4.2 Ring-diagram analysis

4.2.1 Local power spectra

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 $15o × 15o$ ). 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 $k - w$ 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 $P(x, t)$ . The data cube is Fourier transformed to obtain $P(k, w)$ , where $k$ is the horizontal wavevector and $w$ is the angular frequency (approximately a plane-wave decomposition). The three-dimensional power spectrum of the resulting data cube, $2 P (k, w) = |P(k, w)|$ , 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.

Figure 10:

Cuts at constant frequency through the three-dimensional power spectrum. Panels A, B, and C correspond to cuts at frequencies $2.8$ , $3.5$ , and $3.8 mHz$ , respectively. The outermost ring corresponds to the f mode, and the inner rings to $p1$ , $p2$ , $p3$ , and so forth. Displacements of the rings are caused by horizontal flows, while alterations of ring diameters are produced by sound speed perturbations. The spectrum was computed from an image sequence 1664 minutes long beginning on 1999 May 25 for a region near disk center. From Hindman et al. (2004

4.2.2 Measurement procedure

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 $k = ||k||$ in the 3D power spectrum, denoted by $Pk(y, w)$ , where $y$ gives the direction of $k$ (the angle between $^x$ and $k$ ) and $w$ is the angular frequency. The power spectrum is then filtered to remove all but the lowest azimuthal orders $m$ in the expansion of the form

For each p-mode ridge (radial order $n$ ), the filtered power spectrum is fitted at constant wavenumber using the maximum likelihood method (see, e.g., Anderson et al., 1990 ) with a model of the form

Here $A$ is an amplitude, $g$ is the half linewidth, and $w0$ is the frequency at resonance. The ring fit parameters $Ux$ and $Uy$ correspond to a depth-averaged horizontal flow $U = (Ux,Uy)$ causing a Doppler shift $k .U$ . The background noise is modeled by $-3 Bk$ .

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 $Ux$ and $Uy$ , and the central frequency $w0$ . For each radial order $n$ and wavenumber $k$ (or spherical harmonical degree $l$ ) corresponds a new set of values: It is convenient to use the notations $nl U x$ , $nl U y$ for the flow parameters, and $dwnl$ for the difference between the fitted mode frequency and the mode frequency calculated from a standard solar model.

Figure 11:

Logarithm of the power as a function of $kx$ and $ky$ at $w/2p = 3150$ $m Hz$ (left panel) and an unwrapped cylinder at wavenumber $2 2 1/2 k = (kx + ky)$ corresponding to angular degree $l = 586$ as a function of $y$ and $w$ ( $1 mHz < w/2p < 5 mHz$ ; right panel). From Schou and Bogart (1998

4.2.3 Depth inversions

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 ):

Here the kernel $Knlc,r$ is the sensitivity of the frequency of the mode $(n, l)$ to a fractional change in the sound speed at constant density, and $Knlr,c$ is the sensitivity to the fractional change in density at constant sound speed. The function $F$ is a smooth function, $I$ is the mode inertia, and $F/I$ gives the effect of near-surface conditions and non-adiabatic effects on the mode frequencies. We recall that mode inertia is the total kinetic energy of the mode divided by the square of the mode velocity at the solar surface; it has the units of mass and is independent of the choice of normalization of the mode eigenfunction. The kernels $K$ can be computed using normal mode theory (for details see the Living Reviews paper on global helioseismology, Thompson (2005 )). The function $F$ is determined as part of the inversion procedure. The inversion problem is then to determine $dc/c(z)$ and $dr/r(z)$ given a set of observed $dwnl$ .

Horizontal flows in the solar interior are related to a set of ring fit parameters $Unl x$ and $U nl y$ , according to

integral Unxl = dz Knlv (z)vx(z), (48) integral Unl = dz Knl (z)v (z), (49) y v y

where $v x$ and $v y$ are the two horizontal components of the flow velocity, assumed to be functions of depth only. As for the kernels for sound speed and density, the flow kernel $nl K v (z)$ can be obtained from normal mode theory (flow parameters can be converted to frequency splittings and standard global mode methods applied). The inversion problem is then to use the observed $U nxl$ and $Unyl$ to estimate $vx(z)$ and $vy(z)$ .

As we have seen, the general one-dimensional ring diagram inversion problem is to use a set of equations of the form

together with the knowledge of the kernels $Ki$ and a set of observed data $di$ to determine the model function $f(z)$ . The two main approaches that have been used for the inversion of ring diagram parameters are the regularized least squares (RLS) and the optimally localized averages (OLA) methods.

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

Here we have assumed a diagonal error covariance matrix with $si$ being the standard deviation of the error on the measurement $di$ . The regularization operator $R$ takes the function $f (z)$ and returns a scalar. The regularization parameter which controls the relative importance of minimizing the misfit to the data and smoothing the solution is $c$ . Example resolution kernels for RLS inversion are plotted in Figure 12

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 $f~$ of the model $f$ at a particular depth, can be written as a linear combination of the data

so that we can write

with the averaging kernels $k$ given by

The OLA method then attempts to choose the coefficients $ai(z0)$ so as to produce averaging kernels $k$ that are well localized while at the same time controlling the error variance $sum ia2i(z0)s2i$ , on the estimate $f~(z0)$ .

Figure 12:

Representative resolution kernels for (A) RLS and (B) OLA inversions of ring-analysis frequency splittings, plotted as a function of depth. Numbers indicate target depths. The negative sidelobes near the surface, so prevalent in RLS inversion kernels, are almost absent in the OLA kernels. From Haber et al. (2004