On the finite difference solution of two-dimensional induction problems

Summary 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 B-polarization 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.

[1]  R. D. Hibbs,et al.  Electromagnetic induction in three dimensional structures for various source fields. , 1978 .

[2]  C. Hewlett,et al.  Computer Modelling of Electrical Conductivity Structures , 1974 .

[3]  F. W. Jones,et al.  A Discussion of the Finite Difference Method in Computer Modelling of Electrical Conductivity Structures. A Reply to the Discussion by Williamson, Hewlett and Tammemagi , 1974 .

[4]  A. Raiche.  An Integral Equation Approach to Three-Dimensional Modelling , 1974 .

[5]  D. Rankin.  The Perturbation of Alternating Geomagnetic Fields by an Island Near a Coastline: Discussion , 1973 .

[6]  Induction in a layered plane earth by uniform and non-uniform source fields , 1973 .

[7]  F. W. Jones.  Induction in laterally non-uniform conductors: Theory and numerical models , 1973 .

[8]  F. W. Jones,et al.  The Perturbation of Alternating Geomagnetic Fields by Three‐dimensional Island Structures , 1973 .

[9]  F. W. Jones,et al.  The Perturbation of Alternating Geomagnetic Fields by an Island Near a Coastline , 1973 .

[10]  S. Ward,et al.  Analysis of Geoelectromagnetic Data , 1973 .

[11]  F. W. Jones,et al.  Boundary Conditions and Calculation of Surface Values for the General Two-Dimensional Electromagnetic Induction Problem , 1972 .

[12]  F. W. Jones,et al.  A General Computer Program to Determine the Perturbation of Alternating Electric Currents in a Two-Dimensional Model of a Region of Uniform Conductivity with an Embedded Inhomogeneity , 1971 .

[13]  U. Schmucker.  INTERPRETATION OF INDUCTION ANOMALIES ABOVE NONUNIFORM SURFACE LAYERS , 1971 .

[14]  F. W. Jones,et al.  The Perturbations of Alternating Geomagnetic Fields by Conductivity Anomalies , 1970 .