Complex image method for calculating electric and magnetic fields produced by an auroral electrojet of finite length

The electromagnetic field due to ionospheric currents has to be known when evaluating space weather effects at the earth's surface. Forecasting methods of these effects, which include geomagnetically induced currents in technological systems, are being developed. Such applications are time-critical, so the calculation techniques of the electromagnetic field have to be fast but still accurate. The contribution of secondary sources induced within the earth leads to complicated integral formulas for the field at the earth's surface with a time-consuming computation. An approximate method of calculation based on replacing the earth contribution by an image source having mathematically a complex location results in closed-form expressions and in a much faster computation. In this paper we extend the complex image method (CIM) to the case of a more realistic electrojet system consisting of a horizontal line current filament with vertical currents at its ends above a layered earth. To be able to utilize previous CIM results, we prove that the current system can be replaced by a purely horizontal current distribution which is equivalent regarding the total (= primary + induced) magnetic field and the total horizontal electric field at the earth's surface. The latter result is new. Numerical calculations demonstrate that CIM is very accurate and several magnitudes faster than the exact conventional approach.