volume | PIER Journals

Abstract

Development of theory and experiments to retrieve Green's functions from cross correlations of recorded wave fields between two receivers has grown rapidly in the last seven years. The theory includes situations with flow, mechanical and electromagnetic disturbances and their mutual coupling. Here an electromagnetic theory is presented for Green's function retrieval from cross correlations that incorporates general bianisotropic media, which is the most general class of linear media. In the presence of dispersive non-reciprocal media, the Green's function is obtained by cross correlating the recordings at two locations of fields generated by sources on a boundary. The only condition for this relation to be valid is that the medium is non-dissipative. The principle of bianisotropic Green's function retrieval by cross correlation is illustrated with a numerical example.

1. Scherbaum, F., "Seismic imaging of the site response using microearthquake recordings: Part I. Method," Bull. Seism. Soc. Am., Vol. 77, No. 6, 1905-1923, 1987.

2. Scherbaum, F., "Seismic imaging of the site response using microearthquake recordings: Part I. Application to the Swabian Jura, southwest Germany, seismic network," Bull. Seism. Soc. Am., Vol. 77, No. 6, 1924-1944, 1987.

3. Buckingham, M. J., B. V. Berkhout, and S. A. L. Glegg, "Imaging the ocean with ambient noise," Nature, Vol. 356, No. 6367, 327-329, London, March 1992.
doi:10.1038/356327a0

4. Weaver, R. and O. Lobkis, "Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies," Phys. Rev. Lett., Vol. 87, No. 13, 134301-1-134301-4, 2001.
doi:10.1103/PhysRevLett.87.134301

5. Campillo, M. and A. Paul, "Long-range correlations in the diffuse seismic coda waves," Science, Vol. 299, No. 5606, 547-549, 2003.
doi:10.1126/science.1078551

6. Shapiro, N., M. Campillo, L. Stehly, and M. Ritzwoller, "High-resolution surface-wave tomography from ambient seismic noise," Science, Vol. 307, No. 5715, 1615-1618, 2005.
doi:10.1126/science.1108339

7. Draganov, D., K. Wapenaar, W. Mulder, J. Singer, and A. Verdel, "Retrieval of reflections from seismic background-noise measurements," Geoph. Res. Lett., Vol. 34, No. 4, L04305, 2007.
doi:10.1029/2006GL028735

8. Schuster, G., J. Yu, J. Sheng, and J. Rickett, "Interferomet-ric/daylight seismic imaging," Geoph. J. Int., Vol. 157, No. 2, 838-852, 2004.
doi:10.1111/j.1365-246X.2004.02251.x

9. Snieder, R., "Extracting the Green's function from the correlation of coda waves: A derivation based on stationary phase," Phys. Rev. E, Vol. 69, No. 4, 046610-1-046610-8, 2004.
doi:10.1103/PhysRevE.69.046610

10. Wapenaar, K., "Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation," Phys. Rev. Lett., Vol. 93, No. 25, 254301-1-254301-4, 2004.
doi:10.1103/PhysRevLett.93.254301

11. Lerosey, G., J. De Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, "Time reversal of electromagnetic waves," Phys. Rev. Lett., Vol. 92, No. 19, 193904-1-193904-3, 2004.
doi:10.1103/PhysRevLett.92.193904

12. Corbella, I., N. Duffo, M. Vall-llossera, A. Camps, and F. Torres, "The visibility function in interferometric aperture synthesis radiometry," IEEE Trans. on Geoscience and Remote Sensing, Vol. 42, No. 8, 1677-1682, 2004.
doi:10.1109/TGRS.2004.830641

13. Oestges, C., A. Kim, G. Papanicolaou, and A. Paulraj, "Characterization of space-time focusing in time-reversed random fields," IEEE Trans. Ant. and Prop., Vol. 53, No. 1, 283-293, 2005.
doi:10.1109/TAP.2004.836399

14. Snieder, R., "Retrieving the Green's function of the diffusion equation from the response to a random forcing ," Phys. Rev. E, Vol. 74, No. 4, 046620-1-046620-9, 2006.
doi:10.1103/PhysRevE.74.046620

15. Slob, E., D. Draganov, and K. Wapenaar, "GPR without a source," Proceedings of the 11th International Conf. on GPR, ANT. 6, Ohio State University, Columbus, Ohio, , 2006.

16. Wapenaar, K., E. Slob, and R. Snieder, "United Green's function retrieval by cross correlation ," Phys. Rev. Lett., Vol. 97, No. 23, 234301-1-234301-4, 2006.
doi:10.1103/PhysRevLett.97.234301

17. Slob, E., D. Draganov, and K. Wapenaar, "Interferometric electromagnetic Green's functions representations using propagation invariants," Geoph. J. Int., Vol. 169, No. 1, 60-80, 2007.
doi:10.1111/j.1365-246X.2006.03296.x

18. Slob, E. and K. Wapenaar, "GPR without a source: Cross-correlation and cross-convolution methods," IEEE Trans. Geoscience and Remote Sensing, Vol. 45, No. 8, 2501-2510, 2007.
doi:10.1109/TGRS.2007.900995

19. Slob, E. and K. Wapenaar, "Electromagnetic Green's functions retrieval by cross-correlation and cross-convolution in media with losses," Geoph. Res. Lett., Vol. 34, No. 5, L05307, 2007.

20. Weaver, R., "Ward identities and the retrieval of Green's functions in the correlations of a diffuse field," Wave Motion, Vol. 45, No. 5, 596-604, 2008.
doi:10.1016/j.wavemoti.2007.07.007

21. Ruigrok, E., D. Draganov, and K. Wapenaar, "Global-scale seismic interferometry: Theory and numerical examples," Geophysical Prospecting, Vol. 56, No. 3, 395-417, May 2008.
doi:10.1111/j.1365-2478.2008.00697.x

22. Kong, J., "Theorems of bianisotropic media," Proceedings of the IEEE, Vol. 60, No. 9, 1036-1046, 1972.

23. Chow, W. W., J. G.-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys., Vol. 57, No. 1, 61-104, January 1985.
doi:10.1103/RevModPhys.57.61

24. Sihvola, A., "Metamaterials in electromagnetics," Metamaterials, Vol. 1, No. 1, 2-11, 2007.
doi:10.1016/j.metmat.2007.02.003

25. Caloz, C. and T. Itoh, Electromagnetic Metamaterials, John Wiley & Sons, Inc., Hoboken, New Jersey, 2006.

26. Pitarch, J., J. M. C.-Civera, F. L. P.-Foix, and M. A. Solano, "Efficient modal analysis of bianisotropic waveguides by the coupled mode method," IEEE Trans. on Microwave Theory and Techniques, Vol. 55, No. 1, 108-116, 2007.
doi:10.1109/TMTT.2006.888576

27. Cui, T.-J., H.-F. Ma, R. P. Liu, B. Zhao, Q. Cheng, and J. Y. Chin, "A symmetrical circuit model describing all kinds of circuit metamaterials," Progress In Electromagnetics Research B, Vol. 5, 63-76, 2008.
doi:10.2528/PIERB08013009

28. Huang, R., Z.-W. Li, L. B. Kong, L. Liu, and S. Matitsine, "Analysis and design of an ultra-thin metamaterial absorber," Progress In Electromagnetics Research B, Vol. 14, 407-429, 2009.
doi:10.2528/PIERB09040902

29. Weng, Z.-B., Y.-C. Jiao, F.-S. Zhang, Y. Song, and G. Zhao, "A multi-band patch antenna on metamaterial substrate," J. of Electromagn. Waves and Appl., Vol. 22, No. 2/3, 445-452, 2008.
doi:10.1163/156939308784160776

30. Wongkasem, N., A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, "Development of chiral negative refractive index metamaterials for the terahertz frequency regime," IEEE Trans. Antennas and Propagation, Vol. 55, No. 11, 3052-3062, 2007.
doi:10.1109/TAP.2007.909419

31. Meiners, C. and A. F. Jacob, "Numerical and experimental parameter study of helix layers," Trans. Antennas and Propagation, Vol. 56, No. 5, 1321-1328, 2008.
doi:10.1109/TAP.2008.922170

32. Silveirinha, M. G., "Design of linear-to-circular polarization transformers made of long densely packed metallic helices," IEEE Trans. Antennas and Propagation, Vol. 56, No. 2, 390-401, 2008.
doi:10.1109/TAP.2007.915428

33. Santagata, N. M., P. Luo, A. M. Lakhani, D. J. De Witt, B. S. Day, M. L. Norton, and T. P. Pearl, "Organizational structure and electronic decoupling of surface bound chiral domains and biomolecules," IEEE Sensors Journal, Vol. 8, No. 6, 758-766, 2008.
doi:10.1109/JSEN.2008.923187

34. Tajitsu, Y., "Piezoelectricity of chiral polymeric fiber and its application in biomedical engineering," IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 55, No. 5, 1000-1008, 2008.
doi:10.1109/TUFFC.2008.746

35. Yang, X. M., J. Y. Chin, Q. Cheng, X. Q. Lin, and T. J. Cui, "Realization and experimental verification of chiral cascaded circuit," IEEE Microwave and Wireless Components Letters, Vol. 18, No. 5, 308-310, 2008.

36. Dong, J. and C. Xu, "Characteristics of guided modes in planar chiral nihility meta-material waveguides," Progress In Electromagnetics Research B, Vol. 14, 107-126, 2009.
doi:10.2528/PIERB09012201

37. Lindell, I., A. Sihvola, and K. Suchy, "Six-vector formalism in electromagnetics of bi-anisotropic media," Journal of Electromagnetic Waves and Applications, Vol. 9, No. 7-8, 887-903, 1995.
doi:10.1163/156939395X00631

38. Wapenaar, K. and J. Fokkema, "Reciprocity theorems for diffusion, flow and waves," A.S.M.E. Journal of Applied Mechanics, Vol. 71, No. 1, 145-150, 2004.

39. Tai, C., "A study of electrodynamics of moving media," Proceedings of the IEEE, Vol. 52, No. 6, 685-689, 1964.

40. Harrington, R. and A. Villeneuve, "Reciprocity relationships for gyrotropic media," IRE Trans. on Microwave Theory and Techn., Vol. 6, No. 3, 308-310, 1958.
doi:10.1109/TMTT.1958.1124563

41. Altman, C. and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, Kluwer, Dordrecht, 1991.

42. Lindell, I., A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves and Bi-Isotropic Media, Artech House, Boston, 1994.

43. Melrose, D. and R. McPhedran, Electromagnetic Processes in Dispersive Media, Cambridge University Press, Cambridge, 1991.

44. Landau, L. and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York, 1994.

45. Van Manen, D.-J., J. Robertsson, and A. Curtis, "Modeling of wave propagation in inhomogeneous media," Phys. Rev. Lett., Vol. 94, No. 16, 164301-1-164301-4, 2005.
doi:10.1103/PhysRevLett.94.164301

46. Wapenaar, K., "General wave field representations for seismic modeling and inversion ," Geophysics, Vol. 72, No. 5, SM5-SM17, 2007.
doi:10.1190/1.2750646

47. Shen, J., "Negative refractive index in gyrotropically magnetoelectric media," Phys. Rev. B, Vol. 73, No. 4, 045113, 2006.
doi:10.1103/PhysRevB.73.045113

48. Kiehn, R., G. Kiehn, and J. Roberds, "Parity and time-reversal symmetry breaking, singular solitons, and Fresnel surfaces," Phys. Rev. A, Vol. 43, No. 10, 5665-5671, 1991.
doi:10.1103/PhysRevA.43.5665

Dr. Evert C. Slob

Dr. Kees Wapenaar