区间B样条小波有限元GPR模拟双相随机混凝土介质

基于可分离小波理论,由一维区间B样条小波尺度函数的张量积构造二维B样条小波基,并将它作为GPR波动方程求解的插值函数,通过引入转换矩阵,实现小波系数空间与雷达电磁场之间的转换.应用Galerkin算法,推导了二维区间B样条小波有限元GPR波动方程离散格式,求出了2阶1尺度与2阶2尺度BSWI尺度函数的积分值及联系系数,给出了该算法的详细求解过程.编制了BSWI的Matlab模拟程序,应用该程序对两个典型实例进行了正演,结果表明:BSWI能采用较少的单元达到与FEM相似的精度,而BSWI算法尺度提升能提高解的精度,但耗时会急剧增加.最后,将BSWI算法应用于双相随机混凝土模型,说明随机介质模型理论能灵活、有效地描述实际混凝土介质的分布,正演剖面与实测剖面特征更相符,能更真实地模拟雷达波的传播过程,可为提高GPR的探测效果和解释准确性提供理论基础.

The GPR simulation of bi-phase random concrete medium using finite element of B-spline wavelet on the interval

Based on the separable wavelet theory, the scaling functions of one-dimensional B-spline wavelet on the interval (BSWI) is employed to construct the two-dimensional B-spline wavelet bases on the interval. In order to solve the GPR wave equation, these constructed wavelet bases are used as the interpolation functions, and a transformation matrix is introduced to convert between the wavelet coefficient space and the physical space (radar electromagnetic field). In this study, the discrete format of GPR wave equation for two-dimensional interval B-spline wavelet finite element is derived using Galerkin algorithm. And the integral values and the connection coefficients of the second order one-scale and second order two-scale BSWI functions are calculated. The detailed process of the algorithm is given. Then two typical examples are forward modeled using the BSWI method by Matlab program. The result shows that the BSWI with fewer units has the same precision as FEM. Increasing the scale of BSWI algorithm, the precision of the results is also improved, but it is time-consuming. Finally, the BSWI algorithm is applied to model bi-phase random concrete medium, and it proves that the random medium model theory can describe the practical distribution of concrete medium flexibly and effectively. Furthermore, this study found that the forward modeling profile is more accordant with practical profile, which demonstrates that medium model theory can simulate the transmission process of the radar wave more accurately. It also provides a theoretical basis for improving the detection results and interpretation accuracy of GPR.