Simulating the Refractive Index Structure Constant $({C}_{n}^{2})$ in the Surface Layer at Antarctica with a Mesoscale Model

The Taishan station (73°51' S, 76°58' E, at altitude 2621 m) in Antarctica is located on Princess Elizabeth Land between the Chinese Antarctic Zhongshan and Kunlun stations. The Antarctic mobile atmospheric parameter measurement system (Tian et al. 2015) includes a data logger, three-dimensional ultrasonic anemometer, micro-thermometer, temperature and relative humidity (RH) probe, wind speed and direction sensors, 485 communication modules, a power module, and a 3 m tower, which are shown in Figure 1. The main technical indicators of the in situ measurement system are shown in Table 1.

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The refractive index structure constants (C_n², in units of ${{\rm{m}}}^{-2/3}$), measured by the micro-thermometer, are deduced from a pair of horizontally separated micro-temperature probes. For visible and near-infrared wavelengths, the refractive index fluctuation is mainly caused by temperature fluctuation, and C_n² can be directly related to the corresponding temperature structure constant (C_T²) as follows (Marks et al. 1996; Cherubini & Businger 2012):

$$\begin{eqnarray}&&{C}_{n}^{2}={\left(79\times {10}^{-6}\displaystyle \frac{P}{{T}^{2}}\right)}^{2}{C}_{T}^{2},\end{eqnarray} \tag{ 1 }$$

where T is the air temperature (K) and P is the air pressure (hPa). C_T² is defined as the constant of proportionality in the inertial subrange form of the temperature structure function D_T(r). In the inertial subrange of the atmospheric turbulence, C_T² is given in Kolmogorov form (Pant et al. 1999) as follows:

$$\begin{eqnarray}&&{D}_{T}(r)=\langle {[T({\boldsymbol{x}})-T({\boldsymbol{x}}+{\boldsymbol{r}})]}^{2}\rangle ,\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&=\,{C}_{T}^{2}{r}^{2/3},\,{l}_{0}\,\ll r\ll \,{L}_{0},\end{eqnarray} \tag{ 2b }$$

where ${\boldsymbol{x}}$ and ${\boldsymbol{r}}$ denote the position vector, r is the magnitude of ${\boldsymbol{r}}$, $\langle ...\rangle$ represents the ensemble average, and l₀ and L₀ are the inner and outer scales of the atmospheric turbulence, respectively, and have units of m.

The platinum probe has a linear resistance–temperature coefficient, and it responds to an increase in atmospheric temperature with an increase in resistance. The two probes are the legs of a Wheatstone bridge, and the resistance of the probe is very nearly proportional to temperature, and thus temperature change is sensed as a voltage imbalance in the bridge. The micro-thermometer system provides C_T² data by measuring mean square temperature fluctuations (Equation (2)), and thus C_n² data can be acquired from Equation (1). In our case, a 10 μm diameter platinum wire has the equivalent noise of about 0.002 K, and the observed C_n² equivalent resolution is 3 × 10⁻¹⁸ ${{\rm{m}}}^{-2/3}$ (Wu et al. 2015).

The WRF model is a state-of-the-art mesoscale atmospheric model used for both professional forecasting and atmospheric research. It was developed by the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR), both in the United States. Details of the governing equations, transformations, and grid adaptation are given in the User's Guide, http://www2.mmm.ucar.edu/wrf/users/.

In this study, the initial data (at $1^\circ \times 1^\circ$ horizontal resolution) are released every 6 hr by the NCEP (the final analysis or FNL; https://rda.ucar.edu/datasets/ds083.2/). We used grid-nesting techniques, which use different embedded domains of digital elevation models extended on increasingly smaller surfaces, with increasing horizontal resolution but with the same vertical grid (Stein et al. 2000). In a previous study (Masciadri & Lascaux 2012; Lascaux et al. 2015; Masciadri et al. 2017), Masciadri and Lascaux used three embedded domains, where the horizontal resolution of the innermost domain is 500 m, to obtain the most relevant parameters related to optical turbulence (C_n², seeing ε, isoplanatic angle ${\theta }_{0}$, and wavefront coherence time ${\tau }_{0}$). In this work, the three embedded domains (the horizontal resolution of the innermost domain is 500 m) and the vertical resolution are satisfactory to estimate the surface layer C_n² according to the validation experiment. We set the WRF model with a conventional 5:1 nesting scheme for two-way interactive domains; this allows the largest domain to cover an area of approximately 1125 km by 1125 km at a 12.5 km grid resolution. It is then nested down to 2.5 km around the Antarctic Taishan station and then down to 0.5 km at the location of the Antarctic mobile atmospheric parameter measurement system. We also note that the model configuration we selected is suitable for performing C_n² estimates using relatively cheap hardware. The basic simulation parameters settings are listed in Table 2 and detailed in the User's Guide. The Antarctic Plateau map is shown in Figure 2, where the red solid circle stands for the center of the simulation domain.

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Table 2.  WRF Model Grid-nesting Configuration for the Simulation

Domain	Grid Points	${\rm{\Delta }}X$ (km)	Domain Size (km2)
Domain 1	90 × 90	12.5	1125 × 1125
Domain 2	75 × 75	2.5	187.5 × 187.5
Domain 3	45 × 45	0.5	22.5 × 22.5

Note. The second column shows the number of horizontal grid points, the third column shows the horizontal resolution (${\rm{\Delta }}X$), and the fourth column shows the horizontal surface covered by the model domain.

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The WRF model provides a large number of atmospheric parameters (such as the temperature, RH, wind speed, direction, etc.), which depend upon the parameterization schemes that have been chosen for the simulation, most of which are self-explanatory although some require additional information. The main physical schemes are listed in Table 3, and the parameterization schemes are now included in the official WRF release and explained as follows (Skamarock et al. 2006):

• 1.  
  The microphysics process uses the WRF Single-moment 5 class (WSM-5) scheme, which is a slightly more sophisticated version of the WRF Single-moment 3 class (which contains three kinds of water materials: water vapor, cloud water or cloud ice, rainwater or snow) that allows for mixed-phase processes and super-cooled water.
• 2.  
  The longwave radiation uses the Rapid Radiative Transfer Model (RRTM) scheme, an accurate scheme that uses look-up tables for efficiency and accounts for multiple bands and microphysics species. The longwave process is caused by water vapor, ozone, carbon dioxide, and other gases, as well as by the optical depth of cloud.
• 3.  
  The shortwave radiation uses the Goddard scheme, a two-stream multiband scheme using ozone from climatology and cloud effects. The Goddard scheme is suitable for cloud-resolution models.
• 4.  
  The surface layer uses the Eta similarity scheme, which is based on Monin–Obukhov with Zilitinkevich thermal roughness length and standard similarity functions from look-up tables.
• 5.  
  The planetary boundary layer uses the Mellor–Yamada–Janjić (MYJ), scheme, which may be used to forecast turbulent energy using the one-dimensional prognostic turbulent kinetic energy scheme with local vertical mixing.

The site testing experiments were carried out during the 30th CHINARE. Two levels (0.5 m and 2 m above ground) of air temperature, RH, and wind speed and direction, and one level (2 m above ground) of air pressure, surface temperature, C_n², and other atmospheric parameters can be measured by the Antarctic mobile atmospheric parameter measurement system at the same time, which is close to the center grid point of the simulation domain. We should note that the ensemble average time of the micro-thermometer is 10 seconds, while the WRF model outputs result at 10 minute intervals owing to the model configuration limitation. The in situ measured values are averaged over the same interval to match up with the simulated ones for a meaningful comparison.

To actually validate the results between simulation and measurement, it requires the in situ measurements to be matched up with the model output at the corresponding time period. The initial field data (FNL) we used was released at the NCEP Web site 19:00:00 UTC before the start of each simulation, with a simulation time of 72 hr.

The refractive index structure constant (C_n²) is an important parameter in the study of electromagnetic wave propagation in the atmospheric surface layer. Because turbulent fluctuations of the atmospheric refractive index ($n^{\prime}$) at a given wavelength (λ) are related to fluctuations in the temperature (T) and humidity (q) by $n^{\prime} =A(\lambda ,P,T,q)T^{\prime} +B(\lambda ,P,T,q)q^{\prime}$, it is possible to estimate C_n² from the meteorological parameters (Andreas 1988; Andreas et al. 1998; Qing et al. 2016). For Kolmogorov turbulence, which is similar to the atmospheric refractive index structure constant C_n², the temperature structure constant (C_T²), specific humidity structure constant (C_q²), and the temperature–humidity cross-structure constant (C_Tq) can be defined as

$$\begin{eqnarray}&&{C}_{T}^{2}=\displaystyle \frac{\langle {[T({\boldsymbol{x}})-T({\boldsymbol{x}}+{\boldsymbol{r}})]}^{2}\rangle }{{r}^{2/3}},{l}_{0}\ll r\ll {L}_{0},\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&{C}_{q}^{2}=\displaystyle \frac{\langle {[q({\boldsymbol{x}})-q({\boldsymbol{x}}+{\boldsymbol{r}})]}^{2}\rangle }{{r}^{2/3}},{l}_{0}\ll r\ll {L}_{0},\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&{C}_{{Tq}}=\displaystyle \frac{\langle [T({\boldsymbol{x}})-T({\boldsymbol{x}}+{\boldsymbol{r}})][q({\boldsymbol{x}})-q({\boldsymbol{x}}+{\boldsymbol{r}})]\rangle }{{r}^{2/3}},\\ &&\qquad {l}_{0}\ll r\ll {L}_{0}.\end{eqnarray} \tag{ 3c }$$

where ${\boldsymbol{x}}$ and ${\boldsymbol{r}}$ denote the position vector, r is the magnitude of ${\boldsymbol{r}}$, $\langle ...\rangle$ represents the ensemble average, and l₀ and L₀ are the inner and outer scales of the atmospheric turbulence, respectively.

These similarity relations suggest that C_n² also exhibits the MOS theory, and C_n² can be defined in terms of C_T², C_q², and C_Tq as (Gossard 1960; Wesely 1976; Hill et al. 1980)

$$\begin{eqnarray}&&{C}_{n}^{2}={A}^{2}{C}_{T}^{2}+2{{ABC}}_{{Tq}}+{B}^{2}{C}_{q}^{2}\,.\end{eqnarray} \tag{ 4 }$$

For a chosen electromagnetic wavelength and a given set of meteorological conditions, the coefficients A and B are constant. For practical purposes, at the wavelength of interest of 0.55 μm, A = $79.0\times {10}^{-6}$ P/T² and $B=-56.4\times {10}^{-6}$. The first term on the right-hand side of Equation (4) represents the refractive index fluctuations caused by temperature fluctuations and is always positive, the second term represents the correlation of temperature and humidity fluctuations and can be positive or negative, while the third term represents humidity fluctuations and is always positive.

The structure constants C_T², C_q², and C_Tq can be expressed in terms of T_* and q_* as follows (Fairall et al. 1980; Fairall & Larsen 1986):

$$\begin{eqnarray}&&{C}_{T}^{2}={T}_{* }^{2}{z}^{-2/3}{f}_{T}(z/L),\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&{C}_{q}^{2}={q}_{* }^{2}{z}^{-2/3}{f}_{q}(z/L),\end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray}&&{C}_{{Tq}}={\gamma }_{{Tq}}{T}_{* }{q}_{* }{z}^{-2/3}{f}_{{Tq}}(z/L),\end{eqnarray} \tag{ 5c }$$

where z/L is an (dimensionless) indicator of the state of stability of the atmospheric boundary layer with respect to buoyancy, z is the height, and L is the Monin–Obukhov length. ${\gamma }_{{Tq}}$ is the temperature–humidity correlation coefficient. We use a value of 0.5 for ${\gamma }_{{Tq}}$ when ${\rm{\Delta }}T/{\rm{\Delta }}q\leqslant 0$, and a value of 0.8 when ${\rm{\Delta }}T/{\rm{\Delta }}q\geqslant 0$ in this work (Frederickson et al. 2000). The similarity functions ${f}_{T}(z/L)$, ${f}_{q}(z/L)$, and ${f}_{{Tq}}(z/L)$ are determined by experiment and are usually supposed to be ${f}_{T}(z/L)={f}_{q}(z/L)={f}_{{Tq}}(z/L)$. In this paper, the similarity function $f(z/L)$ that we use is determined during a field campaign in Kansas (Wyngaard et al. 1971):

$$\begin{eqnarray}&&f(z/L)=\left\{\begin{array}{l}4.9{(1-7z/L)}^{-2/3},\ z/L\leqslant 0,\\ 4.9(1+2.75z/L),\ z/L\geqslant 0.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

Subsequently, substituting Equation (5) into Equation (4) gives a C_n² expression in terms of T_*, q_*, and L:

$$\begin{eqnarray}&&{C}_{n}^{2}={z}^{-2/3}f(z/L)({A}^{2}{T}_{* }^{2}+2{AB}{\gamma }_{{Tq}}{T}_{* }{q}_{* }+{B}^{2}{q}_{* }^{2})\,.\end{eqnarray} \tag{ 7 }$$

According to MOS theory, it is possible to express the scaling parameters for temperature (T_*), specific humidity (q_*), and wind speed (u_*) as follows:

$$\begin{eqnarray}&&{u}_{* }^{2}={C}_{D}\overline{{U}^{2}},\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}&&{T}_{* }=-\displaystyle \frac{{C}_{H}\overline{U}{\rm{\Delta }}T}{{u}_{* }},\end{eqnarray} \tag{ 8b }$$

$$\begin{eqnarray}&&{q}_{\ast }=-\displaystyle \frac{{C}_{E}\bar{U}{\rm{\Delta }}q}{{u}_{\ast }},\end{eqnarray} \tag{ 8c }$$

where $\overline{U}$ is the average wind speed at height z, ${\rm{\Delta }}T={T}_{s}-T(z)$, and ${\rm{\Delta }}q={q}_{s}-q(z)$. T(z) and q(z) are the average air temperature and average specific humidity at height z, respectively. T_s and q_s are the surface temperature and specific humidity, respectively. The drag coefficient (C_D), sensible heat flux coefficient (C_H), and moisture flux coefficient (C_E) are all from experiment, and their values are given below (Pond et al. 1971; Smith & Banke 1975; Friehe & Schmitt 1976; Friehe 1977):

$$\begin{eqnarray}&&{C}_{D}=1.3\times {10}^{-3},\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&{C}_{H}=\left\{\begin{array}{l}0.91\times {10}^{-3},\ -20\leqslant \overline{U}{\rm{\Delta }}T\leqslant 25,\\ 1.46\times {10}^{-3},\ \overline{U}{\rm{\Delta }}T\geqslant 25\,.\end{array}\right.\end{eqnarray} \tag{ 9b }$$

$$\begin{eqnarray}&&{C}_{E}=1.32\times {10}^{-3}.\end{eqnarray} \tag{ 9c }$$

For the parameterization of the Monin–Obukhov length L, the expression for L is given by the two $\overline{U}{\rm{\Delta }}T$ ranges as follows (Deardorff 1968):

$$\begin{eqnarray}&&{L}^{-1}=\left\{\begin{array}{l}-0.231\left(\displaystyle \frac{{\rm{\Delta }}T}{\overline{{U}^{2}}}+0.212\displaystyle \frac{{\rm{\Delta }}q}{\overline{{U}^{2}}}\right),\,-20\leqslant \overline{U}{\rm{\Delta }}T\leqslant 25,\\ -0.371\left(\displaystyle \frac{{\rm{\Delta }}T}{\overline{{U}^{2}}}+0.132\displaystyle \frac{{\rm{\Delta }}q}{\overline{{U}^{2}}}\right),\,\overline{U}{\rm{\Delta }}T\geqslant 25\,.\end{array}\right.\end{eqnarray} \tag{ 10 }$$

Therefore, the values of U(z), T(z), and q(z), as well as of T_s and q_s, are given by the WRF model first, and then the air–surface difference for temperature (${\rm{\Delta }}T$), specific humidity (${\rm{\Delta }}q$), and average wind speed ($\overline{U}$) is calculated. We can obtain u_*, T_*, and q_* by solving Equations (8)–(9), and obtain z/L by solving Equation (10). Finally, it is simple to estimate C_n² from Equations (6)–(7).

Several statistical operators—the bias, the RMSE, and the correlation coefficient (R_xy)—have been used to assess the accuracy of this approach, which follows the same framework as Lascaux et al. (2015):

$$\begin{eqnarray}&&\mathrm{bias}=\displaystyle \sum _{i=0}^{N}\displaystyle \frac{{{\rm{\Delta }}}_{i}}{N},\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&\mathrm{RMSE}=\sqrt{\displaystyle \sum _{i=0}^{N}\displaystyle \frac{{{\rm{\Delta }}}_{i}^{2}}{N}},\end{eqnarray} \tag{ 11b }$$

$$\begin{eqnarray}&&{R}_{{xy}}=\displaystyle \frac{{\displaystyle \sum }_{i=0}^{N}({X}_{i}-{\overline{X}}_{i})({Y}_{i}-{\overline{Y}}_{i})}{\sqrt{{\displaystyle \sum }_{i=0}^{N}{({X}_{i}-{\overline{X}}_{i})}^{2}{\displaystyle \sum }_{i=0}^{N}{({Y}_{i}-{\overline{Y}}_{i})}^{2}}},\end{eqnarray} \tag{ 11c }$$

with ${{\rm{\Delta }}}_{i}={Y}_{i}-{X}_{i}$, where X_i are the individual measured values, Y_i the individual values simulated by the WRF model at the same time, N is the number of times for which a couple (X_i, Y_i) is available, and ${\overline{X}}_{i}$ and ${\overline{Y}}_{i}$ stand for the average values of the measured and simulated parameters, respectively. From the bias and RMSE, we deduce the bias-corrected RMSE (σ):

$$\begin{eqnarray}&&\sigma =\sqrt{{{\rm{R}}{\rm{M}}{\rm{S}}{\rm{E}}}^{2}-{{\rm{b}}{\rm{i}}{\rm{a}}{\rm{s}}}^{2}};\end{eqnarray} \tag{ 12 }$$

σ represents the error of the model once the systematic error (the bias) is removed.

Because the wind direction is a circular variable, we define ${{\rm{\Delta }}}_{i}$ for the wind direction as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}=\left\{\begin{array}{l}{Y}_{i}-{X}_{i},\,| {Y}_{i}-{X}_{i}| \leqslant 180^\circ ,\\ {Y}_{i}-{X}_{i}-360^\circ ,\,{Y}_{i}-{X}_{i}\gt 180^\circ ,\\ {Y}_{i}-{X}_{i}+360^\circ ,\,{Y}_{i}-{X}_{i}\lt -180^\circ \,.\end{array}\right.\end{eqnarray} \tag{ 13 }$$

The temporal evolution of the bias (model minus measurement), RMSE, σ, and R_xy between the measurement and estimations for the relevant meteorological parameters in the surface layer is shown in Figures 5(a)–(f). The comparisons between measurement and estimations are shown to provide a valuable picture of the surface layer temporal evolution of the bias, RMSE, σ, and R_xy for the relevant meteorological parameters at the Taishan Station in Antarctica.

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Looking at Figure 5(a) for temperature, the bias is well below 4 °C (at some times well below 1 °C), and it is even more satisfactory that the RMSE is basically always lower than 1 °C. Therefore, we obtain promising RMSE, σ, and R_xy values.

Figure 5(b) shows the temporal evolution of the statistical operators for wind speed. We can see that the bias is well below 1 m s⁻¹ (at some times well below 0.5 m s⁻¹), and the RMSE and σ are basically always lower than 2 m s⁻¹, and R_xy is mostly higher than 50% (at some times well above 80%). The values of the bias, RMSE, σ, and R_xy are satisfactory.

Figure 5(c) shows the temporal evolution of the statistical operators for wind direction. We can see that the bias is well below 8°, and the RMSE and σ are basically always lower than 16°, and R_xy is mostly higher than 60% (at some times well above 90%). These are promising results demonstrating the desirable ability of the model to predict the general wind direction.

Figure 5(d) shows the temporal evolution of the statistical operators for RH. We can see that the bias is well below 9%, and the RMSE and σ are basically always lower than 12%, and R_xy is mostly higher than 50% (at some times well above 80%).

Figure 5(e) shows the temporal evolution of the statistical operators for $\mathrm{lg}({C}_{n}^{2})$ estimated using the WRF model in the surface layer, and we obtain satisfactory bias, RMSE, σ, and R_xy values. Moreover, the bias is well below 0.5 ${{\rm{m}}}^{-2/3}$, and the RMSE and σ are basically always lower than 0.8 ${{\rm{m}}}^{-2/3}$, while during the morning and evening transitions, R_xy is close to 0 (but well above 50% most times).

Figure 5(f) shows the temporal evolution of the statistical operators for $\mathrm{lg}({C}_{n}^{2})$ estimated using the measured parameters in the surface layer, and we also obtain promising bias, RMSE, σ, and R_xy values. What's more, the bias is well below 0.6 ${{\rm{m}}}^{-2/3}$; at the same time, the RMSE and σ are basically always lower than 0.9 ${{\rm{m}}}^{-2/3}$, while during morning and evening transitions, R_xy is close to 0 (but well below 0 most times).

The values of the bias, RMSE, σ, and R_xy for the temperature, wind speed and direction, RH, and $\mathrm{lg}({C}_{n}^{2})$ in the surface layer are tabulated in Table 4 for all 20 days. FromTable 4, the bias, RMSE, and σ are very small, especially for the wind speed, RH, and C_n², while the R_xy values are relatively large and the C_n² values estimated from the WRF model in the surface layer are promising. In addition, we present an analysis of the model performances in reconstructing the meteorological parameters day by day. We compute the bias, RMSE, σ, and R_xy for every single day (from the 20 day sample) for the temperature, wind speed and direction, RH, C_n²(Model), and C_n²(Estimate). We also calculate the cumulative distributions of the bias, RMSE, σ, and R_xy, including the median, and first and third quartiles; these are summarized in Table 5. As we have seen, the bias, RMSE, σ, and R_xy of the day-by-day analysis are slightly better than the bias, RMSE, σ, and R_xy of the overall statistics (Table 4).

Table 4.  Values of the Bias, RMSE, σ, and R_xy of the Temperature, Wind Speed and Direction, Relative Humidity (RH), and C_n² in the Surface Layer (Model minus In Situ Measurement, 20 Days Overall, at 2 m High)

Statistical Operator	Temperature	Wind Speed	Wind Direction	Relative Humidity	$\mathrm{lg}({C}_{n}^{2})$(Model)	$\mathrm{lg}({C}_{n}^{2})$(Estimate)
Bias	1.6	0.1	2	2	0.1	0.3
RMSE	3.6	1.5	11	6	0.6	0.5
σ	3.2	1.5	11	6	0.6	0.4
Rxy	83.8%	75.8%	84.3%	71.8%	62.7%	66.8%

Note. Note that the units of the parameters are °C for temperature, m s⁻¹ for wind speed, degrees for wind direction, percent for relative humidity, and m${}^{-2/3}$ for ${\rm{l}}{\rm{g}}({C}_{n}^{2})$ .

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Table 5.  Median Bias, RMSE, σ, and R_xy of the Temperature, Wind Speed, and Direction, Relative Humidity, and C_n² in the Surface Layer for a Single Day (Model minus In Situ Measurement, 20 Day Sample, at 2 m High)

Statistical Operator	Temperature	Wind Speed	Wind Direction	Relative Humidity	$\mathrm{lg}({C}_{n}^{2})$(Model)	$\mathrm{lg}({C}_{n}^{2})$(Estimate)
Bias	$-{0.4}_{-3.0}^{+1.4}$	${0.1}_{-0.4}^{+0.5}$	${2}_{-1}^{+6}$	${1}_{-1}^{+4}$	${0.1}_{-0.1}^{+0.2}$	${0.3}_{+0.2}^{+0.4}$
RMSE	${2.5}_{+2.0}^{+3.9}$	${1.5}_{+1.2}^{+1.7}$	${11}_{+10}^{+12}$	${5}_{+4}^{+6}$	${0.5}_{+0.3}^{+0.6}$	${0.5}_{+0.3}^{+0.7}$
σ	${2.2}_{+1.1}^{+3.0}$	${1.4}_{+1.1}^{+1.6}$	${9}_{+8}^{+11}$	${2}_{+1}^{+3}$	${0.4}_{+0.3}^{+0.5}$	${0.3}_{+0.2}^{+0.5}$
Rxy	$75.0{ \% }_{+70.0 \% }^{+88.0 \% }$	$65.5{ \% }_{+55.0 \% }^{+74.5 \% }$	$81.2{ \% }_{+77.5 \% }^{+85.0 \% }$	$65.5{ \% }_{+53.0 \% }^{+73.0 \% }$	$47.5{ \% }_{+26.5 \% }^{+60.5 \% }$	$65.6{ \% }_{+9.4 \% }^{+84.3 \% }$

Note. The subscripts and the superscripts show the first and third quartiles, respectively. Note that the units of the parameters are °C for temperature, m s⁻¹ for wind speed, degrees for wind direction, percent for relative humidity, and m for ${}^{-2/3}{\rm{l}}{\rm{g}}({C}_{n}^{2})$.

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As mentioned in the introduction, we utilize a contingency table to investigate the relationship between the estimated values and measurements. A contingency table, which is a table with n × n entries that displays the distribution of model and measurement values in terms of frequencies or relative frequencies (Lascaux et al. 2015), allows for the analysis of the relationship between two or more categorical variables.

We list all of the different simple scores we will use in the paper from a generic 3 × 3 contingency table: percent of correct detections (PC), probability of detection (POD; in %), and extremely bad detection (EBD; in %). PC = 100% is the best score, and it corresponds to a perfect estimation. POD = 100% is the best score, which represents the proportion of measured values that have been correctly estimated with this approach. EBD represents the percent of the most distant values estimated with this approach from the measurements, and it is equal to 0% for a perfect estimation. In the case of a perfectly random estimation, all PODs will be equal to 33%, but PC = 33% and EBD = 22.2%. The model will be useful if these values perform better than those random cases (33% or 22.2%). These values are a good reference to evaluate the performance of this approach. We will write POD_i instead of POD ($\mathrm{event}\,i$) when event i is considered (Qing et al. 2016).

In this study, we decided to use climatological tertiles computed from the available measurements. We used 20 days of measurements, from 2014 January 11 to 2014 January 30. Tables 6–11 show the results of the contingency table for the temperature, wind speed and direction, RH, C_n²(Model), and C_n²(Estimate) in the surface layer, respectively.

6.2.1. Temperature

6.2.2. Wind Speed

6.2.3. Wind Direction

6.2.4. Relative Humidity

6.2.5. Optical Turbulence (C_n²)

Finally, we investigate the estimation of the optical turbulence using this approach. The C_n² estimated from the measured meteorological parameters is tabulated in Table 10, and the C_n² estimated from the WRF model output is tabulated in Table 11. We observe that the PC (51.1%) for C_n² estimated from the measured parameters is better than 33%, and the EBD is 6.9%. POD₁ ($\mathrm{lg}({C}_{n}^{2})$ inferior to −14.8 ${{\rm{m}}}^{-2/3}$) is 38.2%, POD₂ ($\mathrm{lg}({C}_{n}^{2})$ between −14.8 ${{\rm{m}}}^{-2/3}$ and −14.3 ${{\rm{m}}}^{-2/3}$) is 33.4%, and POD₃ ($\mathrm{lg}({C}_{n}^{2})$ superior to −14.3 ${{\rm{m}}}^{-2/3}$) is 95.1%, while POD${}_{\mathrm{1,2}}$ is much smaller than 33% (in Table 10).

Table 10.  Same as Table 6, but for C_n² Estimated from the Measured Meteorological Parameters (in ${{\rm{m}}}^{-2/3}$)

	Intervals	Observations
		$\mathrm{lg}({C}_{n}^{2})\leqslant -14.8$	−14.8 ≤ $\mathrm{lg}({C}_{n}^{2})\leqslant -14.3$	$\mathrm{lg}({C}_{n}^{2})\geqslant -14.3$	Total
	$\mathrm{lg}({C}_{n}^{2})\leqslant -14.8$	494	119	5	618
	$-14.8\leqslant \mathrm{lg}({C}_{n}^{2})\leqslant -14.3$	604	288	30	922
Model	$\mathrm{lg}({C}_{n}^{2})\geqslant -14.3$	195	456	689	1280
	Total	1293	863	724	N = 2880

Note. Total events = 2880; PC = 51.1%; EBD = 6.9%; POD₁ = 38.2%; POD₂ = 33.4%; POD₃ = 95.2%.

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Table 11.  Same as Table 6, but for C_n² Estimated from the WRF Model Output (in ${{\rm{m}}}^{-2/3}$)

	Intervals	Observations
		$\mathrm{lg}({C}_{n}^{2})\leqslant -14.8$	$-14.8\leqslant \mathrm{lg}({C}_{n}^{2})\leqslant -14.3$	$\mathrm{lg}({C}_{n}^{2})\geqslant -14.3$	Total
	$\mathrm{lg}({C}_{n}^{2})\leqslant -14.8$	695	230	35	960
	$-14.8\leqslant \mathrm{lg}({C}_{n}^{2})\leqslant -14.3$	241	513	189	943
Model	$\mathrm{lg}({C}_{n}^{2})\geqslant -14.3$	108	165	704	977
	Total	1044	908	928	N = 2880

Note. Total events = 2880; PC = 66.4%; EBD = 4.9%; POD₁ = 66.6%; POD₂ = 56.5%; POD₃ = 75.9%.

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The PC (66.4%) for C_n² estimated from the WRF model output is significantly better than 33%, and the EBD is 4.9%. POD₁ ($\mathrm{lg}({C}_{n}^{2})$ inferior to −14.8 ${{\rm{m}}}^{-2/3}$) is 66.6%, POD₂ ($\mathrm{lg}({C}_{n}^{2})$ between −14.8 ${{\rm{m}}}^{-2/3}$ and −14.3 ${{\rm{m}}}^{-2/3}$) is 56.5%, and POD₃ ($\mathrm{lg}({C}_{n}^{2})$ superior to −14.3 ${{\rm{m}}}^{-2/3}$) is 75.9%. In all cases, the PC and POD_i are much larger than 33% (see Table 11). In view of these points, the results for the C_n² estimated from the WRF model output meteorological parameters are even better than the similar results estimated from the measured meteorological parameters. Hence, this proves the utility of this approach for estimating C_n² in the surface layer.