## Abstract

Based on the statistical study of meteorological balloons equipped with thermosondes, a new model that estimates the profile of the refractive index structure constant ($C_n^2$) is proposed. Utilizing temperature, pressure, and wind shear as inputs, this new approach can estimate vertical profiles of $C_n^2$ with 100 m vertical resolution. We used four outer scale models (Thorpe, HMNSP99, Dewan, and our proposed model) on data acquired from Rongcheng (Shandong Peninsula) and Maoming (Guangdong Province) to estimate the $C_n^2$ profiles and compared the results with the measured $C_n^2$ profile. The proposed method outperformed the other three models, yielding an estimation profile that matched well with the measured median $C_n^2$ profiles, with an average relative error generally less than 3.5% and a mean correlation coefficient larger than 0.72 in Maoming, an average relative error generally less than 3.4% and a mean correlation coefficient larger than 0.84 in Rongcheng. The proposed outer scale model also shows good performance in estimating integrated atmospheric parameters.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Local inhomogeneity of the atmosphere, especially spatial inhomogeneity with respect to temperature, causes optical turbulence, leading to random fluctuations in refractive index [1]. Optical turbulence is associated with a series of other turbulence-related effects, such as beam drift, flicker, and jitter, which significantly affects the normal use of the adaptive optical system [2–4]. The effects of turbulence can be described by the refractive index structure constant ($C_n^2$).

In recent decades, various methods and instruments have been applied to obtain $C_n^2$ [5–9]. $C_n^2$ profiles can be measured directly by thermosondes carried on a sounding balloon. However, direct methods involve a significant amount of labor and resources. In 1964, Hufnagel and Stanley [10] tried to describe $C_n^2$ profiles in different weather conditions by incorporating wind speed as an estimated parameter. However, this method was only suitable for altitudes ranging from 3–24 km. Fried [11] estimated $C_n^2$ using mean models, thus inspiring Wyngaard et al. [12] and Beland et al. [13] to develop $C_n^2$ expressions for the atmospheric surface layer and stratosphere, respectively. When fitted with observation data, these mean profile models represented the statistical average results of atmospheric optical turbulence but did not elucidate the detailed changes and fine structures responsible for the turbulent layer intensity. Therefore, according to Tatarski's turbulence estimation theory [14], various models for calculating $C_n^2$ using conventional meteorological parameters have been developed. For example, Coulman et al. [15] estimated $C_n^2$ profiles from standard meteorological sounding data, which illustrated the change in the outer scale with altitude. Dewan et al. [16] proposed an outer scale model using a horizontal wind gradient based on sounding data with a spatial resolution of 300 m. Ruggiero and DeBenedictis [17] proposed the HMNSP99 outer scale model using sounding data. This model had a similar structure to the Dewan outer scale model, but contained temperature gradients in addition to wind shear. Alternatively, Trinquet and Vernin [18] proposed the AXP model (parameters *A* and *P* are functions of the altitude *h*), which was developed using a statistical study of 162 profiles of meteorological balloons equipped with microthermal sensors, achieving consistent results with the measured $C_n^2$ profiles. Utilizing the Thorpe scale as a measure of the turbulence outer scale, Basu [19] estimated $C_n^2$ profiles by overturning potential temperature to distinguish turbulence.

In the above scale models, the Dewan and Thorpe scales are functions of wind shear and temperature, respectively, creating biases in the fitting accuracy of turbulence data. The outer scale of the HMNSP99 model includes both wind speed and temperature information; however, its validity is limited to certain areas. In our study, a new outer scale as a function of wind shear and potential temperature is described to estimate $C_n^2$ profiles, referred to as the wind shear and potential temperature (WSPT) model. The new model provides accurate estimates and is computationally inexpensive. Section 2 introduces the databases and experiments used in this study. In Section 3, we discuss the concept of the HMNSP99, Dewan, and Thorpe outer scales. Then, we present the expression for the new outer scale, including the detailed steps involved in estimating the turbulence. In Section 4, we compare the median $C_n^2$ profiles estimated by HMNSP99, Dewan, Thorpe, and proposed WSPT outer scales to the measured median $C_n^2$ profiles. Finally, we compare the integrated astroclimatic parameters calculated by the proposed approach and measured data. A summary and discussion are provided in Section 5.

## 2. Experiment and data

Using developed independently by the Anhui Institute of Optics and Fine Mechanics [20], sounding measurements were taken in Maoming (Guangdong) and Rongcheng (Shandong) respectively. Maoming (21.25°N, 110.20°E) is located in the South China Sea, southwest of Guangdong Province, at an altitude of 10.4 m above mean sea level(AMSL), which is a tropical and subtropical transition zone. Rongcheng (37.10°N, 91.13°E) is located in the easternmost region of the Shandong Peninsula, surrounded on three sides by the sea, at an altitude of 80.2 m AMSL, which is a monsoon-influenced temperate humid continental climate. The location and relief of Rongcheng are shown in Fig. 1, where the black point represents the Rongcheng radiosonde station. The balloon was inflated to lift the payload from the Maoming and Rongcheng and Meteorological Bureau to an altitude in excess of 28 km at a ascent rate of 5 m/s. The response frequency of the sensor was 0.1–30 Hz, and the statistical average time was 5 s. To avoid the influence of balloon disturbance on the sounding instrument, a 50-m-long rope was utilized to connect the load and balloon. The balloon-borne radiosonde is shown in Fig. 2. The radiosonde we utilized has been calibrated in many places, and the accurate measurement results with good repeatability have been obtained.

During the experiments,21 valid in Maoming and 8 valid in Rongcheng samples of sounding data series were obtained after removing abnormal data and data with low altitude. The maximum detection height was 28 km above ground level (AGL). Because there are many data in Maoming, it is more representative for us to choose median value of these 21 sets of sounding data in Section 3. At the same time, we selected 8 samples of highly representative individual data in Maoming for targeted analysis in the Appendix A. For completeness, we refer to Appendix B to discuss the individual profiles in Rongcheng [6,20]. The experimental details are listed in Table 1, and the individual sounding experiment data are listed in Table 2.

The radiosonde in our experiments was equipped with a temperature sensor, an air pressure sensor, and a pair of micro-temperature probes. In a single detection test, the profiles of wind speed (*w*), temperature (T), air pressure (P), and relative humidity (*H*) can be obtained at a vertical resolution of 10–40 m. Errors for u are ±0.1 m/s, errors for T are ±0.2°C, errors for Pare ±0.1hPa. Two platinum wire sensors with a diameter of 10 μm were used in the system used for the developed thermosonde. The sensors are positioned 1 m apart to simplify the measurement. Fitted with signal processors and batteries, a polystyrene foam lined with polyurethane insulation provides protection for the sensors. The two sensors are the two arms of the Wheatstone bridge and the resistance of the sensors changes almost linearly with temperature. Therefore, the temperature change is detected as the voltage change of the sensors. The noise level of the sensor and the electronic processor of the signal correspond to a temperature measuring error of 0.002 degree Kelvin (K). Therefore, the voltage amplifier outputs a certain voltage change corresponding to a certain temperature change,

*A*is the calibration coefficient,

*Δ*V is the change of voltage, and

*Δ*T is the change of temperature. The temperature structure function

*D*is influenced by the temperature differences of two micro-temperature probes with distance

_{T}(r)*r*, as defined by Eq. (2) [21]: where

*l*

_{0}≪

*r*≪

*L*

_{0}, with

*l*

_{0}and

*L*

_{0}representing the inner and outer scale, respectively. The eddy size of atmospheric turbulence measured by radiosonde is between the inner scale and the outer scale, and the inner scale is used to restrict the measurement range. In our experiments, r is chosen as 1 m within the inertial subrange. In addition, <…> denotes the ensemble average. The relationship between

*D*and the temperature structure constant $C_T^2$ is expressed as

_{T}(r)## 3. Concept of proposed method

According to Tatarskii’s study [14], $C_n^2$ can be defined as:

where ${c_0}$ is a constant, usually set as 2.8 [22], ${L_0}$ is the outer scale of turbulence in m, and*M*is the vertical gradient of the potential refractive index. The values of

*M*can be calculated from the profiles of temperature ($T$) and air pressure ($P$) [23]:

Tatarskii derived the following expression for $C_T^2$, based on several hypotheses:

where ${c_0}$ is a constant set to 2.8 and*h*represents height above ground in m. Therefore, combining Eq. (4) with Eq. (6), we obtain the revised expression of Eq. (4):

#### 3.1 Dewan model

The Dewan outer scale model is obtained from a large number of experimental data observed by Dewan and Grossbard [24]. Its main feature is the inclusion of wind shear in the expression. The expression of the Dewan outer scale model is as follows:

*S*is the vertical shear of the horizontal wind speed in s

^{-1}, while

*u*and

*v*are the north and east components, respectively, of the horizontal wind speed in m. The model needs to be applied to a vertical profile with a resolution of 300 m.

#### 3.2 HMNSP99 model

In June 1999, Ruggiero and DeBenedictis used balloon data obtained from the Holloman Air Force Base in New Mexico to deduce a relationship between the outer scale, wind shear variable, and temperature gradient. This model was similar to the Dewan model, except that the temperature gradient was added to the expression as follows [17]:

Where $dT/dh$ represents the vertical temperature gradient.

#### 3.3 Thorpe model

The concept of the Thorpe scale was proposed by Thorpe [25] to quantify the extent of water overturning. In this model, a stable sorting method was used to estimate the degree of vertical overturning of large eddies related to shear turbulence in stable stratified water. In 2015, Basu utilized this model to estimate $C_n^2$ as

where ${c_1}$ is an uncertain constant that changes with time and region, while ${\theta _S}$ and ${L_T}$ represent the sorted potential temperature and the Thorpe scale, respectively.#### 3.4 WSPT model

In general, the principal factors affecting atmospheric turbulence are temperature and wind speed. The Thorpe and Dewan outer scale contain only temperature and wind speed information, respectively. The traditional wind speed scale and temperature scale are shown as follows

Because atmospheric turbulence is affected by wind shear and temperature, wind speed or temperature scale can not reflect turbulence. In addition to the temperature gradient, the potential temperature gradient is also commonly used to evaluate atmospheric turbulence. Richardson number(Ri) is a classical approach to diagnose turbulence, can be defined as

where g is the acceleration of gravity, θ_{0}is the potential temperature at ground surface. In a stable stratified atmospheric environment without turbulence, it can be considered that the potential temperature decreases monotonously with height. When there is turbulence, the potential temperature inversion occurs in the turbulent region. Therefore, the occurrence of turbulence can be represented by the gradient of sorted potential temperature $\partial {\theta _s}/\partial h$ and the difference between the original and sorted potential temperature $\varDelta \theta $.

Therefore, in this section we propose a new outer scale model including both wind speed and potential temperature(WSPT) which can be seen a mixing scale of wind speed scale and temperature scale to estimate the $C_n^2$ profiles:

*L*

_{W}is a key factor of the estimation process, and is achieved via the following method:

- 1. Converting the temperature profile obtained from the sounding data into a potential temperature profile.
- 2. Sorting the $\theta $ profile sequence in a monotonically increasing order to obtain the ${\theta _S}$ profile sequence.
- 4. Wind shear
*S*and components of the horizontal wind speeds ($u$ and $v$) are obtained from the sounding data.

Applying these parameters to Eq. (15), we can obtain ${L_W}$_{.} The unknown proportionality constant *c* can be calculated as follows:

- 1. Randomly select one sample of sounding data. In this manuscript, we choose balloon no.1 in Rongcheng to calculate c value. The measured $C_n^2$ is obtained from sounding data, and then, the measured $C_T^2$ is calculated using Eq. (7).
- 2. ${L_W}^{4/3}{({\partial {\theta_S}/\partial h} )^2}\; $can be calculated by the previous process.
- 3. Through Eq. (16),
*c*corresponding to each sounding data can be calculated until a complete set of $c$-values is obtained. Then, the median $c$-value is deduced, which is selected as the proportionality constant for this region. For Rongcheng,*c*is calculated as 0.3.

Following the steps above, the estimated $C_n^2$ can be obtained, as shown in Fig. 3(f). The results of the process described in this section are shown in Fig. 3, using the data measured in Rongcheng on 1 November 2018 as an example. The $C_n^2$ profile estimated by the WSPT model shows superior consistency and accuracy with measured profile.

## 4. Evaluation of the proposed model and performance

#### 4.1 Method of statistical evaluation

To further evaluate the reliability of the WSPT model, we calculated the root mean square error (RMSE), correlation coefficient (${R_{xy}}$), and average relative error (MRE) of the estimation ${\log _{10}}({C_n^2} )$ for the four aforementioned methods, according to:

*n*is the number of spatial sequences, ${X_i}$ is the measured value of ${\log _{10}}({C_n^2} )$ at the i-th height, ${Y_i}$ is the estimated value of ${\log _{10}}({C_n^2} )$ at the i-th height, and ${\overline X} $ and ${\overline Y} $ represent the average of the observation and estimation, respectively. Also relative error(RE) with altitude of the median profile is calculated to evaluate the performances of the mentioned models at different altitudes. RE can be defined as

#### 4.2 Integral statistics of turbulence for new model

It is necessary to obtain integral statistics for the atmospheric coherence length ${r_0}$, seeing $\varepsilon $, coherence time ${\tau _0}$, and isoplanatic angle ${\theta _0}$ to design and evaluate the adaptive optics system. These parameters are determined via the following formulae [26]:

^{−6}m,

*h*is the altitude, ${h_0}$ is the height of the instrument above ground (i.e., the reference height),

*w*is the wind speed, and $\bar{w}$ and $\bar{h}$ are the equivalent wind speed and height, respectively, defined as

#### 4.3 Comparison of the considered outer scale models for median profiles in Maoming

To evaluate the performance of the proposed model, we use the HMNSP99, Dewan, Thorpe, and WSPT models to estimate $C_n^2$ profiles and compare them with the measured $C_n^2$ profiles. The comparative results are shown in Fig. 4(a) (For completeness, we refer to Appendix A to discuss the individual profiles in Maoming). As can be seen from the figure, the atmospheric refractive index structure constant estimated by the new scale is consistent with the measured value in terms of magnitude and the overall trend of the variation. The WSPT model-derived estimation value captures the details of high-altitude turbulence changes with impressive accuracy. Among the four methods, the $C_n^2$ profiles estimated by the WSPT outer scale show the smallest fluctuations, while the largest fluctuations are observed in the Dewan scale. Compared with the other models, the overall estimated value of the Dewan model shows the poorest performance, with an estimated value that is several orders of magnitude larger than the measured value. The performance of the HMNSP99 model is similar to that of the Thorpe model, but the fluctuations in the Thorpe model are smaller. Compared with the other three scales, the $C_n^2$ profiles estimated by the WSPT scale demonstrate similar magnitudes but show the closest variation trend relative to the measured value. Relative error profiles of the above four models is figured in Fig. 4(b). Relative error can demonstrate the variations of the deviation between the estimated value and the measured value with altitude. It can be seen that the deviations between the Dewan model and observed median turbulence is the largest, while that of WSPT model shows the minimum for the whole altitudes. The HMNSP99 and Thorpe model have the similar performance in RE(relative error) above 10 km, while The Thorpe method has the better result below 10 km.

Table 3 shows the RMSE, R_{xy} and MRE for the estimations of median ${\log _{10}}(C_n^2)$ obtained by the mentioned four outer scale models for balloon experiments. The RMSE and MRE of $C_n^2$ estimated by the WSPT model are better compared to other three models. It is worth noting that the R_{xy} of the Thorpe, HMNSP99 and WSPT exceed 0.6, which means that the three models can well reflect the variation trend of measured turbulence. For the median $C_n^2$ values, the correlation coefficient of the ${\log _{10}}(C_n^2)$ estimated by the WSPT scale was the highest compared to the other three models. As shown in Table 4, the average relative error of the median values was generally less than 3.5%, which meets estimation accuracy requirements for $C_n^2$.

The specific results of the integral statistics are listed in Table 4. It is clear that the WSPT model provides reasonable values, close to the measured values. These results demonstrate that our new outer scale model is reliable in reconstructing integrated atmospheric parameters.

#### 4.4 Comparison of the considered outer scale models for median profiles in Rongcheng

Figure 5(a) shows estimated and measured profiles in Rongcheng. Dewan model seems to estimate unreasonable results of turbulence intensities and inaccurate positions of turbulence layers. HMN model shows bad estimation performance under the tropopause. Thorpe and WSPT models have better prediction results, but the predicted value of Thorpe scale slightly bigger than the measured value above 8 km. The relative error with altitude for models is shown in Fig. 5(b), according to definition of relative error, the most reliable and stable model is the model with smallest relative error over the whole altitudes. The Dewan model is worst for all altitudes. In the stratosphere, the other three models have the similar performance with relative error less than 7%. Below 12 km, the WSPT model shows the best estimator with relative error less than 5%.

To evaluate the above four outer scale performances, the statistical results for median ${\log _{10}}(C_n^2)$ between the measurements and model-based estimations are tabulated in Table 5. Among the four models, WSPT model has the highest R_{xy} and secondly lowest RMSE and MRE. In addition, Thorpe scale is shown as a good estimator with superior performance in lowest RMSE and MRE and secondly highest R_{xy}. In particular, the R_{xy} value Dewan models are too small and have very poor correlation with the measured median profile. In conclusion, the WSPT model shows reliable estimated $C_n^2$ profiles and good statistical results in Rongcheng.

The comparison of the astronomical parameters derived from the $C_n^2$ profiles between the WSPT model and balloon observations are tabulated in Table 6. The results demonstrate that the ability in estimating the integrated parameters from the $C_n^2$ profiles.

## 5. Conclusion

We propose a simple and reliable method containing wind shear and potential temperature to estimate $C_n^2$ profiles. Balloon sounding experiments were conducted in Maoming (Guangdong) in 2016 and Rongcheng (Shandong) in 2018 as part of thermosonde campaigns to evaluate the performance of the new model approach. The median profile performance of the new approach is compared with three other outer scale models (HMNSP99, Dewan, and Thorpe). As these models have been established over the same database, we can conclude that the WSPT model shows the best performance with a correlation coefficient larger than 0.72 and an average relative error generally lower than 3.5% in Maoming and a correlation coefficient larger than 0.84 and an average relative error generally lower than 3.4% in Rongcheng. Thus demonstrating superior performance compared to the other three models. Furthermore, the WSPT model is used to calculate the integral statistics of turbulence ${r_0}$, $\varepsilon ,\,{\tau _0}$, and ${\theta _0}$_{,} obtaining a reasonable result relative to the measured equivalents.

It should be noted that he comparison of the results of different scales is indeed r depend on the season and on the geography. In Maoming (Guangdong) in 2016 and Rongcheng (Shandong) in 2018 WSPT shows better performance in RMSE and MRE than Thorpe, Dewan and HMNSP99. Considering the limitations of the data, the WSPT model needs further analysis with more measurements. Further research will focus on the performance of the WSPT model in different regions season and weather conditions.

## Appendix A: analysis for individual sounding balloons in Maoming

Comparisons of the estimated $C_n^2$ profiles obtained by the four outer scale models (Dewan,HMNSP99,Thorpe and WSPT) for individual balloon experiment in Maoming, along with the measured $C_n^2$ profiles are shown in Fig. 6.The performance for estimation by Dewan outer scale is the worst among four models for all sounding balloons. The $C_n^2$ estimated by Dewan model is much smaller than the measured value on the ground with an abrupt change occurring at ground, and much larger than the measured value above the ground. The HMNSP99 and Thorpe model have similar performance for estimation, which are consistent with the measured values in variation change trend. Both of the two models deviate from the measured values to different degrees. The WSPT model has best estimation results in both variation change trend and magnitude. Table 7 shows the RMSE, R_{xy} and MRE for the estimations of ${\log _{10}}(C_n^2)$ calculated by the mentioned four outer scale models for each balloon experiment. For all 8 experiment soundings, the correlation coefficient between the ${\log _{10}}(C_n^2)$ estimated by the WSPT scale and the measured values was the highest compared to the other models.

## Appendix B: analysis for individual sounding balloons in Rongcheng

Figure 7 shows estimated and measured $C_n^2$ profiles in Rongcheng, where Fig. 7(a)-(h) corresponds to the selected 8 data. Dewan model shows unreasonable results of turbulence change trend and inaccurate positions of turbulence layers. Thorpe, HMN and WSPT models have better prediction results, but the predicted value is slightly smaller than the measured value at high altitude.

The results of the statistical analysis are listed in Table 8. ${R_{xy}}$ represents the degree of consistency between the estimated and measured values in the variation trend. The RMSE and MRE reflect the degree of deviation between the estimated and measured values. The Dewan model performs the worst in all aspects. Except for sounding experiment nos. 2, 4, and 7, ${R_{xy}}$, the RMSE, and the MRE of ${\log _{10}}({C_n^2} )$ estimated by the WSPT model outperforms the other three outer scale models. For the no. 2 balloon, the proposed model has the second highest ${R_{xy}}$. For balloon nos. 4 and 7, the RMSE and MRE of the WSPT model are slightly larger than those of the HMNSP99 and Thorpe models; however, ${R_{xy}}$ of the WSPT model shows the best performance among the four outer scale models. In addition, there was little to differentiate the performances of the HMNSP99 and Thorpe models.

## Funding

National Natural Science Foundation of China (91752103); Strategic Priority Research Program of Chinese Academy of Sciences (XDA17010104).

## Disclosures

The authors declare no conflicts of interest.

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