Find your next learning step
Two interferograms can fit perfectly and still fail to identify the true three-dimensional motion. The missing information lives in the viewing geometry.
Open the chapter contents · Remote Sensing root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Build the projection model
Route 2 · Work the two-look geometry
Route 3 · Add independent information
Route 4 · Test limits and uncertainty
Route 5 · Check and continue
CHAPTER 1 OF 12 · BUILD THE PROJECTION MODEL
1. InSAR observes a projection, not a full displacement vector
An interferometric synthetic aperture radar measurement reports change along the radar line of sight. Let ground displacement be the vector d = [E, N, U] in east, north and up components, and let the unit look vector be l = [lE, lN, lU] under a stated sign convention. The observation is
dLOS = lE E + lN N + lU U.
One scalar cannot recover three unknown components. Two viewing geometries give two equations, still fewer than the three components. Any two-component product therefore includes an assumption, an independent measurement or a regularisation rule.
Before reading colours, confirm whether positive means motion toward or away from the sensor. Both conventions exist.
CHAPTER 2 OF 12 · BUILD THE PROJECTION MODEL
2. Why near-polar orbits weaken north-south sensitivity
Many satellite SAR missions fly near-polar ascending and descending tracks and look sideways. Reversing the flight direction strongly changes the east-west look component, while the north component often remains small. The up component stays large because the incidence angle is measured away from vertical.
Ascending and descending observations are therefore useful for separating east-west motion from a second combination dominated by vertical motion. They do not, by themselves, provide a stable general three-dimensional solution. North-south motion lies close to a weak or null direction of the geometry.
“Blindness” is a warning about sensitivity, not an assertion that the north coefficient is exactly zero for every pixel and orbit.
CHAPTER 3 OF 12 · WORK THE TWO-LOOK GEOMETRY
3. Worked two-look example
Use simplified look vectors
lA = [-0.60, -0.10, 0.79] and lD = [0.60, -0.10, 0.79].
Suppose the true displacement is [E,N,U] = [10,20,5] mm. The ascending LOS is -0.60(10) – 0.10(20) + 0.79(5) = -4.05 mm. The descending LOS is 0.60(10) – 0.10(20) + 0.79(5) = 7.95 mm.
Subtracting the equations gives 1.20E = 12, so E = 10 mm. Adding them gives -0.20N + 1.58U = 3.90 mm. That is one equation for N and U. If an analyst assumes N = 0, the estimated U is 2.47 mm, not the true 5 mm. The LOS observations are fitted exactly while the component interpretation is wrong.
| Assumed north motion | Up motion that fits the summed LOS | Fit to both LOS values |
|---|---|---|
| 0 mm | 2.47 mm | Exact |
| 10 mm | 3.73 mm | Exact |
| 20 mm | 5.00 mm | Exact |
Rank deficiency and ill-conditioning are different warnings
A two-by-three design matrix is rank-deficient for a free three-component solution: there are more unknowns than independent observations. An analyst may reduce the problem to two components, for example east and up, by setting north to zero. That reduced two-by-two matrix can still be ill-conditioned if the look geometries are too similar.
Ill-conditioning means small LOS errors produce large component errors. It is quantified by singular values or a condition number, but the number must be calculated from the actual pixel-level look vectors and weighting. A single nominal incidence angle for a whole scene can hide spatial changes in sensitivity.
Good-looking component maps do not repair a small singular value.
CHAPTER 4 OF 12 · WORK THE TWO-LOOK GEOMETRY
4. What ascending plus descending can support
When independent evidence shows north-south motion is negligible, two appropriately different look directions can estimate east and up components. This is common for subsidence basins dominated by vertical motion or deformation known to be mostly east-west and vertical.
The assumption must be visible in the product label and methods. “Vertical displacement” is too strong when it was obtained by projecting LOS under N = 0. Prefer “vertical component under negligible north-motion assumption.”
For a slope, fault or glacier with a known motion direction, one may project onto that direction instead. The geometry improves only if the direction constraint is credible and independent of the LOS values being interpreted.
CHAPTER 5 OF 12 · WORK THE TWO-LOOK GEOMETRY
5. Add information that changes the geometry
Three or more genuinely independent observations can reduce ambiguity, but simply adding more acquisitions from nearly identical tracks mostly improves precision along the same directions. Useful additions include GNSS vectors, along-track or azimuth offsets, multiple satellite headings, levelling for vertical motion or a physically justified displacement direction.
| Added evidence | Geometric benefit | Main caution |
|---|---|---|
| GNSS east-north-up vectors | Anchors weak components and reference frame | Sparse points may not represent every pixel |
| Different satellite heading | Adds a new look direction | Dates, wavelengths and sampling may differ |
| Along-track offsets | Improves sensitivity parallel to flight | Often noisier and lower resolution |
| Levelling | Strong vertical constraint | Limited spatial and temporal coverage |
CHAPTER 6 OF 12 · ADD INDEPENDENT INFORMATION
6. Reference frames and signs belong in the model
LOS displacement is relative to a chosen reference pixel, area or network adjustment. A reference that is moving introduces a common bias. Ascending and descending products may also use different reference areas or time spans, making direct decomposition invalid until they are aligned.
Write the east-north-up convention, look-vector direction, positive LOS sign and reference definition beside the matrix. Confirm whether l points from ground to satellite or satellite to ground. A sign copied from another software package can reverse the inferred horizontal component.
The projection equation is simple enough to test with a synthetic displacement of one millimetre in each coordinate.
CHAPTER 7 OF 12 · ADD INDEPENDENT INFORMATION
7. Time matching and atmospheric errors
Two tracks rarely observe at exactly the same time. If deformation changes between the ascending and descending acquisitions, the equations do not describe one common displacement vector. Seasonal motion, rainfall response or a coseismic step can be mixed into an apparent component difference.
Atmospheric phase screens and unwrapping errors also enter the LOS values before decomposition. The inversion can amplify them along the weak direction. Correcting geometry after a poor interferogram does not restore missing phase information.
Match time intervals where possible, propagate uncertainty and inspect each LOS product before forming components.
CHAPTER 8 OF 12 · ADD INDEPENDENT INFORMATION
8. Evidence checks for a decomposition
Map the look-vector coefficients and the smallest singular value or condition number. Report masks where geometry is weak. Compare predicted LOS values from the decomposed field against the original observations; this checks arithmetic but not uniqueness.
Then withhold independent GNSS or levelling data and test the component estimates. Project the independent three-dimensional vectors into each LOS before comparison. A direct comparison between a horizontal GNSS component and an LOS value mixes quantities.
Finally perturb the LOS observations within their plausible errors and recompute the components. If small perturbations create large north or up changes, the interpretation should remain cautious.
CHAPTER 9 OF 12 · TEST LIMITS AND UNCERTAINTY
9. Limits and common misconceptions
Two tracks do not generally recover three components. A long time series does not change that rank if every acquisition repeats the same two look directions. Regularisation selects one solution among many; it does not create a measurement of the weak component.
North-south sensitivity is usually weak for near-polar satellite geometry, not universally absent. Local incidence and heading matter. “Vertical” products may contain projected horizontal motion when the imposed assumption fails.
The decomposition cannot correct decorrelation, phase unwrapping errors or atmospheric artefacts. Those problems must be assessed in the inputs and propagated into the result.
CHAPTER 10 OF 12 · TEST LIMITS AND UNCERTAINTY
10. Changed-context transfer check
Return to the worked example but suppose independent GNSS gives N = 18 ± 2 mm. Substituting that range into -0.20N + 1.58U = 3.90 yields U near 4.75 mm with uncertainty from both GNSS and LOS. The additional measurement converts an arbitrary assumption into an evidence-backed estimate.
Now change the process to pure subsidence with independent geological support for E and N near zero. A single LOS can estimate U through dLOS/lU, but the result remains conditional on that motion model. State it rather than presenting it as a universal vertical observation.
The transferable habit is to count independent directions before counting images.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Reading route through remote sensing
First verify input phase quality with Interferometric Coherence, Decorrelation and Phase Reliability and InSAR Phase-Unwrapping Residues and Disconnected Components. Then check Atmospheric Phase Screens, Spatial Correlation and False Deformation Ramps.
For geometric comparisons, continue to Incidence Angle, SAR Backscatter and False Change Across Orbits and the Remote Sensing root.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Questions, sources and the next useful move
Can two Sentinel-1 tracks measure north motion? They have some north sensitivity, but the geometry is commonly too weak for a stable unconstrained three-component solution. Does a low residual prove the components are correct? No; many component vectors can project to the same LOS pair. Does another date add a direction? Not if the heading and look geometry are essentially repeated.
Authoritative introductions include NASA’s NISAR explanation of interferometric LOS measurement, the ESA InSAR Principles processing and interpretation guide, and the NISAR solid-earth algorithm and validation plan.
The next useful move is to assemble the design matrix from the actual look vectors, calculate its rank and singular values, and write every imposed component constraint in the map caption before performing the inversion.
Top · Contents · Previous · How X Works