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Abstract
The vertical profile of optical turbulence is a key factor in the performance design of astronomical telescopes and adaptive optics instruments. As site-testing campaigns are extremely expensive, the selection of appropriate spatial resolution data and estimation methods is extremely important. This study investigated the effect of using different methods (Dewan, HMNSP99, Thorpe method) to estimate the refractive index structure constant ($C_{n}^{2}$) using different resolution data (5 m, 25 m, ERA5 data) in Huaihua, Hunan. Compared with Dewan, HMNSP99 for estimating $C_{n}^{2}$ using 5 m and 25 m resolution data, the Thorpe method almost always shows the best performance, with RXY above 0.75 and lower RMSE and MRE between estimated and measured $C_{n}^{2}$. The results of $C_{n}^{2}$ estimation using HMNSP99 at different resolution data varied widely, indicating that HMNSP99 is more sensitive to the data resolution and the temperature gradient is more sensitive to the resolution. Using ERA5 data, the two methods of estimating $C_{n}^{2}$ using Dewan and HMNSP99 have close results. It indicates that the wind shear is the main factor when the spatial resolution of the data is reduced to a certain degree, and the contribution of temperature gradient is small in the high altitude turbulence.
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1. Introduction
Most of the application scenarios of modern optoelectronic engineering are located in or through the atmosphere, such as optical communication, astronomical observation, and high-energy laser technology. As light beams travel long distances in the atmosphere, they are susceptible to optical turbulence [1–3]. Optical turbulence is accompanied by a series of turbulence effects such as beam drift, flicker and jitter, which seriously restrict the working performance of optoelectronic systems [4–6].
Optical turbulence in the atmosphere is usually quantified by the refractive index structure constant ($C_{n}^{2}$), but the limitations of the measurements make it difficult to measure over a long period of time and a large area in a given region. To meet the needs of optical engineering, a number of models for estimating $C_{n}^{2}$ have been proposed [7–10]. Based on the Tatarski theory [11], various relationships between $C_{n}^{2}$ and conventional meteorological parameters were established to estimate $C_{n}^{2}$. For example, Dewan et al. [12] proposed a method to estimate the $C_{n}^{2}$ profile considering wind shear. Ruggiero and DeBenedictis [13] used data from balloon experiments to propose a method known as HMNSP99. The method is similar to Dewan but adds temperature gradients. Thorpe [14] proposed a simple and effective method to identify and describe turbulent eddies, Basu [15] estimated $C_{n}^{2}$ profiles by overturning potential temperature to distinguish turbulence. All three methods above are representative methods for estimating $C_{n}^{2}$.
Sounders are carried by sounding balloons to collect conventional meteorological parameters such as atmospheric temperature and atmospheric pressure at different heights from the ground up to about 30 km. The resolution of the data obtained by sounding balloons varies depending on the acquisition frequency of the instrument. For data with different spatial resolutions, the researchers mentioned that atmospheric regions that are considered dynamically stable when measured on coarse scales become increasingly unstable when observed on increasingly smaller scales [5,16,17]. It can be seen that atmospheric measurements depend on spatial resolution. Luce et al. [18] analyzed turbulent regions with the MU radar and high-resolution (10 m) balloon techniques. Clayson et al. [19] demonstrated that the same turbulence estimation method can be applied to 2 s or 6 s soundings (with resolutions of 10 m and 30 m, respectively). Basu et al. [20] have examined the fidelity of a mesoscale model in simulating high-altitude turbulence. They found that vertical grid spacing of 100 m or finer is needed to faithfully capture the intrinsic variabilities of observed clear air turbulence. Wu et al. [21,22] and Xu et al. [23] used sounding data at 25 m and 10 m resolution, respectively, and multiple methods to estimate $C_{n}^{2}$.
In this study, data with different vertical spatial resolutions were used, as well as three different methods to estimate $C_{n}^{2}$. In addition, the relationships between the effect of different methods to estimate $C_{n}^{2}$ and different vertical data resolutions were analyzed. The paper is organized as follows, in Section 2, the experiment and data used in this paper are briefly described. In Section 3, the methods for applying are introduced. In Section 4, the results of estimating $C_{n}^{2}$ at different vertical resolutions are reported. In Section 5, the $C_{n}^{2}$ estimation results are discussed. Finally, in Section 6, the conclusions are presented.
2. Experiment and data
2.1 Sounding data
In November 2021, we used our self-developed turbulence meteorological sounding instrument [22,24] to conduct sounding measurements of $C_{n}^{2}$ and conventional meteorological parameter profiles in Huaihua (Hunan). The positions and height distributions of the region can be seen in Fig. 1, in which the black circles denote the experimental site.. The instrument is equipped with temperature, humidity and pressure sensors to detect temperature, humidity and pressure parameters. It reports the latitude and longitude of the balloon’s location through GPS at any time, and calculates the average wind speed and wind direction information during this period based on the time domain orientation information between the two GPS data. The instrument for experimental use can contain dual channels, allowing a variety of choices of probes in use depending on the study or measurement objectives. All data used in this paper are measured by the 1m spacing probe to get, in addition to the two thermosondes interval in Fig. 2 is 30 m, because this experiment in the experimental process there are other experimental purposes, so there are rope length different setting requirements, but not related to the content of this paper. The device applied for the thermosonde employed two platinum wire sensors with 18-µm diameter. The sensors were mounted 1 m apart in a horizontal plane. The device has a sampling frequency of 50 Hz, a frequency response range of 0.1 Hz to 30 Hz. A styrofoam box lined with polyurethane insulation housed the signal processing electronics and batteries and provided support for the thermal sensors. The platinum wire has a linear resistance-temperature coefficient, and it responds to an increase in atmospheric temperature with an increase in resistance. The two sensors are legs of a Wheatstone bridge, and the resistance of sensors is very nearly proportional to temperature and thus temperature change is sensed as an imbalance voltage of the Wheatstone bridge. This voltage was then amplified, bandpass filtered, and synchronously detected. The noise level of sensors and the electronic processing of signals corresponded to a temperature difference of 0.002 K. The micro-thermometer can be used to provide the $C_{T}^{2}$ value by measuring the temperature fluctuation, and thus the $C_{n}^{2}$ value can be derived.
Fig. 1. Topographical distribution map. The black circles represent the experimental site: Huaihua radiosonde station.
Fig. 2. Turbulent meteorological soundings used in the balloon-borne radiosonde (left). The balloon-borne radiosonde in the sky (right).
A total of 21 sounding balloons were released during the experiment, each balloon was usually released in the morning and evening of the day, and the weather conditions of the experiment were generally chosen to be sunny or cloudy to avoid the influence of rainfall. A total of 16 valid data were obtained by excluding the anomalous data, and the specific sounding measurement records are shown in Table 1. In addition, as the maximum detection height of each measured datum is different but greater than 30 km, 30 km is selected as the maximum height for analysis. For the convenience of description, all heights in this study refer to the height above sea level.
Table 1. Sounding details record in Huaihua
2.2 ERA5 data
The ECMWF provides global numerical weather forecasts for member and cooperating countries and reanalyses the data for the wider community [25,26]. The fifth generation of the ECMWF atmospheric reanalysis system (ERA-5) provides hourly information on temperature, wind fields, etc. on a 0.25 x 0.25 degree latitude/longitude grid [27]. ERA5 data at the same time point as the sounding data described above were downloaded and then used to obtain profiles using an interpolation function and left for subsequent calculations to compare with the sounding data.
3. Methodology
Based on Kolmogorov’s local homogeneous isotropic theory, Tatarski proposed a parametrized turbulence model [11]:
where $c_{0}$ is a constant usually set to 2.8 [8,15,28–31], $L_{0}$ is the outer scale, and $M$ is the potential refractive index gradient, which is defined as: where its calculation is related to atmospheric pressure $P$ (in hPa), atmospheric temperature $T$ (in K), potential temperature $\theta$ (in K), and height above ground $h$ (in m). The relationship between the temperature structure constant and the refractive index structure constant is [24]: Based on the Tatarski formula, outer scale models for estimating the intensity of atmospheric optical turbulence are proposed as follows.3.1 Dewan model
The Dewan optical turbulence model, which Dewan obtained from a large number of experimental data, with different expressions in the troposphere and stratosphere, respectively [12],
3.2 HMNSP99 model
Ruggiero proposed a new outer scale model (HMNSP99) [13] containing wind shear and temperature gradients using sounding data from Holloman Air Force Base in 1999, with different expressions in the troposphere and stratosphere as follows:
3.3 Thorpe method
The concept of the Thorpe scale ($L_{T}$) was introduced by Thorpe [14] to quantify the scale of water overturning and has been widely used in oceanography. Thorpe scales are also being used in atmospheric science and fluid dynamics. Basu [15] estimated the atmospheric turbulence profile for the first time using the Thorpe method and obtained good results. Thorpe method can be defined as:
where $\theta _{s}$ is the potential temperature profile after sorting, $L_{T}$ is the height difference between the potential temperature before and after sorting, and $c_{T}$ is the constant corresponding to a region.3.4 Statistical evaluation
To assess the impact of sounding data with different resolutions on the atmospheric turbulence intensity, the root-mean-square error ($RMSE$), mean relative error ($MRE$), and correlation coefficient (${R}_{XY}$) are used in this paper. The definitions of the statistical methods are as follows [32,33]:
4. Results and analysis
4.1 Conventional meteorological parameters and vertical gradient quantities at different resolutions
Figure 3 shows the conventional meteorological parameters (temperature, pressure, relative humidity, wind speed and wind direction) at different resolutions (5 m, 25 m and ERA5) in Huaihua on November 26, 2021. In the 5 m and 25 m resolution data, the meteorological parameters in the figure show good agreement in terms of both magnitude, trend, and detail changes. Among the conventional meteorological parameters under ERA5 data, the pressure, wind speed, and wind direction are relatively consistent in magnitude and curve trend, however, there are some slight differences in temperature above tropopause height and relative humidity below the tropopause height in detail.
Fig. 3. Comparison of the conventional meteorological parameters at different resolutions in Huaihua on 26 November 2021. (a) temperature; (b) pressure; (c) relative humidity; (d) wind speed; (e) wind direction.
To further explore the difference of the data at different resolutions, this paper deduces the meteorological parameters ($S$, $\frac {\partial T}{\partial h}$, $\frac {\partial \theta _{s}}{\partial h}$, $N^{2}$), which are based on conventional meteorological parameters. Comparisons of the vertical gradient quantities calculated using the data at different resolutions at the same time as Fig. 3 are shown in Fig. 4. As shown in the figure, the entire profile of wind shear has essentially the same profile trend at 5 m and 25 m data resolutions, but the wind shear values calculated using the high-resolution data are larger than those at the lower resolution. Temperature gradient and $N^{2}$, two vertical gradient quantities, are similar to wind shear in that they have the same trend at different resolutions, but the values calculated using the high-resolution data are larger than those at the low-resolution. Compared to the above vertical gradient quantities, the sorted potential temperature gradient, both in terms of trend and magnitude of the values, are more consistent when calculated using different resolution data. The above vertical gradient quantities calculated for the ERA5 data can be said to be infinitely closer to the smoothed values of the values calculated using the other resolution data.
Fig. 4. Comparison of the vertical gradient quantities at different resolutions in Huaihua on 26 November 2021. (a) $S$; (b) temperature gradient; (c) sorted potential temperature gradient; (d) square of Brunt–Vaisala frequency.
The vertical gradient quantities calculated above will all appear in the next section for each of the method for estimating $C_{n}^{2}$, so knowing clearly how each parameter behaves at different data resolutions will better capture the reasons for the different performance of each method at different data resolutions.
4.2 Estimation of $C_{n}^{2}$ using different methods for different resolution data
Comparisons of the $C_{n}^{2}$ profiles estimated by the three methods (Dewan, HMNSP99, Thorpe method) using data with different resolutions at the same time as Fig. 3 and Fig. 4, are shown in Fig. 5. For completeness, all profile estimates for Huaihua at different resolution data are shown in Fig. 6 and Fig. 7 in Appendix A. Also to evaluate the estimation performance of the three methods mentioned in Section 3 for different resolution data, the statistical results of ${{\log }_{10}}C_{n}^{2}$ between the measurements and method-based estimations are tabulated in Table 2.
Fig. 5. Comparison of the $C_{n}^{2}$ profiles estimated by different methods at different resolutions in Huaihua on 26 November 2021. (a) 5 m; (b) 25 m; (c) ERA5 data.
Table 2. Statistical analysis for the estimation of the $C_{n}^{2}$ in Huaihua using the three methods at different data resolutions
Given the Thorpe method requires the use of higher resolution data, the comparison of the three methods is based on 5 m or 25 m resolution data. It is obvious from Fig. 5(a) and Fig. 5(b), respectively, that the $C_{n}^{2}$ estimated by the Dewan is much larger than the measured value over the whole altitude range, and the $C_{n}^{2}$ estimated by the HMNSP99 is slightly larger than the measured value. The Thorpe method is more consistent with the measured value in terms of trend and magnitude of variation over the whole altitude range. The same results can also be seen from the statistical results in Table 2, where the $RMSE$ and $MRE$ estimated by the Thorpe method are the smallest relatives to the other two methods, and the ${{R}_{XY}}$ is also larger, both above 0.75. Among the estimation performance of the three methods for the same resolution data, the Thorpe method has the best estimation performance and the Dewan method has the worst estimation performance.
In addition, this paper divided the whole layer height into troposphere and stratosphere, and then the estimation results of the whole layer, troposphere and stratosphere were counted separately, and the specific statistics are shown in Table 2. As can be seen from the table, the correlation coefficients of the three methods in the troposphere do not differ much under the same data resolution, and the main difference appears in the stratosphere. The results show that the Dewan has the worst estimation at 5 m and 25 m, which indicates that the high-altitude turbulence is not completely determined by wind shear, and the temperature factor also has some influence on its high-altitude turbulence. The results of $C_{n}^{2}$ estimation using both Dewan and HMNSP99 methods with ERA5 data are shown in Fig. 5(c).
A comparison of Fig. 5(a) and Fig. 5(b) as well as Table 2 shows that the differences in $RMSE$, $MRE$, and correlation coefficients for HMNSP99 using different resolutions of data are larger and better for 25 m. For Dewan using different resolutions of data, the $RMSE$ and $MRE$ decrease and the correlation coefficient increases for 25 m, but the differences are not significant compared to 5 m. When the Thorpe method was estimated using different resolutions, the difference in statistics was not significant.
As shown in Fig. 4, the resolution of the data affects the vertical gradient quantities, leading to differences in the final estimation results. Different resolution data have little effect on the sorted potential temperature gradient, so the Thorpe method has little difference at different resolutions (5 m and 25 m); HMNSP99 contains two factors, temperature gradient and wind shear, and has the largest difference at different resolutions (5 and 25 m), therefore, HMNSP99 is more sensitive to the data resolution. Dewan contains only one factor, wind shear, which is the only one that makes a difference, therefore, there is some difference at different resolutions (5 m and 25 m), but it is not as sensitive as HMNSP99. In addition, it is worth noting that the difference in temperature gradients at different data resolutions is greater in the stratosphere than in the troposphere, which verifies that the difference of HMNSP99 in the stratosphere is greater than that in the troposphere.
5. Discussions
As can be seen in Appendix A, the estimates of $C_{n}^{2}$ using the Dewan method overestimate $C_{n}^{2}$ for both 5 m and 25 m resolution data. This is not only the estimation results of this paper, but also for some researchers who have similarly overestimated $C_{n}^{2}$ using Dewan [21,34]. The Dewan method was obtained by Dewan et al. [12] based on a large number of experimental observations. The main factor in the equation that affects the magnitude of the outer scale is the wind shear, and the outer scale equation is only a primary function of the wind shear, which may overestimate the role of wind shear, and it also does not consider the effect of temperature on the outer scale. Therefore, the $C_{n}^{2}$ result may be overestimated in the estimation process. The HMNSP99 method is a method proposed by Ruggiero et al. [13] using sounding data collected at Holloman Air Force Base, New Mexico, in June 1999. This method can be regarded as a modified form of the Dewan, which adds a temperature gradient factor to the framework of the Dewan method and re-fits the corresponding coefficients. The results of HMNSP99 are closer to the measured values of $C_{n}^{2}$ than those of Dewan and also demonstrate that wind speed is not the only determinant of optical turbulence, and temperature also plays a certain role. In terms of their contribution to the formation of optical turbulence, more relevant studies are needed.
When the data resolution is higher, the data may contain more noise and therefore the amount of fluctuation will be higher. It is also evident in Fig. 4 that as the data resolution decreases, the amount of fluctuation of the gradient factor calculated using sounding data is smaller and more closely resembles a smooth curve. As the data resolution gets lower, the amount of calculated gradient fluctuations decreases and therefore the estimation results will be smaller, thus the overestimation of Dewan and HMNSP99 in all AppendixA (a) is larger than that in (b). Also, the figure shows that the 25 m resolution estimates of Dewan and HMNSP99 are closer to the measured values than the 5 m resolution estimates, and even in some data, the estimates of HMNSP99 match the measured ones very well, which also indicates that the high resolution data may contain more fluctuating values. Some of the estimated values of ERA5 data agree well with the measured values, but some underestimate the measured values, probably because the low resolution of ERA5 data leads to oversmoothing of the values.
Figure 5 shows an example of the data during the sounding site experiment, and Appendix A shows all the experimental data plotted out in the style of Fig. 5. The Thorpe method only considers the effect of temperature on turbulence in the estimation process, so it may produce misjudgment at some heights, which may exaggerate the effect of temperature and lead to a turbulent layer that does not actually exist; it may also ignore the effect of wind shear and lead to the estimated value being smaller than the measured value. In addition, there is an assumption [35] that this vertical inversion is an adiabatic process and there is no heat exchange between the air mass and the surrounding air when using the potential temperature inversion to obtain the Thorpe scale. Secondly, the estimation process using the Thorpe method includes an uncertainty factor as well as different rounding values taken at different vertical resolutions to remove spurious inversions are also slightly different, and thus may cause differences in the estimated profiles for different vertical resolutions. However, in general, Thorpe estimation results do not differ significantly at different vertical resolutions, as can be seen in both the Appendix A and Table 2.
6. Conclusions
In this paper, we estimate $C_{n}^{2}$ using three methods (Dewan, HMNSP99, Thorpe method) with different resolutions of data (5 m, 25 m and ERA5 data) for November 2021 in Huaihua, Hunan Province, and use statistical analysis to assess the effect of analytical data resolution on estimating optical turbulence.
Compared with Dewan and HMNSP99, the Thorpe method almost always show the best performance for estimating $C_{n}^{2}$ using 5 m and 25 m resolution data. It has the highest ${{R}_{XY}}$ and the lowest $RMSE$ and $MRE$ between the estimated and measured $C_{n}^{2}$. The difference between the 5 m and 25 m estimates of $C_{n}^{2}$ using the Thorpe method is not significant, which can certainly indicate that the resolution of 25 m is also sufficient. The resolution of sounding data obtained from the Meteorological Bureau is 5 m, and it is feasible to estimate $C_{n}^{2}$ at this data resolution using the Thorpe method.
The estimation of $C_{n}^{2}$ using both Dewan and HMNSP99 methods is better for low-resolution data than for high-resolution data, indicating that the resolution alone cannot be considered when estimating $C_{n}^{2}$ using these two methods, but needs to be compared with the resolution used in the original fit. Using different resolution data, the results of $C_{n}^{2}$ estimation using HMNSP99 differed more, indicating that HMNSP99 is more sensitive to the data resolution and that the temperature gradient is more sensitive to the resolution. It is worth noting that using ERA5 data, the two methods of estimating $C_{n}^{2}$ using Dewan and HMNSP99 have close results. It indicates that the wind shear is the main factor and the contribution of the temperature gradient is small in high-altitude turbulence when the spatial resolution of the data is reduced to a certain level.
Given the limited amount of data in the sample, future studies will focus on the impact of data at different resolutions on estimating optical turbulence over more areas and under more weather conditions.
Appendix A: analysis for individual sounding balloons in Huaihua at different resolutions
Fig. 6. Comparison of the $C_{n}^{2}$ profiles estimated by different methods at different resolutions in Huaihua. (a) 5 m; (b) 25 m; (c) ERA5 data. Panels in the figure represent balloon launches for sounding experiments 1–8, respectively.
Funding
Foundation of Key Laboratory of Science and Technology Innovation of Chinese Academy of Sciences (CXJJ-21S028); Foundation of Advanced Laser Technology Laboratory of Anhui Province (AHL2021QN02); National Natural Science Foundation of China (41576185, 91752103).
Disclosures
The authors declare no conflicts of interest.
Data Availability
The ERA5 data are available in Ref. [36]. Sounding data underlying this article cannot be shared publicly due to the confidentiality requirements of the project in the study. The data will be shared on reasonable request with the corresponding author.
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