Surface Spherical Harmonics Research Articles

A Linear Operator Method to Compute the Normal Modes with Rotation of any Asymmetric 3D Planet with Pure Vector Spherical Harmonics

In order to compute the free core nutation of the terrestrial planets, such as Earth and Mars, the Moon and lower degree normal modes of the Jovian planets, we propose a linear operator method (LOM). Generalized surface spherical harmonics (GSSHs) are usually applied to the elliptical models with a stress tensor, which cannot be expressed in vector spherical harmonics explicitly. However, GSSHs involve complicated math. LOM is an alternative to GSSHs, whereas it only deals with the coupling fields of the same azimuthal order m, as is the case when a planet model is axially symmetric and rotates about that symmetry axis. We extend LOM to any asymmetric 3D model. The lower degree spheroidal modes of the Earth are computed to validate our method, and the results agree very well with what is observed. We also compute the normal modes of a two-layer Saturn model as a simple application.

Symmetric spherical surface harmonics for predicting average pitting corrosion susceptibility index of carbon steels from EBSD microtexture measurements

This work presents a study on the correlation between the crystallographic texture and the pitting corrosion susceptibility of API 5L steels. The crystallographic textures of API 5L steel samples (codified as S1, S2, S3, and S4) were studied using EBSD measurements. A deterministic model based on symmetric spherical surface harmonics was proposed to describe the strong dependence of the pitting corrosion susceptibility on the crystallographic (grains) orientation of the (body-centered cubic) material. Using the model, pitting corrosion susceptibility index values estimated from EBSD raw data agreed in relative magnitudes (from sample to sample) with the predictions made from X-ray macrotexture measurements, and the results derived from other techniques such as cyclic polarization tests and analyses of SEM and AFM images. This agreement verified the capacity of EBSD to estimate the pitting corrosion susceptibility indices of carbon steels having different microstructures and textures, which were found to be 0.60, 0.53, 0.51, and 0.44 for S1, S2, S3, and S4, respectively. It was established that the higher the values of pitting corrosion susceptibility indices, the higher the tendency to present pitting corrosion of the material; thus, S1>S2>S3>S4. Moreover, the results have shown that the crystallographic texture plays a more decisive role in the behaviour of pitting corrosion in carbon steels compared to other microstructural and chemical parameters such as the grain size and inclusion content. The proposed (deterministic) model could be a novel tool for predicting pitting corrosion of the material, forming part of a novel method for the material selection process.

Incorporation of anisotropic scattering into the method of characteristics

In this study, we incorporate an anisotropic scattering scheme involving spherical harmonics into the method of characteristics (MOC). The neutron transport solution in a light water reactor can be significantly improved because of the impact of an anisotropic scattering source with the MOC flat source approximation. Several problems are selected to verify the proposed scheme and investigate its effects and accuracy. The MOC anisotropic scattering source is based on the expansion of spherical harmonics with Legendre polynomial functions. The angular flux, scattering source, and cross section are expanded in terms of the surface spherical harmonics. Later, the polynomial is expanded to achieve the odd and even parity of the source components. Ultimately, the MOC angular and scalar fluxes are calculated from a combination of two sources. This paper presents various numerical examples that represent the hot and cold conditions of a reactor core with boron concentration, burnable absorbers, and control rod materials, with and without a reflector or baffle. Moreover, a small critical core problem is considered which involves significant neutron leakage at room temperature. We demonstrate that an anisotropic scattering source significantly improves solution accuracy for the small core high-leakage problem, as well as for practical large core analyses.

The analysis of structural magnetic resonance imaging manifestation of hippocampus based on voxel and spherical harmonic surface morphometry in mesial temporal lobe epilepsy

Objective: To compare the hippocampal volume and local surface morphology changes in patients with mesial temporal lobe (mTLE) using the voxel-based morphometry and spherical harmonic methods respectively. Methods: A total of 66 patients (31 males and 35 females, age range from 17 to 48 (28±8) years) with mTLE and 80 age-and gender-matched controls (38 males and 42 females, age range from 19 to 46 (27±7) years) were retrospectively collected from July 2009 to February 2019 at Jinling hospital.. High resolution structural MRI of the whole brain, three-dimensional T1-weighted data(3DT1) were acquired from each subject. The changes of hippocampal volume and surface morphology were evaluated between mTLE groups and controls for observing the hippocampal atrophy pattern by using voxel-based morphometry and spherical harmonic shape descriptions point distribution model respectively. Pearson correlation analysis was conducted for observing the relationship between the morphological changes of hippocampus and disease duration. Results: Compared with the controls, hippocampal volume on the affected side in patients with mTLE was significantly reduced (Z-score:-1.55±0.57 vs 0.38±0.58, P<0.001) and negatively correlated with disease duration (r=-0.297, P=0.016). Furthermore, surface morphology analysis subtly showed that the atrophy of the affected hippocampus in patients with mTLE mainly located in the head, mesial lateral part and posterior tail of the hippocampus. Their displacement values were negatively correlated with disease duration (r=-0.336, P=0.006) and positively associated with the hippocampal grey matter volume (r=0.336, P=0.006). Conclusions: Voxel-based morphometry analysis reveals a global reduction in hippocampal volume, while the morphological measurement method based on surface shape can describe the local morphological changes of hippocampal atrophy.

Double-thin-shell approach to deriving total electron content from GNSS signals and implications for ionospheric dynamics near the magnetic equator

A new technique was developed to estimate the ionospheric total electron content (TEC) from Global Navigation Satellite System (GNSS) satellite signals. The vertically distributed electron density was parameterized by two thin-shell layers (double-shell approach). The spatiotemporal variation of TEC (strictly speaking, partial electron content) associated with each shell was approximated by the functional fitting of spherical surface harmonics. The major improvements over the conventional single-shell approach were as follows: (1) the precise estimation of TEC was achieved; (2) the estimated TEC was less dependent on the choice of shell heights; and (3) the equatorial anomaly was captured more correctly. Furthermore, higher and lower shells exhibited a different pattern of local time vs latitude variation, providing information on the ionosphere–thermosphere dynamics.

Quantifying the individual contributions of GOCE gradients for regional quasi-geoid modeling in a relatively rough topography

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that measures the gravity gradients (GGs) in space with primary objective of determining the gravity field of the Earth with maximum spatial resolution and accuracy. GOCE GGs have been used for the computation of global gravity field models (GGMs) represented by the surface spherical harmonics for about 10 years. However, they may potentially improve the regional quasi-geoid solutions computed from terrestrial gravity and a Global Geopotential Model (GGM) particularly in mountainous areas with rough gravity field. This study exploits the full signal content in the GOCE Level-2 GGs given in Local North-Oriented Frame for improved regional quasi-geoid modeling in Auvergne test area located in France. The least squares collocation method is used in single-input and single-output (SISO) mode where the input is the residual terrestrial gravity data corrected for GGM contribution and residual terrain modeling effects, and also in multiple-input and single-output mode (MISO) where the terrestrial data input is augmented with the four precise GOCE GGs or their combinations. The single outputs for both modes are the quasi-geoid heights at 75 GPS collocated leveling benchmarks distributed in the central area and at 7857 Digital Elevation Model (DEM) points covering the whole area. The combination of the terrestrial gravity with the GOCE along-track gradient, having a lower noise level than the others, gives the best results when a single gradient is used. The further incorporation of the off-diagonal gradient to the along-track gradient results in the best MISO solution showing improvements with respect to the SISO solution only at GPS/leveling benchmarks near a large data gap. Intercomparison of the best MISO and SISO solutions at the DEM points suggests substantial quasi-geoid differences up to 13.6 cm in areas of sparse or no gravity data near the Alps where these results could not be assessed due to the lack of GPS-leveling data.

Crustal density and global gravitational field estimation of the Moon from GRAIL and LOLA satellite data

We employ Newton's integral in the spectral domain to solve two geodetic/geophysical tasks for the Moon. Firstly, we determine 3D bulk density distribution within the lunar crust (inverse problem). For this purpose, we develop a linear mathematical model that parameterises the laterally variable density component by surface spherical harmonics. We exploit GL1500E GRAIL gravitational field model and LOLA topography model to determine bulk density in three types of function: 1) constant, 2) laterally variable, and 3) 3D spatially variable (assuming a linear change in the radial direction). Secondly, we calculate lunar gravitational field models inferred by these three crustal compositions (forward problem) up to spherical harmonic degree 2519 corresponding to a spatial resolution of ∼2.2 km at the lunar equator. Efficacy of these models is assessed with respect to the GRAIL Level 2 gravitational field models. Our spatially variable crustal model represents the best fit globally and also locally in highland areas. We also test the performance of GRAIL models, recent and independent forward models, and our new models against Level 1B GRAIL satellite-to-satellite tracking data focusing on evaluation beyond Level 2 data (i.e., spherical harmonic degrees greater than 650). These medium- and high-frequency signals from our models correlate with the Level 1B observations the best among all global gravitational field models tested. Our high resolution geopotential model with the optimized 3D crustal density variation should be an asset to future lunar lander navigation and geophysical exploration.

Modeling the magnetized Local Bubble from dust data

The Sun is embedded in the so-called Local Bubble (LB) – a cavity of hot plasma created by supernova explosions and surrounded by a shell of cold, dusty gas. Knowing the local distortion of the Galactic magnetic field associated with the LB is critical for the modeling of interstellar polarization data at high Galactic latitudes. In this his paper, we relate the structure of the Galactic magnetic field on the LB scale to three-dimensional (3D) maps of the local interstellar medium (ISM). First, we extracted the geometry of the LB shell, its inner surface, in particular from 3D dust extinction maps of the local ISM. We expanded the shell inner surface in spherical harmonics, up to a variable maximum multipole degree, which enabled us to control the level of complexity for the modeled surface. Next, we applied an analytical model for the ordered magnetic field in the shell to the modeled shell surface. This magnetic field model was successfully fitted to the Planck 353 GHz dust polarized emission maps over the Galactic polar caps. For each polar cap, the direction of the mean magnetic field derived from dust polarization (together with the prior that the field points toward longitude 90° ± 90°) is found to be consistent with the Faraday spectra of the nearby diffuse synchrotron emission. Our work presents a new approach to modeling the local structure of the Galactic magnetic field. We expect our methodology and our results to be useful both in modeling the local ISM as traced by its different components and in modeling the dust polarized emission, which is a long-awaited input for studies of the polarized foregrounds for cosmic microwave background.

Precise and efficient evaluation of gravimetric quantities at arbitrarily scattered points in space

Gravimetric quantities are commonly represented in terms of high degree surface or solid spherical harmonics. After EGM2008, such expansions routinely extend to spherical harmonic degree 2190, which makes the computation of gravimetric quantities at a large number of arbitrarily scattered points in space using harmonic synthesis, a very computationally demanding process. We present here the development of an algorithm and its associated software for the efficient and precise evaluation of gravimetric quantities, represented in high degree solid spherical harmonics, at arbitrarily scattered points in the space exterior to the surface of the Earth. The new algorithm is based on representation of the quantities of interest in solid ellipsoidal harmonics and application of the tensor product trigonometric needlets. A FORTRAN implementation of this algorithm has been developed and extensively tested. The capabilities of the code are demonstrated using as examples the disturbing potential T, height anomaly \(\zeta \), gravity anomaly \(\Delta g\), gravity disturbance \(\delta g\), north–south deflection of the vertical \(\xi \), east–west deflection of the vertical \(\eta \), and the second radial derivative \(T_{rr}\) of the disturbing potential. After a pre-computational step that takes between 1 and 2 h per quantity, the current version of the software is capable of computing on a standard PC each of these quantities in the range from the surface of the Earth up to 544 km above that surface at speeds between 20,000 and 40,000 point evaluations per second, depending on the gravimetric quantity being evaluated, while the relative error does not exceed \(10^{-6}\) and the memory (RAM) use is 9.3 GB.

Solution to the spectral filter problem of residual terrain modelling (RTM)

In physical geodesy, the residual terrain modelling (RTM) technique is frequently used for high-frequency gravity forward modelling. In the RTM technique, a detailed elevation model is high-pass-filtered in the topography domain, which is not equivalent to filtering in the gravity domain. This in-equivalence, denoted as spectral filter problem of the RTM technique, gives rise to two imperfections (errors). The first imperfection is unwanted low-frequency (LF) gravity signals, and the second imperfection is missing high-frequency (HF) signals in the forward-modelled RTM gravity signal. This paper presents new solutions to the RTM spectral filter problem. Our solutions are based on explicit modelling of the two imperfections via corrections. The HF correction is computed using spectral domain gravity forward modelling that delivers the HF gravity signal generated by the long-wavelength RTM reference topography. The LF correction is obtained from pre-computed global RTM gravity grids that are low-pass-filtered using surface or solid spherical harmonics. A numerical case study reveals maximum absolute signal strengths of $$\sim 44$$ mGal (0.5 mGal RMS) for the HF correction and $$\sim 33$$ mGal (0.6 mGal RMS) for the LF correction w.r.t. a degree-2160 reference topography within the data coverage of the SRTM topography model ( $$56^{\circ }\hbox {S} \le \phi \le 60^{\circ }\hbox {N}$$ ). Application of the LF and HF corrections to pre-computed global gravity models (here the GGMplus gravity maps) demonstrates the efficiency of the new corrections over topographically rugged terrain. Over Switzerland, consideration of the HF and LF corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 4.41 to 3.27 mGal, which translates into $$\sim 26$$ % improvement. Over a second test area (Canada), our corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 5.65 to 5.30 mGal ( $$\sim 6$$ % improvement). Particularly over Switzerland, geophysical signals (associated, e.g. with valley fillings) were found to stand out more clearly in the RTM-reduced gravity measurements when the HF and LF correction are taken into account. In summary, the new RTM filter corrections can be easily computed and applied to improve the spectral filter characteristics of the popular RTM approach. Benefits are expected, e.g. in the context of the development of future ultra-high-resolution global gravity models, smoothing of observed gravity data in mountainous terrain and geophysical interpretations of RTM-reduced gravity measurements.

Particle shape quantification using rotation-invariant spherical harmonic analysis

A three-dimensional particle surface can be mathematically represented by a spherical harmonic (SH) coefficient matrix through surface parameterisation and SH expansion. However, this matrix depends not only on the particle shape, but also on the size, position and orientation. This study adopts a rotation-invariant analysis to explore the relationship between SH coefficient matrices and particle shape characteristics. Particle shapes are quantified at different scales (i.e. form, roundness and compactness). These methods are applied to two groups of particles [i.e. Leighton Buzzard sand (LBS) particles and LBS fragments] with distinct shape features. Using rotation invariants, the multi-scale nature of the particle shape is illustrated, and two novel shape descriptors are defined. The results in this paper serve as a starting point for the generation of particle shapes with the prescribed shape features using SHs.

A comparison of modal and spatially matched-filter beamforming for rigid spherical microphone arrays in the context of data-based binaural synthesis

Several approaches to data-based binaural synthesis have been published that capture a sound field by means of a spherical microphone array. The captured sound field is decomposed into plane waves, which are auralized using (far-field) head-related impulse responses (HRIRs). In practice, the decomposition into plane waves is performed by beamforming. A well-known technique for spherical microphone arrays is modal beamforming where the sound field is represented with respect to surface spherical harmonics. A numerically stable approximation is the spatio-temporal matched-filter technique. Here the acoustic reciprocity theorem is used in conjunction with the matched filter technique for beamforming. This paper reviews the matched filter beamformer and compares its properties to the modal beamformer in a realistic scenario. The performance of the two beamforming techniques is compared in the context of data-based binaural synthesis. Time-domain properties of the resulting ear-signals, as well as suitable binaural measures, are investigated for simulated and captured scenarios.

Estimating the slip resistance from spherical nanoindentation and orientation measurements in polycrystalline samples of cubic metals

In this paper, we demonstrate the first application of a recently formulated two-step approach to the estimation of the average slip resistance in a cubic polycrystalline metal sample from a collection of spherical nanoindentation and lattice orientation measurements. In the first step, a crystal plasticity finite element model of the spherical nanoindentation experiment is developed, validated, and employed to capture the functional dependence of the indentation yield strength on the crystal lattice orientation at the indentation site. This functional dependence is captured in a compact representation using surface spherical harmonics (SSH) functions. In the second step, measured values of the indentation yield strength and the crystal lattice orientations at the indentation site from a polycrystalline sample are fit to the function established in the first step to provide a robust estimate for the value of the average slip resistance in the sample. The validity of this approach is demonstrated for cast and annealed Fe-3% Si using measurements published in prior literature.

Application of spherical harmonics analysis on LBS particles and LBS fragments

This paper applies surface parameterization and spherical harmonics analysis to the characterization of particle shapes of Leighton Buzzard sand (LBS) particles and LBS fragments obtained from X-ray micro-tomography (μ CT). The rotation, transition and scale independent spherical coefficients were obtained. The relationship between spherical coefficients and shape parameters of form, roundness and compactness was investigated. The coefficients of degree one determine the principal dimensions of an ellipsoid, which has a similar aspect ratio with the original surface. The coefficients of higher degree characterise more details by increasing the percentage of higher and lower mean curvature on the reconstructed surface. As the spherical degree increases, the reconstructed surface tend to have lower particle roundness, sphericity and convexity, and higher aspect ratio.

Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface

For a spherical interface of radius R separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration, we perform a linear stability analysis in terms of spherical surface harmonics Yn to derive the dispersion relation. The instability behavior is investigated by computing the growth rates and the most-unstable modes as a function of the spherical harmonic degree n. This general methodology is applicable to the entire parameter space spanned by the Atwood number, the viscosity ratio, and the dimensionless number B=(aRρ22/μ22)1/3 R (where aR, ρ2, and μ2 are the local radial acceleration at the interface, and the density and viscosity of the denser overlying fluid, respectively). While the mathematical formulation herein is general, this paper focuses on instability that arises at a spherical viscous fluid/vacuum interface as there is a great deal to be learned from the effects of one-fluid viscosity and sphericity alone. To quantify and understand the effect that curvature and radial acceleration have on the Rayleigh-Taylor instability, a comparison of the growth rates, under homologous driving conditions, between the planar and spherical interfaces is performed. The derived dispersion relation for the planar interface accounts for an underlying finite fluid region of thickness L and normal acceleration aR. Under certain conditions, the development of the most-unstable modes at a spherical interface can take place via the superposition of two adjacent spherical harmonics Yn and Yn+1. This bimodality in the evolution of disturbances in the linear regime does not have a counterpart in the planar configuration where the most-unstable modes are associated with a unique wave number.

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators.

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson-Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online.

Constraining core undertones using stacked IDA gravity records

A search is made for oscillatory gravity signals, possibly associated with core undertones, in the frequency range between the diurnal and semidiurnal tidal bands. IDA tidal data recorded during 6 month intervals following four large earthquakes were used, and the data from different stations were stacked according to spherical harmonic surface amplitude patterns with angular order l≤4. The detection level of a signal in the diurnal-semidiurnal frequency band was roughly 10-20 ngal, but the only signal detected was a 9 ngal signal of 9.54 hr period whose origin remains unexplained.

Detecting Drug Promiscuity Using Gaussian Ensemble Screening

Polypharmacology describes the binding of a ligand to multiple protein targets (a promiscuous ligand) or multiple diverse ligands binding to a given target (a promiscuous target). Pharmaceutical companies are discovering increasing numbers of both promiscuous drugs and drug targets. Hence, polypharmacology is now recognized as an important aspect of drug design. Here, we describe a new and fast way to predict polypharmacological relationships between drug classes quantitatively, which we call Gaussian Ensemble Screening (GES). This approach represents a cluster of molecules with similar spherical harmonic surface shapes as a Gaussian distribution with respect to a selected center molecule. Calculating the Gaussian overlap between pairs of such clusters allows the similarity between drug classes to be calculated analytically without requiring thousands of bootstrap comparisons, as in current promiscuity prediction approaches. We find that such cluster similarity scores also follow a Gaussian distribution. Hence, a cluster similarity score may be transformed into a probability value, or "p-value", in order to quantify the relationships between drug classes. We present results obtained when using the GES approach to predict relationships between drug classes in a subset of the MDL Drug Data Report (MDDR) database. Our results indicate that GES is a useful way to study polypharmacology relationships, and it could provide a novel way to propose new targets for drug repositioning.

Wave field synthesis of a sound field described by spherical harmonics expansion coefficients

Near-field compensated higher order Ambisonics (NFC-HOA) and wave field synthesis (WFS) constitute the two best-known analytic sound field synthesis methods. While WFS is typically used for the synthesis of virtual sound scenes, NFC-HOA is typically employed in order to synthesize sound fields that have been captured with appropriate microphone arrays. Such recorded sound fields are essentially represented by the coefficients of the underlying surface spherical harmonics expansion. A sound field described by such coefficients cannot be straightforwardly synthesized in WFS. This is a consequence of the fact that, unlike in NFC-HOA, it is critical in WFS to carefully select those loudspeakers that contribute to the synthesis of a given sound source in a sound field under consideration. In order to enable such a secondary source selection, it is proposed to employ the well-known concept of decomposing the sound field under consideration into a continuum of plane waves, for which the secondary source selection is straightforward. The plane wave representation is projected onto the horizontal plane and a closed form expression of the secondary source driving signals for horizontal WFS systems of arbitrary convex shape is derived.

Predicting Drug Promiscuity Using Spherical Harmonic Surface Shape-Based Similarity Comparisons

Predicting Drug Promiscuity Using Spherical Harmonic Surface Shape-Based Similarity Comparisons