On the Voltage Response of Homogeneous Earth Models in Central Loop Electromagnetic Sounding

An accurate series-form explicit expression is derived for the voltage induced in the small receiving loop of a central loop electromagnetic sounding system positioned above a homogeneous earth model. The solution is obtained by converting the semi-infinite integral representation for the vertical magnetic field induced at the center of the two loops into a series of simpler integrals, amenable to analytical evaluation. The developed formulation is based on replacing the exponential term in the integrand of the field integral with its power series expansion with respect to the difference of the squares of the wavenumbers in air and in the conducting medium. As a result, the vertical magnetic field and the induced voltage are finally expressed as sums of the spherical Hankel functions depending on geometrical parameters and the wavenumbers. The derived solution exhibits advantages in terms of both time cost and accuracy, when compared to the conventional numerical schemes for evaluating the Sommerfeld-type integrals and to the previously published quasistatic solution to the considered problem.