Shockley surface states at metal surfaces decay exponentially into the
crystal, typically penetrating several interplanar spacings. Within t
his distance defects such as impurity atoms will scatter the surface s
tate electrons and give rise to characteristic standing wave patterns
in the local density of states outside the surface, similar to those r
ecently detected with the STM due to scattering by adatoms. A multiple
-scattering theory is developed to model this situation and applied to
impurities near the Cu(111) surface. The results are discussed within
a two-dimensional scattering model and the prospects for observation
with the STM considered.