Summary A generalized image technique is described for modelling electromagnetic induction in two-dimensional systems, consisting of a thin (Price-type) ocean overlying a perfect conductor in the mantle. The method constructs a Green’s function for currents in the ocean by conformally mapping the perfect conductor boundary into a straight line. Examples are given that show the effects to be expected at an ocean-continent boundary with isotherms rising under the ocean, and at a mid-ocean rise where the high temperatures are believed to be quite shallow. The object of most magnetic variation and magnetotelluric surveys is an understanding of electrical conductivity at great depth (anywhere from 5 to 1000 km depending on the frequency band). When such studies are made near or in the ocean, the conductivity of the water is great enough to modify the fields and complicate the interpretation of the observations (Parkinson 1959); it is therefore important that theoretical studies be capable of accounting for the presence of the ocean in a realistic fashion. Price (1949) develops an approach to this problem in which the oceans are represented by an infinitesimally thin conducting sheet, with integrated conductance defined by 3 = @y, where cr is the conductivity of the water and y the vertical co-ordinate. With this approximation, 5 is allowed to vary from place to place; included in such variations are the ocean depth, salinity and temperature, of which the first is the most important. Electric currents with a component normal to the ocean surface are ignored, but we feel this is entirely reasonable. Although the ocean itself appears to be adequately described by this representation at frequencies above 2 cycles per hour, the incorporation of realistic upper mantle conductivity structures has proved more problematic. Earlier work on this topic (Roden 1964; Bullard & Parker 1970) employed the method of images, which meant that the mantle conductivity was modelled by a perfect conductor (conductivity infinite) with a horizontal surface. Schmucker’s (1971) technique enables us to deal with arbitrary horizontally layered media and, although this method was developed neglecting the Earth’s curvature and only in two dimensions, neither restriction is essential.$ The obvious way to remove the requirement of horizontal layering in the