Spherical Harmonic Expansion Research Articles

Real-time sound pressure interpolation around a rigid sphere using IIR filters based on spherical harmonic expansion

Real-time sound pressure interpolation around a rigid sphere using IIR filters based on spherical harmonic expansion

Small bodies global gravity inversion via the level-set method

We propose an approach to infer large-scale heterogeneities within a small celestial body from measurements of its gravitational potential, provided for instance by spacecraft radio-tracking. The non-uniqueness of the gravity inversion is here mitigated by limiting the solutions to piecewise-constant density distributions, thus composed of multiple regions of uniform density (mass anomalies) dispersed in a background medium. The boundary of each anomaly is defined implicitly as the 0-level surface of a scalar field (called the level-set function), so that by modifying this field the shape and location of the anomaly are varied. The gravitational potential associated with a density distribution is here computed via a line-integral polyhedron method, yielding the coefficients of its spherical harmonics expansion. The density distribution is then adjusted via an iterative least-squares approach with Tikhonov regularization, estimating at every iteration corrections to the level-set function, the density contrast of each anomaly, and the background density, in order to minimize the residuals between the predicted gravity coefficients and those measured. Given the non-convexity of the problem and the lack of prior knowledge assumed (save for the shape of the body), the estimation process is repeated for several random initial distributions, and the resulting solutions are clustered based on global properties independent of the input measurements. This provides families of candidate interior models in agreement with the data, and the spread of the local density values across each family is used to assess the uncertainties associated with the estimated distributions. We present multiple synthetic tests with increasingly more realistic settings (in terms of gravity resolution and precision, and of shape, size and distribution of the internal heterogeneities), showing that our method is generally able to retrieve a ground-truth mass distribution even with noisy data. For further validation, we present an application of the method to real data, namely the Bennu gravity coefficients measured by the OSIRIS-REx team.

Filtered Lebedev quadrature method for robust and efficient beam shape coefficient estimation in acoustic tweezers calibration.

Acoustic tweezers offer a contactless, three-dimensional, and selective approach to trapping objects by harnessing the acoustic radiation force. Precise control of this technique requires accurate calibration of the force, which depends on the object's properties and the spherical harmonics expansion of the incident field through the beam shape coefficients. Previous studies showed that these coefficients can be determined using either the Lebedev quadrature or the angular spectrum methods. However, the former is highly susceptible to noise, while the latter demands extensive implementation time due to the number of required measurement points. A filtered method with a reduced number of points is introduced to address these limitations. Initially, we emphasize the implicit filtering in the angular spectrum method, allowing relative noise insensitivity. Subsequently, we present its unfiltered version, enabling force estimation of a standing field. Finally, we develop a filtered method based on the Lebedev quadrature, requiring fewer points, and apply it to focused vortex beams. Numerical evaluation of the radiation force demonstrates the method's resilience to noise and a reduced need for points compared to previous methods. The filtered Lebedev method paves the way for characterizing high-frequency acoustic tweezers, where measurement constraints necessitate rapid and robust beam shape coefficient estimation techniques.

Evaluation of virtual sensing with spherical array surrounding head for directional noise

Active noise control (ANC) is expected to be applicable to public transportation modalities such as airplanes and trains. In these application systems, passengers may need to avoid wearing devices such as earphones and ear microphones. However, in conventional ANC systems, it is necessary to locate a physical microphone at the ear positions as an error sensor. To overcome this problem, virtual sensing techniques have been studied for estimating sound pressures at ear positions using remote microphones. In this study, we investigated the performance of a virtual sensing method for interpolating the sound pressures at ear positions using a spherical microphone array surrounding the head, based on a spherical harmonic expansion. The experiments were conducted using a rigid sphere instead of a head and dodecahedral spherical microphone array surrounding the sphere. The estimation errors were evaluated for various noise directions at low frequencies (100, 300, 500, and 700 Hz). The experimental results showed that the estimation errors were less than -20 dB for almost all conditions below 500 Hz. By comparison with other virtual sensing methods using a few microphones in the vicinity of the ear positions, we confirmed that our proposed method was effective for any noise direction.

Time domain virtual sensing method based on a rigid-sphere transfer function for active noise control headrests

Active noise control (ANC) is expected to be used in public transportation such as airplanes and trains. While using these applications, passengers prefer not to wear earphones or ear microphones, if possible, to reduce ear strain. However, conventional ANC systems require a physical microphone, which acts as an error sensor, at the ear position. To address this issue, a virtual sensing technique has been studied that uses a microphone mounted on a seat’s headrest to estimate the sound pressure at the ear position. In this study, we propose a virtual sensing technique that interpolates the sound pressure at the ear position using a spherical microphone array surrounding the head. This method improves robustness to variations in noise arrival direction using spherical harmonic coefficients without directional dependence. In general, the interpolation formula using spherical harmonic expansion is obtained in the frequency domain. By contrast, the proposed method designs the interpolation process as an IIR filter by considering the minimum phase and estimates the noise signal directly in the time domain, achieving the real-time processing required for ANC. The effectiveness of the proposed method in reducing noise was investigated for various noise arrival directions.

Generating 3-D auditory stimuli for the evaluation of spatial impression in musical instruments

Spatial hearing plays an important, but not well understood, role in the perception of musical instruments and audio equipment. Psychophysical experiments are necessary to elucidate the processes which convey the spatial impression of sound sources. To this end, we propose a way to generate virtual stimuli for research on the spatial impression of musical instruments, specifically pianos. Our proposal, based on acoustical holography, takes a series of measurements over a dense grid and designs a virtual vibrating plate with the same directivity as the piano, as observed from the measuring grid. The number of active modes in the left-right and front-back directions of this virtual source can be controlled to reduce the radiation pattern complexity and generate a wide range of virtual stimuli. The resulting sound fields are rendered using binaural Ambisonics, using the position of the piano player as the spherical harmonic expansion center. The proposal was used in actual psychophysical experiments looking to shed light into the spatial impression of digital and acoustic pianos. The results show that the proposal does generate spatial sound signals which are consistently associated with different degrees of richness in the spatial impression.

The Upper Mantle Geoid for Lithospheric Structure and Dynamics

AbstractGeoid anomalies offer crucial information on the internal density structure of the Earth, and thus, on its constitution and dynamic state. In order to interpret geoid undulations in terms of depth, magnitude and lateral extension of density anomalies in the lithosphere and upper mantle, the effects of lower mantle density anomalies need to be removed from the full geoid (thus obtaining the residual “upper mantle geoid”). However, how to achieve this seemingly simple filtering exercise has eluded consensus for decades in the solid Earth community. While there is wide agreement regarding the causative masses of degrees >10 in spherical harmonic expansions of the upper mantle geoid, those contributing to degrees <7–8 remain ambiguous. Here we use spherical harmonic analysis and recent tomography and density models from joint seismic‐geodynamic inversions to derive a representative upper mantle geoid, including the contributions from low harmonic degrees. We show that the upper mantle geoid contains important contributions from degrees 5 and 6 and interpret the causative masses as arising from the coupling between the long‐wavelength lithospheric structure and the sublithospheric upper mantle convection pattern. Importantly, the contributions from degrees 3 < l < 8 do not show a simple power‐law behavior (e.g., Kaula's rule), which precludes the use of standard filtering techniques in the spectral domain. Our upper mantle geoid model will be useful in studies of (a) lithospheric structure, (b) dynamic topography and mantle viscosity, (c) lithosphere‐asthenosphere interactions and (d) the global stress field within the lithosphere and its associated hazards.

Research of integrated shimming Halbach magnet for High strength, compact Benchtop NMR device

Due to production and assembly errors of magnets resulting in reduced magnetic field homogeneity, passive shimming (PS) is necessary when Halbach magnets are used in high-resolution Benchtop Nuclear Magnetic Resonance (BNMR) spectrometer. The conventional PS technique, which places independent PS devices inside the magnet aperture, is no longer applicable to small-aperture compact high-field-strength Halbach magnet studies. In this paper, based on spherical harmonic function expansion, we improve the magnetic field homogeneity by optimizing the moving step of the magnet moving arrays composed of Halbach magnets to generate the corresponding harmonic terms to compensate for the magnetic field. With this approach, the homogeneity of a 1 T Halbach magnet was improved from the original 3913 ppm to 8 ppm in a L10 mm × R2.5 mm of 0.64% copper sulfate doped water sample. This work explores the PS mechanism based on the movement of magnetic blocks, which can be applied in BNMR and other compact high-field strength high-homogeneity Halbach magnets application circumstances.

Recovering crustal thickness of the Moon from an extended Vening Meinesz-Moritz inversion

Crustal thickness (Moho surface relief) is closely related to geotectonic, which is the predominant research topic in lunar geophysics. For lunar crustal depth inversion, to address the problem of downward continuation in traditional spectral domain methodologies, we adopt a solution from gravity disturbance (geophysical gravity anomaly). It is available to investigate crustal thickness at arbitrary surface outside the geoid. Using gravity data from the Gravity Recovery and Interior Laboratory (GRAIL) mission and terrain information recovered from the Lunar Orbiter Laser Altimeter (LOLA) mission, global crustal depth of the Moon is recovered by the extended Vening Meinesz-Moritz method. The first application result of this technique on the Moon is validated by four types of released models in the form of spherical harmonic expansion with 310°. Moreover, from the statistics, our lunar Moho solution demonstrates that the average crustal thickness is approximately 41.5 km. The maximum value of 83.1 km and the minimum value of 0.8 km are located in the area of high mountains and Moscoviense mascon, respectively.

A Fast and Super-Resolution Method of Vortex-Based Imaging

With characteristics of particular wavefronts and orthogonal modes, the sparse imaging method based on vortex waves could obtain high-resolution reconstruction. However, building a large-scale observation matrix is time-consuming and heavily burdens the whole imaging procedure. Inspired by the spherical harmonic expansion of plane waves, this paper provides a fast super-resolution imaging method based on vortex waves. Featuring spherical harmonic functions and a transformation, an orthogonal matching pursuit algorithm for high-resolution imaging exploits a low-rank matrix decomposition of the observation matrix. Simulation results show that the proposed method exhibits significant efficiency advantages while retaining high resolution, achieving an efficiency of up to 51.4% when applied to a large-scale observation matrix.

Prospects for detecting anisotropies and polarization of the stochastic gravitational wave background with ground-based detectors

We build an analytical framework to study the observability of anisotropies and a net chiral polarization of the Stochastic Gravitational Wave Background (SGWB) with a generic network of ground-based detectors. We apply this formalism to perform a Fisher forecast of the performance of a network consisting of the current interferometers (LIGO, Virgo and KAGRA) and planned third-generation ones, such as the Einstein Telescope and Cosmic Explorer. Our results yield limits on the observability of anisotropic modes, spanning across noise- and signal-dominated regimes. We find that if the isotropic component of the SGWB has an amplitude close to the current limit, third-generation interferometers with an observation time of 10 years can measure multipoles (in a spherical harmonic expansion) up to ℓ = 8 with 𝒪(10-3 – 10-2) accuracy relative to the isotropic component, and an 𝒪(10-3) amount of net polarization. For weaker signals, the accuracy worsens as roughly the inverse of the SGWB amplitude.

Quantifying molecular deformation in polymer melts by a generalized Zimm plot approach

The Zimm plot has been widely used to characterize the molecular dimensions of polymers from small-angle scattering experiments, where the reciprocal intensity is analyzed as a function of the square of the magnitude of the scattering wavevector Q. This work explores the benefits of analyzing the reciprocal scattering intensity from deformed polymers, extending the original Zimm plot to anisotropic materials. In the small-angle limit, a tensorial extension of the Guinier law is found for the gyration tensor and the reciprocal single-chain structure factor. In the high-Q limit, application of the spherical harmonic expansion technique to the reciprocal structure factor permits direct model-independent analysis of spatially dependent molecular deformation of polymers. Additionally, the contributions from high-order spherical harmonics become insignificant in the reciprocal-intensity representation. The proposed generalized Zimm plot approach is demonstrated computationally with the affine deformation model and the Rouse model, and experimentally with small-angle neutron scattering measurements of deformed polystyrene melts.

Theoretical investigation on the underwater sound absorption of a functionally graded anechoic coating under the phased arc array incidence

Laying anechoic coatings on the surface of underwater vehicles is an effective technology to improve the acoustic stealth performance. Meanwhile, the detection waves may have complex directivity, and it is important to study their influence on the sound absorption of the coating. This article introduces a functionally graded coating under a phased arc array incidence, which simulates a novel anechoic coating on an underwater vehicle to absorb sound waves emitted by active sonars. The incident field of the phased array is constructed by spherical harmonic expansion and the superposition principle. The wave propagation of the functionally graded coating is analyzed through an equivalent laminated model and the reverberation matrix method. Utilizing the angular spectrum method, the reflected sound field and absorption coefficient of the coating under the phased arc array incidence are predicted and the theoretical method is formed. Then the selections of parameters in Fourier transforms of the angular spectrum method are discussed for achieving converged results. A finite element analysis is applied to verify the effectiveness and calculation speed of the proposed method through the establishment of detailed quarter or half model. The effect of phased arc array parameters and the coating on the absorption and reflection performances are further studied. The angular spectrum decomposition of the incident sound field explains the changing trends of the sound absorption curves. The present work provides an important way to deal with this complex underwater sound propagation problem.

Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles

AbstractRecent advances in magnetic microscopy have enabled studies of geological samples whose weak and spatially nonuniform magnetizations were previously inaccessible to standard magnetometry techniques. A quantity of central importance is the net magnetic moment, which reflects the mean direction and the intensity of the magnetization states of numerous ferromagnetic crystals within a certain volume. The planar arrangement of typical magnetic microscopy measurements, which originates from measuring the field immediately above the polished surface of a sample to maximize sensitivity and spatial resolution, makes estimating net moments considerably more challenging than with spherically distributed data. In particular, spatially extended and nonuniform magnetization distributions often cannot be adequately approximated by a single magnetic dipole. To address this limitation, we developed a multipole fitting technique that can accurately estimate net moment using spherical harmonic multipole expansions computed from planar data. Given that the optimal location for the origin of such expansions is unknown beforehand and generally unconstrained, regularization of this inverse problem is critical for obtaining accurate moment estimates from noisy experimental magnetic data. We characterized the performance of the technique using synthetic sources under different conditions (noiseless data, data corrupted with simulated white noise, and data corrupted with measured instrument noise). We then validated and demonstrated the technique using superconducting quantum interference device microscopy measurements of impact melt spherules from Lonar crater, India and dusty olivine chondrules from the CO chondrite meteorite Dominion Range 08006.

A modal expansion method for simulating reverberant sound fields generated by a directional source in a rectangular enclosure

The prediction of reverberant sound fields generated by a directional source is of great interest because practical sound sources are not omnidirectional, especially at high frequencies. For an arbitrary directional source described by cylindrical and spherical harmonics, this paper developed a modal expansion method for calculating the reverberant sound field generated by such a source in both two-dimensional and three-dimensional rectangular enclosures with finite impedance walls. The key is to express the modal source density using the cylindrical or spherical harmonic expansion coefficients of the directional source. A method based on the fast Fourier transform is proposed to enable the fast computation of the summation of enclosure modes when walls are lightly damped or rigid. This makes it possible to obtain accurate reverberant sound fields even in a large room and/or at high frequencies with a relatively low computational load. Numerical results with several typical directional sources are presented. The efficiency and the accuracy of the proposed method are validated by the comparison to the results obtained using the finite element method.

On the efficient simulation of pass-by noise signals from railway wheels

The article presents an approach for calculating pass-by sound pressure radiated from railway wheels in the time domain using moving Green’s functions. The Green’s functions are obtained by using Finite Element (FE) and Boundary Element (BE) methods in the frequency domain, subsequent inverse Fourier transform, followed by convolution with a time series of rolling contact forces to obtain the pass-by time signals. However, traditional BE methods are computationally expensive due to the low structural damping of the wheel, necessitating a high frequency resolution.To overcome this issue, a modal approach is introduced in which the pass-by sound radiated by each wheel mode is calculated separately. This approach incorporates the dynamic response of the wheel in the time-domain processing and thus reduces the cost of the BE solution. A modal source signal is introduced to describe the excitation of each mode at each time step. The sound field radiated by unit modal amplitudes is calculated in BE and subsequently approximated by spherical harmonic (SH) equivalent sources, which allows for efficiently calculating acoustic transfer functions for varying relative positions of the wheel and a stationary receiver. Convolution of the source signal with the moving acoustic transfer function produces the pass-by pressure signal.The article investigates the directivity of the radiation from each mode and finds that most modes, including those with dominant radial deflection, radiate in mostly axial direction at high frequencies. Modes that dominate the pass-by pressure level are identified, both in frequency bands and with respect to the relative positioning of the wheel to the receiver. Finally, it is found that an SH expansion order of approximately 30 is required to satisfy the employed error measures, although lower orders may suffice for an auralisation of the signal.

Incorporation of anisotropic scattering into Fourier moment expanded axial DGFEM SN based 2D/1D coupling method

Fourier moment expanded axial DGFEM (Discontinuous Galerkin Finite Element Method) SN based 2D/1D coupling method is a promising variant for the pin-by-pin neutron transport calculation, due to its excellent computational efficiency and stability. In this study, the incorporation of anisotropic scattering based on spherical harmonic expansion is developed for the axial DGFEM SN based 2D/1D transport approach, to further capture the anisotropic scattering accurately for the heterogeneous fuel rod or strong absorber. Several well-known benchmarks are used to evaluate its computational performance. Numerical results shows that the implementation of anisotropic scattering can significantly improve the accuracy of 2D/1D coupling method. Moreover, compared with the traditional 1D SN based 2D/1D method, the newly developed method could achieve high computational efficiency and convergence rate, as well as a relatively low storage requirement.

External gravitational field of a homogeneous ellipsoidal shell: a reference for testing gravity modelling software

There are numerous applications in geodesy and other geo-sciences in which the gravitational potential effect or other functions of the potential are computed by forward modelling from a given mass distribution. Different volume discretisations, e.g. prisms, tesseroids or mass layers are used. In order to control the numerical realisation of the forward calculation in the practical application, e.g. in reduction tasks, these evaluation programs should be verified against rigorous analytical solutions. In this contribution, a closed analytical solution for the potential of an ellipsoidal shell as a test body is presented. Furthermore, we derive the respective closed formulae for the gravity vector and the gravity gradient tensor. Program implementations of the tesseroid approach are compared on the basis of this ellipsoidal mass arrangement. For the practical usage, fast-converging expansions in spherical harmonics are provided in addition. The derivation of the formulae is based on a closed solution of the potential of a homogeneous ellipsoid for computation points situated on the rotation axis, which then is extended to the external space.

A spectral boundary integral method for simulating electrohydrodynamic flows in viscous drops

A weakly conducting liquid droplet immersed in another leaky dielectric liquid can exhibit rich dynamical behaviors under the effect of an applied electric field. Depending on material properties and field strength, the nonlinear coupling of interfacial charge transport and fluid flow can trigger electrohydrodynamic instabilities that lead to shape deformations and complex dynamics. We present a spectral boundary integral method to simulate droplet electrohydrodynamics in a uniform electric field. All physical variables, such as drop shape and interfacial charge density, are represented using spherical harmonic expansions. In addition to its exponential accuracy, the spectral representation affords a nondissipative dealiasing method required for numerical stability. A comprehensive charge transport model, valid under a wide range of electric field strengths, accounts for charge relaxation, Ohmic conduction, and surface charge convection by the flow. A shape reparametrization technique enables the exploration of significant droplet deformation regimes. For low-viscosity drops, the convection by the flow drives steep interfacial charge gradients near the drop equator. This introduces numerical ringing artifacts that we treat via a weighted spherical harmonic expansion, resulting in solution convergence. The method and simulations are validated against experimental data and analytical predictions in the axisymmetric Taylor and Quincke electrorotation regimes.

A novel GNSS precise point positioning method based on spherical harmonic expansion

A novel GNSS precise point positioning method based on spherical harmonic expansion