  We investigate overdetermined linear inverse problems for which the forward
operator may not be given accurately. We introduce a new tool called the
structure, based on the Wasserstein distance, and propose the use of this to
diagnose and remedy forward operator error. Computing the structure turns out
to use an easy calculation for a Euclidean homogeneous degree one distance, the
Earth Mover's Distance, based on recently developed algorithms. The structure
is proven to distinguish between noise and signals in the residual and gives a
plan to help recover the true direct operator in some interesting cases. We
expect to use this technique not only to diagnose the error, but also to
correct it, which we do in some simple cases presented below.
