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.
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