##
Geçici Elektromanyetik Alan Yayılımının Sonlu Farklar Yöntemleriyle İki Boyutlu Modellenmesi

Geçici Elektromanyetik Alan Yayılımının Sonlu Farklar Yöntemleriyle İki Boyutlu Modellenmesi

##### Dosyalar

##### Tarih

2013-03-14

##### Yazarlar

Denız, Mustafa

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu çalışmada, geçici elektromyanetik yöntemde iki-boyutlu düz çözümde kullanılan difüzyon denkleminin farklı sayısal yöntemlerle çözümleri, çözüm gücü ve hesaplama zamanı bakımından karşılaştırılmıştır. Bu yöntemler sonlu farklar yaklaşımları olan; zamanda geri adımla merkezi fark, zamanda ileri adımla merkezi fark, Du Fort-Frankel yöntemi ve Crank-Nicolson yöntemidir. Model olarak yer yüzeyinde zıt işaretli iki çizgisel kaynağın, yer içinde oluşturduğu elektromanyetik alan yayılımı incelenmiştir. Elektromanyetik alanın ‘transverse electric (TE)’ modu esas alınmıştır. Karşılaştırmalar iki boyutlu tekdüze ortam ve bir boyutlu (1B) model için yapılmıştır. Tek çizgisel kaynak için verilen iki boyutlu (2B) analitik bağıntı, zıt işaretli iki çigisel kaynağa uygulanarak model oluşturulmuştur. Geçici elektromanyetik alanların farklı sonlu farklar teknikleriyle hesaplanması amacıyla yazılan programlar, MATLAB dili ile yazılmıştır. Farklı yöntemler için yapılan karşılaştırmalar hata, oranı, işlem süresi, yöntemlerin duraylılığı ve duyarlılıkları üzerine yapılmıştır. Yazılan MATLAB programının doğruluğu elektromanyetik alan yayılımı için verilen analitik bağıntı yardımıyla sınanmıştır. Hesaplamalar sonuncunda, Crank-Nicolson yöntemi duraylılık ve tutarlılık özellikleriyle difüzyon tipi problemlerin çözümünde en uygun yöntem olarak tespit edilmiştir. Hesaplamalarda işlem hızı göz önüne alındığında, Du-Fort Frankel yöntemi diğer yöntemlere üstünlük sağlamıştır.

This study compares the frequently used finite difference methods for two dimensional (2D) modeling of transient electromagnetic method (TEM). Also electromagnetic boundary conditions are mentioned to define the electromagnetic model. Finite difference methods was used for modeling of electromagnetic field diffusion in two dimensional homogeneous media. Transient electromagnetic method’s theory and it’s apllications were mentioned in small details because of the exact purpose of this study is just comparing of the most-used finite difference methods to simulate of TEM diffusion. Comparisons have been done by accuracy, stability, consistency and process (CPU) time. To make these comparisons MATLAB scripting language was used for computations. Electromagnetic theory is essential to understand how the electromagnetic diffusion behaves in materials. So electromagnetic theory explained by the Maxwell equations which depend on time, spatial parameters and material’s electric and magnetic properties. Electromagnetic diffusion occurs only when electromagnetic wave frequency is very low. Therefore, wave type behavior is negligible on low frequencies where diffusion type behavior is dominant. Hence, the derivation of electromagnetic diffusion equation explained for homogeneous media when electromagnetic source terms are excluded. Because the electromagnetic source is shutted down when the electric field is recorded. TEM method has a wide application area and has been used by many geophysicist. Mineral and geothermal explorations and static shift problem in Magnetotelluric method are a few examples of these applications. In this study, analytical expressions for a double line source generating the 2D electric field has been used as initial condition. Finite difference approximations used in this study are forward-time centered-space scheme (FTSC). Du Fort-Frankel scheme, backward-time centered-space scheme (BTSC) and Crank Nicolson scheme. These approximations are based on the Taylor series expansion’s first term. In other words, the linear approximation has been used, other terms are negligible and not dominant as first one. Each scheme has been analzed in terms of stability, consistency, accuracy and relative computational speed. Crank-Nicholson and Du Fort-Frankel methods have been found to be more accurate and stable than other methods. These methods’s accuracy are second order when fully implicit and fully explicit methods are only first order. Du Fort-Frankel method’s inconsistent behavior needs more care in selecting the time-stepping and grid spacing of the finite difference network. For this reason, Du Fort-Frankel method has been found to be always out of running method. Fully explicit method has failed in terms of both accuracy and stability. Due to it’s stable and consistent behavior, Crank-Nicolson scheme has been determined to be most suitable method. Du Fort-Frankel scheme has been found to be superior than other methods when the CPU time is considered. Because this method does not require solution of the system of linear equations as in Crank-Nicolson method. Implicit schemes’s finite difference equations have been reestablished in sparse matrices. To solve these large linear systems, their specific properties like symmetric or positive definiteness are helpful. This is generally related to finite difference grid and boundary conditions of the model. In this study, linear system has been found to be symmetric and positive definite. Hence Conjugate-Gradient method which is known as an optimization method has been used to solve this large linear system. MATLAB codes have been developed to solve sparse systems. All scripts and functions were coded by author to optimize the solution of the specific problem which is electromagnetic propagation in two-dimensional homogeneous media. The electromagnetic diffusion for two different homogeneous media has been shown as snapshots for three different values of time. Two stations has been selected for comparisons with the analytical solution. In these comparisons, Crank-Nicolson and Du Fort-Frankel method have been found to be more accurate with less than two per cent relative error. Other methods have been found to have much more relative error relative to Crank-Nicolson and Du Fort-Frankel method. Regular grids have been used to estimate how accuratetly values of left and right boundaries are calculated. Both Crank-Nicolson and Du Fort-Frankel methods have been found to be superior in terms of accuracy with less than sixteen per cent relative error. Even in the worst case scenario where relative error is largest at the right and left boundaries has been found to be almost acceptable. This case has been analzed for two different values of Courant number’s, applicable values for the all methods except for only fully explicit method. For different values of Courant’s number, ratio of time-stepping value to grid spacing, each finite difference method has been compared in terms of accuracy. For each finite difference scheme, mean absolute errors as a function of Courant’s number have been compared. Due to it’s unstable behavior, fully explicit scheme failed to calculate electromagnetic diffusion at higher values of 0.25 of Courant’s number. Electromagnetic field propagation, triggered by opposite signed double line source, was analzed from earth’s surface to earth’s interior. TE mode was predicted for electromagnetic field propagation. In the comparisons, 2D homogeneous resistivity models were used. Primary electromagnetic field for a line source was calculated using the analytical solution for a homogeneous medium. Programs with aim of calculating transient electromagnetic field by different finite difference methods have been coded by MATLAB scripting language environment. Comparisons of finite difference methods were made in terms of stability, consistency, accuracy and process (CPU) time. MATLAB programs, used for computation of electromagnetic field of 2D homogeneous medium, has been confirmed by the analytic equation. Crank-Nicolson and Du Fort-Frankel method has been found to be most suitable methods for computing electromagnetic diffusion in two-dimensional homogeneous media. Each methods has it’s own disadvantage. Crank-Nicolson method has much more calculation than the Du Fort-Frankel method so this method is relatively slower. Du Fort-Frankel method is inconsistent in some situations. Therefore, it is always good to simulate electromagnetic field propagation with hybrid methods which Crank-Nicolson and Du Fort-Frankel. This type of approach guarantees stability and consistency with Crank-Nicolson method and process speed with Du Fort-Frankel method. After gaining stability using smaller time-steps with Crank-Nicolson, Du-Fort-Frankel method can be used with larger-time-steps for computation. Since both stability and speed takes place. Hence, Courant’s number is vital to model two-dimensional electromagnetic diffusion as it is in any other diffusion type problem. Furthermore, this study can be developed by using more accurate explicit methods. Adding alternate direction implicit (ADI) and locally one dimensional (LOD) methods to comparisons can provide more accurate results.

This study compares the frequently used finite difference methods for two dimensional (2D) modeling of transient electromagnetic method (TEM). Also electromagnetic boundary conditions are mentioned to define the electromagnetic model. Finite difference methods was used for modeling of electromagnetic field diffusion in two dimensional homogeneous media. Transient electromagnetic method’s theory and it’s apllications were mentioned in small details because of the exact purpose of this study is just comparing of the most-used finite difference methods to simulate of TEM diffusion. Comparisons have been done by accuracy, stability, consistency and process (CPU) time. To make these comparisons MATLAB scripting language was used for computations. Electromagnetic theory is essential to understand how the electromagnetic diffusion behaves in materials. So electromagnetic theory explained by the Maxwell equations which depend on time, spatial parameters and material’s electric and magnetic properties. Electromagnetic diffusion occurs only when electromagnetic wave frequency is very low. Therefore, wave type behavior is negligible on low frequencies where diffusion type behavior is dominant. Hence, the derivation of electromagnetic diffusion equation explained for homogeneous media when electromagnetic source terms are excluded. Because the electromagnetic source is shutted down when the electric field is recorded. TEM method has a wide application area and has been used by many geophysicist. Mineral and geothermal explorations and static shift problem in Magnetotelluric method are a few examples of these applications. In this study, analytical expressions for a double line source generating the 2D electric field has been used as initial condition. Finite difference approximations used in this study are forward-time centered-space scheme (FTSC). Du Fort-Frankel scheme, backward-time centered-space scheme (BTSC) and Crank Nicolson scheme. These approximations are based on the Taylor series expansion’s first term. In other words, the linear approximation has been used, other terms are negligible and not dominant as first one. Each scheme has been analzed in terms of stability, consistency, accuracy and relative computational speed. Crank-Nicholson and Du Fort-Frankel methods have been found to be more accurate and stable than other methods. These methods’s accuracy are second order when fully implicit and fully explicit methods are only first order. Du Fort-Frankel method’s inconsistent behavior needs more care in selecting the time-stepping and grid spacing of the finite difference network. For this reason, Du Fort-Frankel method has been found to be always out of running method. Fully explicit method has failed in terms of both accuracy and stability. Due to it’s stable and consistent behavior, Crank-Nicolson scheme has been determined to be most suitable method. Du Fort-Frankel scheme has been found to be superior than other methods when the CPU time is considered. Because this method does not require solution of the system of linear equations as in Crank-Nicolson method. Implicit schemes’s finite difference equations have been reestablished in sparse matrices. To solve these large linear systems, their specific properties like symmetric or positive definiteness are helpful. This is generally related to finite difference grid and boundary conditions of the model. In this study, linear system has been found to be symmetric and positive definite. Hence Conjugate-Gradient method which is known as an optimization method has been used to solve this large linear system. MATLAB codes have been developed to solve sparse systems. All scripts and functions were coded by author to optimize the solution of the specific problem which is electromagnetic propagation in two-dimensional homogeneous media. The electromagnetic diffusion for two different homogeneous media has been shown as snapshots for three different values of time. Two stations has been selected for comparisons with the analytical solution. In these comparisons, Crank-Nicolson and Du Fort-Frankel method have been found to be more accurate with less than two per cent relative error. Other methods have been found to have much more relative error relative to Crank-Nicolson and Du Fort-Frankel method. Regular grids have been used to estimate how accuratetly values of left and right boundaries are calculated. Both Crank-Nicolson and Du Fort-Frankel methods have been found to be superior in terms of accuracy with less than sixteen per cent relative error. Even in the worst case scenario where relative error is largest at the right and left boundaries has been found to be almost acceptable. This case has been analzed for two different values of Courant number’s, applicable values for the all methods except for only fully explicit method. For different values of Courant’s number, ratio of time-stepping value to grid spacing, each finite difference method has been compared in terms of accuracy. For each finite difference scheme, mean absolute errors as a function of Courant’s number have been compared. Due to it’s unstable behavior, fully explicit scheme failed to calculate electromagnetic diffusion at higher values of 0.25 of Courant’s number. Electromagnetic field propagation, triggered by opposite signed double line source, was analzed from earth’s surface to earth’s interior. TE mode was predicted for electromagnetic field propagation. In the comparisons, 2D homogeneous resistivity models were used. Primary electromagnetic field for a line source was calculated using the analytical solution for a homogeneous medium. Programs with aim of calculating transient electromagnetic field by different finite difference methods have been coded by MATLAB scripting language environment. Comparisons of finite difference methods were made in terms of stability, consistency, accuracy and process (CPU) time. MATLAB programs, used for computation of electromagnetic field of 2D homogeneous medium, has been confirmed by the analytic equation. Crank-Nicolson and Du Fort-Frankel method has been found to be most suitable methods for computing electromagnetic diffusion in two-dimensional homogeneous media. Each methods has it’s own disadvantage. Crank-Nicolson method has much more calculation than the Du Fort-Frankel method so this method is relatively slower. Du Fort-Frankel method is inconsistent in some situations. Therefore, it is always good to simulate electromagnetic field propagation with hybrid methods which Crank-Nicolson and Du Fort-Frankel. This type of approach guarantees stability and consistency with Crank-Nicolson method and process speed with Du Fort-Frankel method. After gaining stability using smaller time-steps with Crank-Nicolson, Du-Fort-Frankel method can be used with larger-time-steps for computation. Since both stability and speed takes place. Hence, Courant’s number is vital to model two-dimensional electromagnetic diffusion as it is in any other diffusion type problem. Furthermore, this study can be developed by using more accurate explicit methods. Adding alternate direction implicit (ADI) and locally one dimensional (LOD) methods to comparisons can provide more accurate results.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2013

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2013

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2013

##### Anahtar kelimeler

Geçici Elektromanyetik Alan,
Sonlu Farklar,
Transient Electromagnetic Field,
Finite Differences