PSI - Issue 84

Mario Costantini et al. / Procedia Structural Integrity 84 (2026) 859–866

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2. Advanced Micro-Doppler Analysis from Satellite SAR Data Modern VHR spotlight SAR acquisitions illuminate scenes for seconds, challenging the standard assumption of stationary targets. Any motion during this dwell moves the scattering phase center, creating a residual phase term, manifesting as a time-varying Doppler contribution, referred as Micro-Doppler. We treat such vibrations as a specific instance of the broader Doppler-variation sensing problem. Let denote SAR slow time (azimuth time) (Cumming et al., 2005). After standard focusing terms are removed, a motion-induced residual range component ( ) produces an additional phase contribution: ( ) ≈ 4 ( ) (1) The associated motion-induced instantaneous Doppler (in Hz) is: ( )= 21 ( ) ≈ 2 ( ) , (2) where ( ) is the line-of-sight (LOS) velocity component and is the radar wavelength. This relationship clarifies the general scope: estimating time-varying Doppler ( ) is equivalent to estimating the LOS velocity evolution; integrating ( ) yields LOS displacement, when phase stability and modelling assumptions support it. In the following we analyze what happens in situations involving moving targets: • Constant velocity (translation): ( ) = 0 produces a shift in the Doppler centroid proportional to the target velocity; this can manifest as azimuth shift and/or blurring in the SAR image. • Uniform acceleration and polynomial motion: ( ) varies approximately linearly (or as a lower-order polynomial), producing chirp-like Doppler behavior; polynomial phase estimators have been proposed for related problems (Wang et al. 2012, Anghel et al. 2016). • Periodic motion (e.g., vibration): ( ) is (almost) sinusoidal, generating azimuth shifts with sinusoidal patterns; this is the classic case of vibration Micro-Doppler (Chen et al. 2006, Ruegg et al. 2007). • Intermittent or non-stationary micro-movement: the movement components produce non-stationary Doppler content, with complex patterns of azimuth shifts. 2.1. Sub-Pixel offset Tracking (SPOT) The SPOT method leverages the coupling between a target's range velocity and its azimuth position in a focused SAR image. A vibration in the range direction with velocity ( ) induces an azimuth shift Δ ( ) proportional to the radial velocity: ( ) = − ( ) 0 (3) where 0 is the slant range and is the platform velocity. By dividing the full synthetic aperture into a sequence of overlapping sub-apertures, a time-series of low-resolution images is generated. The SPOT algorithm tracks the geometric center or peak intensity of a scatterer across these frames. The resulting time-series of azimuth shifts reconstructs the velocity profile of the vibration. While often used for vibration, SPOT generalizes to any Doppler variation producing trackable location changes during the aperture. For quasi-stationary scatterers, it detects periodic components (vibration) or slow trends (drift), while for moving targets, it captures azimuth migration linked to Doppler behavior. SPOT relies on bright, stable scatterers; in cluttered urban scenes, it becomes sensitive to parameter choices and degrades if the structural response is weak or aspect-dependent (Duque et al. 2015, Duque et al. 2019).