We have performed a number of simulations using the 4-layers fit of
Roggermann et al. (1995) to various models published in the
literature, rescaling them for different values of .
Namely the Greenwood *good seeing* model (Greenwood 1977) is denoted
by G-77; the fit with Maui (Hawaii) U.S. Air Force Optical Station (Miller
et al. 1976) is denoted by SLC-N and the Hufnagel-Valley
model (Hufnagel 1974) with an upper atmospheric wind of 54 miles per hour
() is denoted by HV-54.

We have generated series of 65536 tilts for each of the four layers in a grid of 6 different values for each of the 3 different atmosphere models adopted. For each layer a normal distributed tilt with a dispersion given by (Sarazin & Roddier 1990):

has been calculated and propagated up to the mesospheric Sodium layer.
The effective tilt height has been computed together with the residual
jitter for each of the 65536 time-points taken into consideration.

**Figure 3:** The histograms for and for *D*=8 m,
m in the HV-54 model. The run is composed by 65536 realizations.
As it can be seen follows roughly a Gaussian distribution, while
the distribution is much more platicurtic

As it can be easily seen in Fig. 3 (click here) the distribution of the effective height is far from a Gaussian shape; however the final residual tilt distribution is very well fitted by a Gaussian distribution, so that it can be identified by the single figure of the dispersion . Following Sandler et al. (1994) one can determine the Strehl ratio degradation contribution due only to the problem pointed out in this paper using the relationship:

The results of the simulation are collected in Tables 1 (click here) and 2 (click here) and in
Fig. 4 (click here) for the cases of *D* = 3.58 m and *D* = 8.00 m
telescopes; the fluctuations of the mean effective height
are due to the different realizations of the same multi-layer models.
In order to get a deeper insight on the meanings of the published
numbers it is here recalled that the diffraction limit capabilities for
a nm wavelengths, are respectively 33.9 mas
and 15.2 mas for the two telescope aperture taken into consideration.

**Table 1:** The results for the perfomed simulation in the
*D*=3.58 m case

**Table 2:** The same as for Table 1 (click here), in the *D* = 8.00 m telescope case

**Figure 4:** Behaviour of *SR* vs. for *D* = 3.58 m (upper panel) and
*D* = 8.00 m (lower panel) telescope

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