mode depend on that of other modes, the whole procedure is repeated until

(essentially) all the desired modes have converged. For the ¬rst iteration

and for any modes outside the desired range a model is used the determine

the mode parameters.

While not inherently a part of the MDI algorithm, individual mode fre-

quencies are generally not ¬tted directly. Rather, the individual mode fre-

quencies are expanded using so-called a coe¬cients

jmax

(l)

νnlm = νnl + aj (n, l)Pj (m),

j=1

(l)

where Pj are suitably chosen polynomials of order j (Schou et al. 1994),

νnl is the mean multiplet frequency and the aj are the a coe¬cients. Also,

rather than ¬tting individual linewidths and amplitudes the inherent values

are assumed to be independent of m. The correlation in the background

noise between di¬erent m™s is determined from the data far from the modes.

Fitting the m™s simultaneously and limiting the number of free parameters

greatly stabilizes this method and allows it to operate at low S/N.

17.4.2.2 The GONG algorithm

The GONG algorithm is comparatively simple. Each mode (n, l, m) is ¬tted

separately using a rather large frequency interval around the initial guess

frequency. As part of the ¬t a number of adjacent modes (mostly l leaks) are

¬tted together with the target mode. The existence of m leaks is ignored,

and the resulting peak is ¬tted as single mode, e¬ectively averaging the un-

derlying frequencies. No consistency is imposed between e.g. the frequency

of a mode when it is the target mode and the frequency when it appears as

a leak in another ¬t. Due to the mix of di¬erent m™s in a given peak this

would have required the use of a leakage matrix.

17.4.2.3 Ridge ¬tting

Finally in the region where both l and m leaks blend together so-called ridge

¬tting algorithms have been used. These have ranged from fairly simple

cross-correlation algorithms (e.g., Korzennik 1990) to more sophisticated

Schou

254

algorithms making use of leakage matrices (Rhodes et al. 2001). As discussed

in the next section the results from these methods have, due to systematic

errors, not been extensively used. However, recent progress (Rhodes et al.

2001) indicated that this may be possible in the near future, leading, one

may hope, to signi¬cantly improved knowledge of the near-surface layers.

It should be mentioned that the technique known as ring diagrams is in

many ways quite similar to this. In that technique a patch of the Sun is

tracked and passed through a three-dimensional Fourier transform. The

ridges in the resulting power spectra are then ¬tted to obtain frequency

shifts, etc. For some results from this technique, see Haber et al. (2002).

17.4.3 Analysis problems

All the stages of the analysis can contribute to systematic errors in the

¬nal results. While it may be argued that all the problems are in the data

analysis (the Sun does not make mistakes and many instrumental problems

can be corrected in the data analysis), it is nonetheless useful to divide these

problems into various categories including unknown physics or parameters,

instrumental problems and approximations in the analysis. In addition,

obvious problems of unknown origin seem ever present. Below I will brie¬‚y

address some of these problems and give more details for a few selected ones.

17.4.3.1 Bad physics and parameters

One such problem is that the assumed direction of the solar rotation axis,

which is needed for proper remapping, appears to be o¬ by around 0.1—¦

(Giles 1999). This is the most likely cause for the annual variation in the

f-mode frequencies shown in Figure 17.2, which has been the cause of some

confusion.

Other problems include the unknown ratio of the horizontal to vertical

displacement near the solar surface, the exact place in the solar atmosphere

where the oscillations are observed and the degree to which magnetic ¬elds

a¬ect the observations.

17.4.3.2 Instrumental problems

While much has been done to measure or estimate the instrumental proper-

ties from ¬rst principles, there are many problems left. Among these are the

cubic distortion introduced by the optics, distortion caused by the CCD not

being perpendicular to the optical axis, non-square CCD pixels, errors in

the absolute P-angle from the mounting of the instrument, the point-spread

function (the PSF which describes how a point source is imaged) and its

spatial/temporal variations, Doppler sensitivity variations and so forth.

Helioseismic data analysis 255

Fig. 17.2. Examples of systematic errors. Left panel shows the di¬erence between

the e¬ective radius estimated from f modes and that of a solar model. The results

were derived from 72-day time series of MDI Medium-l data covering the period

1996 May 1 to 2002 January 17. The right-hand panel shows the normalized resid-

uals in a1 from an inversion of 360 days of MDI data covering the period 1996 May

1 through 1997 April 25.

As an example consider the combination of a cubic distortion, a radius

error and PSF uncertainties that has caused di¬culties for MDI.

Consider a ray of light entering the optical system at an angle θ away

from the optical axis. In an ideal optical system the distance y from the

optical axis at which the light intersects the image plane is given by y = ax,

where x = tan θ and a is related to the focal length. One of the lowest order

aberrations is the cubic distortion where y = ax + bx3 and b/a describes the

strength of the cubic distortion. Depending on the sign of b/a this is known

as either pincushion or barrel distortion, named after the appearance of a

square photographed through such an optical system.

To understand the e¬ect of the cubic distortion note that it corresponds to

a radially varying image scale and that the overall image scale is determined

by measuring the apparent solar radius. However, since the data processing

weights di¬erent parts of the image di¬erently the e¬ective image scale of

the data actually used is di¬erent from the overall average and the main

e¬ect of the distortion is the same as that of an incorrect radius.

In the case of the ridge ¬tting what one e¬ectively does is to determine

the center of gravity of the ridge. The knowledge of the leakage matrix and

assumptions regarding the smoothness of the frequency as a function of l

then allows one to determine the frequency of the target mode, by correcting

for the o¬set of the center of gravity of the leaks relative to the target mode.

The results of numerical calculations of this shift for a number of cases are

shown in Figure 17.3. As can be seen the shift is close to linear in l. To

Schou

256

Fig. 17.3. The center of gravity of the ridges (in units of spherical-harmonic degree)

as estimated from leakage matrices with various assumptions. The solid lines (on

top of each other) are for the ideal case without and with a gaussian PSF. The

dotted and dashed lines are the corresponding cases for a radius error of 0.057%.

The dash-dotted and dash-triple dotted lines are for a cubic distortion similar to

that in MDI.

see why this is the case consider that since the individual modes cannot be

observed the only way to determine l is to used the apparent wavelength

of the oscillations. With a fractional image scale error of one might thus

expect l to be shifted by an amount l. This is close to what is seen.

Also illustrated in Figure 17.3 is the e¬ect of a cubic distortion. As can

be seen it is indistinguishable from a radius error at l ¤ 200 where the

leaks can be measured. At higher degrees the results diverge and even show

the opposite e¬ect when a PSF is introduced. On the other hand the solid

lines show that the PSF has little in¬‚uence when the geometry is correct.

It is clear that the underlying problem needs to be understood in order to

estimate the corrections at higher degrees. The amount of cubic distortion

has recently been determined from a ray trace of the optics. Assuming that

there are no other similar problems, the necessary parameters may therefore

be determined from the medium-degree ¬ts and ridge ¬tting may see more

extensive use in the future.

17.4.3.3 Algorithm problems

These are generally problems where a known e¬ect is ignored or incorrectly

modeled or where approximations have to be made in order to make the

computational burden reasonable.

Among the e¬ects that are commonly ignored, in addition to the instru-

mental problems mentioned above, are the horizontal displacement in the