M6. Synthetic Aperture Radar

1. The SAR Principle

A real aperture of length $L$ at wavelength $\lambda$ and range $R$resolves along-track detail no finer than $\delta_a^{\text{real}} = R\lambda/L$. For a 12 m antenna at 850 km, $\delta_a^{\text{real}} = 4$ km — useless for imaging.

A moving radar collects returns from many positions along its flight path. Coherently combining them synthesizes an equivalent aperture of length

giving azimuth resolution:

Counterintuitively, the smaller the real antenna, the finer the synthetic resolution — because a narrower real beam is compensated by a longer usable synthetic aperture. A 12 m SAR antenna gives 6 m azimuth resolution, independent of range or wavelength.

2. Phase History and Azimuth Compression

A stationary point target at closest-approach range $R_0$ is seen at instantaneous range$R(\eta) = \sqrt{R_0^2 + (v_p\eta)^2} \approx R_0 + v_p^2\eta^2/(2R_0)$ where $\eta$is azimuth time. The received phase is

This is a linear FM chirp in azimuth time with rate $K_a = 2 v_p^2/(\lambda R_0)$ and bandwidth $B_a = K_a T_{\text{SA}} = 2v_p/L$ (time-on-target). The matched filter in azimuth gives compression $1/B_a$ seconds — equivalently $L/2$ meters.

3. The Range-Doppler Algorithm (RDA)

The classic SAR processor. Steps:

1. Range compression (fast-time matched filter with transmitted chirp).
2. Azimuth FFT to enter range-Doppler domain.
3. Range Cell Migration Correction (RCMC): the target crosses range bins during the synthetic aperture; correct the shift $\Delta R(f_\eta)=\lambda^2 R_0 f_\eta^2/(8 v_p^2)$.
4. Azimuth compression with matched filter in Doppler domain.
5. Inverse azimuth FFT back to image domain.

Computational cost: $O(N^2\log N)$ — tractable on modest hardware. RDA is ideal for stripmap mode; for large squint angles or spotlight, more advanced algorithms are needed.

4. Chirp Scaling and ω-k Algorithms

The Chirp Scaling Algorithm (CSA) (Raney et al. 1994) avoids the interpolation of RCMC by scaling the range chirp in Doppler domain, yielding a better-posed processing flow. The $\omega$-k (or “range migration”) algorithm works entirely in the 2-D frequency domain using a Stolt interpolation

producing exact, squint-free compression for any geometry. It is the standard for spotlight SAR and modern spaceborne systems.

5. SAR Imaging Modes

Each mode is a different compromise on the azimuth-resolution × swath product. TOPSAR (terrain observation by progressive scans) combines a ScanSAR-like wide swath with uniform image quality using backward-then-forward beam steering — the current workhorse for global monitoring.

6. Interferometric SAR (InSAR)

Two SAR images acquired from slightly different positions (perpendicular baseline $B_\perp$) have slightly different path lengths to the same ground point. The interferometric phase

encodes elevation $h$ to within a $2\pi$ wrapping. For Sentinel-1 at C-band with $B_\perp=150$ m, one fringe corresponds to a height ambiguity of ~60 m. Phase unwrapping (branch-cut, minimum-cost-flow, SNAPHU) reconstructs the absolute phase.

Differential InSAR (DInSAR) subtracts two interferograms with identical baselines but different times, leaving only surface deformation. A single $2\pi$ fringe corresponds to$\lambda/2 \approx 2.8$ cm of line-of-sight displacement at C-band — the basis for earthquake, volcano, and glacier monitoring.

Persistent Scatterer Interferometry (PSI) and SBAS (Small BAseline Subset) time-series methods achieve millimeter/year precision over urban areas from multi-year stacks.

7. Polarimetric SAR

Transmitting and receiving alternately horizontal (H) and vertical (V) polarizations gives the full $2\times 2$ scattering matrix:

Pauli decomposition rewrites this into physically meaningful components:

interpreted as: surface (single bounce, blue), double-bounce (dihedral, red), volume (cross-pol, green). A Pauli RGB composite of fully polarimetric SAR shows cities red (dihedrals), forests green (random orientations), ocean dark (weak and copolar), bare ground blue.

More advanced: Cloude-Pottier $(H, A, \bar\alpha)$ decomposition, Freeman-Durden three-component, Yamaguchi four-component. These are the foundation of biomass estimation and land-cover classification from space.

8. SAR Applications

• Earthquake deformation: DInSAR of the 1992 Landers quake (Massonnet et al. 1993) launched modern geodesy from space.
• Volcano inflation: Kilauea, Etna monitored in near-real-time with Sentinel-1.
• Glacier & ice-sheet motion: offset tracking and InSAR at tens of m/yr accuracy.
• Forest biomass: L-band (ALOS-2, NISAR) polarimetric inversion, BIOMASS P-band (2025).
• Flood mapping: water surfaces are dark and glassy; rapid-disaster response.
• Oil-spill detection: slicks dampen capillary waves, appearing dark in VV.
• Ship detection: metal hulls bright against ocean; automatic classification via deep learning.
• Archaeology: subsurface moisture differences reveal buried walls in hyper-arid regions.

See the Earth Observation course for data-access practicalities.

9. Major Spaceborne SAR Systems

10. Speckle and Multi-Looking

Coherent radar imagery suffers from speckle: the interference of many elemental scatterers within a resolution cell produces Rayleigh-distributed amplitude, appearing as salt-and-pepper noise. Multi-looking (incoherent averaging of adjacent pixels) reduces speckle variance by $1/N$at the cost of resolution. The equivalent number of looks (ENL) is the standard speckle metric.

Modern despeckling: non-local means (NL-SAR), total-variation denoising, and CNN-based methods (SAR2SAR) outperform classical Lee/Frost filters while preserving edges.

11. Azimuth and Range Ambiguities

Every SAR must sample Doppler at PRF > $B_a = 2 v_p/L$ to avoid azimuth ambiguities (aliased returns from the azimuth antenna sidelobes). Simultaneously, PRF < $c/(2 R_{\text{swath}})$to avoid range ambiguities. These inequalities combine to a forbidden zone; the design sweet spot is called the diamond diagram. Typical Sentinel-1 IW PRFs are 1-1.5 kHz, swath ~250 km, resolution 5×20 m.

12. Geocoding and Terrain Correction

Raw SAR images are in slant-range / azimuth-time coordinates — not map coordinates. Geocoding projects the image onto a DEM-defined ground grid. Range-Doppler geocoding solves for each ground pixel the satellite position $\mathbf{s}$ satisfying

Topographic distortions — foreshortening, layover, and shadow — are geometric artifacts inherent to side-looking radar and must be corrected or masked.

Simulation: Azimuth Compression, InSAR, Pauli RGB

The Python simulation builds an azimuth phase history for a C-band SAR at Sentinel-1 parameters (850 km, 7.6 km/s, 12 m antenna), shows the resulting chirp and its compressed azimuth response confirming $\delta_a = L/2$, performs 1-D InSAR phase unwrapping, and generates a synthetic polarimetric Pauli RGB with urban, forest, and ocean regions.

13. Ground Moving Target Indication (GMTI) in SAR

Moving targets are mis-located in SAR imagery because their Doppler signature differs from stationary scatterers at the same range. A target with radial velocity $v_r$ is displaced in azimuth by $\Delta x = R v_r/v_p$ and defocused by its acceleration. Dual-channel SARs (ATI — along-track interferometry) use the phase difference between two along-track antennas to measure $v_r$ directly, enabling imaging plus tracking from a single sensor.

14. SAR Calibration and Radiometric Accuracy

Quantitative SAR requires absolute calibration. Corner reflectors deployed at known ground positions serve as calibration targets; their theoretical RCS$\sigma_c = 4\pi\ell^4/(3\lambda^2)$ for an equilateral triangular trihedral of leg$\ell$ provides a reference. Sentinel-1 maintains 1 dB absolute radiometric accuracy via permanent transponders and regular campaigns.

15. Autofocus Algorithms

Imperfect platform motion knowledge (GPS noise, atmospheric path, clock jitter) defocuses SAR images. Autofocus algorithms estimate the residual phase error directly from the data:

• Map drift: correlate two half-aperture images.
• PGA (Phase Gradient Autofocus): average gradient of isolated bright scatterers in sub-images.
• Minimum entropy: optimize phase to minimize image entropy.
• MCA (Multichannel autofocus): exploit redundancy across polarizations or subapertures.

PGA is the de facto standard for airborne spotlight SAR. Modern spaceborne SARs with GPS + star-tracker attitude seldom need autofocus, but it remains essential for UAV-mounted and squinted imaging.

16. Bistatic and Tomographic SAR

Separating transmitter and receiver on different platforms yields bistatic SAR — TanDEM-X was the first operational bistatic InSAR, generating the WorldDEM global DEM with 12 m posting. An inter-satellite baseline of ~150 m at X-band gives decimeter elevation precision.

SAR tomography (TomoSAR) uses N-baseline stacks to reconstruct the 3-D distribution of scatterers within a resolution cell — revealing forest vertical structure and individual building floors. The inversion is a sparse compressed-sensing problem.

18. Inverse SAR (ISAR)

The target moves relative to a stationary radar: pitch, roll, and yaw modulations provide the effective “aperture synthesis”. Ships at sea naturally rotate through waves; aircraft maneuver. ISAR images classify ships and aircraft from long range; it is the operational imaging mode of AWACS and maritime patrol radars.

19. Squinted and Sliding-Spotlight Modes

Sliding spotlight combines stripmap's unlimited azimuth coverage with spotlight's fine resolution by continuously steering the beam forward. The virtual rotation center is behind the target, so the beam dwells longer on each point. It is the standard high-resolution mode of Sentinel-1 and TerraSAR-X.

Squinted imaging (non-zero Doppler centroid) gains along-track access time but complicates processing because range-Doppler coupling becomes severe. Omega-k processing handles arbitrary squint angles exactly.

20. Compressed-Sensing and AI SAR

SAR scenes are typically sparse in appropriate bases, enabling compressed-sensing reconstruction from under-sampled data: fewer pulses, lower PRF, or azimuth-gap acquisitions. Neural-network imagers (U-Net variants) now produce competitive or superior SAR images from raw data, skipping the classical processing chain. These techniques matter most for low-cost small-satellite SAR constellations where onboard compute is limited.