Example of the detectability interpolation and smoothing for the catalogue-level simulation in the second distance bin with d in the range of [21.5, 46.4] kpc. Detection of satellites within this distance range is binned as a function of M_V and r_h as shown in subplot A. A total of 1497 mock satellites are plotted as scatters in the M_V − r_h map in subplot B. The 2D interpolation is applied to generate the full detection efficiency map, as shown in subplot C. We apply a Gaussian kernel to smooth the map, with the 50% detectability limit plotted in red in subplot D, and the fitted limit plotted in yellow.