Set Of Spherical Harmonic Coefficients Research Articles

Assessment of recent global geopotential models based on the Auvergne test area data

The gravitational field is important for many natural phenomena related to earth dynamics, especially mass transport. Its precise determination is essential for earth sciences such as geodesy, geophysics and oceanography. Determining the earth’s gravitational field is the same as determining the earth’s potential. Since this potential is a harmonic function outside the earth, spherical harmonic series are often used to represent the gravity field. Global Geopotential Models (GGMs) are sets of spherical harmonic coefficients representing the earth’s gravity field at different wavelengths. GGMs developed by scientists are published by the International Centre for Gravity Earth Models (ICGEM). When the structure of the GGMs is examined, it is seen that they consist of different degrees and various data groups. The accuracy and resolution provided by each GGM vary depending on the degree of the model and the data used for developing the GGM. Also, geodetic quantities such as potential, geoid undulation, deflection of the vertical components, gravity, and anomaly values can be derived from GGMs within the framework of mathematical principles. In this study, gravity, geoid heights, and free-air gravity anomaly values at test points using different GGMs produced in recent years have been investigated. The study area contains 98000 test points chosen from the Auvergne test area in France. Within the scope of the study, the geoid undulations, gravity values, and free-air gravity anomalies for all points derived from seven recent GGMs have been compared with ground-truth data and the statistical results have been obtained.

Spatial Wiener filter to reduce spatial aliasing with spherical microphone arrays.

Three dimensional acoustic soundfields can be represented by a set of spherical harmonic coefficients that are extracted from pressure signals recorded by a spherical microphone array. The extraction method used and truncation order chosen introduce errors of spatial aliasing in the coefficients and truncation error in a reconstructed signal. A spatial Wiener filter (SWF) extraction method is proposed in this paper, using second order statistics of typical soundfield characteristics (signal power, estimated source locations, and internal microphone noise) and accounts for the presence of coefficients beyond the truncation order to reduce spatial aliasing in the extracted coefficients. The SWF can also distinguish between "wanted" and "unwanted" sources, reducing the contributions of unwanted sources to the extracted coefficients. The SWF is compared against the state of the art methods; regularized inverse (or generalized inverse), and orthonormal extraction methods, which are explored under a similar framework to the SWF. The authors compare these methods and show the benefit of the SWF for plane waves, with varying assumptions about the source characteristics. The SWF can also extract coefficients beyond the traditional truncation limit of a given array, unlike other methods.

A Theoretical Design Approach for Passive Shimming of a Magic-Angle-Spinning NMR Magnet.

This paper presents a passive shimming design approach for a magic-angle-spinning (MAS) NMR magnet. In order to achieve a 1.5-T magic-angle field in NMR samples, we created two independent orthogonal magnetic vector fields by two separate coils: the dipole and solenoid. These two coils create a combined 1.5-T magnetic field vector directed at the magic angle (54.74° from the spinning axis). Additionally, the stringent magnetic field homogeneity requirement of the MAS magnet is the same as that of a solenoidal NMR magnet. The challenge for the magic-angle passive shimming design is to correct both the dipole and solenoid magnetic field spherical harmonics with one set of iron pieces, the so-called ferromagnetic shimming. Furthermore, the magnetization of the iron pieces is produced by both the dipole and solenoid coils. In our design approach, a matrix of 2 mm by 5 mm iron pieces with different thicknesses was attached to a thin-walled tube, 90-mm diameter and 40-mm high. Two sets of spherical harmonic coefficients were calculated for both the dipole and solenoid coil windings. By using the multiple-objective linear programming optimization technique and coordinate transformations, we have designed a passive shimming set that can theoretically reduce 22 lower-order spherical harmonics and improve the homogeneity of our MAS NMR magnet.

An analytical approach towards passive ferromagnetic shimming design for a high-resolution NMR magnet

This paper presents a warm-bore ferromagnetic shimming design for a high-resolution NMR magnet. The design is based on spherical harmonic coefficient reduction techniques. We have applied the technique to a 700 MHz magnet completed at the MIT Francis Bitter Magnet Laboratory. The 700 MHz magnet was composed of a low-temperature superconducting (LTS) 600 MHz NMR magnet (L600), purchased from a magnet vendor, and a 100 MHz high-temperature superconducting (HTS) insert (H100), designed and built at FBML. Based on the measured magnetic field data, we derived a total of 22 low-order spherical harmonic coefficients. Another set of spherical harmonic coefficients was calculated for iron pieces attached to the surface of a 50 mm wide and 72 mm high tube. To improve the magnet homogeneity, we applied a multiple objective linear programming method to minimize unwanted spherical harmonic coefficients. A result is a ferromagnetic shimming set of 74 iron pieces. Analysis shows that with this ferromagnetic shimming set, the magnet field homogeneity may be significantly improved, here by a factor of eight.

Anm: a geometrical model for the composite structure of mortar and concrete using real-shape particles

AbstractThe composite geometrical structure of mortar composites can be represented by a model consisting of sand embedded in a cement paste matrix and the structure of concrete by gravel embedded in a mortar matrix. Traditionally, spheres have often been used to represent aggregates (sand and gravel), although the accuracy of properties computed for structures using spherical aggregates as inclusions can be limited when the property contrast between aggregate and matrix is large. In this paper, a new geometrical model is described, which can simulate the composite structures of mortar and concrete with real-shape aggregates. The aggregate shapes are either directly or statistically taken from real particles, using a spherical harmonic expansion, where a set of spherical harmonic coefficients,anm, is used to describe the irregular shape. The model name of Anm is taken from this choice of notation. The take-and-place parking method is employed to put multiple irregular particles together within a pre-determined empty container, which becomes a representative volume element. This representative volume element can then be used as input into some kind of computational material model, which uses other numerical techniques such as finite elements to compute properties of the Anm composite structure.

EGM_TIM_RL05: An independent geoid with centimeter accuracy purely based on the GOCE mission

AbstractAfter more than 4.5 years in orbit, the Gravity field and steady‐state Ocean Circulation Explorer (GOCE) mission ended with the reentry of the satellite on 11 November 2013. This publication serves as a reference for the fifth gravity field model based on the time‐wise approach (EGM_TIM_RL05), a global model only determined from GOCE observations. Due to its independence of any other gravity data, a consistent and homogeneous set of spherical harmonic coefficients up to degree and order 280 (corresponding to spatial resolution of 71.5 km on ground) is provided including a full covariance matrix characterizing the uncertainties of the model. The associated covariance matrix realistically describes the model quality. It is the first model which is purely based on GOCE including all observations collected during the entire mission. The achieved mean global accuracy is 2.4 cm in terms of geoid heights and 0.7 mGal for gravity anomalies at a spatial resolution of 100 km.

SEARCHES FOR LARGE-SCALE ANISOTROPY IN THE ARRIVAL DIRECTIONS OF COSMIC RAYS DETECTED ABOVE ENERGY OF 1019eV AT THE PIERRE AUGER OBSERVATORY AND THE TELESCOPE ARRAY

Spherical harmonic moments are well-suited for capturing anisotropy at any scale in the flux of cosmic rays. An unambiguous measurement of the full set of spherical harmonic coefficients requires full-sky coverage. This can be achieved by combining data from observatories located in both the northern and southern hemispheres. To this end, a joint analysis using data recorded at the Telescope Array and the Pierre Auger Observatory above $10^{19}$ eV is presented in this work. The resulting multipolar expansion of the flux of cosmic rays allows us to perform a series of anisotropy searches, and in particular to report on the angular power spectrum of cosmic rays above $10^{19}$ eV. No significant deviation from isotropic expectations is found throughout the analyses performed. Upper limits on the amplitudes of the dipole and quadrupole moments are derived as a function of the direction in the sky, varying between 7% and 13% for the dipole and between 7% and 10% for a symmetric quadrupole.

Altitude‐adjusted corrected geomagnetic coordinates: Definition and functional approximations

AbstractAnalysis of the functional approximations used to transform between geographic and Altitude‐Adjusted Corrected Geomagnetic (AACGM) coordinates reveals that errors of >50 km can occur in the auroral and polar regions. These errors are the result of efforts to better approximate AACGM coordinates near the magnetic equator and the South Atlantic Anomaly. In these regions AACGM coordinates are not defined and alternate coordinates have been used. This augmentation and emphasis on the solution in regions near the equator result in spherical harmonic approximating functions that are less accurate than need be in the auroral and polar regions. In response, a new set of spherical harmonic coefficients have been derived that better represent AACGM coordinates in these regions. These new AACGM coefficients are limited to below 2000 km in altitude in order to ensure accuracy. For altitudes above 2000 km, a magnetic field‐line tracing solution is recommended. A software package developed to take advantage of the new AACGM coefficients provides the capability of tracing magnetic field lines at any altitude, for improved accuracy. In addition, linear interpolation between 5 year epochs is used to produce coordinates that vary smoothly over the entire period from 1965 to present. The intent of this work is to provide a more accurate procedure for determining AACGM coordinates in the auroral and polar regions for the study of magnetospheric and ionospheric processes.

A spectral assessment review of current satellite‐only and combined Earth gravity models

AbstractThe realization over the last decade of dedicated gravity field satellite missions enabled the production of a series of new satellite‐only and combined models for the Earth's gravity field. Using different sensors, measurement techniques, and algorithmic procedures, the final product in each case is a set of spherical harmonic coefficients representing the series expansion of the gravitational potential up to a certain maximum degree, depending on the mission characteristics and the range of the available data. The present review performs a detailed quantified analysis of a representative selection of currently available CHAMP (Challenging Minisatellite Payload), GRACE (Gravity Recovery and Climate Experiment), GOCE (Gravity Field and Steady‐State Ocean Circulation Explorer), and combined Earth gravity models. In this comparative analysis, we also include the so‐called topographic/isostatic gravity models that represent the contribution of global digital elevation maps for the topography and ocean bathymetry. Applying a range of available spatial and spectral accuracy and assessment measures, such as correlation per degree and order, smoothing per degree and order, signal‐to‐noise ratio, gain, degree variances, and error degree variances, one gains a quantified “peek” inside the quality of these models spanning over their whole spectrum. The applied error and assessment measures are defined both in an absolute and relative manner with respect to other similar models or some reference Earth gravity models. Furthermore, the nature of the performed analysis (degree‐wise, order‐wise, and cumulative) permits the identification of distinct spectral bandwidths in these models, enables the quantification of some standard features of the observed field, such as its “long wavelength”, “short wavelength”, or “very high frequency part”, and specifies the attenuation of the gravity signal with increasing altitude from the Earth's surface. An examination of the assessment quantities reveals certain bandwidths of these models with characteristic statistical features. A band‐limited synthesis of these bandwidths in the space domain quantifies the corresponding contributions in terms of selected gravity field functionals, including second‐order derivatives at GOCE altitude.

A Target Field Approach to Open MRI Magnet Design

The design of an fMRI magnet for the study of the human motor cortex poses a number of challenges due to the necessity of maintaining the subject in a natural, erect position, with free access to the environment. This paper presents the design of an asymmetric superconducting magnet composed of racetrack coils. The coil configuration is derived from the one presented in [1] from which the non-planar coils have been eliminated, with significant simplifications of the manufacturing process. The position of the coils is defined along the principles described in [1] and the current in the coils is determined by means of an approach similar to the Inverse Boundary Method [2-4] in which the goal is established as a set of spherical harmonic coefficients. The major challenge in the design lies in limiting the coil currents to sustainable values without compromising the field homogeneity. The current limit requires an increase in the number of winding layers that, in turn, induces a change in the physical dimensions, making the system nonlinear. It is shown how this can be counteracted by means of an iterative procedure that converges to acceptable results. In the final configuration the magnet accommodates a sitting individual and produces a field intensity of 1.4 T. It consists of planar coils only with current densities and maximum field intensities compatible with currently available superconductors.

Working models for the gravitational field of Phobos

Models for the gravitational field of Mars moon Phobos were developed using the latest shape model and assuming homogeneous density distribution. Three methods were applied in our study. Comparisons were made between these methods and all were shown to yield consistent results. Notably, the most accurate shape model of Phobos to date, complete up to degree and order 17 was used for the first time in our analysis. A set of spherical harmonic coefficients up to degree and order 17 were derived for the gravitational field of Phobos. Also considered was the gravitational field on the surface of Phobos. Typical characteristics as well as some pronounced surface features of this irregular-shaped small body could be conveniently identified. The results are readily applicable for such purposes as spacecraft orbit analysis and assessing the dynamical environment of Phobos.

Contribution of satellite laser ranging to combined gravity field models

In the framework of satellite-only gravity field modeling, satellite laser ranging (SLR) data is typically exploited to recover long-wavelength features. This contribution provides a detailed discussion of the SLR component of GOCO02S, the latest release of combined models within the GOCO series. Over a period of five years (January 2006 to December 2010), observations to LAGEOS-1, LAGEOS-2, Ajisai, Stella, and Starlette were analyzed. We conducted a series of closed-loop simulations and found that estimating monthly sets of spherical harmonic coefficients beyond degree five leads to exceedingly ill-posed normal equation systems. Therefore, we adopted degree five as the spectral resolution for real data analysis. We compared our monthly coefficient estimates of degree two with SLR and Gravity Recovery and Climate Experiment (GRACE) time series provided by the Center for Space Research (CSR) at Austin, Texas. Significant deviations in C 20 were noted between SLR and GRACE; the agreement is better for the non-zonal coefficients. Fitting sinusoids together with a linear trend to our C 20 time series yielded a rate of (−1.75 ± 0.6) × 10 −11/yr; this drift is equivalent to a geoid change from pole to equator of 0.35 ± 0.12 mm/yr or an apparent Greenland mass loss of 178.5 ± 61.2 km 3/yr. The mean of all monthly solutions, averaged over the five-year period, served as input for the satellite-only model GOCO02S. The contribution of SLR to the combined gravity field model is highest for C 20, and hence is essential for the determination of the Earth’s oblateness.

Candidates for multiple impact craters?: Popigai and Chicxulub as seen by the global high resolution gravitational field model EGM2008

Abstract. In 2008 the new Earth Gravitational Model (EGM2008) was released. It contains a complete set of spherical harmonic coefficients of the Earth's gravitational potential (Stokes parameters) to degree 2190 and order 2159 and selected orders to degree 2190, that can be used for evaluation of various potential quantities with both the unprecedented accuracy and high spatial resolution. Two such quantities, the gravity anomaly and second-order radial derivative of the disturbing potential, were computed over selected areas with known impact craters. The displays of these derivatives for two such sites clearly show not only the strong circular-like features known to be associated with them but also other symmetrical structures which appear to make them multiple impact sites. At Popigai, Siberia, the series of circular features fall in a line from the "primary crater" in the southeast (SE) direction. At Chicxulub, Yucatán, there appears to be one more crater close to the "primary" in the northeast (NE) direction, as well as possibly others in the vicinity of the main crater (SW). Gravity information alone is not, however, proof of impact craters but it is useful in identifying candidate sites for further study, for examination by geologists and geophysicists. In the case of Chicxulub, a very recent single seismic profile suggests that a more likely explanation for the observed circular like gravity signal from EGM2008 NE of the "primary" is a pre-impact basin.

Proposal of new models of the bottom-side B0 and B1 parameters for IRI

The time series of hourly electron density profiles N( h) obtained from 27 ionosonde stations distributed world-wide have been used to obtain N( h) average profiles on a monthly basis and to extract the expected bottom-side parameters that define the IRI profile under quiet conditions. The time series embrace the time interval from 1998 to 2006, which practically contains the entire solar cycle 23. The Spherical Harmonic Analysis (SHA) has been used as an analytical technique for modeling globally the B0 and B1 parameters as general functions on a spherical surface. Due to the irregular longitudinal distribution of the stations over the globe, it has been assumed that the ionosphere remains approximately constant in form for a given day under quiet conditions for a particular coordinate system. Since the Earth rotates under a Sun-fixed system, the time differences have been considered to be equivalent to longitude differences. The time dependence has been represented by a two-degree Fourier expansion to model the annual and semiannual variations and the year-by-year analyses of the B0 and B1 have furnished nine sets of spherical harmonic coefficients for each parameter. The spatial–temporal yearly coefficients have been further expressed as linear functions of Rz12 to model the solar cycle dependence. The resultant analytical model provides a tool to predict B0 and B1 at any location distributed among the used range of latitudes (70°N–50°S) and at any time that improves the fit to the observed data with respect to IRI prediction.

SHEF: a vHTS geometrical filter using coefficients of spherical harmonic molecular surfaces

SHEF (spherical harmonic coefficient filter), a geometrical matching procedure constituting a preliminary step in the virtual high throughput screening of large databases of small drug-like molecules, is demonstrated. This filter uses a description of both the binding site of the target and the ligand surfaces using spherical harmonic polynomial expansions. Using this representation, which is based on limited sets of spherical harmonic coefficients, considerably reduces the complexity of surface complementarity calculation. As a first test, 188 known protein-ligand complexes were used, and the results of docking the abstracted ligands into the bare proteins using SHEF were compared to the original X-ray structures. The ability of SHEF to retrieve known ligands "hidden" in a virtual library of 1,000 randomly selected drug-like compounds is also demonstrated.

Rotational dynamics of a solar system body under solar radiation torques

The rotational dynamics of a small solar system body subject to solar radiation torques is investigated. A set of averaged evolutionary equations are derived as an analytic function of a set of spherical harmonic coefficients that describe the torque acting on the body due to solar radiation. The analysis also includes the effect of thermal inertia. The resulting equations are studied and a set of possible dynamical outcomes for the rotation rate and obliquity of a small body are found and characterized.

An Oceanographer's Guide to GOCE and the Geoid

Abstract. A review is given of the geodetic concepts necessary for oceanographers to make use of satellite gravity data to define the geoid, and to interpret the resulting product. The geoid is defined, with particular attention to subtleties related to the representation of the permanent tide, and the way in which the geoid is represented in ocean models. The usual spherical harmonic description of the gravitational field is described, together with the concepts required to calculate a geoid from the spherical harmonic coefficients. A brief description is given of the measurement system in the GOCE satellite mission, scheduled for launch shortly. Finally, a recipe is given for calculation of the ocean dynamic topography, given a map of sea surface height above a reference ellipsoid, a set of spherical harmonic coefficients for the gravitational field, and defining constants.

Sumatra tsunami affects observations by GRACE satellites

Changes in the Earth's gravity field are caused by the redistribution of mass within the Earth and on or above its surface. While previous studies [Tapley, 2004; Wahr, 2004] showed that the Gravity Recovery and Climate Experiment (GRACE) satellite mission, executed by NASA, had successfully determined oceanic mass redistribution, the relative motions between two GRACE satellites caused by the 2004 Sumatra tsunami is still uncertain.This present study combines a numerical model of the tsunami and GRACE orbit data to estimate the realistic effect of oceanic mass redistribution on the inter‐satellite range‐rate change between two GRACE satellites. The GRACE mission is designed to map out the Earth's gravity field to high accuracy. Instead of measuring the Earth gravity field or mass variations directly GRACE estimates a set of spherical harmonic coefficients denoting the Earth gravity field each month by measuring range changes between two spacecrafts [Chambers et al., 2004].

An isostatically compensated gravity model using spherical layer distributions

A new isostatic model for the Earth's gravity field is presented based on a simple hypothesis of layers approximating constant density contrasts. The spherical layer distribution used to describe the hydrostatic equilibrium of the Earth's masses leads to a new set of spherical harmonic coefficients for the gravitational potential. First attempts to quantify the information content of these coefficients led to the outcome that they seem to explain the observed gravity field for a certain wavelength band, while they are insufficient for short and very long wavelengths. A synthesis of the derived coefficients over specific degree ranges provided a computation of band-limited geoid undulations on a global scale. The association of these potential quantities with known tectonic structures, such as the topography of the core-mantle boundary, strengthens the belief that the interpretation of Earth gravity models, especially those arising from global digital elevation models, should be considered in close relation with deep-Earth structure.

Evaluation of band-limited topographical effects in airborne gravimetry

Cutting-edge scalar airborne gravimeters can measure the Earth's gravity field with an accuracy of 1–2 mGal for spatial resolutions as high as 2 km (half wavelength). These specifications make gravity data from airborne systems applicable for geoid determination. Airborne gravity data must however be filtered prior to their intended applications in order to reduce the high-frequency observation noise caused mainly by aircraft dynamics. As a result, only discrete samples of the band-limited representation of the continuous gravity field can be obtained. Such gravity data can however be used successfully for accurate band-limited geoid determination. The second method of Helmert gravity reduction is considered in this contribution by studying procedures that would allow for evaluation of band-limited topographical effects on gravity and the geoid that are consistent with the relatively high accuracy and high spatial resolution of band-limited airborne gravity observations available today and in the foreseeable future. Using a 3×3 arcsec digital elevation model, values of the topographical effects are computed using the second method of Helmert reduction along selected flight lines in the Canadian Rocky Mountains. The values of the topographical effects are processed by the same low-pass filter that is applied to airborne gravity and compared to values computed by newly developed algorithms. This new approach can be used for computation of the topographical effects from a set of spherical harmonic coefficients. It results in a fast and efficient algorithm which can be used worldwide (with the exception of the polar regions) for the reduction of airborne data used for geoid determination. The next step of gravity reduction, the downward continuation, is not treated in this contribution.