Extracting spatial covariance given only mean and variance

Question

I have the results of an MCMC experiment in the form of a set of means $\mu(\phi,\theta)$ and standard deviations $\sigma(\phi,\theta)$ as a function of spatial coordinates $\phi$ and $\theta$ (in this case latitude and longitude). Unfortunately, the actual distribution that generated these maps was not saved due to space constraints. Is it possible to use this data to get a decent estimate of spatial covariance (perhaps by assuming a particular form for the spatial covariance), and if so, how?

Answer 1

Unfortunately, no. Your mean function has no information you can use: covariance is shift-invariant, meaning adding a fixed function $\mu(\phi, \theta)$ will not affect it. Likewise, any pattern of spatial correlation, from spatially uniform residuals at one extreme to iid errors at the other, could be compatible with the variance estimates that you see.

If you have some prior belief about the mean function ("this estimate looks too spiky"), you could begin to speculate about the covariance patterns that might have led to that type of estimation error, but that's grasping at straws, and I wouldn't know how to proceed technically.