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.

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New boundary conditions applicable on the sides and top of numerical grids used in the solution of E-polarization problems in electromagnetic induction are proposed. It is demonstrated that these new boundary conditions improve the accuracy of the numerical solution and also that their use allows such problems to be solved on grids of much smaller overall size than were formerly necessary.