  The spatial dependence of total column ozone varies strongly with latitude,
so that homogeneous models (invariant to all rotations) are clearly unsuitable.
However, an assumption of axial symmetry, which means that the process model is
invariant to rotations about the Earth's axis, is much more plausible and
considerably simplifies the modeling. Using TOMS (Total Ozone Mapping
Spectrometer) measurements of total column ozone over a six-day period, this
work investigates the modeling of axially symmetric processes on the sphere
using expansions in spherical harmonics. It turns out that one can capture many
of the large scale features of the spatial covariance structure using a
relatively small number of terms in such an expansion, but the resulting fitted
model provides a horrible fit to the data when evaluated via its likelihood
because of its inability to describe accurately the process's local behavior.
Thus, there remains the challenge of developing computationally tractable
models that capture both the large and small scale structure of these data.
