Radar interferometry

In depth

Each radar pixel records the amplitude of the echo and also its phase. For a target at slant range $R$, the round trip adds a phase proportional to $R$ measured in wavelengths, with the factor of two for the double path included, plus a contribution $\varphi_{s}$ from how the scatterers in the pixel combine.

where $\varphi$ is the phase of the pixel, in $\mathrm{rad}$; $\lambda$ is the radar wavelength, in $\mathrm{m}$; $R$ is the slant range to the target, in $\mathrm{m}$; and $\varphi_{s}$ is the phase added by the way the scatterers within the pixel combine, in $\mathrm{rad}$.

The scattering term is random from pixel to pixel but is the same on both dates if the surface has not changed, so subtracting the phases of two images cancels it. What remains, the interferogram phase, is the sum of several parts [1, 2, 3].

where every term is a phase in $\mathrm{rad}$; $\Delta\varphi$ is the interferogram phase, the difference between the phases of the two images; $\varphi_{\mathrm{flat}}$ is the flat-Earth phase, from viewing a smooth reference surface from two positions; $\varphi_{\mathrm{topo}}$ is the topographic phase, from the height of the ground; $\varphi_{\mathrm{defo}}$ is the deformation phase, from movement of the ground along the line of sight; $\varphi_{\mathrm{atm}}$ is the atmospheric phase, from the change in signal delay between the dates; and $\varphi_{\mathrm{noise}}$ is the noise, from decorrelation and the instrument.

The flat-Earth term is computed from the orbits and removed. The topographic term arises because the two passes view the ground from slightly different positions, separated by the baseline shown in Figure 1, and it grows with the component of the baseline perpendicular to the line of sight $B_\perp$ and the height of the point $h$.

where $\varphi_{\mathrm{topo}}$ is the topographic phase, in $\mathrm{rad}$; $\lambda$ is the radar wavelength, in $\mathrm{m}$; $B_\perp$ is the perpendicular baseline, the component of the separation between the two orbit positions at right angles to the line of sight, in $\mathrm{m}$; $h$ is the height of the point above the reference surface, in $\mathrm{m}$; $R$ is the slant range, in $\mathrm{m}$; and $\theta$ is the incidence angle, between the radar beam and the vertical at the ground.

Subtracting a terrain model removes it, leaving the term of interest. If the ground moves by $d$ along the line of sight between the two dates, the range changes by the same amount.

where $\varphi_{\mathrm{defo}}$ is the deformation phase, in $\mathrm{rad}$; $\lambda$ is the radar wavelength, in $\mathrm{m}$; and $d$ is the displacement of the ground along the line of sight between the two dates, in $\mathrm{m}$, positive away from the radar.

One full cycle of phase, a fringe in the colour image, therefore corresponds to a motion of half a wavelength, 2.77 cm at the 5.55 cm wavelength of Sentinel-1, as the second panel of Figure 1 shows. Phase is known only modulo $2\pi$, so the fringes have to be counted, or unwrapped, to turn them into total motion. A phase precision of a tenth of a cycle corresponds to a range change of under 3 mm at that wavelength [1, 2].

Three things limit the method. The two images must stay coherent, meaning that the scattering term is unchanged, and vegetation, snow and ploughing destroy it over time, so it works best on buildings, rock and bare ground. Water vapour changes the signal delay between the dates and adds the atmospheric term, which is why single interferograms are not enough for the smallest motions. And a large perpendicular baseline decorrelates the signal. Time-series methods address the second and third by selecting stable scatterers or small-baseline pairs and combining many images, which brings the precision of the slow motion down to millimetres per year [4, 5, 6]. The first radar map of an earthquake's displacement field was of the 1992 Landers earthquake [7], and the idea of mapping small elevation changes by differencing interferograms had been set out a few years earlier [8].