  While the influence of impurities on the local density of states (LDOS) in a
metal is notoriously non-local due to interference effects, low order moments
of the LDOS in general can be shown to depend only on the local structure of
the Hamiltonian. Specifically, we show that an analysis of the spatial
variations of these moments permits one to ``work backwards'' from scanning
tunneling microscopy (STM) data to infer the local structure of the underlying
effective Hamiltonian. Applying this analysis to STM data from the high
temperature superconductor, Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, we find that
the variations of the electro-chemical potential are remarkably small (i.e.,
the disorder is, in a sense, weak) but that there are large variations in the
local magnitude of the d-wave gap parameter.

Superconductivity;Strongly Correlated Electrons