From 2D to 3D: Recovering Turbulent Density Dispersions from Noisy Data

Turbulence plays a central role in shaping the structure and dynamics of the interstellar medium (ISM), governing the star formation rate (SFR) and the initial mass function (IMF). A key consequence of turbulence is the generation of density fluctuations, which regulate the amount of dense gas available for star formation. Accurate measurements of the three-dimensional (3D) turbulent density dispersion are therefore essential for understanding molecular-cloud structure and star formation. However, observations typically provide only two-dimensional (2D) column densities and are often affected by measurement/detector noise. The Brunt method estimates the 3D density dispersion from 2D column-density maps, but it does not account for finite signal-to-noise ratio (SNR). Here, we extend the method to recover the 3D turbulent density dispersion from noise-contaminated observations. Using numerical simulations spanning a range of density perturbation amplitudes and noise types, we identify a characteristic noise wavenumber, k noise, corresponding to the intersection of the signal and noise spectra. Restricting the Brunt reconstruction to wavenumbers below k noise yields a denoised density-dispersion estimate that closely reproduces the noise-free result. We provide a practical prescription to determine k noise directly from the measurement SNR and image resolution. Alternatively, if the noise spectrum is known, it can be subtracted directly from the observed spectrum, eliminating the need to estimate k noise. The proposed correction recovers the noise-free density dispersion with errors of < 5% for SNR>=3 and < 15% for SNR>=1, enabling substantially more reliable estimates of turbulent density fluctuations from noisy column-density data.