The InSAR absolute phase amid singularities

TL;DR

This work defines an observational, universal absolute InSAR phase by unwrapping the interferometric phase along a chosen absolute path connecting the primary and secondary configurations. It shows that, except for ideal point targets, the absolute phase is generally path-dependent and not simply proportional to a range difference, with absolute phase singularities arising where coherence vanishes along the path. The paper develops a rigorous framework using differential forms, complex-plane geometry, and topology to assess path dependence, patchability, and reconstructability, including continuous and discrete unwrapping, loop tests, and branch-cut strategies. It also extends the concept to differential and across-track InSAR, illustrating with multiple target scenarios (point scatterers, multiple surfaces, intermittent decorrelation, and canopy canopies) and highlighting how singularities complicate reconstruction while pointing to multifrequency and ancillary-information approaches to offset estimation. The results offer principled guidelines for evaluating InSAR processing chains and interpreting observations in low-coherence regimes, where the absolute phase may lose a direct physical interpretation as a simple range difference.

Abstract

The radar interferometric absolute phase is essential for estimating topography and displacements. However, its conventional definition based on the range difference is idealized in that it cannot be applied to complex, dynamic targets. Here, a universal observational definition is proposed, which is easiest to describe for differential interferometry: The absolute phase is determined by temporally unwrapping the phase while continuously varying the intermediate acquisition time between primary and secondary acquisitions. This absolute phase is typically not directly observable because a continuous series of observations is required. The absolute phase of a point target is proportional to the range difference, matching the conventional definition. For general targets undergoing a cyclic change, the absolute phase may be nonzero and then cannot be interpreted as a range difference. When a phase singularity (vanishing coherence) occurs at an intermediate time, the absolute phase becomes undefined, a situation termed an absolute phase singularity. Absolute phase singularities complicate absolute phase reconstruction through multifrequency techniques and through unwrapping multidimensional interferograms. They leave no trace in an interferogram, but unwrapping paths need to avoid those across which the absolute phase jumps by nonzero integer multiples of $2 π$. Mathematical analyses identify conditions for unwrapping-based reconstruction up to a constant, accounting for absolute phase singularities, undersampling and noise. The general definition of the absolute phase and the mathematical analyses enable a comprehensive appraisal of InSAR processing chains and support the interpretation of observations whenever low coherence engenders phase and absolute phase singularities.

Figures (10)

• Figure 1: The differential InSAR absolute phase $\phi^{\mathrm{a}}(s)$ as a function of dimensionless time $s$ for three surface segments. The InSAR observations do not resolve the segments and are continuous in $s$. The segments' normalized positions $b_1$, $b_2$, and $b_3$ are shown in the three bottom panels and also indicated in the insets. Model details can be found in Sec. \ref{['sec:threeseg']}.
• Figure 2: a) A coherence field $\gamma$ on $W^{+}$ induces a wrapped phase field $\varphi$ through composition with the argument function. b--e) An example coherence field with b) wrapped phase and c) coherence magnitude, along with two paths $\Gamma$ and $\tilde{\Gamma}$ from $w_1$ to $w'$. d) The two paths are mapped into the complex plane through $\gamma$. e) The absolute phase, whose value at $w'$ is the unwrapped phase difference along $\Gamma$.
• Figure 3: Differential InSAR model of three surface segments with positions $\mathbf{b} = [b_1, b_2, b_3]$. a) The purple loop with basepoint $\mathbf{b}_1 = [0, 0, 0]$ (white cross) in parameter space describes the temporal evolution from Fig. \ref{['fig:threesurfts']}, it encircles a phase singularity (black line) in the plane $b_1 = 0$. b--d) Wrapped phase with white contours indicating vanishing real and imaginary part (b), coherence (c) and absolute phase in the modeling framework of Sec. \ref{['sec:diffmodeling']} (d) shown in the plane $b_1 = 0$ for fixed primary $\mathbf{b}_1$, along with phase singularities (black dots) and absolute phase singularities (gray lines).
• Figure 4: Modeled differential InSAR absolute phase for intermittent decorrelation from \ref{['eq:decorrelation']} with $\beta = 16$. The thick black line represents the ensemble, the thin lines the speckle-affected realizations, color coded depending on $\phi^{\mathrm{a}}(1)$. The polar plot on the right shows the coherence $\gamma$, the outer circle corresponding to $\rho = 1$.
• Figure 5: BorealScat tomographic observations of a forest canopy. a) The multilooked intensity on 2018-07-24 00:00 decreases downward from canopy top at $\sim$25 m, except the subcanopy corner reflector (CR). b) 24-hour interferogram (July 24 to July 25, midnight) with wrapped phase (hue) at coherence magnitude (brightness). c) Absolute phase estimate from successive 5-min interferograms, with points A and B shown as markers in a--c) differing by $2 \pi$. The path in $\mathbb{C}$ traced by A and B over the 24-hour interval is shown in the insets.