FIT1D: A new SMURF command for ACSIS data

More time ago than I am willing to admit, I started coding a Starlink routine to fit spectral lines in ACSIS cubes. Got a long way until SCUBA-2 commissioning and calibration put a halt to it, but I have finally managed to finish the program, technically a beta-version, as part of the upcoming Kapuahi release of Starlink.

Usage:

% fit1d  in  out  rms  [config]  [userval]  [pardir]  [parndf]  [parcomp]

What distinguishes FIT1D from other profile fitting routines is that it specifically attempts to deal with two issues: non-gaussian profile shapes and the fact that ACSIS data-cubes have many, potentially very different, profiles to fit. Regarding the latter, there are many fitting routines that produce fitted profiles for data-cubes, but FIT1D also produces cubes with the fitted parameters themselves and can use such files as input to give control over the fit of, in principle, each individual profile in the cube. Thus it is e.g. possible to fit broad lines on the nucleus of a galaxy and narrow lines everywhere else. More about that below.

A section has been added to the SMURF documentation in SUN/258 about FIT1D, which I will try to summarize here. FIT1D is generic in that it can fit profiles along any axis of an up to 7-dim hyper-cube, but will be discussed here in the context of a default RA-Dec-Vel ACSIS cube. Note that the routine assumes that data have been baseline-subtracted, using e.g. MFITTREND, i.e. that the profiles have a zero-level at 0. 

Gauss-Hermite shapes as a function of the 3rd-order
      skewness coefficient 'h3' and the 4th-order the kurtosis (peakiness)
      coefficient 'h4'. The red box indicates the limits on acceptable
      values for h3 and h4 as defined in the defaults configuration file. Note
      that the fitted profile by default is restricted to positive values
      and will omit the shown negative features.

Non-gaussian Profiles.

FIT1D essentially re-implements the fitting code for non-gaussian profiles from the GIPSY package (Kapteyn Institute, Groningen, The Netherlands). Function types that can be fitted are Gaussian, Gauss-Hermite, and Voigt profiles. In particular, Gauss-Hermite functions are a powerful extention when fitting profiles that are skewed, peaky, or only approximately gaussian. The figure above shows Gauss-Hermite profiles as a function of the skewness coefficient h3 and the kurtosis (peakiness) coefficient h4. See  for SUN/258 further details, but note that the default setting in the configuration file is for FIT1D to suppress the negative features in the fitted profiles and to leave only the positive part of Gauss-Hermites.

Because of their ability to fit distorted shapes, Gauss-Hermites are particularly well suited to "capture" the maximum amount of emission from a cube. The fits can be remarkably accurate as is shown in the the figure below showing a 3-component fit (i.e. up to 3 spectral lines) using gausshermite2 functions (i.e. fitting both h3 and h4). Collapsing the resulting cube with fitted profiles can thus result in an accurate and almost noise-free white-light or total-emission map.

Fit1d - Black: original profiles; Red: results of a
    3-component Gauss-Hermite2 fit (fitting both h3 and h4)

FIT1D derives its ability to fit a complex line-shape both from the Gauss-Hermite function but also from that it can fit multiple (sub) components to get the best match possible. However, that can make the interpretation of the fits in terms of the physical characteristics and quantities difficult, hence for those you may also want to make a fit of the line-shape using a single standard Gaussian function. 

Component Parameter files

Besides a data-cube with the fitted profiles FIT1D also outputs so-called Component parameter files as NDF extensions in the header of the output file. These can also be copied out as independent data-cubes. There is a file for each component (i.e. line) that was fitted along the profile up to the number of components requested by the user. Each plane of a Component parameter file has an image of the value of a fitted parameter across the field-of-view. For instance, the one resulting from a gaussian fit has images respectively showing the fitted Amplitude, Position (velocity), and FWHM as well as a plane with an id-number of the function used.

Much of the (anticipated) use of FIT1D derives from the fact that Component parameter files can be used as input as well: either to provide initial estimates or fixed values to the fitting routine.  The difference between values specified in the Component parameter files
and ones declared in a User parameter values file is that the former can vary across the field-of-view whereas the latter will result in the same value being used for all profiles. E.g. for use with spectral-line surveys the User parameter values file can be used to provide initial estimates of the frequencies or velocities at which lines are expected or to fix fits at those frequencies.

By manipulating Component parameter files e.g. resulting from an initial fit, the user can customize or correct subsequent fits. In extrema, a Component parameter file could be made from scratch based on a model and be used to create a spectral-line data-cube with that model (config option: model_only=1) or be used as initial estimates for a fit. Of more practical use, Component parameter files can be used to correct problems associated with a fit since the art of fitting is not in the fitting algorithm, but in providing accurate initial estimates. For instance, the left image below shows a section of an Amplitude plane of a fit where there are problems in a few locations. Setting these location to bad values and using FILLBAD to interpolate over them, the corrected Component parameter file was used as initial estimate for a subsequent fit, resulting in the image on the right

Fit1d - Left: Section of a parameter file showing
      originally fitted amplitudes; Right: Amplitudes after using a
      corrected parameter file from the original fit as initial estimates
      for a subsequent fit.

More creative options are possible: after an initial fit with a gaussian, the function id can be changed to a gausshermite1 in part of the field and the resulting file used as initial estimates for a subsequent fit to account for skewed profiles there. Similarly, the initial guess of the FWHM can be made wide on e.g. nucleus of a galaxy while leaving it more narrow outside. As another example, the fit of multiple components can be limited to only part of the field by setting the parameter file for the second and higher components to bad values outside the relevant region (multiple component parameter files can be used as input: one for each component to be fitted).

In conclusion: please remember that this is a beta-release and that you may run into unanticipated issues. Also chosen limits in the configuration file may need tweaking. If an initial fit looks poor, try adjusting minamp (in units of rms!) or, in particular, minfwhm (in units of pixels!) in the configuration file (see: $SMURF_DIR/smurf_fit1d.def). Also use range to limit the fit to a relevant region of the spectrum.

The released implementation of FIT1D can fit up to 7 components per profile per run, but the output of multiple runs each covering a range in velocities or frequencies can be combined. The fit itself is fully multi-threaded and will be much faster on a modern multi-core computer: a 3-component gausshermite2 fit of 1.1 million spectra (a 2 Gb input file) took 15 minutes on a dual-core, 16 Gb memory machine versus 4 minutes on one with 12 cores and 75 Gb of memory.

Happy fitting!

Remo