  Doppler broadening plays an important role in laser rovibrational
spectroscopy of trapped deuterated molecular hydrogen ions (HD$^+$), even at
the millikelvin temperatures achieved through sympathetic cooling by
laser-cooled beryllium ions. Recently, Biesheuvel \textit{et al.}
\cite{Biesheuvel2016} presented a theoretical lineshape model for such
transitions which not only considers linestrengths and Doppler broadening, but
also the finite sample size and population redistribution by blackbody
radiation, which are important in view of the long storage and probe times
achievable in ion traps. Here, we employ the rate equation model developed by
Biesheuvel \textit{et al.} to theoretically study the Doppler-broadened
hyperfine structure of the $(v,L):(0,3)\rightarrow(4,2)$ rovibrational
transition in HD$^+$ at 1442~nm. We observe prominent yet hitherto unrecognized
ghost features in the simulated spectrum, whose positions depend on the Doppler
width, transition rates, and saturation levels of the hyperfine components
addressed by the laser. We explain the origin and behavior of such features,
and we provide a simple quantitative guideline to assess whether ghost features
may appear. As such ghost features may be common to saturated Doppler-broadened
spectra of rotational and vibrational transitions in trapped ions composed of
partly overlapping lines, our work illustrates the necessity to use lineshape
models that take into account all the relevant physics.
