The Bremmer series for a multi-dimensional acoustic scattering problem

Gustafsson, Mats

doi:10.1088/0305-4470/33/9/314

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…It is known that this lower order variation can be accounted for in an exact decomposition for the isotropic media [32,36], the latter introduced the notation of 'true amplitude one-way wave equations'. An alternative approach is to include the correction term in the sources of the problem [11,7]. In the present paper we show that the explicit asymptotic admittance operator can include or ignore the vertical variation of the decomposition operator with minimal changes to the solution.…”

Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

“…The method of wave-splitting has a long history with a wide area of applications; an overview of some of the history is given in [8]. For the isotropic case wave-splitting has been used extensively to construct fast propagation methods [12,7,27,35].Recently, there has been an interest in methods that are almost frequency independent in calculation complexity based on wave-splitting [29,25]. Wave-splitting has also given raise to algorithms to reconstruct material parameters, for example the generalized Bremmer coupling series approach and the downward continuation approach [11,23,30] see also [33].…”

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confidence: 99%

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Asymptotic wave-splitting in anisotropic linear acoustics

Jonsson¹,

Norgren²

2010

Wave Motion

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Linear acoustic wave-splitting is an often used tool in describing sound-wave propagation through earth's subsurface. Earth's subsurface is in general anisotropic due to the presence of water-filled porous rocks. Due to the complexity and the implicitness of the wave-splitting solutions in anisotropic media, wave-splitting in seismic experiments is often modeled as isotropic. With the present paper, we have derived a simple wave-splitting procedure for an instantaneously reacting anisotropic media that includes spatial variation in depth, yielding both a traditional (approximate) and a 'true amplitude' wave-field decomposition. One of the main advantages of the method presented here is that it gives an explicit asymptotic representation of the linear acoustic-admittance operator to all orders of smoothness for the smooth, positive definite anisotropic material parameters considered here. Once the admittance operator is known we obtain an explicit asymptotic wave-splitting solution.1 dia. This solution enables us to obtain an explicit asymptotic representation of the wave-splitting operators in such anisotropic media.Wave-splitting, or wave-field decomposition, is a tool to decompose the wave-field into 'up'-and 'down'-going wave field constituents in configurations with a certain directionality [19,11,18,24], as e.g., a seismic experiment for probing earth's subsurface e.g., [23]. The wave-splitting procedure results in two one-way equations for the wave-field constituents. Wavesplitting has been used to model and analyze wave propagation in both inverse problems and migration models. The method of wave-splitting has a long history with a wide area of applications; an overview of some of the history is given in [8]. For the isotropic case wave-splitting has been used extensively to construct fast propagation methods [12,7,27,35].Recently, there has been an interest in methods that are almost frequency independent in calculation complexity based on wave-splitting [29,25]. Wave-splitting has also given raise to algorithms to reconstruct material parameters, for example the generalized Bremmer coupling series approach and the downward continuation approach [11,23,30] see also [33]. Another application area of wave-splitting is in the context of boundary conditions and time-reversal mirrors [17,6]. Wave-splitting methods have been implemented in several physically different contexts and for a range of constitutive relations: wave-splitting for wave equations [32,34]. The electromagnetic equations are wave-field decomposed both for isotropic [22,5,24], anisotropic lossless (the spectral theoretical approach) [16] and wave-splitting has been extended to the homogeneous lossless stratified bi-anisotropic case [26,20]. Wave-splitting methods have been applied to linear-elastodynamic equations as for propagation on beams see e.g., [15] as well as in the half-space in homogeneous stratified anisotropic media [9] and up-/down symmetric media [13].One limitation to the present methods of wave-splitting is that it has been al...

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Asymptotic wave-splitting in anisotropic linear acoustics

Jonsson¹,

Norgren²

2010

Wave Motion

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“…(B.13)Thus the generalized eigenvalue problem reduces to an eigenvalue problem for the isotropic case, i.e., s ± reduces to diagonal matrices. If we consider the wave-splitting problem by earlier developed techniques see e.g [8,30]. we finda ±I z −1 = ±I z −1 s ± .…”

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confidence: 92%

“…For time-domain wave-splitting see [11], where both the wave equation and the Maxwell's equations are considered with both applications and theory. Wave-splitting in connection with Bremmer series for linear acoustics [14,8,9] and uniform asymptotics and normal modes [7,15] has been used to analyze the wave-field constituents. An extension to include dispersion is presented in [28] and wave-splitting on structural elements in [21].…”

Section: Introductionmentioning

confidence: 99%

Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations

Jonsson¹

2009

Inverse Problems &Amp; Imaging

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The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system's matrix. A constructive proof of the existence of directional wave-field decomposition with respect to the normal of the boundary is presented.In the process of defining the wave-field decomposition (wave-splitting), the resolvent set of the time-Laplace representation of the system's matrix is analyzed. This set is shown to contain a strip around the imaginary axis. We construct a splitting matrix as a Dunford-Taylor type integral over the resolvent of the unbounded operator defined by the electromagnetic system's matrix. The splitting matrix commutes with the system's matrix and the decomposition is obtained via a generalized eigenvalue-eigenvector procedure. The decomposition is expressed in terms of components of the splitting matrix. The constructive solution to the question on the existence of a decomposition also generates an impedance mapping solution to an algebraic Riccati operator equation. This solution is the electromagnetic generalization in an anisotropic media of a Dirichlet-to-Neumann map. Keywords directional wave-field decomposition, wave-splitting, anisotropy, electromagnetic system's matrix, generalized eigenvalue problem, algebraic Riccati operator equation, generalized vertical wave number.where J e = external electric current density [A/m 2 ] and K e = external magnetic current density [V/m 2 ]. The external currents are applied, prescribed, sources. The causality of the field is taken into account by requiring that all field quantities are bounded functions of B. L. G. Jonsson

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