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Ground penetrating radar (GPR) forward is one of the geophysical research directions.Through the forward of geological model,the database of radar model can be enriched and the characteristics of typical geological radar echo images can be understood,which in turn can guide the data interpretation of GPR measured profile,thereby improving the GPR data interpretation level.In this article,the interpolating wavelet scale function by using iterative interpolation method is presented,and the derivative of scale function is used in spatial differentiation of discrete Maxwell equations. The forward modeling formula of GPR based on the interpolation wavelet scale method is derived by using fourth-order Runge-Kutta method (RK4) for calculating the higher time derivative.Compared with the conventional finite difference time domain (FDTD) algorithm based on the central difference method,the interpolation wavelet scale algorithm improves the accuracy of GPR wave equation in both space and time discretization.Firstly,the FDTD algorithm and the interpolation wavelet scale method are applied to the forward modeling of a layered model with analytic solution. Single channel radar data and analytical solution fitting indicate that the interpolation wavelet scale method has higher accuracy than FDTD,with the same mesh generation used.Therefore,auxiliary differential equation perfectly matching layer (ADE-PML) boundary condition is used on an interpolation wavelet scale,and the comparisons between reflection errors obtained using CPML (FDTD),RK4ADE-PML (FDTD),and RK4ADE-PML (interpolating wavelet scales) in a homogeneous medium model show that the absorption effect of RK4ADE-PML (interpolating wavelet scales) is better than the other two absorbing boundaries.Finally,interpolation wavelet scale method,with both UPML,FDTD and RK4ADE-PML loaded,is used for two-dimensional GPR forward modeling,showing good absorption effect for evanescent wave.From all the experimental results,the following conclusions are obtained.1) Using the derivative of the interpolating wavelet scale function instead of central difference schemes for the spatial derivative discretization of Maxwell equations and time derivative calculated using the fourth-order Runge Kutta algorithm,the interpolating wavelet scale algorithm has higher accuracy than regular FDTD algorithm due to the improvement in the spatial and time accuracy of GPR wave equation.2) The best absorption layer parameters of interpolating wavelet scale RK4ADE-PML are selected, when the maximum value of the reflection error is the minimum.The maximum reflection error can reach-93 dB,which increases 20 dB compared with that of UMPL boundary in FDTD algorithm.And the higher simulation accuracy of interpolating wavelet scale algorithm than FDTD algorithm is confirmed after calculating single channel radar data.3) Comparing wave field snapshots of GPR forward modeling,radar pictures from wide-angle method and section method indicates that interpolating wavelet scale RK4ADE-PML reduces reflection error of absorption boundary,improves both spatial and time accuracy,is more effective than UPML boundary in eliminating false reflection of large angle incidence, and has better absorption effect for evanescent wave and low-frequency wave.
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Keywords:
- ground penetrating radar /
- interpolating wavelet scales method /
- auxiliary differentiation /
- fourth-order Runge-Kutta
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[9] Wei B, Li X Y, Wang F, Ge D B 2009 Acta Phys. Sin. 58 6174 (in Chinese)[魏兵, 李小勇, 王飞, 葛德彪2009物理学报58 6174]
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[12] Zhuan S X, Ma X K 2012 Acta Phys. Sin. 61 110206 (in Chinese)[颛孙旭, 马西奎2012物理学报61 110206]
[13] Xu L J, Liu S B, Mo J J, Yuan N C 2006 Acta Phys. Sin. 55 3470 (in Chinese)[徐利军, 刘少斌, 莫锦军, 袁乃昌2006物理学报55 3470]
[14] Vivek K, Mani M 2006 J. Comput. Appl. Math. 230 803
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[16] Marta D L L P, Stewart C, Robert P 2012 J. Comput. Phys. 231 6754
[17] Rodrigo B B, Marco A C S, Raul R E S 2013 Finite Elem. Anal. Des. 75 71
[18] Martin R, Komatitsch D, Gedney S D, Bruthiaux E 2010 CMES 56 17
[19] Zhang W, Shen Y 2010 Geophysics 75 141
[20] Zhao J G 2014 Jilin University 44 675 (in Chinese)[赵建国2014吉林大学学报44 675]
[21] Li J X 2007 Ph. D. Dissertation (Tianjin:Tianjin University) (in Chinese)[李建雄2007博士学位论文(天津:天津大学)]
[22] Deslauriers G, Dubuc S 1989 Constr. Approx. 5 49
[23] Dubuc 1986 Math. Anal. Appl. 114 185
[24] Satio N, Beylkin G 1993 IEEE Trans. Signal Process. 41 319
[25] Sweldens W 1996 Appl. Comput. Harmon. Anal. 3 186
[26] M Sc Hao J L 2011 Ph. D. Dissertation (Zur Erlangung des akademischen Grades eines)
[27] Ge D B, Yan Y B 2005 Finite-Differeence Time-Domain Method for Electromagnetic Waves (Xi'an:Xidian University Press) p31(in Chinese)[葛德彪, 闫玉波2005电磁波时域有限差分方法(西安:西安电子科技大学出版社)第31页]
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[1] Li J 2014 Ph. D. Dissertation (Changchun:Jilin University) (in Chinese)[李静2014博士学位论文(长春:吉林大学)]
[2] Feng D S, Chen J W, Wu Q 2014 Chin. J. Geophys. 57 1322 (in Chinese)[冯德山, 陈佳维, 吴奇2014地球物理学报57 1322]
[3] Irving J, Knight R 2006 Comput. Geosci. 32 1247
[4] Liu S X, Zeng Z F 2007 Chin. J. Geophys. 50 320 (in Chinese)[刘四新, 曾昭发2007地球物理学报50 320]
[5] Diamanti N, Giannopoulos A 2009 J. Appl. Geophys. 67 309
[6] Teixeira F L 2008 IEEE Trans. Antennas and Propag. 56 2150
[7] Feng D S, Chen C S, Dai Q W 2010 Chin. J. Geophys. 53 2484 (in Chinese)[冯德山, 陈承申, 戴前伟2010地球物理学报53 2484]
[8] Li J, Zeng Z F, Liu S X 2012 Comput. Geosci. 49 121
[9] Wei B, Li X Y, Wang F, Ge D B 2009 Acta Phys. Sin. 58 6174 (in Chinese)[魏兵, 李小勇, 王飞, 葛德彪2009物理学报58 6174]
[10] Di Q Y, Wang M Y 1999 Chin. J. Geophys. 42 818 (in Chinese)[底青云, 王妙月1999地球物理学报42 818]
[11] Li Z H, Huang Q H, Wang Y B 2009 Chin. J. Geophys. 52 1915 (in Chinese)[李展辉, 黄清华, 王彦宾2009地球物理学报52 1915]
[12] Zhuan S X, Ma X K 2012 Acta Phys. Sin. 61 110206 (in Chinese)[颛孙旭, 马西奎2012物理学报61 110206]
[13] Xu L J, Liu S B, Mo J J, Yuan N C 2006 Acta Phys. Sin. 55 3470 (in Chinese)[徐利军, 刘少斌, 莫锦军, 袁乃昌2006物理学报55 3470]
[14] Vivek K, Mani M 2006 J. Comput. Appl. Math. 230 803
[15] Pedro P, Margarete O D, Paulo J S G F, Sônia M G, Anamaria G, José R P 2007 IEEE Trans. Magn. 43 1013
[16] Marta D L L P, Stewart C, Robert P 2012 J. Comput. Phys. 231 6754
[17] Rodrigo B B, Marco A C S, Raul R E S 2013 Finite Elem. Anal. Des. 75 71
[18] Martin R, Komatitsch D, Gedney S D, Bruthiaux E 2010 CMES 56 17
[19] Zhang W, Shen Y 2010 Geophysics 75 141
[20] Zhao J G 2014 Jilin University 44 675 (in Chinese)[赵建国2014吉林大学学报44 675]
[21] Li J X 2007 Ph. D. Dissertation (Tianjin:Tianjin University) (in Chinese)[李建雄2007博士学位论文(天津:天津大学)]
[22] Deslauriers G, Dubuc S 1989 Constr. Approx. 5 49
[23] Dubuc 1986 Math. Anal. Appl. 114 185
[24] Satio N, Beylkin G 1993 IEEE Trans. Signal Process. 41 319
[25] Sweldens W 1996 Appl. Comput. Harmon. Anal. 3 186
[26] M Sc Hao J L 2011 Ph. D. Dissertation (Zur Erlangung des akademischen Grades eines)
[27] Ge D B, Yan Y B 2005 Finite-Differeence Time-Domain Method for Electromagnetic Waves (Xi'an:Xidian University Press) p31(in Chinese)[葛德彪, 闫玉波2005电磁波时域有限差分方法(西安:西安电子科技大学出版社)第31页]
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