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

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

Overview Problem statement Approach: Generate synthetic data Estimate system parameters Summary Future work

Problem statement (1) 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. Polarimetric radar system model: S = scattering matrix R F = one-way Faraday rotation matrix R = receive distortion matrix (of the radar system) T = transmit distortion matrix (of the radar system) A(r, θ ) = real factor representing the overall gain of the radar system exp(j ϕ ) = complex factor representing the round-trip phase delay and system dependent phase effects on a signal N = additive noise terms With δ i (i = 1,2,3,4) = crosstalk values f 1 , f 2 = channel amplitude and phase imbalance terms Ω = Faraday rotation angle

Problem statement (2) General assumptions: A(r, θ ), exp(j ϕ ), N are considered negligible Reciprocal crosstalk: δ 3 = δ 1 , δ 4 = δ 2 Backscatter reciprocity: S HV = S VH 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

Approach Overview Generate synthetic data = create ground truth for Ω and system parameters = superimpose on airborne data Estimate system parameters Estimate arg(f 1 /f 2 ) Estimate remaining system parameters from nonlinear system of equations

Generate Ground Truth (1) AIRSAR data available for P, L, C band Predict Faraday rotation angle Ω Faraday rotation ranging from 2.5° (C-band) to 320 ° (P-band) Input needed for models: altitude frequency year/date look/azimuth angle time latitude/longitude 2 models needed: IRI – 2007 (International Reference Ionosphere) IGRF (International Geomagnetic Reference Field) ⟶ gives Ω to within +/- 10° 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

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

Estimate system parameters (1) Convert system model into a form suggested by Freeman (2008): Assume Ω is known to within ΔΩ : t = tan( Ω ) Ω truth = Ω + ΔΩ Δ t = tan( Ω + ΔΩ ) – tan( Ω ) Faraday rotation Ω can be assumed known to within +/- 10°, -10° < ΔΩ < 10°

Estimate system parameters (2) Invert for S xx S xx polarimetric cross products: Find a representative quantity that allows retrieval of additional information: A-priori estimates for system distortion parameters available from ground testing + pre-flight calibration Sweep values of phase of f 1 , f 2 and calculate A for each (f 1 , f 2 ) - pair For a perfectly calibrated system: A = 0 The relative phase of f 1 , f 2 can be retrieved by finding the minimum of A If a similar relationship can be defined for other parameters, take advantage of them

Estimate system parameters: f 1 , f 2 known (1)

Estimate system parameters: f 1 , f 2 known (2) Solve nonlinear equation system with Secant method or Newton-Raphson method (Fletcher-Reeves or Polak-Ribière) Initialization of unknowns: - 15 S xx S xx polarimetric cross products initialized with respective M xx M xx - δ 1 , δ 2 good guess - Ω to within +/- 10° - f 1 , f 2 assumed known Requires first and second derivatives Derived derivatives analytically Excerpt of sample output:

Estimate system parameters: f 1 , f 2 known (3) With cost function for L-band: Ω can be retrieved for an interval Ω truth = Ω guess +/- 10° for initial conditions in the following range: Magnitude of δ 1 , δ 2 : | δ truth | +/- 8 dB Phase of δ 1 , δ 2 : < δ truth +/- 40 ° S xx S xx can be retrieved to within -30 dB δ 1 , δ 2 are not well retrieved

Estimate system parameters: f 1 , f 2 known (4) With cost function for P-band: Ω can be retrieved for an interval Ω truth = Ω guess +/- 10° for the same range of initial conditions Observed some sensitivity to initial conditions (estimation sometimes diverges) δ 1 , δ 2 are not well retrieved

Estimate system parameters: f 1 , f 2 known (5) Computation time (Number of iterations = 500) Secant: ~ 4 hrs Newton-Raphson: ~ 15 hrs Make cost function sensitive to δ 1 , δ 2 : Adjust optimization algorithm Secant method: adjust α ⟶ only minor improvement Newton-Raphson: algorithm with Fletcher-Reeves converges better than Polak-Ribi è re Create cost function to be sensitive to different parameter Retrieval of Faraday rotation angle is opposed to retrieval of δ 1 , δ 2 ⟶ split task into two procedures ⟶ different cost functions

Estimate system parameters: Sensitive to δ 1 , δ 2 (1) Original cost function is kept for retrieval of Ω Find the optimal cost function to retrieve δ 1 , δ 2 : Investigate individual channels while sweeping different parameters to assess sensitivity 32 channels, 20 unknowns: allows to discard 12 channels weighting of the remaining channels to emphasize desired sensitivity

Estimate system parameters: Sensitive to δ 1 , δ 2 (2) Plots of magnitude and phase of δ 1 , δ 2 sweeped over respective parameter show: Cost function is more sensitive, no more divergence Magnitude of δ 1 has most sensitivity; magnitude of δ 2 and phase of δ 1 , δ 2 are more difficult to retrieve Cost function has to be further improved ⟶ work in progress

Estimate system parameters: full system model (1) Full system model: solve for everything in one shot Setup: 32 nonlinear equations, 16 complex polarimetric cross products of the Muller matrix 24 unknowns: - 15 S xx S xx poarimetric cross products - Real and imaginary part of δ 1 , δ 2 - Faraday rotation angle Ω - Real and imaginary part of f 1 , f 2 Cost function without simplifications, all channels contribute Initialization of unknowns: - 15 S xx S xx polarimetric cross products initialized with respective M xx M xx - δ 1 , δ 2 , f 1 , f 2 good guess - Ω to within +/- 10°

Estimate system parameters: full system model (2) Findings for L-band: Promising retrieval for Ω , f 1 , f 2 and S xx S xx for initial conditions in the following range: | δ truth | +/- 8 dB < δ truth +/- 40 ° To retrieve δ 1 and δ 2 still need a modified cost function

Summary and Future work Developed method to produce pseudo-spaceborne data ⟶ solid ground truth Investigated two different approaches to directly estimate parameters of the system model Procedure to improve cost function outlined and enhanced sensitivity established for δ 1 , δ 2 First results are promising for retrieval of Ω , S xx S xx and f 1 , f 2 as well as δ 1 , δ 2 Find optimal cost function to retrieve δ 1 , δ 2 Integrate the two approaches to make retrieval more robust

Thank you very much for your attention. Questions?

Retrieval in L-band for different points in AIRSAR data With cost function in L-band: Retrieval in location Pixel: 200/2560 Line: 800/1156

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

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