  We consider the problem of local tunneling into cuprate superconductors,
combining model based calculations for the superconducting order parameter with
wavefunction information obtained from first principles electronic structure.
For some time it has been proposed that scanning tunneling microscopy (STM)
spectra do not reflect the properties of the superconducting layer in the
CuO$_2$ plane directly beneath the STM tip, but rather a weighted sum of
spatially proximate states determined by the details of the tunneling process.
These "filter" ideas have been countered with the argument that similar
conductance patterns have been seen around impurities and charge ordered states
in systems with atomically quite different barrier layers. Here we use a
recently developed Wannier function based method to calculate topographies,
spectra, conductance maps and normalized conductance maps close to impurities.
We find that it is the local planar Cu $d_{x^2-y^2}$ Wannier function,
qualitatively similar for many systems, that controls the form of the tunneling
spectrum and the spatial patterns near perturbations. We explain how, despite
the fact that STM observables depend on the materials-specific details of the
tunneling process and setup parameters, there is an overall universality in the
qualitative features of conductance spectra. In particular, we discuss why STM
results on Bi$_2$Sr$_2$CaCu$_2$O$_8$ and Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$ are
essentially identical.
