逆Laplace变换新算法及其在时间域电磁响应计算中的应用

New algorithm for inverse Laplace transform and its application in calculation of time domain electromagnetic response

时间域电磁响应的正演计算多是由频率域响应经逆Laplace变换而得到.逆Laplace变换的计算精度和效率是时间域电磁响应计算中方法选择的重要指标.论文分析了几种逆Laplace变换的算法机制,并优选出Talbot算法计算了水平电偶源层状模型的时间域电磁响应.逆Laplace变换常用的算法有折线法、数字滤波算法和Gaver-Stehfest算法(简称G-S算法).折线法需要精细地确定分割步长以提高精度,数字滤波算法系数很多,适应频率范围受计算问题所限,而G-S算法受计算机字长和问题对象的影响大.本文在64位计算平台中计算比较了G-S算法、Euler算法和Talbot算法的节点数对于精度的影响,发现Talbot算法受节点数影响小,计算精度高,适应频率范围宽.最后利用21点Talbot算法计算了水平电偶源轴向偶极装置均匀大地模型径向电场的阶跃响应和冲激响应,计算精度及响应时间范围均优于G-S算法.计算了水平电偶源赤道偶极装置均匀大地模型垂直磁场的阶跃响应和冲激响应,冲激响应峰值时刻对于电阻率的变化响应灵敏,与轴向偶极径向电场响应能力相当,但垂直磁场随收发距增大,衰减较快.根据层状模型阶跃响应晚期渐近值计算的视电阻率,水平电偶源轴向偶极径向电场有能力发现大埋深高阻或低阻薄层,收发距应大于中间目标层埋深的5~6倍方可完整探测,类似的,采用水平电偶源赤道偶极装置测量垂直磁场也能达到与之相当的探测能力.计算结果证实了21点Talbot算法适应不同地电模型、不同观测方式的时间域电磁响应计算.

The forward calculation of electromagnetic response in time domain is mainly derived by inverse Laplace transform from frequency domain. The computational accuracy and efficiency of inverse Laplace transform is an important index for the choice of electromagnetic response calculation methods in time domain. In this paper,the algorithm mechanism of several inverse Laplace transform is analyzed, and the Talbot algorithm is selected to calculate the electromagnetic response in the time domain of the layered model excited by horizontal electric dipole source. The commonly used inverse Laplace transform algorithms are polygonal approximations method, digital filtering algorithm and GaverStehfest algorithm（ referred to as G-S algorithm）. In order to improve the accuracy,the method of polygonal approximations needs to determine the segmentation step precisely,the digital filtering algorithm has many coefficients,and the adaptive frequency range is limited by the computational problem. As well as, the G-S algorithm is affected by the computer word length and the object of the problem. In this paper,the G-S algorithm,Euler algorithm and Talbot algorithm are compared,the influence of the number of nodes on the precision is calculated and compared in the 64 bit computing platform. It is found that the Talbot algorithm is less affected by the node number and the calculation precision is high. Finally,a 21 points Talbot algorithm is used to calculate the step response and impulse response of the axial dipole radial electric field of the homogeneous earth model with the axial dipole array by horizontal electric dipole source. The results are better than the G-S algorithm in terms of computational accuracy and response time range. The step response and impulse response of the equatorial dipole array vertical magnetic field of uniform earth model excited by horizontal electric dipole source are also calculated,and the peak value of the impulse response is sensitive to the change of the resistivity,however,the vertical magnetic field decreases rapidly with the increase of the receiving and transmitting distance. The step response of the layered model apparent resistivity late asymptotic value in axial radial electric dipole array excited by horizontal electric dipole source has the ability to discover the deep high or low resistivity thin layer,and the receiving distance should be 5 ～ 6 times larger than the target depth can complete detection. Similarly,the measurement of equatorial dipole-dipole vertical magnetic field excited by the horizontal electrical dipole source can achieve samegoal for exploration. The calculation results confirmed that the 21 point Talbot algorithm is suitable for the electromagnetic response calculation of different geoelectric models and different observation methods in time domain.