基于小波有限元的探地雷达正演模拟及偏移处理

项目名称： 基于小波有限元的探地雷达正演模拟及偏移处理

项目编号： No.40804027

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 建筑科学

项目作者： 冯德山

作者单位： 中南大学

项目金额： 20万元

中文摘要： 小波有限元具有善于分析非平稳信号、自适应局部刻画能力强的特性，在大梯度、奇异性问题求解中具有广阔的应用前景。Daubechies小波基由于具有正交性和紧支撑等独特优点，被选用为小波有限元进行GPR正演的基函数。项目研究取得了一系列成果。具体包括：(1) 开展了小波有限元基函数的选取研究；(2) 总结了Daubechies小波滤波器系数、尺度函数、小波函数、导数值及"联系系数"的高精度求解方法；(3) 基于wavelet-Galerkin方法推导了GPR二维波动方程离散格式；(4) 研究了大型刚度矩阵方程组的小波域压缩及高效求解方法；(5) 应用后验误差估计方法实现了Daubechies小波有限元单元节点自适应配置及求解精度的提升；(6) 提出了基于速度估计的改进型线性变换有限差分GPR偏移算法，提高了复杂非均匀变速介质中GPR剖面的偏移精度；(7) 提出了小波域二、三维GPR波动方程逆时偏移算法，实现雷达资料快速、高精度二、三维偏移处理；(8) 提出结合透射边界条件和Sarma边界条件的混合边界条件，极大地改善了有限元GPR正演截断边界处的反射，并编写了基于该算法的有限元正演程序。

中文关键词： 小波有限元；探地雷达；Daubeiches小波；正演模拟；偏移

英文摘要： wavelet finite element is good at non-uniform signal analysis, and has the strong ability of self - adaptive part portray. It has broad prospects in solving large gradient, singularity problems. Because Daubechies wavelets basic fuction has the unique advantages of orthogonality and compactly supported，it could be selected as the basic function of wavelet finite element for GPR forward. Based on this, the project has got series achievement as below: (1) carry out the selected research for the basic function of wavelet finite element；(2) summary the high-precision solving method of Daubechies wavelet filter coefficient, scale coefficient, wavelet coefficient, derivative values and contact coefficient；(3) based on the wavelet-Galerkin method, derive the discrete format of GPR two- dimension wave equation；(4) research the wavelet domain compression and efficient solving method of huge stiffness matrix equations；(5) the method of after- application error testing realize the self-adaptive configuration of element nodes of Daubechies wavelet finite element and upgrade the sovling accuracy；(6) put forward the improved linear transformation finite difference GPR migration methods based on the speed estimation，and improve the migration accuracy of GPR profile in the complex non-uniform variable-speed media；(7) put forward the the reverse time migration algorithm of wavelet domain two or three-dimension GPR wave equation, realize the two or three-dimension GPR migration processing rapidly and accuratly；(8) put forward the mixed boundary conditions combined transmission boundary conditions and Sarma boundary conditions，greatly improve the cut-off boundary of FEM reflection in GPR forwad, and compile the FEM forward simulation based on this algorithm.

英文关键词： wavelet finite element;ground penetrating radar;Daubechies wavelets; forward simulation;migration processing