Computation of the Sommerfeld Integral tails using the matrix pencil method

The oscillating infinite domain Sommerfeld integrals (SI) are difficult to integrate using a numerical procedure when dealing with structures in a layered media, even though several researchers have attempted to do that. Generally, integration along the real axis is used to compute the SI. However, significant computational effort is required to integrate the oscillating and slowly decaying function along the tail. Extrapolation methods are generally applied to accelerate the rate of convergence of these integrals. However, there are difficulties with the extrapolation methods, such as locations for the breakpoints. In this paper, we illustrate a simplified approach for accurate and efficient calculation of the integrals dealing with the tails of the SI. In this paper, we fit the tail by a sum of finite (usually 10 to 20) complex exponentials using the matrix pencil method (MPM). The integral of the tail of the SI is then simply calculated by summing some complex numbers. No numerical integration is needed in this process, as the integrals can be done analytically. Good accuracy is achieved with a small number of evaluations for the integral kernel (60 points for the MPM as compared with hundreds or thousands of functional evaluations using the traditional extrapolation methods) along the tails of the SI. Simulation results show that to obtain the similar accuracy in the evaluation of the SI, the MPM is approximately 10 times faster than the traditional extrapolation methods. Moreover, since the MPM is robust to the effects of noise, this method is more stable, especially for large values of the horizontal distances. The method proposed in this paper is thus a new and better technique to obtain accurate results for the computation of the Green´s function for a layered media in the spatial domain.