The Dopplergrams, after having a rough calibration applied, are detrended by subtracting the per-pixel mean for the interval being analyzed. For this work we use a period of 1664 minutes (27^{h} 44^{m}), during which the Sun always rotates through at least 15 degrees of Carrington longitude. Samples for analysis are selected at exact intervals of 15 degrees in Carrington longitude of the sub-observer point, so the sampling interval is fixed in the Carrington frame, but varies slightly in time as the spacecraft moves in its elliptic orbit about the Sun. The images are interpolated into local maps using Postel's azimuthal equidistant projection centred on points rotating at a rate suitable to their central latitude in a standard model of surface differential rotation (Snodgrass, 1984). This effectively tracks the surface, thereby removing the very large displacements in the mode frequencies due to rotation that would be present in a fixed window from the image. The local maps have a resolution that preserves the maximum resolution at disc centre, of course, and they extend radially 16 degrees from their centres. The centres are spaced at 15-degree intervals in heliographic latitude and longitude at the central instant of the tracking interval of 1664 min. We thus construct a mosaic of 189 overlapping three-dimensional data cubes of Doppler residuals versus time and surface spatial location for each analyzed interval; there are up to 24 such intervals over the course of a Carrington rotation. These data cubes are Fourier transformed to yield the power spectra to which the p-mode ridges can be fit.
The spectra are fit to multi-parametric models using two different procedures; one, described by Haber et al. (1998) is rapid and efficient for extracting the fundamental parameters u_{x} and u_{y} reflecting mode advection that can be inverted to determine the vertical profile of transverse velocities. The other, described by Basu et al. (1998) is a more traditional one providing a larger number of parameters in the ridge fits. The velocity parameters from the former set of fits were inverted with a regularized least-squares (RLS) procedure, the latter using optimally localized averaging (OLA). The one-dimensional inversions yield estimates of the average radial profiles of transverse (zonal and meridional) velocities below each of the local tracked elements making up the mosaic.
We have followed separate procedures for determining the mean latitudinal
profiles of the near-surface velocity flows as they evolve through the
solar cycle.
a. In the first case, the ridge fits and RLS inversions are
performed for each point in the mosaic for each daily tracking interval;
the resulting flow maps are then averaged together for the points from
several different mosaics corresponding to the same viewing angle
Most of the results shown here are based on rotational averages of
all points at the same latitude, regardless of viewing angle.
b. In the latter case, we average together
the spectra from the corresponding points on the daily mosaics, again at
the same viewing angle, and then perform the mode fits and the OLA
inversions to the averaged spectra. Typically we have averaged together
spectra or fits from the central meridian points at each of 15 latitude
centers in the range ±52.5° on eight consecutive
mosaics, corresponding to one-third of a Carrington rotation or about
nine days.
The 8-day averages on central meridian are of course inherently noisier than the 24-day averages of all points, involving as they do as little as 1/45 of the data - at low latitudes there are 15 mosaic tiles along a parallel extending up to 52.5° from central meridian. However, there are both symmetric and anti-symmetric systematic effects in the power spectra dependent on disc location, presumably related to variations in both the effective focal length and plate scale over the field of view and the astigmatism. When spectra from the same 8 or 9 days of points on the central meridian only are analyzed using both of the two procedures for averagin, fitting, and inversion, the agreement is excellent (figure 2).