## 2 Proposed Model: QPE Analytical Approximation Model
### QPE Approximation  
QPE can be expressed in radians as [23]  
\[\text{QPE}=\pi\Delta f_{r}\left(\frac{T_{int}}{2}\right)^{2} \tag{4}\]  
in which \(f_{r}\) is the Doppler rate of the echo signal, and \(T_{int}\) represents the integration time, also known as the synthetic-aperture time. In stripmap SAR geometry, the integration time is the time it takes the point of interest to enter and exit the entire 3 dB edge of the SAR antenna's beam illumination.  
The variables \(\mathbf{R}_{st}\), \(\mathbf{V}_{st}\), and \(\mathbf{A}_{st}\) represent the relative position, velocity, and acceleration of the spaceborne SAR satellite, respectively. In the ECI system, for the antenna aiming point on the ground, the commonly used \(f_{r}\) expression in stripmap mode when the squint angle equals zero is  
\[f_{r}=\frac{2}{\lambda}\left[\frac{\mathbf{V}_{st}\cdot\mathbf{V}_{st}}{ \left|\mathbf{R}_{st}\right|}+\frac{\mathbf{A}_{st}\cdot\mathbf{R}_{st}}{ \left|\mathbf{R}_{st}\right|}-\frac{\left(\mathbf{V}_{st}\cdot\mathbf{R}_{st} \right)^{2}}{\left|\mathbf{R}_{st}\right|^{3}}\right] \tag{5}\]  
The \(f_{r}\) expression in Equation (5) is an approximation expression, and it can achieve high precision if more terms remain. In the analytical approximation model, only the first two terms in Equation (5) are analyzed. This simplifies the model while retaining a relatively high precision.  
The orbit determination system measurements contain errors, which will lead to errors in \(\mathbf{R}_{st}\), \(\mathbf{V}_{st}\), and \(\mathbf{A}_{st}\). The terms that define the differences between the state vectors derived from measurements and the true state vectors are  
\[\Delta\mathbf{R}_{st} =\mathbf{R}_{st,e}-\mathbf{R}_{st} \tag{6}\] \[\Delta\mathbf{V}_{st} =\mathbf{V}_{st,e}-\mathbf{V}_{st}\] (7) \[\Delta\mathbf{A}_{st} =\mathbf{A}_{st,e}-\mathbf{A}_{st} \tag{8}\]  
where the subscript \(e\) indicates that the vectors contain measurement errors. Thus, the difference between the measured \(f_{r}\) and true \(f_{r}\) value is  
\[\Delta f_{r}=f_{r,e}-f_{r} \tag{9}\]  
The parameter \(f_{r}\) in Equation (5) is often used to determine the precise value for SAR raw data simulation or imaging algorithms. For the approximate expression of QPE, considering the magnitude of each term in \(f_{r}\), some terms can be omitted while still achieving an acceptable approximation of QPE. The VLS method in Section 2.2 maintains the real value of the relative range \(\rho=\left|\mathbf{R}_{st}\right|\). Thus, Equation (4) combined with Equation (9) gives the QPE approximation expression  
\[\text{QPE} =\pi\left(f_{r,e}-f_{r}\right)\left(\frac{T_{int}}{2}\right)^{2}\] \[\approx\frac{2\pi}{\lambda\rho}\left(2\Delta\mathbf{V}_{st}\cdot \mathbf{V}_{st}+\Delta\mathbf{A}_{st}\cdot\mathbf{R}_{st}\right)\left(\frac{ T_{int}}{2}\right)^{2} \tag{10}\]  
For convenience, the terms \(\Delta\mathbf{V}_{st}\cdot\mathbf{V}_{st}\) and \(\Delta\mathbf{A}_{st}\cdot\mathbf{R}_{st}\) are named _velocity_ and _acceleration vector term_ in the following sections.  
The approximation of QPE in Equation (10) is the basis of the QPE analytical approximation function and maximum QPE derivation.