A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem
A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the late...
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| Autor*in: |
Zhang, Tian Shan [verfasserIn] Schultz, Adam [verfasserIn] |
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E-Artikel |
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| Erschienen: |
Oxford, UK: Blackwell Publishing Ltd ; 1992 |
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Online-Ressource |
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| Reproduktion: |
2007 ; Blackwell Publishing Journal Backfiles 1879-2005 |
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| Übergeordnetes Werk: |
In: Geophysical journal international - Oxford . Wiley-Blackwell, 1922, 111(1992), 2, Seite 0 |
| Übergeordnetes Werk: |
volume:111 ; year:1992 ; number:2 ; pages:0 |
| Links: |
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| DOI / URN: |
10.1111/j.1365-246X.1992.tb00580.x |
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| 520 | |a A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. | ||
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10.1111/j.1365-246X.1992.tb00580.x doi (DE-627)NLEJ239653440 DE-627 ger DE-627 rakwb Zhang, Tian Shan verfasserin aut A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem Oxford, UK Blackwell Publishing Ltd 1992 Online-Ressource nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. 2007 Blackwell Publishing Journal Backfiles 1879-2005 |2007|||||||||| forward problem Schultz, Adam verfasserin aut In Geophysical journal international Oxford . Wiley-Blackwell, 1922 111(1992), 2, Seite 0 Online-Ressource (DE-627)NLEJ243927827 (DE-600)2006420-2 1365-246X nnns volume:111 year:1992 number:2 pages:0 http://dx.doi.org/10.1111/j.1365-246X.1992.tb00580.x text/html Verlag Deutschlandweit zugänglich Volltext GBV_USEFLAG_U ZDB-1-DJB GBV_NL_ARTICLE AR 111 1992 2 0 |
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10.1111/j.1365-246X.1992.tb00580.x doi (DE-627)NLEJ239653440 DE-627 ger DE-627 rakwb Zhang, Tian Shan verfasserin aut A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem Oxford, UK Blackwell Publishing Ltd 1992 Online-Ressource nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. 2007 Blackwell Publishing Journal Backfiles 1879-2005 |2007|||||||||| forward problem Schultz, Adam verfasserin aut In Geophysical journal international Oxford . Wiley-Blackwell, 1922 111(1992), 2, Seite 0 Online-Ressource (DE-627)NLEJ243927827 (DE-600)2006420-2 1365-246X nnns volume:111 year:1992 number:2 pages:0 http://dx.doi.org/10.1111/j.1365-246X.1992.tb00580.x text/html Verlag Deutschlandweit zugänglich Volltext GBV_USEFLAG_U ZDB-1-DJB GBV_NL_ARTICLE AR 111 1992 2 0 |
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10.1111/j.1365-246X.1992.tb00580.x doi (DE-627)NLEJ239653440 DE-627 ger DE-627 rakwb Zhang, Tian Shan verfasserin aut A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem Oxford, UK Blackwell Publishing Ltd 1992 Online-Ressource nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. 2007 Blackwell Publishing Journal Backfiles 1879-2005 |2007|||||||||| forward problem Schultz, Adam verfasserin aut In Geophysical journal international Oxford . Wiley-Blackwell, 1922 111(1992), 2, Seite 0 Online-Ressource (DE-627)NLEJ243927827 (DE-600)2006420-2 1365-246X nnns volume:111 year:1992 number:2 pages:0 http://dx.doi.org/10.1111/j.1365-246X.1992.tb00580.x text/html Verlag Deutschlandweit zugänglich Volltext GBV_USEFLAG_U ZDB-1-DJB GBV_NL_ARTICLE AR 111 1992 2 0 |
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10.1111/j.1365-246X.1992.tb00580.x doi (DE-627)NLEJ239653440 DE-627 ger DE-627 rakwb Zhang, Tian Shan verfasserin aut A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem Oxford, UK Blackwell Publishing Ltd 1992 Online-Ressource nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. 2007 Blackwell Publishing Journal Backfiles 1879-2005 |2007|||||||||| forward problem Schultz, Adam verfasserin aut In Geophysical journal international Oxford . Wiley-Blackwell, 1922 111(1992), 2, Seite 0 Online-Ressource (DE-627)NLEJ243927827 (DE-600)2006420-2 1365-246X nnns volume:111 year:1992 number:2 pages:0 http://dx.doi.org/10.1111/j.1365-246X.1992.tb00580.x text/html Verlag Deutschlandweit zugänglich Volltext GBV_USEFLAG_U ZDB-1-DJB GBV_NL_ARTICLE AR 111 1992 2 0 |
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10.1111/j.1365-246X.1992.tb00580.x doi (DE-627)NLEJ239653440 DE-627 ger DE-627 rakwb Zhang, Tian Shan verfasserin aut A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem Oxford, UK Blackwell Publishing Ltd 1992 Online-Ressource nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. 2007 Blackwell Publishing Journal Backfiles 1879-2005 |2007|||||||||| forward problem Schultz, Adam verfasserin aut In Geophysical journal international Oxford . Wiley-Blackwell, 1922 111(1992), 2, Seite 0 Online-Ressource (DE-627)NLEJ243927827 (DE-600)2006420-2 1365-246X nnns volume:111 year:1992 number:2 pages:0 http://dx.doi.org/10.1111/j.1365-246X.1992.tb00580.x text/html Verlag Deutschlandweit zugänglich Volltext GBV_USEFLAG_U ZDB-1-DJB GBV_NL_ARTICLE AR 111 1992 2 0 |
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A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem |
| abstract |
A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. |
| abstractGer |
A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. |
| abstract_unstemmed |
A solution of the three-dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid-mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ(0)(r) +σ(1) (r, Θ, ϕ).It can be shown that the zeroth-order approximation of the problem is just the one-dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth-order equation. The first-order perturbation solution depends on the solution to the zeroth-order problem. The three-dimensional perturbation to the initial one-dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three-dimensional solution is calculated from the coefficients of this expansion by numerical integration.Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three-dimensional model has also been calculated up to second-order approximation. |
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A 3-D perturbation solution for the EM induction problem in a spherical earth—the forward problem |
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