时间域航空电磁2.5维非线性共轭梯度反演

对于时间域航空电磁法二维和三维反演来说，最大的困难在于有效的算法和大的计算量需求.本文利用非线性共轭梯度法实现了时间域航空电磁法2.5维反演方法，着重解决了迭代反演过程中灵敏度矩阵计算、最佳迭代步长计算、初始模型选取等问题.在正演计算中，我们采用有限元法求解拉式傅氏域中的电磁场偏微分方程，再通过逆拉氏和逆傅氏变换高精度数值算法得到时间域电磁响应.在灵敏度矩阵计算中，采用了基于拉式傅氏双变换的伴随方程法，时间消耗只需计算两次正演，从而节约了大量计算时间.对于最佳步长计算，二次插值向后追踪法能够保证反演迭代的稳定性.设计两个理论模型，检验反演算法的有效性，并讨论了选择不同初始模型对反演结果的影响.模型算例表明：非线性共轭梯度方法应用于时间域航空电磁2.5维反演中稳定可靠，反演结果能够有效地反映地下真实电性结构.当选择的初始模型电阻率值与真实背景电阻率值接近时，能得到较好的反演结果，当初始模型电阻率远大于或远小于真实背景电阻率值时反演效果就会变差.

2.5D inversion of time domain airborne electromagnetic data using nonlinear conjugate gradients

Inversion of time domain airborne electromagnetic (AEM) data are known for the difficulties in large computational requirement and effective algorithms, especially for two and three-dimensional problems. We have developed a 2.5 dimensional (2.5D) inverse algorithm for the time domain AEM data using the nonlinear conjugate gradient method with improved accuracy and efficiency. This paper focuses on solving the computation of the sensitivity matrix, the optimal step length and the initial model selection in this algorithm. In the forward modeling, we employ the finite element method (FEM) to solve the Maxwell's equations in the Laplace and Fourier domains. The time domain responses are then obtained by the high-precision inverse Laplace and Fourier transforms. The sensitivity matrix is calculated by using the adjoint equation method with the Laplace and Fourier transforms, which requires only two forward modeling per iteration and reduces the time cost significantly. The backtracking method in the optimal step length computation ensures the stability of this iterative inverse algorithm. Then, we present two model studies and discuss the effects of different initial models. The synthetic studies demonstrate our inversion algorithm is stable and can yield reliable results, which reflect the underground electrical structure reasonably. It also turns out that the inversion result is good if the initial model is close to the true background. The inversion model becomes worse if the initial model is several times larger or smaller than the true values of the background resistivity.