Numerical solution of integral equation for ground-to-ground radiowave propagation

The radiation of a vertical current element on planar, radially inhomogeneous ground is usually formulated as an integral equation for the attenuation factor. In the special case of homogeneous flat ground this integral equation can be transformed into a closed form expression. However, numerical techniques have to be applied to solve the integral equation for any inhomogeneous ground so as to achieve accurate numerical results. Four such methods are described to solve such an integral equation and the numerical results are compared with the closed form solution for the case of the homogeneous flat ground