A Multi-Model Ensemble Pattern Method to Estimate the Refractive Index Structure Parameter Profile and Integrated Astronomical Parameters in the Atmosphere

Abstract

In this study, we devised a constraint method, called multi-model ensemble pattern (MEP), to estimate the refractive index structure parameter ($C_n^2$) profiles based on observational data and multiple existing models. We verified this approach against radiosonde data from field campaigns in China’s eastern and northern coastal areas. Multi-dimensional statistical evaluations for the $C_n^2$ profiles and integrated astronomical parameters have proved MEP’s relatively reliable performance in estimating optical turbulence in the atmosphere. The correlation coefficients of MEP and measurement overall $C_n^2$ in two areas are up to 0.65 and 0.76. A much higher correlation can be found for a single radiosonde profile. Meanwhile, the difference evaluation of integrated astronomical parameters also shows its relatively robust performance compared to a single model. The prowess of this reliable approach allows us to carry out regional investigation on optical turbulence features with routine meteorological data soon.

1. Introduction

Optical turbulence (OT), caused by atmospheric inhomogeneities and fluctuations, is one of the most critical factors that limit the transmission and performance of imaging systems [1]. Researchers involved in light propagation in the atmosphere, especially laser physicists and astronomers, have been concerned with this issue for decades [2,3,4,5,6,7,8,9,10,11]. A turbulent atmosphere impacts light wave propagation in various aspects, such as phase changes and intensity fluctuations. These distortions lead to significant blurring, scintillations, broadening, arrival angle fluctuations, and laser beam wander [1,2,8]. Hence, parameterization and characterization of OT are essential for designing and operating photoelectric systems.

Among all the parameters assessing the influence on optoelectronic systems from the turbulent atmosphere, the refractive index structure parameter ($C_n^2$) is commonly used to characterize the optical turbulence in the atmosphere. The past decades have witnessed researchers’ efforts to measure, parameterize, and estimate $C_n^2$. Up to now, different techniques (direct or indirect) using optical or non-optical principles have developed to obtain $C_n^2$ [12]. Among these techniques, a pair of micro-thermometers (MT) is the most common equipment used to obtain $C_n^2$ by invoking several hypotheses [1]. Utilizing a balloon-borne MT (in situ measurements), usually accompanied by measurements of routine meteorological parameters, is extensively employed for getting the $C_n^2$ profile in photoelectric applications, for example, site testing [6,13,14] and astronomical observatories routine scheduling [3,15]. Other remote sensing methods and instruments, for example, Multi-Aperture Scintillation Sensor (MASS), Slope Detection And Ranging (Slodar) and Solar Differential Image Motion Monitor+ (S-Dimm+), are also of vital importance for the development of modeling and the refinement of empirical dependencies for astronomy [16,17,18,19,20].

Meanwhile, methods parameterizing and estimating $C_n^2$ profiles are established to meet the need of engineering practices. Empirical, physically-based, statistical, and data-driven learning methods to estimate $C_n^2$ were subsequently developed. Simple empirical methods, such as the submarine laser communication (SLC) model [21], are only involved in a single elevation parameter. Physically-based models referring to thermodynamics or dynamics factors exist in lots of literature. Owing to their abundant physical connotations, these models are competitive in characterizing OT in terms of its physical mechanism. Hufnagel developed the Hufnagel model based on meteorology and stellar scintillation data [22]. The Hufnagel-Valley5/7 (HV5/7) model [22,23] is one of the most popular forms related to wind velocity in the free atmosphere. Ruggiero and DeBenedictis proposed the Hmnsp99 outer scale model, referring to gradients of temperature and wind shear [24]. Dewan developed a similar turbulence outer scale method utilizing wind shear [25]. Thorpe investigated the relationship between potential temperature inversion and the Thorpe scale; Basu proposed a simple approach to estimate $C_n^2$ profiles with the coarse-resolution potential temperature profiles [21,26]. The Ellison scale was developed to quantify the scales of water body overturns. This theory was also used to calculate $C_n^2$ [27,28]. Recently, several modified models, such as the wind shear and potential temperature (WSPT) model [29] and wind shear and temperature gradient (WSTG) model [30], were also applied to estimate $C_n^2$ profiles under different experimental environments. Other methods were developed in a statistical view, for example, statistical models devised by Vanzandt [31] and Trinquet [32]. Along with the development of computer science, deep learning tools have shown their advantage in handling high-dimensional and nonlinear issues. Researchers also applied this useful tool in estimating $C_n^2$ [33,34,35].

However, no one of the existing estimating approaches are superior to any of the others, to the best of our knowledge. Each existing approach has its own merits and limitations [21]. The universality and robustness of most existing approaches and models should be improved. However, the turbulent atmosphere with random, nonlinear, and infinite-element features makes it difficult to completely specify the precise mathematical expression of $C_n^2$ from the routine macroscopic meteorological parameters—for now, at least. The existing physical-based approaches were established on several hypotheses and statistical evidence, more or less. Here, we propose a multi-model ensemble pattern (MEP) method to estimate $C_n^2$ based on several existing physically-based methods. The purpose of this study is to take advantage of different existing approaches. The proposed model performance is not always the best. However, it can ensure that the $C_n^2$ and integrated astronomical parameters estimated by the MEP are competitive compared to the best of the existing models if it is not.

This paper is organized as follows: Section 2 describes the experimental site, instruments, and radiosonde data. Section 3 presents the theory of several existing approaches to estimate $C_n^2$ that we adopted and the proposed MEP method. Section 4.1 depicts the results of $C_n^2$ using different models. Section 4.2 exhibits the evaluation of different models in calculating integrated astronomical parameters. The summary and conclusions are given in Section 5.

2. Experimental Principles and Scientific Data

2.1. Experimental Principles

According to the Gladstone law [12,36] and neglecting the water vapor concentration contribution, the refractive index structure parameter $C_n^2$ (m${}^{-2/3}$) can be computed via pressure P (hPa), absolute temperature T (K), and temperature structure parameter $C_T^2$ (K${}^2$m${}^{-2/3}$) as follows:

$$C_n^2={\left(79\times {10}^{-6}\frac{P}{T^2}\right)}^2{C}_T^2.$$

(1)

$C_T^2$ can be calculated by the temperature structure function $D_T^2$ based on the Kolmogorov–Obukhov turbulence assumption [1]. $D_T^2$ is defined as:

$$D_T\left(r\right)=〈{\left[T\left(\overrightarrow{x}\right)-T\left(\overrightarrow{x}+\overrightarrow{r}\right)\right]}^2〉=C_T^2{r}^{2/3}\left(l_0\ll r\ll L_0\right),$$

(2)

where triangle brackets denote an ensemble average; $\overrightarrow{x}$ and $\overrightarrow{x}+\overrightarrow{r}$ are the positions of the temperature probes; $l_0$ and $L_0$ represent the inner and outer scales, respectively; and r represents the distance of two probes that should be in the inertial sub-region. Radiosonde balloons equipped with micro-thermometers (MT) and routine meteorological sensors are used worldwide to obtain optical turbulence and meteorology parameters profiles.

In our case, the temperature probes (red rectangular boxes) used are shown in Figure 1b. The two platinum probes were isolated 1 m (r = 1 m) horizontally. T and P data necessary for calculating $C_n^2$ were measured by onboard temperature and pressure sensors. A Global Positioning System (GPS) was used to obtain the position information, and wind velocity was calculated from GPS data with a precision of 0.3 m/s. The Anhui Institute of Optics and Fine Mechanics (AIOFM) designed the whole system. The instruments’ performance was summarized in Ref. [37]. The platinum wire probe resistance was 10 $\mathsf{\Omega }$ with 10 μm diameter. The minimum detectable value of $C_T^2$ was $4.0\times {10}^{-6}$K${}^2$m${}^{-2/3}$. The sampling frequency of the processor was up to 100 Hz, and the data were averaged with a time interval of 1 s. The precision of the temperature and pressure sensors were 0.2 K and 1.5 hPa. The balloons ascend with a vertical velocity of approximately 5 m/s. The data were re-processed with a space interval of 10 m.

2.2. Scientific Data

The field observations were carried on two areas (Figure 1a) during April 2018. One observation was undertaken in the eastern coastal area of China (hereinafter ECACN), and the other in the northern coastal area of China (hereinafter NCACN). After removing several incomplete datasets with low termination altitude or missing data, we chose 16 and 20 profiles of ECACN and NCACN, respectively. The data collections of two areas are summarized in

Table 1. Radiosonde data collection of two areas.

Areas	Morning Launches	Evening Launches	Total Launches
ECACN	0	16	16
NCACN	9	11	20

. More details are documented in Appendix B 

Table A1. ECACN radiosonde details (BJT: Beijing time).

Site	Flight Number	Date	Release Time	Flight Duration	Termination Altitude
			(BJT)	/s	(AGL)/m
	1	5 Apirl 2018	19:30	5010	29,020
	2	9 Apirl 2018	19:30	5027	31,210
	3	10 Apirl 2018	19:30	4767	30,410
	4	11 Apirl 2018	19:30	4788	27,880
	5	12 Apirl 2018	19:30	5014	29,810
	6	15 Apirl 2018	19:30	4088	25,740
	7	17 Apirl 2018	19:30	4355	26,410
	8	18 Apirl 2018	19:30	4686	28,150
ECACN	9	19 Apirl 2018	19:30	5037	30,120
	10	20 Apirl 2018	19:30	5371	31,620
	11	22 Apirl 2018	19:30	4820	28,220
	12	24 Apirl 2018	19:30	5176	30,710
	13	25 Apirl 2018	19:30	5051	30,910
	14	26 Apirl 2018	19:30	5088	29,410
	15	27 Apirl 2018	19:30	5443	31,750
	16	28 Apirl 2018	19:30	5144	31,170

and

Table A2. NCACN radiosonde details.

Site	Flight Number	Date	Release Time	Flight Duration	Termination Altitude
			(BJT)	/s	(AGL)/m
	1	3 Apirl 2018	19:30	4464	28,660
	2	4 Apirl 2018	7:30	4210	27,970
	3	4 Apirl 2018	19:30	4747	29,460
	4	5 Apirl 2018	7:30	4808	29,320
	5	8 Apirl 2018	19:30	4271	27,370
	6	9 Apirl 2018	7:30	4855	28,880
	7	9 Apirl 2018	19:30	4780	29,660
	8	10 Apirl 2018	19:30	5275	29,780
	9	12 Apirl 2018	7:30	4591	27,810
	10	13 Apirl 2018	7:30	4633	28,710
NCACN	11	14 Apirl 2018	19:30	5069	29,680
	12	16 Apirl 2018	7:30	5360	29,380
	13	16 Apirl 2018	19:30	5292	30,050
	14	17 Apirl 2018	7:30	5176	28,850
	15	20 Apirl 2018	7:30	5155	29,660
	16	21 Apirl 2018	19:30	4853	29,400
	17	25 Apirl 2018	19:30	5012	30,750
	18	26 Apirl 2018	7:30	4714	28,790
	19	26 Apirl 2018	19:30	4901	30,660
	20	27 Apirl 2018	19:30	4798	28,530

.

3. Methodology of MEP

3.1. Theory of the Adopted Models

Seven different approaches (HV: Hufnagel-Valley 5/7; H9: Hmnsp99; DN: Dewan; TE: Thorpe; EN: Ellison; WT: WSPT; WG: WSTG) estimating $C_n^2$ with routine meteorological parameters were adopted in our study. We have summarized theories of these approaches in Appendix A to avoid interrupting the fluency of this article. More details can be found in the corresponding literature.

In data processing, all approaches except for HV involved gradient variables (the measured meteorological parameters or their derived parameters). Several approaches (TE, EN, and WT) calculated $C_n^2$ related to the sizes of localized overturns of the potential temperature. It was hard to distinguish these overturns for coarse resolution data because potential temperature profiles have an increasing tendency with height most of the time. Hence, we adopted the original resolution data in these approaches. Meanwhile, the other approaches (H9, DN, and WSTG) were calculated in the vertical resolution of 60 m. All seven approach estimations were re-processed on the scale of 60 m for consistency and convenience. Meanwhile, data exceeding 1 km above the ground level (AGL) were selected. Hence, the feature of $C_n^2$ and integrated astronomical parameters represent the free atmosphere results in our case.

3.2. MEP Method

Before introducing the principles of MEP, several theoretical basics should be elaborated first. For two variables, r and f, $r_n$ is the reference variable (MT measured $C_n^2$ in this study), and $f_n$ is the corresponding pattern result (estimated as $C_n^2$ in this study). The correlation coefficient (R) and their root-mean-square difference between two fields ($E^{\prime }$, also known as the centered root-mean-square difference) are defined as:

$$R=\frac{\frac{1}{N}{\displaystyle \sum _{n=1}^N}\left(f_n-\overline{f}\right)\left(r_n-\overline{r}\right)}{{\sigma }_f{\sigma }_r},$$

(3)

$$E^{\prime }={\left\{\frac{1}{N}\sum _{n=1}^N{\left[\left(f_n-\overline{f}\right)-\left(r_n-\overline{r}\right)\right]}^2\right\}}^{1/2},$$

(4)

where ${\sigma }_r$ (${\sigma }_r=1/N\sqrt{{\displaystyle \sum _{n=1}^N}{\left(r_n-\overline{r}\right)}^2}$) and ${\sigma }_f$ (${\sigma }_f=1/N\sqrt{{\displaystyle \sum _{n=1}^N}{\left(f_n-\overline{f}\right)}^2}$) denote the reference variable standard deviation and pattern result standard deviation, respectively; $\overline{r}$ and $\overline{f}$ represent the average of two variables. Thus, we can deduce the relationship between the reference standard deviation ${\sigma }_r$, pattern standard deviation ${\sigma }_f$, and correlation coefficient R as:

$${E^{\prime }}^2={{\sigma }_f}^2+{{\sigma }_r}^2-2{\sigma }_f{\sigma }_{r}R,$$

(5)

Taylor devised the Taylor diagram to provide a concise statistical summary of how well the patterns match each other in terms of the above four statistics (${\sigma }_r$, ${\sigma }_f$, R and $E^{\prime }$) [38]. In our study, we have normalized the statistics (${\hat{\sigma }}_f={\sigma }_f/{\sigma }_r$, ${\hat{\sigma }}_r=1$, ${\hat{E}}^{\prime }=E^{\prime }/{\sigma }_r$) referring to ${\sigma }_r$ for convenience. According to Taylor’s work, a skill function was also developed to assess the models’ performance as follows:

$$S\left(\alpha ,\beta \right)=\frac{2^{\alpha }{\left(1+R\right)}^{\beta }}{{\left({\hat{\sigma }}_f+1/{\hat{\sigma }}_f\right)}^{\alpha }{\left(1+R_0\right)}^{\beta }},$$

(6)

where $R_0$ represents the maximum of R in a set of the same model and we set $R_0=1$; $\alpha$ and $\beta$ are penalty coefficients that can adjust the proportion of skill function via model variance and correlation coefficient. A more considerable value of $\alpha$ or $\beta$ means that the corresponding statistic ($\hat{\sigma }$ or R) has a more significant influence on the result of $S(\alpha ,\beta )$.

Further, a weight function is defined as:

$$W_j\left(\gamma \right)=\frac{{S_j}^{\gamma }}{{\displaystyle \sum _{i=1}^N}{S_i}^{\gamma }}.$$

(7)

Note that all skill values are in the range of 0–1. We set a penalty parameter $\gamma$ to distinguish the model’s performance. Consequently, the multi-model ensemble pattern (MEP) method process is divided into three steps, as shown in Figure 2. We have summarized them as follows:

1.

Using routine meteorological parameters estimating $C_n^2$ with multiple models;

2.

Obtaining models skills $S(\alpha ,\beta )$ against MT results in Equation (6);

3.

Calculating weights $W_j\left(\gamma \right)$ of different models and MEP results.

Meanwhile, parameters ($\alpha ,\beta$) are used for modulating the weights of different statistics, and $\gamma$ is used to distinguish the different models’ performance. These penalty parameters can be changed as the research focus changes in practice. For example, we can increase the $\alpha$ value to increase the weight of data fluctuation in the evaluation system. It is the same for the $\beta$ for correlations, and we chose the latter condition in our case. Moreover, a considerable $\gamma$ means a more significant influence on the evaluation of skills. In our case, we set $\alpha =2,\beta =6,\gamma =4$.

3.3. Statistical Analysis

In addition to the correlation coefficient (R), the root mean square error ($RMSE$), bias ($Bias$), and mean absolute error ($MAE$) were calculated to evaluate the performance of the different approaches. The definitions of these statistics are as follows:

$$RMSE=\sqrt{\frac{1}{N}\sum _{n=1}^N{\left(r_n-f_n\right)}^2},$$

(8)

$$Bias=\frac{1}{N}\sum _{n=1}^N\left(r_n-f_n\right),$$

(9)

$$MAE=\frac{1}{N}\sum _{n=1}^N\left|r_n-f_n\right|.$$

(10)

4. Results and Discussion

4.1. Measured and Estimated $C_n^2$ Profiles

We employed 16 and 20 radiosonde datasets of two areas when evaluating the performance of the proposed and adopted methods. We used $lo{g}_{10}\left(C_n^2\right)$ instead of $C_n^2$ to generate readable data and curves. We provide two $C_n^2$ profiles of all approaches against the MT of each dataset in the primary text. See Appendix B for all days details in two sites (Figure A1, Figure A2, Figure A3 in ECACN; Figure A4, Figure A5, Figure A6, Figure A7 in NCACN).

Figure 3a,b displays MT and estimations $C_n^2$ profiles from the ECACN radiosonde campaign. The overall trends of estimations are consistent with MT. The $C_n^2$ magnitude of estimations and MT are mainly distributed in the range of ${10}^{-15}-{10}^{-19}{m}^{-2/3}$. Distinct differences can also be seen between different estimations. HV has a very high correlation with MT within the troposphere. However, it underestimates $C_n^2$ significantly above approximately 20 km, which indicates that a more turbulent and complex atmospheric state might exist above the troposphere in this area. TE, EN, and WT have better performance in magnitude owing to the calibration of unknown proportionality constants according to MT measurements. H9, DN, and WG have a similar trend in the overall trend, while these estimations fluctuate a little bit more around the mean value against TE, EN, WT, and ME. By combining the corresponding Taylor diagrams in Figure 3c,d, we can also easily find that the values of normalized standard deviations of HV, H9, DE, and WG are much bigger than MT most of the time. Meanwhile, closer normalized standard deviation values to 1 (or MT) of TE, EN, and ME means that these approaches have similar behavior in $C_n^2$ fluctuation magnitude. In addition, ME also shows its advantage in correlation evaluation. Among all 16 launches, correlation coefficients between ME and MT are mainly distributed around 0.6–0.8 and the best one is up to approximately 0.9.

Figure A1, Figure A2, Figure A3 in Appendix B display all the $C_n^2$ profiles in ECACN. Scatter figures of all approaches against MT for all launches were plotted to further study the overall statistical features. Figure 4 shows all launches estimated $C_n^2$ statistical feature in ECACN. The relevant statistics are summarized in

Table 2. ECACN 16 $C_n^2$ profiles statistics.

Statistics	HV	H9	DN	TE	EN	WT	WG	ME
R	0.64	0.61	0.53	0.60	0.63	0.53	0.60	0.65
$Bias$	0.60	0.57	0.08	−0.30	0.18	−0.005	−0.15	−0.11
$MAE$	0.81	0.73	0.76	0.56	0.53	0.54	0.64	0.51
$RMSE$	1.13	0.92	0.95	0.70	0.67	0.68	0.81	0.64

. Although the overall $Bias$ of ME is slightly larger than WT, R, $MAE$ and $RMSE$ of ME present the best performance of all approaches.

Validation was also done to the radiosonde data from the campaign carried out in NCACN. Twenty sounding datasets were selected in this area. Figure 5 exhibits two launches $C_n^2$ profiles and their corresponding Taylor diagrams. The characteristics of different approaches estimation profiles are similar to those in ECACN. MEP correlation coefficients of a single launch in NCACN are mainly distributed around 0.7–0.9, and the best one is more than 0.95. Figure A4, Figure A5, Figure A6, Figure A7 in Appendix B display all the $C_n^2$ profiles in NCACN. Figure 6 shows all 20 launches estimated $C_n^2$ against the MT statistical feature in NCACN. The relevant statistics are summarized in

Table 3. NCACN 20 $C_n^2$ profiles statistics.

Statistics	HV	H9	DN	TE	EN	WT	WG	ME
R	0.69	0.72	0.62	0.74	0.75	0.64	0.73	0.76
$Bias$	0.52	0.61	0.06	−0.22	0.26	0.02	−0.13	−0.09
$MAE$	0.74	0.71	0.73	0.49	0.49	0.52	0.56	0.45
$RMSE$	1.01	0.89	0.90	0.62	0.64	0.66	0.69	0.58

. The overall correlation criteria R of MEP is the best of all approaches, up to 0.7632. Meanwhile, the deviation criteria $Bias$, $MAE$, and $RMSE$ of MEP are the smallest. The above results of all the approaches in ECACN and NCACN have proved the potential of MEP in estimating $C_n^2$ utilizing radiosonde data.

4.2. Integrated Astronomical Parameters from Measured and Estimated $C_n^2$ Profiles

Evaluating the optical turbulence influence on optoelectronic facilities (ground-based observatories, laser transmission, and free atmosphere optical communication systems) in the atmosphere is one of the primary aims of researchers. Hence, we also calculated the integrated astronomical parameters (Fried parameter $r_0$, seeing $ϵ$, isoplanatic angle ${\theta }_{AO}$ and scintillation rate ${\sigma }_I^2$) to evaluate the performance of the proposed method. These parameters are defined as [5,9,12]:

$$r_0={\left[0.423{\left(\frac{2\pi }{\lambda }\right)}^{2}sec\phi \underset{h_0}{\overset{\infty }{\int }}{C}_n^2\left(h\right)dh\right]}^{-3/5},$$

(11)

$$\epsilon =5.25{\lambda }^{-1/5}{\left[\underset{h_0}{\overset{\infty }{\int }}{C}_n^2\left(h\right)dh\right]}^{3/5},$$

(12)

$${\theta }_{AO}=0.057{\lambda }^{6/5}{\left[\underset{h_0}{\overset{\infty }{\int }}{C}_n^2\left(h\right)h^{5/3}dh\right]}^{-3/5},$$

(13)

$${\sigma }_I^2=19.12{\lambda }^{-7/6}\underset{h_0}{\overset{\infty }{\int }}{C}_n^2\left(h\right)h^{5/6}dh.$$

(14)

$\phi$ is the solar zenith angle set as 0${}^{\circ }$; $\lambda$ is a given wavelength (we set $\lambda$ = 550 nm); h denotes the elevation above ground level (AGL) of the sites; $h_0$ represents the initial elevation (we set $h_0$ = 1000 m). Therefore, the conclusions of the integrated astronomical parameters included in this study can only represent the influence of the free atmosphere.

Details of these integrated astronomical parameters of all launches in ECACN are listed in the Appendix B,

Table A3. ECACN integrated astronomical parameters details ($r_0$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	6.45	15.41	5.95	19.31	4.25	4.08	10.48	5.92	5.30
	2	13.15	14.38	6.52	21.05	7.33	11.32	11.79	6.67	9.05
	3	21.22	16.91	4.80	22.64	8.72	15.68	12.40	6.98	10.10
	4	10.47	17.37	4.51	20.13	8.13	15.24	12.73	5.64	7.78
	5	5.63	18.97	6.43	20.46	7.50	13.76	10.94	6.34	9.34
	6	9.72	12.37	3.77	20.69	7.05	13.10	9.98	6.33	9.03
	7	10.63	12.53	4.51	20.72	7.28	14.47	11.04	6.64	7.56
	8	8.66	15.11	4.57	21.32	7.64	14.30	12.02	7.14	9.65
	9	15.61	19.53	5.68	21.35	7.69	14.93	16.86	6.92	9.21
$r_0$/cm	10	23.28	20.11	6.23	23.69	8.21	15.39	14.21	7.22	11.84
	11	18.69	19.12	7.52	21.14	8.01	14.69	12.23	7.03	9.71
	12	6.43	16.28	2.31	20.37	7.21	14.47	9.76	6.78	8.82
	13	9.19	15.40	4.08	22.02	7.38	14.27	9.58	6.82	8.06
	14	8.12	14.94	7.06	18.54	6.05	6.97	11.01	7.19	7.47
	15	12.90	13.44	6.36	19.82	7.41	14.82	12.43	6.70	8.66
	16	9.72	13.12	5.57	18.49	7.11	12.14	12.34	6.78	8.27
	Median	10.10	15.41	5.63	20.70	7.39	14.38	11.91	6.78	8.93
	Mean	11.87	15.94	5.37	20.73	7.31	13.10	11.86	6.69	8.74

,

Table A4. ECACN integrated astronomical parameters details ($ϵ$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	1.72	0.72	1.87	0.58	2.61	2.73	1.06	1.88	2.10
	2	0.85	0.77	1.71	0.53	1.52	0.98	0.94	1.67	1.23
	3	0.52	0.66	2.32	0.49	1.28	0.71	0.90	1.59	1.10
	4	1.06	0.64	2.47	0.55	1.37	0.73	0.87	1.97	1.43
	5	1.97	0.59	1.73	0.54	1.48	0.81	1.02	1.75	1.19
	6	1.14	0.90	2.95	0.54	1.58	0.85	1.11	1.76	1.23
	7	1.05	0.89	2.47	0.54	1.53	0.77	1.01	1.67	1.47
	8	1.28	0.74	2.43	0.52	1.46	0.78	0.92	1.56	1.15
	9	0.71	0.57	1.96	0.52	1.45	0.74	0.66	1.61	1.21
$ϵ$/${}^{″}$	10	0.48	0.55	1.79	0.47	1.35	0.72	0.78	1.54	0.94
	11	0.59	0.58	1.48	0.53	1.39	0.76	0.91	1.58	1.15
	12	1.73	0.68	4.81	0.55	1.54	0.77	1.14	1.64	1.26
	13	1.21	0.72	2.72	0.50	1.51	0.78	1.16	1.63	1.38
	14	1.37	0.74	1.57	0.60	1.84	1.59	1.01	1.55	1.49
	15	0.86	0.83	1.75	0.56	1.50	0.75	0.89	1.66	1.28
	16	1.14	0.85	1.99	0.60	1.56	0.92	0.90	1.64	1.34
	Median	1.10	0.72	1.98	0.54	1.50	0.77	0.93	1.64	1.25
	Mean	1.11	0.71	2.25	0.54	1.56	0.96	0.96	1.67	1.31

,

Table A5. ECACN integrated astronomical parameters details ($\theta {}_{AO}$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	0.33	1.23	0.69	1.69	0.57	0.99	0.84	0.57	0.69
	2	0.78	1.09	0.47	1.79	0.57	1.13	0.68	0.60	0.74
	3	2.01	1.47	0.31	1.89	0.62	1.20	0.70	0.58	0.77
	4	0.83	1.55	0.28	1.77	0.60	1.14	0.70	0.55	0.68
	5	0.52	1.91	0.42	1.74	0.59	1.02	0.64	0.55	0.73
	6	0.57	0.87	0.38	1.76	0.60	1.08	0.68	0.56	0.73
	7	0.85	0.88	0.31	1.67	0.60	1.18	0.74	0.55	0.54
	8	0.69	1.18	0.39	1.75	0.58	1.16	0.71	0.61	0.77
	9	1.36	2.07	0.46	1.77	0.58	1.16	1.00	0.61	0.75
$\theta {}_{AO}$/${}^{″}$	10	1.13	2.27	0.46	1.90	0.60	1.16	0.88	0.60	0.91
	11	1.10	1.95	0.47	1.77	0.59	1.12	0.76	0.60	0.73
	12	0.47	1.36	0.32	1.75	0.61	1.17	0.68	0.63	0.80
	13	0.46	1.23	0.47	1.86	0.59	1.14	0.67	0.63	0.66
	14	0.40	1.16	0.68	1.58	0.48	0.53	0.74	0.59	0.58
	15	0.65	0.98	0.61	1.73	0.58	1.15	0.83	0.62	0.71
	16	0.60	0.94	0.49	1.60	0.57	0.97	0.74	0.56	0.66
	Median	0.67	1.23	0.46	1.76	0.59	1.14	0.72	0.60	0.73
	Mean	0.80	1.38	0.45	1.75	0.58	1.08	0.75	0.59	0.72

,

Table A6. ECACN integrated astronomical parameters details (${\sigma }_I^2$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	1.54	0.24	0.76	0.15	1.10	0.75	0.41	0.99	0.81
	2	0.40	0.28	1.04	0.13	0.80	0.30	0.49	0.87	0.54
	3	0.10	0.19	2.02	0.12	0.66	0.24	0.47	0.85	0.49
	4	0.44	0.17	2.44	0.14	0.73	0.26	0.46	1.03	0.67
	5	1.03	0.13	1.22	0.14	0.78	0.31	0.56	0.97	0.55
	6	0.63	0.40	1.95	0.14	0.82	0.31	0.57	0.95	0.58
	7	0.37	0.39	2.10	0.14	0.79	0.26	0.49	0.95	0.86
	8	0.61	0.25	1.61	0.14	0.78	0.26	0.49	0.82	0.51
	9	0.21	0.12	1.12	0.13	0.77	0.26	0.27	0.82	0.54
${\sigma }_I^2$	10	0.18	0.11	1.08	0.12	0.72	0.25	0.34	0.82	0.38
	11	0.22	0.13	0.99	0.14	0.75	0.27	0.44	0.84	0.54
	12	1.01	0.21	3.18	0.14	0.78	0.26	0.57	0.82	0.52
	13	0.82	0.24	1.44	0.12	0.78	0.27	0.58	0.81	0.66
	14	1.02	0.26	0.72	0.17	1.18	1.05	0.49	0.84	0.85
	15	0.47	0.33	0.87	0.14	0.82	0.26	0.40	0.84	0.60
	16	0.63	0.35	1.16	0.16	0.84	0.37	0.46	0.92	0.67
	Median	0.54	0.24	1.19	0.14	0.78	0.26	0.48	0.85	0.56
	Mean	0.61	0.24	1.48	0.14	0.82	0.35	0.47	0.88	0.61

. The median values represent regional features and are of referential value for photoelectric applications. Median values of $r_0$, $ϵ$, ${\theta }_{AO}$, and ${\sigma }_I^2$ calculated from MT are 10.10 cm, 1.10${}^{″}$, 0.67${}^{″}$, and 0.54${}^{″}$, respectively. These parameters calculated from ME are 8.93 cm, 1.25${}^{″}$, 0.73${}^{″}$, and 0.56${}^{″}$. The relative errors of median values are rather small. All the integrated astronomical parameters are depicted in Figure 7, and the relevant statistical feature of these parameters are summarized in

Table 4. ECACN integrated astronomical parameters statistics (@$\lambda$ = 550 nm).

	Statistics	HV	H9	DN	TE	EN	WT	WG	ME
	R	0.46	0.32	0.70	0.61	0.46	0.62	0.46	0.69
$r_0$	$Bias$	−4.07	6.50	−8.87	4.56	−1.23	0.005	5.17	3.13
	$MAE$	5.01	6.60	8.87	4.89	4.17	3.27	5.31	4.01
	$RMSE$	6.11	8.13	9.86	6.48	4.79	4.28	7.17	5.32
	R	0.14	0.34	0.53	0.53	0.44	0.65	0.40	0.51
$ϵ$	$Bias$	0.39	−1.14	0.57	−0.45	0.15	0.15	−0.56	−0.20
	$MAE$	0.42	1.17	0.57	0.54	0.39	0.29	0.60	0.38
	$RMSE$	0.58	1.37	0.70	0.58	0.52	0.39	0.69	0.42
	R	0.45	−0.42	0.56	0.49	0.45	0.29	0.03	0.38
${\theta }_{AO}$	$Bias$	−0.59	0.35	−0.95	0.21	−0.28	0.05	0.21	0.08
	$MAE$	0.66	0.43	0.97	0.30	0.41	0.28	0.31	0.28
	$RMSE$	0.74	0.59	1.03	0.45	0.47	0.40	0.47	0.40
	R	0.14	−0.08	0.51	0.72	0.62	0.40	0.31	0.50
${\sigma }_I^2$	$Bias$	0.37	−0.87	0.47	−0.21	0.25	0.14	−0.28	−0.004
	$MAE$	0.38	1.01	0.47	0.33	0.29	0.27	0.40	0.28
	$RMSE$	0.53	1.18	0.60	0.37	0.39	0.38	0.46	0.34

. HV and DN overestimated $r_0$ and ${\theta }_{AO}$ and underestimated $ϵ$ and ${\sigma }_I^2$ can be easily found both from the figures and their $Bias$ from the table against MT. The ME correlation coefficients of $R_0$, $ϵ$ and ${\sigma }_I^2$ are quite good. Meanwhile, the deviations are rather small compared to the other approaches.

The same computation process was done for the 20 radiosonde data from NCACN. The parameters calculated in NCACN are listed in the Appendix B,

Table A7. NCACN integrated astronomical parameters details ($r_0$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	2.95	17.18	3.74	19.16	6.84	8.21	12.00	5.21	7.12
	2	8.63	14.05	3.57	20.75	7.36	13.01	10.20	6.15	8.27
	3	9.04	14.56	3.85	21.19	7.50	14.53	10.75	6.64	8.16
	4	5.67	13.40	2.67	20.68	6.90	8.99	9.81	5.77	7.40
	5	4.93	12.47	3.96	17.86	7.79	15.88	8.42	6.19	8.03
	6	15.11	13.97	4.86	18.12	7.82	14.03	9.47	5.81	8.84
	7	7.19	15.95	5.82	18.41	7.76	16.78	10.38	6.20	5.66
	8	19.46	10.79	3.86	19.55	9.41	17.60	15.12	6.57	10.77
	9	9.86	10.72	4.16	20.76	8.94	15.73	10.18	6.40	10.73
$r_0$	10	3.32	14.17	1.61	20.21	7.85	13.02	11.18	6.46	6.27
	11	13.17	11.67	3.99	19.27	7.60	14.78	9.66	6.57	7.11
	12	15.67	15.55	5.16	17.36	7.84	15.65	12.09	5.93	9.40
	13	13.35	18.65	6.17	20.62	8.16	16.50	14.61	6.91	9.29
	14	7.06	18.80	4.77	20.02	7.42	11.02	13.74	6.24	8.79
	15	10.77	21.23	4.92	21.43	8.19	12.19	12.68	6.11	9.73
	16	12.67	20.45	3.74	24.38	8.56	14.97	14.64	6.84	8.83
	17	16.23	13.34	6.61	20.65	8.50	16.84	12.93	6.82	9.93
	18	13.03	13.83	6.00	21.46	8.35	14.32	12.10	6.31	10.11
	19	15.06	13.40	6.60	20.38	8.86	16.90	13.65	7.10	9.62
	20	6.06	18.60	5.22	21.95	8.43	16.24	13.98	6.75	10.36
	Median	10.31	14.11	4.46	20.50	7.84	14.88	12.04	6.36	8.83
	Mean	10.46	15.14	4.56	20.21	8.00	14.36	11.88	6.35	8.72

,

Table A8. NCACN integrated astronomical parameters details ($ϵ$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	3.77	0.65	2.97	0.58	1.63	1.35	0.93	2.13	1.56
	2	1.29	0.79	3.12	0.54	1.51	0.85	1.09	1.81	1.34
	3	1.23	0.76	2.88	0.52	1.48	0.76	1.03	1.67	1.36
	4	1.96	0.83	4.17	0.54	1.61	1.24	1.13	1.93	1.50
	5	2.26	0.89	2.81	0.62	1.43	0.70	1.32	1.80	1.39
	6	0.74	0.80	2.29	0.61	1.42	0.79	1.17	1.91	1.26
	7	1.55	0.70	1.91	0.60	1.43	0.66	1.07	1.79	1.97
	8	0.57	1.03	2.88	0.57	1.18	0.63	0.74	1.69	1.03
	9	1.13	1.04	2.67	0.54	1.24	0.71	1.09	1.74	1.04
$ϵ$/${}^{″}$	10	3.35	0.78	6.92	0.55	1.42	0.85	0.99	1.72	1.77
	11	0.84	0.95	2.79	0.58	1.46	0.75	1.15	1.69	1.56
	12	0.71	0.71	2.16	0.64	1.42	0.71	0.92	1.87	1.18
	13	0.83	0.60	1.80	0.54	1.36	0.67	0.76	1.61	1.20
	14	1.57	0.59	2.33	0.56	1.50	1.01	0.81	1.78	1.26
	15	1.03	0.52	2.26	0.52	1.36	0.91	0.88	1.82	1.14
	16	0.88	0.54	2.97	0.46	1.30	0.74	0.76	1.62	1.26
	17	0.69	0.83	1.68	0.54	1.31	0.66	0.86	1.63	1.12
	18	0.85	0.80	1.85	0.52	1.33	0.78	0.92	1.76	1.10
	19	0.74	0.83	1.68	0.55	1.25	0.66	0.81	1.57	1.16
	20	1.83	0.60	2.13	0.51	1.32	0.68	0.80	1.65	1.07
	Median	1.08	0.79	2.50	0.54	1.42	0.75	0.92	1.75	1.26
	Mean	1.39	0.76	2.71	0.55	1.40	0.81	0.96	1.76	1.31

,

Table A9. NCACN integrated astronomical parameters details ($\theta {}_{AO}$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	0.49	1.52	0.35	1.82	0.62	0.97	0.91	0.54	0.72
	2	0.64	1.05	0.62	1.76	0.63	1.05	0.73	0.59	0.72
	3	0.61	1.11	0.36	1.98	0.66	1.23	0.69	0.63	0.71
	4	0.47	0.97	0.33	1.84	0.59	0.80	0.68	0.54	0.65
	5	0.41	0.88	0.24	1.86	0.63	1.30	0.55	0.57	0.67
	6	1.02	1.04	0.31	1.56	0.62	1.11	0.56	0.46	0.67
	7	0.35	1.31	0.56	1.63	0.63	1.38	0.68	0.58	0.53
	8	1.26	0.72	0.21	1.63	0.64	1.30	0.88	0.56	0.78
	9	0.51	0.71	0.26	1.71	0.65	1.19	0.63	0.57	0.75
$\theta {}_{AO}$/${}^{″}$	10	0.15	1.06	0.19	1.81	0.62	0.96	0.68	0.53	0.59
	11	0.97	0.80	0.29	1.86	0.64	1.16	0.65	0.61	0.59
	12	0.93	1.25	0.37	1.65	0.61	1.26	0.74	0.50	0.73
	13	1.02	1.83	0.39	1.96	0.61	1.28	0.78	0.65	0.77
	14	0.38	1.87	0.27	1.93	0.59	0.95	0.83	0.57	0.66
	15	0.51	2.75	0.75	1.80	0.63	0.92	0.79	0.58	0.76
	16	1.45	2.40	0.41	2.06	0.68	1.23	0.99	0.65	0.77
	17	1.29	0.97	0.45	1.77	0.63	1.31	0.66	0.59	0.77
	18	0.74	1.02	0.44	1.89	0.62	1.10	0.71	0.65	0.75
	19	1.06	0.97	0.43	1.70	0.62	1.28	0.77	0.59	0.73
	20	0.61	1.82	0.36	1.93	0.66	1.29	0.74	0.63	0.87
	Median	0.63	1.06	0.36	1.82	0.63	1.21	0.72	0.58	0.72
	Mean	0.74	1.30	0.38	1.81	0.63	1.15	0.73	0.58	0.71

,

Table A10. NCACN integrated astronomical parameters details (${\sigma }_I^2$ @$\lambda$ = 550 nm).

	Flight					Method
Parameter	Number	MT	HV	H9	DN	TE	EN	WT	WG	ME
	1	2.28	0.18	1.94	0.14	0.80	0.50	0.36	1.16	0.69
	2	0.63	0.30	1.33	0.14	0.74	0.32	0.50	0.94	0.60
	3	0.69	0.27	1.90	0.12	0.69	0.25	0.52	0.82	0.62
	4	1.19	0.33	2.85	0.14	0.85	0.54	0.58	1.06	0.74
	5	1.66	0.39	2.96	0.15	0.72	0.22	0.79	0.99	0.68
	6	0.29	0.30	2.07	0.17	0.73	0.29	0.72	1.26	0.64
	7	1.38	0.22	0.99	0.16	0.73	0.20	0.53	0.97	1.10
	8	0.19	0.53	3.69	0.16	0.64	0.21	0.33	0.96	0.49
	9	0.79	0.54	2.83	0.14	0.65	0.25	0.63	0.93	0.50
${\sigma }_I^2$	10	5.37	0.29	8.17	0.14	0.73	0.35	0.52	1.01	0.96
	11	0.36	0.45	2.42	0.14	0.74	0.26	0.62	0.88	0.85
	12	0.31	0.23	1.62	0.17	0.75	0.23	0.47	1.19	0.56
	13	0.32	0.14	1.40	0.13	0.72	0.22	0.40	0.80	0.54
	14	1.32	0.14	2.49	0.13	0.81	0.41	0.38	0.99	0.65
	15	0.74	0.09	0.77	0.13	0.68	0.39	0.41	0.96	0.51
	16	0.23	0.10	1.58	0.11	0.61	0.24	0.29	0.79	0.54
	17	0.22	0.34	1.18	0.14	0.69	0.21	0.51	0.91	0.52
	18	0.45	0.31	1.25	0.13	0.72	0.29	0.48	0.86	0.52
	19	0.27	0.33	1.26	0.15	0.69	0.22	0.41	0.86	0.57
	20	0.85	0.14	1.66	0.12	0.64	0.22	0.41	0.84	0.44
	Median	0.66	0.30	1.78	0.14	0.72	0.25	0.49	0.95	0.58
	Mean	0.98	0.28	2.22	0.14	0.72	0.29	0.49	0.96	0.64

. Median values of $r_0$, $ϵ$, ${\theta }_{AO}$, and ${\sigma }_I^2$ calculated from MT are 10.31 cm. 1.08${}^{″}$, 0.63${}^{″}$, and 0.66${}^{″}$, respectively. The parameters calculated from ME are 8.83 cm, 1.26${}^{″}$, 0.72${}^{″}$, and 0.58${}^{″}$. All the integrated astronomical parameters are portrayed in Figure 8, and the relevant statistical feature of these parameters are summarized in

Table 5. NCACN integrated astronomical parameters statistics (@$\lambda$ = 550 nm).

	Statistics	HV	H9	DN	TE	EN	WT	WG	ME
	R	−0.22	0.51	−0.006	0.64	0.59	0.40	0.42	0.60
$r_0$	$Bias$	−4.68	5.90	−9.75	2.46	−3.90	−1.42	4.11	1.74
	$MAE$	6.28	5.98	9.75	4.04	4.19	3.69	4.86	3.63
	$RMSE$	7.61	7.17	10.89	4.87	5.37	4.44	6.03	4.27
	R	−0.16	0.62	0.12	0.58	0.64	0.24	0.53	0.58
$ϵ$	$Bias$	0.63	−1.32	0.84	−0.007	0.58	0.43	−0.37	0.08
	$MAE$	0.72	1.40	0.84	0.60	0.60	0.57	0.76	0.55
	$RMSE$	1.09	1.60	1.20	0.80	0.95	0.94	0.88	0.75
	R	0.01	0.03	−0.03	0.38	0.47	0.32	0.20	0.45
${\theta }_{AO}$	$Bias$	−0.56	0.36	−1.07	0.11	−0.41	0.01	0.16	0.03
	$MAE$	0.67	0.41	1.07	0.29	0.43	0.29	0.29	0.29
	$RMSE$	0.85	0.52	1.13	0.36	0.51	0.33	0.38	0.32
	R	−0.07	0.81	−0.05	0.31	0.39	0.11	0.24	0.59
${\sigma }_I^2$	$Bias$	0.69	−1.24	0.84	0.26	0.69	0.48	0.02	0.34
	$MAE$	0.76	1.31	0.84	0.64	0.69	0.64	0.69	0.60
	$RMSE$	1.36	1.54	1.42	1.16	1.31	1.24	1.13	1.11

. The features of these parameters’ statistics are similar to the results in ECACN. The ME correlation coefficients of all parameters are even better in general compared to ECACN. Meanwhile, the deviation statistics are relatively small overall compared to the other approaches. A comprehensive comparison in two experimental areas between MT and the best estimations of all the parameter statistics were summarized in

Table 6. Performance of MEP/ME against the best one (within parentheses) (the integrated astronomical parameters were calculated for the wavelength of light at $\lambda$ = 550 nm. The values retain two decimal places).

Areas	Parameters	R	$\mathit{Bias}$	$\mathit{MAE}$	$\mathit{RMSE}$
	$C_n^2$	0.65(0.65:ME)	−0.11(−0.005:WT)	0.51(0.51:ME)	0.64(0.64:ME)
	$r_0$	0.69(0.70:DN)	3.13(0.005:WT)	4.01(3.27:WT)	5.32(4.28:WT)
ECACN	$ϵ$	0.51(0.65:WT)	−0.20(0.15:EN)	0.38(0.29:WT)	0.42(0.39:WT)
	$\theta {}_{AO}$	0.38(0.56:DN)	0.08(0.05:WT)	0.28(0.28:WT)	0.40(0.40:ME)
	${\sigma }_I^2$	0.50(0.72:TE)	−0.004(−0.004:ME)	0.28(0.27:WT)	0.34(0.34:ME)
	$C_n^2$	0.76(0.76:ME)	−0.09(0.02:WT)	0.45(0.45:ME)	0.58(0.58:ME)
	$r_0$	0.60(0.64:TE)	1.74(−1.42:WT)	3.63(3.63:ME)	4.27(4.27:ME)
NCACN	$ϵ$	0.58(0.64:EN)	0.08(−0.007:TE)	0.55(0.55:ME)	0.75(0.75:ME)
	$\theta {}_{AO}$	0.45(0.47:EN)	0.03(0.01:WT)	0.29(0.29:ME)	0.32(0.32:ME)
	${\sigma }_I^2$	0.59(0.59:ME)	0.34(0.02:WG)	0.60(0.60:ME)	1.11(1.11:ME)

. Although MEP is not always the best estimation among these eight approaches, its gap with the optimal approach is minimal. All the above show MEP’s considerable universality in studying optical turbulence characteristics.

5. Conclusions

In this study we propose a multi-model ensemble pattern method to estimate the $C_n^2$ based on several existing physical-based approaches. Balloon radiosonde data were collected in two areas of China to validate this method. Multiple dimensions evaluation including $C_n^2$ and the integrated astronomical parameters ($r_0$, $ϵ$, ${\theta }_{AO}$ and ${\sigma }_I^2$) of all approaches were done against the MT measured results. Statistical analysis of these methods’ performance mainly focuses on the overall trend (R) and deviation (Bias, MAE, and RMSE). The best performance of all approaches against the MEP is summarized in Table 6. The $C_n^2$ correlation coefficients of MEP are up to 0.65 and 0.76. The overall agreements of the $C_n^2$ profiles in two areas are quite good. A single profile has an even higher correlation coefficient of more than 0.95. Several statistical assessments of deviations of $C_n^2$ are relatively small. These indicate that MEP has the capacity to estimate $C_n^2$ well. Meanwhile, the evaluations of integrated astronomical parameters also show its promising potential for calculating these parameters as $C_n^2$ does. Although MEP was not always the best method in all parameters statistical evaluations, it showed competitive performance in these evaluations. Hence, the MEP method presented good stability and universality, and even the validation radiosonde data were collected in different areas, which meant significantly different atmospheric conditions. The MEP appreciably contains more information than a single method, including thermodynamic and dynamic factors of the optical turbulence. Moreover, the MEP method could be less sensitive to different parametric settings caused by each method, producing a more robust $C_n^2$ estimate.

It should be noted that a single approach performed relatively well after well-designing the relevant parameters according to field radiosonde measurement in previous practice [21,28,29,30]. However, the designed parameters might be less effective for other sites. This weakness makes a single model challenging to extend without sufficient prior data. Reliable and universal methods estimating $C_n^2$ from routine meteorological parameters are critical to evaluate the optical turbulence influence on adaptive optics systems. The most obvious example is a forecasting study in which the astronomer can not obtain optical turbulence in advance directly. Generally, researchers can forecast $C_n^2$ via weather forecasting models, combining different estimating approaches [8,12,39,40]. In addition, it also provides us with an applicable method to study regional optical turbulence characteristics from historical meteorological data. To be certain, more validation work should be done up until that point.