Deriving volume density profiles of filaments from observed surface densities

Accurate characterization of filamentary structures in star-forming clouds is essential for understanding star formation. Traditional methods fit observed surface density profiles $\Sigma(r)$ with slope $\gamma$ and width $H$ using the Plummer function, assuming $\beta=\gamma+1$ and $h\approx H$ for the volume density slope and width. These assumptions break down for shallow profiles, with the slope and width relations deviating progressively more for compact and extended filaments, respectively. We present a new fitting method that explicitly accounts for finite cylindrical geometry and establishes self-consistent empirical relationships between the parameters of $\Sigma(r)$ and those of the volume density profile $\rho(r)$ with slope $\beta$ and width $h$. The method was validated on model profiles and applied to selected filaments in the California molecular cloud. The slope difference $\delta\equiv\beta-\gamma$ falls below unity for shallow ($\beta\lesssim 2$) and compact profiles; $h$ and $H$ can differ by over an order of magnitude for extended filaments with shallow slopes. Accurate parameter recovery requires high resolvedness $R\equiv H/O\gtrsim 10$ (where $O$ is the beam width); at lower resolvedness, slopes are severely overestimated and filaments remain unresolved even when $H\gg O$. The traditional Plummer function yields systematically overestimated slopes. Accurate deconvolution requires a priori knowledge of the true parameters, creating a fundamental circular problem whose only robust solution is obtaining sufficiently high angular resolution. Current far-infrared observations typically lack sufficient resolution, and some previously reported filament properties may require reinterpretation.