A radar image is built from range and Doppler, not from a camera’s angles. That one fact reshapes everything you see: mountains lean toward the sensor, slopes brighten or vanish, and a summit is drawn where its echo lands, not where it stands. These four exercises let you steer the geometry yourself — two on real Umbra scenes, two on the equations behind them.
Read these before you open the exercises. Each is a paragraph, not a lecture; the exercises are where they stick.
The look angle θL is measured at the satellite, from straight down (nadir) to the line of sight. The incidence angle θi is measured at the ground, between the incoming ray and the local vertical. On a flat Earth they are equal; on the real, curved Earth the surface tilts away under the beam, so θi is always a few degrees larger than θL. Incidence — not look angle — is what controls brightness, shadowing, and how slant range maps to the ground.
The radar measures distance along the line of sight — slant range. Range resolution is set by the pulse bandwidth alone, ρr = c/(2B), and it is constant across the swath. But projected onto the ground it is stretched by 1/sin θi, so the near edge (small θi) always resolves more coarsely than the far edge. This is why a single SAR strip is sharper on its far side.
Because pixels are ordered by range, a slope facing the radar is compressed — foreshortening, and it happens on every slope. Push the foreslope past the incidence angle and the order flips: the top is closer in range than the base, so it is drawn in front of it — layover. On the back side, when the slope is steeper than 90°−θi, the beam skips over it entirely and leaves a shadow with no return. Layover and shadow are geometric and irreparable — you plan the acquisition to avoid them, you don’t fix them afterward.
A geocoded-ellipsoid (GEC) product places every pixel by range, assuming the ground sits on a smooth ellipsoid. Real terrain has height h above that reference, and a point that high is nearer in range than the ellipsoid assumes, so the geocoder puts it toward the radar by a horizontal displacement that grows with height and with how shallow the look is. That is the whole reason terrain correction (an RTC or DEM-based geocode) exists.
Eleven real X-band spotlight scenes. Draw the flight direction, the line of sight, and the layover direction straight onto the amplitude image; the tool marks you against that scene’s own recorded geometry.
Nineteen acquisitions over Fuego, Nevado del Ruiz, Gamalama and EEFIT Quimbaya, each with seven pixel-registered layers. Click any point on the topography and the tool shows where the geocoded product actually puts it — and why.
Set the orbit altitude, look angle, beamwidth and bandwidth. Watch the swath width, the incidence angle from near to far edge, and the ground-range resolution respond in real time.
The pulse rate must beat the Doppler bandwidth yet stay under the echo-return limit. Move the PRF into and out of the window, then shrink the antenna until no clean PRF exists at all.
Imagery: Umbra X-band spotlight amplitude; optical basemap © Esri World Imagery; topography from the Copernicus GLO-30 DEM. Built for CIVE 6359 — SAR Remote Sensing, University of Houston, RiSE Lab. The two scene exercises need to be served over HTTP (they read pixel heights from the image); opening this page from a web address handles that automatically.