Least-Squares Deconvolution of stellar spectra

A new cross-correlation technique

Least-Squares Deconvolution or LSD is a new cross-correlation technique for computing average profiles from thousands of spectral lines simultaneously. Under several rough approximations (additive line profiles, wavelength independent limb-darkening, self-similar local profile shape, weak magnetic fields), unpolarised/polarised stellar spectra can indeed be seen as a line pattern convolved with an average line profile. In this context, extracting this average line profile amounts to a linear deconvolution problem. We treat it as a matrix problem and look for the least-squares solution. In practice, LSD is very similar to most other cross-correlation techniques, though slightly more sophisticated in the sense that it cleans the cross-correlation profile from the autocorrelation profile of the line pattern.

LSD of Stokes V, Q and U profiles

LSD is particularly well suited for measuring line profile Zeeman signatures generated by magnetic fields at the surfaces of stars through the Zeeman effect. In rapidly rotating cool active stars for instance, circularly polarised (Stokes V) signatures of line profiles are very small (with a typical amplitude of about 0.1% peak-to-peak); they are in particular smaller than photon noise for the vast majority of spectral lines and stellar targets (even on the biggest telescopes), exposure times being limited to no more than a few % of the rotational cycle. Linearly polarised (Stokes Q and U) signatures of magnetic chemically peculiar stars are just as small and difficult to measure. All assumptions inherent to LSD are found to be verified at noise level accuracy in this particular context.

LSD extracts line profile polarisation information from thousands of spectral lines simultaneously. Below is an example in the particular case of the RS CVn system HR 1099 (K1 subgiant and G5 dwarf). The LSD Stokes V profile (upper curve, full line) clearly indicates a Zeeman signature, in conjunction with the broad LSD unpolarised spectral line of the K1 subgiant (lower curve). This signature could not have been detected when averaging together no more than a few unblended spectral lines (upper curve, dotted line).


Monitoring such Zeeman signatures throughout a complete rotational cycle allows one for instance to map the details of the associated stellar surface magnetic topology.

LSD of Stokes I profiles

Surprisingly enough, LSD is also found to work remarkably well when applied to unpolarised (Stokes I) spectra, even though assumptions inherent to LSD are clearly not verified at noise level accuracy (except for very weakly exposed stellar spectra). In particular, we observe that LSD yields very clean average Stokes I profiles with a flat continuum and a considerably enhanced signal to noise ratio independently of stellar rotation rate.

Below is an example of how much LSD can enhance spectrum quality, in the particular case of a weakly exposed spectrum of the classical T Tauri star SU Aur (signal to noise ratio peaking at about 55 in the red). The signal to noise ratio in the LSD Stokes I profile (full line) is at least 8 larger than in any individual photospheric feature of the original spectrum (the dotted line illustrates the particular case of the strong Ca I line @ 643.9075 nm), with no loss of spectral resolution. One obvious advantage of LSD is therefore to detect profile distortions due to surface brightness inhomogeneities in faint and/or rapidly rotating stars with a much higher accuracy.

Among all other possible applications, one can mention accurate chemical abundance determination, as well as precise radial or rotational velocity measurements on stars too faint or rotating too rapidly to be studied with more conventional methods.


As one can see from the graph below (obtained from spectropolarimetric data recorded at the Anglo-Australian Telescope), we find that the multiplex gain in signal to noise ratio associated with LSD increases roughly with the square root of the number of spectral lines used and can be as large as a factor of 32 in the particular case of a K1 star. This amounts to a sensitivity increase of 7.5 mag, at no cost other than using an echelle spectrograph that can collect 250 nm in a single exposure.


Related publications

Shorlin S.L.S, Wade G.A., Donati J.-F., Landstreet J.D., Petit P., Sigut T.A.A., Strasser S., ``A sensitive search for magnetic fields in B, A and F stars'' (2002) A&A 392, 637

Donati J.-F., Mengel M., Carter B.D., Cameron A.C., Wichmann R., ``Surface differential rotation and prominences of the Lupus post T Tauri star RX J1508.6-4423'' (2000) MNRAS 316, 699

Barnes J.R., Cameron A.C., James D.J., Donati J.-F., ``Doppler images from dual-site observations of southern rapidly rotating stars I: differential rotation on PZ Tel'' (2000) MNRAS 314, 162

Wade G.A., Donati J.-F., Landstreet J.D., Shorlin S.L.S., ``Spectropolarimetric measurements of magnetic Ap and Bp stars in all four Stokes parameters'' (1999) MNRAS 313, 823

Wade G.A., Donati J.-F., Landstreet J.D., Shorlin S.L.S., ``High precision magnetic field measurements of Ap and Bp stars'' (1999) MNRAS 313, 851

Donati J.-F., ``Magnetic cycles of HR 1099 and LQ Hydrae'' (1999) MNRAS 302, 457

Donati J.-F., Cameron A.C., Hussain G.A.J., Semel M., ``Magnetic topology and prominence patterns on AB Doradus'' (1999) MNRAS 302, 437

Barnes J.R., Cameron A.C., Unruh Y.C., Donati J.-F., Hussain G.A.J., ``Latitude distributions and lifetimes of starspots on G dwarfs in the Alpha Persei cluster'' (1998) MNRAS 299, 904

Donati J.-F., Cameron A.C., ``Differential rotation and magnetic polarity patterns on AB Doradus'' (1997) MNRAS 291, 1

Donati J.-F., Semel M., Carter B.D., Rees D.E., Cameron A.C., ``Spectropolarimetric observations of active stars'' (1997) MNRAS 291, 658

© Jean-François Donati, last update on 2003 Jun. 16