## 2 Linearization Method
### Range Migration Algorithm in SAR Imagery  
The Range Migration Algorithm (RMA) here has been previously described [19,20]. Consider an ideal SAR image and geometry in Figure 2 with the following properties: (1) the aperture is a straight line; (2) radar data are sampled at uniform intervals; (3) the image scene is broadside and centered on the aperture; (4) the radar antenna illumination is a constant amplitude plane wave.  
At the beginning of RMA processing, the collected SAR signal history is range compressed, match filtered, and compensated for range skew if the data are from a pulse radar. This is not necessary for FMCW radar data. Then, range curvature is removed for scatterers at the center of the scene. This range curvature is recognized in the range profile as an arc with a quadratic phase characteristic.  
The RMA operates on the two-dimensional complex phase space \(S(k_{x},k_{r})\) where \(k_{r}\) is the range phase space and \(k_{x}\) is the along-track phase space, if range compressed pulse data would require transformation to the phase domain by a Fourier Transform (FT) on the pulse-by-pulse data. The sampled data of an FMCW radar are already in terms of sampled phase, \(k_{r}\), but should be windowed for a suitable method such as a Hamming window. If these data represent real only phase history, it is necessary to transform this into complex data by techniques such as a Hilbert transform or by an Inverse Fourier Transform (IFT) on the one-sided result of a Fourier Transform (FT). Finally, a one-dimensional FT taken in cross-range will yield the \(k_{x}\) along-track phase space.  
The second step is the application of a two-dimensional matched filter. This operation corrects for range curvature of all scatterers at the same range as the scene center. The phase of this two-dimensional matched filter is given by:  
\[\Phi_{mf}(k_{r},k_{x})=-R_{s}k_{r}+R_{s}\sqrt{k_{r}^{2}-k_{x}^{2}} \tag{9}\]  
Figure 2: SAR Imagery and scene geometry (**a**) SAR image processing (Purdue), (**b**) SAR image processing (sandia.gov, accessed on 20 October 2020).  
where \(R_{s}\) is the slant-range to the center of the scene as shown in Figure 3 above. For scatterers at slant-ranges other than \(Rs\), some curvature will remain.  
For an ideal linear FM/CW radar, a point scatterer on the center line of the scene at \(R_{s}\) is a sinusoid in the \(k_{r}\) coordinate at every point along track corroroduct \(k_{x}\) in the (\(k_{x}\),\(k_{r}\)) domain. However, point scatterers at other ranges would have some residual curvature, i.e., the sinusoid stretches (in cases where the target slant-range is less than \(R_{s}\)), or compresses (if the slant-range is greater that \(R_{s}\)) as the user traverses in the \(k_{x}\) direction. On top of this curvature is the phase-error caused by a nonlinear VCO in the FMCW radar considered here.  
The Stoltz transformation corrects this their residual curvature described above [21, 22]. This transform is a convenient mechanism to also correct for phase-error. The Stoltz transform is a one dimensional mapping of (\(k_{x}\),\(k_{r}\)) domain space into (\(k_{x}\),\(k_{\gamma}\)) space by  
\[k_{\gamma}=\sqrt{k_{r}^{2}-k_{x}^{2}} \tag{10}\]  
This is implemented by interpolation, so the result is uniformly spaced samples \(k_{r}\) remain uniformly spaced in \(k_{\gamma}\). If not, the resultant pixel size would vary in range across the image.  
The final step in Range Migration Algorithm (RMA) is a two-dimensional Fourier transform that result in an image in range and cross-range. This would result in a well-focused image for an ideal radar.