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Abstract
The distribution of optical turbulence ($C_n^2$ profiles) is the fundamental parameter closely related to the design and application of optoelectronic systems. Since systematic direct measurements of optical turbulence for many climates and seasons are not available, it is useful to estimate $C_n^2$ effectively from the routine meteorological parameters. The $C_n^2$ profiles are estimated by routine meteorological parameters based on the Tatarskii model, and the estimated results are compared with the corresponding radiosonde measurements from the field campaigns at Rongcheng (${122.37^ \circ}{\rm{E}}$, ${37.15^ \circ}{\rm{N}}$), Taizhou (${121.42^ \circ}{\rm{E}}$, ${28.62^ \circ}{\rm{N}}$), and Dachaidan (${95.35^ \circ}{\rm{E}}$, ${37.74^ \circ}{\rm{N}}$) in China. The agreement between the estimation model and the measurement is very close, except for a portion of the atmosphere where it showed considerable difference. Additionally, statistical operators are used to quantify the performance of the estimated model, and the statistical results also show that the estimated and measured $C_n^2$ profiles are consistent well. Furthermore, the integrated parameters (such as the Fried parameter, ${r_0}$) from radiosonde measurement are 7.92 cm, 5.39 cm, and 3.68 cm at Rongcheng, Taizhou, and Dachaidan, respectively. Therefore, the $C_n^2$ profiles and their characteristics in these typical climate sites provide useful information to assess the effect of laser transmission in the atmosphere, which are usually used in the design of optoelectronic systems and astronomical site testing.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Atmospheric turbulence is the major reason for serious decline of imaging quality of the optoelectronic systems (e.g., astronomical observation, laser communication, and target detection) and a series of turbulent effects such as beam drift, flicker, and jitter resulting from the influence of atmospheric turbulence, which seriously affects the normal use of optical systems. The intensity of atmospheric optical turbulence is usually characterized by the refractive index structure constant ($C_n^2$, ${{\rm{m}}^{- 2/3}}$), which is the key parameter usually used to assess the effect of optical turbulence in the atmosphere [1,2]. Therefore, the knowledge of $C_n^2$ profile is essential to evaluate the performance of photoelectric systems such as astronomical observation, free-space optical communication, etc. At present, there are various instruments for measuring atmospheric optical turbulence, such as the balloon-borne micro-thermometer [3], scintillation detection and ranging (SCIDAR) [4], multi-aperture scintillation sensor (MASS) [5], and scintillator [6] have been used to measure atmospheric turbulence. These instruments are expensive and difficult to operate in spatial-temporal domain, especially in harsh and complex environments.
In many cases, it is impractical and expensive to deploy instrumentation to characterize the atmospheric turbulence, making estimation a less expensive and convenient alternative. The Tatarskii model provides a rigorous scientific basis to estimate $C_n^2$ profiles from routine meteorological parameters. Hufnagel summarized the upper altitude atmospheric turbulence parameter model based on the measured data, but this mode had a limited range of use [7]. Furthermore, the $C_n^2$ profiles indicated that the Hufnagel model agreed well with the radiosonde measurement that was obtained on three balloon flights from the runway of Holloman Air Force Base [8]. Van Zandt developed the National Oceanic and Atmospheric Administration (NOAA) model, which integrated the fine structure of wind shear and was relatively complex [9]. Furthermore, Abahamid studied the behavior of atmospheric turbulence in the boundary layer and free atmosphere using balloon-borne radiosondes from different sites, which is based on the optical turbulence parameterization of the Tatarskii model [10]. As done in [11], the Tatarskii model is adopted to study the relative contributions of temperature and relative humidity to the refractive index gradient by using three years of high-resolution radiosonde data over the tropical station Gadanki. Even though there is considerable diversity among the reported results, quasi-universality of the estimated $C_n^2$ profile with meteorological parameters is not clearly discernible. Each existing approach has its own merits and limitations, but none of them are known to be superior. Moreover, the optical turbulence has regional differences, so it is necessary to carry out the measurement and model research in different regions. In this study, we try to state the $C_n^2$ profiles characteristics at Rongcheng, Taizhou, and Dachaidan typical climate sites, and investigate the accuracy of the estimation results by comparing them with the corresponding radiosonde measurements.
This paper is broken down as follows: The detailed description of balloon-borne measurement is given in Section 2. The methodologies of estimating $C_n^2$ are introduced in Section 3. The results of $C_n^2$ deduced from the meteorological parameters and radiosonde measurements and the integrated parameters derived from $C_n^2$ profiles are presented in Section 4. In addition, some statistical operators are used to assess the accuracy of the estimation results. Finally, conclusions are drawn in Section 5.
2. BALLOON-BORNE MEASUREMENT SYSTEM
A. $C_n^2$ Data from the Micro-Thermometer
For visible and near-infrared wavelengths, optical turbulence is mainly caused by temperature fluctuation. In this study, $C_n^2$ values deduced from the micro-thermometer are used for comparison, and they are measured by a pair of horizontally micro-temperature probes. The temperature structure function ${D_T}(r)$ is one of statistics describing temperature disturbance at two points separated by distance $r$ [12] and is given by
Considering the Gladstone formula, there is a straightforward relationship between the temperature structure constant $C_T^2$ and the refractive index structure constant $C_n^2$, and $C_n^2$ is expressed as
where $P$ is the dry-air pressure in hPa.In this study, the balloon-borne micro-thermometers are used to measure the temperature structure constant using fine-wire probes separated by a 1 m horizontal distance and probes employed the platinum wire with 10 µm diameter, which is based on the existing research and measuring device. The micro-thermometer has the frequency response range of 0.05–30 Hz, and the noise level of sensors and the electronic processing of signals corresponded to a temperature difference of 0.002 K, which follows the internationally used technical indicators [3,14–22]. The micro-thermometer system is shown in Fig. 1. A pair of probe wires form two legs of a Wheatstone bridge. The platinum wire has a linear resistance-temperature coefficient, and it responds to an increase in atmospheric temperature with an increase in resistance. Thus, the temperature change can be transformed into the resistance change to obtain a rapidly varying voltage. Subsequently, the signals are amplified, filtered, and averaged. The micro-thermometer system provides $C_T^2$ data by measuring mean square temperature fluctuations from Eq. (1); thus, the $C_n^2$ value can be acquired from Eqs. (2) and (3).
B. Radiosonde Measurement and Data Collection
The location and terrain height distribution of three field campaigns sites are shown in Fig. 2. The Rongcheng site has a warm temperate continental monsoon humid climate, and the average annual temperature is 11.3°C. The site has the characteristics of obvious monsoon and concentrated precipitation, its climate is greatly influenced by the surrounding Yellow Sea in the north, east, and south, and it has outstanding marine climate characteristics. The Taizhou site is hot and rainy in summer and cool in winter, and it has a considerable seasonal variation that is a tropical climate in summer influenced by tropical marine air masses and subtropical climate in winter influenced by polar continental air mass. Annual precipitation varies from 1185 mm to 2029 mm, and the average precipitation for many years has been 1632 mm. The average air temperature is lower than 10°C in winter, higher than 22°C in summer, and between 10°C and 22°C in spring and autumn. The Dachaidan site is located on the northern edge of the Qaidam Basin, characterized by strong ultraviolet rays, large temperature difference, dry and little rain, wind, and cold. The Dachaidan site has a typical desert climate of the inland plateau with average annual precipitation is 83.5 mm, while the annual average air temperature is 1.4°C.
Fig. 2. Map of three field campaigns sites. Rongcheng site, ${37.15^ \circ}{\rm{N}}$, ${122.37^ \circ}{\rm{E}}$, at altitude 77.5 m. Taizhou site, ${28.62^ \circ}{\rm{N}}$, ${121.42^ \circ}{\rm{E}}$, at altitude 5 m. Dachaidan site, ${37.74^ \circ}{\rm{N}}$, ${95.35^ \circ}{\rm{E}}$, at altitude 3180 m.
An image of a balloon-borne radiosonde measurement experiment is shown in Fig. 3. High-resolution balloons carried with micro-thermometers were launched at Rongcheng, Taizhou, and Dachaidan radiosonde sites, and the measurements were conducted during different periods of time at different sites, including the high-resolution profiles of temperature, pressure, wind speed, wind direction, and $C_n^2$ acquired in three sites. It was worth highlighting that the balloon-borne micro-thermometers were launched at these radiosonde sites measuring $C_n^2$ profiles for the first time. The measurement wind speed and temperature had an accuracy of $0.3\; {\rm{m}} \cdot {{\rm{s}}^{- 1}}$ and 0.2°C, respectively. All the parameters are measured approximately every seven to eight meters, and the altitude range of micro-thermometers measurements is from the surface to 30 km above sea level. In the present paper, excluding the radiosonde measurements that are abnormal due to various factors such as weather and damage caused by strong winds, the remaining available 24 radiosonde measurements with each date and hour at three sites are listed in Table 1.
Table 1. The 24 Available Measurements Taken by Balloon-Borne Micro-Thermometers Launched at Three Field Campaign Sitesa
3. METHODOLOGIES OF ESTIMATING OPTICAL TURBULENCE
A. Estimation Model
The parameterization models of optical turbulence are developed to convert standard radiosonde meteorological parameters into vertical profiles of $C_n^2$, the refractive index structure constant [23]. According to Kolmogorov’s theory, which was under the assumption of local homogeneity and stationarity of the refractive index fluctuations, the parameterization model uses the Tatarskii model for estimating $C_n^2$ profile as below [1]:
where $a$ is a dimensionless constant that is most commonly used at a value of 2.8 [10]. The potential refractive index gradient $M$ is expressed asHMNSP99 outer scale mode has been used to estimate the $C_n^2$ profiles model, which takes the temperature lapse rate ($\frac{{\text{d}T}}{{\text{d}h}}$) and wind shear ($S$) into account [24]. The model contains more atmospheric parameters playing an important role in estimating $C_n^2$ profiles, and its specific expression is
B. Statistical Operators
The statistical operators of the bias and the root mean square error (RMSE) are used to analyze the correlation between the measured and estimated values of the radiosonde [25], and the definition of the statistical operators is given as follows:
where ${X_i}$ represents the individual measured values, ${Y_i}$ represents the individual values estimated by the model at the same height, and $N$ is the number of samples for a couple (${X_i}$, ${Y_i}$) at the same height. ${\bar X _i}$ and ${\bar Y_i}$ stand for the average value of measured and the model parameters, respectively. From the bias and the RMSE, it is possible to get the bias-corrected RMSE ($\sigma$) as below which represents the intrinsic uncertainty not affected by the bias, and it provides fundamental information on the systematic and statistical uncertainties.4. RESULTS
Figure 4 shows the $C_n^2$ profiles between estimated by model and measured by the micro-thermometer at the Rongcheng site. Obviously, the profiles of estimated and measured are consistent in trend and magnitude in general. Moreover, the profiles reveal the $C_n^2$ level drops steeply near the ground, and then the $C_n^2$ values gradually increase with altitudes from the boundary layer to the low stratosphere and gradually decrease with altitudes in the free atmosphere thereafter. It is well visible that the differences between the model and measurement are relatively minor from 8 to 18 km. It is worth highlighting that the $C_n^2$ peaks correspond to the tropopause with altitudes around 17 km. However, the measured values in the troposphere are slightly smaller than the estimated values, and this could be the result of the wider variety of climatic conditions during balloon measurement and accuracy of the estimated model.
The $C_n^2$ profiles estimated by the model and measured by the micro-thermometer at the Taizhou site are illustrated in Fig. 5. The $C_n^2$ profiles reveal a steep drop around the ground from the magnitude of ${10^{- 14}} \;{{\rm{m}}^{- 2/3}}$ to ${10^{- 16}} \;{{\rm{m}}^{- 2/3}}$. This is followed by an increase of turbulence with altitudes from around 3 km to the tropopause (about 16 km) and then a decrease gradually above the tropopause. Almost all the results of the comparison are in fair visual agreement, particularly in the lower atmosphere. Thus, estimated and measured the profiles present the good agreement as a whole. Note that the optical turbulence near the ground is strong, and there exist the strong turbulent layers from 10 to 18 km. It is notable that the differences between the model and the measurement are relatively small from 3 to 18 km, including more obvious above 18 km. Differences in fine structure may be attributed to the surroundings that have a mutual effect on atmospheric conditions and hence on optical turbulence.
Fig. 7. Bias, the RMSE, and the bias-corrected RMSE ($\sigma$) of ${\log}(C_n^2)$ between the model and the radiosonde at Rongcheng, Taizhou, and Dachaidan sites.
Fig. 8. Average of temperature, wind speed, and $\log (C_n^2)$ between model and radiosonde at three typical climate sites, Rongcheng, Taizhou, and Dachaidan.
On the other hand, there is a little more disagreement between the estimated results and the micro-thermometer results in the typical desert climate Dachaidan site, which is depicted in Fig. 6. The $C_n^2$ profiles decrease steeply in the surface layer, gradually increase up to the low stratosphere, and then decrease gradually in the upper stratosphere. The comparison between the estimated and measured profiles is consistent in the trend in general. It is obvious that the agreement of estimated values with measured values in trend is especially good at the lower altitudes (below 14 km). These differences can be attributed to the local orographic disturbances and a wider variety of atmospheric conditions during balloon measurement. In general, the $C_n^2$ profiles obtained by the model are in accordance with the $C_n^2$ profiles obtained by the radiosonde measurement. However, there is still some room to improve the estimated model.
Shown in Fig. 7 are the comparisons of statistical operators between the $C_n^2$ profiles produced by estimations and measurements. Figure 7 shows the statistical operators of ${\log}(C_n^2)$, the bias within $1 \; {{\rm{m}}^{- 2/3}}$, and RMSE within $1.2 \;{{\rm{m}}^{- 2/3}}$ at Rongcheng and Taizhou sites. It is obvious that bias exceeds $1 \;{{\rm{m}}^{- 2/3}}$ at about 8 km and decreases with altitudes. As can be seen in Figs. 4 and 5, the estimated values in the stratosphere are slightly smaller than the measured values. Also the bias is within $1 \;{{\rm{m}}^{- 2/3}}$ from the ground up to 14 km, and the values of bias vary rapidly, which are less than $1.3 \;{{\rm{m}}^{- 2/3}}$ above 14 km at Dachaidan. Notably, the RMSE values are within $1 \;{{\rm{m}}^{- 2/3}}$ from the ground to 14 km, and the $\sigma$ values at three sites are within $1 \;{{\rm{m}}^{- 2/3}}$. Therefore, the estimated values are generally coherent with the radiosonde measurements at three sites. It should be pointed out that the estimated values have a relatively large drift compared with measurements above 14 km, which can be obviously seen in Fig. 6.
Fig. 9. Integrated parameters derived from $C_n^2$ profiles between model and radiosonde at Rongcheng, Taizhou, and Dachaidan sites.
Table 2. Comparison of the Integrated Parameters (${r_0}$, $\varepsilon$, $\theta$, and $\tau$) for Astronomical Applications between Model and Radiosonde Measurement at Rongcheng Site
Table 3. Same as Table 2, but for Taizhou Site
Table 4. Same as Table 2, but for Dachaidan Site
Overall, the average profiles of temperature and wind speed from the radiosonde measurement, and the average $C_n^2$ profiles between estimations and measurements, are clearly illustrated in Fig. 8. From the temperature profiles, the tropopause heights for Rongcheng, Taizhou, and Dachaidan are about 17 km, 16 km, and 14 km, respectively. Moreover, there are significant differences in wind speed trend with several climates that are distinguishable, while the wind speed profile for Dachaidan is weaker than that at Rongcheng and Taizhou. At the capping inversion layer near the top of the boundary layer, strong temperature gradients exist, and an increase in the turbulence strength is noted in this region. In the free atmosphere, the $C_n^2$ values are near the order of ${10^{- 17}} \;{{\rm{m}}^{- 2/3}}$, and the $C_n^2$ values are in the order of ${10^{- 16}} \;{{\rm{m}}^{- 2/3}}$ in the troposphere. Obviously, it is well visible that the magnitude of the $C_n^2$ values from Dachaidan is larger than that at Rongcheng and Taizhou. In general, the measured values are larger than the estimated values at Rongcheng, Taizhou, and Dachaidan, which describe the characteristics of turbulence at these three sites.
The integrated parameters (Fried parameter ${r_0}$, seeing $\varepsilon$, isoplanatic angle $\theta$, and wavefront coherence time $\tau$) derived from $C_n^2$ profiles are considered the main turbulence parameters for astronomy and optical telecommunication, and the comparison of integrated parameters between model and radiosonde are shown in Fig. 9, and tabulated in Tables 2–4. The mean ${r_0}$ values from radiosonde measurement are 7.92 cm, 5.39 cm, and 3.68 cm at the Rongcheng, Taizhou, and Dachaidan sites, and the corresponding mean ${r_0}$ values from the model are 8.05 cm, 5.82 cm, and 5.66 cm, respectively. Obviously, the mean ${r_0}$ from radiosonde measurement at Dachaidan is lower than other sites, which shows that the optical turbulence from Dachaidan is stronger than that at Rongcheng and Taizhou. It appears evident that the results we have achieved are impressive and provide potential values for the application of adaptive optics systems.
5. CONCLUSION
This research effort set out to demonstrate several significant characteristics of $C_n^2$ profiles at three typical climate sites (Rongcheng, Taizhou, and Dachaidan) by the balloon-borne radiosonde field campaigns, and this research effort achieved satisfactory results for this goal. Foremost, this research provides a potential atmospheric optical parameters database for electro-optical systems. The estimation $C_n^2$ profiles based on the Tatarskii optical turbulence parameterization have been compared with the corresponding radiosonde measurements, and the typical characteristics of optical turbulence in these three sites have been analyzed and discussed with the statistical operators. Accordingly, the integrated astroclimatic parameters between model and measurements have been discussed under these three typical climate sites. The results obtained by the radiosonde measurement and the estimation are summarized as follows.
The measured $C_n^2$ profiles are slightly smaller than the estimated values at Rongcheng in the troposphere, while the estimated values are smaller than the measured values at Taizhou in the stratosphere. The $C_n^2$ profiles estimated by the model are much smaller than those measured by the radiosonde measurement at Dachaidan in the stratosphere. In fact, the estimated $C_n^2$ profiles are not completely consistent with the measured values at the corresponding altitudes, which results from the intermittent optical turbulence.
Although it is difficult to describe the optical turbulence fine structure precisely, the statistical analysis results indicate that the estimation model has a reasonable performance. Additional improvement areas are under consideration as well, and the $C_n^2$ profiles obtained from the model can provide a rough approximation of optical turbulence profiles characteristics. Despite these inherent difficulties, we will still see that the results we have achieved are impressive and, more importantly, provide potential value for the application of electro-optical systems.
There is still a lack of $C_n^2$ fine structure changes at some specific altitudes, which requires a large amount of radiosonde data to continuously modify and improve the estimation model. The research that modified the $C_n^2$ estimation model is still ongoing. Additionally, whether it is feasible and reliable to adopt the same parameterization scheme and outer scale model for sites with different climate types is worthy of studying deeply.
Funding
Foundation of Key Laboratory of Science and Technology Innovation of Chinese Academy of Sciences (CXJJ-19S028); National Key Research and Development Program of China (02-ZD-001-026-01); Strategic Priority Research Program of Chinese Academy of Sciences (XDA17010104).
Disclosures
The authors declare no conflicts of interest.
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