PSI - Issue 84

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

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2.2. Doppler Phase Coherence (DPC)

The DPC methodology (patent pending) primarily utilizes focused Single Look Complex (SLC) SAR imagery or range-compressed, and Range Cell Migration (RCM) corrected data )[Cumming et al. (2005)]. Operating on the azimuth Discrete Fourier Transform (DFT), the analysis focuses on its residual phase. While theoretically null for stationary scenes, target motion or topographic discrepancies generate a non-zero phase signature representing the target's Doppler history. The core of this framework relies on deriving a phase model, ( , ̅) , with ̅ the parameters of the model to be determined, to account for specific target heights or kinematic profiles. Unknown parameters are estimated by maximizing the Doppler Phase Coherence (DPC), ( ̅) , a metric quantifying the alignment between the observed Doppler history and the theoretical model after removing the target's position-specific linear phase ramp. The maximization problem is formalized as follows: ( ̅) = ̅ | 1 ∑ ( ) [ ( )+ 2 + ( ; ̅)] =−01 | (4) Here, ( ) represents the azimuth DFT phase of a local SAR image patch, of size along azimuth, denotes the azimuth index of the point within the patch. While the current formulation assumes a dominant point-scatterer confined to a single resolution cell, the method is extensible to distributed targets by imposing geometric constraints across multiple resolution cells. DPC retrieves complex kinematic parameters by linking the physical motion in the spatial domain and the resulting phase model. While analytical solutions exist for certain motion types, arbitrary trajectories are modeled by decomposing the antenna aperture and Doppler history into infinitesimal segments. Crucially, the estimation exploits the full Doppler history simultaneously, offering greater robustness than sub-aperture techniques. A key innovation in this study involves optimizing weights, ( ) , in equation (4), as standard SAR azimuth spectrum amplitude-based weights proved suboptimal for DPC maximization. The resulting DPC value acts as a statistical likelihood measure; values approaching unity indicate high fidelity to the motion hypothesis, effectively detecting specific dynamic behaviors within the scene. 2.3. Practical limits and measurable Doppler-variation range For a dwell time , nominal frequency resolution scales as ~ 1⁄ ; a 20 s dwell suggests Δ ∼ 0.05 Hz, which is particularly favourable for the low-frequency modes (< few Hz) typical of civil infrastructure (Biondi et al. 2020b, Vattulainen et al. 2024). The upper measurable frequency depends on slow-time sampling and the extraction strategy. In practice, detectability is constrained by SNR/SCR, clutter, mixed scatterers, and aspect-dependent scattering (Ruegg et al. 2007, Djurovic et al. 2017, Biondi et al. 2020b, Vattulainen et al. 2024). Generally, frequency estimates are more robust than amplitude estimates under these conditions, motivating specific quality controls and data aggregation across multiple measurement points (Biondi et al. 2020b, Vattulainen et al. 2024). 3. Experiments and results 3.1. Controlled validations and accuracy expectations While the SPOT algorithm was already used and its potentiality already presented (Vattulainen et al. 2026, Lotti et al. 2026), we conducted a validation campaign to strictly assess DPC efficacy using trihedral corner reflectors in Trento and Glasgow, subjected to known vertical sinusoidal oscillations. Analysing VHR TerraSAR-X and Umbra acquisitions, the results demonstrate rigorous alignment between estimated motion parameters and ground-truth logs, even in complex dual-frequency scenarios (final entry of Table 1). The impact of uncompensated motion is evident in the Trento TerraSAR-X dataset (first entry in Table 1), where the vibrating CR exhibits azimuth displacement and "ghost" replicas (Fig. 1a). DPC application retrieves the vibration signal, suppressing these artifacts and restoring the target's true position. Additionally, Fourier domain analysis confirms the phase follows the expected sinusoidal pattern consistent with the mechanical oscillation (Fig. 1b).