[45] This paper is mainly of a review character and compiles the existing methods of description of the field in a spherical anisotropic waveguide channel Earth-ionosphere with the dependence of the medium properties on two spatial coordinates at its excitation by an arbitrary oriented short antenna. A two-dimensionally inhomogeneous waveguide is modeled by a sequence of homogeneous pieces. To accelerate the iteration process while looking for the eigenvalues, it is proposed to use an effective height of the waveguide channel calculated as a height below which vacuum is located at the least dependence of the elements of the matrix of the reflective properties of the ionosphere on the spectral parameter for the main mode. It is shown that the use of a bivector at calculation of the reflective characteristics of the ionosphere makes it possible to avoid an appearance of a singularity. For acceleration of the calculations it is proposed to calculate the coefficients of reexcitation of normal waves over the vacuum region, the control of the reexcitation matrix elements accuracy being performed by the above described method.
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