Before calling a radar difference a surface change
A first-principles guide to incidence angle, reference area and cross-orbit comparison, with calculations in linear power and decibels.
Choose the question you need to answer
Identify the geometry
Calculate the angle effect
Compare orbits
Use terrain correction and limits
1. A darker radar return may be geometry, not change
Two SAR images can show the same ground target with different backscatter even when the target has not changed. Incidence angle, look direction, local slope, polarisation and the reference area used by calibration all influence the number.
This matters when images come from different positions within a swath, different relative orbits, or ascending and descending passes. A raw pixel difference is then a mixture of surface response and viewing geometry.
The companion SAR decibel lesson explains logarithmic power comparisons. This article asks a different question: are the two powers normalised to a comparable geometry?
2. Define the angle and the reference plane
Incidence angle is the angle between the radar line of sight and a chosen normal. An ellipsoid incidence angle uses the reference ellipsoid. A local incidence angle uses the local terrain normal from a digital elevation model.
On flat ground the two can be close. On a slope facing toward or away from the radar they can differ substantially. The same nominal orbit angle therefore does not guarantee the same local illuminated geometry.
Write the angle source in the processing record. “Incidence angle 40°” is incomplete if the reader cannot tell whether it is ellipsoid-based, local, per-pixel or a scene-centre value.
3. Keep beta nought, sigma nought and gamma nought distinct
Beta nought, sigma nought and gamma nought normalise radar brightness to different reference areas. Software can output all three, but their names are not decorative synonyms.
A common flat-terrain relation is gamma nought equals sigma nought divided by cos θ. In decibels, gamma⁰dB=sigma⁰dB−10log10(cos θ). This removes an ellipsoid-incidence component under the model.
In rugged terrain, radiometric terrain correction estimates the illuminated or scattering area using sensor geometry and a DEM. A simple cosine division is not a complete replacement for terrain flattening.
4. Work a no-change example in linear power
Suppose a stable target has gamma⁰=0.10, which is −10 dB. If sigma⁰=gamma⁰cos θ, then at 30° the observed sigma⁰ is 0.10×0.8660=0.0866. At 45° it is 0.10×0.7071=0.07071.
| Angle | cos θ | Stable gamma⁰ | Expected sigma⁰ | Sigma⁰ in dB |
|---|---|---|---|---|
| 30° | 0.8660 | 0.1000 | 0.08660 | −10.62 |
| 45° | 0.7071 | 0.1000 | 0.07071 | −11.51 |
| 60° | 0.5000 | 0.1000 | 0.05000 | −13.01 |
A naive comparison from 30° to 45° reports about −0.88 dB of change. From 30° to 60° it reports about −2.39 dB. In this constructed example the surface response did not change at all.
5. Apply the correction in decibels without mixing domains
At 30°, 10log10(cos θ)≈−0.625 dB. Therefore gamma⁰dB=sigma⁰dB−(−0.625)=sigma⁰dB+0.625. Applying that to −10.625 dB returns −10 dB.
At 45° the correction is about +1.505 dB, and at 60° it is +3.010 dB. Each corrected value again returns −10 dB in the flat-reference model.
Do not divide a decibel number by cos θ. Either correct in linear power and reconvert, or use the additive decibel relation. Decibels encode a logarithm; ordinary linear arithmetic on the printed dB value changes the meaning.
6. Work a measured-looking pair and preserve the residual
Assume one pixel is −10.60 dB at 30° and another is −11.20 dB at 45°. The raw difference is −0.60 dB, which suggests a decrease.
Angle normalisation gives approximately −9.98 dB for the first image and −9.70 dB for the second. The residual comparison is now about +0.28 dB. Geometry did not merely shrink the difference; it reversed its sign.
The residual is still not proof of a surface change. Speckle, moisture, roughness, vegetation structure, calibration, thermal noise and registration can all contribute. Normalisation removes one stated mechanism, not every alternative.
7. Across orbits, look direction changes the scattering problem
Ascending and descending passes usually view terrain from different azimuth directions. A slope, row structure, building facade or forest edge can present a different orientation to the radar even when local incidence angles are numerically similar.
Layover, foreshortening and shadow also depend on geometry. Some pixels do not represent the same ground support or may contain unreliable information in one acquisition.
For change detection, prefer the same relative orbit, acquisition mode, polarisation and comparable incidence-angle range when possible. If cross-orbit comparison is necessary, treat it as a separate measurement design with geometry masks and validation.
8. Put a comparison-ready checklist beside the numbers
| Field | Image A | Image B | Decision |
|---|---|---|---|
| Relative orbit | Same or different? | Same or different? | Match or model explicitly |
| Pass direction | Ascending/descending | Ascending/descending | Check azimuth sensitivity |
| Polarisation | VV/VH/HH/HV | VV/VH/HH/HV | Must be comparable |
| Backscatter convention | beta⁰/sigma⁰/gamma⁰ | beta⁰/sigma⁰/gamma⁰ | Use one stated convention |
| Incidence information | Ellipsoid/local | Ellipsoid/local | Use consistent angle source |
| Terrain correction | Method and DEM | Method and DEM | Align processing and masks |
Add acquisition time, season, precipitation history and spatial registration to the same ledger. Radar responds to water and structure; a geometry-correct product can still record a real but unwanted environmental difference.
9. Terrain flattening needs area, not only an angle
In mountainous terrain a pixel’s ground-projected area and radar-illuminated area can differ sharply. Radiometric terrain correction estimates an area-normalisation factor from orbit geometry and a DEM, then maps the return to a chosen gamma⁰ convention.
The result depends on DEM quality, oversampling, resampling and masking. Shadowed or severely distorted areas should not be forced into an apparently precise comparison.
Use the processor’s local-incidence-angle, scattering-area and shadow or distortion layers where available. Retaining these auxiliaries makes the correction auditable.
10. Limits and misconceptions
Gamma⁰ is not automatically “the true backscatter.” It is a normalised coefficient tied to a reference-area convention and processing chain.
Equal angles do not guarantee equal geometry, especially across opposite look directions or rugged terrain. Different angles do not always invalidate a comparison if a validated model and suitable normalisation are used.
Terrain correction cannot repair temporal mismatch, polarisation mismatch, mixed land cover, saturation, speckle or misregistration. It also cannot identify the cause of a residual change.
11. Questions for a defensible change claim
Are both images calibrated to the same coefficient and units? Were linear values averaged before conversion to decibels? Are incidence angles and local slopes available per pixel? Were shadow and layover masked?
Do the images share relative orbit, pass direction, mode and polarisation? If not, what experiment shows the correction transfers across those geometries? Is the residual larger than processing and sampling uncertainty?
Was the changed area checked against independent evidence, field observations or another sensor? A clean numerical subtraction is only the start of an evidence chain.
12. Continue through the remote-sensing evidence system
Start with the Remote Sensing root. Use the SAR decibel guide, the co-registration false-change lesson, and the stored-value conversion guide as linked checks.
Official technical reading: NASA Earthdata’s Sentinel-1 radiometric-calibration documentation distinguishes beta⁰, sigma⁰ and gamma⁰; Copernicus Data Space S1GRD examples expose terrain-corrected gamma⁰ with local incidence angle, scattering area and shadow mask.
Return to the Gold Standard of Education and the How X Works hub. The transferable rule is to normalise the measurement geometry before interpreting a difference as change.