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In this paper we put forward a possibility of refractive index profile retrieval using field measurements at an array of radio receivers through the variational adjoint approach. The derivation of the adjoint model begins with the electromagnetic parabolic equation for a smooth, perfectly conducting surface and horizontal polarization conditions. To deal with the ill-posed difficulties of the inversion, the regularization idea is introduced into the establishment of the cost function. Based on the variational assimilation idea, the retrieval iterative format is constructed. Numerical experiments demonstrate the feasibility of theoretic algorithm for refractivity estimation. However, using the split-step Fourier algorithm to solve the forward model and the adjoint model, the intrinsic error of the solutions will increase with the extension of the propagation range, which reduces the inversion accuracy at long distance propagation. Through adopting a good initial refractivity profile and introducing the background fields in the cost function the inversions could generally be improved.
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Keywords:
- atmospheric refractivity /
- electromagnetic parabolic equation /
- variational adjoint approach /
- regularization
[1] Yan H J, Fu Y, Hong Z J 2006 Introduction to Modern Atmospheric Refraction (Shanghai: Science and Educational Press) (in Chinese) p16 [严豪健、符 养、洪振杰 2006 现代大气折射引论 (上海:上海科技教育出版社) 第16页]
[2] Halvey R A 1983 Proc. IEEE Part F 130 643
[3] [4] [5] Richter J H 1969 Radio Sci. 4 1261
[6] [7] Gingras D F, Gerstoft P, Gerr N L 1997 IEEE Trans. Antennas Propagat. 45 1536
[8] Tabrikian J, Krolik J L 1999 IEEE Trans. Antennas Propagat. 47 1727
[9] [10] [11] Gerstoft P, Gingras D F, Rogers L T, Hodgkiss W S 2000 IEEE Trans. Antennas Propagat. 48 345
[12] [13] Valtr P and Pechac P 2005 Novel Method of Vertical Refractivity Profile Estimation Using Angle of Arrival Spectra 28th General Assembly of International Union. of Radio Science New Delhi, India
[14] [15] Zhao X F, Huang S X, Shi W L, Xiang J 2011 Chin. Phys. B 20 099201
[16] Huang S X, Han W, Wu R S 2004 Sci. Chin. D 47 630
[17] [18] [19] Huang S X, Wu R S 2005 Mathematical Physics Problems in Atmospheric Science 2nd Edition (Beijing: Meteorology Press) (in Chinese) p460 [黄思训、伍荣生 2005 大气科学中的数学物理问题 第二版 (北京:气象出版社) 第460页]
[20] [21] Huang S X, Teng J J, Lan W R, Xiang J 2005 Chin. J. Theor. Appl. Mech. 37 399 (in Chinese) [黄思训、滕加俊、兰伟仁、项 杰 2005 力学学报 37 399]
[22] [23] Hursky P, Porter M B, Cornuelle B D, Hodgkiss W S, Kuperman W A 2004 J. Acoust. Soc. Am. 115 607
[24] [25] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 1734 (in Chinese) [盛 峥、黄思训 2010 物理学报 59 1734]
[26] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 3912 (in Chinese) [盛 峥、黄思训 2010 物理学报 59 3912]
[27] [28] [29] Dockery G D 1988 IEEE Trans. Antennas Propagat. 36 1464
[30] Kuttler J R, Dockery G D 1991 Radio Sci. 26 381
[31] [32] [33] Barrios A E 1994 IEEE Trans. Antennas Propagat. 42 90
[34] [35] Akbarpour R, Webster A R 2005 IEEE Trans. Antennas Propagat. 53 3785
[36] [37] Barrios A E, Anderson K D, Lindem G E 2006 Advanced Propagation Model (APM) Analysis of VHF Signals in the Southern California Desert Technical Report 1945 San Diego, USA
[38] Valtr P, Pechac P, Kvicera V, Grabner M A 2010 Radioengineering 19 117
[39] [40] Zhao X F, Huang S X 2010 Radioengineering 19 601
[41] [42] Morales J L, Nocedal J 2001 ACM Trans. on Mathemath. Software 27 83
[43] [44] Yardim C 2007 Ph.D. Dissertation (Electrical Engineering, University of California, San Diego, USA)
[45] [46] [47] Tikhonov A N, Arsenin V Y 1977 Solutions of ill-posed problems Winston and Sons, Washington, USA
[48] Hansen P C, Oleary D P 1993 Siam J. Sci. Comput. 14 1487
[49] [50] [51] Li Jun, Huang S X 2001 Sci. Chin. Ser. D 44 847
[52] Kalnay E 2005 Atmospheric Modeling, Data Assimilation and Predictability (Beijing: Meteorology Press) (in Chinese) p115
[53] -
[1] Yan H J, Fu Y, Hong Z J 2006 Introduction to Modern Atmospheric Refraction (Shanghai: Science and Educational Press) (in Chinese) p16 [严豪健、符 养、洪振杰 2006 现代大气折射引论 (上海:上海科技教育出版社) 第16页]
[2] Halvey R A 1983 Proc. IEEE Part F 130 643
[3] [4] [5] Richter J H 1969 Radio Sci. 4 1261
[6] [7] Gingras D F, Gerstoft P, Gerr N L 1997 IEEE Trans. Antennas Propagat. 45 1536
[8] Tabrikian J, Krolik J L 1999 IEEE Trans. Antennas Propagat. 47 1727
[9] [10] [11] Gerstoft P, Gingras D F, Rogers L T, Hodgkiss W S 2000 IEEE Trans. Antennas Propagat. 48 345
[12] [13] Valtr P and Pechac P 2005 Novel Method of Vertical Refractivity Profile Estimation Using Angle of Arrival Spectra 28th General Assembly of International Union. of Radio Science New Delhi, India
[14] [15] Zhao X F, Huang S X, Shi W L, Xiang J 2011 Chin. Phys. B 20 099201
[16] Huang S X, Han W, Wu R S 2004 Sci. Chin. D 47 630
[17] [18] [19] Huang S X, Wu R S 2005 Mathematical Physics Problems in Atmospheric Science 2nd Edition (Beijing: Meteorology Press) (in Chinese) p460 [黄思训、伍荣生 2005 大气科学中的数学物理问题 第二版 (北京:气象出版社) 第460页]
[20] [21] Huang S X, Teng J J, Lan W R, Xiang J 2005 Chin. J. Theor. Appl. Mech. 37 399 (in Chinese) [黄思训、滕加俊、兰伟仁、项 杰 2005 力学学报 37 399]
[22] [23] Hursky P, Porter M B, Cornuelle B D, Hodgkiss W S, Kuperman W A 2004 J. Acoust. Soc. Am. 115 607
[24] [25] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 1734 (in Chinese) [盛 峥、黄思训 2010 物理学报 59 1734]
[26] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 3912 (in Chinese) [盛 峥、黄思训 2010 物理学报 59 3912]
[27] [28] [29] Dockery G D 1988 IEEE Trans. Antennas Propagat. 36 1464
[30] Kuttler J R, Dockery G D 1991 Radio Sci. 26 381
[31] [32] [33] Barrios A E 1994 IEEE Trans. Antennas Propagat. 42 90
[34] [35] Akbarpour R, Webster A R 2005 IEEE Trans. Antennas Propagat. 53 3785
[36] [37] Barrios A E, Anderson K D, Lindem G E 2006 Advanced Propagation Model (APM) Analysis of VHF Signals in the Southern California Desert Technical Report 1945 San Diego, USA
[38] Valtr P, Pechac P, Kvicera V, Grabner M A 2010 Radioengineering 19 117
[39] [40] Zhao X F, Huang S X 2010 Radioengineering 19 601
[41] [42] Morales J L, Nocedal J 2001 ACM Trans. on Mathemath. Software 27 83
[43] [44] Yardim C 2007 Ph.D. Dissertation (Electrical Engineering, University of California, San Diego, USA)
[45] [46] [47] Tikhonov A N, Arsenin V Y 1977 Solutions of ill-posed problems Winston and Sons, Washington, USA
[48] Hansen P C, Oleary D P 1993 Siam J. Sci. Comput. 14 1487
[49] [50] [51] Li Jun, Huang S X 2001 Sci. Chin. Ser. D 44 847
[52] Kalnay E 2005 Atmospheric Modeling, Data Assimilation and Predictability (Beijing: Meteorology Press) (in Chinese) p115
[53]
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