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 z for each eigenmode. Variations in sound speed produce frequency shifts proportional to the square of the wave number.
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 = vkcos(
where 0 is
the angle between the k vector and the direction of motion,
we ``unwrap'' the surface for a given value of k, that is, we make
a cylindrical cut through the power spectrum parallel to the w axis,
giving P(w, 0). The peaks in this two-dimensional
power spectrum are a series of nearly straight lines at the eigenfrequencies,
with a slight sinusoidal dependence on 0.
(See Fig. 2 and Fig. 3). We
determine the frequency wn( 0)
eigenmode and then Fourier analyze it. The amplitude and phase of the
first order term in the Fourier expansion then give the magnitude and
direction, respectively, of the fluid motion. Higher-order terms of
course provide information on 2nd-order effects of sound speed variation.
This analysis is performed for each clearly identifiable mode and repeated
for numerous values of k so that the distortion parameters can be
built up as functions of radial order n and horizontal wavenumber
k for inversion.
Of course the waves are quantized only in the vertical direction
by the trapping cavity under our plane-wave assumption, and are continuous
in k; the choice of how thick to make the cylindrical cuts and how
to space them is arbitrary and essentially dictated by the data resolution.
But for spherical-harmonic analysis of oscillations of high spatial frequency
in the horizontal direction it is generally impossible to isolate individual
eigenmodes by degree and azimuthal order as well.
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.
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.