## 5 Discussion  
We designed an analytical approximation model of the QPE introduced by orbit determination errors. Our aim is to reduce the calculations and the processing time while revealing the core of the process by which error is transferred to QPE calculations. The technical parameter iteration period during early-stage development for an onboard real-time SAR imaging mission will benefit from the proposed methods. SAR imaging requires the relative position, velocity, and acceleration vectors between the radar antenna and the targets to estimate the Doppler centroid and Doppler rate. Currently, in many ground-based processing systems for existing SAR missions, high-accuracy orbit determination data are applied to estimate the Doppler parameters. Such data have a time delay from the SAR raw data acquisition period, which is usually in the range of several hours or more. For real-time onboard SAR imaging missions, it is not realistic to wait for such high-accuracy orbit determination data to meet the \"real-time\" requirement. Real-time onboard orbit determination data have to be applied in the real-time onboard SAR imaging procedure. Current spaceborne SAR missions are usually based on a professional remote sensing satellite platform, and they are not as sensitive to perturbation. For a spaceborne SAR payload mounted on a small satellite platform to perform a constellation mission, such as MirrorSAR proposed by the DLR [27], there will be some differences (i.e., it is not as steady as a large remote sensing satellite platform) in the movement of the actual target satellite. In these situations, the errors in the real-time onboard orbit determination data should be evaluated in order to examine whether they will jeopardize the new real-time SAR imaging mission concept. There exist other terms of phase error in the context of SAR data processing [28]. Besides QPE, there are linear, higher-order (more than cubic), sinusoidal, and random phase errors, among others. We selected QPE as the error of interest because it has defocus and loss-of-resolution effects on the  
Figure 5: QPE result from numerical simulation (blue) compared with that from the analytical approximation model (red). The expected value and standard deviation of QPE are shown in (**a**) and (**b**), respectively. The dashed magenta line in (**b**) is the value of \(\sigma\left[\mathrm{QPE}\right]_{max}\) from Equation (47). image. An acceptable QPE should be seen as a tool that is used out of necessity but is not adequate for new SAR missions. Research on the prediction of SAR imaging quality from the precision of the orbit ephemeris has been carried out [29]: the upper limit expressions of multiple phase errors and their sensitivity to the orbit ephemeris covariance have been previously established. However, this method has two major limitations when applied to spaceborne real-time SAR imaging missions. Firstly, those upper limits only reflect the extreme situation, which will not always occur during a mission. We need to examine the whole situation of the phase error, as well as its probability form. Currently, the numerical simulation method can produce the probability of the phase error, but it is time-consuming. Secondly, in spaceborne real-time SAR imaging missions, it is impossible to get the orbit ephemeris directly, and it is calculated from onboard orbit determination data. The direct link between the precision of the onboard orbit determination data and the phase error in SAR imaging is still unclear. The proposed analytical approximation model is a potential solution to the two limitations above. With the precision of the onboard orbit determination system, which can be measured on the ground, the proposed analytical approximation model gives the direct QPE probability distribution result, which can simplify the analysis. Compared with existing methods, the proposed analytical approximation model reduces the total calculations and processing time while revealing the core of the process by which errors are transferred to QPE calculations.