3. Integrated full-disk MDI velocity signal
Using a pair of tunable Michelson interferometers along with a Lyot and a set of fixed filters, the MDI instrument spatially resolves the solar image onto a 10241024 CCD camera in narrow bands (9.4 pm) along the Ni I 676.8nm line profile. The velocity data used here is from the calibrated level-1.4 MDI LOI-proxy Doppler images. The LOI-proxy refers to the image binning mask chosen for comparison between MDI and the Luminosity Oscillations Imager (LOI), which is a part of the Variability of solar Irradiance and Gravity Oscillations (VIRGO) instrument aboard SOHO. Aboard SOHO, the MDI processor averages the 10241024 full-disk observations into 180 spatial bins. An example of the MDI LOI-proxy mask is shown in Fig. 1 for Doppler data with the large spatial scale gradients removed (also see Hoeksema et al. 1998). Instrument details and project goals of the VIRGO instrument are outlined in Fröhlich et al. (1995). Initial results from VIRGO and VIRGO/LOI are presented by Fröhlich et al. (1997) and Appourchaux et al. (1997) respectively.
Temporal gaps in the MDI LOI-proxy data, mostly due to telemetry drop outs, result in a signal coverage of approximately 97% for the period investigated here. These gaps are filled with a high order autoregressive model. As with the GOLF signal, the time series gap filling has a negligible effect on the final calculated power spectra and is used here to simplify the temporal filtering.
3.1. Modeling the GOLF signal
The observed GOLF signal is estimated here by using MDI LOI-proxy velocity images along with the GOLF velocity response functions discussed in Sect. 2.1. Following from Eq. (1) the simulated GOLF velocity signal, hereafter referred to as GOLF-sim, can be expressed as
where is the sensitivity function for the Na D blue wings with the - magnetic modulation, is defined by Eq. (2) and is MDI LOI-proxy velocity data. Here we have chosen the negative magnetic modulation. The modulation choice is arbitrary since the signal difference between the sensitivity functions is negligible in terms of the comparison presented here.
The GOLF velocity response function varies spatially across the solar disc depending on the orbital velocity and the observed inclination angle of the solar polar axis, B0. The individual velocity response functions are interpolated between calculated daily response functions. The daily velocity response functions are created using the average offset velocity and B0 for each day. All the spatial masking is done in the LOI-proxy bin frame. So, for example, the GOLF velocity sensitivity functions are calculated at a pixel resolution of a 128128 array and then rebinned using the LOI-proxy mask. An example of a velocity sensitivity function binned using the MDI LOI-proxy mask is shown in Fig. 1.
3.2. Additional spatial masks
In addition to the velocity response functions used to create the GOLF-sim signal, three other spatial masks are applied to the MDI LOI-proxy velocity images for comparison. A particular type of masking, referred to as zero-sum, greatly decreases the instrumental background noise in both MDI velocity and intensity data (Scherrer 1997). A zero-sum mask is any mask that has a full-disk integrated value of zero. The effect of the zero-sum masking is quite noticeable and is discussed in more detail in Sect. 4.1.
Besides GOLF-sim, three additional masked signals are used in the comparison: north-south, central region and central zero-sum. The north-south mask is divided at the image north and south mid-line into two equal parts with equal weight but opposite sign. The central region mask is defined by Gaussian weights centered on the image disc center. The Gaussian is defined such that the 1/e drop-off is at the image radius defining the inner 10% of the image area. The central zero-sum mask is the same as the central region mask except it has a mean of zero. The raw integrated full-disk LOI-proxy velocity signal is referred to hereafter as LOI-proxy.
Since each spatial mask has a different response to the observed solar surface velocity field, particular masks are more optimal for various l and m multiplets relative to other masks (e.g. Christensen-Dalsgaard 1984; Kosovichev 1986; Appourchaux & Andersen 1989). Consequently, the measured signal-to-background ratio for a given mode is expected to be dependent on the spatial mask used. Following Christensen-Dalsgaard (1984, 1989), the acoustic mode sensitivity is estimated as a function of degree l and the azimuthal order m for the four spatial masks used in this investigation (see Fig. 3). The numerical details for this calculation are fully described in Henney (1999). In Fig. 3, notice that the central zero-sum is more sensitive to modes than the other spatially masked signals. In addition, note that the north-south spatial mask is sensitive to the odd modes, whereas the other masks are sensitive to the even modes. The sensitivity of the GOLF-sim mask to the odd modes is a result of the non-homogeneous response of the GOLF instrument over the solar disc (see Fig. 1). Furthermore, the temporal variation of B0 breaks the full-disk symmetry response of the central and north-south masks. This effect produces an increased sensitivity of the central masks and the north-south mask to the odd and even modes respectively.
3.3. Power spectra fitting
The signal-to-background ratio for low degree () and low frequency (Hz) acoustic modes observed in the GOLF and the MDI velocity signals are compared in the following section. The individual mode parameters from the observed power spectra are fit following Anderson et al. (1990) by using maximum likelihood estimator where the model, , is defined as
The mode multiplet amplitude, half-width, frequency and the local background power are represented by , w, and c respectively. In addition, the central frequency for each mode and the amount of rotational splitting is represented by and s respectively.
The error associated to each parameter is estimated from the width of the corresponding distribution obtained by a Monte-Carlo simulation. For each individual multiplet we use the value of the parameters estimated from the fit to generate a set of 500 artificial spectra according to the technique described by Anderson et al. (1990). This approach produces a probability distribution function for each parameter from which the error can be estimated. We have verified that the width of the distributions is in excellent agreement with the mean values of the internal error estimates.
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999