## 1. Introduction

[2] The (geo)electric field occurring at the Earth's surface in connection with geomagnetic storms produces geomagnetically induced currents (GIC) in technological systems, such as electric power transmission grids, oil and gas pipelines, telecommunication cables, and railway equipment [*Boteler et al.*, 1998]. GIC are the end of the space weather chain (Sun-solar wind-magnetosphere-ionosphere-Earth's surface) and thus constitute an essential part of today's active space weather research. GIC are a possible source of problems to the systems. Power system transformers may be saturated by GIC, leading to troubles in the operation of the system or even to permanent damage [*Molinski et al.*, 2000]. The best known event is the blackout of electricity in Québec, Canada, for several hours during the large magnetic storm in March 1989 [*Czech et al.*, 1992]. Pipelines may suffer from problems associated with corrosion and its control due to GIC and the accompanying pipe-to-soil voltages [*Gummow*, 2002]. Telecommunication and railway equipment can experience harmful overvoltages.

[3] The first GIC observations were already made in the early telegraph systems over 150 years ago; the history and several references are presented by *Boteler et al.* [1998]. Large GIC occur most frequently in the auroral regions that extend across North America and the Nordic countries. However, during intense space storms, high values of GIC may occur at lower latitudes, also depending on the network configuration and resistances. The problems GIC may cause to a power system vary with the electric power transmitted in the grid at the particular moment of a large GIC. The increasing number of technological systems and the growing dependence of society on their reliability make research on space weather, including GIC, very important today.

[4] It is usually convenient to make a theoretical calculation of GIC in a system in two separate steps: (1) The horizontal geoelectric field at the Earth's surface is determined. (2) This field is an “external” input to compute the currents in the particular technological system.

[5] In general, the second step, which is basically an application of electric circuit theory, is faster and easier to perform: A matrix formalism is appropriate for a power system [*Lehtinen and Pirjola*, 1985], and a pipeline network can be treated by using distributed-source transmission line theory [*Boteler*, 1997]. The first step requires models about ionospheric-magnetospheric currents and about the Earth's conductivity structure as the surface electric and magnetic fields are, besides being primarily produced by space currents, secondarily affected by currents induced in the conducting ground. At high latitudes, where the most significant geomagnetic effects occur, geoelectromagnetic disturbances are principally due to an auroral electrojet current system, which contains an intense east-west current (the electrojet) in the ionosphere. The electrojet is typically some hundreds of kilometers wide and on the order of thousands of kilometers long. The ionospheric current system is coupled, through field-aligned currents, to current systems farther out in the Earth's magnetosphere. The electrojet flows at a height of approximately 110 km and can reach a magnitude of millions of amperes.

[6] The simplest model for calculating the surface electric and magnetic fields due to an auroral electrojet is an infinitely long horizontal line current above a uniform or layered Earth [*Price*, 1962; *Albertson and Van Baelen*, 1970; *Hermance and Peltier*, 1970]. Examination of this problem provides a basic solution that can often be extended to include more realistic and sophisticated electrojet features. As a sheet current of a specified width can be constructed of adjacent line currents, generalizations to sheet currents are obtained by a trivial superposition from line current results. Furthermore, *Boteler et al.* [2000] demonstrate the equivalence of a sheet current with a line current at a greater height. Calculations of the electric and magnetic fields produced by a line current above the Earth are usable in other areas, too, such as studies of the coupling between power lines and other conductors and geophysical exploration of the ground structure.

[7] A straightforward calculation of the electric and magnetic fields at the surface of a layered Earth caused by an overhead infinitely long line current, which is based on solving Maxwell's equations and on using boundary conditions, leads to integral expressions of the fields [*Price*, 1962; *Albertson and Van Baelen*, 1970]. The fast Fourier (FFT) and fast Hankel (FHT) transforms provide efficient methods of computing the integrals. An alternative approach is to write the integrals as series expansions. This was used by *Carson* [1926] to provide an expression for the electric field at the surface of a uniform Earth. Recently, exact series expansions have been presented by *Pirjola* [1998] for both the electric and magnetic fields on a uniform Earth. Approximate series expansions may also be derived in the case of a layered Earth [*Pirjola et al.*, 1999].

[8] In the complex image method (CIM) the secondary contribution from the ground to the surface fields is calculated by replacing the real ground by a perfect conductor located at a complex depth, which depends on the Earth's real conductivity structure and on the frequency [*Wait and Spies*, 1969; *Thomson and Weaver*, 1975; *Bannister*, 1986]. This greatly simplifies the problem and leads to fast and handy numerical computations. When the primary source is an infinitely long line current (as in this paper), CIM results in simple closed-form expressions for the surface electric and magnetic fields. While CIM was originally introduced and is widely used for engineering purposes, its applicability to space weather and GIC research as well has been emphasized and demonstrated recently [*Boteler and Pirjola*, 1998; *Pirjola and Viljanen*, 1998; *Viljanen et al.*, 1999; *Pirjola et al.*, 2000].

[9] *Boteler and Pirjola* [1999] provide a summary of and a comparison between the exact method (EXA), the series expansion method (SER), and the complex image method (CIM) of calculating the surface fields due to an infinitely long line current. In this paper, we consider the same geophysical model and the three methods, paying special attention to the mathematical approximations needed. This paper provides formulas readily usable for computing the surface fields with the three methods. The derivation of the equations is not included here, but pertinent references are given. Numerical computations are presented which show the good accuracy of SER and CIM. The great practical benefits of applying CIM do not become fully clear in connection with an infinitely long line current because EXA calculations can also be made fast then. The value of CIM would be much higher if a more complicated ionospheric-magnetospheric current system is considered. Thus this paper serves more as a summary of the three methods and as a proof of the accuracies of SER and CIM than as a plain demonstration of the efficiency of CIM.