Multiple Multipole Expansions Research Articles

Spectroscopic Optical Coherence Tomography by Using Multiple Multipole Expansion

This paper presents a pre-processing method to remove multiple scattering artifacts in spectroscopic optical coherence tomography (SOCT) using time–frequency analysis approaches. The method uses a multiple multipole expansion approach to model the light fields in SOCT. It is shown that the multiple scattered fields can be characterized by higher order terms of the multiple multipole expansion. Hence, the multiple scattering artifact can thus be eliminated by applying the time–frequency transform on the SOCT measurements characterized by the lower order terms. Simulation and experimental results are presented to show the effectiveness of the proposed pre-processing method.

A method to measure the equivalent acoustic impedance of a scatterer

The procedures of simplifying scatterers’ geometry and finding its equivalent acoustic impedance are proposed by using scattering holography and multiple multipole expansions. The scattered sound field depends on the incident field, the geometry, and the physical property of the scatterer, which can be measured by using tools such as a set of microphones and a 3‐D scanner. Once the sound field over a surface is measured, the whole sound field of 3‐D space can be reconstructed by acoustic holography. However, predicting near‐field sound by holography is inefficient in practice, because huge numbers of measurement points are required to describe the complex sound field in the vicinity of a scatterer. With this regard, multiple multipole expansions are employed to reduce the number and the size of elements of a scatterer necessary to express the measured scattered field. Therefore it is equivalent to simplifying the geometry of a scatterer, and enables us to predict the sound pressure and velocity, as well as acoustic impedance, on the scatterers’ surface. Using the proposed method, it is demonstrated that the equivalent acoustic impedance can be obtained. [Work supported by the BK21 project initiated by the Ministry of Education and Human Resources Development of Korea.]

Near Field Analysis of the Modal Scattering at Radiating Apertures

Summary A rigorous method for the determination of the modal scattering at radiating apertures is presented. Near-field boundary conditions to be satisfied on the outer surface of the conducting structure containing the aperture (e.g. a horn antenna) are completely taken into account. The resulting scattering matrix of the radiating aperture can be integrated into any mode-matching technique as a termination for the cascaded structure commonly used to match the aperture to the exciting waveguide. The method is based on a multiple-multipole expansion of the electromagnetic field outside the radiating object in conjunction with a point-matching technique. Horn antennas with elliptical and circular apertures have been analyzed using the suggested method. Excellent agreement with measurement results has been achieved for both the radiation pattern and the aperture scattering parameters.

Separation of upgoing and downgoing wavefields using the boundary element method

Summary The boundary element method has been applied for decomposing seafloor measurements into upgoing and downgoing waves. Two complementary solutions in the forms of a linear matrix system and a multiple multipole expansion are obtained without any assumptions about the subsurface. If the elastic parameters below the seafloor are known, these solutions are explicit. They do not require any knowledge of the source signature. Basic components—such as Green’s function computation—can be replaced by alternative acoustic or elastic solutions. Results are useful for deghosting, free-surface multiple attenuation and P/S splitting of ocean-bottom cable data.

Acousto-elastic multiple scattering: A comparison of ultrasonic experiments with multiple multipole expansions

Solid rods are submerged in a water tank and illuminated by an ultrasonic source transducer. Along the water surface, transducers are used to measure the propagating wave fields. Even for only three rods, the scattered wave field is very complex. Although the major events can be readily interpreted as direct reflections, the remaining wave field consists of complicated interactions of the wave fields with the scatterers and the water surface. To aid the interpretation, the experiment is modeled numerically using the multiple multipoles method which decomposes the wave fields by their respective origins. Hence, particular events can be correlated with particular scatterers as well as with particular paths of propagation. The numerical method has already been used as a versatile tool to model acoustic or elastic multiple-scattering problems where homogeneous scatterers were embedded in a homogeneous full space. In the present paper, the scheme is expanded to multiple scattering between solids submerged in a fluid. For each scatterer, the waves induced in the fluid are expressed by sets of multipole solutions with different origins. Thus by construction, the scattered wave fields are decomposed by scatterer. The numerical solutions match the ultrasonic experiments very well. Furthermore, the different events can be identified as reflections, internal multiples, or diffractions through the solid rods.

Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions

Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions

Formulations of the extended boundary condition method for incident gaussian beams using multiple-multipole expansions

Novel formulations for improving the numerical stability of the extended boundary conditions method and extending its application to highly elongated dielectric objects illuminated by a Gaussian beam are described. The derived formulations are obtained as special cases of a general approach, which considers arbitrary complete system functions on the particle surface. The surface currents are approximated by multiple spherical expansions or, for a fixed azimuthal mode, by the lowest-order multipoles located inside the particle surface. Results illustrating the application of the procedures developed are presented.

Multiple multipole expansions for elastic scattering

This paper presents a new approach to solve scattering of elastic waves in two dimensions. Traditionally, wave fields are expanded into an orthogonal set of basis functions. Unfortunately, these expansions converge rather slowly for complex geometries. The new approach enhances convergence by summing multiple expansions with different centers of expansions. This allows irregularities of the boundary to be resolved locally from the neighboring center of expansion. Mathematically, the wave fields are expanded into a set of nonorthogonal basis functions. The incident wave field and the fields induced by the scatterers are matched by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the nonorthogonal expansions, more matching points are used than actually needed, resulting in an overdetermined system which is solved in the least-squares sense. Since there are free parameters, such as location and number of expansion centers, as well as kind and orders of expansion functions used, numerical experiments are performed to measure the performance of different discretizations. An empirical set of rules governing the choice of these parameters is found from these experiments. The resulting algorithm is a very general tool to solve relatively large and complex two-dimensional scattering problems.

Scattering of acoustic and elastic waves using hybrid multiple multipole expansions—Finite element technique

In this paper, two different methods to solve scattering problems in acoustic or elastic media are coupled to enhance their usefulness. The multiple multipole (MMP) expansions are used to solve for the scattered fields in unbounded homogeneous regions. The finite element (FE) method is used to calculate the scattered fields in bounded, heterogeneous regions. Applying the boundary conditions, the different regions and methods are coupled together in the least squares sense as the MMP method requires. By construction, the scattered field in the homogeneous regions is decomposed into P and S modes for the elastic case. In some examples, the scattered fields are calculated and compared to the analytical solutions. Additionally, the seismograms are calculated for a few scattering problems with one or several heterogeneous scatterers of complex geometries. The hybrid MMP–FEM technique proves to be a rather general and useful tool to solve complex, two-dimensional scattering problems.

Wave scattering using a hybrid MMP–FE technique

Among the methods amendable to solve acoustic or elastic scattering problems are the finite elements method (FEM) and multiple multipole (MMP) expansions [M. G. Imhof, J. Acoust. Soc. Am. 97, 754–763 (1995)]. Among the difficulties the FEM encounters are long calculation times, large memory requirements, and the need for absorbing boundary conditions. On the other hand, the FEM is perfectly suited for heterogeneous media because it automatically accounts for changes in material properties. In contrast, the computational cost of MMP expansions is independent of the source–receiver geometry. An unbounded domain poses no difficulty since the expansion functions account automatically for the radiation condition. MMP expansions often converge with very few terms and therefore reduce the computational effort. Unfortunately, expansion functions for heterogeneous media are difficult to find. Scattering problems with heterogeneous scatterers embedded in a homogeneous background are difficult (FEM) or impossible (MMP) to solve with either of these techniques alone. Therefore, the MMP expansions are coupled with the FE method. For a cylindrical scattering problem, the resulting solution is compared to the analytical one. Results for multiple inhomogeneous scatterers embedded in a homogeneous media are presented.

Multiple multipole expansions for acoustic scattering

The paper presents a new approach to solve multiple scattering of acoustic waves in two dimensions. Traditionally wave fields are expanded into an orthogonal set of basis functions. Often these functions build a multipole. Unfortunately, these multipoles converge rather slowly for complex geometries. The new approach enhances convergence by including multiple multipoles into each region, allowing irregularities of the boundary to be resolved locally. The wave fields are expanded into a set of nonorthogonal basis functions. The incident wave field and the fields induces by the scatterers are matched in the least-square sense by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the nonorthogonal expansions we choose more matching points than actually needed resulting in an overdetermined system which is solved in the least-squares sense. This allows an estimate of how well the expansion converges and can help to tune the scheme to enhance accuracy or reduce runtime. The idea of the multiple multipole can be extended by using more complicated basis functions which are closer to the solution sought. The resulting algorithm is a very general tool to solve relatively large and complex two-dimensional scattering problems.