## 1 Introduction  
Inverse synthetic aperture ladar (ISAL) is a kind of active imaging radar that combines the advantages of synthetic aperture techniques [1] with coherent laser radar to obtain high-resolution images [2]. Compared with conventional microwave radar, laser radar creates a larger Doppler bandwidth and improves the imaging quality in terms of azimuth resolution. Owing to laser heterodyne (coherent) detection, ISAL could monitor the amplitude and phase data of extremely weak echoes [3], which enables imaging for long-distance moving targets with a high resolution. However, the laser decoherence induced by atmospheric turbulence reduces the heterodyne detection efficiency and introduces extra phase noise, resulting in degradation of the image resolution. This is one of the main obstacles to the application of ISAL. Therefore, it is necessary to study systematically the effect mechanism of atmospheric turbulence on ISAL and find ways to overcome the adverse effects in order to successfully implement ISAL. Over the past decade, most previous simulation and experiment works on ISAL and SAL imaging in atmospheric turbulence have been demonstrated [4; 5; 6; 7; 8; 9; 10; 11]. In 2014, the performance characterization of phase gradient autofocus for ISAL was studied by [PERSON] [12], who showed that the phase gradient algorithm (PGA) can still maintain its advantage of automatic focusing under atmospheric turbulence and low signal-to-noise ratio (SNR) conditions. In 2016, based on a Monte Carlo random factor, [PERSON] [6] used a Kolmogorov phase screen to simulate turbulence and rank one phase error estimation to compensate SAL images. [PERSON] [3] demonstrated the correction capability of the PGA algorithm under a low carrier-to-noise ratio (CNR) and atmospheric turbulence through experimentation. In the same year, [PERSON] [13] found that the influence of atmospheric turbulence was related to the time through an experiment with ISAL at 1.1 km. That aside, there are few studies on eliminating or mitigating the adverse effects of atmospheric turbulence on ISAL, such as from [PERSON] and [PERSON] [14; 15]. In 2019, they evaluated the performance of three model error correction algorithms to mitigate atmospheric blurring in reconstructed imagery [14]. In 2020, model-based reconstruction algorithms with model error corrections are proposed to mitigate the deleterious effects of atmospheric turbulence and restore image quality [15]. In 2022, [PERSON] studied the imaging performance and anti-interference ability of an SAL with photonic lantern coupling under atmospheric turbulence [16]. At the same time, [PERSON] employed the PGA algorithm to study the ISAL imaging of rough targets in slant atmospheric turbulence through the use of an atmospheric random phase screen [11]. It was found that ISAL images with rough targets could be resolved not only under under weak turbulence but strong turbulence. Generally, turbulence-induced phase fluctuation is a continuously varying random process. Nevertheless, methods used in previous studies did not consider to describe the dynamic evolution of turbulence. Aside from that, the decoherence induced by turbulence could reduce the heterodyne efficiency and could introduce an extra random phase, which would deteriorate the phase stability and make coherent synthesis impossible. Thus, we develop corresponding methods to overcome the shortcomings of existing studies. Considering the dynamic evolution of turbulence, we employ the moving ILPS method [17] to simulate the time-varying receiving electric field. Based on the principle of heterodyne detection, the influence of atmospheric turbulence on heterodyne detection can be investigated from both the spatial and temporal domains. The spatial decoherence of echo causes distortions in the laser wavefront, degrading the optical field from a single mode to multi-mode, and causing a mismatch between the reference and return signal modes, resulting in a rapid decrease in the SNR.