  Drumhead surface states that link together loops of nodal lines arise in
Dirac nodal-line semimetals as a consequence of the topologically non-trivial
band crossings. We used low-temperature scanning tunneling microscopy and
Fourier-transformed scanning tunneling spectroscopy to investigate the
quasiparticle interference (QPI) properties of ZrSiTe. Our results show two
scattering signals across the drumhead state resolving the energy-momentum
relationship through the occupied and unoccupied energy ranges it is predicted
to span. Observation of this drumhead state is in contrast to previous studies
on ZrSiS and ZrSiSe, where the QPI was dominated by topologically trivial bulk
bands and surface states. Furthermore, we observe a near $\mathbf{k}
\rightarrow -\mathbf{k}$ scattering process across the $\Gamma$-point, enabled
by scattering between the spin-split drumhead bands in this material.
