  This paper presents a case study from a single, six-hour observing period to
illustrate the application of techniques developed for interferometric radio
telescopes to the spectral analysis of observations of ionospheric fluctuations
with sparse arrays. We have adapted the deconvolution methods used for making
high dynamic range images of cosmic sources with radio arrays to making
comparably high dynamic range maps of spectral power of wavelike ionospheric
phenomena. In the example presented here, we have used observations of the
total electron content (TEC) gradient derived from Very Large Array (VLA)
observations of synchrotron emission from two galaxy clusters at 330 MHz as
well as GPS-based TEC measurements from a sparse array of 33 receivers located
within New Mexico near the VLA. We show that these techniques provide a
significant improvement in signal to noise (S/N) of detected wavelike
structures by correcting for both measurement inaccuracies and wavefront
distortions. This is especially true for the GPS data when combining all
available satellite/receiver pairs, which probe a larger physical area and
likely have a wider variety of measurement errors than in the single-satellite
case. In this instance, we found the peak S/N of the detected waves was
improved by more than an order of magnitude. The data products generated by the
deconvolution procedure also allow for a reconstruction of the fluctuations as
a two-dimensional waveform/phase screen that can be used to correct for their
effects.
