Toward Unbiased Abundance Measurements in Inhomogeneous $\mathrm{H\,II}$ Regions
Probing the chemical content of the interstellar medium (ISM) in nearby galaxies provides key insight into their chemical evolution and informs our interpretation of galaxies at higher redshift. However, nonlinear structure in the ISM, including density and temperature inhomogeneities, can bias chemical abundance measurements and systematically affect empirical calibrations derived from them. In this work, we investigate biases in $T_e$-derived oxygen abundance determinations and explore the physical properties that correlate with them. We combine $\mathrm{[O\,II]}\lambda\lambda3726, 3729$ measurements from SITELLE with a full suite of optical emission lines obtained with MUSE. From auroral emission lines ($\mathrm{[N\,II]}\lambda5755$, $\mathrm{[S\,III]}\lambda6312$, and $\mathrm{[O\,II]}\lambda\lambda7320, 7330$) and nebular emission lines (including $\mathrm{[N\,II]}\lambda6584$ and $\mathrm{[S\,III]}\lambda9069$), we derive electron densities, temperatures, and chemical abundances for a sample of $\mathrm{H\, II}$ regions in five galaxies. We find that densities derived from the $\mathrm{[O\,II]}$ auroral-to-nebular ratio are $\sim10^3$ cm$^{-3}$, which is higher than the standard $\mathrm{[S\,II]}$ densities derived from nebular doublet ratios. We demonstrate that combining the $\mathrm{[N,II]}$ electron temperature with the density inferred from the $\mathrm{[O\,II]}$ auroral-to-nebular line ratio yields singly ionized oxygen abundances consistent with literature expectations for a prescription insensitive to density inhomogeneities. We also find that the $\mathrm{[S\,III]}$ temperature provides a reliable estimate of $T_{e,\mathrm{[O\,III]}}$, enabling robust measurements of doubly ionized oxygen abundances. Overall, these results indicate that the abundance discrepancy factor could be higher in more chemically evolved $\mathrm{H\, II}$ regions.