## Abstract

A model for estimating astronomical seeing at Kunlun Station (Dome A, Antarctica) is proposed. This model is based on the Tatarskii equation, using the wind shear and temperature gradient as inputs, and a seeing model depending directly on the weather data is provided. The seeing and near-ground weather data to build and validate the proposed seeing model were measured at Dome A during the summer of 2019. Two calculation methods were tested from the measured weather data relating the wind shear and temperature gradient to a combination of the two levels for the boundary layer. Both methods performed well, with correlation coefficients higher than 0.77. The model can capture the main seeing trends in which the seeing becomes small when weak wind speed and strong temperature inversion occur inside the boundary layer.

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

## 1. Introduction

“Seeing” is a critical parameter representing the optical turbulence intensity in the atmosphere, which affects the quality of astronomical images. The Antarctic Plateau is almost completely covered by snow and ice, but it has attracted interest from the astronomical community owing to its excellent seeing [1,2]. Estimating seeing in Antarctica would be significant, as many experiments of observational astronomy have been planned there [3].

Dome A (80.37° S, 77.53° E), located in East Antarctica, has the highest elevation at 4089 m on the Antarctic Plateau. On Earth, the best seeing is possible from Dome A [1,2]. Excellent seeing was directly measured by the Kunlun Differential Image Motion Monitor (KL-DIMM [4]) at Dome A during summer and winter in 2019 [2]. The optical turbulence above Dome A was mainly concentrated in the first tens of meters above the ground [5]. Similarly, at the South Pole [6] and Dome C [7] in Antarctica, only a very small fraction of the optical turbulence was found to be above the boundary layer, based on the analysis of sounding data. In addition, turbulence occurs sufficiently infrequently at high altitudes based on the computation of the Richardson number [8]; despite the occurrence of turbulence, its implications for seeing would be minor because of the relatively low air density at high altitudes [9]. Moreover, estimating the optical turbulence at high altitudes (or “free-atmosphere” seeing) has an expensive equipment cost and low data accuracy. Thus, this study focused on the boundary layer, which is a major contributor to seeing in Antarctica, and the free-atmosphere seeing was assumed to be a constant (similar to a previous report [10]).

The Monin–Obukhov similarity (MOS) theory [11] has been used to estimate the refractive index structure constant ($C_n^2$) over snow and ice from weather data [12–15]. However, the MOS theory cannot be used to estimate the optical turbulence above Dome A owing to its limited applicability under very stable conditions [16–18] because an extremely stable atmosphere was found above the snow surface of Dome A [19,20]. Applying the MOS theory was attempted at Dome A, but its estimated $C_n^2$ was unacceptably high. Fortunately, the Tatarskii equation can be used in stably stratified conditions [21].

In this study, a model for estimating the astronomical seeing at Dome A from weather data (wind speed, temperature, etc.) is proposed. An equation for seeing directly depending on weather data is given based on the Tatarskii equation. The Tatarskii equation, which was built in the context of boundary-layer turbulence [22,23], has been used to estimate the profile of $C_n^2$ in the atmosphere, and the seeing can be calculated by integrating $C_n^2$ [24,25]. In addition, the Tatarskii equation was tested at the Antarctic Taishan Station [26]. Therefore, the proposed seeing model has a rigorous physical basis.

The remainder of this paper is organized as follows. In Section 2, the experiment is described briefly, and data are provided. In Section 3, a model for estimating seeing is proposed. In Section 4, the methods for applying and evaluating the proposed model are introduced. The estimation results of the proposed model are analyzed in Section 5. In Section 6, a discussion is provided. Finally, Section 7 concludes the paper.

## 2. Experiment and data

The measured seeing and near-ground weather data at Dome A were acquired from previous work [2] and are available for download. The seeing was observed every minute by the KL-DIMM between January 25, 2019 and March 11, 2019. KL-DIMM can endure low temperatures (−80°C). Weather data, including temperature and wind speed, were simultaneously measured using a new second-generation Kunlun Automated Weather Station (KLAWS-2G [20]), which has a mast height of 15 m. The main technical indicators of the observational instruments are shown in Table 1, and details can be found on the website of the Kunlun Observatory, Dome A, Antarctica.

## 3. Theory

#### 3.1 Optical turbulence

In this study, the boundary layer $C_n^2$ was estimated by the Tatarskii equation, which is expressed as follows [22].

where ${L_0}$ is the outer scale of turbulence, and*M*is the vertical gradient of the potential refractive index. The value of ${M^2}$ can be calculated from the pressure

*P*(unit: hPa) and temperature $T$ (unit: K) [27].

*z*(in m) is the height above the ground, and $\theta $ (unit: K) is the potential temperature defined by In June 1999, Ruggiero and DeBenedictis used the sounding data obtained from the Holloman Air Force Base in New Mexico [28] and concluded that the outer scale can be regarded as a function of wind shear

*S*(unit: s

^{−1}) and temperature gradient $dT/dz$ (unit: K·m

^{−1}):

*b*, $c$).

#### 3.2 Boundary-layer thickness

The boundary-layer thickness (${H_{BL}}$, in m), which determines the affected scope of boundary-layer optical turbulence, generally increases with an increase in the wind speeds within the boundary layer. At Dome C, the values of ${H_{BL}}$ appear to increase linearly with the wind speed, and a linear relationship between ${H_{BL}}$ and wind speed at 3.6-m ${U_{3.6}}$ (m·s^{−1}) can be established [30]:

The entrance pupil of the KL-DIMM was approximately 9 m above the ground [2], which would be affected by only a portion of the boundary layer. Based on Eq. (6), the thickness of the boundary layer affecting the observation of KL-DIMM can be expressed as

^{−1}suggests that the KL-DIMM is within the boundary layer (${H_{BLA}}$ > 0 m). Thus, the data associated with a ${U_{3.6}}$ weaker than 19/12 m·s

^{−1}were omitted when using the measurements in the linear fitting (see further details in Section 4).

#### 3.3 Seeing model

The KL-DIMM observed seeing (${\varepsilon _{TOT}}$), which is contributed by the free atmosphere (${\varepsilon _{FA}}$) and the boundary layer (${\varepsilon _{BLA}}$), can be calculated by [32]

Here, the boundary-layer seeing can be calculated as follows (above 9 m, the height of the entrance pupil of the KL-DIMM):

The observational data of sonic radar at Dome A indicated that the optical turbulence within the boundary layer varies by only a factor of a few and then drops by several orders of magnitude within a few meters [5]. In addition, an experiment over an asphalted area showed that the vertical atmospheric turbulence was almost in a homogeneous state and was hardly affected by the temperature and wind speed changes [33]. Then, the optical turbulence intensity inside the boundary layer can be considered to be uniform in the vertical direction (similar to previous research [10]). Thus, Eq. (9) can be rewritten as

Finally, substituting Eq. (5) into Eq. (1) and substituting the resulting equation into Eq. (10) gives an equation for seeing that depends directly on the weather data.## 4. Method for applying and evaluating the proposed seeing model

#### 4.1 Determination of seeing model

According to Eq. (11), the model for estimating seeing depends on *S* and $dT/dz$, and their values can be calculated by multiple-level measurements of KLAWS-2G. To test the robustness of the proposed model, two calculation methods were developed from the measured weather data relating the wind shear and temperature gradient to a combination of the two levels for the boundary layer:

The three undetermined coefficients ($A$, *B*, and $C$) in Eqs. (12a) and (12b) can be determined using a linear fitting, and the fitting data were obtained from the KL-DIMM (seeing) and KLAWS-2G (temperature and wind speed) between January 25, 2019, and February 25, 2019 (UTC). Such fitting only makes sense when the KL-DIMM is within the boundary layer; then, only the measurements under the condition of ${U_{3.6}}$ > 19/12 m·s^{−1} (when ${H_{BLA}}$ > 0 m) and ${\varepsilon _{TOT}}$ > 0.31 arcsec (the value of free-atmosphere seeing is assumed to be 0.31 arcsec) were used. Inspired by previous research [26], the fitting function was divided into two cases ($\partial \theta /\partial z$ < 0 and $\partial \theta /\partial z$ > 0). In addition, the Tatarskii equation shows weakness in the ability to estimate $C_n^2$ under neutral atmospheric conditions ($\partial \theta /\partial z$ ∼ 0) [26]. Therefore, in this study, the data meeting the condition $|{\partial \theta /\partial z} |$ < 0.05 K·m^{−1} also was not used. Finally, the coefficients ($A$, *B*, and $C$) in Eqs. (12a) and (12b) were fitted, as listed in Table 2.

Finally, ${\varepsilon _{BLA}}$ can be estimated by substituting the weather data into Eq. (12) using the values of *A*, *B*, and *C* in Table 2. Then, one can obtain ${\varepsilon _{TOT}}$ using Eq. (8), assuming ${\varepsilon _{FA}}$ = 0.31 arcsec.

#### 4.2 Method of statistical evaluation

The statistical operators average bias ($Bias$), root mean square error ($RMSE$), and Pearson correlation coefficient (${R_{xy}}$), which have been used to assess the accuracy of the estimated astronomical seeing in previous studies [34,35], are expressed as

*N*is the number of times for which a couple (${X_i}$, ${Y_i}$) is available, where $\overline {{X_i}}$ and $\overline {{Y_i}} $ are the corresponding average values.

## 5. Results

#### 5.1 Statistical analysis

To verify the prediction ability of the proposed seeing model, weather data for further eight days, measured by KLAWS-2G from February 26 through March 5, 2019 (UTC), were used. The missing measurements during these eight days were fewer. The total-atmosphere seeing values (${\varepsilon _{TOT}}$) were estimated using the model and Eqs. (8) and (12) with ${\varepsilon _{FA}}$ = 0.31 arcsec. Hence, the model estimated ${\varepsilon _{TOT}}$ cannot be smaller than 0.31 arcsec, which is one of the limitations of the model. Figures 1(a) and (b) show the scattering plots related to Eqs. (12a) and (12b), respectively. The estimated seeing using both models shows some negative bias when the measured seeing is large. Such deviations may be caused by the inaccurate estimation of free-atmosphere seeing.

Table 3 lists the $Bias$, $RMSE$, and ${R_{xy}}$ for seeing. Equation (12b) shows slightly better performance than Eq. (12a), and its ${R_{xy}}$ reaches 0.81. This could be caused by the larger height range (2 to 12 m), which provides a better representation of the boundary-layer atmosphere.

#### 5.2 Temporal evolution

Table 3 shows Eq. (12b) using the data at 2 and 12 m high has better prediction performance; thus, the temporal evolutions of ${\varepsilon _{TOT}}$ from Eqs. (8) and (12b) are shown in Figs. 2 and 3 (four days per figure), and the corresponding temporal evolutions of wind speed and temperature are also displayed to help analyze the correlation between the weather data and the seeing. The empty area in Fig. 2(a), (c), and (e) indicate that the measured data are missing.

Figures 2 and 3 show the periods when seeing is relatively large and small, respectively. Comparing Fig. 2 with Fig. 3 reveals that a better seeing can be obtained when weak wind speed and strong temperature inversion occur. The seeing model can capture the main characteristics of the seeing trends, as shown in Figs. 2(e) and 3(e).

## 6. Discussion

Figure 1 and Table 3 show that the seeing models using Eqs. (12a) and (12b) perform similarly. This may be because of the stable boundary-layer atmosphere at Dome A, and the estimated optical turbulence intensity inside the boundary layer could be less likely to be fragmented, thus tending to be uniform with height [5]. The estimations show little effect from the selected levels of weather data. In this study, it seemed feasible to use the weather data (2 and 6 m) below the entrance pupil (9 m) of the KL-DIMM to estimate the seeing (${R_{xy}}$ = 0.77). The intensity of turbulence at Dome C has also been observed to be distributed almost uniformly within the boundary layer see a previoius article [30]. In summary, these results suggest that the assumption that the optical turbulence intensity inside the boundary layer is uniform is reasonable.

As discussed elsewhere [36], the temperature profile is mostly linear with height in the weakly stable regime, whereas the temperature profile in the very stable regime is exponential in shape. Thus, Figs. 2(d) and 3(d) show the mean vertical profiles of the weakly and very stable boundary layer, respectively. Comparing Figs. 2(f) and 3(f) reveals that a better seeing can be obtained in a very stable state. Infrared radiative cooling behaves like a diffusive process, giving the temperature profile an exponential shape [37]. Therefore, the low wind conditions and exponential shape of the temperature profile shown in Fig. 3 suggest very weak turbulence activity and a radiative-dominated stable boundary layer [38]. The proposed seeing model, Eq. (11), can capture such features. This is because the boundary-layer thickness (${H_{BLA}}$) is proportional to the wind speed (${U_{3.6}}$), as defined in Eq. (7), and the boundary-layer seeing value (${\varepsilon _{BLA}}$) is inversely related to the temperature gradient ($dT/dz$), as the fitted model coefficients presented in Table 2 show ($C$ is negative at $\partial \theta /\partial z$ > 0.05 K·m^{−1}).

## 7. Conclusion

A model including the wind shear and temperature gradient for estimating astronomical seeing was proposed based on the Tatarskii equation. The seeing and near-ground weather data at Dome A, which were measured by the KL-DIMM and KLAWS-2G, were selected as inputs for the seeing model. Two methods of calculating the wind shear and temperature gradient were utilized to test the prediction ability of the proposed model, both of which provided good performance, with ${R_{xy}}$ values higher than 0.77.

The results show that the seeing became small when weak wind speed and strong temperature inversion occurred inside the boundary layer. The proposed seeing model can quantitatively express their relationship and achieve good estimation compared with the measurements.

To the authors’ knowledge, this study is the first to build a direct relationship between seeing and weather data over the Antarctic Plateau. Such a simplified calculation of seeing should be attempted at other sites, such as Domes C and F. In addition, the MOS theory becomes less credible when the boundary layer is under a stable atmospheric condition [16–18]. However, the air above the Antarctic Plateau is known to be a stable atmosphere [3]. In this study, the seeing model based on the Tatarskii equation was found to be a reliable alternative to the MOS theory because the Tatarskii equation has been recommended for use in stably stratified conditions [21].

## Funding

Foundation of Key Laboratory of Science and Technology Innovation of Chinese Academy of Sciences (CXJJ-19S028); National Natural Science Foundation of China (41576185, 91752103).

## Acknowledgments

We wish to thank Bin Ma, Zhaohui Shang, Yi Hu, et al. (National Astronomical Observatories, Chinese Academy of Sciences) for their shared data [2].

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are available in Ref. [2].

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