# An Improved Back-Projection Algorithm for GNSS-R BSAK Imaging Based on CPU and GPU Platform
## 5 Discussion  
In this section, the necessity of GPU accelerating BPA of GNSS-R BSAR is analysed. The computational cost of the BPA range compression step can be expressed as \(N_{a}N_{r}\log_{2}N_{r}+N_{a}N_{r}\), where \(N_{a}\) and \(N_{r}\) are the number of range and azimuth samples, respectively. The computational cost of the back-projection step can be expressed as \(N_{m}N_{n}N_{a}\), where \(N_{m}\) and \(N_{n}\) are the numbers of samples of the imaging scene. Thus, the total computational cost for the improved BPA can be represented by \(N_{a}N_{r}\log_{2}N_{r}+N_{a}N_{r}+N_{m}N_{n}N_{a}\). The computational cost of an improved RPA is \(3N_{a}N_{r}\log_{2}N_{r}+2N_{a}N_{r}\log_{2}N_{a}+7N_{a}N_{r}\)[27]. It can be seen that the calculation amount of BPA is much greater than RDA. However, BPA is more suitable for GNSS-R BSAR. The reason is as follows:  
1. The geometry of GNSS-R BSAR is complex. The BPA can focus the echo data without the influence of geometry, while the frequency domain algorithms are greatly affected by the geometric structure [28]. Especially in the multisatellite fusion or multistation fusion mode [19; 23], more complex geometry will lead to most of the frequency domain algorithm, which is no longer applicable.
2. The frequency domain algorithms rely on the accuracy of the equivalent squint model and cannot achieve a long-time synthetic aperture [41]. However, increasing the synthetic aperture time can not only improve the azimuth resolution and signal-to-noise ratio but also improve the characteristic information of the target in the imaged area [24]. Especially in one station fixed mode, the synthetic aperture time can reach thousands of seconds. The BPA algorithm is not affected by the synthetic aperture time.  
Figure 11: Cross-sections for Target No. 6, (**a**) is range profile, and (**b**) is azimuth profile.  
In practical applications, whether airborne large scene imaging or one-station fixed mode long synthetic aperture time imaging, the BPA algorithm will achieve a huge amount of calculations, which will greatly affect the efficiency of BPA applications. Especially in the application of deformation monitoring, there is a demand for response time. Therefore, it is necessary to use GPU to accelerate BPA for reducing the imaged time.  
In this paper, we proposed GPU accelerated BPA, which has a good acceleration ability in GNSS-R BSAR's one-stop fixed mode. The simulation and experimental verification results verify the feasibility and effectiveness of the proposed algorithm. The simulated imaging is performed on an area of 4 km \(\times\) 4 km, and the synthetic aperture time is 300 s. The point targets in the image generated by the proposed algorithm are well focused. Comparing the proposed algorithm with the BPA run only on CPU, the image quality is the same, while the running speed is increased by 163.96 times. In the experiment, the synthetic aperture time was increased to the 1800 s for an image area of 600 m \(\times\) 600 m. The collected raw data were processed by the proposed algorithm and the BPA run only on the CPU, respectively. The imaging results are similar for the two algorithms, the running time of the proposed algorithm is 128.04 s, while the running time on the CPU is 19,974.24 s; the time taken by the proposed algorithm is greatly reduced.