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A reciprocal monostatic target is commonly represented by a symmetric scattering matrix, so the cross-polar channels are equal under the stated convention. That equality is a physical and geometric assumption, not a formatting rule. A measured difference can carry information about non-reciprocal or non-symmetric scattering, but it can also be produced by bistatic geometry, propagation, channel imbalance, phase error or noise.
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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 geometry before invoking reciprocity
Write the complex Sinclair matrix with co-polar entries SHH and SVV and cross-polar entries SHV and SVH. In a calibrated monostatic backscatter configuration with reciprocal media and consistent antenna conventions, the familiar reduction SHV=SVH applies.
Bistatic transmitter and receiver directions, moving or magneto-optic media, convention changes and antenna-coordinate choices require separate treatment. Do not force a general four-channel measurement into the reciprocal three-component form before checking the acquisition contract.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Split the cross channels into symmetric and antisymmetric parts
Define s=(SHV+SVH)/2 and a=(SHV−SVH)/2. Then SHV=s+a and SVH=s−a. Reciprocity makes a zero, while a non-zero antisymmetric component identifies exactly what is discarded by cross-channel averaging.
This decomposition is algebraic. Interpreting a physically still requires calibration, signal-to-noise ratio, propagation and geometry evidence. A small non-zero estimate is expected when the channels contain independent noise.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a complex cross-channel ledger
Suppose SHV=0.30+0.10i and SVH=0.10−0.10i in the same calibrated basis. Then s=0.20 and a=0.10+0.10i. The cross-channel powers are 0.10 and 0.02, while |s|²=0.04 and |a|²=0.02.
| Quantity | Complex value | Power | Role |
|---|---|---|---|
| SHV | 0.30+0.10i | 0.10 | Measured H-to-V channel |
| SVH | 0.10−0.10i | 0.02 | Measured V-to-H channel |
| s | 0.20 | 0.04 | Symmetric component |
| a | 0.10+0.10i | 0.02 | Antisymmetric component |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Relate reciprocity to polarimetric vector dimension
Under reciprocity, Pauli and lexicographic target vectors commonly use three complex components because one cross channel represents both directions. Without reciprocity, the full scattering matrix has four complex entries and the antisymmetric degree of freedom must remain.
A decomposition derived for reciprocal monostatic data can misallocate antisymmetric energy if applied unchanged. Check the derivation of every invariant, colour composite and target vector rather than assuming a four-to-three reduction is harmless.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change the phase calibration and create an apparent violation
Start with a truly reciprocal cross channel of amplitude 0.20 and zero phase in both directions. If one receive path acquires a 20-degree phase bias, the two recorded complex values differ even though their magnitudes match.
| Case | |SHV| | |SVH| | Phase difference | Safe reading |
|---|---|---|---|---|
| Ideal reciprocal | 0.20 | 0.20 | 0° | Consistent with reciprocity |
| Channel phase bias | 0.20 | 0.20 | 20° | Calibrate before physics |
| Gain and phase bias | 0.22 | 0.18 | 20° | Amplitude and phase correction |
| Low SNR | Uncertain | Uncertain | Unstable | Use uncertainty limits |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Use helicity without turning it into a label
Circular-polarisation and helicity-sensitive combinations can separate handed responses and reveal imbalances that a reciprocal linear-basis summary hides. The sign depends on propagation, handedness and basis conventions, so the convention must travel with the result.
Helicity is not a universal material label. Oriented structures, multiple scattering, chiral response, rotation and non-reciprocal mechanisms can influence handed components. Interpretation needs scene geometry and comparison targets.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the established polarimetric routes
The degree-of-polarisation guide separates pure transformations from ensemble mixing. The Pauli decomposition guide shows how basis conventions affect colour reading.
Use the Kennaugh and Huynen guide for real matrix representations, deorientation for roll effects and Cloude decomposition for eigenchannel assumptions.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Separate target physics from propagation and processing
Faraday rotation, ionospheric variability, antenna crosstalk, timing mismatch, gain drift and imperfect coregistration can change the cross-channel relationship. Multilooking also replaces one deterministic matrix with second-order ensemble information.
Use corner reflectors or other calibration references, cross-talk estimates, noise subtraction, reciprocal residual maps and support-sensitivity tests. Inspect complex residuals, not only a colour composite or a scalar ratio.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
Reciprocity does not mean SHH=SVV, and symmetry of the scattering matrix is not the same as geometric symmetry of the object. A reciprocal target can have very different co-polar responses.
A non-zero antisymmetric estimate is not automatically exotic physics. Finite precision, low signal, basis mismatch and bistatic acquisition can explain it. Conversely, averaging SHV and SVH at ingestion can erase the evidence needed to diagnose a real violation.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record monostatic or bistatic geometry, frequency, transmit and receive coordinates, propagation convention, full complex matrix, calibration targets, gain and phase corrections, cross-talk, noise floor, reciprocal residual, symmetric and antisymmetric components, circular-basis convention and uncertainty.
ESA’s SAR Polarimetry Basic Concepts states the reciprocal-scatterer reduction and develops Sinclair and decomposition formalisms. NASA JPL’s Synthetic Aperture Radar Polarimetry provides a full treatment of scattering matrices, bases and polarimetric remote sensing.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is SHV equal to SVH in every radar?
No. The familiar equality belongs to a reciprocal monostatic backscatter setting with consistent calibration and conventions.
Can I average the two cross channels?
Only after verifying reciprocity for the intended analysis. Averaging suppresses the antisymmetric component and can hide calibration or physical information.
Does helicity prove a chiral material?
No. It is a convention-dependent polarimetric response that can arise from several structural, propagation and scattering mechanisms.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a calibration transfer check
Take reciprocal cross channels of 0.25 exp(i10°). Apply a 10 percent gain error and a −15° phase error to SVH only. Compute s and a before and after correction, and state which residual would remain if the target were truly non-reciprocal.
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