Up: Error due to atmospheric
Subsections
In this section, solar images with similar characteristics as those recorded
at the astrolabe through the atmospheric turbulence are simulated. The optical
response of the whole system atmosphere and instrument is then calculated
using a fractal model to generate the randomly perturbed wavefronts. It is
used to build solar images in various seeing conditions and with different
exposure times. Let us first present the synthetic image of the Sun which is
needed for the simulation.
Several models of the solar limb darkening function exist. The model we use to
build the synthetic image of the Sun is given by (Allen 1973):
| ![\begin{eqnarray}
O_{\lambda}(\alpha)=O_{\lambda}(0)\left[ 1-u_{2}-v_{2}+u_{2}\cos
(\alpha)+v_{2}\cos^{2}(\alpha)\right] \end{eqnarray}](/articles/aas/full/1999/13/ds1705/img3.gif) |
(1) |
where
is the intensity of the solar continuum at an
angle
from the centre of the disk.
represents the angle
between a Sun's radius vector and the line of sight.
corresponds to
the centre of the disk and
to the solar edge. The
ratio
, which varies with the
wavelength
defines the limb darkening. For the used observation
wavelength (
m), the constants u2 and v2 are respectively equal to 0.93 and
-0.23.
The plot in
Fig. 2a shows the solar limb darkening function where
the intensity was normalized between 0 and 255 for our needed calculations.
The Fig. 2b
represents a solar image sample where the field size is equal
to 128 by 128''2.
![\begin{figure}
\includegraphics [height=1.7331in,width=3.2396in]{ds1705f2.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg12.gif) |
Figure 2:
Solar limb darkening function a)
and the synthetic image of the sun b) |
The psf of the whole system instrument and atmosphere is needed to generate
the solar image sequences as recorded through the atmosphere in various
observation conditions. We present first, the fractal model used to generate
the two-dimensional randomly perturbed wavefronts and then the deduced optical responses.
The turbulence follows a fractal structure. The model used to generate the
randomly perturbed wavefront function
is based on a mid point
displacement algorithm (Lane et al. 1992; Beaumont et al.
1996). To simulate a L by L phase screen, the first step is to choose
three points
,
and
disposed at the summits of
an isoceles triangle with base and height equal to 2L (Fig. 3). Phase
function values at these points are Gaussian random variables with zero means
and with variances given by:
|  |
(2) |
and
|  |
(3) |
where
| ![\begin{eqnarray}
\left[1 -1.485\left( \frac{r}{\mathit{L}_{0}}\right)
^{\frac{1}...
...6.281\left(
\frac{r}{\mathit{L}_{0}}\right) ^{\frac{7}{3}}\right] \end{eqnarray}](/articles/aas/full/1999/13/ds1705/img20.gif) |
(4) |
denotes a mean average.
is the phase structure
function deduced from the Kolmogorov's law and the Von Karman model (Ziad
1993). r0 is the Fried's parameter which qualifies the observations
(seeing) (Fried 1966) and
the spatial coherence outer scale
of the wavefronts (Borgnino 1990).
![\begin{figure}
\includegraphics [
height=2.38in,
width=2.8634in
]{ds1705f3.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg23.gif) |
Figure 3:
The fractal interpolation grid used to generate the
perturbed
wavefronts |
Equations (2), (3) and (4) show that
,
and
follow the theoretical phase structure function deduced from the Kolmogorov
turbulence law since:
Three new points
,
and
of the phase screen
are generated midway between each existing points by a process of linear
interpolation and addition of independent random variables
,
and
, such as:
with variances given by:
and
where
is a semi-empirical coefficient equal to 0.25.
The iterative procedure interpolation/displacement is then repeated in each
new generated triangle until the desired numbers of samples are obtained. The
needed L by L phase screen is extracted from the middle of the base of the
original isoceles triangle (Fig. 3).
The Fig. 4 shows the simulation of a
perturbed wavefront sample using the model. It is simulated for a Fried's
parameter r0 equal to 4 cm and a spatial coherence outer scale supposed infinite.
![\begin{figure}
\includegraphics [
height=7.3411cm,
width=9cm
]{ds1705f4.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg35.gif) |
Figure 4:
Perturbed wavefront sample generated for
r0= 4 cm and infinite |
![\begin{figure}
\includegraphics [
height=3cm,
width=14cm
]{ds1705f5.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg36.gif) |
Figure 5:
Speckle images obtained from the fractal model. The image field is
equal to 64 by 64''2 and the Fried's
parameter equal respectively to 2 a), 4 b), 6 c)
and 10 cm d) |
The psf S of the whole system telescope and atmosphere is given by the
square modulus of the Fourier Transform (FT) of the complex amplitude
limited to the pupil area:
| ![\begin{eqnarray}
S(\overrightarrow{\mathbf{\theta}})=\left[ FT_{\rm pupil}\left(...
...i\left(
\overrightarrow{\mathbf{\xi}}\right) \right) \right] ^{2} \end{eqnarray}](/articles/aas/full/1999/13/ds1705/img38.gif) |
(5) |
where

Equation (8) assumed the near field approximation (Roddier 1981).
is the turbulent phase generated with the fractal model,
is the angular frequency vector considered in the pupil plane
and
the angular coordinate vector in the
focal plane.
corresponds to an image of speckles. In
Fig. 5 are represented four typical short exposure speckle images obtained
using the procedure developed in the previous section. They are simulated for
Fried's parameters r0 equal respectively to 2, 4, 6 and
10 cm
and with a pupil aperture D equal to
10 cm. The spatial coherence outer scale of the used randomly perturbed wavefronts
has an infinite value. Large boiling and displacements of the speckle
centroids are observed for small values of r0 while for r0 equal to
the pupil aperture, the speckle image is similar to the diffraction pattern of
the instrument.
![\begin{figure}
\includegraphics [
height=3.5cm,
width=8.5cm
]{ds1705f6.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg43.gif) |
Figure 6:
Simulated solar images recorded with a short exposure time and
r0 equal to 3 a)
and 7 cm b). Solar limbs extracted from images
recorded in these seeing conditions (dashed line: r0 = 3 cm and solid
line: r0 = 7 cm) c) |
We are then able to simulate psf's of the whole system telescope and
atmosphere in various seeing conditions and also for different integration
times (average of short exposure samples).
Solar images as recorded through the terrestrial atmosphere are simulated.
Assuming isoplanatism, a solar image is obtained by the convolution of the
synthetic image of the Sun (see Sect. 2.1) with the psf of the imaging
system (atmosphere and telescope):
|  |
(6) |
where
is the convolution symbol and
the isoplanatic angle.
We use this formalism to build solar images recorded in various seeing
conditions and with different CCD integration times. The spatial coherence
outer scale of the generated perturbed wavefronts is infinite. The pupil
aperture D and the observation wavelength are respectively taken equal to
10 cm
and
. These kind of images will be useful in the next section to develop the
error study on solar diameter measurement. They have similar properties as
those recorded with the solar astrolabe (Laclare et al. 1996;
Laclare & Merlin 1991).
In this case, solar images are simulated using various Fried's parameters
r0 (Fig. 6) but integration times short enough to freeze the turbulent
motion of the atmosphere i.e. the integration time t is less than the
characteristic evolution time of the atmosphere
(Aime et al.
1986). For a given simulated image sequence, Fried' s parameter r0
remains constant but images are obtained with independent psf's.
![\begin{figure}
\includegraphics [
height=3.5cm,
width=8.5cm
]{ds1705f7.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg51.gif) |
Figure 7:
Simulated solar image recorded in same seeing conditions (r0 = 5
cm) but with different exposure times ( a) and b)). Solar limbs extracted from images recorded with exposure time
respectively lower (solid line) and higher (dashed line) than the atmospheric
characteristic time c) |
Long exposure solar images are obtained by the average over a great number of
short exposure frames. The integration time t of the long exposure images is
then equal to
where n is the number of averaged images and
the atmospheric characteristic time. Figure 7 shows the simulated solar images
recorded in same seeing conditions (
) but with different integration times.
![\begin{figure}
\includegraphics [
height=5cm,
width=17cm,
]{ds1705f8.eps}\end{figure}](/articles/aas/full/1999/13/ds1705/Timg54.gif) |
Figure 8:
Analysis of an image sequence composed of short exposure frames
recorded for a Fried's parameter equal to 3 cm. a) Typical simulated solar
limb extracted from a solar image. b) Second derivative of the solar limb
function and definition of the inflexion point.
c) Reconstructed trajectories
of both direct and reflected solar images approximated by straight lines using
least-square fits |
Up: Error due to atmospheric
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