  The problem of covariance estimation for replicated surface-valued processes
is examined from the functional data analysis perspective. Considerations of
statistical and computational efficiency often compel the use of separability
of the covariance, even though the assumption may fail in practice. We consider
a setting where the covariance structure may fail to be separable locally --
either due to noise contamination or due to the presence of a~non-separable
short-range dependent signal component. That is, the covariance is an additive
perturbation of a separable component by a~non-separable but banded component.
We introduce non-parametric estimators hinging on the novel concept of shifted
partial tracing, enabling computationally efficient estimation of the model
under dense observation. Due to the denoising properties of shifted partial
tracing, our methods are shown to yield consistent estimators even under noisy
discrete observation, without the need for smoothing. Further to deriving the
convergence rates and limit theorems, we also show that the implementation of
our estimators, including prediction, comes at no computational overhead
relative to a separable model. Finally, we demonstrate empirical performance
and computational feasibility of our methods in an extensive simulation study
and on a real data set.
