3 Application to data

3.1 ($m,\nu$) echelle diagram for the LOI/SOHO data

Example of these diagrams can be seen in Figs. 2, 3 and 4 for 1 year of LOI data for l=1,2 and 5, respectively. It is important when one makes such diagrams to tune the spacing for the degree to study. For example, one can see in Fig. 2 that the ridges of power of l=0,1,2 and 3 have different shapes than in Fig. 3. Other modes with a significant different spacing can be seen more like diagonal ridges crossing the $(m,\nu)$ diagrams; this is a powerful tool to identify other degrees. In Fig. 4, the l=5 $(m,\nu)$ echelle diagram is clearly contaminated by an other degree, i.e. l=8, which appears at different frequencies depending on m. For l=5, m=-5, the l=8, m=+8 is quite strong; while for l=5, m=+5, this is l=8, m=-8 which shows up. This kind of `anti'-splitting behaviour is typical of any aliasing degrees. It is more prominent in the LOI data because of the undersampling effect due to the large size of the LOI pixels.

  

Figure 2: $(m,\nu)$ echelle diagram for 1 year of LOI data for l=1. The scale is in ppm$\mu$Hz-1/2. The spacing is tuned for l=1 ($\Delta\nu$ = 134.8 $\mu$Hz). (Left) The full diagram centered on l=1. The l=3 modes are located about 15 $\mu$Hz at the left hand side of the l=1. The l=2 modes are on the right edge, while the l=0 are on the opposite side. The l=4 modes can be seen around +40 $\mu$Hz. (Right) The same diagram but enlarged around l=1. The frequency shift or splitting of the modes due to the solar rotation can be seen: the 2 patches of power for m=-1 and m=+1 are slightly displaced from each other. The artificial correlation between m=-1 and m=+1 is not as clear

  

Figure 3: $(m,\nu)$ echelle diagram for 1 year of LOI data for l=2. The scale is in ppm$\mu$Hz-1/2. The spacing is tuned for l=2 ($\Delta\nu$ = 135.1 $\mu$Hz). (Left) The full diagram centered on l=2 with a spacing of 135. $\mu$Hz. The l=0 modes are located about 10 $\mu$Hz at the right hand side of the l=2 modes. The l=3 modes are on the right edge, while the l=1 modes are on the opposite side. The l=4 modes are easily seen at about -25 $\mu$Hz; the l=5 starts to appear at +25 $\mu$Hz. (Right) The same diagram but enlarged around l=2 with the l=0 modes at the right hand side. The splitting of the modes is clear. Note the absence of the l=0 modes for $m=\pm 1$

  

Figure 4: $(m,\nu)$ echelle diagram for 1 year of LOI data for l=5. The scale is in ppm$\mu$Hz-1/2. The spacing is tuned for l=5 ($\Delta\nu$ = 136.2 $\mu$Hz). Here we can clearly see the splitting of the l=5 modes. Unfortunately, the l=5 data of the LOI are heavily polluted by the presence of the l=8 modes. But since the spacing for the l=8 modes is different compared with that of the l=5 modes, the l=8 modes are seen as ridges going from lower left to upper right in each m panel; while the l=5 modes are seen more like straight ridges in each m panel. It can also be seen that the |m|=5 Fourier spectra contain information both about $m=\pm 5$. This is the result of the large LOI pixel size creating oversampling. The l=8 modes are also distinguished because the "splitting" of this alias is in the opposite direction compared with that of the l=5 modes

3.2 A useful detail

Before using the other diagrams on real data, we need to point out a very important property coming from the way the m signals are built. If the weights W_l,m applied to the images, to extract an l,m mode, have the same properties as of the spherical harmonics then we have:

W_l,m=W_l,-m^*

which means that both the Fourier spectra of +m and -m can be obtained from using a single filter W_l,+m. The Fourier spectrum of +m will be in the positive part of the frequency range, while that of -m will be in the negative part. This approach was first used by Rhodes et al.  (1979) for measuring solar rotation, and mentioned theoretically by Appourchaux & Andersen (1990) for the case of the LOI. Due to the property of the Fourier transform, we recover, in the negative part of the frequency range, not the Fourier spectrum of -m but its conjugate . This fact is very important, if one does not take care of the sign of the imaginary part of -m, it will lead to very serious problem. Needless to say that fitting data without taking into account this detail will have devastating effects. Obviously this important detail cannot be detected in the power spectra.

3.3 Leakage matrix measurement for a single degree

3.3.1 LOI/SOHO data

According to theoretical computation of the p-mode sensitivities of the LOI pixels (Appourchaux & Andersen 1990) and using the real shape of the LOI pixels (Appourchaux & Telljohann 1996), the leakage matrices of l=1 and 2 are given by:

$$\begin{eqnarray} \tens{\cal{C}}^{(1,1)}=\left(\begin{array} {ccc} 1 & 0 & \alpha... ...& 1 & 0\\ \alpha_{4} & 0 & \alpha_{1} & 0 & 1\\ \end{array}\right)\end{eqnarray}$$ (11)

with: $\alpha=0.474$, $\alpha_{1}=\alpha_{3}=-0.308$, $\alpha_{2}=0.576$, $\alpha_{4}=-0.216$. These leakage matrices are mean value over one year. The leakages vary throughout the year because the distance between SOHO and the Sun varies. There is no B angle effect as the mean B is zero over 1 year. All the leakage elements are real as the LOI filters have the same symmetry as the spherical harmonics.

Figures 5 and 6 display the cross echelle diagram for l=1 and 2, respectively. From Figs. 5 and 6, we can directly verify using Eq. (6) the sign, and sometimes even the magnitude of the leakage elements. The best cross check that our theoretical computations are correct is to apply the inverse of the leakage matrices to the original data (See Sect. 2.2). Figures 7 and 8 show how the artificial correlations (or m leaks) can be removed from the LOI data; for the latter we also cleaned the original data from the presence of the l=0 modes. In this latter case, $\vec{y}$ and $\vec{x}$ (cf. Eq. (1)) are both 6-element vectors where the first element is the observed Fourier spectrum for l=0, and the other 5 are the Fourier spectra for l=2 and the associated m's. Both vectors are related to each other by the leakage matrix $\tens{\cal{C}}^{(0,2)}$ of dimension 6 $\times$ 6. We shall see later on with the GONG data that cleaning, similar to that of the LOI, can be achieved not only for 2 degrees but also for 3 degrees (l=1,6 and 9).

Unfortunately, for the LOI data the leakage matrices of l=4, l'=7 and l=5, l'=8 cannot be inverted. It means that the 2l+2l'+2 Fourier spectra of $l \ge 4$ are linearly dependent. This is the result of the pixel undersampling and has two dramatic consequences: the leakage matrix cannot be verified as for the other degrees, and fitting the data as described in Part I is not valid anymore as the leakage matrix needs to be invertible. This problem is specific to the LOI data. One way around the problem is to restrict the fitting to a subset of the spectra having no linear dependence.

  

Figure 5: Real part of the cross echelle diagrams for 1 year of LOI data for l=1. The scale is in ppm2$\mu$Hz-1. The spacing is tuned for l=1 ($\Delta\nu$ = 134.8 $\mu$Hz). (Left) For l=1, m=-1. The lower panel represents the echelle diagram of m=-1. The 2 other panels are the cross spectra with m=0 and m=1. As predicted, there is no visible correlation between m=-1 and m=0 while there is between m=-1 and m=+1. Here we visualize the first line of the real part of the matrix $\tens{\cal{V}}^{(1,1)}$. (Right) For l=1, m=0. The bottom panel is the same as the middle panel of the left diagram but with a different color scale. The middle panel is the power spectra of m=0 already shown in Fig. 2; the l=6 modes are visible as a diagonal ridge crossing the ridge of the l=1 modes

  

Figure 6: Real part of the cross echelle diagrams for 1 year of LOI data for l=2. The scale is in ppm2$\mu$Hz-1. The spacing is tuned for l=2 ($\Delta\nu$ = 135.1 $\mu$Hz). (Top, left) For m=-2. The middle panel (for m'=0) has negative "power" close to the frequencies of the m=-2, indicating that $\alpha_{3}$ is negative. There is some indication that $\alpha_{1}$ may be negative. The upper panel (for m'=2) has 2 patches of negative "power" close to the frequencies of m=-2 and m=+2, showing that $\alpha_{4}$ is also negative. These 2 patches are clearly separated because of the mode splitting. (Top, right) For m=-1. Here it is obvious that $\alpha_{2}$ is positive. (Opposite) For m=0. There is now 2 patches of negative power in 2 opposite panels (m'=-2 and m'=2) due to the negativity of $\alpha_{3}$. It is likely that $\alpha_{1}$ is also negative

  

Figure 7: Real part of the cross echelle diagrams of 1 year of LOI data for l=1. The scale is in ppm2$\mu$Hz-1. The spacing is tuned for l=1 ($\Delta\nu$ = 134.8 $\mu$Hz). The inverse of the theoretical leakage matrix has been applied to the original data with $\alpha$ = 0.474. (Left) For l=1, m=-1. The artificial correlation between m=-1 and m=+1 has been entirely removed (See Fig. 5 for comparison). (Right) For l=1, m=0. There is no improvement as there was no correlation before

  

Figure 8: Real part of the cross echelle diagrams of 1 year of LOI data for l=2 after having applied the inverse of the leakage matrix. The scale is in ppm2$\mu$Hz-1. The spacing is tuned for l=2 ($\Delta\nu$ = 135.1 $\mu$Hz). (Top, left) For m=-2. (Top, right) For m=-1. (Opposite) For m=0. Using the inverse of the leakage matrix we have removed all artificial correlations (See Fig. 6 for comparison). The spectra have been cleaned both from the m leaks and from l=0 modes because we used the leakage matrix of l=2 and l=0 only; the leakages from all the other degrees were ignored

3.3.2 GONG data

The leakage matrices of the GONG data have been computed by R. Howe (1996, private communication). We also computed similar leakage matrices using the equations developed in Part I. The integration was made in the $\theta,\phi$ plane with $\theta_{\rm max}=65.2^{\circ}, \phi_{\rm max}=53.5^{\circ}$ without apodization. We also took into account the effect of subtracting the velocity of the Sun seen as a star which affects GONG instrument's sensitivity to modes only detected by integrated sunlight instrument (mainly the modes for which l+m is even). The theoretical leakage matrices of l=1 and 2 for GONG are also given by Eq. (11) but with: $\alpha=-0.55$, $\alpha_{1}=-0.268$, $\alpha_{2}=0.451$, $\alpha_{3}=-0.122$, and $\alpha_{4}=-0.290$. For l=1, the leakage between m=-1 and m=+1 is negative due to the subtraction of the mean velocity which affects these modes. For l=2, the leakage between m=-1 and m=+1 has about the same value as for the LOI; these modes are not affected by the subtraction.

The cross echelle diagrams of the l=2 GONG data are very close to those of the LOI (See Fig. 6). Figure 9 shows an example of a GONG cross echelle diagram after having applied the inverse of the leakage matrix. It is clear that the p-mode correlations are removed. For l=1 we have used the cross echelle diagram of m=-1 and m=1 for inferring quantitatively the off-diagonal leakage element $\alpha$. We first applied the inverse of a leakage matrix to the l=1 data, and then constructed the cross echelle diagram of m=-1 and m'=1 for these data. We then collapsed this diagram by adding up all the modes with n=10-26, and finally we corrected the collapsed diagram from the solar noise background. The collapsed diagram is shown in Fig. 10 (Left) for no correction ($\alpha=0$) and for an $\alpha$ of -0.53; the corrected surface of the collapsed diagram as a function of $\alpha$ is shown in Fig. 10 (Right). When the corrected surface is close to 0, there is no artificial correlation remaining.

Cleaning the data from artificial correlations has also been done in a different way by Toutain et al. (1998). Using Singular Value Decomposition, they recomputed pixel filters for the MDI/LOI proxy so as to remove the m leaks and the other aliasing degrees. Here we have shown, that data cleaning is also possible without having the pixel time series, but using the Fourier spectra. This latter cleaning technique is more useful as one has to reduce a smaller amount of data.

  

Figure 9: Real part of the cross echelle diagrams for 360 days of GONG data for l=2 after having applied the inverse of the leakage matrix. The scale is in m$^{2}\,$s$^{-2}\,$Hz-1. The spacing is tuned for l=2 ($\Delta\nu$ = 135.1 $\mu$Hz). (Top, left) For m=-2. (Top, right) For m=-1. (Opposite) For m=0. As for the LOI data the artificial correlations are removed. The remaining correlations are artifact due to the statistical fluctuations that are enhanced when the signal-to-noise ratio is low as for m=0

  

Figure 10: (Left) Collapsed diagram of the real part of the cross spectrum of m=-1 and m=+1 for the GONG data after having applied the inverse of the leakage matrix, (top) no correction, (bottom) $\alpha =-0.53$. (Right) Surface of the collapsed diagram as a function of $\alpha$, the surface is 0 for $\alpha =-0.53$

3.4 Leakage matrix measurement for many degrees

3.4.1 LOI/SOHO data

Figure 11 shows the inter echelle diagram for l=1, l'=6. This diagram is more difficult to interpret, but using specific spacing for l or l', one can unambiguously identify which degree contributes to the correlation. For the LOI data, the l=5 and l=8 (similarly l=4 and l=7) are strongly contaminated by each other. This is again due to the spatial undersampling which prevents the LOI data to be cleaned from aliasing degrees making the leakage matrix not invertible. Fortunately, the l=1 LOI data are far less affected by the presence of l=6 compared with that of GONG (See in the next section); this is the result of an effective apodization function (limb darkening) which creates a narrower spatial response than that of GONG (line-of-sight projection).

  

Figure 11: Real part of the inter echelle diagrams for 1 year of LOI data for l=1 and l'=6. The scale is in ppm2$\mu$Hz-1. The spacing is tuned for l'=6 ($\Delta\nu$ = 136.7 $\mu$Hz). (Left) For l=1, m=-1. Given the structure of the ridges, it means that the l=1 modes leak into the even m modes of l'=6, while the l'=6 modes do not leak into the l=1. (Right) For l=1, m=0. Again using the structure of the ridges, the odd m' of l'=6 leaks weakly into l=1, m=0

Figure 8 has already shown that we could clean the l=2 LOI data from the l=0 mode. This was done by taking into account the leakage matrix $\tens{\cal{C}}^{(0,2)}$. Unfortunately as mentioned above for the $l \ge 4$, the LOI data cannot be cleaned from aliasing degrees because the leakage matrice $\tens{\cal{C}}^{(1,6)}$, $\tens{\cal{C}}^{(4,7)}$ and $\tens{\cal{C}}^{(5,8)}$ are singular.

3.4.2 GONG data

Figure 12 shows the inter echelle diagram for l=1, l'=6. The correlations are clearly visible. If they are not taken into account they are likely to bias the frequency and splittings of the l=1 (See Rabello-Soares & Appourchaux 1998, in preparation). By using the inverse of the leakage matrix of l=1,6 and 9 (i.e. $\tens{\cal{C}}^{(1,6,9)}$), one can clean the data from these spurious correlations arising both from the two aliasing degrees (l=6 and 9) and from the m leaks. Figure 13 shows the "cleaned" inter echelle diagram which should be compared with Figs. 12. The correlations have been almost entirely removed. Some correlations between the l=1, m=-1 modes and the l=6 modes are still present. This is due to the fact that the computation of the leakage for $l=1, m=\pm1$ are very sensitive to the subtraction of the velocity of the Sun seen as a star. This is not the case for the l=1, m=0 modes as they are insensitive to this effect. Nevertheless, our imperfect knowledge of the leakage matrix can be adjusted in order to remove the residual correlations. It helped to reduce the systematic errors of the l=1 splitting at high frequencies, mainly where the l=6 and 9 modes start to cross over the l=1 modes (Rabello-Soares & Appourchaux 1998, in preparation).

  

Figure 12: Real part of the inter echelle diagrams of 1 year of GONG data for l=1 and l'=6. The scale is in m$^{2}\,$s$^{-2}\,$Hz-1. The spacing is tuned for l'=6 ($\Delta\nu$ = 136.7 $\mu$Hz). (Left) For l=1, m=-1. The strongest correlations are for $m'=\pm 6$ due to l'=6. (Right) For l=1, m=0. The l=6 modes are strong in all the panels. For $m'=\pm 1$ other modes can be seen as an "asterisk" ridge: the ridge going from bottom right to upper left represents the l=1 modes; the ridge going from bottom left to upper right represents the l=9 modes; the vertical ridge represents the l=6 modes. At the far left, the ridge parallel to that of the l=1 represents the l=3 modes

  

Figure 13: Real part of the inter echelle diagrams of 1 year of GONG data for l=1 and l'=6 after having applied the inverse of the leakage matrix $\tens{\cal{C}}^{(1,6,9)}$. The scale is in m$^{2}\,$s$^{-2}\,$Hz-1. The spacing is tuned for l'=6 ($\Delta\nu$ = 136.7 $\mu$Hz). (Left) For l=1, m=-1. The faint yellow oblique ridges are due to the l=1 modes. (Right) For l=1, m=0. The correlations due to the l=1 and 6 modes are almost entirely removed

It is possible to apply this cleaning technique to higher degree modes ($l \approx 50-100$. In principle, this is feasible, although the plethora of data involved may be fairly substantial. For higher degrees, the ridges of the modes in the $(m,\nu)$ echelle diagrams are almost parallel to each other. The idea would be not to use the full Fourier spectra, as we did for the low degree GONG data, but to use only a small frequency range of about $\pm$ 30 $\mu$Hz around the target degree. By doing so, we will not only clean the data from the aliasing degrees but also from the m leaks. In this case, the mode covariance matrix is diagonal. If we assume, wrongly, that the noise covariance matrix is diagonal, the statistics of each cleaned spectrum, for high degree modes, could be approximated by a $\chi^{2}$ with 2 degree of freedom. Neglecting the off-diagonal elements of the noise covariance matrix will lead to larger mode linewidth, thereby producing underestimated splittings; this systematic bias will decrease as the degree. This approximation will produce much less systematic errors than the approximation used by Hill et al. (1996) for the original data. The use of the suggested approximation for the GONG data may help, at the same time, to reduce systematic errors, and to increase computing speed for higher degree modes. The degree where one needs to switch between this approximation and the correct analysis needs to be determined.

3.5 Noise covariance matrix measurement

3.5.1 LOI/SOHO data

The leakage and ratio matrices have similar properties (See Part I). For example, for a given l there is no correlation between the m for which m+m' is odd for either matrix. As outlined in Part I, for a given degree, the ratio matrix is also close to the leakage matrix. For example, Fig. 14 shows that for the LOI data the measured ratio matrix element between m=-1 and m=+1 is about +0.55 which is rather close to the $\alpha=0.474$ given in Eq. (11). The ratio matrix is also useful if one wants to reduce the number of noise parameters to be fitted. This is useful when the noise background, mainly of solar origin, varies slowly over the p-mode range. In this case the ratio can be assumed to be constant over the p-mode range. For example for l=1 we can fit 2 noise parameters instead of 3. When the noise varies in the p-mode range, it is more advisable to fit as many noise parameters as required.

Figure 15 shows the cross spectrum ratios of l=2 for the LOI data. As this is commonly the case for the LOI, the ratios are independent of frequency. In addition, as outlined in Part I, the ratio matrix is very close to the leakage matrix. Using the values of $\alpha$ given for the LOI in Eq. (11) and Fig. 15, one can see that this is the case for the LOI. The ratio matrix has even the same symmetry property as the leakage matrix (not shown here). In the case of the LOI, we sometimes use the independence of the ratio with frequency to reduce the number of free parameters. It is less straightforward to measure the noise correlation for the GONG data than for the LOI data. We recommend to measure it in between the p modes because that is what the fitting routines will determine.

3.5.2 GONG data

Figures 14 and 15 show the cross spectrum ratios of l=2 for the GONG data. The ratio matrices (as the leakage matrices) are not symmetrical. Figure 14 shows also the effect of the subtraction of the full disk integrated velocity in the GONG data. In this case, the correlation is extremely high (about -0.8) and very different from the $\alpha=-0.55$ computed for GONG. Apart from the l=1, these matrices tend to be very close to those of the leakage matrices (see previous section). It is also clear that the ratios depend upon frequency, probably due to the effect of mesogranulation affecting the spatial properties of the noise with frequency. In this case, the dependence of the noise correlation upon frequency has to be taken into account in the fit either by adding free parameters or by measuring these correlations in the cross spectrum ratios (see Rabello-Soares & Appourchaux 1998, in preparation).

  

Figure 14: Real part of the cross spectrum ratios for l=1 smoothed to 10 $\mu$Hz. (Left) LOI data: for m=-1 and m' = 1. The noise correlation is about 0.55. (Right) GONG data: for m=-1 and m=1. The noise correlation is about -0.77. The large anti-correlation is due to the subtraction of the full-disk velocity which tends to reduce the amplitude of the $m=\pm 1$ modes

  

Figure 15: Real part of the cross spectrum ratios of l=2 smoothed to 10 $\mu$Hz. (Left) For the LOI data: top, for m=-2 and m'=0. The noise correlation is about -0.45; bottom, for m=-1 and m=1. The noise correlation is about 0.60. (Right) For the GONG data: top, for m=-2 and m'=0. The noise correlation is rather small and about -0.10; bottom, for m=-1 and m'=1. The typical value is about 0.4