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Subsections

Mosaicing

A mosaic image is produced from interferometer data taken from several different, adjacent pointings on the celestial sphere. A single-pointing of data can be treated as a mosaic. One does not want to force the treatment of all single-pointing data as mosaics, ie, one does not always want to image the entire primary beam (PB) or correct for the PB. The mosaic machinery also allows for combining data from telescopes with different PB's and different imaging models, an interferometer and a single dish, for example.

Operations on Mosaic Datasets (Visibilities)

One will need to perform ``bulk'' operations on data from all pointings from one telescope (ie, calibration). One will need to perform ``piecemeal'' operations on all data from a single pointing from a single telescope (ie, gridding/FFTing). For convenience of ``bulk'' operations, all the data from one telescope should be stored together in one dataset. Then the ``piecemeal'' operations could be performed by selecting out subsets of data. In addition to all the generic data selection possible with single-pointing interferometers, one must be able to select on sky pointing position (a single dish selector) and Telescope. Something like the YegSet Iterator which would step through all pointings one at a time would be nice. It is unclear whether the data from multiple telescopes should be stored in one dataset or in one dataset for each telescope. The number of different telescopes used will usually be one or two, so association of mosaic datasets from different telescopes by hand will not be too cumbersome.

Images

The final product of the MOSAIC imaging algorithm is a plain old image. The main difference between a MOSAIC image and an image produced by CLEAN, say, is that it is inappropriate to apply a PB to the final MOSAIC image: the parameters OBSRA, OBSDEC, and TELESCOP are not uniquely specified. While it is unclear how they would be used, if these parameters are present at all, they should be in arrays attached to the image indicating ALL pointings of ALL telescopes. There may or may not be an associated SENSITIVITY image, which is essentially a generalized primary beam, and can be used to determine the theoretical thermal noise in each pixel.

The fundamental equation at work in mosaicing is derived by making a pixel by pixel least square fit to the true sky brightness from a number of overlapping images weighted by the PB:

I_mosaic( $\displaystyle \bf {x}$ ) = $\displaystyle {\frac{\sum_{i=1}^{Npointings} I_{i} A_{i}({\bf {x}}- {\bf {x}}_{i})}{\sum_{i=1}^{Npointings} (A_{i}({\bf {x}}- {\bf {x}}_{i}))^{2}}}$

(1)

where I_i( $\bf {x}$ ) is an image made at pointing i, and A_i( $\bf {x}$ - $\bf {x}_{i}^{}$ ) is the primary beam of the telescope used for pointing i with pointing center $\bf {x}_{i}^{}$ . The square root of the denominator is the sensitivity image.

It has not been specified what kind of images the I_i( $\bf {x}$ ) are. In some cases, I_i will be independently deconvolved images. In this case, they may have some permanence outside the mosaicing method. More common, at least when the MMA comes on line, will be the case where I_i are dirty images produced for the sole purpose of making a mosaiced image. The mosaiced dirty image may then be deconvolved with a single effective point spread function. In this case, the I_i have no permanence. And finally, the most complicated case:

I_i( $\displaystyle \bf {x}$ ) = I^D_i( $\displaystyle \bf {x}$ ) - PSF_i * (A_i( $\displaystyle \bf {x}$ - $\displaystyle \bf {x}_{i}^{}$ ) $\displaystyle \hat{I}$ ( $\displaystyle \bf {x}$ )) (2)

where I^D_i( $\bf {x}$ ) is the dirty image made from data from pointing i, PSF_i is the point spread function for pointing i, and $\hat{I}$ ( $\bf {x}$ ) is a model image of the entire mosaic field. An unnormalized mosaic (ie, not divided by the denominator in Equation 1) of this quantity is proportional to the gradient of $\chi^{2}_{}$ with respect to the model image, and indicates how the model image must be altered in the next iteration of a maximum entropy mosaicing scheme. In this case, the I^D_i( $\bf {x}$ ) and the PSF_i (or more conveniently, the Fourier transform of the PSF_i) should persist from iteration to iteration for efficiency sake, though they could be recreated on each iteration if memory or disk space were in short supply. The intermediate images I_i( $\bf {x}$ ) can disappear as soon as they are accumulated into the gradient mosaic. The I^D_i and PSF_i do not need to persist after the final mosaic image is formed. While it may be convenient to place each of the I^D_i and PSF_i into collections of images, they are not really images of images. The details of how each of the I_i are combined should be implicit in the coordinate headers of the I_i and the coordinate header of some template image (ie, the final mosaic image or the gradient image).

Pathological Cases

The following two ImagingModels require knowledge of some parameters which seem to reside in the TelModel. The extent of the dependence of the ImagingModel on the TelModel is analyzed for the case of varying antenna voltage patterns and varying antenna pointing position.

Different voltage patterns for each antenna

If the primary beams are different for each antenna, then a single visibility is related to the sky brightness distribution via a Fourier sum:

V_{j, k}( $\displaystyle \bf {u}$ ) = $\displaystyle \int$ I( $\displaystyle \bf {x}$ )E_j( $\displaystyle \bf {x}^{\prime}_{}$ - $\displaystyle \bf {x}_{p}^{}$ )E^*_k( $\displaystyle \bf {x}^{\prime}_{}$ - $\displaystyle \bf {x}_{p}^{}$ )e^{-i2^.}d $\displaystyle \bf {x}$ (3)

where E_j( $\bf {x}$ ) is the complex voltage pattern for the j antenna ( PB(x) = | E(x)|²), and $\bf {x}^{\prime}_{}$ is related to $\bf {x}$ by rotation for azimuth-elevation mount antennas. For the general case of E_j $\ne$ E_k for j $\ne$ k, it is more efficient to use a direct Fourier sum than an FFT to calculate a model visibility from a model sky brightness distribution since the calculation must be performed one visibility point at a time. Correction for different voltage patterns may be important for high dynamic range MMA mosaic observations. The voltage patterns would probably be determined holographically. An imaging algorithm would then adjust the image iteratively using a gradient of $\chi^{2}_{}$ with respect to I( $\bf {x}$ ) which looks something like

$\displaystyle \Delta$ I( $\displaystyle \bf {x}$ ) = 2 $\displaystyle \sum_{i=1}^{Npointings}$ $\displaystyle \sum_{j,k;j<k}^{}$ FT(V_{i;j, k} - $\displaystyle \hat{V}_{i;j,k}^{}$ )E_j( $\displaystyle \bf {x}$ - $\displaystyle \bf {x}_{i}^{}$ )E_k( $\displaystyle \bf {x}$ - $\displaystyle \bf {x}_{i}^{}$ )

(4)

where $\hat{V}_{i;j,k}^{}$ is the model visibility calculated from the current model of the sky brightness distribution as per Equation 3. The basic requirement from Equations 3 and 4 is that the ImagingModel must have access to the complex voltage patterns for each antenna. Actually, ``VoltagePattern.Apply'' is the only method required external to ImagingModel. Under the current understanding, the voltage patterns would reside in the TelModel. If this is so, then the VPMosImagingModel requires very little of the TelModel.

Different pointing centers for each antenna

Pointing errors are thematically similar to different voltage patterns. A single visibility is related to the true sky brightness distribution as

V_{j, k}( $\displaystyle \bf {u}$ ) = $\displaystyle \int$ I( $\displaystyle \bf {x}$ )E( $\displaystyle \bf {x}^{\prime}_{}$ - $\displaystyle \bf {x}_{p_{j}}^{}$ )E^*( $\displaystyle \bf {x}^{\prime}_{}$ - $\displaystyle \bf {x}_{p_{k}}^{}$ )e^{-i2^.}d $\displaystyle \bf {x}$ , (5)

where the differences between Equation 5 and Equation 3 are the inclusion of antenna-dependent pointing positions $\bf {x}_{p_{j}}^{}$ and the omission of antenna dependent voltage patterns. Like the voltage pattern case, a direct Fourier sum must be used. The equation for the gradient image will be omitted to save on indices, but the basic idea is the same as for the antenna-dependent voltage pattern case. The PEMosImagingModel must have access to a method which supplies either a pointing correction or an absolute pointing position for a given antenna and at a given time. Again, this seems to require that the PEMosImagingModel have knowledge of the TelModel.

It is unclear whether these imaging methods for these two pathological cases will actually be implemented. However, it is known that simulations of pointing errors and voltage pattern errors (as caused by surface errors, for example) is required for the design of the MMA and other high frequency instruments.

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