## Data

We have analyzed data from *`dense-pack'* mosaics of tracked regions
of Dopplergrams from each of the four annual MDI Dynamics Campaigns of
1996-99. The dense-pack mosaics consist of regions of 16° extent
centred at 7.5° spacings in latitude and longitude, mapped with Postel's
projection, tracked for periods of 1664 minutes centred on times when the
Carrington longitude of the solar disc central meridian is a multiple of
15°, apodized around circular regions of 15° diameter, and Fourier
transformed to obtain local ring-diagram power spectra. For this study we
have averaged together successive sets of 8 spectra obtained for the same
heliocentric location, e.g. regions at latitude 0° and Carrington
longitudes 120°, 105°, ..., 15° on successive `days'.
Frequencies as a function of wavenumber *k* are determined from
fits to the average spectra.
The ring-fitting procedure does not yet provide satisfactory uncertainty
estimates. In order to estimate the statistical uncertainties in the
frequencies we have separately fit six sets of 8-day averaged spectra each,
covering two full Carrington rotations in 1996, when the sun was almost
uniformly magnetically quiet. The cumulative distribution
of fits as a function of the scatter among the six sets for the region at
latitude 0 and (central meridian) longitude 0 is shown in Figure 0.
For the sake of comparison with succeeding years we have used only frequency
estimates for which the scatter in 1996 is less than 0.5 or 0.75 microHz,
depending on the number of modes being compared.

## Results

The frequencies obtained for the same region (lat 0, lon 0) in 1997 and 1999
exhibit very small differences from those in 1996, with no systematic trends
(Figure 1a, c). However, there is a clear trend of increasingly large
negative frequency shifts with increasing mode frequency between 1996
and 1998 (Fig. 1b). This effect is exclusively seen in the *p*-modes
(radial order ≥ 1) and especially in those of higher order (Fig 2.);
there is no systematic difference in *f*-mode frequencies at
disc center among the years studied.