A mathematical structure for a theory of electromagnetic induction in the Earth in low latitude

Summary

In low latitude the spatial distribution functions of the source field over the surface and the dimensions of the source, are important in any theory of electromagnetic induction developed for studying the conductivity structure of the Earth. The author has built up a mathematical structure for a theory of electromagnetic induction in anyn-layered earth model in low latitude. No simple solution is assumed for the horizontal distribution function of the source field and no assumption is made about the horizontal gradients of the source. The mathematical structure involves the concept of downward continuity of the field equations inside then-layered earth model. The resulting mathematical functions derived for anyn-layered earth model are complex. Hence a new matrix algebra of complex numbers is introduced by the author and this is built into the theory. From the upward continuity of the field equations, an inequality equation is derived in order to determine the heighth ₀ at which the induction field of the earth becomes negligible compared with the source field. The comparison of such heights at two or more stations under the same influence of the source field can be used for the resolution of the lateral distribution of the earch conductivity structure at these stations. The application of the theory will follow in a subsequent paper.