摘要
光学、电磁学和地震学都需要计算Hankel变换。除少数核函数的Hankel变换有解析式,绝大多数的通常采用数值积分的方法对其近似计算。虽然数字滤波法是常规的计算方法,但由于滤波系数的不统一,导致计算结果并不完全一致,甚至在某些情况结果是错误的。为了解决这个问题,采用直接数值积分的方法计算Hankel变换。具体过程是:首先将Hankel变换分解为积分限从零到Bessel函数的第一个零点值的积分段和后续各相邻零点值作为积分限的积分段的和,然后用高斯求积计算每段积分。对高斯求积结果组成的序列采用连分式求和,无论是计算结果的精确性还是收敛速度都要优于直接求和。这里首先详细阐述了联用高斯求积与连分式求和计算Hankel变换的基本原理和相关算法,然后用Matlab编程对比了直接求和与连分式求和的计算效果,最后将此方法应用于电偶源在均匀半空间激发的地下场计算的两个实例中。应用表明:将高斯求积与连分式求和进行联用是计算Hankel变换的一种有效方法,可以广泛应用于地球物理应用中。
It is necessary to evaluate Hankel transforms in optics,electromagnetics and seismology.The numerical integra-tion method is generally used to compute approximately most Hankel transforms,except few kernel functions＇Hankel trans-forms having analytic expression.Although the digital filtering method is the conventional numerical aspects,the results from the method are not completely concordant because of inconsistent filtering coefficients,and in some instances the calculations are even wrong.To solve the problem,the direct numerical integration is adopted to compute Hankel transforms.Specific process is as the following：to begin with,the method is to break the Hankel transforms into integral parts＇summations.These integral parts are made up of integrals which range of integration are from zero to the first zero value and subsequent adjacent zero value of Bessel function,then using Gaussian quadrature method evaluates each integral parts.The continued fraction method is applied to sum the sequence of partial integration terms,which is better than the direct summation method both the computational result precision and convergent speed.The paper first elaborates the basic principle and the correlation algo-rithms of computation Hankel transforms used the method of united Gaussian quadrature and continued fraction summation, then contrasts calculation results between the direct summation method and the continued fraction method using Matlab pro-gramming,finally the method has been applied to evaluate the electromagnetic field of two instances stimulated by electric di-pole in homogeneous half space.The application shows that it is effective way to evaluate Hankel transforms using the method of united Gaussian quadrature and continued fraction summation,which is widely used by applications of geophysics.
出处
《物探化探计算技术》
CAS
CSCD
2015年第1期1-9,共9页
Computing Techniques For Geophysical and Geochemical Exploration
基金
国家级基金(OSR-02-01)
吉林大学学部基金(4305050102RV)
