Up: The art of fitting
Subsections
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
Hz. The spectrum is cut into
pieces of 135
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.
 |
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 Hz-1/2
(ppm Hz-1/2).
The spacing is tuned for l=0 ( = 134.8 Hz). The l=0 modes are at the center,
the l=2 modes are about
10 Hz on the left hand side, the l=1 modes are about +65
Hz on the right hand side, the l=3 can be faintly seen at
about
12 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 Hz and +15 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
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
.
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 (
) can be related to the
individual Fourier spectra of the normal modes (
) by the leakage
matrix
by:
|  |
(1) |
where
and
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
of
(
),
can be derived from the sub-matrix
whose elements can be expressed as:
|  |
|
| (2) |
where E is the expectation,
is the model of the line shape of the power spectrum of the (l'',m'') mode,
is the covariance matrix of the noise, and
with the
having a mean of zero. The
factor 2
comes from the fact that the real part of
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
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
. Drawing from the usefulness of the
diagrams of Grec and Brown, we
created four
new diagrams for visualizing an estimate of
, all having
various diagnostics power:
- 1.
echelle diagram: estimate of the diagonal elements
of
(l=l')
- 2.
- cross echelle diagram: estimate of the off-diagonal elements
of
(l=l')
- 3.
- inter echelle diagram: estimate of the off-diagonal elements
of
(
) - 4.
- cross spectrum ratio: estimate of the ratio of the elements of
.
Each diagnostic is described hereafter in more detail.
The
echelle diagram is made of 2l+1 echelle
diagrams of each l,m power spectra or
. 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:
|  |
(3) |
where
symbolizes the estimate of
. 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
diagrams; this is a powerful tool to identify
other degrees.
Nevertheless, the diagnostics power of the
echelle diagram is rather limited for deriving the leakage matrix:
it
can be shown using Eq. (2) that the diagonal elements of
can be expressed as:
|  |
(4) |
As we can see with Eq. (4), the sign information
of the elements of
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.
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
.
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:
|  |
(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
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
can be expressed as:
|  |
|
| (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
carries
information over the sign of the leakage elements
and
. 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
(
) 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
: 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.
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
. 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:
|  |
(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
. 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
and
.
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
has to be used,
producing diagrams that should have no artificial correlation due
to the p modes.
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:
|  |
(8) |
But instead of visualizing
, we prefer to
look
directly at the correlation by computing the ratio of the cross
spectra
as:
|  |
(9) |
This ratio is called the cross spectrum ratio . The cross
spectrum ratio gives an estimate of the ratio matrix
(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.
Up: The art of fitting
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