自适应多尺度第二代小波配点法探地雷达数值模拟

基于第二代小波变换的提升方案构造了插值小波，将雷达波场函数进行了二维小波变换，得到所有尺度上与计算网格相联系的小波系数和尺度系数.对所有尺度上的小波系数进行分析，根据解的局部性与小波系数阈值的控制，实现网格压缩和配点的自适应调节.保留大于给定阈值的小波系数及对应网格点，令小于给定阈值的小波系数为零，并舍弃其对应网格点.达到光滑区域采用较少的计算网格点，在奇异性较大的区域采用较多的计算网格点的目的.通过对自适应网格进行邻域校正、重构检查等附加修正，推导了场值更新的显式时间迭代方案.最后，以均匀、阶梯与复杂三个典型GPR模型为例，与常规数值计算结果对比表明：自适应小波配点法（AWCM）利用第二代小波的多尺度分解和快速变换的特点，可以使计算网格随着时间步适应解的移动和变化，允许计算资源更有效地使用，具有高压缩率，达到跟踪奇异性的目的，特别适合于探地雷达正演中波传问题的模拟.

Adaptive multi-scale second-generation wavelet collocation method numerical simulation of Ground Penetrating Radar

In this work, an interpolating wavelet is constructed by the lifting scheme of the second generation wavelet transformation, by employing the two-dimensional wavelet transform into the wave field function of radar, both the wavelet coefficients and scale coefficients that corresponding to the computational grid of different scales can be all calculated. And the wavelet coefficients of different scales are sequentially analyzed according to the locality of solution and the threshold of wavelet coefficient which facilitates the implementation of mesh compression and self-adaptation adjustment to collocation point. Since the wavelet coefficients that larger than a given threshold are reserved, as well as its corresponding grid points, while others that less than the given threshold are set to zero and discarded. Subsequently, the strategy of using fewer computational grid points in smoother region and more computational grid points in singularity region can be implemented and have effect. In addition, an explicit scheme of time iteration for updating the field values is deduced by using an additional corrections techniques to the adaptive mesh, for instance, the adding adjacent zone and the reconstruction check. Finally, the validation of the proposed method is carried out by using three typical simulation tests, which involving a homogeneous, a stair and a complicated model. Compared with the conventional method, the results demonstrate that, the advantage in multi-scale decomposition and fast transform of second generation wavelet, the proposed method can make the computing grid adapted to the movement or change of the solution with time step, and make the computing resources be utilized more efficiently, which characterizes proposed methods high compression rate and tracks the singularity region more accurately. Thus, the proposed adaptive wavelet collocation method (AWCM) is considered as the suitable solution for the simulation of wave-propagation problem in GPR.