ИСТИНА

The integral equations approach is widely used to solve various electrodynamics problems, in particular the ones in the electromagnetic sounding. The main computational challenge of this approach is an evaluation of coefficients of a corresponding system of linear equations, i.e. the volumetric integrals of the Green's tensor of the layered media. The Green's tensor components are the improper integrals containing the Bessel functions. While the integration in vertical direction can be performed analytically, the integration over the horizontal domains involves the fifth-order integrals over the fast-oscillating functions. Motivated by the Anderson's idea for the Hankel transformation computing, we propose a new approach to construct the quadrature formulas for such multiple integrals. Changing the order of integration and making the appropriate substitution allows converting the fifth-order integral to the convolution with the special kernel. Following Anderson we then compute the spectrum of this kernel and build the quadrature formula based on Shannon's interpolation. It is important to stress that both the nodes and the weights in the obtained formula significantly depend on the integration domains. At the same time their computational cost is independent of the integration domains. Numerical implementation of the proposed integration method allows selecting a tradeoff between a computational time and an accuracy. Moreover, in a number of cases (e.g. large number of cells in vertical direction) the computational cost of the quadrature rule parameters is negligible compared to that of the integrand in the quadrature nodes.