| Daubechies小波有限元求解GPR波动方程 |
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| 引用本文： | 冯德山,杨炳坤,王珣,杜华坤.Daubechies小波有限元求解GPR波动方程[J].地球物理学报,2016,59(1):342-354. |
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| 作者姓名： | 冯德山 杨炳坤 王珣 杜华坤 |
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| 作者单位： | 1. 中南大学地球科学与信息物理学院, 长沙 410083;2. 中南大学有色金属成矿预测与地质环境监测教育部重点实验室, 长沙 410083;3. 福建省建筑科学研究院, 福州 350025 |
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| 基金项目： | 国家自然科学基金项目(41574116,41074085),中南大学创新驱动项目(2015CX008),中南大学升华育英人才计划,教育部新世纪优秀人才支持计划(NCET-12-0551),中南大学教师研究基金(2014JSJJ001),湖湘青年创新创业平台培养对象项目共同资助. |
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| 摘 要： | 基于可分离小波理论,由一维Daubechies尺度函数的张量积构造二维Daubechies小波基,并将它作为GPR波动方程求解的插值函数,导出了二维Daubechies小波有限元GPR方程离散格式;通过引入转换矩阵,实现小波系数空间与雷达场值之间转换.引入自由度凝聚技术,有效解决了小波有限元求解中小波单元内部自由度过多的问题,节约了计算量并方便与传统有限元法耦合.然后,详细阐述了Daubechies小波有限元联系系数计算方法,有效解决了小波有限元求解偏微分方程的难点与核心问题.最后,以两个典型GPR模型为例,对比了Daubechies小波有限元与传统有限元的雷达正演剖面图与单道波形图,结果表明:在相同的剖分方式及节点数目条件下,Daubechies小波有限元的紧支性与正交性一定程度上提高了求解效率,它与有限元法求解结果能较好地吻合,验证了Daubechies小波有限元算法的正确性.
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| 关 键 词： | 探地雷达 Daubechies小波有限元 自由度凝聚技术 联系系数 波动方程 正演模拟 |
| 收稿时间： | 2014-10-31 |
Daubechies wavelet finite element method for solving the GPR wave equations |
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| FENG De-Shan,YANG Bing-Kun,WANG Xun,DU Hua-Kun.Daubechies wavelet finite element method for solving the GPR wave equations[J].Chinese Journal of Geophysics,2016,59(1):342-354. |
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| Authors: | FENG De-Shan YANG Bing-Kun WANG Xun DU Hua-Kun |
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| Institution: | 1. School of Geosciences and Info-Physics, Central South University, Changsha 410083, China;2. Key Laboratory of Metallogenic Prediction of Non-Ferrous Metals and Geological Environment Monitor(Central South University), Ministry of Education, Changsha 410083, China;3. Fujian Academy of Building Research, Fuzhou 350025, China |
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| Abstract: | Based on the separable wavelet theory, we construct the two-dimensional Daubechies wavelet bases by means of one-dimensional Daubechies scaling functions, which is used for interpolation functions of solving the GPR wave equation, thus present the discrete format of two-dimensional Daubechies wavelet finite element GPR equation. By introducing a transformation matrix, the transformation between the wavelet coefficient space and the GPR electromagnetic field is implemented. By introducing the degree of freedom condensation technique, it effectively solves the problem of too much freedom in internal wavelet unit during the solution process of the wavelet finite element, reducing the amount of calculation and can be coupled easily with traditional finite element method. Then the calculation formulas of connection coefficient used in Daubechies wavelet finite element are elaborated, which effectively resolve the difficulty and core problem in solving partial differential equations by wavelet finite element. Finally, with two typical GPR models as example, comparing the radar forward sections and the single waveforms between Daubechies wavelet finite element method and the traditional finite element method, and the result shows that under the conditions of the same dividing method and the number of nodes, the compact support and orthogonality of Daubechies wavelet finite element improves the solving efficiency to some extent, and it can be fitted well with the solving result of finite element method, validating the correctness of the Daubechies wavelet finite element method, which provides a new idea for solving the GPR wave equation. |
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| Keywords: | Ground Penetrating Radar Daubechies wavelet finite element method Degree of freedom condensation technique Connection coefficient Wave equation Forward modeling |
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