Abstract
A general approach to the construction of differential boundary conditions for vector fields satisfying the Helmholtz equation is proposed on the basis of the field expansion in multipole series and the application of annihilating operators to them. The resulting differential constraints can be used as boundary conditions in solving external boundary value problems. Examples of their application to the solution of forward geoelectric problems in three-dimensionally inhomogeneous media are examined. Their use at a finite distance from the source of an anomaly is shown to yield more accurate results than those obtained under the assumption that the anomalous field at this distance vanishes. Another effect of their application is a substantial decrease in the dimensions of the modeling domain and therefore in the time required to solve the forward problem. The “safe” distance for using the Dirichlet-type boundary conditions is estimated.
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References
C. A. Brebbia and S. Walker, “Simplified Boundary Elements for Radiation Problems,” Appl. Math. Model, No. 2, 135–137 (1978).
R. L. Mackie and T. R. Madden, “Conjugate Gradient Relaxation Solutions for Three-Dimensional Magnetotelluric Modeling,” Geophysics 58, 1052–1057 (1993).
G. A. Newman and D. L. Alumbaugh, “Three-Dimensional Magnetotelluric Inversion Using Non-Linear Conjugate Gradients,” Geophys. J. Int. 140, 410–424 (2000).
I. K. Reddy, D. Rankin, and R. J. Phillips, “Three-Dimensional Modeling in Magnetotelluric and Magnetic Variational Sounding,” Geophys. J. R. Astron. Soc. 51, 313–325 (1977).
J. Riordan, Combinatorial Identities (Wiley, New York, 1968; Nauka, Moscow, 1982).
Y. Sasaki, “Full 3-D Inversion of Electromagnetic Data on PC,” J. Appl. Geophys. 46(1), 45–54 (2001).
J. T. Smith, “Conservative Modeling of 3-D Electromagnetic Fields, Part I: Properties and Error Analysis,” Geophysics 61(5), 1308–1318 (1996).
V. V. Spichak, “FDM3D Software Package for Numerical Modeling of 3-D Electromagnetic Fields,” in Algorithms and Programs for Solving Forward and Inverse Problems of Electromagnetic Induction in the Earth (IZMIRAN, Moscow, 1983) [in Russian].
V. V. Spichak, “Differential Boundary Conditions for Electric and Magnetic Fields in an Unbounded Conductive Medium,” in Electromagnetic Sounding of the Earth (IZMIRAN, Moscow, 1985), pp. 14–22 [in Russian].
V. V. Spichak, Magnetotelluric Fields in Three-Dimensional Geoelectric Models (Nauchnyi Mir, Moscow, 1999) [in Russian].
J. T. Weaver and C. R. Brewitt-Taylor, “Improved Boundary Conditions for the Numerical Solution of E-Polarization Problems in Geomagnetic Induction,” Geophys. J. R. Astron. Soc. 54, 309–317 (1978).
P. Weidelt, “Electromagnetic Induction in Three-Dimensional Structures,” Geophysics, 42(1), 85–109 (1975).
C. K. Wilcox, “An Expansion Theorem for Electromagnetic Fields,” Comm. Pure Appl. Math. 9, 115–134 (1956).
C. K. Wilcox, “Spherical Means and Radiation Conditions,” Arch. Ration. Mech. Anal. 3, 133–148 (1959).
M. S. Zhdanov, N. G. Golubev, V. V. Spichak, and I. M. Varentsov, “The Construction of Effective Methods for Electromagnetic Modeling,” Geophys. J. R. Astron. Soc. 68, 589–607 (1982).
M. S. Zhdanov and V. V. Spichak, “Integrals of the Stratton-Chu Type for Heterogeneous Media with Applications to Geoelectric Problems,” in Mathematical Modeling of Electromagnetic Fields (IZMIRAN, Moscow, 1983), pp. 4–25 [in Russian].
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Original Russian Text © V.V. Spichak, 2006, published in Fizika Zemli, 2006, No. 3, pp. 17–24.
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Spichak, V.V. A method for constructing high-order differential boundary conditions for solving external boundary value problems in geoelectromagnetism. Izv.-Phys. Solid Earth 42, 193–200 (2006). https://doi.org/10.1134/S1069351306030025
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DOI: https://doi.org/10.1134/S1069351306030025