Among the various techniques currently used in local helioseismology (Ring- diagram analysis: Hill (1988); Time-distance analysis: Duvall et al. (1993); Acoustic holography: Lindsey & Braun (1998)), the Fourier-Hankel spectral decomposition was one of the earliest techniques used by helioseismologists to explore the subsurface regions of the Sun. It was first employed by Braun et al. (1987), and later followed up by a series of work (Braun et al., 1988; Braun & Duvall, 1990; Braun et al., 1990, 1992) to study the interaction of solar acoustic modes (p-modes) with isolated large scale magnetic structures like sunspots. Sunspots are local discontinuities in the near surface regions of the Sun. The p-modes sample these discontinuities and leave signature of their interaction on the oscillation spectra. Whereas, the spectra of the modes that have not encountered the sunspot remains unaltered. Fourier- Hankel method can be used to separate these two wave fields. This is done by decomposing the wave field into radially inward and outward propagating waves on an annular region centered in the sunspot. Although, Hankel functions are good approximations, later studies also explored this method with the help of associated Legendre functions to account for the curvature of the solar surface.
The application of Fourier-Hankel/Legendre technique was later extended to other helioseismic studies. Braun & Fan (1998) carried out meridional flow measurements using this technique by decomposing the wavefield into equatorward and poleward wavefields and measuring the frequency shifts between them. Further work carried out by Krieger et al. (2007) and Doerr et al. (2010) in this direction have resulted in the development of a fast and efficient code for the computation of Fourier-Hankel/Legendre transform which can be easily applied to a large amount of data (Glogowski, 2011). Preliminary results for meridional flow obtained from this work are comparable to the results obtained using ring- diagram analysis.
This document provides a description of the Fourier-Hankel/Legendre module on the SDO/HMI JSOC data-analysis pipeline for processing HMI data.
Extract the archive to the local directory. Edit the Makefile with the appropriate paths to CFITSIO, HDF5 (requires hdf5-1.8.14 compiled with the --enable-parallel option), PostgreSQL, and DRMS libraries.
In case the libraries are not located in your default path. Configure it with the respective paths.
./configure --with-hdf5=<Path to the h5pcc executable> --with-cfitsio=<CFITSIO path> --with-drms=<DRMS Path> make
Make produces three executables: postel_remap, tld, fld.
The Fourier-Legendre analysis are usually performed on small areas of the solar surface to infer the average properties beneath the selected region. For instance, to estimate the meridional flow as a function of latitude, rectangular patches are extracted from the Dopplergrams in the North-South direction along a specific longitude (usually the Stonyhurst φ = 0◦), centered at different latitudes. The Fourier-Hankel/Legendre decomposition gives the amplitude of the poleward and equatorward directed flow field. The difference in frequency between these two oppositely directed components can be attributed to a Doppler shift as a result of a flow in a particular direction. Assuming that the origin of this flow is be due to the meridional circulation, the frequency shifts can be inverted to get the magnitude of the meridional flow at different depths as a function of latitudes. For sunspot seismology, annular regions are selected centered in the spot and the decomposition gives the amplitude of the ingoing and outgoing waves within this annulus. The absorption coefficients and scattering phase-shifts can then be estimated from the amplitude of the two components. It should be noted that, for the meridional flow measurements the center is fixed on one of the poles, while in the case of sunspot it is the spot center. In both cases, the decomposition is carried out on a equi-angluar grid and therefore the first step is to select the region of interest and map it onto an appropriate grid. The Fourier-Hankel/Legendre decomposition is then carried on this tracked/mapped cube.