Direct Estimation of Faraday Rotation and other system distortion parameters from polarimetric SAR data Mariko Burgin 1 , ...
Overview <ul><li>Problem statement </li></ul><ul><li>Approach: </li></ul><ul><ul><li>Generate synthetic data </li></ul></u...
Problem statement (1) <ul><li>Low- frequency radars – radars operating at L-band and lower – must be well calibrated not o...
Problem statement (2) <ul><li>General assumptions: </li></ul><ul><ul><li>A(r, θ ), exp(j ϕ ), N are considered negligible ...
Approach Overview <ul><li>Generate synthetic data </li></ul><ul><li>=  create ground truth for  Ω  and system parameters <...
Generate Ground Truth (1) AIRSAR data available for P, L, C band Predict Faraday rotation angle  Ω <ul><li>Faraday rotatio...
Generate Ground Truth (2) Pseudo-spaceborne “measured” values + ground truth are known Faraday rotation angle for AIRSAR d...
Estimate system parameters (1) <ul><li>Convert system model into a form suggested by Freeman (2008): </li></ul><ul><li>Ass...
Estimate system parameters (2) <ul><li>Invert for S xx S xx  polarimetric cross products: </li></ul><ul><li>Find a  repres...
Estimate system parameters: f 1 , f 2  known (1)
Estimate system parameters: f 1 , f 2  known (2) <ul><li>Solve nonlinear equation system with Secant method or Newton-Raph...
Estimate system parameters: f 1 , f 2  known (3) <ul><li>With cost function for L-band: </li></ul><ul><li>Ω can be retriev...
Estimate system parameters: f 1 , f 2  known (4) <ul><li>With cost function for P-band: </li></ul><ul><li>Ω can be retriev...
Estimate system parameters: f 1 , f 2  known (5) <ul><li>Computation time (Number of iterations = 500) </li></ul><ul><li>S...
Estimate system parameters: Sensitive to  δ 1 ,  δ 2  (1) <ul><li>Original cost function is kept for retrieval of  Ω </li>...
Estimate system parameters: Sensitive to  δ 1 ,  δ 2  (2) <ul><li>Plots of magnitude and phase of  δ 1 ,  δ 2  sweeped ove...
Estimate system parameters: full system model (1) <ul><li>Full system model:  solve for everything in one shot </li></ul><...
Estimate system parameters: full system model (2) <ul><li>Findings for L-band: </li></ul><ul><li>Promising retrieval for  ...
Summary and Future work <ul><li>Developed method to produce pseudo-spaceborne data ⟶ solid ground truth </li></ul><ul><li>...
Thank you very much for your attention. Questions?
 
Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval ...
Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval ...
Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval ...
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WE3.L09 - DIRECT ESTIMATION OF FARADAY ROTATION AND OTHER SYSTEM DISTORTION PARAMETERS FROM POLARIMETRIC SAR DATA

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WE3.L09 - DIRECT ESTIMATION OF FARADAY ROTATION AND OTHER SYSTEM DISTORTION PARAMETERS FROM POLARIMETRIC SAR DATA

  1. 1. Direct Estimation of Faraday Rotation and other system distortion parameters from polarimetric SAR data Mariko Burgin 1 , Mahta Moghaddam 1 Anthony Freeman 2 1 The University of Michigan, Ann Arbor, MI, USA 2 JPL, California Institute of Technology, Pasadena, CA, USA IGARSS 2010, Honolulu, Hawaii July 25 – 30, 2010
  2. 2. Overview <ul><li>Problem statement </li></ul><ul><li>Approach: </li></ul><ul><ul><li>Generate synthetic data </li></ul></ul><ul><ul><li>Estimate system parameters </li></ul></ul><ul><li>Summary </li></ul><ul><li>Future work </li></ul>
  3. 3. Problem statement (1) <ul><li>Low- frequency radars – radars operating at L-band and lower – must be well calibrated not only for system distortion terms, but also for Faraday rotation. </li></ul><ul><li>Polarimetric radar system model: </li></ul><ul><li>S = scattering matrix </li></ul><ul><li>R F = one-way Faraday rotation matrix </li></ul><ul><li>R = receive distortion matrix (of the radar system) </li></ul><ul><li>T = transmit distortion matrix (of the radar system) </li></ul><ul><li>A(r, θ ) = real factor representing the overall gain of the radar system </li></ul><ul><li>exp(j ϕ ) = complex factor representing the round-trip phase delay and system </li></ul><ul><li> dependent phase effects on a signal </li></ul><ul><li>N = additive noise terms </li></ul><ul><li>With δ i (i = 1,2,3,4) = crosstalk values </li></ul><ul><li>f 1 , f 2 = channel amplitude and phase imbalance terms </li></ul><ul><li>Ω = Faraday rotation angle </li></ul>
  4. 4. Problem statement (2) <ul><li>General assumptions: </li></ul><ul><ul><li>A(r, θ ), exp(j ϕ ), N are considered negligible </li></ul></ul><ul><ul><li>Reciprocal crosstalk: δ 3 = δ 1 , δ 4 = δ 2 </li></ul></ul><ul><ul><li>Backscatter reciprocity: S HV = S VH </li></ul></ul>KNOWN UNKNOWN Measured: 16 polarimetric cross products from the Muller matrix in the form M xx M xx , x є {H, V} Retrieval: 15 polarimetric cross products from the scattering matrix in the form S xx S xx , x є {H, V} plus f 1 , f 2 , δ 1 , δ 2 , Ω
  5. 5. Approach Overview <ul><li>Generate synthetic data </li></ul><ul><li>= create ground truth for Ω and system parameters </li></ul><ul><li>= superimpose on airborne data </li></ul><ul><li>Estimate system parameters </li></ul><ul><ul><li>Estimate arg(f 1 /f 2 ) </li></ul></ul><ul><ul><li>Estimate remaining system parameters from nonlinear system of equations </li></ul></ul>
  6. 6. Generate Ground Truth (1) AIRSAR data available for P, L, C band Predict Faraday rotation angle Ω <ul><li>Faraday rotation ranging from 2.5° (C-band) to 320 ° (P-band) </li></ul><ul><li>Input needed for models: </li></ul><ul><li>altitude </li></ul><ul><li>frequency </li></ul><ul><li>year/date </li></ul><ul><li>look/azimuth angle </li></ul><ul><li>time </li></ul><ul><li>latitude/longitude </li></ul><ul><li>2 models needed: </li></ul><ul><li>IRI – 2007 (International Reference Ionosphere) </li></ul><ul><li>IGRF (International Geomagnetic Reference Field) </li></ul><ul><li>⟶ gives Ω to within +/- 10° </li></ul>16 polarimetric cross products of the scattering matrix in the form S xx S xx Pseudo-spaceborne Ω ~ 7.7 km data without Faraday rotation ~ 1300 km data with Faraday rotation airborne Longitude Faraday rotation angle Ω in degrees March 1, 2001 – 12:00 UTC 100 80 60 40 20 0 -20 -40 -60 -80
  7. 7. Generate Ground Truth (2) Pseudo-spaceborne “measured” values + ground truth are known Faraday rotation angle for AIRSAR data sets: L-band: Ω = 1.5° P-band: Ω = 12° Assumed system distortion parameters (arbitrary): Build 16 “measured” Muller matrix polarimetric cross products Pseudo-spaceborne Ω ~ 7.7 km data without Faraday rotation ~ 1300 km data with Faraday rotation airborne
  8. 8. Estimate system parameters (1) <ul><li>Convert system model into a form suggested by Freeman (2008): </li></ul><ul><li>Assume Ω is known to within ΔΩ : </li></ul><ul><li>t = tan( Ω ) Ω truth = Ω + ΔΩ </li></ul><ul><li>Δ t = tan( Ω + ΔΩ ) – tan( Ω ) </li></ul><ul><li>Faraday rotation Ω can be assumed known to within +/- 10°, </li></ul><ul><li>-10° < ΔΩ < 10° </li></ul>
  9. 9. Estimate system parameters (2) <ul><li>Invert for S xx S xx polarimetric cross products: </li></ul><ul><li>Find a representative quantity that allows retrieval of additional information: </li></ul><ul><ul><li>A-priori estimates for system distortion parameters available from ground testing + pre-flight calibration </li></ul></ul><ul><ul><li>Sweep values of phase of f 1 , f 2 and calculate A for each (f 1 , f 2 ) - pair </li></ul></ul><ul><ul><li>For a perfectly calibrated system: A = 0 </li></ul></ul><ul><ul><li>The relative phase of f 1 , f 2 can be retrieved by finding the minimum of A </li></ul></ul><ul><ul><li>If a similar relationship can be defined for other parameters, take advantage of them </li></ul></ul>
  10. 10. Estimate system parameters: f 1 , f 2 known (1)
  11. 11. Estimate system parameters: f 1 , f 2 known (2) <ul><li>Solve nonlinear equation system with Secant method or Newton-Raphson method (Fletcher-Reeves or Polak-Ribière) </li></ul><ul><li>Initialization of unknowns: - 15 S xx S xx polarimetric cross products initialized with respective M xx M xx </li></ul><ul><li> - δ 1 , δ 2 good guess </li></ul><ul><li> - Ω to within +/- 10° </li></ul><ul><li> - f 1 , f 2 assumed known </li></ul><ul><li>Requires first and second derivatives </li></ul><ul><ul><li>Derived derivatives analytically </li></ul></ul><ul><ul><li>Excerpt of sample output: </li></ul></ul>
  12. 12. Estimate system parameters: f 1 , f 2 known (3) <ul><li>With cost function for L-band: </li></ul><ul><li>Ω can be retrieved for an interval Ω truth = Ω guess +/- 10° for initial conditions in the following range: </li></ul><ul><ul><li>Magnitude of δ 1 , δ 2 : | δ truth | +/- 8 dB </li></ul></ul><ul><ul><li>Phase of δ 1 , δ 2 : < δ truth +/- 40 ° </li></ul></ul><ul><li>S xx S xx can be retrieved to within -30 dB </li></ul><ul><li>δ 1 , δ 2 are not well retrieved </li></ul>
  13. 13. Estimate system parameters: f 1 , f 2 known (4) <ul><li>With cost function for P-band: </li></ul><ul><li>Ω can be retrieved for an interval Ω truth = Ω guess +/- 10° for the same range of initial conditions </li></ul><ul><li>Observed some sensitivity to initial conditions (estimation sometimes diverges) </li></ul><ul><li>δ 1 , δ 2 are not well retrieved </li></ul>
  14. 14. Estimate system parameters: f 1 , f 2 known (5) <ul><li>Computation time (Number of iterations = 500) </li></ul><ul><li>Secant: ~ 4 hrs </li></ul><ul><li>Newton-Raphson: ~ 15 hrs </li></ul><ul><li>Make cost function sensitive to δ 1 , δ 2 : </li></ul><ul><li>Adjust optimization algorithm </li></ul><ul><ul><li>Secant method: adjust α ⟶ only minor improvement </li></ul></ul><ul><ul><li>Newton-Raphson: algorithm with Fletcher-Reeves converges better than Polak-Ribi è re </li></ul></ul><ul><li>Create cost function to be sensitive to different parameter </li></ul><ul><ul><li>Retrieval of Faraday rotation angle is opposed to retrieval </li></ul></ul><ul><ul><li>of δ 1 , δ 2 ⟶ split task into two procedures </li></ul></ul><ul><ul><li>⟶ different cost functions </li></ul></ul>
  15. 15. Estimate system parameters: Sensitive to δ 1 , δ 2 (1) <ul><li>Original cost function is kept for retrieval of Ω </li></ul><ul><li>Find the optimal cost function to retrieve δ 1 , δ 2 : </li></ul><ul><ul><li>Investigate individual channels while sweeping different parameters to assess sensitivity </li></ul></ul><ul><ul><li>32 channels, 20 unknowns: </li></ul></ul><ul><ul><ul><li>allows to discard 12 channels </li></ul></ul></ul><ul><ul><ul><li>weighting of the remaining channels to emphasize desired sensitivity </li></ul></ul></ul>
  16. 16. Estimate system parameters: Sensitive to δ 1 , δ 2 (2) <ul><li>Plots of magnitude and phase of δ 1 , δ 2 sweeped over respective parameter show: </li></ul><ul><li>Cost function is more sensitive, no more divergence </li></ul><ul><li>Magnitude of δ 1 has most sensitivity; magnitude of δ 2 and phase of δ 1 , δ 2 are more difficult to retrieve </li></ul><ul><li>Cost function has to be further improved ⟶ work in progress </li></ul>
  17. 17. Estimate system parameters: full system model (1) <ul><li>Full system model: solve for everything in one shot </li></ul><ul><li>Setup: </li></ul><ul><ul><li>32 nonlinear equations, 16 complex polarimetric cross products of the Muller matrix </li></ul></ul><ul><ul><li>24 unknowns: - 15 S xx S xx poarimetric cross products </li></ul></ul><ul><ul><li>- Real and imaginary part of δ 1 , δ 2 </li></ul></ul><ul><ul><ul><ul><ul><li>- Faraday rotation angle Ω </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>- Real and imaginary part of f 1 , f 2 </li></ul></ul></ul></ul></ul><ul><li>Cost function without simplifications, all channels contribute </li></ul><ul><li>Initialization of unknowns: - 15 S xx S xx polarimetric cross products initialized with respective M xx M xx </li></ul><ul><ul><ul><ul><li>- δ 1 , δ 2 , f 1 , f 2 good guess </li></ul></ul></ul></ul><ul><li>- Ω to within +/- 10° </li></ul>
  18. 18. Estimate system parameters: full system model (2) <ul><li>Findings for L-band: </li></ul><ul><li>Promising retrieval for Ω , f 1 , f 2 and S xx S xx for initial conditions in the following range: </li></ul><ul><ul><li>| δ truth | +/- 8 dB </li></ul></ul><ul><ul><li>< δ truth +/- 40 ° </li></ul></ul><ul><li>To retrieve δ 1 and δ 2 still need a modified cost function </li></ul>
  19. 19. Summary and Future work <ul><li>Developed method to produce pseudo-spaceborne data ⟶ solid ground truth </li></ul><ul><li>Investigated two different approaches to directly estimate parameters of the system model </li></ul><ul><li>Procedure to improve cost function outlined and enhanced sensitivity established for δ 1 , δ 2 </li></ul><ul><li>First results are promising for retrieval of Ω , S xx S xx and f 1 , f 2 as well as δ 1 , δ 2 </li></ul><ul><li>Find optimal cost function to retrieve δ 1 , δ 2 </li></ul><ul><li>Integrate the two approaches to make retrieval more robust </li></ul>
  20. 20. Thank you very much for your attention. Questions?
  21. 22. Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval in location </li></ul><ul><li>Pixel: 200/2560 </li></ul><ul><li>Line: 800/1156 </li></ul>
  22. 23. Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval in location </li></ul><ul><li>Pixel: 1200/2560 </li></ul><ul><li>Line: 700/1156 </li></ul>
  23. 24. Retrieval in L-band for different points in AIRSAR data <ul><li>With cost function in L-band: </li></ul><ul><li>Retrieval in location </li></ul><ul><li>Pixel: 1000/2560 </li></ul><ul><li>Line: 1200/1156 </li></ul>

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