Recovering the effective impact parameters in integral-field absorption-line tomography
Integral-field-unit (IFU) spectroscopy of extended background sources, such as gravitational arcs, delivers many closely spaced absorption sightlines through the circumgalactic medium (CGM) of foreground galaxies. However, the source extent and smearing through the point spread function (PSF) cause each spaxel to integrate light from a region larger than itself. The usually adopted geometric spaxel centre is therefore not representative of where the absorbed flux originates. Informed by high-resolution imaging, we introduce a flux-contribution formalism that reconstructs the spatial origin of the flux in each spaxel by defining flux-weighted coordinates and impact parameters. Using a data-driven simulation built on a real gravitational-arc configuration, we show that spaxel centres are displaced from the flux-weighted positions by $|\Delta\mathbf{x}| \propto \theta_{\rm PSF}(1-f)$, with $f$ the spaxel-source overlap fraction, and that the resulting impact-parameter bias scales with the lensing magnification as $\mu^{-1/2}$. These offsets are largest for off-arc spaxels and flatten the radial column-density profiles. They also inject a coherent geometry-driven scatter that mimics intrinsic CGM structure and that is removed neither by a tighter spaxel selection nor by a finer spatial binning. For identical uncertainties, the spaxel-centre coordinates leave residuals of about twice the noise level ($\chi^2_\nu \sim 4$), whereas the flux-weighted coordinates recover the input profile to within the uncertainties ($\chi^2_\nu \sim 1$) and remain robust to a PSF misspecification of $\pm 10\%$. The method we propose provides a better-motivated and readily adoptable framework for tomographic studies, whether as the primary analysis or as a robustness check on spaxel-centre results.