频率和收发距广普适用的均匀层状介质电磁场直接积分算法

Research on direct integration algorithm of electromagnetic field in homogeneous layered media for a wide range of frequencies and transceiver distances

在地球物理勘探中,均匀层状介质电磁场解析表达式含有汉克尔积分,其核函数为0阶和1阶Bessel函数,随着其宗量的增大,Bessel函数呈现出快速振荡和慢衰减的特性,由此造成汉克尔积分难以高效、高精度计算,特别是高频和大收发距情况下难度更大。针对这个问题,提出一种高效、高精度的直接积分方法(Direct integration method),其基本思想是Bessel函数可以在两个区间并分别用不同的多项式展开,每一区间的汉克尔积分可离散为多个单元积分之和,每个单元被积函数可采用三次样条插值函数表示,由此可求得单元积分的解析解,通过叠加,从而求得汉克尔积分的数值解。在此基础上,利用均匀全空间电偶极子电磁场的解析解,正确选取积分范围并合理剖分积分单元。数值解与解析解的对比表明该算法正确可靠。与数字滤波类算法的计算结果对比表明,该算法广泛适用于不同频率和不同收发距条件下电磁场的计算,具有较强的普适性。

In geophysical exploration,the analytic expression of electromagnetic field in homogeneous layered media is Hankel integral,whose kernel function is Bessel function of order 0 and 1.With the increase of its synthesis,Bessel function presents the characteristics of fast oscillation and slow attenuation,which makes Hankel integral difficult to calculate with high efficiency and accuracy,especially in the case of high frequency and large receiving distance.In order to solve this problem,this paper proposes a direct integration method which is efficient and of high precision.The basic idea is that Bessel function can be divided into two intervals and expanded by different polynomials.Hankel integral of each interval can be divided into the sum of multiple unit integrals.The integrated function of each unit is represented by a cubic spline interpolation function,from which the analytic solution of the integral can be obtained.The numerical solution of Hankel integral can be obtained by superposition.On this basis,using the analytical solution to the electromagnetic field of the electric dipole in the uniform whole space,the correct selection of the integral range and the appropriate partition of the integral element are studied.The comparison between the numerical solution and the analytical solution shows that the algorithm is correct and reliable.The comparison between the algorithm proposed in this paper and the digital filtering algorithm shows that the algorithm is widely applicable to the calculation of electromagnetic fields with different frequencies and different receiving distances.Therefore it is used very universally.