|Figure 7: Contour plot of the relative SNR for various binary star magnitude combinations. Contours are at 10% the peak level. The values decrease from left to right|
However, if one is collecting data across the whole telescope pupil using a telescope with a large obstruction ratio, one also has to take into account the effect of the pupil transfer function upon the SNR within the correlation plane. This is particularly so if one is operating in the generalised SCIDAR mode since one is then imaging the scintillation pattern viewed through two overlapping pupils. The degree of pupil overlap is directly proportional to both the angular separation of the binary and the effective distance below the telescope pupil one defocuses to.
|Figure 8: Example transfer functions of overlapping pupils, for the generalised SCIDAR case, as a function of altitude. Curves are given for binary star separations of 2, 4, 6 & 8 arcseconds and assume a telescope diameter of 1 m and 1 cm pix-1 image plane sampling|
These restrictions upon binary star separation are shown graphically in Fig. 9 for an assumed pupil sampling of 1 cm pix-1, a telescope pupil diameter of 1 m and a conjugate plane distance of 3.5 km.
|Figure 9: Plot of altitude resolution as a function of binary separation showing the limits imposed for a reasonable altitude sampling (top) and a maximum altitude reached of 25 km above image plane (bottom), for a 1 m telescope assuming 1 cm pix-1 image plane sampling|
It must be noted that both the upper limit upon and the SNR within the correlation plane can be increased, if using a telescope of a large enough diameter (e.g. D > 3.5 m) to enable an artificial circular pupil greater than 1m to be imposed within the image of the segment of the mirror observed. In this case, assuming a similar pupil sampling as previously, the restriction upon then only originates from the size of the CCD, such that in our case .
To enable this we define a ``solution plane", which contains both the true
(and unknown) profile and the best-fit solution
profile, , and a ``data plane", within which
resides both the true (observed) data, , and the correlation
function obtained from the profile,
A schematic outline of this representation is shown in Fig. 10.
|Figure 10: Schematic description of the method used to estimate the RMS error on any particular Cn2(h) profile obtained|
The approach taken here is to numerically perturb the profile about its position within the solution plane by incremental distances by applying random Gaussian noise, until the RMS distance within the data plane between its corresponding correlation function and is equal to the RMS distance . When this occurs the RMS distance between and its perturbed value can then be considered as describing the radius of a circle within the solution plane upon which the ``true" value of Cn2(h) lies and therefore represents a reasonable approximation of the RMS error .
In doing so one is assuming that the solution is close to the global minimum within the solution plane, or at least within a local minimum which is not very different from the true solution. This assumption is justified both by the relatively low value of we achieve prior to termination of the iterative routine and also by the strongly linear relationship that has been found in all cases between and as the degree of perturbation applied to is increased.
Having performed this calculation for a number of different Cn2(h) profiles, it has been found that the value of is greater than the value of in most cases. For example for the case shown in Fig. 5, and are found to be 4.3% and 7.2% respectively. This highlights the need for high SNR data for a reliable Cn2(h) solution.
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