Summary The magnetotelluric impedance tensor is defined in terms of seven independent parameters that are invariant under a rotation of the horizontal axes on the surface of the Earth, plus an angle that defines the orientation of the axes of reference. The invariants are algebraically related to but nevertheless different from those recently proposed by Szarka & Menvielle (1997). They have been chosen in such a way as to have clear representations on a Mohr circle diagram and also to reveal geoelectric properties of the Earth near the site where the impedance data are measured. The first two invariants define the properties of a 1‐D earth when the next four invariants are negligibly small. If the next two are also non‐negligible, the earth is 2‐D with a strike direction that can be recovered. The last three invariants indicate different degrees of three‐dimensionality and the discussion of them with reference to small‐scale galvanic distortion in an otherwise 1‐ or 2‐D structure largely retraces the insightful pioneering work of Bahr (1988). The properties of the invariants are illustrated with numerical calculations for a synthetic model consisting of a small conductive anomaly in the form of a cube at the surface of an otherwise 2‐D earth that is divided by a vertical fault into regions with a strong resistivity contrast. Results are presented for synthetic data that contain only numerical noise, and for data to which 2 per cent random Gaussian noise has been added. The theoretical properties of the invariants are verified by the pure numerical data, and are confirmed statistically by the noisy data.

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The numerical solution by finite differences of two-dimensional problems in electromagnetic induction is reexamined with a view to generalizing the method to three-dimensional models. Previously published work, in which fictitious values were used to derive the finite difference equations, is discussed and some errors in the theory which appear to have gone undetected so far, are pointed out. It is shown that the previously published Bpolarization formulas are incorrect at points where regions of different conductivity meet, and that the E-polarization formulas are inaccurate when the step sizes of the numerical grid around the point are uneven. An appropriately-modified version of the two-dimensional theory is developed on the assumption that the Earth's conductivity is a smoothly-varying function of position, a method which naturally lends itself to three-dimensional generalization. All the required finite-difference formulas are derived in detail, and presented in a form which is suitable for programming. A simple numerical calculation is given to illustrate the application of the method and the results are compared with those obtained from previous work.

It is shown that many geophysical problems involving the induction of earth currents by external magnetic variations can be solved by a method of images in which the earth is replaced by the image of the inducing source located at some complex depth beneath the earth's surface. An expression for this complex depth in a horizontally stratified earth is derived, and the theory is developed in a general form that may be applied to any inducing magnetic field of external origin. The technique is applicable when the modulus of the image depth is somewhat less than a characteristic length of the field and also when it greatly exceeds the characteristic length. The theory is applied to dipole, line current, and elementary sources over a three‐layer earth and compared with the exact solutions of previous authors. The magnitude of the error involved in the case of elementary sources is found to be nearly always less than 10%.

A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an Eor a B-polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green's tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions.