= Solution
Use repeat-pass <interferometric synthetic-aperture radar>. An orbiting radar transmits coherent microwave pulses and records the amplitude and phase of the returning field. Form an <interferogram> from registered complex images acquired before and after the <earthquake>, remove the orbital contribution and the topographic phase predicted by a <digital elevation model>, and correct atmospheric and other long-wavelength phase delays. Where <interferometric coherence> is adequate, unwrap the residual phase to estimate <line-of-sight displacement>:
$$
\Delta d_{\rm LOS}=-\frac{\lambda}{4\pi}\Delta\phi,
$$
with sign set by the instrument's displacement convention. Each $2\pi$ phase cycle represents $\lambda/2$ of range change.
This maps a distributed <earthquake> displacement field day or night and through cloud, including areas inaccessible on foot. Roughly $10$–$30\ \mathrm m$ ground sampling and a scene spanning many kilometres on both sides of the <fault> can resolve regional deformation; a narrow surface break benefits from finer pixels or complementary sub-metre optical surveys. Sensitivity to displacement can be millimetres to centimetres in coherent areas, but that does not equal <spatial resolution>. A single viewing direction measures one component; ascending and descending tracks and <GNSS> constrain more components. At the rupture itself, large gradients and changed ground cover can destroy <interferometric coherence>; radar pixel-offset tracking or before/after optical correlation can then recover metre-scale <fault offsets>.
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