Column-density estimated from C¹⁸O versus ¹³CO (frame a) and H¹³CO⁺ versus HCO⁺ (frame b) line ratios, plotted against the true column density N′(H₂) (accounting for the effect of density-dependent fractional abundances). Results are shown for both ideal observations and the combined ngVLA and single-dish data. The diagonal dashed line corresponds to the expected one-to-one relation. For each dataset (listed in the legends) there are four contour levels that are equidistant from close to zero up to 80% of the maximum point density. The background greyscale images correspond to the estimates calculated from ngVLA line ratios. In frame a, the estimates for optically thin C¹⁸O emission at T_ex = 15 K are shown for comparison. Plots use data within 10″ of the pointing centres and a resolution of 0.7″.