联用高斯求积与连分式求和计算Hankel变换及在地球物理上的应用
详细信息 本馆镜像全文    |  推荐本文 | | 获取馆网全文
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 integration method is generally used to compute approximately most Hankel transforms,except few kernel functions' Hankel transforms 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 algorithms 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 programming,finally the method has been applied to evaluate the electromagnetic field of two instances stimulated by electric dipole 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.
引文
[1]NABIGHIAN M N.Electromagnetic method in applied geohpysics.volume 1,theory.society of exploration geophysicists[M].Beijing:Geological Publishing House,1992.
    [2]闫桂峰.数学物理方法[M].北京:清华大学出版社,2006.YAN G F.Methods of mathmatical physics[M].Beijing:Tsinghua Press,2006.(In Chinese)
    [3]朴化荣.电磁测深原理[M].北京:地质出版,1990.PIAO H R.The principle of electromagnetic sounding[M].BeiGeological Publishing House,1990.(In Chinese)
    [4]ANDERSON W L.A hybrid fast Hankel transform algorithm for electromagnetic modeling[J].Geophysics,1989,54:263-266.
    [5]ANDERSON W L.Comment on‘Optimized fast Hankel transform filter’by Niels Boie Christensen[J].Geophysical Prospecting,1991,39:445-447.
    [6]GUPTASARMA D.Optimisation of shorter digital filters for increasing accuracy[J].Geophysical Prospecting,1982,30:501-514.
    [7]Guptasarma D,SINGH B.New digital linear filters for Hankel J0 and J1transforms[J].Geophysical Prospecting,1997,45:745-762.
    [8]JOHANSEN H K,SORENSEN K.I Fast Hankel transforms[J].Geophysical Prospecting,1979,27:876-901.
    [9]KONG F N.Hankel transform filters for dipole antenna radiationin a conductive medium[J].Geophysical Prospecting,2007,55:83-89.
    [10]CORNILLE P.Computation of Hankel transform[J].SIAM Rev.1972,14:278-285.
    [11]CHAVEA D.Numerial integration of related Hankel transforms by quadrature and continued fraction expansion[J].Geophysics,1983,48:1671-1686.
    [12]翁爱华,王雪秋.利用数值积分提高一维模型电偶源电磁测深响应计算精度[J].西北地震学报,2003,25(3):193-197.WEN A H,WANG X Q.Utilizing direct integration to enhance calculation accuracy of 1Delectromagnetic response for current dipole source[J].Northwestern Seismological Journal,2003,25(3):193-197.(In Chinese)
    [13]张辉,李桐林.利用高斯求积和连分式展开计算电磁张量格林函数积分[J].地球物理学进展,2005,20(3):667-670.ZHANG H,LI T L.Computation of Green's tensor integrals for electromagnetic problems using Gaussian quadrature and continued fraction[J].Progress in Geophysis,2005,20(3):667-670.(In Chinese)
    [14]BAKER G A,JR.Essentials of Pade approximants[M].New York:Academic Press,1975.
    [15]HANGGI P,ROESEL F.Evaluation of infinite series by use of continued fraction expansions:a numerical study[J].Comp.Phys,1980,37:242-258.
    [16]HOHMANN G W.Three dimensional induced polarization and electromagnetic modeling[J].Geophysics,1975,40(2):309-324.
    [17]陈桂波,汪宏年.各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟[J].物理学报,2009,58(6):3849-3857.CHENG G B,WANG H N.Three-dimensional numerical modeling of marine controlled-source electromagnetic responses in a layered anisotropic seabed using integral equation method[J].ACTA Physica Sinca,2009,58(6):3849-3857.(In Chinese)
    [18]张辉.复电阻率三维电磁场正反演研究[D].长春:吉林大学地球探测科学与技术学院,2006.ZHANG H.Research of complex resistivity 3Delectromagnetic forward and inversion[D].Changchun:college of Geo-Exploration Science and Technology,Jilin University,2006.(In Chinese)
    [19]徐凯军.2.5维复电阻率电磁场正反演研究[D].长春:吉林大学地球探测科学与技术学院,2007.XU K J.Study on 2.5Dcomplex resistivity electromagnetic forward and inversion[D].Changchun:college of Geo-Exploration Science and Technology,Jilin University,2007.(In Chinese)
    [20]CAGNIARD L.Basic theory of the magnetotelluric methods of geophysical prospecting[J].Geophysics,1953,18:605-635.

版权所有:© 2023 中国地质图书馆 中国地质调查局地学文献中心