# High-Resolution and Wide-Swath 3D Imaging for Urban Areas Based on Distributed Spaceborne SAR
## 2 Materials and Methods
### Distributed SAR 3D Imaging Geometry and Theory  
This paper proposes a 3D imaging technology scheme of a distributed HRWS SAR with a multi-channel arrangement. The tomography uses multi-baseline satellite SAR to measure the same area at different angles and form a synthetic aperture in the elevation direction to obtain high-dimensional geometric resolution and achieve a real 3D imaging effect. The imaging geometry of distributed SAR tomography is illustrated in Figure 3. There is an SLC data set with N pieces of mutual registration, which is spatially arranged with a baseline \(b_{\perp n}\),\(n=0,\ldots,N-1\) relative to the reference image. Afterwards, phase correction processing is required on the dataset. SAR phase correction is a technique used to compensate for phase errors in SAR imaging systems. Phase errors can arise from various sources such as antenna imperfections, atmospheric disturbances, or platform motion. These errors can result in image distortions, blurring, or misalignment of the SAR imagery. The phase correction process involves estimating and compensating for the phase errors in the received radar signal. This is typically conducted by referencing a known phase reference or utilizing a calibration signal. By applying the phase correction algorithm, the phase errors can be effectively mitigated, leading to improved image quality and accuracy. The range-Doppler algorithm is used in this paper. This algorithm estimates the phase errors by analyzing the range and Doppler frequency information of the received radar signal. Based on these estimates, the algorithm applies a corrective phase shift to each range cell of the SAR image, thereby compensating for the phase errors [29]. Figure 1: The distributed HRWS SAR 3D imaging system. Figure 3: The geometry of the distributed SAR 3D imaging. Figure 2: The overall flow of the distributed HRWS SAR 3D imaging system. After phase correction of the data set is performed, the received signal \(g_{n}\) in each azimuth-range unit can be represented as backscattering intensity \(\gamma\) superposition of all signals in cross-range direction [30]:  
\[g_{n}=\int\gamma(s)e^{-j\pi\xi_{n}s}ds \tag{1}\]  
where \(s\) denotes the range of elevation of the scene; \(\xi_{n}=2b_{\perp n}/\lambda r\) represents the spatial frequency; \(b_{\perp n}\), \(\lambda\), and \(r\) denote the length of vertical baseline, wavelength, and range, respectively; and \(\gamma(s)\) represents the distribution of the backscattering coefficient in elevation. The above formula indicates that the received signal of the antenna is a Fourier sample which has the backscattering intensity along the azimuth \(s\) with frequency \(\xi_{n}\). Therefore, the 3D imaging problem is transformed into the inverse-Fourier transform of non-uniform sampling. In urban applications, the general scattering process has main scattering and a layover effect, so it can be considered that the intensity of backscattering \(\gamma\) is composed of a limited number of \(\delta\)-Dirac functions with different scattering phase centers. The typical composition is the ground, building surface, and top, as illustrated by the red wavy line/arrow in Figure 3.  
If L-order gridding discrete sampling (\(s_{l},l=1,2,\cdots,L\)) is performed on the elevation direction \(s\), the linear equations composed of M equations can be obtained. \[g=A\gamma+\varepsilon \tag{2}\]  
where \(g=\left[g_{1},g_{2},\cdots,g_{m}\right]^{T}\) denotes the observation vector, \(A_{ml}=\exp(j2\pi\xi_{m}s_{l})\) represents the elements of the observation matrix, \(\gamma=\left[\gamma_{1},\gamma_{2},\cdots,\gamma_{L}\right]^{T}\) is a discrete sampling of the elevation backscattering distribution, and \(\varepsilon\) represents the noise vector. The above formula is a system of underdetermined equations since \(L\gg M\). To simplify the problem, the noise vector is generally assumed to be stationary zero-mean Gaussian white noise, and the distribution model of the observation vector can be obtained [31].