Spherical Harmonic Expansion Coefficients Research Articles

Application of the cos-Fourier expansion to data transformation between different latitude–longitude grids

In geodesy and geophysics, data of global coverage are often given on a latitude–longitude grid of constant intervals in latitude and longitude, that we call the gridded data (GD), or as a spherical harmonic expansion (SHE). Generally, as the total number of SHE coefficients is only about half of that of the GD, the GD and the SHE do not have one-to-one correspondence, i.e. the GD can be recovered from the SHE computed using the GD. However, in the case when the GD can be exactly represented using a SHE, like the geoid computed from a geopotential model given in terms of a SHE, the GD and the SHE do have one-to-one correspondence. But this is just a special case. Hence, the SHE does not always preserve the complete resolution of the GD. In fact, the GD may have higher resolution than the SHE at higher latitudes. In this paper, we suggest to use the tensor-product of the cosine series over colatitude and the Fourier series over longitude, referred to as the cos-Fourier expansion (CFE), to represent the GD. The CFE always has a one-to-one correspondence with the GD, and preserves the resolution of the GD. The CFE is then applied for transforming GD between different grids. The results of a particular application for generating the ocean function (OF) over a 2 . 5 ∘ × 2 . 5 ∘ grid from that originally given over a 5 ′ × 5 ′ grid are presented to show the feasibility of the method. Furthermore, a comparison between the CFE and SHE is made using digital land and sea-floor elevation data that demonstrates the above-mentioned characteristics of the CFE and SHE.

Investigation of a secular variation impulse using satellite data: The 2003 geomagnetic jerk

Observatory monthly means provide an excellent opportunity to study the temporal changes of the magnetic field at a given location. Unfortunately, the uneven distribution of the present observatory network makes it difficult to determine the global field change pattern. Recently, we have developed an approach to extract satellite monthly means at a regular network of “virtual observatories” at 400 km altitude, based on CHAMP magnetic measurements. Using monthly means for 2001–2005 from those “virtual observatories” we investigate the space–time structure of the short-period variation of the Earth's magnetic field by means of a Spherical Harmonic Expansion, followed by a separation into external (magnetospheric) and internal part. This allows, for the first time, to study the secular variation globally and directly from satellite magnetic data. Analyzing the time series of the magnetic field at the “virtual observatories” as well as those of the spherical harmonic expansion coefficients, we detect a secular variation impulse (an abrupt jump in the second time derivative of the magnetic field) in the CHAMP satellite data during the first months of the year 2003. The jerk occurred simultaneously in the northern and southern hemispheres in a rather limited area near 90° E, with maximum jerk strength at about ± 30° latitude, a region also characterized by a strong secular acceleration (second time derivative of the magnetic field). We show that the 2003 geomagnetic jerk is not worldwide in occurrence and that there is an evidence for this event in the length-of-day variation.

An evaluation of spherical designs for molecular-like surfaces

The use of spherical harmonics in the molecular sciences is widespread. They have been employed with success in, for instance, the crystallographic fast rotation function, small-angle scattering particle reconstruction, molecular surface visualisation, protein–protein docking, active site analysis and protein function prediction. The calculation of spherical harmonic expansion coefficients requires integration over the full sphere and can be a computationally cumbersome and also numerically sensitive (with respect to the integration weights) procedure. It is shown here how the use of spherical t-designs and pre-computed near-equal weight integration layouts can significantly reduce the computational effort in the determination of spherical harmonic expansion coefficients for molecular surfaces, thus giving rise to a robust and highly efficient algorithm for the construction of molecular-like objects.

Real spherical harmonic expansion coefficients as 3D shape descriptors for protein binding pocket and ligand comparisons

An increasing number of protein structures are being determined for which no biochemical characterization is available. The analysis of protein structure and function assignment is becoming an unexpected challenge and a major bottleneck towards the goal of well-annotated genomes. As shape plays a crucial role in biomolecular recognition and function, the examination and development of shape description and comparison techniques is likely to be of prime importance for understanding protein structure-function relationships. A novel technique is presented for the comparison of protein binding pockets. The method uses the coefficients of a real spherical harmonics expansion to describe the shape of a protein's binding pocket. Shape similarity is computed as the L2 distance in coefficient space. Such comparisons in several thousands per second can be carried out on a standard linux PC. Other properties such as the electrostatic potential fit seamlessly into the same framework. The method can also be used directly for describing the shape of proteins and other molecules. A limited version of the software for the real spherical harmonics expansion of a set of points in PDB format is freely available upon request from the authors. Binding pocket comparisons and ligand prediction will be made available through the protein structure annotation pipeline Profunc (written by Roman Laskowski) which will be accessible from the EBI website shortly.

A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space Integrals

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.

Analysis of the Kamionkowski-Loeb method of reducing cosmic variance with CMB polarization

Part of the CMB polarization signal in the direction of galaxy clusters is produced by Thomson scattering of the CMB temperature quadrupole. In principle this allows measurement of the CMB power spectrum harmonic C{sub 2}(z) with higher accuracy (at z>0) than the cosmic variance limit imposed by sample variance on one CMB sky. However the observed signals are statistically correlated if the comoving separation between the clusters is small enough. Thus one cannot reduce the sample variance by more than roughly the number of separate regions available which produce uncorrelated signals, as first pointed out by Kamionkowski and Loeb. In this paper we analyze in detail the procedure outlined by Kamionkowski and Loeb, computing the correlation of the polarization signals by considering the variation of the spherical harmonic expansion coefficients of the temperature anisotropy on our past light cone. Given a hypothetical set of Stokes parameter measurements of the CMB polarization in the directions of galaxy clusters, distributed at random on a given redshift shell, we show how to construct an estimator of the angular power spectrum harmonic C{sub 2} at that redshift. We then compare the variance of this estimator with the cosmic variance of the CMB multipole onmore » our sky which probes the same scale. We find that in fact the cosmic variance is not reducible below the single sky CMB value using the cluster method. Thus this method is not likely to be of use for reconstruction of the primordial power spectrum. However the method does yield a measurement of C{sub 2} as a function of redshift with increasing accuracy at higher redshift, and thus potentially a probe of the mechanism which may have suppressed the quadrupole. We also examine to what extent the redshift dependence of C{sub 2} can be used to probe the time changing potential anisotropy as the universe evolves into the vacuum dominated phase (the late-time integrated Sachs-Wolfe effect). We find that this effect is not observable in the time dependence of C{sub 2} since it is swamped by cosmic variance, but there is an observable signature in the correlation functions of the Stokes parameters.« less

Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics

A cone-beam image reconstruction algorithm using spherical harmonic expansions is proposed. The reconstruction algorithm is in the form of a summation of inner products of two discrete arrays of spherical harmonic expansion coefficients at each cone-beam point of acquisition. This form is different from the common filtered backprojection algorithm and the direct Fourier reconstruction algorithm. There is no re-sampling of the data, and spherical harmonic expansions are used instead of Fourier expansions. As a special case, a new fan-beam image reconstruction algorithm is also derived in terms of a circular harmonic expansion. Computer simulation results for both cone-beam and fan-beam algorithms are presented for circular planar orbit acquisitions. The algorithms give accurate reconstructions; however, the implementation of the cone-beam reconstruction algorithm is computationally intensive. A relatively efficient algorithm is proposed for reconstructing the central slice of the image when a circular scanning orbit is used.

Testing for non-Gaussianity of the cosmic microwave background in harmonic space: An empirical process approach

We present a new, model-independent approach for measuring non-Gaussianity of the cosmic microwave background (CMB) anisotropy pattern. Our approach is based on the empirical distribution function of the normalized spherical harmonic expansion coefficients ${a}_{\mathcal{l}m}$ of a nearly full-sky CMB map, like the ones expected from forthcoming satellite experiments. Using a set of Kolmogorov-Smirnov type tests, we check for Gaussianity and independency of the ${a}_{\mathcal{l}m}.$ We test the method on two non-Gaussian toy models of the CMB, one generated in spherical harmonic space and one in pixel (real) space. We also provide some rigorous results, possibly of independent interest, on the exact distribution of the spherical harmonic coefficients normalized by an estimated angular power spectrum.

Greens function modified spherical harmonic analysis

In an effort to develop a model for impulse responses in reverberant spaces, techniques for source radiation modeling were explored. One method of decomposing a source field involves an analysis based on spherical harmonic expansions of the field. Spherical harmonic decomposition is a powerful tool, but it is a mathematical approach which is completely independent of the type of source that creates the field. As a consequence, this technique cannot, in general, be used to gain insight into the underlying physical nature of the source. Moreover, because this is a brute force technique, the analysis can be extremely inefficient. In this presentation, modifications to the spherical harmonic analysis and synthesis equations are made to overcome these problems. These are based on the (theoretical) source types associated with the individual spherical harmonic expansion coefficients, and determination of the location of the primary components of the (physical) source. These modifications can greatly improve the computational efficiency and allow the source to be described by a set of coefficients which is more directly related to actual source types present. Illustrative examples will be shown. [Work supported by the Applied Research Laboratory at Penn State University under the Educational and Foundational Research Program.]

Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces

A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrization is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimize a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coefficients. This rotational property is applied for the first time to the 3-dimensional molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody heavy chain variable domains. Special attention is given to computational efficiency. The spherical harmonic expansion coefficients are determined using fast surface sampling and integration schemes based on the tessellation of a regular icosahedron. Low resolution surfaces can be generated and displayed in under 10 s and a pair of surfaces can be maximally superposed in under 3 s on a contemporary workstation. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 383–395, 1999

Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces

A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrization is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimize a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coefficients. This rotational property is applied for the first time to the 3-dimensional molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody heavy chain variable domains. Special attention is given to computational efficiency. The spherical harmonic expansion coefficients are determined using fast surface sampling and integration schemes based on the tessellation of a regular icosahedron. Low resolution surfaces can be generated and displayed in under 10 s and a pair of surfaces can be maximally superposed in under 3 s on a contemporary workstation. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 383–395, 1999

Computer simulation studies of anisotropic systems. XXVII. The direct pair correlation function of the Gay–Berne discotic nematic and estimates of its elastic constants

Monte Carlo simulations have been performed for a discotic liquid crystal composed of Gay–Berne particles. On the basis of these simulations for the nematic phase, a subset of the spherical harmonic expansion coefficients of the direct pair correlation function (DPCF) were determined from the pair distribution function (PDF) by solving the Ornstein–Zernike (OZ) equation. This was achieved by generalizing the Wiener–Hopf factorization scheme for the numerical solution of the OZ equation. Only the expansion coefficients gl1,l2,l(r) (lα⩽4) of the PDF in the laboratory frame were used when solving the OZ equation; this means that the DPCF so obtained is equivalent to that for a nematic in which the director is randomly distributed. From the DPCF, the scaled Oseen–Zöcher–Frank elastic constants K11*, K22*, and K33*, as well as the surface constant K13*, have been calculated from the subset of expansion coefficients. Generally, we find that K33*<K11*<K22*, in agreement with what is expected and found for discotic nematics. These results are quantitatively but not qualitatively different from those calculated with the help of analytical approximations for the same spherical harmonic expansion coefficients of the direct pair correlation function. For example, the values of the bulk elastic constants determined via the OZ equation are about three times larger than the bulk elasticity obtained with the low density approximation.

Statistical Interpolation Using Cyclostationary EOFs

Investigated here is the space–time estimation or statistical interpolation of a global variable based on a few observations. Estimation of a global data field is essentially a problem of optimally estimating spherical harmonic expansion coefficients. The optimal estimation technique used here is similar to that in Kim et al. An important exception is that cyclostationary empirical orthogonal functions (CSEOFs) are used to develop the estimation technique instead of regular empirical orthogonal functions (EOFs). The use of CSEOFs is motivated by the fact that many climatic variables are (approximately) cyclostationary. That is, the statistics of a climatic variable vary periodically with a distinct nested periodicity. The developed technique is applied to estimating the global field of monthly surface temperature anomalies, which is a notable example of cyclostationary processes. The CSEOFs, an essential ingredient for formulating a cyclostationary estimation technique, account for the monthly variation of the surface temperature statistics, namely much larger variance in the winter than in the summer. Further, cyclostationary statistics contain information on how different months are correlated. This allows one to use all 12 months of measurements, thereby optimizing the estimation technique both in space and time. As the test results indicate, estimation error is much reduced when using the cyclostationary technique.

Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory

Summary High-frequency seismograms of local earthquakes are considered to consist of incoherent body waves scattered by random inhomogeneities in the earth medium. For the study of seismogram envelopes, we can describe the scattering process on the basis of energy transport theory by representing the random inhomogeneities as distributed point-like scatterers. By introducing the concept of directional distribution of energy density, we propose a formulation of the multiple non-isotropic scattering process in 3-D space. We suppose that the axially symmetric non-isotropic scattering is described by a directional scattering coefficient, the medium is characterized by one wave velocity, and the source radiation is spherical. In addition to the Fourier transformation in space and the Laplace transformation in time used to solve for energy density in isotropic scattering media, we use a spherical harmonic series expansion in solid angle. Then, the energy transport equation given as an integral can be written as simultaneous linear equations, where coefficients are given by the Wigner 3-j symbols and the spherical harmonic expansion coefficients of the directional scattering coefficient. The lowest term of the spherical harmonic series corresponds to isotropic scattering. When the directional scattering coefficient is written as a finite-length spherical harmonic series, the simultaneous linear equations can be solved for a finite number of unknowns and we can obtain the spatio-temporal distribution of energy density. The numerical calculation for a case with strong forward scattering shows a uniform distribution of energy density around the hypocentre at large lapse times.

Density-functional theory of the nematic phase: Results for a system of hard ellipsoids of revolution.

A second-order density-functional theory is used to study the isotropic-nematic transition in a system of hard ellipsoids of revolution. The direct pair-correlation functions of the coexisting isotropic liquid that enter in the theory as input information are obtained from solving the Ornstein-Zernike equation using the Percus-Yevick closure relation. The spherical harmonic expansion coefficients of the correlation functions obtained from this solution are in good agreement with those found from computer simulations. We find that a system spontaneously transforms to a nematic phase when the structural parameter denoted by c${\mathrm{^}}_{22}^{(0)}$ attains a value close to 4.40. This value of c${\mathrm{^}}_{22}^{(0)}$ depends, although very weakly, on the value of the length-to-width ratio of the molecules. The transition parameters we found are in very good agreement with the results generated by computer simulations. By using the harmonic coefficients of the direct pair-correlation functions, we have calculated the Frank elastic constants of the nematic phase.

Program to calculate the least-squares estimates of the spherical harmonic expansion coefficients of an equally angular-gridded scalar field

The program SPHAN estimates the spherical harmonic coefficients of a scalar field by least-square fitting of data values measured on an equal angular grid on a globe.

Transformation of spherical harmonic and Cartesian magnetic field expansion coefficients

Transformation properties of magnetic field expansion coefficients from spherical to Cartesian coordinates are given. Since the expansions are equivalent because they satisfy the Laplace equation, the transformation is unique. Spherical-to-Cartesian and Cartesian-to-spherical transformations are listed to fifth order.

Non-spherical bridge function theory of molecular fluids

A new theory based on the Ornstein-Zernike integral equation and applicable to any hard-body molecular fluid is proposed. The theory approximates the bridge function of the molecular fluid by the hard-sphere bridge function with an angular-dependent hard-sphere diameter. The theory is tested on the homonuclear hard-dumbell system over a wide range of elongations and densities. The predicted compressibility factors are numerically indistinguishable from available computer simulation results and the theory also gives a very good description of the pair distribution function, as represented both by the spherical harmonic expansion coefficients and by the site-site distribution functions.

Planetary magnetic fields: A comparative view

The sums of the squares of the non-axial (m ≠ 0) spherical-harmonic expansion coefficients (g n m, h n m) yield a readily interpretable exponential spectrum when normalized so as to reflect the global energy content per degree of freedom and plotted against n for each magnetized planet. Indeed, the resulting spectra for Earth, Jupiter, and Uranus suggest an equipartition of magnetic energy among the available degrees of freedom and yield reasonable values for the nominal core radii r c (= 0.432 R E, 0.756 R J, 0.464 R U), which can be interpreted as the mean values of r within the respective fluid cores. Results for Saturn are indeterminate because the most reliable field models are axisymmetric. Results for Mercury are ambiguous because spatial coverage during the Mariner-10 encounters was too limited. Results for Neptune await encounter by Voyager 2. The implications of magnetic-energy equipartition for dynamo theory are unclear but presumably significant, since equipartition seems to suggest the importance of the α-effect (turbulent-eddy formation) for the symmetry-breaking required by Cowling's Theorem in the context of an α-ω dynamo. It seems significant, moreover, that the equatorial projection of the dipole moment belongs to the same exponential spectrum as the analogous projections of the quadrupole and higher moments. If (as seems probable) the spectrum persists during a magnetic reversal (characterized by the condition g 1 0 = 0), then the dipole moment at the time of polarity transition would be perpendicular to the planetary rotation axis and about 20% as strong as the time-averaged moment, which is inclined ∼10° to the rotation axis. Observation of a 58.6° inclination for Uranus' magnetic axis suggests that |g 1 0| is abnormally small at the present epoch (i.e., that Uranus is in the process of undergoing a magnetic reversal). A more usual (∼10°) inclination typical of Earth and Jupiter at times not near polarity transitions would correspond to a dipole moment μ ∼ 1 G-R U 3 and an offset ∼ 0.07 R U for Uranus and (by implication, according to Blackett's Law and similar considerations) a dipole moment μ ∼ 1 G-R N 3 for Neptune.

A test of a modified algorithm for computing spherical harmonic coefficients using an FFT

The Dilts (1985) algorithm for computing the spherical harmonic expansion coefficients for a function on a sphere, on the basis of a two-dimensional FFT, is presently modified, tested, and found to eliminate problems of overflow and large storage requirements associated with the encounter of harmonic degree values greater than 16. Results from timing tests show the Dilts program to be impractical, however, for the computation of spherical harmonic expansion coefficients for large harmonic degree values.