Ring-diagram analysis is a powerful tool to infer the speed and direction of horizontal flows below the solar surface by observing the Doppler shifts of ambient acoustic waves from power spectra of solar oscillations computed over patches of the solar surface (typically 15° × 15°). Thus ring analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun).
The ring diagram method is based on the computation and fitting of local power spectra and was first introduced by Hill (1988). A small patch is tracked as it moves across the disk. In this process, images are remapped onto a projection grid, such as the cylindrical equal-area projection, Postel’s azimuthal equidistant projection, or the transverse cylindrical equidistant projection (see, e.g., Corbard et al., 2003). A series of tracked images form a data cube, i.e., the surface Doppler velocity as a function of the two spatial coordinates and time . The data cube is Fourier transformed to obtain , where is the horizontal wavevector and is the angular frequency (approximately a plane-wave decomposition). The three-dimensional power spectrum of the resulting data cube, , yields the basic input data for ring diagrams. As in the global power spectrum, the main features are the ridges corresponding to the normal modes of the Sun. As there are two wavenumber directions, the ridges now appear as rings when seen in a cut through the spectrum at constant frequency (Figure 10). Flows in regions, over which the power spectrum is computed, introduce Doppler shifts in the oscillation spectrum, and changes in sound speed alter the locations of the rings. Thus by fitting the positions and shapes of the rings in the power spectrum the subsurface flows and sound-speed can be estimated.
One method of fitting local power spectra was described by Schou and Bogart (1998) (see also Haber et al., 2002; Howe et al., 2004). The first operation is to construct cylindrical cuts at constant in the 3D power spectrum, denoted by , where gives the direction of (the angle between and ) and is the angular frequency. The power spectrum is then filtered to remove all but the lowest azimuthal orders in the expansion of the formet al., 1990) with a model of the form
Other fitting techniques are described by Patrón et al. (1997) and Basu et al. (1999). Essentially all ring-diagram fitting has been done assuming symmetric profiles for the ridges in the power spectrum. Basu and Antia (1999) concluded that including line asymmetry in the model power spectrum improves the fits, but does not substantially change the inferred flows.
The most important parameters used in ring-diagram analysis are the flow parameters and , and the central frequency . For each radial order and wavenumber (or spherical harmonical degree ) corresponds a new set of values: It is convenient to use the notations , for the flow parameters, and for the difference between the fitted mode frequency and the mode frequency calculated from a standard solar model.
The goal of inverting ring fit parameters is to determine the sound speed, density, and mass flows in the region underneath the tile over which the local power spectrum was computed. Generally, the assumption is that the sound speed, density, and flows are only functions of depth within the region covered by a particular tile. The forward problem for ring diagrams has traditionally been done by analogy with global mode helioseismology.
Inversions for sound speed and density are done using the mode frequencies determined from the ring fitting (e.g., Basu et al., 2004). The changes in the mode frequencies can be related to changes in the sound speed and density according to (Dziembowski et al., 1990):
Horizontal flows in the solar interior are related to a set of ring fit parameters and , according to
As we have seen, the general one-dimensional ring diagram inversion problem is to use a set of equations of the form
The RLS method (see, e.g., Hansen, 1998; Larsen, 1998) selects the model that minimizes a combination of the misfit between the observed data and the data predicted by the model and some function of the model, for example the integral of the square of the second derivative. In particular RLS minimizes a function of the form
The OLA method (Backus and Gilbert, 1968) attempts to produce localized averaging kernels while simultaneously controlling the error magnification. For a discussion of OLA in ring diagram inversions see Basu et al. (1999) and Haber et al. (2004). The essential idea in OLA methods is that the estimate of the model at a particular depth, can be written as a linear combination of the data
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