The probability of identifying the cosmic web environment of galaxies around clusters motivated by the Weave Wide Field Cluster Survey

Upcoming wide-field spectroscopic surveys will observe galaxies in a range of cosmic web environments in and around galaxy clusters. In this paper, we test and quantify how successfully we will be able to identify the environment of individual galaxies in the vicinity of massive galaxy clusters, reaching out to $\sim5R_{200}$ into the clusters' infall region. We focus on the WEAVE Wide Field Cluster Survey (WWFCS), but the methods we develop can be easily generalised to any similar spectroscopic survey. Using numerical simulations of a large sample of massive galaxy clusters from \textsc{TheThreeHundred} project, we produce mock observations that take into account the selection effects and observational constraints imposed by the WWFCS. We then compare the `true' environment of each galaxy derived from the simulations (cluster core, filament, and neither core nor filament, {``NCF''}) with the one derived from the observational data, where only galaxy sky positions and spectroscopic redshifts will be available. We find that, while cluster core galaxy samples can be built with a high level of completeness and moderate contamination, the filament and NCF galaxy samples will be significantly contaminated and incomplete due to projection effects exacerbated by the galaxies' peculiar velocities. We conclude that, in the infall regions surrounding massive galaxy clusters, associating galaxies with the correct cosmic web environment is highly uncertain. However, with large enough spectroscopic samples like the ones the WWFCS will provide (thousands of galaxies per cluster, {out to $5R_{200}$}), and the correct statistical treatment that takes into account the probabilities we provide here, we expect we will be able to extract robust and well-quantified conclusions on the relationship between galaxy properties and their environment.