## 2 Dechirp Influence on Squint Sliding-Spot Imaging Mode
### The Geometric Forward Calculation Process  
As in Figure 6, the SAR satellite images the ground area using the squint sliding-spot mode, and after the squint imaging process described in Section 2, a slant range image is output, which does not contain geographic information. Based on the image coordinate of any target point, we obtain the slant range and the satellite position at the azimuth time of the target. Then we can calculate the azimuth squint angle of the antenna toward the target, and combine with the slant range and the target elevation to calculate the real central pointing vector at the target. Finally, the intersection point of this pointing vector and the earth surface is obtained as the geographic coordinates of the target.  
(a) The image coordinate \((cr,ca)\) of the target point \(T\) is known, and \(cr\) is the range row number. Because the dechirped data are eventually focused to the location with the zero doppler frequency, the zero doppler moment of the target image point after dechirp is taken as the imaging moment of the point. Then a one-to-one correspondence between the azimuth line number \(ca\) of the image and the azimuth time \(\eta\) can be established. Calculate the slant range of \((cr,ca)\) as  
\[r=r_{min}(ca)+cr*c/(2*f_{S}) \tag{7}\]  
Here \(r_{min}(ca)\) is the real near range of line \(ca\) in the slant range image obtained after dechirp and cropping, which is determined by the original near range before dechirp and the offset \(D(\eta)\) in Equation (3) jointly. Because the slant range offset of each image line is different during the dechirp process, \(r_{min}(ca)\) changes with \(ca\). Suppose the surface elevation of the target \(T\) is \(H\), \(f_{S}\) is the range sampling frequency of SAR, \(\gamma\) is the range side-looking angle, which refers to the angle between the antenna pointing and the satellite orbital plane. Now we begin to solve the latitude and longitude coordinates \((T_{lat},T_{lon})\) of \(T\).  
(b) Suppose SAR locates at point \(S\) at time \(ca\), and its position coordinates in the WGS 84 coordinate system are \(\left(S_{x},S_{y},S_{z}\right)\). Extend the moment forward and backward at equal time intervals to obtain the adjacent moments \(ca-c1\) and \(ca+c1\); \(c1\) is the change row amount corresponding to the time interval, generally taking 3 to 5. Suppose SAR locates at point \(L\) and \(R\) at \(ca-c1\) and \(ca+c1\); its position coordinates are \(\left(R_{x},R_{y},R_{z}\right)\) and \(\left(L_{x},L_{y},L_{z}\right)\), respectively. The near slant range of the corresponding lines are, respectively, \(r_{min}(ca-c1)\) and \(r_{min}(ca+c1)\). Points \(L\), \(R\), and \(T\) form a triangle \(\Delta LTR\) whose vertex is \(T\) and base is \(LR\). Since the \(c1\) value is very small, \(|LR|\ll|ST|\), so \(S\) can be approximately taken as the midpoint of \(LR\), then the half distance of the base is \(c=|LR|/2\), and the length of the triangle midline on the base is \(|ST|=r\). As described above, the range history of any target on the slant range image after dechirp is symmetrical and straight, so the difference \(d\) between the two triangle waist lengths is the difference between two slant offsets of the line \(ca-c1\) and \(ca+c1\), and the calculation equation is  
\[d=|LT|-|RT|=r_{min}(ca-c1)-r_{min}(ca+c1) \tag{8}\]  
(c) According to the triangle midline theorem, knowing the half distance \(c\) of the base, the difference \(d\) between the lengths of the two waists, and the length \(r\) of the base midline, we can calculate the angle \(\theta\) between the midline and the vertical line of the base as \(\theta=arccos\left[\frac{d\cdot\sqrt{2}+c^{2}-d^{2}/4}{2\cdot r\cdot c}\right] -\frac{\pi}{4}\) (Figure 8), which is exactly the squint angle of SAR at point \(S\). The squint angle refers to the angle between the antenna pointing and the plane perpendicular to the satellite's motion direction.