Electromagnetic field of a dipole on a two‐layer earth

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Abstract
The electromagnetic field due to a horizontal electric dipole placed on top of a two‐layer earth is represented in terms of fields due to a dipole over a half‐space earth and its image source fields. Integral representations of image source fields are evaluated with uniform asymptotic approximations. Leading order ordinary saddle‐point approximation, giving rise to the geometrical optics approximation (GOA), is shown to be inaccurate. This is especially true when the angle of observation is close to the critical angle, which corresponds to the presence of a branch‐point singularity near the saddle point. In the uniform asymptotic approximation, the integrand of the image source integral is split into a branch‐point free part and another part containing the branch‐point singularity. The branch‐point free part can be approximated with a spherical wave function, while the part containing the branch point can be approximated with parabolic cylinder functions. Vertical magnetic field components and the horizontal electric field component near the surface are illustrated and compared with the geometrical optics approximation, giving the direct numerical result as well as experimental measurement. It is shown that the uniform asymptotic approximation yields excellent agreement with numerical and experimental results compared to the geometrical optics approximation.

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