It has been demonstrated that a full three-dimensional plane-wave analysis of solar oscillations can be readily performed and the results inverted to yield at least plausible inferences about the dynamics of the upper convective envelope (Hill, 1988).

In a plane geometry, the dynamical equations governing acoustic-gravity
waves can be written very simply. It is quite straightforward to show
that a motion relative to the observer of the field
in which the waves propagate results in
an apparent frequency shift proportional to the dot product of the velocity
vector and the horizontal wave-number vector
**v**(x,y,z)**·** * k(x,y,z)*,
suitably averaged in depth

The mapping from heliographic coordinates implicitly involves tracking of the solar surface, at least to the extent that the heliographic coordinate system reflects the mean motion of the surface. Since the photosphere rotates differentially, the question of which coordinates to use is not trivial. There are three basic choices: to assume rigid body rotation (using the Carrington coordinates for example); to assume uniform rotation of the field of interest, but at a rate appropriate to some representative latitude within the field; or to track each latitude independently, using a shearing coordinate system. While developing and validatiing the analysis techniques, we have so far restricted our attention to datasets of no more than a few days in length, so the differential shear over the comparatively small fields of view is not a matter of concern. We have however chosen to use as the base rate for tracking the differential rotation rate suitable to the center of the selected field. This means that the Carrington coordinates describing the field are only applicable at one instant. We describe the field by the Carrington coordinates of its center at the (virtual) time of crossing of the central meridian observed from Earth.

As the data are mapped, they are also apodized in a circularly symmetric pattern, in order to eliminate the effects of non-isotropic higher spatial frequencies in the data resulting from our use of a square field.

The time series of mapped and apodized data is then Fourier transformed
in three dimensions and the three-dimensional *k-w* power
spectrum is computed by standard methods. It is the ease and speed
of this calculation that provides the impetus for plane-wave analysis.
The eigenmodes organize the power spectrum into surfaces of local peak
power of the approximate form *w* ~ sqrt *(k)*. Two cuts through
a typical power spectrum are shown in Fig. 1. The first
is equivalent to the plane *ky = 0*, corresponding to a standard one-dimensional *kw* (or *lv*) diagram. The second
is a cut through the space at a particular temporal frequency. The
characteristic rings of power representing successive
eigenmodes of decreasing vertical order with increasing *k* have
inspired the name ``Ring Diagram Analysis'' to this branch of heliseismology.

Up to this point, with the exception of the choice of mapping and tracking,
the analysis is quite standard. The real opportunity for efficiency (and
different systematic errors) comes in the parametrization of the
distortions of the individual eigenmode surfaces. Because the first-order
effect due to advection is to change the frequency of a given mode by an
amount proportional to
** v·k** =

We have independently analyzed the same selections from the test dataset
using the methods described in
Haber *et al.* (1995)
and those described here. We have used and compared independent programs
to perform all preliminary stages of the analysis from mapping through
determination of the power spectrum, and then have used independent methods
of mode identification and parametrization applied to the same power spectra.
The actual preliminary results are presented
elsewhere (Bogart *et al.*, 1995). Here
we focus only on the validation tests.

The primary validation test has been to take a data set consisting of
a single day's data (that of June 22, 1993), select a particular region,
and then to track the region at each of several different rates in various
directions. The tracking rates were not known to those performing the
mode parametrizations. Our assumption is that without performing an
inversion and without knowing details of the real velocity profile with
depth, it must still be true that a uniform translation of the observer
will raise or lower the frequencies by a fixed amount for each
mode, so that the *differences* between the inferred vector
velocities in two cases for any mode should be the same as the differences
in the two artificially generated field velocities. The results from
the new method are shown in Figure 5, and those from
both methods are given in Table 1. It is clear
that both methods generally yield roughly similar results and match
the expected velocity differences within about 20 m/s.
We consider this a very promising confirmation of the essential
power and accuracy of the technique. There are a few noticeable exceptions:
The ring diagram technique seriously underestimated the meridional
velocity component in one case (C), and both were far from the expected
zonal component in case (A). Since this case involved changes in the
higher order components in the differential rotation law, this could
perhaps expose an error in the tracking program, but we do not have
an explanation at this time. More work remains to be done to
explore any possible systematic differences between the two methods.
It is also of course essential to approach the problem of inversion
for the vertical structure of the velocity field as well as 2nd-order
effects of sound-speed variations.

We will extract maps of each calibrated observable centered on a set of selection points. The mapped subimages for the full-disc field will have an extent of 30 degrees, i.e. extend at least 15 degrees from the selection center in all directions. The mapped regions will be tracked at a constant rotation rate applicable to the latitude of the selection center and referred to the heliographic coordinates of the center at the time of its meridian crossing as observed from Earth. The selection centers from the full-disc data will be located at approximately 15 degree spacings extending as far as ±60 degrees latitude. (The size of the fields will of course be smaller and their spacing correspondingly closer for the high-resolution fields.)

For each time series of tracked regions at a given selection center, we will compute the spatial-temporal power spectra for time samples of about 3 days' (4096 minutes) duration; for a longer data series, multiple power spectra will be computed so that the whole data series is sampled. These power spectra will be analyzed using the procedure described in this note to construct tables of 1st- and probably 2nd-order coefficients in the azimuthal expansion of frequency shifts as functions of radial order and horizontal wave number up to the spatial Nyquist frequency. We believe that the frequency-shift coefficients can be used in that form as the starting point for inversions, but this remains to be established.

- Bogart R. S., Sá, L., Haber, D.,
& Hill, F. 1995.
``Preliminary Results from Plane-Wave Analysis of Helioseismic Data''
*4th SOHO Workshop on Helioseismology*, Pacific Grove, California, Apr. 2-6, 1995 - Haber, D., Toomre, J., Hill, F., &
Gough D. 1995. ``Local Area Analysis of High-Degree Solar Oscillations:
New Ring Fitting Procedures''
*4th SOHO Workshop on Helioseismology*, Pacific Grove, California, Apr. 2-6, 1995 - Hill, F. 1988.
*Ap. J.,***333**, 996. - Richardus, P. & Adler, R. K. 1972.
*Map Projections for Geodecists, Cartographers, and Geographers,*North-Holland Publishing Co., Amsterdam.

- Table 1: comparison of inferred velocities and
actual velocities in an artificially mistracked region. The method
described here is labeled the Fourier method, that described in Haber
*et al.*(1995) the Ring method.Case Expected Fourier Ring (A) +352* -35 | +273 -41 | +221 -31 (B) -7 -104 | -10 -99 | +1 -81 (C) 0 +174 | -8 +165 | +2 +110 (E) +74 0 | +62 +3 | +60 +0 * or possibly +278; there may have been an error in the differential rotation correction involved in this case.

## Figures

- Figure 1: two standard cuts through the plane-wave
power spectrum of a one-day time series of data from the High-L
Helioseisomometer. The tracked region is large, extending 30 degrees
from its center. The cut shown in (a) is equivalent to a cut along a plane
of a particular direction of
; it is actually made by taking azimuthal averages over all values for each**k***k*. (b), a ring-diagram, is a cut through a particular temporal frequency*w*plane, so the two axes are*kx*and*ky*.

- Figure 2: A cylindrical cut through the power-spectrum
of Figure 1, corresponding to a particular value of
*k*. The vertical axis corresponds to temporal frequency*w*, the horizontal axis to the azimuthal angle= arctan (~~0~~*ky/kx)*

- Figure 3: cylindrical cut through a power-spectrum
of High-L Helioseisomometer data deliberately mistracked at a rate equal
to twice the solar rotational velocity,
*i.e.*with an apparent east-west motion of 4 km/sec transverse to the line of sight. This enormous artificial velocity enhances the amplitude of the sinusoidal dependence of the ridge locations in azimuth (equivalent to ring displacement) to the point at which it is plainly visible. The phase is appropriate to a nearly pure east-west motion.

- Figure 4: Effect of different mappings. Ring diagrams
are shown for the same 4-hour time series of data: (a) using the standard
Postel's projection; (b) using a cylindrical equal-area projection. The
greater distortion and asymmetry in the rings in case (b) should be evident.

- Figure 5: Average velocity vectors inferred in the
test cases using the Fourier method of analysis. For each of the other
four test cases, we exhibit the differential velocity relative to the one
test case (labeled D) which had no spurious motion; these should be identical
to the individual spurious motions, which are given in parentheses. See also
Table 1.

- Figure 1: two standard cuts through the plane-wave
power spectrum of a one-day time series of data from the High-L
Helioseisomometer. The tracked region is large, extending 30 degrees
from its center. The cut shown in (a) is equivalent to a cut along a plane
of a particular direction of