Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes-theory and example

Singer, B. Sh.; Fainberg, E. B.

doi:10.1046/j.1365-246x.1999.00844.x

Cited by 13 publications

(13 citation statements)

References 27 publications

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“…Recently, the primary methods of 3-D MT forward modelling are the finite difference method (FDM), the finite element method (FEM) and the integral equation method (IEM). Avdeev et al (1997) and Singer & Fainberg (1999) successfully applied this technique to MT forward modelling. Wannamaker (1984) first applied the IEM to 3-D MT forward calculation for only a simple model with a few heterogeneous blocks because the matrix is dense, which was difficult to solve at that time.…”

Section: Formulation Of 3-d Mt Forward Calculationmentioning

confidence: 99%

A regularized three-dimensional magnetotelluric inversion with a minimum gradient support constraint

Zhang¹,

Koyama²,

Utada³

et al. 2012

Geophysical Journal International

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SUMMARY Most 3‐D magnetotelluric (MT) inversions are classified as a regularized inversion with a smoothness constraint. These inverse algorithms provide smooth solutions but cannot clearly image sharp geo‐electrical interfaces. In this paper, we introduce the minimum gradient support (MGS) functional to regularize the 3‐D MT inverse problem. This functional has a property whereby the functional seeks a structure with minimum volume containing large conductivity gradients. Therefore, the MGS functional can be used to search for a model with a sharp boundary. We apply the MGS functional to 3‐D MT inversion to obtain a clear and accurate image of geo‐electrical interfaces. In addition, the modified scattering equation approach introduced in the modified iterative dissipative method (MIDM) is applied to forward calculation, which is based on integral equation (IE) formulation and allows us to efficiently reduce the time required for forward calculation with high accuracy. The quasi‐Newton iterative method is used to optimize the objective functional. It is a kind of iterative method with simplified calculation of the inverse Hessian matrix using Broyden–Fletcher–Goldfarb–Shanno (BFGS) update. The convergence of this iterative method is guaranteed with inexact line searches. We also modify the adaptive approach for optimum selection of the regularization parameter so as to fit the inverse algorithm of this study. Three synthetic models are investigated, and the obtained results are compared with those obtained by a smoothing inversion. Based on the comparison, we confirm that the MGS inversion can provide higher resolution when geo‐electrical interfaces are sharp. This property will help us to determine reliable electrical structures by the MT exploration method.

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Section: Formulation Of 3-d Mt Forward Calculationmentioning

confidence: 99%

A regularized three-dimensional magnetotelluric inversion with a minimum gradient support constraint

Zhang¹,

Koyama²,

Utada³

et al. 2012

Geophysical Journal International

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“…In conclusion, it is worth noticing that despite the fact that throughout our consideration the layer was assumed to be thin, the terms correcting Maxwell's solution for finite thickness of the layer cannot be derived from the generalized thin-sheet approximation (Dmitriev, 1969;Ranganayaki and Madden, 1980). As discussed by Singer and Fainberg (1999), the generalized thin-sheet approximation is the first-order approximation with respect to small parameters h √ ωμ 0 σ ∼ √ h/v s t and h/λ τ , where λ τ is the characteristic length of the field variations along the surface of the layer. These parameters are generally independent.…”

Section: Discussionmentioning

confidence: 96%

Transient field solution for a layer of finite thickness on a resistive basement

Singer

Green²

2003

GEOPHYSICS

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A closed-form asymptotic solution is derived for the magnetic field of the currents induced by a transient airborne magnetic source in a conductive layer of finite thickness. The conductive layer rests on top of a resistive half-space. Like the well-known solution found by J. C. Maxwell for a thin conductive sheet surrounded by an insulator, the secondary magnetic field is expressed in terms of an image source receding from the layer. However, the new solution also accounts for the layer thickness h and the conductivity of the half-space.One of the conclusions from the new solution is that the mirror plane that specifies the position of the image is located below the upper interface of the conductive layer at depth h/3. This indicates the correct position at which the equivalent thin sheet should be placed when Maxwell's solution is applied to a layer of finite thickness.If the basement that underlies the layer is highly resistive, Maxwell's solution becomes accurate when induced currents are almost uniformly spread across the layer. It remains accurate as long as currents induced in the basement can be neglected. Eventually, the secondary magnetic field of these currents will prevail over the field of currents in the layer. Maxwell's solution loses its accuracy long before this occurs. Depending on parameters of the model, the validity time range of Maxwell's solution may be narrow or even nonexistent.The generalized image solution is applicable in thewhere v s is the image recession speed, and σ s and σ b are the layer and basement conductivities, respectively. This range is significantly wider than that of Maxwell's solution. At early times, the secondary magnetic field is controlled by the position of the nearest interface of the conductive layer. Accounting for this observation, a simple modification of the new solution may be used to extend the applicability range towards even earlier times. The generalized solution is faster by several orders of magnitude than a numeric solution based on successive wavenumber-tospace and frequency-to-time domain fast Hankel transforms.

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“…This optimization procedure relies on the same ‘energy norm’ which is used to evaluate the norms of the integral operator in the continuous equation and the matrix of the system of linear equations. Taking into consideration the fact that the norms of the continuous and discrete operators depend only on the maximum lateral contrast of the conductivity distribution, the accuracy of the approximants can be reliably evaluated and used to terminate the computation, once the necessary relative accuracy of the numerical solution has been reached. All the expressions for matrix elements are reduced to presentations that were used for the MIDM thin layer codes, previously described by Singer & Fainberg (1999). Therefore, we use the same routines that were designed with a view to achieving the highest accuracy of the kernel matrix evaluation, which can be achieved for the given grid.…”

Section: Discussionmentioning

confidence: 99%

“…Dawson & Weaver 1979), we may use auxiliary 2-D 2.5-D or 3-D computations to determine the normal field. In case of a 3-D background model, the normal field can be found from the IE using a coarse grid and then detailed to a finer grid as described by Singer & Fainberg (1999) for thin layer models.…”

Section: N O R M a L B A C Kg Ro U N D A N D A N O M A L O U S F I mentioning

confidence: 99%

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Electromagnetic integral equation approach based on contraction operator and solution optimization in Krylov subspace

Singer¹

2008

Geophysical Journal International

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S U M M A R YThe paper presents a new code for modelling electromagnetic fields in complicated 3-D environments and provides examples of the code application. The code is based on an integral equation (IE) for the scattered electromagnetic field, presented in the form used by the Modified Iterative Dissipative Method (MIDM). This IE possesses contraction properties that allow it to be solved iteratively. As a result, for an arbitrary earth model and any source of the electromagnetic field, the sequence of approximations converges to the solution at any frequency.The system of linear equations that represents a finite-dimensional counterpart of the continuous IE is derived using a projection definition of the system matrix. According to this definition, the matrix is calculated by integrating the Green's function over the 'source' and 'receiver' cells of the numerical grid. Such a system preserves contraction properties of the continuous equation and can be solved using the same iterative technique. The condition number of the system matrix and, therefore, the convergence rate depends only on the physical properties of the model under consideration. In particular, these parameters remain independent of the numerical grid used for numerical simulation. Applied to the system of linear equations, the iterative perturbation approach generates a sequence of approximations, converging to the solution. The number of iterations is significantly reduced by finding the best possible approximant inside the Krylov subspace, which spans either all accumulated iterates or, if it is necessary to save the memory, only a limited number of the latest iterates. Optimization significantly reduces the number of iterates and weakens its dependence on the lateral contrast of the model. Unlike more traditional conjugate gradient approaches, the iterations are terminated when the approximate solution reaches the requested relative accuracy. The number of the required iterates, which for simple iterations, is approximately proportional to the square root of the lateral contrast of conductivity, in case of optimization, becomes approximately proportional to the logarithm of the lateral contrast. The effect of optimization increases with the complexity of the model under consideration.The new code is applied to a number of problems of subsurface, borehole and marine geophysics. Some of the results are compared with independent computations using different approaches.

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Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes-theory and example

Cited by 13 publications

References 27 publications

A regularized three-dimensional magnetotelluric inversion with a minimum gradient support constraint

A regularized three-dimensional magnetotelluric inversion with a minimum gradient support constraint

Transient field solution for a layer of finite thickness on a resistive basement

Electromagnetic integral equation approach based on contraction operator and solution optimization in Krylov subspace

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