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Spherical Harmonics (spherical + harmonics)

Distribution by Scientific Domains

Physics and Astronomy	30%

Selected Abstracts

Spherical harmonics in a non-polar co-ordinate system and application to Fourier series in 2-sphere,

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007
H. M. Nasir
Abstract A new non-polar spherical co-ordinate system for the three-dimensional space is introduced. The co-ordinate system is composed of six local co-ordinate systems mapped from six faces of a cube on to the 2-sphere. Weakly orthogonal and orthogonal spherical harmonics are constructed in this co-ordinate system. The spherical harmonics are easily computable functions consisting of polynomials and square root of polynomials. Examples of finite Fourier series computations are given in terms of the new spherical harmonics to demonstrate their immediate applicability. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Basis functions for the consistent and accurate representation of surface mass loading

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2007
Peter J. Clarke
SUMMARY Inversion of geodetic site displacement data to infer surface mass loads has previously been demonstrated using a spherical harmonic representation of the load. This method suffers from the continent-rich, ocean-poor distribution of geodetic data, coupled with the predominance of the continental load (water storage and atmospheric pressure) compared with the ocean bottom pressure (including the inverse barometer response). Finer-scale inversion becomes unstable due to the rapidly increasing number of parameters which are poorly constrained by the data geometry. Several approaches have previously been tried to mitigate this, including the adoption of constraints over the oceanic domain derived from ocean circulation models, the use of smoothness constraints for the oceanic load, and the incorporation of GRACE gravity field data. However, these methods do not provide appropriate treatment of mass conservation and of the ocean's equilibrium-tide response to the total gravitational field. Instead, we propose a modified set of basis functions as an alternative to standard spherical harmonics. Our basis functions allow variability of the load over continental regions, but impose global mass conservation and equilibrium tidal behaviour of the oceans. We test our basis functions first for the efficiency of fitting to realistic modelled surface loads, and then for accuracy of the estimates of the inferred load compared with the known model load, using synthetic geodetic displacements with real GPS network geometry. Compared to standard spherical harmonics, our basis functions yield a better fit to the model loads over the period 1997,2005, for an equivalent number of parameters, and provide a more accurate and stable fit using the synthetic geodetic displacements. In particular, recovery of the low-degree coefficients is greatly improved. Using a nine-parameter fit we are able to model 58 per cent of the variance in the synthetic degree-1 zonal coefficient time-series, 38,41 per cent of the degree-1 non-zonal coefficients, and 80 per cent of the degree-2 zonal coefficient. An equivalent spherical harmonic estimate truncated at degree 2 is able to model the degree-1 zonal coefficient similarly (56 per cent of variance), but only models 59 per cent of the degree-2 zonal coefficient variance and is unable to model the degree-1 non-zonal coefficients. [source]

Localized spectral analysis on the sphere

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2005
Mark A. Wieczorek
SUMMARY It is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius ,0, and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space,bandwidth product N0= (L+ 1) ,0/,. The first N0, 1 windows are near perfectly concentrated, and the best-concentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere. [source]

Approximation method for high-degree harmonics in normal mode modelling

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2002
R. E. M. Riva
Summary For some loading applications, the normal modes approach to the viscoelastic relaxation of a spherical earth requires the use of spherical harmonics up to a high degree. Examples include postseismic deformation (internal loading) and sea level variations due to glacial isostatic adjustment (external loading). In the case of postseismic modelling, the convergence of the solution, given as a spherical harmonic expansion series, is directly dependent on loading depth and requires several thousands of terms for shallow earthquake sources. The particular structure of the analytical fundamental solutions used in normal mode techniques usually does not allow a straightforward calculation, since numerical problems can readily occur due to the stiffness of the matrices used in the propagation routines. Here we show a way of removing this stiffness problem by approximating the fundamental matrix solutions, followed by a rescaling procedure, in this way we can virtually go up to whatever harmonic degree is required. [source]

Fast single domain,subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2009
Jure Ravnik
Abstract In this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity,vorticity formulation of the incompressible Navier,Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary-domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved. The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non-linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid-driven cavity and the onset of natural convection in a differentially heated enclosure. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Multi-dimensional inhomogeneity indicators and the force on uncharged spheres in electric fields

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2009
Dirk Langemann
Abstract Uncharged droplets on outdoor high-voltage equipment suffer a non-vanishing total force in non-homogeneous electric fields. Here, the model problem of a spherical test body is considered in arbitrary dimensions. A series expansion of inhomogeneity indicators is proven, which approximates the total force in local terms of the undisturbed electric field. The proof uses the ideas of generalized spherical harmonics without referring to the particular choice of the orthonormal system. The fast converging series expansion establishes a relationship between the solutions of two partial differential equations on different domains. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Spherical harmonics in a non-polar co-ordinate system and application to Fourier series in 2-sphere,

Non-linear redshift distortions: the two-point correlation function

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 2 2001
Somnath Bharadwaj
We consider a situation where the density and peculiar velocities in real space are linear, and we calculate ,s, the two-point correlation function in redshift space, incorporating all non-linear effects which arise as a consequence of the map from real to redshift space. Our result is non-perturbative and it includes the effects of possible multi-streaming in redshift space. We find that the deviations from the predictions of the linear redshift distortion analysis increase for the higher spherical harmonics of ,s. While the deviations are insignificant for the monopole ,0, the hexadecapole ,4 exhibits large deviations from the linear predictions. For a COBE normalized , cold dark matter (CDM) power spectrum, our results for ,4 deviate from the linear predictions by a factor of two on the scale of ,10 h,1 Mpc. The deviations from the linear predictions depend separately on f(,) and b. This holds the possibility of removing the degeneracy that exists between these two parameters in the linear analysis of redshift surveys which yields only . We also show that the commonly used phenomenological model, where the non-linear redshift two-point correlation function is calculated by convolving the linear redshift correlation function with an isotropic pair velocity distribution function, is a limiting case of our result. [source]

Comparison of the ENEAR peculiar velocities with the PSCz gravity field

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 3 2001
Adi Nusser
We present a comparison between the peculiar velocity field measured from the ENEAR all-sky Dn,, catalogue and that derived from the galaxy distribution of the IRAS Point Source Catalog Redshift Survey (PSCz). The analysis is based on a modal expansion of these data in redshift space by means of spherical harmonics and Bessel functions. The effective smoothing scale of the expansion is almost linear with redshift reaching 1500 km s,1 at 3000 km s,1. The general flow patterns in the filtered ENEAR and PSCz velocity fields agree well within 6000 km s,1, assuming a linear biasing relation between the mass and the PSCz galaxies. The comparison allows us to determine the parameter where , is the cosmological density parameter and b is the linear biasing factor. A likelihood analysis of the ENEAR and PSCz modes yields in good agreement with values obtained from Tully,Fisher surveys. [source]

Electronic spectra of [(CH3)2NH2]5Cd2CuCl11 crystals

PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 11 2004
V. Kapustianik
Abstract The temperature evolution of Cu2+ ion environment in the solid solutions of ((CH3)2NH2)5Cd2CuCl11 is studied on the basis of absorption spectroscopy data. For the detailed analysis of experimental data the special program package Crys Tool 1.0 based on quantum-mechanical models, first of all on the model of normalized spherical harmonics (NSH), has been employed. It has been found that similarly to the crystal of ((CH3)2NH2)5Cd3Cl11 (DMACC) the investigated solid solution contains tetragonally distorted octahedral metal,halogen complexes of two types and the degree of their distortion is changed considerably at the temperatures of phase transitions (PTs). The parameters of crystal field, angular overlap model, as well as the copper,chlorine distances, show continuous changes at T1 = 176 K that should be related to the second-order transition, whereas the jump-like anomalies of the spectral parameters at T2 = 115 K (on cooling) are characteristic of the first-order PTs. Introduction of the copper ions into the structure of the host DMACC crystal induces the shifts of these PTs toward low temperatures by 3.5 and 5 K, respectively. The observed structural changes around T0 = 313 K are connected with a complex co-operative effect involving weakening of the hydrogen bonds and modification of the Jahn,Teller distortion with temperature. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]