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Subsections

2 Diagnostics for helioseismic data analysis

The echelle diagram was first introduced by Grec (1981). It is based on the fact that the low-degree modes are essentially equidistant in frequency for a given l; the typical spacing for l=0 is 135 $\mu$Hz. The spectrum is cut into pieces of 135 $\mu$Hz which are stacked on top of each other. Since the modes are not truly equidistant in frequency, the echelle diagram shows up power as distorted ridges; an example is given in Fig. 1 for the LOI/SOHO instrument seeing the Sun as a star.

  
\begin{figure}
\centerline{
\psfig {file=ds7200f1.ps,width=7.2cm,angle=90}
}\end{figure} Figure 1: Amplitude spectra echelle diagram for 1 year of LOI data seeing the Sun as a star. The scale is in part-per-million$\mu$Hz-1/2 (ppm$\mu$Hz-1/2). The spacing is tuned for l=0 ($\Delta\nu$ = 134.8 $\mu$Hz). The l=0 modes are at the center, the l=2 modes are about 10 $\mu$Hz on the left hand side, the l=1 modes are about +65 $\mu$Hz on the right hand side, the l=3 can be faintly seen at about 12 $\mu$Hz from the left hand side of the l=1. Other modes such as l=4 and l=5 can also be seen faintly seen at -35 $\mu$Hz and +15 $\mu$Hz, respectively. The distortion of the ridges are due to sound speed gradients in the solar core
Another useful diagram was introduced by Brown (1985), the so-called $(m,\nu)$ diagram which shows how the frequency of an l,m mode depends upon m. Most often this diagram is only shown for a single n and for intermediate degrees $l \ge 10$.

The purpose of these diagrams is always to show an estimate of the variance of the spectra. In our case we also want to visualize not only the variance but also the covariance of the Fourier spectra. Here we briefly recall from Part I that the observed Fourier spectra ($\vec{y}$) can be related to the individual Fourier spectra of the normal modes ($\vec{x}$) by the leakage matrix $\tens{\cal{C}}^{(l,l')}$ by:
\begin{displaymath}
\vec{y}=\tens{\cal{C}}^{(l,l')}\vec{x}\end{displaymath} (1)
where $\vec{x}(\nu)$ and $\vec{y}(\nu)$ are 2 complex vectors made each of 2l+2l'+2 components: 2l+1 components for l, 2l'+1 components for l'. Here only two l values have been used for simplicity, there is no difficulty to extend the expression of the leakage matrix for a number of degrees greater than 2. The covariance matrix $\tens{V}^{(l,l')}_{m,m'}$ of $\vec{z}_{\vec{y}}$ ($\vec{z}_{\vec{y}}^{\rm 
T}=(\mbox{Re}(\vec{y}^{\rm T}),\mbox{Im}(\vec{y}^{\rm T}))$), can be derived from the sub-matrix $\tens{\cal{V}}^{(l,l')}$ whose elements can be expressed as:
   \begin{eqnarray}
2\tens{\cal{V}}^{(l,l')}_{m,m'}=E[y_{l,m}(\nu)y_{l',m'}^{*}(\nu...
 ...,m''}^{(l,l'')*}f_{m''}^{l''}(\nu)+2\tens{\cal{B}}_{m,m'}^{(l,l')}\end{eqnarray}
(2)
where E is the expectation, $f_{m''}^{l''}(\nu)$ is the model of the line shape of the power spectrum of the (l'',m'') mode, $\tens{\cal{B}}^{(l,l')}$ is the covariance matrix of the noise, and with the $y_{l,m}(\nu)$ having a mean of zero. The factor 2 comes from the fact that the real part of $\tens{\cal{V}}^{(l,l')}$ represents both the covariance of the real or imaginary parts of the Fourier spectra (See Part I, Sect. 3.3.2); the same property applied to the imaginary part of $\tens{\cal{V}}^{(l,l')}$ which represents the covariance between the real part and the imaginary part of the Fourier spectra. Equation (2) contains all the information that we need for visualizing an estimate of the real and imaginary parts of $\tens{\cal{V}}^{(l,l')}$. Drawing from the usefulness of the diagrams of Grec and Brown, we created four new diagrams for visualizing an estimate of $\tens{\cal{V}}^{(l,l')}$, all having various diagnostics power:

1.
$(m,\nu)$ echelle diagram: estimate of the diagonal elements of $\tens{\cal{V}}^{(l,l)}$ (l=l')
2.
cross echelle diagram: estimate of the off-diagonal elements of $\tens{\cal{V}}^{(l,l)}$ (l=l')
3.
inter echelle diagram: estimate of the off-diagonal elements of $\tens{\cal{V}}^{(l,l')}$ ($l \neq l'$)
4.
cross spectrum ratio: estimate of the ratio of the elements of $\tens{\cal{B}}^{(l,l')}$.
Each diagnostic is described hereafter in more detail.

2.1 $(m,\nu)$ echelle diagram

The $(m,\nu)$ echelle diagram is made of 2l+1 echelle diagrams of each l,m power spectra or $\vert y_{l,m}(\nu)\vert^{2}$. The 2l+1 echelle diagrams are stacked on top of each other to show the dependence of the mode frequency upon m. These diagrams give an estimate of the diagonal of the covariance matrix of the observations as:
\begin{displaymath}
2\widetilde{\tens{\cal{V}}}_{m,m}^{(l,l)}(\nu)=\vert y_{l,m}(\nu)\vert^{2} \end{displaymath} (3)
where $\widetilde{\tens{\cal{V}}}^{(l,l)}$ symbolizes the estimate of $\tens{\cal{V}}^{(l,l)}$. It is important when one makes these diagrams to tune the spacing for the degree to study. The spacing for a given l can be computed from available p-mode frequencies. Since the spacing varies with the degree, 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.

Nevertheless, the diagnostics power of the $m,\nu$ echelle diagram is rather limited for deriving the leakage matrix: it can be shown using Eq. (2) that the diagonal elements of $\tens{\cal{V}}^{(l,l)}$ can be expressed as:
   \begin{eqnarray}
\tens{\cal{V}}^{(l,l)}_{m,m}=\sum_{m''=-l}^{m''=l} 
\vert\tens{...
 ...''}^{(l,l)}\vert^{2}f_{m''}^{l}(\nu)+\tens{\cal{B}}_{m,m}^{(l,l)}.\end{eqnarray} (4)
As we can see with Eq. (4), the sign information of the elements of $\tens{\cal{C}}^{(l,l)}$ is lost; second, their magnitude being typically less than 0.5, the leakage elements cannot be easily seen in the power spectra. Another kind of diagram that preserves the sign of the leakage elements had to be devised.

2.2 Cross echelle diagram

The cross echelle diagram of an l,m mode is made of 2l+1 echelle diagrams of the cross spectrum of m and m' or $y_{l,m}(\nu)y_{l,m'}^{*}(\nu)$. The 2l+1 real (or imaginary) parts of the cross spectra are stacked on top of each other to show the dependence upon m of the mode frequency. These diagrams give an estimate of the rows (or columns) of the covariance matrix of the observations as:
\begin{displaymath}
2{\widetilde{\tens{\cal{V}}}_{m,m'}^{(l,l)}(\nu)}=y_{l,m}(\nu)y_{l,m'}^{*}(\nu).\end{displaymath} (5)
Of course when m=m' we get the echelle diagrams of the previous section. Only l+1 cross echelle diagrams are shown as the matrix $\tens{\cal{V}}^{(l,l')}$ is hermitian by definition.

The imaginary part of the cross spectra has some diagnostic power: it represents the correlation between the real and imaginary parts of the Fourier spectra. When the leakage matrix is real, which is generally the case, there is no correlation between the real and imaginary parts. Nevertheless the imaginary part could be helpful to find errors in the filters applied to the images (See Part I, Sect. 3.3.1).

It can be shown that the elements of $\tens{\cal{V}}^{(l,l)}$ can be expressed as:
   \begin{eqnarray}
\tens{\cal{V}}_{m,m'}^{(l,l)}=\tens{\cal{C}}_{m,m'}^{(l,l)}f_{m...
 ..._{m,m''}^{(l,l)*}f_{m''}^{l''}(\nu)+\tens{\cal{B}}_{m,m'}^{(l,l)}.\end{eqnarray}
(6)
As we can see with Eq. (6), these diagrams preserve the sign of the leakage matrix elements. In general, the cross spectra for m,m', representing $\tens{\cal{V}}_{m,m'}^{(l,l)}$ carries information over the sign of the leakage elements $\tens{\cal{C}}^{(l,l)}_{m,m'}$ and $\tens{\cal{C}}^{(l,l)*}_{m',m}$. The other additional terms expressed as product of leakage elements are sometimes more difficult to interpret.

But the power of these diagrams is not only restricted to checking the sign of the elements of the leakage matrix. They are also real tools to get a first order estimate of the leakage matrix. We have shown in Part I, that there is no difference between fitting data for which the leakage matrix is the identity, and data for which the leakage matrix is not . We showed in Part I, that the covariance matrix of $\vec{\tilde{x}}$ ($\vec{\tilde{x}}=\tens{\cal{C}}^{-1}\vec{y}$) can be written, similarly to that of Eq. (2)), by the sum of 2 matrices: the first one represents the mode covariance matrix and is diagonal, and the second term represents the covariance matrix of the noise. Therefore, applying the inverse of the leakage matrix to the data should, in principle, remove all artificial mode correlations between the Fourier spectra of $\vec{\tilde{x}}$: this can be verified using the cross echelle diagrams. This is the most powerful test for deriving the leakage matrices.

The cross echelle diagrams are useful to verify the correlation within a given degree, but other degrees are known to leak into the target degree, such as l=6, 7 and 8 into l=1, 4 and 8, respectively. The purpose of the next diagram is to assess the magnitude of these leakages.

2.3 Inter echelle diagram

The inter echelle diagram of an l,m mode for the degree l' is made of 2l'+1 echelle diagrams of the cross spectrum of l,m and l',m' or $y_{l,m}(\nu)y_{l',m'}^{*}(\nu)$. The 2l'+1 real part of the cross spectra are stacked on top of each other to show the dependence upon m'. These diagrams give an estimate of the rows (or columns) of the covariance matrix of the observations as:
\begin{displaymath}
2{\widetilde{\tens{\cal{V}}}^{(l,l')}_{m,m'}(\nu)}=y_{l,m}(\nu)y_{l',m'}^{*}(\nu).\end{displaymath} (7)
Similarly as for the cross echelle diagram, it will help to visualize the covariance matrix between different degrees, and to derive leakage elements of the full leakage matrix $\tens{\cal{C}}^{(l,l')}$. One can derive an equation similar to Eq. (6) for different degrees showing that the inter echelle diagram carries information over the sign of the leakage elements $\tens{\cal{C}}^{(l,l')}_{m,m'}$ and $\tens{\cal{C}}^{(l',l)*}_{m',m}$.

As mentioned above applying the inverse of the leakage matrix will help to verify to the first order that there is no artificial correlation due to the p modes. When different degrees are involved the full leakage matrix $\tens{\cal{C}}^{(l,l')}$ has to be used, producing diagrams that should have no artificial correlation due to the p modes.

2.4 Cross spectrum ratio

All the previous diagrams are helpful to understand and visualize the mode covariance matrices. Unfortunately, due to the high signal-to-noise ratio, these diagrams cannot be used to evaluate the correlation of the noise in the Fourier spectra. In between the p modes, these correlations can be more easily visualized as we have:
   \begin{eqnarray}
\tens{\cal{V}}_{m,m'}^{(l,l)}\approx\tens{\cal{B}}_{m,m'}^{(l,l)}.\end{eqnarray} (8)
But instead of visualizing $\tens{\cal{B}}^{(l,l')}$, we prefer to look directly at the correlation by computing the ratio of the cross spectra as:
\begin{displaymath}
\widetilde{\frac{\tens{\cal{B}}^{(l,l')}_{m,m'}}{\tens{\cal{...
 ..._{l,m}(\nu)y_{l,m'}^{*}(\nu)}{\vert y_{l,m}(\nu)\vert^{2}}\cdot\end{displaymath} (9)
This ratio is called the cross spectrum ratio . The cross spectrum ratio gives an estimate of the ratio matrix $\tens{\cal{R}}^{(l,l')}$ (See Part I) which gives a better understanding of how much the noise background is correlated between the Fourier spectra. Nevertheless, in the p-mode frequency range, the cross spectrum ratio is more difficult to interpret as the noise correlation is affected by the presence of the modes. By looking away from the modes (at high or low frequency) or by looking between the modes, one could obtain a reasonable good estimate of the noise correlation.


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