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Kirpichnikova, Natalya Yakovlevna - One of the best experts on this subject based on the ideXlab platform.

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 461, 2017, pp. 148–173.REF Compliant by Deposit in Stirling's Repositor

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 461, 2017, pp. 148–173

Kirpichnikova, Anna S - One of the best experts on this subject based on the ideXlab platform.

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 461, 2017, pp. 148–173.REF Compliant by Deposit in Stirling's Repositor

  • Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Kirpichnikova, Anna S, Kirpichnikova, Natalya Yakovlevna
    Abstract:

    This paper continues a series of publications on the shortWave diffraction of the Plane Wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the Wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the Incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping Waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortWave diffraction of a Plane Incident Wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the Incident Wavelength. The Incident Wave propagates along the axis of revolution. The total Wave field U is the sum of the Incident U inc and reflected U ref Waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortWave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the Wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total Wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the Wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 461, 2017, pp. 148–173

Kojiro Irikura - One of the best experts on this subject based on the ideXlab platform.

  • boundary shape Waveform inversion for two dimensional basin structure using three component array data of Plane Incident Wave with an arbitrary azimuth
    Bulletin of the Seismological Society of America, 1997
    Co-Authors: Shin Aoi, Tomotaka Iwata, Hiroyuki Fujiwara, Kojiro Irikura
    Abstract:

    We extend a new Waveform inversion scheme (Aoi et al. , 1995) for estimating underground structure with an irregular-shaped basement as a target to cases where Plane Waves with an arbitrary azimuth impinge on the structure, i.e., from a direction not necessarily perpendicular to the major axis of the structure. We proved the validity of this scheme by numerical experiments. We had already achieved the formulation and numerical experiments for the cases where an SH Wave impinges on a 2D basin structure and had shown that we could estimate the entire basin structure with seismic Waveforms from only a few surface stations by using whole Waveforms that include the surface Waves. However, when the epicenter is located out of the Plane including the observation stations, even the cases of 2D structure cannot be treated as a simple 2D ( SH or P-SV ) problem because of the Wave with an azimuth that is not 0°. Therefore, by dealing with 3D Wave fields in the present study, we extend the inversion scheme in order to apply it to Incident Waves with an arbitrary azimuth. The differential seismograms, which represent the sensitivity of change in the Waveform, show different patterns in three components, and we demonstrate that inversion with three components, compared with the inversion with only one of them, leads to a linearized equation system with a smaller condition number and a more stable computation. Furthermore, we detect certain parts that are estimated with much less difficulty than others, depending on the direction from which the Incident Wave impinged. In the latter case, we can estimate the entire structure by employing simultaneously several data from Incident Waves arriving from different directions. We thus demonstrate by numerical experiments that the extension of our inversion method to cases where the Incident Wave with an arbitrary azimuth impinges on the structure enables us to estimate with increased accuracy an underground structure under more general conditions of the epicenter locations.

Shin Aoi - One of the best experts on this subject based on the ideXlab platform.

  • boundary shape Waveform inversion for two dimensional basin structure using three component array data of Plane Incident Wave with an arbitrary azimuth
    Bulletin of the Seismological Society of America, 1997
    Co-Authors: Shin Aoi, Tomotaka Iwata, Hiroyuki Fujiwara, Kojiro Irikura
    Abstract:

    We extend a new Waveform inversion scheme (Aoi et al. , 1995) for estimating underground structure with an irregular-shaped basement as a target to cases where Plane Waves with an arbitrary azimuth impinge on the structure, i.e., from a direction not necessarily perpendicular to the major axis of the structure. We proved the validity of this scheme by numerical experiments. We had already achieved the formulation and numerical experiments for the cases where an SH Wave impinges on a 2D basin structure and had shown that we could estimate the entire basin structure with seismic Waveforms from only a few surface stations by using whole Waveforms that include the surface Waves. However, when the epicenter is located out of the Plane including the observation stations, even the cases of 2D structure cannot be treated as a simple 2D ( SH or P-SV ) problem because of the Wave with an azimuth that is not 0°. Therefore, by dealing with 3D Wave fields in the present study, we extend the inversion scheme in order to apply it to Incident Waves with an arbitrary azimuth. The differential seismograms, which represent the sensitivity of change in the Waveform, show different patterns in three components, and we demonstrate that inversion with three components, compared with the inversion with only one of them, leads to a linearized equation system with a smaller condition number and a more stable computation. Furthermore, we detect certain parts that are estimated with much less difficulty than others, depending on the direction from which the Incident Wave impinged. In the latter case, we can estimate the entire structure by employing simultaneously several data from Incident Waves arriving from different directions. We thus demonstrate by numerical experiments that the extension of our inversion method to cases where the Incident Wave with an arbitrary azimuth impinges on the structure enables us to estimate with increased accuracy an underground structure under more general conditions of the epicenter locations.

Hiroyuki Fujiwara - One of the best experts on this subject based on the ideXlab platform.

  • boundary shape Waveform inversion for two dimensional basin structure using three component array data of Plane Incident Wave with an arbitrary azimuth
    Bulletin of the Seismological Society of America, 1997
    Co-Authors: Shin Aoi, Tomotaka Iwata, Hiroyuki Fujiwara, Kojiro Irikura
    Abstract:

    We extend a new Waveform inversion scheme (Aoi et al. , 1995) for estimating underground structure with an irregular-shaped basement as a target to cases where Plane Waves with an arbitrary azimuth impinge on the structure, i.e., from a direction not necessarily perpendicular to the major axis of the structure. We proved the validity of this scheme by numerical experiments. We had already achieved the formulation and numerical experiments for the cases where an SH Wave impinges on a 2D basin structure and had shown that we could estimate the entire basin structure with seismic Waveforms from only a few surface stations by using whole Waveforms that include the surface Waves. However, when the epicenter is located out of the Plane including the observation stations, even the cases of 2D structure cannot be treated as a simple 2D ( SH or P-SV ) problem because of the Wave with an azimuth that is not 0°. Therefore, by dealing with 3D Wave fields in the present study, we extend the inversion scheme in order to apply it to Incident Waves with an arbitrary azimuth. The differential seismograms, which represent the sensitivity of change in the Waveform, show different patterns in three components, and we demonstrate that inversion with three components, compared with the inversion with only one of them, leads to a linearized equation system with a smaller condition number and a more stable computation. Furthermore, we detect certain parts that are estimated with much less difficulty than others, depending on the direction from which the Incident Wave impinged. In the latter case, we can estimate the entire structure by employing simultaneously several data from Incident Waves arriving from different directions. We thus demonstrate by numerical experiments that the extension of our inversion method to cases where the Incident Wave with an arbitrary azimuth impinges on the structure enables us to estimate with increased accuracy an underground structure under more general conditions of the epicenter locations.