**Figure 2:**
Schematic diagram of the optical system for SCIDAR. **a)** shows
the optical system for pupil plane SCIDAR. In this mode the telescope
entrance pupil is imaged by the telescope mirror system *L _{1}* and the field
lens

Figure 1 shows that in a simplified one-dimensional
representation of the observing procedure for a binary object observed
at zenith (a), a peak in the correlation plane occurs at a pupil
separation , (b), and thus by
simple geometry, denotes the presence of a single turbulent layer
at an altitude above the telescope pupil.
This simple situation can be generalised to the case where multiple turbulent
layers exist, each one having an associated altitude calculable from the peak
position within the average correlation.
Also for the general case where the binary star is observed at some
zenith angle *z*, then the geometrical conversion above becomes .

In the conventional operation of SCIDAR, the conjugate plane
of the imaging system is
placed at the altitude of the pupil of the telescope - referred to here as
pupil-plane SCIDAR.
However, low altitude turbulence cannot be
determined in this mode since the scintillation requires a large distance over
which to develop a signal strong enough to be detected. Consequently the
correlation signal close to zero altitude is occupied only by a central
core corresponding to the sum of all other higher altitude peaks.
This is shown diagrammatically as the central peak in Fig. 1a.
Recently, however, a solution to this problem was proposed by Fuchs et al.
(1994)
in which one changes the conjugate height of the imaging system from that of the
telescope pupil, to one some distance below it. In effect, this creates an extra
distance for the scintillation pattern to develop to a detectable level
and also shifts the central core to below the pupil (zero) altitude.
Experiments performed by us in this *generalised* SCIDAR mode appear
to confirm the validity of this approach and the results are discussed later
in Sect. 4.

Figure 3:
Example average scintillation pattern image obtained in the generalised
SCIDAR mode at the JKT, in La Palma from 18000 1.6 ms exposure frames. The
scale is in relative intensity units |

For example, for the Jacobus Kapteyn Telescope (JKT) in La Palma, the
telescope has a 1m pupil with an F/15 focal ratio and therefore an ideal
focal length for the field lens (*L _{2}*), to give this required sampling,
would be 159 mm. This would produce a 10.6 mm diameter pupil image at the
intensifier, or equivalently a 2.04 mm pupil image on the CCD.

For the generalised mode shown in Fig. 2b, the field lens (*L _{2}*) is
replaced
by one of a different focal length,

The digital output from the DALSA Camera is acquired by a Bitflow Raptor PCI frame capture board controlled by a fast PC and the data is stored directly onto Digital Linear Tape. The frame exposure times normally used are 1.6 ms and 2.7 ms, with frame acquisition rates of 305 and 188 frames per second respectively.

Figure 3 shows an example of the average scintillation pattern of 18000 frames obtained using this system on the JKT whilst running in the generalised SCIDAR mode. The sheared image of the telescope pupil is clearly visible.

(9) |

(where <

The finite width of these peaks results directly
from the Fresnel length associated with each turbulent layer, given
by (Vernin & Azouit
1983).
This in turn effectively limits the degree of resolution in *C*_{n}^{2}(*h*)
attainable from the SCIDAR data.
For the data collected thus far, ranges from
around 700 m (for a pupil-conjugate distance of say 3 km) to around 2000 m
(for the maximum altitudes considered).
Assuming a pupil plane sampling, , of 1 cm pix^{-1} and
binary star separations between 4-9'', then
the altitude sampling available, given by
,
corresponds to around 200-500 m, and thus represents a reasonable sampling
rate of the Fresnel length at all altitudes.

A slice taken through the correlation plane, parallel to the direction of the binary star, , can then be expressed as (Vernin & Roddier 1975),

(10) |

The log amplitude covariance of the scintillation fluctuations then follows as,

(11) |

(12) |

(13) |

(14) |

The particular method one might employ for this problem (and also to
some extent the confidence of the solution) depends very much upon the
*C*_{n}^{2}(*h*) altitude sampling that is required.
Tests performed upon a number of high resolution *C*_{n}^{2}(*h*) profiles
obtained by balloon-borne instruments (Vernin & Muñoz-Tuñôn
1992)
have shown that the main parameters
which determine AO anisoplanatic performance can be calculated with reasonable
accuracy provided that the altitude sampling, , is less than 1km.
For the data collected thus far, the altitude resolution available is between
around 200-500 m, and
therefore comfortably within this constraint.
However it has been found that a simple least-squares inversion of Eq. (12)
for these data is incapable of producing a reliable solution unless one
rectangularises the two-dimensional integral equation such that multiple pupil
samples
map onto each altitude sample and remains fixed at
1 cm pix^{-1} but is extended to 2-3 km.

Although, ultimately, it is the Fresnel length that limits the altitude
resolution attainable, in an effort to realise the altitude sampling that is
made available
by that within the pupil, a number of iterative inversion methods have been
explored.
One that has proved successful is based upon a maximum entropy routine
which also allows for the controlled propagation of noise between the data and
solution
(Skilling & Bryan 1984).
In practice, we first calculate a crude (large )
least-squares inversion of *C*_{n}^{2}(*h*) using a rectangularised version
of Eq. (12) and then use an interpolated version of this as the first guess
estimate for the iterative routine.
As an aid to deciding when the routine should be terminated, the following
RMS difference error metric is calculated following each iteration:

(15) |

In Fig. 5, (a) shows the correctly scaled correlation slice,
, of Fig. 4a, and in (b) the corresponding
*C*_{n}^{2}(*h*)
profile obtained.
Also shown in Fig. 5a, for comparison is
which gives some indication of the accuracy of
the solution obtained where, in this case, .
It is interesting to note that for the particular example shown here the
apparent
single correlation peak at a height of 7.5 km in Fig. 5a has
been resolved into two distinct turbulent layers within the corresponding
*C*_{n}^{2}(*h*) profile of Fig. 5b at heights of 7.2 km and 8.1 km.

The apparent width of the dominant turbulent layers shown here probably does not
reflect the true physical depth over which the optical turbulence exists.
It is more likely that this apparent spread of *C*_{n}^{2}(*h*) values is due
to the effective resolution imposed by the Fresnel length, as discussed
in Sect. 2.3.1.
Therefore, although it appears that one is oversampling
each peak in the *C*_{n}^{2}(*h*) profile, the suspicion is that in fact the layers
of turbulence themselves could be substantially thinner.

Confidence in the inversion methods used has been obtained by testing them on realistic simulated SCIDAR data generated using a Fresnel wavefront propagation code developed within our group (Adcock et al. 1996).

Figure 6:
a) Simulated spatial correlation data along with b) a
comparison of the model input C_{n}^{2}(h) profile (bar-plot) and that
obtained using our integral inversion method |

Figure 4b shows such a plot generated from the same data as in
Fig. 4a,
with = 23 ms (corresponding to =0.42 ms^{-1}).
The zero velocity position within this plot is denoted at the centre by a
cross.
In comparing this with the spatial-correlation plot of Fig. 4a one
can see that, as indicated by the the *C*_{n}^{2}(*h*) of Fig. 5b, the
apparent
single peak at the greatest distance from the central core has been resolved
into two layers at heights of 7.2 km and 8.1 km with associated velocities
of 14.6 ms^{-1} (wind direction Position Angle, PA = 103) and
9.8 ms^{-1} (PA = 130) respectively.
It is also worth noting that the correlation peak closest to the central core
(corresponding to a height close to the telescope pupil) has shifted only very
slightly from its central position in the spatial-correlation (*V* =
0.01 ms^{- 1}).
This is indicative perhaps of the presence of almost static turbulence within
the telescope dome since the ground wind speed recorded at this time was
13 ms^{-1}.

The algorithm to allow automative calculation of *V*(*h*) is based upon the
assumption that the optical turbulence is
highly discretised (the justification for this is clarified later in
Sect. 4). Having first determined the altitudes of each discrete layer and
the orientation of the binary star observed, a template is then
constructed for each layer, each one representing their individual
spatial correlation signal. The cross-correlation of each of these templates
with the spatio-temporal plot then allows the speed and direction of each layer
to be identified.

Copyright The European Southern Observatory (ESO)