This note examines the theoretical problem of continuation of electromagnetic fields by the method of spectral analysis and synthesis, without making any restricting assumptions in regard to the conductivity of the medium and the frequency of the field. It is found that the space frequency spectrum of the continued field at a level z is obtained by multiplying that of the observed field on z=0 by exp (±θz), where θ22=λx2+λy2−k2, λx and λy are the spatial frequencies in two orthogonal directions and k is the propagation constant. For spherical geometry, the expansion of the observed field needs to be done in terms of spherical harmonics, and the multiplying factors for inward and outward continuation are, respectively,
(a/r)12?[Jn+12(kr)/Jn+12(ka)] and (a/r)12?[Hn+12(1)(kr)/Hn+12(1)(ka)], where J and H are Bessel functions, r is the radius of sphere of continuation and a is the radius of the sphere of observation. While the method is valid for continuation in air or inside a homogeneous medium, it fails in the more practical case when one wants to continue inside an inhomogeneous earth. This is because, in certain regions of an in homogeneous earth, the field consists of two unknown parts, one of which increases with z or r while the other decreases.
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