Fast computation of Hankel Transform using orthonormal exponential approximation of complex kernel function (original) (raw)
Related papers
Journal of Geophysics and Engineering, 2014
This paper analyses the details of a procedure for the numerical integration of Hankel transforms in the calculation of the electromagnetic fields generated by a large horizontal loop over a 1D earth. The method performs the integration by deforming the integration path into the complex plane and applying Cauchy's theorem on a modified version of the integrand. The modification is the replacement of the Bessel functions J 0 and J 1 by the Hankel functions H (1) 0 and H (1) 1 respectively. The integration in the complex plane takes advantage of the exponentially decaying behaviour of the Hankel functions, allowing calculation on very small segments, instead of the infinite line of the original improper integrals. A crucial point in this problem is the location of the poles. The companion paper shows two methods to estimate the pole locations. We have used this method to calculate the fields of very large loops. Our results show that this method allows the estimation of the integrals with fewer evaluations of the integrand functions than other methods.
Geophysical Prospecting, 2006
The analytical solution and algorithm for simulating the electric potential in an arbitrarily anisotropic multilayered medium produced by a point DC source is here proposed. The solution is presented as a combination of Hankel transforms of integer order and Fourier transforms based on the analytical recurrent equations obtained for the potential spectrum. For the conversion of the potential spectrum into the space domain, we have applied the algorithm of the Fast Fourier Transform for logarithmically spaced points. A comparison of the modelling results with the power-series solution for two-layered anisotropic structures demonstrated the high accuracy and computing-time efficiency of the method proposed.
2014
The helicopter-borne electromagnetic (HEM) frequency-domain exploration method is an airborne electromagnetic (AEM) technique that is widely used for vast and rough areas for resistivity imaging. The vast amount of digitized data flowing from the HEM method requires an efficient and accurate inversion algorithm. Generally, the inverse modelling of HEM data in the first step requires a precise and efficient technique provided by a forward modelling algorithm. The exact calculation of the sensitivity matrix or Jacobian is also of the utmost importance. As such, the main objective of this study is to design an efficient algorithm for the forward modelling of HEM frequency-domain data for the configuration of horizontal coplanar (HCP) coils using fast Hankel transforms (FHTs). An attempt is also made to use an analytical approach to derive the required equations for the Jacobian matrix. To achieve these goals, an elaborated algorithm for the simultaneous calculation of the forward compu...
Application of Wavelets in Numerical Evaluation of Hankel Transform Arising in Seismology
Industrial and Applied Mathematics, 2015
The computation of electromagnetic (EM) fields for 1-D layered earth model requires evaluation of Hankel transform. In this paper we propose a stable algorithm for the first time that is quite accurate and fast for numerical evaluation of the Hankel transform using wavelets arising in seismology. We have projected an approach depending on separating the integrand tf(t)J ν (pt) into two components; the slowly varying components tf(t) and the rapidly oscillating component J ν (pt). Then either tf(t) is expanded into wavelet series using wavelets orthonormal basis and truncating the series at an optimal level or approximating tf(t) by a quadratic over the subinterval using the Filon quadrature philosophy. The solutions obtained by proposed wavelet method applied on three test functions indicate that the approach is easy to implement and computationally very attractive. We have supported a new efficient and stable technique based on compactly supported orthonormal wavelet bases.
IEEE Transactions on Microwave Theory and Techniques, 2000
When the fast Hankel transform (FHT) filter technique is used to calculate the spatial-domain Green's functions for planar multilayered media, it can be difficult to obtain accurate numerical results because of branch-cut singularities and the surface-wave pole singularities. Although the singularities can be efficiently avoided through deforming the integration path of the Hankel transform from the real axis to the first quadrant in the complex plane, the argument of the integral kernel becomes a complex number so that the FHT filter algorithm cannot be directly applied. The modified FHT filter algorithm is proposed to overcome this problem by expressing the Bessel function with a complex argument as a sum of terms of product of Bessel function with the real part of the argument and Bessel function with the imaginary part of the argument. The FHT filter technique can then be applied to each expansion term. Numerical results confirm that the proposed approach has high accuracy and good efficiency in the near and intermediate fields. More importantly, it successfully extends the applicability of the conventional FHT method for general multilayered geometries.
Fast algorithm for the computation of the zero-order Hankel transform
Journal of the Optical Society of America, 1983
The Hankel transform may be defined as the two-dimensional Fourier transform of a circularly symmetric function. A new Hankel-transform algorithm based on this definition is described. The proposed algorithm efficiently generates a rectangularly sampled two-dimensional output array by using the circular symmetry properties of the input array and two-dimensional vector radix fast-Fourier transform techniques. It accomplishes this by partitioning the input matrix into smaller and smaller processing blocks while removing redundant blocks from data manipulations. For applications that require the output data to be sampled on a two-dimensional rectangular raster, the convenience and the computational speed of the resulting algorithm offer advantages over the one-dimensional Hankel-transform algorithms currently available. (3) (4) Unfortunately, the complexity and the lengthy computation times required for these methods appear to limit their application. Another approach, by Siegman, 4 uses a change of variables within the HT integral to yield a two-sided cross-correlation integral that can be computed by a series of FFT's as follows: F(p) = 2w Jo f(r)Jo(2rrp)rdr, r = roeax, p = poe"Y
The fields from a finite electrical dipole—A new computational approach
GEOPHYSICS, 1994
Controlled‐source, frequency‐domain, and time‐domain electromagnetic methods require accurate, fast, and reliable methods of computing the electric and magnetic fields from the source configurations used. Except for small magnetic dipole sources, all electric and magnetic sources are composed of lengths of straight wire, which may be grounded. If the source‐receiver separation is large enough, the composite electrical dipoles may be considered to be infinitely small, and in a 1-D earth model the fields are expressed as Hankel transforms of an input function, which depends only on the model parameters. The Hankel transforms can be evaluated using the digital filter theory of fast Hankel transforms. However, the approximation of the infinitely small dipole is not always valid, and fields from a finite electrical dipole must be calculated. Traditionally, this is done by numerical integration of the fields from an infinitesimal dipole, thus increasing computation time considerably. The ...
Modelling and inversion of electromagnetic data using an approximate plate model
Geophysical Prospecting, 2002
A B S T R A C T This paper presents a computational method for the interpretation of electromagnetic (EM) profile data in the frequency domain using a thin plate model within a two-layer earth. The modelling method is based on an integral equation formulation, where the conductor is represented by a lattice structure composed of twodimensional surface elements. Several approximations are used to simplify the theoretical basis and to decrease the computation time. The simple parametric model allows efficient use of optimization methods. We employ a linearized inversion scheme based on singular value decomposition and adaptive damping. The new forward computation method and the parameter optimization are combined in the computer program, emplates. The modelling examples demonstrate that the approximate method is capable of describing the characteristic behaviour of the EM response of a thin plate-like conductor in conductive surroundings. The efficacy of the inversion is demonstrated using both synthetic and field data. An optional depth compensation method is used to improve the interpreted values of the depth of burial. The results show that the method is cost effective and suitable for interactive interpretation of EM data.
A Modified Fast Hankel Transform algorithm for calculating planar multilayered Green's function
2010 International Conference on Electromagnetics in Advanced Applications, 2010
When the Fast Hankel Transform filter technique is used to calculate the dyadic multilayered Green's functions, it can be difficult to obtain accurate numerical results because of the branch-cut singularity and the surface wave poles singularity. The Modified Fast Hankel Transform filter algorithm is proposed to overcome this problem by expressing the Bessel function with a complex argument as a sum of terms of product of Bessel function with the real part of the argument and Bessel function with the imaginary part of the argument. Then the Fast Hankel Transform filter technique is applied to each expansion term. Numerical results confirm that the proposed approach has high accuracy and efficiency and successfully extends the applicability of the conventional Fast Hankel Transform method to general multilayered geometries.
Exponential Finite Difference Method for Simulation of Electromagnetic Response of Layered Earth
This paper presents an efficient algorithm Exponential Finite Difference Method (EFDM) for simulation of electromagnetic response of layered earth by considering the exponential basis function. EFDM demands a parameter μ to be chosen judiciously to obtain optimum results. The estimators of optimum μ can be obtained from eigenvalue analysis of the coefficient matrix; however, near optimal values can be constructed using model parameters. Since the electromagnetic response has oscillatory behavior, EFDM handles it better and gives more accurate results in comparison to the Classical Finite Difference Method (CFDM). Using EFDM we can choose coarser grids to obtain same accuracy of result as CFDM provides with a given grid. As a result EFDM reduces the time and cost of computation in comparison to CFDM.