= Solution
<Interferometric synthetic-aperture radar> compares the <phase> of coherent radar returns acquired before and after the <earthquake> from closely repeated viewing geometry. After image registration and removal of orbital and topographic phase, the residual <interferometric phase> constrains relative ground displacement along the radar <line of sight>. Defining $u_{\mathrm{LOS}}$ positive toward the satellite and choosing the interferogram's phase order accordingly,
$$
\Delta\phi_{\mathrm{def}}=\frac{4\pi}{\lambda}u_{\mathrm{LOS}}\pmod{2\pi}.
$$
One complete colour cycle therefore represents $\lambda/2$ of relative <line-of-sight displacement>. Reversing the image order reverses the displayed phase sign. This observes a projection of both horizontal and vertical motion, not uplift alone.<Interferometry by NASA>[NISAR mission explanation: remove orbit/topography contributions and interpret each complete fringe as half a radar wavelength of line-of-sight range change.]{ref}
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3-q4-interferograms.png]
{title=Qualitative interferograms for surface-breaking and blind reverse rupture}
{description=Original wrapped-phase cartoons and corresponding unwrapped line profiles. They illustrate a displacement discontinuity versus a continuous deformation lobe, rather than numerical elastic fault models. The chosen phase sign is positive toward the radar. Every complete colour cycle represents half a radar wavelength of line-of-sight displacement; real fringe shape depends on viewing and rupture geometry.}
If the main reverse rupture reaches the surface, coherent areas on opposite sides can show a displacement step: fringes become offset, truncated or abruptly juxtaposed along a trace. A narrow belt can lose <interferometric coherence> where cracking, intense gradients or changed scatterers prevent reliable phase comparison. If the rupture remains blind, a simple elastic model instead predicts a continuous surface displacement field, often a broad asymmetric lobe with continuous curved or closed fringes and a smooth displacement gradient across the projected tip.
These are diagnostic tendencies, not a unique test. An ordinary wrapped-phase boundary is just a $2\pi$ colour reset and is not itself a rupture. A step of an integer number of cycles can be hidden in wrapped phase; loss of <interferometric coherence> can hide either kind of near-field deformation. Vegetation, atmosphere and viewing direction also affect the pattern, and a displacement perpendicular to the radar <line of sight> is poorly sensed. Inspect coherence and unwrapped displacement, compare ascending and descending views where possible, and test a <fault> model against field observations. A sharp <fold scarp> above a blind rupture or a secondary shallow fault can also create localized gradients, so smooth fringes do not prove a buried tip and broken fringes alone do not prove a surface-breaking main rupture.
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