches in the reference antenna cannot be set to zero). The number of patches that one needs to have in the PB will depend on the size of the iso-planatic patches and that in turn will depend on the stability of the ionosphere, observing frequency, and PB size.

For the iso-planatic SeflCal, for an array with N antennas, the problem is overdetermined by a factor of N/2. If the source model is inaccurate or the S/N ratio is poor, for SelfCal to suceed, this overdeterminacy is essential. For non-isoplanatic case, the limit on the maximum number of patches is set by this requirement of overdeterminacy and overdeterminacy decreases as the number of patches increase. For large M (M will be large when the ionosphere is bad and the iosoplanatic patches are small), accurate initial model and high S/N ratio will be required.

For arrays like GMRT and VLA, the primary beams of most of the antennas will overlap at the typical ionospheric height (300 Km) for low frequency (<300 MHz) observations. This means that in the above scheme, the gains associated with the patches where the PBs overlap, will be solved for more than once (for each of the PBs in the overlapping region). Since the total number of unknows should be less than the number of baselines, this restricts the total number of patches in the PB to less than N(N - 1) + 2/2N. This also adds the extra computation of solving for redundant unknowns in the overlapping regions.

In the scheme proposed by Subrahmanya, the overlap of PBs at the ionospheric height, is taken into account by dividing the ionosphere, intercepted by the entire array, into isoplanatic patches and solving for the associated gains of these patches. Thus, for an array with N antennas with the ionosphere divided into M isoplanatic patches, the total number of unknowns would be N+M-2. This will relax the limit on the maximum possible number of isoplanatic patches to N(N - 1)/2 + + 2 - N. However, in general, the ionosphere cannot be regarded as a thin layer and the ionospheric structure in vertical direction has to be considered.

Formulation of the problem

For the case of isoplanatic ionosphere, the measured visibility can be written as

V_pq^iso = V^mode^{i((t) - (t))}

(6)

where $\phi_{q}^{}$ is the antenna based phase. Since the the antenna based phases are a function of time alone, the assumption of iso-planaticity is implicit.

Let the model sky image be represented by

B(x, y) = $\displaystyle \sum_{k}^{}$ A_k $\displaystyle \delta$ (x - x_k, y - y_k)

(7)

and the model visibility given by

V^mod = $\displaystyle \sum_{k}^{}$ $\displaystyle \xi_{k}^{}$

(8)

$\displaystyle \xi_{k}^{}$ = A_ke^{i(ux_k + vy_k + w1-x_k²-y_k²)}.

(9)

In the non-isoplanatic case, the observed visibilities can be expressed as

V_pq^non = e^{i((t) - (t))} $\displaystyle \sum_{k}^{}$ $\displaystyle \xi_{k}^{}$ e^{i((p) - (q))}

(10)

where $\psi_{k}^{}$ (p) is the total antenna based phase (apart from the phase due to the source) in the PB of antenna p in the direction (x_k, y_k) (the isoplanatic case is a special case of this equation where $\psi_{k}^{}$ (p) = $\psi_{k}^{}$ (q)). If $\theta_{l}^{}$ (p) represent the phase associated with the l^th cell, the total phase $\psi_{k}^{}$ (p) can be written as

$\displaystyle \psi_{k}^{}$ (p) = $\displaystyle \sum_{l}^{}$ w_l(p;x_k, y_k) $\displaystyle \theta_{l}^{}$

(11)

where w_l(p;x_k, y_k) $\theta_{l}^{}$ are weights associated with some interpolation scheme used for interpolation.

If the model visibility is represented as

V^mod_pq = e^{i(
- )} $\displaystyle \sum_{k}^{}$ $\displaystyle \xi$ e^{i(w_l^qk - w_l^pk)}

(12)

the unknowns are the $\phi$ 's and $\theta$ 's and can be solved for by minimizing the equation

S = $\displaystyle \sum_{p}^{}$ $\displaystyle \sum_{q}^{}$ | V^non_pq - V^mod_pq|².

(13)

The operations required

Non-Isoplanatic Selfcalibration

In the case of iso-planatic SelfCal, the algorithm walks through the time slices and generates the antenna gains as a function of time by solving for different time slices. For the non-isoplanatic case, the algorithm has to not only walk through the time slices, but also the space slices - the iso-planatic patches and generate antenna gain solutions as a function of direction in the sky for each time slice.

The most basic additional requirement for handling non-isoplanatic selfcalibration is to be able to get the estimate of the visibilities as a function of the direction in the sky. This requirement is explicit in Equation 10.

To start the non-isoplanatic imaging cycle, a model image is needed. At very low frequencies, the phases will be stable for less than a minute. With many isoplanatic patches across the PB and source structure which is often somewhat homogeneous (ie, no dominant point source), getting a model image to begin the selfcalibration cycle is non-trivial. A point-source model is usually inappropriate. A reasonable solution would be to observe the region of interest at higher frequencies where phase stability is not a problem and use the higher frequency image as a starting model.

Once a model image exists and each patch of sky's contribution to the model visibilities can be calculated, a least-square minimization can be done on Equation 13 to get solutions of $\theta_{l}^{}$ and $\phi_{p}^{}$ . Since the visibilities are required as a function of the direction in the sky, Solve requires the ImagingModel's PredictYegs method and therefore couples the TelModel with the ImagingModel explicitly.

One requirement on TelModel is that the gains must always be iteratively adjusted; you cannot apply the gains to a visibility set and then throw the gains away since the visibility set will then be phase stable for only one patch of sky.

Non-Isoplanatic Imaging

Once direction dependent gains exist, we can go about the work of imaging. Two approaches seem fruitful: a direct method, which applies the ``local'' gains for each patch, inverts the visibilities, and cleans; and an inverse approach, which tries to find the image which reproduces the data (raw) visibilities after unapplying the gains.

The direct approach is similar in spirit to MX and the SDE task FLY. It will likely be built onto FLY since FLY correctly deals with the non-coplanar baseline problem already. FLY generates a wide-field image from a collection of overlapping 2-D facets, each tangent to the celectial sphere, reducing the computation required to solve the full 3-D problem. In general, the facet size will be much smaller than the isoplanatic patch size, and the facet size is resolution dependent while the isoplanatic patch size is not. So, a patch will probably cover many facets.

In an N facet FLY, the visibilities are gridded onto N different Fourier planes, each conjugate to a facet plane, and Fourier transformed. The brightest feature in all facets is found. CLEANING in each facet proceeds independently until the maximum residual in each facet reaches the global maximum times the synthesized beams peak sidelobe level. At this point, the clean components which have been found in each facet are Fourier transformed and subtracted from the data visibilities. This ends one round of the major cycle. The major cycle is repeated until the desired image residual is achieved.

FLY can easily be modified to treat nonisoplanatic imaging. I assume that the gains for the isoplanatic patches have been determined from selfcal. As noted above, an isoplanatic patch probably spans severl FLY facets. However, the gains from adjacent patches may be different enough to warrent some interpolation of the gains in sky position. Some form of error analysis must be carried out to find out how small the image sections must be to produce the desired image fidelity. Smaller chunks will lead to less efficiency. We will ignore the interpolation problem in this discussion.

To image, apply the gains for the i^th isoplanatic patch to the visibilities. Grid them onto the Fourier planes and transform to produce the facets within the i^th isoplanatic patch. Loop over all isoplanatic patches, applying the appropriate gains. Again, conservative CLEANING proceeds, stopping in each facet when the maximum residual gets close to the global image peak before CLEANING started times the maximum sidelobe level. At this point, the clean components from each facet are Fourier transformed back to their respective visibility planes, the gains from the appropriate isoplanatic patch are un-applied to the component visibilities just generated, and the newly corrupted component visibilities are subtracted from the raw, uncalibrated visibilities. This is the end of the major cycle, which is repeated until the desired image residual is achieved. Both the Gainapply and Gain-unapply methods of TelModel are required by ImagingModel,

The inverse nonisoplanatic imaging method will not be developed in detail since it is not clear to us how to solve the noncoplanar baseline problem within its single-image context. It may be useful for VLBI or linear arrays. The inverse method requires the same services of TelModel or ImagingModel which the calibration and the direct non-isoplanatic imaging algorithm require.

Summary of Required Operations

For each time interval, there is a mapping for each antenna of the ionosphere's isoplanatic patches onto the celestial sphere. Both ImagingModel and TelModel must know about this mapping.
TelModel's Solve method must use the method PredictYegs for each patch on the source.
There can be no ``split'' to apply all the gains to the raw visibilities; rather, the gains must be updated and the raw data visibilities must be kept.
ImagingModel must use the TelModel.Gain-unapply and TelModel.Gainapply. (The inverse method only requires Gain-unapply.)

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