Harmonic Expansion Research Articles

GNSS standard point positioning method based on spherical harmonic expansion of signal propagation path relating errors

The errors affecting standard point positioning mainly include the ionospheric delay error, tropospheric delay error, clock bias, and multipath effect. The errors are divided into two categories depending on the relation to the signal propagation path: errors related to the propagation path and errors unrelated to the propagation path. The ionospheric delay error, tropospheric delay error, and multipath effect are related to the satellite signal propagation path. Based on this characteristic, a new GPS standard point positioning algorithm based on spherical harmonic expansion (SPP-SH) is proposed, which uses spherical harmonic expansion to represent the errors related to the propagation path. We hope to continuously promote the further development of standard point positioning (SPP) by proposing this new algorithm and to further reduce the redundancy of SPP algorithms and improve the efficiency and reliability of the algorithms. The most important purpose of this manuscript is to validate the proposed SPP-SH algorithm based on GPS data provided by IGS stations. Pseudorange data of multiple continuous epochs are used to achieve a sequential sliding point positioning solution, and the truncated singular value decomposition and iteration method correcting characteristic values are used to solve the ill-conditioned equations in the SPP-SH algorithm. The SPP-SH algorithm is implemented and verified using IGS station data. The positioning accuracy of the SPP-SH algorithm for both dual-frequency data and single-frequency data is better than that of a traditional SPP algorithm. By analyzing the positioning results of the stations at different latitudes, when the spherical harmonic expansion is expanded to the same degree, the positioning accuracy in low latitude areas is poor because the station is seriously affected by factors such as the ionosphere and water vapor, while the positioning accuracy at high latitude stations is the best. The proposed SPP-SH algorithm is also realized in the kinematic scene based on the dual-frequency data collected dynamically. The kinematic data is collected by an Android phone which supports receiving dual-frequency GPS observations. To validate the reliability of the SPP-SH algorithm, two strategies are designed to validate the results, which proves that the newly proposed method can be used for positioning with low accuracy in kinematic conditions.

Effect of Higher-Order Harmonics on the Steady-State Neutron Flux in Accelerator-Driven Subcritical Reactors

Higher-order neutron fluxes (i.e., higher-order harmonics) are widely applied in perturbation theory and modal kinetics, and they are important for research on the physical characteristics of accelerator-driven subcritical reactors (ADSRs). This paper presents a computational scheme for reconstructing the neutron flux in the steady state according to the biorthogonal properties of the forward and adjoint neutron fluxes, which can be used to analyze how higher-order harmonics affect the steady-state neutron flux under λ - and prompt α -modes. Simulation results indicated that a modal synthesis method based on λ - and prompt α -modes can effectively reconstruct the steady-state neutron flux and core power in an ADSR with a power reconstruction accuracy of within 5%. The higher-order harmonics can be classified into three types according to their physical characteristics: the first type contributes significantly to the steady-state neutron flux, the second type contributes almost nothing to the steady-state neutron flux, and the third type contributes nothing to the steady-state neutron flux. The external neutron source contributes only to specific harmonic expansion orders, which are characterized by significant axial and radial symmetry for both the λ - and prompt α -modes.

Acoustic imaging with spherical microphone array and Kriging

Acoustic imaging can be performed using a spherical microphone array (SMA) and conventional beamforming (CBF) or spherical harmonic beamforming (SHB). At low frequencies, the mainlobe width depends on the SMA radius for CBF and on the order of the spherical harmonics expansion for SHB, which is related to the number of microphones. In this letter, Kriging is used to virtually increase the SMA radius and/or the number of microphones. Numerical and experimental investigations show the effectiveness of Kriging to reduce the mainlobe width and thus improve the acoustic images obtained with a SMA and CBF or SHB.

Dimensional estimates for measures on quaternionic spheres

In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansions. This estimate is an analog of a result previously obtained by the authors for the complex spheres.

Interaction of Two Charged Dielectric Spheres with a Point Charge

We consider the problem of interaction of three charged particles, the size of one of which can be disregarded. The equations for the expansion coefficients of the electric field potential are derived using the method of expansion in spherical harmonics. Expressions are obtained for the Cartesian components of the interaction force and the torque due to this force. It is shown that in spite of the axial symmetry breaking after the addition of the third particle, if the free charge is distributed uniformly over the surface of a spherical particle, all vector components of the torque acting on this particle are equal to zero. By separating the contributions from image charges in explicit form, we have derived the expressions for the surface charge density and the force of interaction of the particles. The conditions for the emergence of attraction between similarly charged spherical particles depending on the position of the point particle are investigated.

Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity

Abstract We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao et al. [Monge-Ampère equation on exterior domains, Calc. Var PDE. 52 (2015), 39–63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion, and the iteration method.

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Objective. We provide a systematic framework for quantifying the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field. Approach. We define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V m−1 as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics. Main results. We report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that at rest, responses are dominated by their dipole terms (), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell, only the dipole term is nonzero. However, while minor, other terms are relevant for states different from resting. In particular, we show how and terms can modify the function to induce asymmetries in the response. Significance. This work provides a practical framework for the representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for developing micro- and mesoscale models and optimizing brain stimulation solutions.

Quantification of Global Cloud Properties With Use of Spherical Harmonic Functions

AbstractSpherical harmonic (SH) expansion is a useful tool to study any variable that has valid values at all latitudes and longitudes. The variable can be quantified as a sum of different spherical harmonic components, which are the spherical harmonic functions multiplied by their expansion coefficients. We find that the SH components of cloud radiative effect (CRE) have correlations with El Niño‐Southern Oscillation (ENSO) and the Hadley Circulation (HC). In particular, the expansion degree 2 () SH power spectrum component anomaly of CRE is strongly correlated with ENSO. The two dipole patterns appearing in the SH component anomaly map can be reasonably explained by a known mechanism of ENSO's impact on cloud properties. The and SH power spectrum components are correlated with HC intensity, whereas the and components are correlated with HC latitudinal widths. In ENSO warm and cold phases, the HC‐correlated SH components have opposite anomalies, which suggests the impact of ENSO on HC. This study illustrates that the SH expansion technique provides a different perspective to study the impacts of large‐scale atmospheric circulation on global cloud properties and radiative effects.

Efficient few-body calculations in finite volume

Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.

Sound Insulation Performance of the Stiffened Laminated Plate with Different Reinforcement Forms

A semi-analytical methodology on the sound transmission of stiffened laminated plate under different reinforcement forms is developed to explore its sound insulation performance under the action of a plane sound wave load, in accordance with the classical laminated plate theory and in consideration of stiffener flexion and torsional motions. The formula of acoustic transmission loss is obtained by utilizing spatial harmonic expansion and the virtual work principle. Subsequently, the predicted values of proposed methodology are validated by the existing models. Eventually, some characteristic parameters are considered to investigate their effects on sound insulation behavior.

Logarithmic supertranslations and supertranslation-invariant Lorentz charges

We extend the BMS(4) group by adding logarithmic supertranslations. This is done by relaxing the boundary conditions on the metric and its conjugate momentum at spatial infinity in order to allow logarithmic terms of carefully designed form in the asymptotic expansion, while still preserving finiteness of the action. Standard theorems of the Hamiltonian formalism are used to derive the (finite) generators of the logarithmic supertranslations. As the ordinary supertranslations, these depend on a function of the angles. Ordinary and logarithmic supertranslations are then shown to form an abelian subalgebra with non-vanishing central extension. Because of this central term, one can make nonlinear redefinitions of the generators of the algebra so that the pure supertranslations (ℓ > 1 in a spherical harmonic expansion) and the logarithmic supertranslations have vanishing brackets with all the Poincaré generators, and, in particular, transform in the trivial representation of the Lorentz group. The symmetry algebra is then the direct sum of the Poincaré algebra and the infinite-dimensional abelian algebra formed by the pure supertranslations and the logarithmic supertranslations (with central extension). The pure supertranslations are thus completely decoupled from the standard Poincaré algebra in the asymptotic symmetry algebra. This implies in particular that one can provide a definition of the angular momentum which is manifestly free from supertranslation ambiguities. An intermediate redefinition providing a partial decoupling of the pure and logarithmic supertranslations is also given.

Least-squares collocation: a spherical harmonic representer theorem

SUMMARY The functional analysis of the least-squares collocation (LSC) for gravity potential modelling using m measurements is revisited starting from an explicit spherical harmonic expansion. A spherical harmonic representer theorem (SHRT) is given: the model of the potential is a linear combination of m kernels or covariances. This theorem is independent of the specific forms of the data-fitting loss and the regularizer, showing that it is a stronger result than the LSC theory. The corresponding reproducing kernel Hilbert space is explicitly specified. When the least-squares data-fitting loss and the quadratic regularizer are employed, the SHRT gives exactly the LSC method for variable prediction. The nominal prediction precision assessment of the SHRT and that of the LSC are also explicitly compared; this contributes to the unification of the deterministic and stochastic analyses of the LSC theory.

Inverse Source Solutions With Huygens’ Surface Conforming Distributed Directive Spherical Harmonics Expansions

Time-harmonic inverse source solutions are commonly working with electric and magnetic surface current densities defined and discretized on appropriately chosen Huygens’ surfaces. An efficient meshless alternative are spherical harmonics expansions of low-order distributed along the chosen Huygens’ surface, which still possess pretty good spatial localization properties. Similar to expansions with electric and magnetic surface current densities, distributed expansions with Cartesian spherical harmonics (SHCs) or standard vector spherical harmonics (SHVs) are redundant when they are placed on Huygens’ surfaces. This redundancy is reduced by allowing only spherical harmonics, which can be excited by surface current density distributions in a specific plane, and by forming directive spherical harmonics. This results in a considerable reduction of the necessary number of unknowns for representing the sources and improved conditioning of the discretized integral equations. Inspired by the Huygens’ radiator concept, appropriate directive harmonics on the basis of SHCs and SHVs are derived, implemented, and compared in terms of their performances. The validity of the presented techniques is confirmed via inverse source solutions for synthetic and real measurement data.

A nonlocal electron transport model in the diffusion scaling of hydrodynamics

We analyze a model of nonlocal electron transport named P1-diffusion based on a spherical harmonic expansion in velocity space and a diffusion scaling, which makes it compatible with assumptions from magneto-hydrodynamics (MHD). An iterative, fully implicit (CFL-free, as defined by the Courant Friedrich Levy condition) and asymptotic preserving discretization is proposed, which necessitates the inversion of a possibly large number of—but small—linear systems. It is found accurate with respect to reference solutions from a Vlasov–Fokker–Planck–Maxwell code (based on a Polynomial expansion of order N, or PN expansion) on a series of tests, which are representative of the conduction zone in laser-created plasmas. Thereby, the present approach is a good candidate for being embedded in multi-D MHD codes.

Gravity degree–depth relationship using point mass spherical harmonics

SUMMARYRelationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree–depth relationship. Our relationship resembles an earlier one derived by Bowin, with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. overestimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to $d = (1-e^{\frac{-1}{n+1}})R$, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modelling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.

A THREE-DIMENSIONAL APPROACH TO IMPROVE THE INTERNATIONAL REFERENCE IONOSPHERE (IRI) MODEL

Abstract. A three-dimensional model based on time and location is introduced to enhance the accuracy of vertical total electron content (VTEC) global ionosphere maps (GIMs) of ionosphere empirical models. The chosen empirical model is the International Reference Ionosphere 2016 (IRI-2016) model and observations are the GIMs VTEC values from Center for Orbit Determination in Europe (CODE). For this purpose, IRI-2016 is considered as a background model and the VTECs from GIMs are considered as the correction term. The correction term is modelled using spherical harmonics expansion base functions with a degree equal to 15 in Earth-fixed coordinate system with a polynomial B-spline base function for time resolution. The improved VTEC maps are constructed on 1st January 2018 with a Kp index of 4. The root mean square error (RMSE) improvements range from 1.65 TECU to 2.81TECU about 70 percent to 85 percent and the bias improvements are from 0.28 TECU to 1.00 TECU with an average of 0.75 TECU, respectively.

Model Variations of the Crust Thickness of Mars and Venus Using the Love Numbers Method

Based on topography and gravitational field data, model variations in the crust thickness of Mars and Venus were calculated using the Love numbers method. The method takes into account the adjustment of the planetary interior to loads on the surface and in the interior. Numerical modeling was carried out using the expansion in spherical harmonics of the topography and gravitational field data up to the 90th degree and order for Mars and up to the 70th degree and order for Venus. The topography of the crust–mantle boundary suggests partial Airy isostatic compensation. The model of the Martian crust is consistent with the interval of crustal thickness values under the site of the InSight station in the southwestern part of Elysium Planitia obtained from the results of a seismic experiment. The comparison with the available global models of the crust of Mars and Venus was carried out.

A Novel Approach for Establishing the Global Ionospheric Model With High Spatiotemporal Resolution

The global ionospheric model is the most effective way to study the structure and variation of the global ionosphere. However, the current global ionosphere maps (GIMs) have the defect of low spatiotemporal resolution and cannot reflect the short-term nonlinear changes and small-scale structures of the vertical total electron content (VTEC). This article proposes a new method for establishing a global ionospheric model with high spatiotemporal resolution. The spherical harmonic (SH) expansions are used to model the VTEC observations, and the Kalman filter is used to estimate the model’s coefficients with high accuracy. The method calculates SH coefficients every 5 min, increasing the spatial resolution to 7.2°. Precise determination methods for the state noise covariance matrix and the observation noise covariance matrix in the Kalman filter are also proposed. The model with a high spatiotemporal resolution was established using the observations of 300 global navigation satellite system (GNSS) tracking stations worldwide. The results show that the model with high spatiotemporal resolution can more finely reflect the nonlinear changes of VTEC in a short period and the small-scale structure of the ionosphere. The accuracy of the high-resolution model is validated using high-precision differential slant total electron content (dSTEC) observations from three sets of tracking stations and compared with the final product of three international GNSS service ionosphere associate analysis centers (IGS IAACs). The results show that the accuracy of the high spatiotemporal resolution model in this article is better than the products of IGS IAACs. The research in this article provides a new idea for establishing GIMs with higher spatiotemporal resolution and accuracy.

Diffuseness quantification in an acoustic test chamber using the isotropy derived by the angular spectrum of sound waves based on the spherical harmonics expansion

Diffuseness quantification in an acoustic test chamber using the isotropy derived by the angular spectrum of sound waves based on the spherical harmonics expansion

Explicit Conversion Formulas Between Spherical Wave Expansions With Scalar and Vector Expansion Coefficients

Explicit formulas for converting spherical wave expansions with vector basis functions and scalar expansion coefficients into spherical wave expansions with scalar basis functions and vector expansion coefficients and vice versa are presented. The formulas are given in terms of Wigner-3- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula> -symbols. The conversion formulas are derived by spherical harmonics expansions of products of two spherical harmonics and by levering on recurrence relations of the associated Legendre functions. The expansions with vector coefficients are redundant and explicit formulas are given for the linear combination of vector coefficients of which the expanded fields cancel identically to zero. The redundancy in the expansion can be used to find compact expansions with a minimum expansion order. For every vector spherical wave function, two expansions using scalar spherical harmonics are presented: on the one hand, a radial-component free expansion and on the other hand, a minimum-order expansion. The correctness of all expansions is verified with a simple computer code up to a mode order of 65. The agreement of the expanded near and far fields in both types of expansions is better than 14 digits of accuracy for most cases.