###### Abstract  
Nowadays, the Earth observation based on the Moon has attracted attention from many researchers and relevant departments. There also exists a considerable amount of interest in monitoring large-scale and long-term geoscience phenomena using the Moon-based SAR (MBS). However, the Earth's observation from MBS has long transmission time, and the relative motion of MBS with its Earth ground target (EGT) is much different to the space-borne SAR, the above reasons indicate that the traditional stop-and-go model is no longer suitable for MBS in frequency domain imaging. Here a dual-path separate calculation method for single pulse is presented in this paper for a better match of a real scenario, and then the slant range is fitted to a high-order polynomial series. The MBS's location, the synthetic aperture time and other factors have effects on length of the dual-path and fit bias. Without thoroughly investigated phase de-correlation processing in frequency domain, and to avoid computational costs in traditional back-projection (BP) algorithm, the paper first proposes a fast back-projection (FBP) algorithm in time domain for MBS, a platform that has long transmission time and long synthetic aperture time. In the FBP algorithm, the original method, that projected echo on all pixels in the imaging area, is changed to projected echo on a centerline instead. A suitable interpolation for points on the centerline is adopted to reduce the projected error; the synthetic aperture length and imaging area are also divided into subsections to reduce computation cost. The formula indicates that the range error could be control once the product of sub-imaging area's length and sub-aperture's length stay constant. Through the theoretical analysis, the detailed range difference mainly at apogee, perigee, ascending, and descending nodes indicate the necessity to separately calculate the dual-path for MBS's single pulse transmission in Earth-Moon motion, with real ephemeris been adopted; then, the high-order polynomial fitting will better describe the motion trajectory. Lastly, the FBP algorithm proposed is simulated in a specific scenario under acceptable resolution, and the result shows its feasibility for image compression. M 2022 2021 2022 20  
the Earth and Moon centroids are approximately 3.6 \(\times\) 10\({}^{5}\), 4.0 \(\times\) 10\({}^{5}\), and 3.8 \(\times\) 10\({}^{5}\) km, respectively. Such a long distance brings about longer echo delay time and longer synthetic aperture time, which makes the traditional stop-and-go model, in which the position of the EGT and the platform (e.g., aircraft) are considered the same when echo is transmitted and received, no longer suitable for MBS. Among Earth-Moon periodic motion, the dual-path delay time of the echo can exceed 2.3 s, thus the distance between echo transmitted and received of MBS changes about 2 km as Moon's rotation speed fluctuates between about 960 km/s to about 1100 km/s, so to continue to reckon the transmission path of MBS using the stop-and-go model is inconsistent with reality. In the analysis of MBS scenario, the slant history in dual path are separately calculated. Second, although the ratio of average distance between Earth and Moon to the average Moon radius is only a trivial value of 0.0046, for a more accurate description of slant range and phase shift between MBS and EGT, the MBS should not simply be considered to coincide with the Moon centroid, because any locations of MBS in the near side will bring varying isocrange distribution, and thus bring varying visibility of EGT caused by signal reflective properties or terrain occlusion. Third, since the echo of MBS will pass through the entire ionosphere and atmosphere, the two spheres will undoubtedly have adverse effects not only on calculation of slant history and phase history especially in frequency domain, but also on signal power attenuation.