## 1 Introduction
Third, in the evaluation of the far-field beam quality, the scaling law assumes that the facula is a Gaussian distribution, which cannot accurately describe the far-field beam distribution. In recent years, many researchers are actively exploring new methods to calculate the far-field intensity distribution on HEL propagation. In 2016, [PERSON] [17] obtained the scaling rule of arbitrary aperture shape, beam type, and array beam under vacuum condition. However, [PERSON] [18] found that the formula was not suitable for beam quality evaluation under turbulent conditions through wave optics numerical simulation in 2018. [PERSON] [19] found that the spot distribution under the combined effect of turbulence and thermal blooming can be obtained by convoluting the far-field spot image of the simple turbulence effect and the far-field spot image of the simple thermal blooming effect. Compared with the simulation of wave optics, this method is a new HEL scaling law model for fast, accurate, and enhanced modeling of combined thermal blooming and turbulence effects on HEL propagation. However, [PERSON]'s method is an empirical exploration and at present it lacks a theoretical basis, as well as a more comprehensive analysis considering platform jitter, initial aberration of the laser system, and other factors. If we regard the transmission channel of the laser as an optical system, the calculation of the spot shape after the laser is transmitted through the atmosphere can be regarded as the calculation of the PSF of the system. In the field of astronomical imaging and adaptive optics (AO), the PSF calculation is extremely important and needs to be accurately known because it provides crucial information about optical systems for design, characterization, and optical diagnostics. It is generally difficult to obtain an accurate PSF. Scholars generally use empirical, theoretical, or parameter fitting methods, e.g., [PERSON] [20] proposed a semi-empirical technique that can improve the determination of AO off-axis PSF. The results show that this simple method reduces error in the prediction of the basic radial variation in the PSF. [PERSON] [21] derived an analytic formulation of the Anisoplanatic Point-Spread Function. The analytic formulation can captures the dependencies of anisoplanatism on aperture diameter, observing wavelength, angular offset, zenith angle, and turbulence profile. [PERSON] [22] developed a model of the AO long-exposure PSF based on end-to-end simulated PSFs using the OOMAO. [PERSON] [23] has proposed that the irradiance of partially coherent light propagating under the influence of multiple random effects is the convolution of irradiance propagating in vacuum and the point spread function of the system representing random effects. Based on [PERSON]'s new scaling law model results and the calculation method of PSF proposed by scholars, in this paper we try to explore and validate the theory model of the convolution method for HEL propagation with turbulence and thermal blooming effects. In this study, the process of laser propagating to the target is equivalent to the imaging process of point source, which provides theoretical support for the method. At the same time, considering the aberration of laser beam and the jitter effect of the platform, the method is extended. Finally, to test our theory and model, the improved method is compared with the simulation results of a four-dimensional program of wave optics.