  The quasiparticle scattering interference (QSI) is intimately related to the
nature of the quasiparticle and of its interplay with a variety of electronic
orders and superconductivity. Here starting from the microscopic octet
scattering model, the nature of QSI in cuprate superconductors is studied in
the $T$-matrix approach. In particular, a new method of the inversion of matrix
is developed to accurately derive the $T$-matrix for various kinds of
impurities, and then the obtained $T$-matrix is employed to calculate the local
density of states (LDOS). It is shown that the overall features of the LDOS
modulation can be described qualitatively by taking into account the
quasiparticle scattering from a single impurity on the kinetic-energy-driven
homogeneous superconducting-state, where the QSI scattering wave vectors ${\bf
q}_{i}$ and the related QSI peak dispersions are internally consistent within
the octet scattering model. However, the pronounced QSI peaks in the
momentum-space LDOS modulation pattern for a single impurity are smeared
heavily in the case for multiple impurities, and then the momentum-space LDOS
modulation for multiple impurities exhibits a speckle pattern. Moreover, the
momentum-space LDOS modulation for Gaussian-random-distribution of on-site
impurity at a relatively large deviation displays a similar behavior of the
LDOS modulation for multiple impurities. The theory also indicates that the
impurity weight linearly increases with the increase of the impurity-scattering
strength for the weak scattering strength and tends to saturate in the strong
scattering strength, while it decreases with the increase of the
impurity-scattering screening length for the short screening-length and
saturate in the long screening-length.
