Ionospheric Power-Spectrum Tomography in Radio Interferometry
  A tomographic method is described to quantify the three-dimensional
power-spectrum of the ionospheric electron-density fluctuations based on
radio-interferometric observations by a two-dimensional planar array. The
method is valid to first-order Born approximation and might be applicable to
correct observed visibilities for phase variations due to the imprint of the
full three-dimensional ionosphere. It is shown that not the ionospheric
electron density distribution is the primary structure to model in
interferometry, but its autocorrelation function or equivalent its
power-spectrum. An exact mathematical expression is derived that provides the
three dimensional power-spectrum of the ionospheric electron-density
fluctuations directly from a rescaled scattered intensity field and an incident
intensity field convolved with a complex unit phasor that depends on the w-term
and is defined on the full sky pupil plane. In the limit of a small field of
view, the method reduces to the single phase screen approximation. Tomographic
self-calibration can become important in high-dynamic range observations at low
radio frequencies with wide-field antenna interferometers, because a
three-dimensional ionosphere causes a spatially varying convolution of the sky,
whereas a single phase screen results in a spatially invariant convolution. A
thick ionosphere can therefore not be approximated by a single phase screen
without introducing errors in the calibration process. By applying a Radon
projection and the Fourier projection-slice theorem, it is shown that the
phase-screen approach in three dimensions is identical to the tomographic
method. Finally we suggest that residual speckle can cause a diffuse intensity
halo around sources, due to uncorrectable ionospheric phase fluctuations in the
short integrations, which could pose a fundamental limit on the dynamic range
in long-integration images.
