Local earthquake tomography (LET) is a popular method for inverting arrival time picks of local-regional earthquakes for P- and S-wave velocity and hypocenter parameters in seismically active regions. This popularity is due to some robust and well-documented open-source codes that are sometimes used as black boxes. The availability of a very complete time-pick database on the Western Alps gives us a chance to thoroughly investigate the influence of the numerous processes and parameters involved when applying LET to the Western Alps or similar targets. From a subset of high-quality manual picks (1989-2014), we compute preliminary P and S velocity models that are used to predict arrival times for later events and allow the selected fusion of picks downloaded from different seismological agencies for a consistent 33-year database (1989-2021). Different model reconstructions are proposed by selecting different subsets of the arrival time dataset. Aside data input into the LET, influence of initial stratified or three-dimensional velocity models and related initial earthquake locations is investigated together with grid discretization, Laplacian smoothing and damping parameters in the standard penalty approach commonly used by LET codes. These numerical parameters account for the expected limited resolution of seismic waves due to their finite-frequency content. Parameter selection is handled by the user, whereas frequency impact is only implicit in the onset picks. The earthquake distribution allows a reconstruction down to 40-km depth over an area of ~500x500 km2. Robust features such as the high-velocity Ivrea body anomaly, and a deep low-velocity anomaly associated with crustal thickening underneath the mountain belt survive whatever the tomography strategy and parameters. Finally, a comparison with previous LET reconstructions suggests that finite-frequency content be explicitly incorporated through wave equation tomography to improve spatial resolution. This would fully exploit observables collected from seismograms, albeit with a significant increase in computer costs.