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A model-based polarimetric decomposition asks a disciplined inverse question: can the measured covariance matrix be represented as a non-negative mixture of idealised surface, double-bounce and volume scattering models? The Freeman-Durden decomposition supplies one influential answer. Its coloured power maps are useful only when the assumptions, total-power ledger, orientation handling and residual mismatch remain visible.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. State the three-component model
The Freeman-Durden model represents an averaged polarimetric covariance matrix as a sum of surface, double-bounce and volume covariance models, each multiplied by a fitted power.
Surface scattering is associated with a rough surface response, double-bounce with two interacting surfaces, and volume with an ensemble of randomly oriented dipoles. These are model components, not direct photographs of objects.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Prepare the covariance matrix before fitting
Calibration, noise correction, multilooking and the chosen polarimetric basis affect the matrix. Terrain slope and target orientation can move energy between channels and violate the reflection-symmetry assumptions used by the basic model.
Document preprocessing before comparing component maps. A decomposition cannot distinguish physical scattering from uncorrected channel imbalance by algebra alone.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Keep a total-power ledger
For a physically admissible fit, the component powers are non-negative and their model matrices should reproduce the observed covariance within the stated residual tolerance.
| Quantity | Illustrative fitted value | Ledger reading |
|---|---|---|
| Observed span | 10 power units | Total measured polarimetric power |
| Surface Ps | 6 | Admissible positive contribution |
| Double-bounce Pd | 5 | Admissible positive contribution |
| Volume Pv | −1 | Impossible physical power; algebra sums to 10 but the model failed |
The example deliberately shows why a perfect scalar sum is too weak. Matrix structure, non-negativity and residuals must all be checked.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. See where negative powers come from
Sequential model subtraction can leave a residual that no longer belongs to the cone generated by the remaining component models. Solving the equations then returns a negative coefficient.
| Possible cause | Observable symptom | Useful response |
|---|---|---|
| Orientation or slope | Power swaps between surface and double-bounce | Apply and report justified orientation compensation |
| Wrong volume model | Negative surface or double-bounce over distributed targets | Test an alternative volume covariance |
| Noise or calibration error | Small negative powers near the floor | Revisit calibration and uncertainty |
| Missing mechanism | Structured residuals or repeated negatives | Use a richer model or report mismatch |
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Enforce non-negativity without hiding residuals
A constrained fit can require Ps,Pd,Pv≥0. If the unconstrained estimate is (6,5,−1), setting Pv to zero and refitting the other components changes both their values and the matrix residual.
Report the constrained objective or residual norm. Renormalising the remaining colours to preserve the same total can make the map look tidy while concealing that the original model did not fit.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Distinguish decomposition from classification
The outputs are continuous fitted powers. Turning the largest component into a land-cover label discards uncertainty, secondary mechanisms and model residuals.
A bright double-bounce channel can support an urban interpretation in context, but it does not prove “building” without geometry, incidence angle, orientation and neighbouring evidence.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Transfer the model across incidence and support
Change the multilook window or move across a boundary between vegetation and road. The covariance mixture changes, so fitted volume and surface powers can shift even when neither material changes.
The changed-context check is to compare component fractions, residuals and negative-power frequency across window sizes and incidence-angle bands. Stable colours without stable residuals are not strong evidence.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect entropy, orientation and coherence
Use polarimetric entropy-alpha and mixed pixels for an eigenvalue view that does not assume the same three physical models. Use polarimetric deorientation when target alignment distorts component allocation.
The guide to polarimetric coherence and calibration drift supplies the instrument and channel-stability checks that precede any physical interpretation.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
Negative fitted power is not negative electromagnetic energy. It is evidence that the selected component matrices, preprocessing or noise assumptions cannot represent the observation with non-negative coefficients.
Non-negative output does not prove the model is correct. Different admissible mixtures can fit similar matrices, and a constrained algorithm may move mismatch into the residual.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record sensor, wavelength, incidence angle, orbit, calibration, noise floor, basis, look support, orientation compensation, volume model, fit order, non-negativity rule, span closure, residual norm and uncertainty or sensitivity tests.
NASA JPL’s Advanced Polarimetric Concepts describes model-based decomposition and the Freeman-Durden three-component scheme. ESA SNAP’s Polarimetric Decomposition Operator documentation lists the implemented decomposition families and their power constraints.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should every negative power be clipped to zero?
Not silently. Diagnose its size and cause, refit if justified, and retain a residual or failure flag.
Does the largest power identify the land cover?
No. It identifies the dominant fitted model component under the selected assumptions.
Why can orientation change the result?
Rotating target geometry redistributes covariance terms that the basic model assigns to different mechanisms.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a constrained-fit check
Start from the illustrative powers (6,5,−1). Compare three responses: raw reporting, zero clipping with no refit, and a non-negative refit. State which quantities remain conserved and which residual evidence must be added.
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