Let D be a bounded, simply connected domain in the plane and let $F( {\theta ;k,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l } )$ be the far field pattern arising from the scattering of a time harmonic, acoustic plane wave $u_l ( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} } ) = \exp ( {ik\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l } )$, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l $ is a unit vector and k is the wave number and the time harmonic factor $e^{ - i\omega t} $ has been factored out. It is assumed, in addition, that the total field satisfies homogeneous Dirichlet boundary conditions on $\partial D$. In this paper, a method is presented for recovering $\partial D$ given the far field patterns $F( {\theta ;k,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l ...

The limited aperture problem of inverse acoustic scattering: dirichlet boundary conditions

Data recovery in inverse scattering: From limited-aperture to full-aperture

Kirchhoff data mapping (KDM) is a procedure for transforming data from a given input source/receiver configuration and background earth model to data corresponding to a different output source/receiver configuration and background model The generalization of NMO/DMO, datuming and offset continuation are three examples of KDM applications This paper describes a ‘platform’ for KDM for scalar wavefields The word, platform, indicates that no calculations are carried out in this paper that would adapt the derived formula to any one of a list of KDMs that are presented in the text Platform formulae are presented in 3D and in 25D For the latter, the validity of the platform equation is verified — within the constraints of high-frequency asymptotics — by applying it to a Kirchhoff approximate representation of the upward scattered data from a single reflector and for an arbitrary source/receiver configuration The KDM formalism is shown to map this Kirchhoff model data in the input source/receiver configuration to Kirchhoff data in the output source/receiver configuration, with one exception The method does not map the reflection coefficient Thus, we verify that, asymptotically, the ray theoretical geometrical spreading effects due to propagation and reflection (including reflector curvature) are mapped by this formalism, consistent with the input and output modelling parameters, while the input reflection coefficient is preserved In this sense, this is a ‘true-amplitude’ formalism As with earlier Kirchhoff inversion, a slight modification of the kernel of KDM provides alternative integral operators for estimating the specular reflection angle, both in the input configuration and in the output configuration, thereby providing a basis for amplitude-versus-angle analysis of the data

A platform for Kirchhoff data mapping in scalar models of data acquisition

We propose a three-step solution methodology to increase the discrete set of acoustic far-field pattern (FFP) measurements, available in a small range of observation angles (small aperture). The first two steps of the proposed procedure allow the extension of the data to an aperture larger than !=2. They use a regularized Newton algorithm where the total variation of the FFP is incorporated as a regularization term. The third step consists in applying the standard Tikhonov regularization technique to recover the full aperture of the FFP from the previously extended field. Numerical results obtained using synthetic data illustrate the potential of the proposed procedure for reconstructing the full aperture of the FFP from data given in an aperture as small as backscattering measurements.

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A Multi-Step Procedure for Enriching Limited Two-Dimensional Acoustic Far-Field Pattern Measurements.

We are concerned with the inverse scattering problems associated with incomplete measurement data. It is a challenging topic of increasing importance in many practical applications. Based on a prototypical working model, we propose a machine learning based inverse scattering scheme, which integrates a CNN (convolution neural network) for the data retrieval. The proposed method can effectively cope with the reconstruction under limited-aperture and/or phaseless far-field data. Numerical experiments verify the promising features of our new scheme.

Machine learning based data retrieval for inverse scattering problems with incomplete data

: The limited aperture problem of physical inverse scattering is examined i.e., the problem of identifying a target from an analysis of band- limiting viewing apertures. The target may be completely identified from an analysis of high frequency, bandlimited data. If the directions of viewing angles are limited as well, it is shown that the target surface can be identified where the target normal lies within the range of viewing directions; outside of this range, no information is available. It is shown that this phenomenon is totally a feature of the Fourier transform of characteristic (one- zero) functions and independent of the inverse scattering formalism. Numerical examples are given for the case of a perfectly reflecting circular cylinder.

Robert D. Mager

Papers

An approach to the limited aperture problem of physical optics Farfield inverse scattering