  We revisit the interpretation of quasiparticle scattering interference in
cuprate high-$T_c$ superconductors. This phenomenon has been very successful in
reconstructing the dispersions of d-wave Bogoliubov excitations, but the
successful identification and interpretation of QPI in scanning tunneling
spectroscopy (STS) experiments rely on theoretical results obtained for the
case of isolated impurities. We introduce a highly flexible technique to
simulate STS measurements by computing the local density of states using
real-space Green's functions defined on two-dimensional lattices with as many
as 100,000 sites. We focus on the following question: to what extent can the
experimental results be reproduced when various forms of distributed disorder
are present? We consider randomly distributed point-like impurities, smooth
"Coulombic" disorder, and disorder arising from random on-site energies and
superconducting gaps. We find an apparent paradox: the QPI peaks in the
Fourier-transformed local density of states appear to be sharper and better
defined in experiment than those seen in our simulations. We arrive at a no-go
result for smooth-potential disorder since this does not reproduce the QPI
peaks associated with large-momentum scattering. An ensemble of point-like
impurities gets closest to experiment, but this goes hand in hand with impurity
cores that are not seen in experiment. We also study the effects of possible
measurement artifacts (the "fork mechanism"), which turn out to be of
relatively minor consequence. It appears that there is an unknown mechanism at
work which renders the QPI peaks much sharper than they are based on present
theoretical understanding.
