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Each resolution element encodes the phase related to the propagation distance from the radar to the ground as well as the intrinsic phase of the backscattering process. The resolution element comprises an arrangement of scatterers – trees, buildings, people, etc. – that is spatially random from element to element, and leads to a spatially random pattern of backscatter phase in an image. As such, since we can only measure the phase in an image within one cycle (i.e. we do not measure the absolute phase), it is not possible to observe the deterministic propagation component directly. Interferometric synthetic aperture radar (InSAR) (Rosen et al., 2000; Hanssen, 2001) techniques use two or more SAR images over the same region to obtain surface topography or surface motion.

The figure illustrates the InSAR imaging geometry. At time t1, a radar satellite emits a pulse at S1, then receives an echo reflected from a ground pixel A and measures the phase φ1 of the received echo. All scatterers within the associated ground resolution element contribute to φ1. As a result, the phase φ1 is a statistical quantity that is uniformly distributed over interval (0, 2π) and we cannot directly use φ1 to infer the distance r1 between S1 and A. Later at time t2, the satellite emits another pulse at S2 and makes a phase measurement φ2. If the scattering property of the ground resolution element has not changed since t1, all scatterers within the resolution element contribute to φ2 the same way as they contribute to φ1. Under the assumption that |r1 − r2| << |r1| (the parallel-ray approximation), the phase difference between φ1 and φ2 can be used to infer the topographic height z of the pixel A (Hanssen, 2001, Section 3.2)

If we know the topographic height z, we can further measure any small ground deformation occurring at pixel A between t1 and t2. Figure 3 3 illustrates the InSAR imaging geometry in this case. At time t1, a radar satellite measures the phase φ1 between the satellite and a ground pixel A along the LOS direction. Later at time t2, the ground pixel A moves to A′ and the satellite makes another phase measurement φ2 between the satellite and the ground pixel. After removing the known phase φ′ due to the surface topography, the unwrapped InSAR phase ∆φ = φ2 − φ1 − φ′ is proportional to the ground deformation ∆d between t1 and t2 along the satellite LOS ground direction as:

∆φ=φ2 −φ1 −φ′ = 4π∆d / λ

where λ is the radar wavelength. In this equation we assume that there is no error in the InSAR phase measurement. Below we discuss in depth various error sources in InSAR deformation measurements and their impact on InSAR image quality.

Note that InSAR techniques only measure one-dimensional LOS motion. However, deformation is better characterized in three dimensions: east, north and up. Given an LOS direction unit vector e = [e1, e2, e3], we can project the deformation in east, north and up coordinates along the LOS direction as:

∆d = e1 ∆deast + e2 ∆dnorth + e3 ∆dup

Because radar satellites are usually polar orbiting, the north component of the LOS unit vector e2 is often negligible relative to the east and vertical components. When InSAR measurements along two or more LOS directions are available, we can combine multiple LOS deformation measurements over the same region to separate the east and vertical ground motions, given that the term e2 ∆dnorth is negligible.

For this mission, interferometric observations of any given point on the ground are acquired every 12 days. To the extent that the ground does not change appreciably in most places over 12 days, ground motion can be measured.