Improved Helioseismic Analysis of Medium-$\ell$ Data from the Michelson Doppler Imager

Abstract

We present a comprehensive study of one method for measuring various parameters of global modes of oscillation of the Sun. Using velocity data taken by the Michelson Doppler Imager (MDI), we analyze spherical harmonic degrees $\ell\leq300$. Both current and historical methodologies are explained, and the various differences between the two are investigated to determine their effects on global-mode parameters and systematic errors in the analysis. These differences include a number of geometric corrections made during spherical harmonic decomposition; updated routines for generating window functions, detrending time series, and filling gaps; and consideration of physical effects such as mode-profile asymmetry, horizontal displacement at the solar surface, and distortion of eigenfunctions by differential rotation. We apply these changes one by one to three years of data, and then reanalyze the entire MDI mission applying all of them, using both the original 72-day long time series and 360-day long time series. We find significant changes in mode parameters, both as a result of the various changes to the processing, as well as between the 72-day and 360-day analyses. We find reduced residuals of inversions for internal rotation, but seeming artifacts remain, such as the peak in the rotation rate near the surface at high latitudes. An annual periodicity in the $f$-mode frequencies is also investigated.

Appendix

Detailed information on how to access MDI data from the global helioseismology pipeline can be found on the website of the Joint Science Operations Center (JSOC) at jsoc.stanford.edu/MDI/MDI_Global.html . This page contains documentation describing how the datasets used in this article were made and how they can be remade. In this appendix we describe how to access the relevant archived data. In what follows we assume some familiarity with the Data Record Management System (DRMS), detailed documentation for which is linked from the above website.

Mode-parameter files (as ASCII tables) for every analysis discussed in this article are available in the Electronic Supplementary Material. For the original analysis, they (and a helpful Readme file) can also be found at sun.stanford.edu/~schou/anavw72z/ . For all other analyses, they can also be retrieved from JSOC. The fields of a mode-parameter file are the following: $\ell$, $n$, $\nu_{0}$, $A$, $w$, $b$, $x$, $\{\tan (\gamma)\}$, $\sigma_{\nu_{0}}, \sigma_{A}, \sigma_{w}, \sigma_{b}, \sigma_{x}, \{\sigma_{\tan(\gamma)}\}, a_{1}, a_{2}, \ldots, a_{N}, \sigma_{1}, \sigma_{2}, \ldots, \sigma_{N}$. The parameter $\tan(\gamma)$ and its error will not be present for fits done with symmetric profiles. The value of $N$ is either 6, 18, or 36. Any parameter with zero error has not been fit for. The parameter $x$ is not fit for in these analyses and is retained for historical reasons.

The data for the different “corrections” are labeled by the strings corr1 to corr9 corresponding to the numbering scheme in Table 2. The final correction in this set refers to the first way of applying the Woodard effect (holding $B_{1}$ and $B_{2}$ constant). These data have all been generated in the first author’s name space, with mode parameters found in su_tplarson.corr_vw_V_sht_modes. The primekeys are T_START, LMIN, LMAX, NDT, and TAG, where T_START is the beginning of the corresponding time series, most easily specified by the MDI day number suffixed by “d” (see Table 1). For all records in this series, $\mathsf{LMIN}=0$, $\mathsf{LMAX}=300$, and $\mathsf {NDT}=103\mbox{,}680$, so these primekeys need never be specified. The TAG keyword is the label string, so TAG and T_START uniquely specify every record.

The second way of applying the Woodard effect and the asymmetric fits are both represented in the official MDI name space (mdi). For the former, mode parameters can be found in mdi.vw_V_sht_modes and for the latter in mdi.vw_V_sht_modes_asym. The primekeys are the same as given above, with the exception that these series do not have the TAG keyword and that $\mathsf{NDT}=518\mbox{,}400$ for the 360-day fits. In addition, the results used in this article have the VERSION keyword (not a primekey) in these series set to version2. If these data are reprocessed in the future, VERSION will get a new value, but old versions can easily be retrieved.

The data series containing time series and window functions in the mdi name space have also been archived and can be retrieved; details on these data products are given on the above website. The corresponding data in the su_tplarson name space have not been archived, but can be recreated if needed. The procedure for doing so can be found on the website. The original time series and window functions have been archived in the dsds namespace, but have not yet been ported to the standard DRMS format for global helioseismology data products. They can, however, still be retrieved by request.