الفهرس Only 14 pages are availabe for public view
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Abstract
The magnetic field of a vertical magnetic dipole located at a height (h) above a two-layer conducting earth is expressed in analytical form in the far-region of both the air and the overburden slab. In these expressions, the displacement currents in the earth’s layers are taken into consideration. The analytical results are given by using the new approach which was presented first by Chew [53], and with the aid of the complex image theory [55].
Firstly, we began with the wave equation of the scalar magnetic Hertz potential , to find the electric and magnetic fields as Sommerfeld integral representations with unknown functions. Under the consideration of the initial, boundary, transition, and radiation conditions at the two boundaries (air, overburden slab and ground) to the integral representations we can determine these functions.
Secondly, to evaluate the values of these integrals, we have used the new approach, which hinges on the use of some integral identities and identification of the stationary phase point of the integrand. The location of the stationary phase point is easily remembered from its physical interpretation. With this new approach, the integration was divided into two factors; the first one is the slow factor, while the other one is the rapid factor. Then, by using the complex image theory we can replace the reflection coefficient by an exponential function which tends to the summation of algebraic series.
With this simple method, we can find the value of the Sommerfeld integral representations in few steps instead of the previous complicated one which was dependent on several transformations.