## 4 Spatial Variance Analysis and Correction for GTLR-BiSAR
### Range Variance Analysis and Correction Strategy  
After the correction of azimuth variance of RD data, the supporting area of the targets cluster with the same \(R_{r0}\) migrates to the desired positions in RD domain. Nevertheless, the supporting area of the targets cluster is characterized with different curvatures along the range direction shown in Figure 8. Facing the range variance problem, we firstly analyze the bulk, total and diff RCMs in detail. After that, we deduce the chirp scaling function for GTLR-BiSAR system to equalize the curves on the basis of the former RCMs analysis. Finally, the correction strategy of the range variance is completed. The vertical and horizontal directions of Figure 8 are defined as the azimuth frequency and the bistatic range gate, in accordance with the definitions in Figure 5. The horizontal dashed line \(f_{a}\) and \(f_{a\_ref}\) are the Doppler frequency and azimuth reference frequency, the vertical black dotted lines represent the near \(R_{bi\_near}\), middle \(R_{bi\_ref}\), and far \(R_{bi\_far}\) bistatic ranges. The solid green, red, and blue curves are utilized to express the RCMs of targets located in the near, middle and far bistatic range gates, respectively. The three red dotted curves represent the RCMs of targets located in the middle bistatic range gate. By shifting the red dotted curve to the near and far range gates, we distinguish the range variance characteristics visually. We assume the distance from vertical black dotted lines to the responding solid curves as \(RCM_{total}\). The distance from vertical black dotted lines to the red dotted curves is supposed as \(RCM_{bulk}\). The difference between \(RCM_{total}\) and \(RCM_{bulk}\) is defined as \(RCM_{diff}\). According to Equation (21) in Section 2, we obtain the different \(RCM\) expressions as Equation (28). Take \(P\) as an example (the blue point in Figure 8), \(P\) is located in the far bistatic range curve. \(P_{c}\) is located in the middle bistatic range curve. Connect \(P_{c}\) with \(P\), the black dashed line \(P_{c}\)-\(P\) intersects the red dotted curve at \(P^{\prime}\). If we can shift \(P\) to \(P^{\prime}\) to complete the range variance removal, the middle and far curves are  
Figure 8: The range variance curves in RD domain. characterized with the same \(RCM_{bulk}\), the unified \(RCM\) correction can be performed to improve the processing efficiency. \[RCM_{total} =\frac{R_{r0p}}{D(f_{a},V_{r})}-\frac{R_{r0p}}{D(f_{a\_ref},V_{r})} \tag{28}\] \[RCM_{bulk} =\frac{R_{r0\_ref}}{D(f_{a},V_{r})}-\frac{R_{r0\_ref}}{D(f_{a\_ref},V_{r})}\] \[RCM_{diff} =RCM_{total}-RCM_{bulk}\] \[=\frac{R_{r0p}}{D(f_{a},V_{r})}-\frac{R_{r0p}}{D(f_{a\_ref},V_{r}) }-\frac{R_{r0\_ref}}{D(f_{a},V_{r})}+\frac{R_{r0\_ref}}{D(f_{a\_ref},V_{r})}\]  
The \(RCM_{diff}\) shift can be achieved by multiplying a chirp scaling function (CSF) [23] in monostatic SAR processing. On this basis, we deduce the CSF for the GTLR-BiSAR system. According to Equation (21) in Section 2, the target \(P\) locates in the range gate (in unit of time) as follows:  
\[\tau=\frac{1}{c}[R_{Tp}+\frac{R_{r0p}}{D(f_{a\_ref},V_{r})}+\frac{R_{r0\_ref}} {D(f_{a\_ref},V_{r\_ref})}-\frac{R_{r0\_ref}}{D(f_{a\_ref},V_{r\_ref})}] \tag{29}\]  
set \(P_{c}\) (the red point in Figure 8) as the original time, then we get:  
\[ \begin{split}\tau^{\prime}&=\tau-\frac{1}{c}[R_{T\_ ref}+\frac{R_{r0\_ref}}{D(f_{a},V_{r\_ref})}]\\ &=\frac{1}{c}(R_{Tp}-R_{T\_ref})+\frac{1}{c}[\frac{R_{r0\_p}}{D( f_{a\_ref},V_{r})}-\frac{R_{r0\_ref}}{D(f_{a\_ref},V_{r\_ref})}]\end{split}  \tag{30}\]  
Express \(RCM_{diff}\) with \(\tau^{\prime}\) as follows:  
\[ \begin{split}\Delta\tau^{\prime}(\tau^{\prime},f_{a})=\frac{RCM _{diff}}{c}&=[\frac{D(f_{a\_ref},V_{r})}{D(f_{a},V_{r})}-1]\tau^{ \prime}-[\frac{D(f_{a\_ref},V_{r})}{D(f_{a},V_{r})}-1]\frac{R_{Tp}-R_{T\_ref}} {c}\\ &\qquad+\frac{R_{r0\_ref}}{c}[\frac{D(f_{a\_ref},V_{r})}{D(f_{a},V_{r})D(f_{a\_ref},V_{r\_ref})}-\frac{1}{D(f_{a},V_{r\_ref})}]\end{split}