4 Scidar/Meso-nh statistical comparison

4.1 Model configuration

The spatio-temporal model configuration is the same as described in Paper I. The orographic model, centered above Paranal (70.40 W, 24.61 S), has a 500 m horizontal resolution over a total surface of 60 km $\times$ 20 km, that is 120 $\times$ 40 grid points (Fig. 1 in Masciadri et al. 1998). A physical time step $t_{\rm s}=2.5$ s was used and the following vertical resolution was chosen: an increasing vertical step from 50 m to 245 m between ground level to 3 km and a constant resolution of 600 m up to 20 km.

4.2 Statistical analysis

To test the capacity of the Meso-Nh model to predict seeing, we used Scidar as the reference. Scidar has been tested in many ways, in various sites and compared with other techniques such as DIMM and balloons. Moreover, in Sect. 2 we proved that during the PARSCA93 campaign the correlation between Scidar and DIMM-ESO was good. We compared Scidar and Meso-Nh seeing values during each selected night and we analyzed the correlation between the measurements and simulations. During the worst seeing night we noticed that the seeing increased suddenly after about 3 hours due to the occurrence of a $C_{\rm N}^2 \$ layer at 4 km, also seen by the Scidar. Thus, we decided to make 4 hours simulations for each night and we analyzed all the forecasts ranging between 30 min and 4 hours. The first 30 min are discarded because we verified that, for all the simulations, the flow is not yet adapted to the orography. Being aware that our statistical sample is poor (8 nights only) we tried to extract the most complete information with the available data using different techniques. We used two methods that will be named Method A and Method B. For both methods we calculated a linear regression fit and computed the following statistical estimators:

• The correlation coefficient.

   
  

  $$r=\frac{\sum_{i}x_{i}y_{i}}{\sqrt{\sum_{i}x_{i}^{2}\sum_{i}y_{i}^{2}}}.$$ (1)

  where x_i and y_i are the seeing estimated with Scidar and Meso-Nh.

• The correlation coefficient for centered variables

   
  

  $$r_{\rm c}=\frac{\sum_{i}\left( x_{i}-\overline{x}\right) \le... ...ight) ^{2}\sum_{i}\left( y_{i}-\overline{y}\right) ^{2}}} \cdot$$ (2)

  This gives a more significant estimation of correlation because it is independent of the mean values which are always positive, like seeing.

• The estimator P of the correlation coefficient.

  To estimate the reliability of the correlation coefficient we computed the probability P that two uncorrelated distributions of the analyzed data give a correlation coefficient greater than that observed.

• The standard deviation $\sigma$ between Scidar measurements and the regression straight line is

   
  

  $$\sigma^{2}=\frac{1}{N-1}\sum_{i}\left( y_{i}-ax_{i}\right) ^{2} .$$ (3)

  

Table 2: Method A: statistical analysis summary. $\alpha$ = calibration coefficient, a = regression line slope, r = correlation coefficient, $r_{\rm c}$ = centered correlation coefficient, $\sigma$ = standard deviation of the Meso-Nh distribution data with respect to the regression straight line, P = probability that two uncorrelated distribution x_i and y_i, belonging to the same parent distributions of the analyzed data, give a correlation coefficient larger than that observed

  

Table 3: Method B(I): statistical analysis summary. The first 100 m (first two model levels) are rejected. $\beta$ = calibration coefficient. For a, r, $r_{\rm c}$, $\sigma$ and P, same as Table 2. ${\varepsilon_{\rm Tot}}^{*}$ is identical to the [$2^{\rm h}-3^{\rm h}$] column of Table 2

4.2.1 Method A: High temporal resolution

In this method we compare the seeing deduced from Scidar measurements with the Meso-Nh simulations above Paranal every 2.5 s. In order to have a better estimation of the correct adaptation time, we considered the seeing averaged over different time intervals and we computed the statistical parameters defined before for each interval. In Table 2 we report the statistical results obtained over 6 different time intervals over a complete 4$^{\rm h}$ simulation time: [$1^{\rm h}-2^{\rm h}$], [$2^{\rm h}-3^{\rm h}$], [$3^{\rm h}-4^{\rm h}$], [$1^{\rm h}-3^{\rm h}$], [$2^{\rm h}-4^{\rm h}$] and finally [$1^{\rm h}-4^{\rm h}$]. This method gives good temporal statistics but does not give any information about the vertical structure of the optical turbulence.

4.2.2 Method B: High spatial vertical resolution

As we are interested not only in the seeing prediction but also in the turbulent profile prediction, for each night we compared the $C_{\rm N}^2 \$ averaged profiles from the Scidar with those obtained from the Meso-Nh output every 30 min. So doing, we have direct access to the temporal evolution of $C_{\rm N}^2 \$ profile, that is the optical turbulence evolution at all the model levels.

For each night, we splitted the atmosphere into two regions: we computed the contribution of the boundary layer (BL) defined here between ground level and 5 km and that of the free atmosphere (FA) above 5 km. The same splitting has been used for both Scidar and Meso-Nh. As we are not completely confident in the ability of the Scidar to measure the optical turbulence in the surface layer (first hundred of meters) nor in the Meso-Nh model, we used two sets of Meso-Nh outputs, with and without the surface layer in order to evaluate the sensitivity of the numerical model to the orographic effect. We thus defined a $\varepsilon _{\rm BL}$, a $\varepsilon _{\rm FA}$ and a $\varepsilon _{\rm TOT}$ in the following way

 

$$\varepsilon _{\rm BL}=5.30\lambda ^{-1/5}\left( \int\limits_{h_{0}}^{5000}C_{\rm N}^2(h){\rm d}h\right) ^{3/5}$$ (4)

 

$$\varepsilon _{\rm FA}=5.30\lambda ^{-1/5}\left( \int\limits_{5000}^\infty C_{\rm N}^2(h){\rm d}h\right) ^{3/5}$$ (5)

 

$$\varepsilon _{\rm Tot}=5.30\lambda ^{-1/5}\left( \int\limits_{h_{0}}^\infty C_{\rm N}^2(h){\rm d}h\right) ^{3/5}$$ (6)

for $\lambda =0.5\ 10^{-6}$ m.

• Method B(I): h₀ = 2676 m, we removed the first 106 m [3 $\leq$ N $\leq$ 40].
• Method B(II): h₀ = 2560 m, we kept all the vertical levels [1 $\leq$ N $\leq$ 40].

We underline that the ground level altitude is 2560 m and not 2640 m (the true Paranal altitude) because of an average effect due to the horizontal model resolution used. The second method B is less well statistically defined than the method A. We can average, in fact, only 4 $C_{\rm N}^2 \$ profiles for each night related to the 1$^{\rm h}$, 2$^{\rm h}$, 3$^{\rm h}$ and 4$^{\rm h}$ outputs, but we can analyze the model sensitivity in the first 100 m. In Table 3 and Table 4 are reported the statistical results for two different configurations. We estimated that this test was necessary because we often found that a strong $C_{\rm N}^2 \$ layer was produced by the model at this low altitude. At the moment we have no a priori reasons to reject or accept this contribution because we know that the Scidar sensitivity at this altitude is poor. Scidar is based on scintillation measurements and it is particularly sensitive to the high troposphere turbulence. The Generalized configuration is sensitive to the low levels turbulence too. During this campaign only a Classic version of Scidar was employed.

  

Table 3: Method B(II): statistical analysis summary. Same as Table 3 but with all the 40 vertical levels

4.3 Significance of statistical estimators

Having a small amount of data, the correlation coefficient $r_{\rm c}$ is a poor estimator for deciding whether an observed correlation is statistically significant or not. $r_{\rm c}$ tells how good is the fit to a straight line but ignores of the individual distributions x_i and y_i.

We therefore compute the probability P that two uncorrelated distributions x_i and y_i (belonging to the same parental distribution) give a correlation coefficient greater than that found. A large P means that $r_{\rm c}$ is poor, while a small P means that $r_{\rm c}$ is good. A classic test (Press et al. 1989), adapted for a small amount of data, (N < 20) gives the results reported in the Tables 2-4.

The standard deviation of data from the linear regression line might estimate in a complementary way, the data dispersion around the optimized regression line. We report the values in Tables 2-4.

4.4 Calibration of simulations

In both methods A and B we made a calibration of the Meso-Nh outputs using the following procedure. We compute the mean value of Scidar seeing measured $\langle \varepsilon _{\rm SCI}\rangle$ and the Meso-Nh simulated seeing $\langle \varepsilon _{\rm MNH}\rangle .$ All the simulated values are multiplied by the calibration coefficient $\alpha =\frac{\langle \varepsilon _{\rm SCI}\rangle }{\langle \varepsilon _{\rm MNH}\rangle }$, before doing any statistical analysis as described in Method A and B. In Method B, the calibration coefficient is named $\beta$.