Finding a Spherically Symmetric Cosmology from Observations in
  Observational Coordinates -- Advantages and Challenges
  One of the continuing challenges in cosmology has been to determine the
large-scale space-time metric from observations with a minimum of assumptions
-- without, for instance, assuming that the universe is almost
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW). If we are lucky enough this
would be a way of demonstrating that our universe is FLRW, instead of
presupposing it or simply showing that the observations are consistent with
FLRW. Showing how to do this within the more general spherically symmetric,
inhomogeneous space-time framework takes us a long way towards fulfilling this
goal. In recent work researchers have shown how this can be done both in the
traditional Lema\^{i}tre-Tolman-Bondi (LTB) 3 + 1 coordinate framework, and in
the observational coordinate (OC) framework. In this paper we investigate the
stability of solutions, and the use of data in the OC field equations including
their time evolution and compare both approaches with respect to the
singularity problem at the maximum of the angular-diameter distance, the
stability of solutions, and the use of data in the field equations. This allows
a more detailed account and assessment of the OC integration procedure, and
enables a comparison of the relative advantages of the two equivalent solution
frameworks. Both formulations and integration procedures should, in principle,
lead to the same results. However, as we show in this paper, the OC procedure
manifests certain advantages, particularly in the avoidance of coordinate
singularities at the maximum of the angular-diameter distance, and in the
stability of the solutions obtained. This particular feature is what allows us
to do the best fitting of the data to smooth data functions and the possibility
of constructing analytic solutions to the field equations.
