Relating the Statistics of the Angle of Linear Polarization (AoLP) to Measurement Uncertainty of the Stokes Vector

Transcription

1 !! Relating the Statistics of the Angle of Linear Polarization (AoLP) to Measurement Uncertainty of the Stokes Vector! Meredith Kupinski Assistant Research Professor University of Arizona, College of Optical Sciences Polarimetric Techniques & Technology Workshop Lorentz Center, Leiden, the Netherlands March 26, 214

2 University of Arizona!2

3 Assumptions & Notation q 1 q 1 q q = Q/I -. pr(u) =N (u, -1 u = U/I u 2 u) increase u,q AoLP increase u decrease d DoLP increase u 1 pr(q) =N (q, 2 q) u,q = q = u = 1 2 atan(u/q) d = q u 2 + q 2 linear Stokes parameters uncorrelated Gaussian RVs!3

4 Utility of AoLP Statistical Analysis Hypothesis Testing: Are two samples different in AoLP? Quantitative: Given measurements what are confidence intervals on AoLP? Are these measurements due to multiple scattering in the cloud or noise?! [degrees] Intensity AoLP [ o ] An 86 nm GroundMSPI cloud measurement acquired 8/16/213 13:27(PST) at 32 o N, 11 o W. The resulting scattering angle over the FOV was 14 o -19 o.

5 Statistical Properties of AoLP pdf of DoLP DOLP 1 Analytic form for AoLP moments Rayleigh-Rice. -9 AOLP 9 pdf of AoLP involves error functions!

6 Choices for Estimating Moments of AoLP Sample Statistics 1 4 Analytic Form 1 4 first moment q 2 q [degrees] ũ [degrees] ũ ũ = u u,q second moment = AoLP Propagation of Error q q Analytic Form 1 1 q = q u,q ũ [degrees] ũ 6

7 Propagation of Error for Estimating AoLP Variance Two methods differ for very low DoLP and high measurement precision Propagation of error underestimates variance at low DoLP σ θ (K =) σ PEθ σ θ σ u,q =. Statistical testing is most useful when signals are weak DoLP [%]

8 Convergence Properties of Analytic AoLP variance K number of terms Analytic solution for AoLP moments is a Fourier Series of modulated Bessel functions. Converges around a few hundred terms ˆσθ(K) K=2 K=1 K=2 K= ˆσ PEθ

9 Multi-angle Spectro-Polarimetric Imager (MSPI) Collaboratively designed/built with JPL 1st acquisition June 21 Focal Plane Assembly board Dual PhotoElastic Modulators (PEM) Back side of mirror 2 Designed to prevent common polarimeter jitter artifacts! D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, "Dual- photoelastic- modulator- based polarimetric imaging concept for aerosol remote sensing," Appl. Opt. 46, (27). Entrance aperture with baffles MSPI Specs ******************************************** MSPI Hardware Schematic pushbroom acquisition wavebands: 3, 38, 44, 47*,, 66*, 86*, 93 (* polarization band) polarimetric uncertainty: <.% field of view: ±1 IFOV.2 ground resolution: 12m 2.2km Input light 9 PEMs + quarter- wave plates Polarization analyzer Time- modulated signal

10 AoLP Uncertainty Thresholding 4 % of image 48 % of image 6 % of image AoLP [scattering coords.] AoLP [scattering coords.] AoLP [scattering coords.] Confidence intervals ±2 σ AoLP <=3 mostly sky pixels DoLP[%] σ AoLP <=6 more cloud pixels DoLP[%] σ AoLP <=9 Intermediate to sky and 9 o DoLP[%] 1 AoLP [ o ] [degrees] DoLP [%] u,q=. AoLP [ o ] [degrees] M E A S U R E M E N T S A PRIORI CALCULATE

11 Detection of Multiple Scattering 18 Sky σ AoLP =9 ± 6 Binary Mask of Pixel Locations AoLP [scattering coords.] Confidence intervals ±2 Intensity % of image DoLP[%] 11

12 CONCLUSIONS New method for calculating moments of AoLP from uncorrelated Gaussian measurements of the linear Stokes parameters Compared new analytic method to propagation of error and sample methods of estimating moments Example of AoLP statistical analysis for identifying multiple scattering events in MSPI cloud images 12

13 UA senior design project to improve performance and packaging Teachers create original curriculum for student-led experiments Dissemination at American Indian Sci. Eng. Soc. (AISES) conference 13

14 ACKNOWLEDGEMENTS Russell Chipman, UA Professor Optical Sciences Scott Tyo, UA Professor Optical Sciences Eric Clarkson, UA Professor of Medical Imaging Dave Diner, NASA-JPL Senior Scientist, PI-MSPI instrument Christine Bradley, UA Graduate Research Assistant National Science Foundation (NSF) Science, Engineering and Education for Sustainability (SEES) Fellowship Program This work is supported by NSF !14

15 REFERENCES J. Naghizadeh-Khouei and D. Clarke, On the statistical behavior of the position angle of linear polarization, Astronomy and Astrophysics 274, 968 (1993) R. A. Chipman, OSA Handbook of Optics, chap. Polarimetry, pp (McGraw- Hill, 199). F. Goudail and A. Beniere, Estimation precision of the DoLP and of the AoLP in the presence of different sources of noise, Appl. Opt. 49(4), (21) J. L. Quinn, Bayesian analysis of polarization measurements, Astronomy and Astrophysics 38, A6 (212) P. R. Bevington and D. K. Robinson. Data reduction and error analysis for the physical sciences, volume 2. McGraw-Hill New York, F. Olver, D. Lozier, R. Boisvert, and C. Clark NIST Handbook of Mathematical Functions, Cambridge University Press 26,(21) D. Diner, F. Xu, J. Martonchik, B. Rheingans, S. Geier, V. Jovanovic, A. Davis, R. Chipman, S. McClain Exploration of a Polarized Surface Bidirectional Reflectance Model Using the Ground-Based Multiangle SpectroPolarimetric Imager, Atmosphere 3(4), (212)!1