The Southern Wide-field Gamma-ray Observatory (SWGO) is an international collaboration working on a ground-based gamma-ray observatory that will be located in the southern hemisphere.
A crucial step in the analysis is to identify the showers produced by gamma rays and separate them from the abundant background of hadronic showers. In this work, we propose to adapt the observable $S_b$ used successfully to composition studies in the Pierre Auger Observatory to the
SWGO detector. This observable takes into account the signal and the position of each triggered detector. It characterizes the shape of the lateral distribution of the signal, which depends on the nature of the primary particle. The value of $S_b$ is therefore suited to identify gamma induced showers and to reject the more frequent hadronic showers. This analysis has been performed using a scaling factor to link the observable $S_b$ between the central and outrigger array. Also we
explore how this new observable improves the separation of primary proton and gamma-induced air showers in terms of the merit factor.