A synthetic aperture radar (SAR) system requires external absolute calibration so that radiometric measurements can be exploited in numerous scientific and commercial applications. Besides estimating a calibration factor, metrological standards also demand the derivation of a respective calibration uncertainty. This uncertainty is currently not systematically determined. Here for the first time it is proposed to use hierarchical modeling and Bayesian statistics as a consistent method for handling and analyzing the hierarchical data typically acquired during external calibration campaigns. Through the use of Markov chain Monte Carlo simulations, a joint posterior probability can be conveniently derived from measurement data despite the necessary grouping of data samples. The applicability of the method is demonstrated through a case study: The radar reflectivity of DLR’s new C-band Kalibri transponder is derived through a series of RADARSAT-2 acquisitions and a comparison with reference point targets (corner reflectors). The systematic derivation of calibration uncertainties is seen as an important step toward traceable radiometric calibration of synthetic aperture radars.
References
[1]
Van Zyl, J.J.; Kim, Y. Synthetic Aperture Radar Polarimetry; John Wiley & Sons, Inc: Hoboken, NJ, USA, 2011.
[2]
Dobson, M.C.; Ulaby, F.T.; Letoan, T.; Beaudoin, A.; Kasischke, E.S.; Christensen, N. Dependence of radar backscatter on coniferous forest biomass. IEEE Trans. Geosci. Remote Sens. Electr 1992, 30, 412–415.
[3]
Freeman, A. SAR calibration: An overview. IEEE Trans. GeoscI. Remote Sens 1992, 30, 1107–1121.
[4]
Curlander, J.C. Synthetic Aperture Radar: Systems and Signal Processing; John Wiley & Sons, Inc: Hoboken, NJ, USA, 1991.
[5]
Schwerdt, M.; Br？utigam, B.; Bachmann, M.; D？ring, B.; Schrank, D.; Hueso Gonzalez, J. Final TerraSAR-X calibration results based on novel efficient methods. IEEE Trans. Geosci. Remote Sens 2010, 48, 677–689.
[6]
Shimada, M.; Isoguchi, O.; Tadono, T.; Isono, K. PALSAR radiometric and geometric calibration. IEEE Trans. Geosci. Remote Sens 2009, 47, 3915–3932.
[7]
Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement. ISO/IEC Guide 98-3:2008;; ISO copyright office: Geneva, Switzerland, 2008.
[8]
Schwerdt, M.; Hueso Gonzalez, J.; Bachmann, M.; Schrank, D.; D？ring, B.; Tous Ramon, N.; Walter Antony, J.M. In-Orbit Calibration of the TanDEM-X System. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada, 24–29 July 2011; pp. 2420–2423.
[9]
Luscombe, A.P.; Thompson, A.A. RADARSAT-2 Calibration: Proposed Targets and Techniques. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada, 24–29 July 2001; pp. 496–498.
[10]
Luscombe, A. Image Quality and Calibration of RADARSAT-2. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Cape Town, South Africa, 12–17 July 2009; pp. 757–760.
[11]
Chakraborty, M.; Panigrahy, S.; Rajawat, A.S.; Kumar, R.; Murthy, T.V.R.; Haldar, D.; Chakraborty, A.; Kumar, T.; Rode, S.; Kumar, H.; et al. Initial results using RISAT-1 C-band SAR data. Curr. Sci 2013, 104, 490.
[12]
Gelman, A.; Carlin, J.B.; Stern, H.S.; Rubin, D.B. Bayesian Data Analysis, 2nd ed ed.; Chapman & Hall/CRC: Boca Raton, FL, USA, 2004.
[13]
Gregory, P. Bayesian Logical Data Analysis for the Physical Sciences; Cambridge University Press: Cambridge, UK, 2005.
[14]
Bayarri, M.J.; Berger, J.O. The interplay of bayesian and frequentist analysis. Stat. Sci 2004, 19, 58–80.
[15]
D？ring, B.J.; Schwerdt, M. The radiometric measurement quantity for SAR images. IEEE Trans. Geosci. Remote Sens. 2013, doi:10.1109/TGRS.2012.2234128.
[16]
Bolstad, W.M. Introduction to Bayesian Statistics, 2nd ed ed.; John Wiley & Sons, Inc: Hoboken, NJ, USA, 2007; p. 464.
[17]
Patil, A.; Huard, D.; Fonnesbeck, C.J. PyMC: Bayesian stochastic modelling in Python. J. Stat. Softw 2010, 35, 1–81.
[18]
D？ring, B.J.; Looser, P.; Jirousek, M.; Schwerdt, M.; Peichl, M. Highly Accurate Calibration Target for Multiple Mode SAR Systems. Proceedings of the European Conference on Synthetic Aperture Radar, Aachen, Germany, 7–10 June 2010; 8.
Oerry, A.W.; Brock, B.C. Radar Cross Section of Triangular Trihedral Reflector with Extended Bottom Plate. Technical Report May;; Sandia National Laboratories: Albuquerque, NM, USA, 2009.
[21]
Gray, A.L.; Vachon, P.W.; Livingstone, C.E.; Lukowski, T.I. Synthetic aperture radar calibration using reference reflectors. IEEE Trans. Geosci. Remote Sens 1990, 28, 374–383.
[22]
Dettwiler, M. RADARSAT-2 Product Format Definition. Technical Report 1/9;; MacDonald, Dettwiler and Associates Ltd: Richmond, Canada, 2011.
[23]
Schwerdt, M.; D？ring, B.; Zink, M.; Schrank, D. In-Orbit Calibration Plan of Sentinel-l. Proceedings of the European Conference on Synthetic Aperture Radar, Aachen, Germany, 7–10 June 2010; pp. 350–353.
[24]
Weise, K.; W？ger, W. A Bayesian theory of measurement uncertainty. Meas. Sci. Technol. 1992, doi:10.1088/0957-0233/4/1/001.
[25]
Kacker, R.; Jones, A. On use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent. Metrologia 2003, doi:10.1088/0026-1394/40/5/305.
[26]
Willink, R.; White, R. Disentangling Classical and Bayesian Approaches to Uncertainty Analysis. Technical Report No. CCT/12-08;; BIPM: Sevres, France, 2012.
