The variance of the pixel noise is assumed to be the same for the pixels with the same shape. The mean of the pixel noise is 0. For the LOI with its 12 pixels, there are 3 different shapes giving 3 independent noises.
After generating the synthetic signals according to Eq. (39), the data are fitted by minimizing the likelihood of Eq. (32). Figure 1 shows an example of Fourier spectra generated synthetically. The typical signal-to-noise ratio in the power spectra is about 20-30. The frequency resolution is equivalent to 4 months of data. We performed 1000 simulations of the spectra.
|Figure 1: (Left) Power spectra of a synthetic l=1 as it would be observed by the LOI. The frequency resolution corresponds to 4 months of data. The signal-to-noise ratio is about 20-30. The traces from bottom to top corresponds to m=-1, 0, +1. (Right) Fourier spectra for l=1 (same data). The first, third and fifth traces from the bottom represents the real part of the spectrum of m=-1,0 and 1, respectively; the other traces are the imaginary parts. The leakage between m=-1 and m=+1 is 0.45 in the Fourier spectra|
|Figure 2: Histograms for the fitted parameters: (Plain line) Data, (Dashed line) Normal distribution with the same mean and as the fitted parameters. (Top) Frequency (in Hz), splitting a1 (in Hz), ( in Hz); (Middle) (Amplitude) for m=-1, 0, 1; (Bottom) (pixel noise). For each histogram, the target value, the mean fitted value and the 1- fitted valued are displayed. The Kolmogorov-Smirnov test (Kol.) is displayed for each histogram; a number close to 0 show that the distribution is not normal|
|Figure 3: Histograms for the error bars: (Plain line) Data, (Dashed line) Normal distribution with the same mean and . (Top) Frequency error (in Hz), splitting a1 error (in Hz), error ( in Hz); (Middle) (Amplitude) error for m=-1, 0, 1; (Bottom) (pixel noise) error. For each histogram, the target value, the mean fitted value and the 1- fitted valued are displayed. The Kolmogorov-Smirnov test is displayed for each histogram; a number close to 0 show that the distribution is not normal|
Hereafter, we have investigated the influence of a wrongly assumed leakage matrix on the fitted parameters of l = 1. We made 100 realizations and change the leakage parameter between m=-1 and m=+1 by % from a nominal value for the LOI of 0.45. Figure 4 shows the influence of varying the assumed leakage element on the fitted parameters. It is quite interesting to note that the inferred central frequency is insensitive to mistakes in the leakage matrix. The linewidth becomes underestimated when the error is larger than 20%, while the amplitudes become overestimated. The most important result is the fact that the systematic error made on the splitting is not linear but quadratic. This systematic error can become as large as the error bars. For example, with 1 year of LOI data and averaging over 10 modes, the error bars on the mean splitting is about 15 nHz; this should be compared to a systematic error of 10 nHz for an error of 10% of the l=1 leakage elements.
Another test similar to that of the l=1 was performed with the l=2 mode. We have assumed that all the off-diagonal elements of the leakage matrix were wrong by the same fixed amount. Figure 5 shows the results only for the splitting coefficients (from a1 to a4). The other parameters linewidth, amplitudes and noises behave in the same manner as for l=1. The systematic error on the splitting has also the same quadratic dependence as for l=1. For l=2 the splitting error bars are typically smaller than for l=1. In this case the systematic errors become larger than the error bars, and therefore start to influence the inverted solar rotation.
It means that it is quite easy to underestimate the splitting whenever we under- or overestimate the leakage element. As a matter of fact, this behaviour was also found in the GONG data for l=1 and 2 (Rabello-Soares & Appourchaux 1998, in preparation). On the other hand, errors in the leakage matrix will not result in overestimating the splitting. If the splitting is overestimated, the most likely source should be the presence of other degrees not taken into account in the analysis.
We also checked the correlation of the splitting coefficients derived for l = 2. Figures 6 and 7 show respectively the variance and the covariance of the splitting coefficients as a function of the leakage elements error. It can be concluded that the splitting coefficients become correlated only when a large overestimation of about 50% is made for the off-diagonal leakage elements. This result is only valid when fitting Fourier spectra. For other methods, such as fitting power spectra, possible correlation amongst the splitting coefficients could have drastic consequences for the inverted solar rotation profiles.
|Figure 4: Influence of the fitted parameters to relative changes of the assumed leakage element between m=-1 and m=+1 for l=1. (Top) Frequency, splitting a1, ; (Middle) (Amplitude) for m=-1, 0, 1; (Bottom) (pixel noise). The target parameters are the same as for Fig. 2. Please note the parabolic shape for the splitting|
|Figure 5: Influence of the fitted splitting parameters to relative changes of the assumed of the assumed off-diagonal leakage element for l=2. (Top, left) a1, target value: 410 nHz; (Top, right) a2, target value: -30 nHz; (Bottom, left) a3, target value: -10 nHz; (Bottom, right) a4, target value: +50 nHz|
|Figure 6: Diagonal elements of the covariance matrix of the splitting coefficient, for l=2. They are given as a function of the relative change of the assumed off-diagonal leakage element. (Top, left) For a1; (Top, right) For a2; (Bottom, left) For a3; (Bottom, right) For a4|
|Figure 7: Off-diagonal elements of the covariance matrix of the splitting coefficient, for l=2. They are given as a function of the relative change of the assumed off-diagonal leakage element. (Top, left) For a1 and a2; (Top, right) For a1 and a3; (Middle, left) For a1 and a4; (Middle, right) For a2 and a3; (Bottom, left) For a2 and a4; (Bottom, right) For a3 and a4|
|Figure 8: Comparison of low-degree splittings measured using 1000 Monte-Carlo simulations by 3 different fitting techniques. The nominal splitting is 410 nHz. The + is the technique commonly used by GONG, the asterisk is the technique used by Appourchaux et al. (1995), the diamond is the technique described in this paper and also used for the SOI/MDI data. The error bars are the formal error bars for a single realization|
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