## 4 Evaluation and Results
### Evaluation Model  
For the purpose of validating the proposed analytical approximation model for QPE, a Monte Carlo-based numerical simulation is introduced as a reference for validating the proposed method. In the simulation approach, large samples of error data are generated first according to the specified probability distribution parameters. Then, these error samples are input to a simulation program. For each sample of error data in the orbit determination data, the simulation program calculates the Doppler parameters of the SAR imaging procedure with and without the given error sample set and records the difference between these two situations. The probability distribution of the difference from the Doppler parameter simulation is analyzed to provide a reference for the validation of the analytical approximation model.  
In this simulation program, there are no approximations such as those in the analytical approximation model. For beam steering, the simulation uses the total zero Doppler steering (TZDS) method [26] of TerraSAR-X, which has a joint yaw and pitch steering to minimize the Doppler centroid residence. The ellipsoid reference in WGS 84 is chosen to be the Earth's surface mode in the simulation.  
The QPE approximation has the same slant range and integration time data as the Monte Carlo simulation in order to get a more precise approximation result for \(\
u\in[0,2\pi]\). The QPE maximum value calculation works with the mean slant range and integration time for a much simpler computation.  
The Monte Carlo method generates 30,000 error samples of the position and velocity vectors per true anomaly, and there are 1000 true anomaly calculation points in \(\
u\in[0,2\pi]\).  
The inputs of the simulation are provided in Table 1. These parameters can be divided into four groups. The first group includes the Earth's physical parameters \(E_{a}\) and \(E_{b}\), which represent the semi-major and semi-minor axes of the Earth ellipsoid, respectively. The second group includes the satellite platform orbit parameters, namely, the semi-major axes of the orbit \(a\), the argument of periapsis \(\omega\), the orbital inclination \(i\), and the eccentricity \(e\). The third group contains the SAR payload parameters, i.e., the center frequency of the transmitting signal, the center off-nadir angle, and the azimuth length of the antenna. The last group represents the probability distribution of the errors in the real-time orbit determination data. The errors have a normal distribution with an expected value of zero, thus only the standard deviations are listed in the table. \(\sigma_{p}\) and \(\sigma_{v}\) represent the standard deviations of the errors in the position and velocity data from the real-time orbit determination system. A comparison of the 3D positioning residuals (RMS) using broadcast ephemerides results in around 1.5 m in the reference [21]. We selected a 3 m standard deviation in the real-time positioning errors. The standard deviation in the real-time velocity error is approximated by the results in [20].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(E_{a}\) & 6,378,137.000 m & \(E_{b}\) & 6,356,752.314 m \\ \(a\) & 6,778,140.000 m & \(\omega\) & 90.00\({}^{\circ}\) \\ \(i\) & 97.42\({}^{\circ}\) & \(e\) & 0.0011 \\ SAR Center Frequency & 9.60 GHz & \(\sigma_{p}\) & 3 m \\ Center off-nadir angle & 33.8\({}^{\circ}\) & \(\sigma_{v}\) & 0.1 m/s \\ Antenna Azimuth Length & 1.92 m & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation Parameters.