6.4 Data Reduction Pipeline
6.4.3 Imaging
The third step was to produce an image from the visibility data. Since there is no spatial coherence be-tween electric fields generated by two surface elements within a radio source, their cross-correlation func-tion is equal to the Fourier transform of the brightness distribufunc-tion of the source, for quasi-monochromatic waves (Van-Cittert-Zernike theorem; Born and Wolf 1999)
Vν(~r1, ~r2)≈ Z Z
A(~s)Iν(~s)e−2
πiν~s·(~r1−~r2)
c dΩ (6.1)
whereVν(~r1) =hE~ν(~r1)E~∗ν(~r2)iis the cross-correlation function between the electric fields of two surface elements, ‘*’ indicates the complex conjugate,~r1−~r2is the vector separation between the two antennas,
CHAPTER 6. DATA REDUCTION TO DETERMINE POLARIZATION LEAKAGE 48
phase /degamp / Jy
frequency
amp / Jyphase /deg
frequency
Spectra for one scan on OQ208 Pol RCP. Baseline Nl−Wf
Figure 6.3: Example spectra on the North Liberty (VLBA) - Westford (IVS) baseline at 2225 MHz on OQ 208 showing amplitude in jansky vs frequency and phase in degrees vs frequency. Left: phase and amplitude without calibration: the phases rotates from 0◦to−180◦wraps to 180◦and rotates to 100◦across the 16 MHz band. Right:
the phases are now corrected by fringe fitting the data to remove a residual delay and now lie on a horizontal line close to 0◦.
A(~s) is the effective collecting area, Iν(~s) is the brightness distribution,~sis the unit vector toward the source anddΩ is an element of solid angle subtended by the source as visible from the antennas.
For imaging, it is more convenient to express the baseline length in terms of wavelengths at the centre frequency of the RF band in the direction toward the East (coordinateu) and the North (coordinatev).
The plane defined by this coordinate system is called the (u, v) plane and each antenna pair measures a point in the (u, v) plane (Clark 1999 and Thompson 1999). In this coordinate system,
ν~s·(~r1−~r2)
c =ul+vm (6.2)
dΩ = dldm
√1−l2−m2 (6.3)
where l andm are the source coordinates expressed as direction cosines in this reference frame. Figure 6.4 shows this coordinate system. In case of a source whose extension is small (like for geodesy) l ≈0 andm≈0 thereforedΩ =dldm.
Thus, Equation 6.1 can be written as Vν(u, v) =
Z Z
A(l, m)Iν(l, m)e−2πi(ul+vm)dldm (6.4)
which is equivalent to a Fourier transform, therefore, we can express the source brightness distribution as a function of the visibility measurements by inverting Equation 6.4.
A(l, m)Iν(l, m) = Z Z
Vν(u, v)e2πi(ul+vm)dudv (6.5)
Since the antenna spacings are sparse and irregular, the (u,v) plane coverage is sparse and irregular as well. For computational economy AIPS uses fast Fourier transforms (FFT) to calculateA(l, m)Iν(l, m).
The FFT requires the data to lie on a regular grid and that the number of sample points be a power of 2.
Thus AIPS grids the visibility measurements onto the (u,v) plane and fills every pixel of the (u,v) plane with values. If the cell is empty then the value is zero. If the cell contains more than one measurement then AIPS uses a weighted average of the measurements as the cell value. To do that AIPS folds each visibility onto a regular grid with a gridding convolution function centred on the coordinates of the measurements, in other words, AIPS multiplies the data by a gridding convolution function and translate the convolution function to the centre of every (u,v) cell, integrates the product of the convolution function with the discrete function describing the surrounding visibilities measurements and writes the result into the corresponding (u,v) cell. In still other words, AIPS convolves the convolution function with the discrete function describing the surrounding visibility measurements and multiply it by a comb function.
To avoid aliasing one wants to select a convolution function so that its Fourier transform remains unity within the image (i.e. 256 x 256 pixel) and has small or absent side lobes beyond the edges of the image.
Sidelobes could allow a confusing source that lies outside the image to produce an aliased response within the image. To see whether a source in the image is an aliased response from a source outside the image one can change the cellsize as it would cause the aliased source to move within the image (Cornwell 1995).
The image obtained is called the dirty image (Id), since it corresponds to the sky brightness (also called the real image),I, convolved with the synthesised beam (also called the dirty beam3),B and it is Id=B∗I. The image can be deconvolved sinceB andIdare known i.e. AIPS solves the equation forI.
B is calculated by the imaging task IMAGR by replacing the measured amplitudes with a value equal to one and phases zero degrees and Fourier transforming it to form the point spread function (PSF)4 in the image domain. The simplest way to deconvolve the image and PSF would be to take the Fourier transform of the above equation, and to divide the Fourier transform ofId by the Fourier transform ofBand then
3The dirty beam is the diffraction pattern of the array, which is given by the Fourier transform of the (u,v) plane coverage.
4The PSF is what I would see for a 1 Jansky (Jy) point source at the field centre
CHAPTER 6. DATA REDUCTION TO DETERMINE POLARIZATION LEAKAGE 50
Figure 6.4: Scheme describing the (u,v) coordinate system and how the source coordinate are related to the (u,v) coordinate. At the top left the source of flux densityIν(l, m) is visble and at the bottom right there is the two-telescope array observing it. Thompson 1999
Fourier transform the result back to the image domain. However this procedure fails because the Fourier transform ofB has zeroes where there are no measurements, therefore the division is undefined in some areas. Instead this deconvolution can be performed using common algorithms such as CLEAN (H¨ogbom 1974) or MEM (Maximum Entropy Method) (Burg 1967). I used the CLEAN algorithm, which iteratively takes the peak in the image, translates and scales the PSF to the position of the peak and subtracts a fraction of it from the whole image to partially remove the peak and its sidelobes. A δ-component with the same flux density is added in the clean map at the same position. I stopped the cleaning process when the largest negative value in the image was larger than the remaining positive peak and restored the image by adding back in a Gaussian profile at the position of the components that were subtracted away.
To reduce further the calibration errors in phase and amplitude, I iteratively improved my model of the sky brightness distribution using a self calibration cycle consisting of determining the antenna gains, imaging and deconvolving (CLEAN)(Cornwell and Fomalont 1999). This self-calibration cycle, when iterated, uses the visibility measurements and solves simultaneously for antenna-based gains and for the source structure.
Figure 6.5 shows the before-and-after images after two cycles of phase self calibration and one of amplitude self calibration. The artifacts in the image were reduced by the improvement in the amplitude calibration. Typically I used 12 iterations of self calibration to obtain a dynamic range between 200 : 1 to almost 400 : 1 depending on the setup, for the 1.6 Jansky (Jy) source OQ 208. The imaging process is required to calculate the leakage. The solutions (SN table) produced by the task IMAGR are required for solving simultaneously for intrinsic polarization and polarization leakage. Clearly the better the image the more precise the determination of the leakage.