Column-Density Estimation from the Equivalent Widths of Absorption Doublets

Na I D absorption is widely used to estimate the dust extinction toward Galactic and extragalactic sources, but the inferred column densities are unreliable once the lines saturate. We present an analytical framework for estimating column densities from the equivalent widths ($EW$s) of absorption-line doublets. The method requires only that the two doublet members be measured separately -- the velocity structure need not be resolved -- so a resolving power $\lambda/\Delta\lambda\gtrsim10^{3}$ suffices for Na I D, two orders of magnitude below velocity-resolved methods. For the Gaussian curve of growth, the classical inversion reduces to a universal saturation correction depending only on the doublet ratio, $R=EW_2/EW_1$: at fixed $R$, the inferred column density is linear in the measured $EW$. The formulation yields the exact Gaussian inversion, $N_{\tilde\kappa}$, which provides the Doppler parameter; a closed-form second-order approximation; and a strict lower bound, $N_{\rm min}$, valid for an arbitrary velocity structure. We recommend reporting $N_{\rm min}$: an optimal, assumption-free lower bound, within a few per cent of $N_{\tilde\kappa}$ for $R\gtrsim1.4$ and at most a factor $3.5$ below it at $R=1.1$. The ratio $N_{\tilde\kappa}/N_{\rm 2nd}$ provides a per-object saturation diagnostic. We validate the framework against published profile-fitting Na I D column densities toward Type Ia supernovae, using only the integrated $EW$s. An explicit closed-form formula converts the two $EW$s into an extinction estimate through the observed $\log N(\mathrm{Na\,I})$--$A_V$ relation. A public implementation with full uncertainty propagation accompanies the paper.