## 1 Introduction  
Real-time synthetic aperture radar (SAR) imaging has always been a research focus in certain applications. Real-time imaging enhances the effectiveness and widens the application range of the SAR technique. In airborne SAR, real-time sensing plays an important role, and there are many well-established systems [1; 2; 3; 4; 5] and techniques [6; 7; 8; 9; 10; 11]. Real-time processing techniques are also available for unmanned aerial vehicle (UAV) SAR applications [12; 13; 14]. For spaceborne SAR, because of its unique configuration and onboard computation capabilities, real-time SAR imaging systems and techniques are still being developed for both digital [15; 16; 17] and optical [18] methods. For the purpose of implementing onboard real-time spaceborne SAR imaging, a set of Doppler parameters with high precision and accuracy must be calculated, and accurate position and velocity vectors are needed for the Doppler parameter calculation. In typical methods used to process SAR echo data, a high-accuracy orbit ephemeris is applied in the ground-based processing system, and state vectors, i.e., position and velocity vectors of the SAR satellite, are determined with high accuracy. However, this kind of high-accuracy orbit ephemeris is only generated hours, days, or even weeks after a single SAR observation [19]. Thus, this kind of high-accuracy orbit ephemeris cannot be obtained during real-time SAR imaging. Nevertheless, we are still able to achieve onboard orbit determination using GNSS-based receivers and appropriate algorithms [20; 21]. It is acknowledged that this approach to orbit determination has relatively low accuracy in the state vectors of SAR satellites compared with the high-accuracy orbit ephemeris. However, it remains the best, and possibly the only, onboard orbit determination method that is available for real-time SAR imaging. SAR mission designers must account for the fact that errors in such onboard orbit determination data will adversely affect the quality of the SAR imaging product through inaccurate Doppler parameter calculation. Although estimation algorithms have been developed to generate the Doppler centroid and Doppler rate with high accuracy for SAR imaging, they are time-consuming in practice. Real-time spaceborne SAR imaging certainly does not belong on a mission that is not time-sensitive. The product of real-time SAR imaging may be the source data for other mission types, such as onboard deformation monitoring or target recognition. Given these requirements, the estimation algorithm should not be built into the processing workflow because of its long processing time. Thus, the concept of quadrature phase error (QPE) is introduced to evaluate the influence of onboard orbit determination errors on spaceborne SAR imaging products. The azimuth impulse response width of a SAR image has a certain relationship with QPE, and it provides a more intuitive evaluation standard of the errors. Errors of an onboard orbit determination system are often given in the form of probabilities, along with the type of probability distributions and the corresponding parameters. The most common case for these errors is a normal distribution with an expected value of zero and a standard deviation \(\sigma\). Therefore, the probability distribution of QPE needs to be examined. In practice, there exist two major methods that have been applied to QPE analysis, i.e., the extreme value method and the numerical simulation method. The extreme value method is the simplest way to investigate QPE; it includes the maximum possible error in the calculation and ignores the probability distribution of the errors. This method faces challenges in revealing the QPE probability distribution, which means it only reflects the extreme situation in practice. For the numerical simulation methods, Monte Carlo simulations are often applied [22]. However, this approach requires a huge number of generated error samples and repeated experiments.