## 1 Introduction  
Atmospheric turbulence is one of the most prominent features in the atmospheric boundary layer [1]. The main causes of atmospheric turbulence are solar radiation, wind shear, and heat convection, which lead to a change in the temperature field and velocity field [2]. Atmospheric turbulence will cause random fluctuations in the atmospheric refractive index. The Earth's atmosphere is in a state of turbulent motion all the time, and the turbulence phenomenon is closely related to astronomy, meteorology, aviation, and other fields. In recent years, lasers have been widely used in communication and remote sensing, which has also greatly promoted research into light wave transmission in turbulent atmospheres. When light waves are transmitted in the atmosphere, the relevant features are affected by atmospheric turbulence and show an irregular changing state [3]. The randomness of atmospheric turbulence, which makes the refractive index of the atmosphere appear in random fluctuations, results in an optical turbulence effect in the transmission of light waves in the atmosphere [4]. This mainly includes light intensity fluctuations, arrival angle fluctuations, spot drifts, beam expansions, etc., which all lead to laser wavefront distortions and coherence damage, resulting in some difficulties in atmospheric remote sensing and satellite communication [5; 6; 7; 8]. At the same time, the atmospheric turbulence movement is accompanied by the transfer and exchange of energy, momentum, and matter, which will have an impact on temperature and humidity. In some arid areas, the atmospheric turbulence effect will even directly affect the amount of rainfall, having a serious impact on the climate [9]. The physical quantity measurements commonly used to describe atmospheric turbulence effects include the scintillation index, the refractive index structure constant, and coherence length [10; 11; 12]. Among them, the atmospheric refractive index structure constant \(C_{n}^{2}\) is an important physical quantity characteristic that can represent atmospheric turbulence intensity and directly describes the degree of variation in refractive index uneven random fluctuations. The distribution profile of the atmospheric refractive index structure constant with height is called the atmospheric turbulence profile. The acquisition of the atmospheric turbulence profile is the basis for the study of the vertical distribution characteristics of turbulence intensity, along with the statistics regarding the temporal and spatial changes in turbulence. Therefore, the accurate detection of the basic characteristics of the spatial and temporal structure of atmospheric turbulence and the acquisition of the real-time distribution of atmospheric turbulence intensity is of great significance for the evaluation of atmospheric laser transmission, optical communication, and the meteorological prediction model [13; 14]. Lidar, as an active remote-sensing technology, uses a laser as the light source and obtains atmospheric distribution information along the laser beam transmission path by detecting the backscattering signals of the interaction between the laser and the atmosphere. At present, the lidar technologies commonly used to detect atmospheric turbulence mainly include differential image motion lidar (DIM), differential column image motion lidar (DCIM), and differential wavefront lidar (DW). DIM lidar is designed according to the principle of differential image motion measurement (DIMM). The atmospheric coherence length \(r_{0}\) is obtained by receiving the dither variance of the centroid of the spot on the aperture. In 2016, [PERSON] et al. developed an atmospheric turbulence profile lidar based on differential image motion and conducted some experiments, but its resolution was limited [15]. In recent years, the S-DIMM+ method was proposed on the basis of DIMM, and the Shack-Hartmann wavefront sensor with a wide field of view was used to detect the atmospheric turbulence profile [16; 17]. In 2018, [PERSON] et al.