Left: posterior probability distributions of infall velocity, V_in, peak optical depth, τ₀, velocity dispersion, σ, excitation temperature of the “front” layer, T_f, and excitation temperature of the “rear” layer, T_r for the 37.7 GHz CH₃OH transitiontowards W31C. The maximum posterior possibility point in the parameter space is shown in orange lines and points. The contours represent the 0.5, 1.0, 1.5, and 2.0σ confidence intervals. Right: observed and modelled spectra for the 37.7 GHz CH₃OH transition. The blue spectrum represents the observed data, and the red curve indicates the fitted line from the two-layer model (see Sect. 4.1). The vertical black dashed line represents the systemic velocity of V_LSR = −3.43 km s⁻¹.