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Description
Recent studies have demonstrated that non-zero closure phases ($\Phi_{123}$), obtained by circularly summing three interferometric phases, can arise from phase noise and geophysical signals [1]. Existing literature identifies three mechanisms contributing to geophysical closure phases: changes in the dielectric properties of the medium, including soil moisture variations, wet biomass accumulation, and snow metamorphism; volume scattering in conjunction with perpendicular baselines; and differential skewed motion [1-4]. Existing propagation models account for closure phases by considering dielectric variations in soil and vegetation independently [5-7]. In this work, we develop a composite propagation model for a three-layer medium consisting of vegetation, the ground surface, and subsurface soil.
To model the total signal response from interactions with all components, we consider both attenuation and phase alterations of the traveling wave through a lossy medium, the vegetation canopy. Retaining the same geometric approximation as in the previous propagation models, the signal corresponding to a point at range $r_0$ and azimuth $y_0$ can be expressed as
$$p(r_0, y_0, t) =\int_0^h\xi_v(z,t)e^{-2jk_{z,v}z}dz + e^{-2jk_{z,v} h}\int_0^{\infty}\xi_s(z,t)e^{-2jk_{z,s}z}dz + e^{-2jk_{z,v}h}\xi_{g}(t)p_{g}(r_0, y_0, t),$$
where $\xi(z, t)$ is a complex-valued volumetric scattering profile, which we treat as a random process, and $p_{g}(r_0, y_0, t)$ represents the contribution of surface scattering. Assuming that the scattering behaves as a stationary random process, a height-dependent coherence term, $\rho(t-t_0, z)$, can be incorporated into the second order characteristics of the scattering distribution:
$$E[\xi(z, t)\xi^*(z_0, t_0)] = f(z)\delta(z-z_0)\rho(t-t_0, z),$$
where $f(z)$ is the vertical scattering density profile. Assuming that the scattering components are mutually uncorrelated and that the ground surface has the same dielectric properties as the underlying soil, the expected value of interferogram product between two acquisitions with different vegetation and soil dielectric constants is given by:
$$I_{12} = \int_0^h f_v(z)\rho_v(t-t_0, z)e^{-2j(k_{z1,v}-k_{z2,v}^*)z}dz + e^{-2j(k_{z1,v}h_1-k_{z2,v}^*h_2) } \bigg(\int_0^{\infty} f_s(z)\rho_s(t-t_0, z)e^{-2j(k_{z1,s}-k_{z2,s}^*)z}dz + \rho_g(t-t_0)I_{z,g}\bigg),$$
where $\rho_g$ is the surface scattering coherence function and $I_{z,g}$ represents the expected value of the interferometric phase contribution estimated using the AIEM.
In the bare-soil and vegetation-only models, the phase, and consequently the closure phase, depends on the vertical scattering profile but not on the total NRCS. In contrast, the phase in the composite model depends on the relative strengths of the individual scattering contributions. To examine the general behavior of the composite model and assess the influence of each component on the resulting closure phases, we simulate the model under a set of simplifying assumptions. Most biogeophysical parameters are held fixed, while soil and leaf moisture are treated as the primary drivers of phase variations and are assumed to vary simultaneously. In addition, to evaluate the model under more realistic conditions, we predict closure-phase time series using in situ measurements, considering both scenarios with and without decorrelation effects.
This three-layer propagation model quantifies the contributions and interactions of individual layers, offering new physical insights into the origin of closure phases and highlighting the importance of complex multilayer interactions.
Reference
[1] Francesco De Zan, Mariantonietta Zonno, and Paco Lopez-Dekker, “Phase inconsistencies and multiple scattering in sar interferometry,” IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 12, pp. 6608–6616, 2015.
[2] Simon Zwieback, Xingyu Liu, Sofia Antonova, Birgit Heim, Annett Bartsch, Julia Boike, and Irena Hajnsek, “A statistical test of phase closure to detect influences on DInSAR deformation estimates besides displacements and decorrelation noise: Two case studies in high-latitude regions,” IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 9, pp.5588–5601, 2016.
[3] Francesco De Zan, Paloma Saporta, and Giorgio Gomba, “Spatiotemporal analysis of C-band interferometric phase anomalies over Sicily,” in EUSAR 2022; 14th European Conference on Synthetic Aperture Radar. VDE, 2022, pp. 1–4.
[4] Yan Yuan and Paco L´opez-Dekker, “Statistical properties of closure phases in sar interferometry,” IEEE Transactions on Geoscience and Remote Sensing, 2026.
[5] Francesco De Zan, Alessandro Parizzi, Pau Prats-Iraola, and Paco L´opez-Dekker, “A sar interferometric model for soil moisture,” IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 1, pp. 418–425, 2013.
[6] Francesco De Zan and Giorgio Gomba, “Vegetation and soil moisture inversion from sar closure phases: First experiments and results,” Remote sensing of environment, vol. 217, pp. 562–572, 2018.
[7] Yan Yuan, Marcel Kleinherenbrink, and Paco L´opez-Dekker, “On crop growth and insar closure phases,” IEEE Transactions on Geoscience and Remote Sensing, vol. 62, pp. 1–12, 2024.