  We consider the estimation of carbon dioxide flux at the ocean-atmosphere
interface, given weighted averages of the mixing ratio in a vertical
atmospheric column. In particular we examine the dependence of the posterior
covariance on the weighting function used in taking observations, motivated by
the fact that this function is instrument-dependent, hence one needs the
ability to compare different weights. The estimation problem is considered
using a variational data assimilation method, which is shown to admit an
equivalent infinite-dimensional Bayesian formulation.
  The main tool in our investigation is an explicit formula for the posterior
covariance in terms of the prior covariance and observation operator. Using
this formula, we compare weighting functions concentrated near the surface of
the earth with those concentrated near the top of the atmosphere, in terms of
the resulting covariance operators.
  We also consider the problem of observational targeting, and ask if it is
possible to reduce the covariance in a prescribed direction through an
appropriate choice of weighting function. We find that this is not the
case---there exist directions in which one can never gain information,
regardless of the choice of weight.
