given by $$N({\mathbf{r}},\omega)=(-1/\pi) \mbox{Im} \left[
{\mathcal{G}}_{11}({\mathbf{r}},{\mathbf{r}},\omega) +
{\mathcal{G}}_{22}({\mathbf{r}},{\mathbf{r}},-\omega)\right].$$

![(Color online) Same as Fig. \[fig1\] but including local impurity-induced reduced hopping $\delta t=-0.24t$.[]{data-label="fig2"}](Fig2.eps){width="1.0\columnwidth"}

In explicit calculations we use the band $\xi_\k = -2t\,
(\cos k_x+\cos k_y) - 4t' \cos k_x \cos k_y - \mu$ with a $d$-wave order parameter $ \Delta_\k = \frac{\Delta_0}{2}\ (\cos k_x-\cos
k_y)\, . $ To model the Fermi surface in the cuprates we take $t'
= -0.35 t$ and $\mu=-1.1 t$, with energy measured in units of nearest neighbor hopping $t$. We also choose $\Delta_0=0.2t$. The modulations in the hopping are included on the bonds near the impurity site assuming maximum modulations $\delta t$ on the nearest neighbor bonds, with a Gaussian decay on other bonds with decay length of one lattice constant. This is roughly consistent with the renormalization of the effective hopping found in recent studies of impurities in a host described by the $t-t'-J$ model.[@gabay] Note this is an atomic scale modulation caused by the in-plane impurity due solely to magnetic correlations in the host. By contrast, we also study a modulation of the order parameter, but assume that this is caused by an out-of-plane influence such as a dopant atom, which gives rise 