The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in t...The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.展开更多
This paper discusses use of approximations and the Integral Mean Value Theorem to show that 6 coefficients approximately describe the distortions of near surface inhomogeneities on the MT field of a horizontally layer...This paper discusses use of approximations and the Integral Mean Value Theorem to show that 6 coefficients approximately describe the distortions of near surface inhomogeneities on the MT field of a horizontally layered earth model. When these 6 coefficients are considered together with those of the magnetic field of a horizontally layered earth model,the analytic and approximate wave impedance equations can be derived for the MT response of a horizontally layered earth model with near-surface 2-D and 3-D inhomogeneities. These approximate wave impedance equations are used with inverted MT data for 2-D and 3-D forward modelling. Although these 6 coefficients cannot be determined before inversion,initial estimates can be used. The 6 coefficients and the asistivity and thickness of each layer of a horizontally layered earth can be obtained by using published inversion methods. The 6 coefficients give important informaion (depths and resistivities) on the near-surface inhomogenelties.The authors inverted 2-D and 3-D theoretical models for Fast Approximate Inversion of Magnetotelluric (FAIMT) data for a horizontally layered earth with near-surface inhomogeneities compares favorably with traditional invrsion methods, especially for inverting regional or basin structures. This method simplifies computation and gives a reasonable 1 -D geological model with fewer nonuniquenas problems.展开更多
文摘The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.
文摘This paper discusses use of approximations and the Integral Mean Value Theorem to show that 6 coefficients approximately describe the distortions of near surface inhomogeneities on the MT field of a horizontally layered earth model. When these 6 coefficients are considered together with those of the magnetic field of a horizontally layered earth model,the analytic and approximate wave impedance equations can be derived for the MT response of a horizontally layered earth model with near-surface 2-D and 3-D inhomogeneities. These approximate wave impedance equations are used with inverted MT data for 2-D and 3-D forward modelling. Although these 6 coefficients cannot be determined before inversion,initial estimates can be used. The 6 coefficients and the asistivity and thickness of each layer of a horizontally layered earth can be obtained by using published inversion methods. The 6 coefficients give important informaion (depths and resistivities) on the near-surface inhomogenelties.The authors inverted 2-D and 3-D theoretical models for Fast Approximate Inversion of Magnetotelluric (FAIMT) data for a horizontally layered earth with near-surface inhomogeneities compares favorably with traditional invrsion methods, especially for inverting regional or basin structures. This method simplifies computation and gives a reasonable 1 -D geological model with fewer nonuniquenas problems.
