  The local density of states power spectrum of optimally doped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO) has been interpreted in terms of
quasiparticle interference peaks corresponding to an "octet'' of scattering
wave vectors connecting k-points where the density of states is maximal. Until
now, theoretical treatments have not been able to reproduce the experimentally
observed weights and widths of these "octet'' peaks; in particular, the
predominance of the dispersing "q$_1$'' peak parallel to the Cu-O bond
directions has remained a mystery. In addition, such theories predict
"background'' features which are not observed experimentally. Here, we show
that most of the discrepancies can be resolved when a realistic model for the
out-of-plane disorder in BSCCO is used. Weak extended potential scatterers,
which are assumed to represent cation disorder, suppress large-momentum
features and broaden the low-energy "q$_7$''-peaks, whereas scattering at order
parameter variations, possibly caused by a dopant-modulated pair interaction
around interstitial oxygens, strongly enhances the dispersing "q$_1$''-peaks.
