[RM-6715] [linemeas] compute velocity uncertainty

Created on 2021-09-27 10:00:19 by Didier Vibert. % Done: 50

Since the fitting of the line width, w, is done by least-square minimization by looping on a grid of velocity dispersion, we can estimate the width of the peak in pdf(w) to infer var(w), and deduce var(v), where w^2 = sigma_LSF^2 + v ^2 .

Or we could use the analytical expression derived in https://www.osapublishing.org/abstract.cfm?URI=ao-46-22-5374 (eq 19)
var(w) = 2 * sigma^2 * w / ( sqrt(pi) * A^2 * dx )
where

• A is the Gaussian amplitude
• w is the Gaussian width (sigma)
• sigma is the noise standard deviation at the line position (assuming it is uncorrelated pixel 2 pixel, and with constant variance)
• dx is the pixel size (same units than w, typically wavelength in Angstroms)

Note: when/if we will use a non-linear-least square minimizer (eg Levenberg-Marquardt), we could use the estimated returned covariance matrix to get the variance on sigma