基于BTTB矩阵的快速高精度三维磁场正演

本文改进了一种快速、高精度空间域三维正演算法，用来计算地下场源在水平观测面产生的磁异常ΔT场及其梯度场，以解决传统空间域正演计算效率低的问题.算法采用长方体对场源区域进行剖分，观测点与场源剖分单元体中心点在水平面上的投影重合.改进的算法具有以下三个特点：（1）采用无解析奇点的解析解公式计算磁异常，保证计算精度.（2）通过构造特殊的分块托普利兹（BTTB，Block-Toeplitz Toeplitz-Block）矩阵，利用其结构特性压缩核矩阵，并且用预先计算并存储中间变量，优化计算核矩阵的过程以提高计算效率.（3）基于BTTB矩阵的特殊性质，将核矩阵与磁化率向量的乘积转化为二维离散卷积的形式，因此能利用快速傅里叶变换进一步提高计算效率.模型实验显示，当剖分个数较多时，改进的快速正演算法比传统解析解方法快约5个数量级，比现有的8点高斯-快速傅立叶变换（Gauss-FFT）正演算法快约两个数量级，而且绝对误差极小（最大约为10^-6 nT），同时将反演时核矩阵的内存占用降低约5个数量级，证明了该正演算法具有高精度、高效率、低存储量的优点.最后设计了一个合成模型实验，将改进后的快速正演算法运用到磁异常ΔT反演中，反演所得三维磁化率与真实模型特征一致，且大幅降低反演计算时间和内存占用，验证了快速正演算法的实用性.

Fast and high accuracy 3D magnetic anomaly forward modeling based on BTTB matrix

Given the low efficiency of the traditional three-dimensional(3 D)magnetic forward modeling method in the spatial domain,we propose a fast and high-accuracy 3 D forward algorithm to calculate the magnetic anomalyΔT and its gradient on a horizontal observation surface generated by the underground source.The vertical cuboid is selected as a fundamental mesh element to discretize the source region.The observation points align with the center points of the discretized cuboids on the horizontal plane.The improved algorithm has three advantages:(1)The analytical solution formula without singularity is used to calculate the magnetic anomaly generated by the field source on the horizontal observation plane to ensure accuracy.(2)The BTTB(Block-Toeplitz Toeplitz-Block)matrix is constructed for compressing the kernel matrix according to its structural properties,and intermediate variables are pre-calculated and stored to improve efficiency.(3)The product of the susceptibility vector and the kernel matrix with the structure of the BTTB matrix is transformed into a two-dimensional discrete convolution by using the FFT(Fast Fourier Transform)to improve the computational efficiency further.The model tests show that the improved algorithm is about five orders of magnitude faster than the traditional analytic method and about two orders of magnitude faster than the existing Gauss-FFT algorithm with 8 Gaussian nodes.In the model test,the maximum absolute error is about 10^-6 nT,and the memory usage of the kernel matrix at inversion is reduced by about five orders of magnitude.These indicate that the improved algorithm has the advantages of high precision,high efficiency,and low storage.Finally,the forward algorithm is applied to an inversion of magnetic anomalyΔT.The result shows that the 3 D magnetic susceptibility obtained by the inversion is consistent with the input model,and the computational time and memory usage are greatly reduced,which verifies the practicability of the improved algorithm.