FQUO: Fit a Broadening Function to a Fourier Quotient Spectrum

FQUO fits the broadening Gaussian that when convolved with a template minimizes least-squares from a given spectrum. The input buffers must hold already the 1-d Fourier transforms of the template and the data spectra (use command FFT with the keyword ONEDIM), so the user can do his own personal recipe to prepare the data before the FFT (de-trending, edge-masking, etc) using other VISTA commands like ROWPOLY. The data buffer is replaced by the fit, the Fourier transform of the properly scaled, broadened and shifted template.

By default (MODE=2) the least-squares minimization is performed comparing directly the data A with the convolution (product in Fourier space) of the template B and the broadening Gaussian G. If MODE=1 is used instead, the least-squares are minimized comparing the Fourier-quotient A/B with G.

The three parameters to fit are the so called gamma-parameter (G), the velocity dispersion (S), and the relative velocity (V) with respect to the template. Since this is a non-linear fit, the user must give a first guess for these parameters. The amplitude G of the Gaussian estimates a relative line-strength, the width S measures a relative broadening, and any relative shift shows up in the displacement V of the Gaussian. The template and the data should normally be in a logarithmic wavelength scale (command ALIGN with keyword log) before they are transformed in Fourier space. In this case, the units for the broadening S and the shift V are km/sec. Otherwise (data not in equally-spaced velocity bins before FFT), S and V will be measured in units that are given by the next combination of the value of the fits card CDELT1 and the speed of light: (1.0e-10)*CDELT1/C.

The initial guess can be given in three ways. Keywords G=, S=, V=, and GUESS=g,s,v, give the guessed values directly in the command line. Keyword GUESS=buf reads these values from buffer 'buf', where buf contains in column 1 the guess for gamma, in column 2 the guess for sigma, and in column 3 the guess for V. GUESS=buf is particularly useful with multi-row spectra (long-slit) or when taking previous results saved with the keyword LOAD=buf.

Sometimes one or more of the parameters are known a priori and do not want be be fitted. In these cases, the user can FIX a parameter at its input value with keywords FIXG, FIXS and/or FIXV.

The user can constrain the fit parameters parameters to be within certain limits with keywords G=g,mng,mxg, S=s,mns,mxs, and/or V=v,mnv,mxv. For example S=s,mns,mng will be fitting Sigma taking value 's' as first guess and keeping the fitted sigma between values 'mns' and 'mxs'. When only two values are specified in the keyword, like S=mns,mxs, the fit will be performed taking the average value as first guess and still constraining the final fit parameter within the limiting values.

Keywords KS=ks and KE=ke, can be used to limit the fit within Fourier wave-numbers ks to ke. Keyword WBUF=buf can be used to apply different weights to different wave-numbers according to the values in buffer 'buf'.

The data, template and weight images can have a single row (1-d spectroscopy) or any combination of one ore more rows (long-slit spectroscopy). If the template (weight) is a single row image, it will be used to fit all the rows of the data image. Otherwise the template (weights) will be the taken from the corresponding row in the data buffer. When the template (weights) image does not cover all the data rows, the template (weights) will be 'extrapolated' using the first or last row of the template (weights) image. Normally all the rows of the data image will be fit but the user can limit the fit to a single row with Keyword ROW=r (ignored in 1-D spectroscopy).