Spherical Harmonic Degree Research Articles

BINARITY AND PULSATION IN ALGOL-TYPE BINARY SYSTEM SX DRACONIS

Photometric observations of SX Dra were carried out to determine the properties of the components and pulsational characteristics of the more massive pulsational component. Physical parameters of the component stars were obtained by modeling B and V light curves using the Wilson-Devinney code. Results indicate that SX Dra is a semi-detached system with the secondary component filling its Roche lobe. The O–C data showed parabolic and sinusoidal variation. Cyclic variation in the O–C diagram could be the result of the possible magnetic activity of the secondary component instead of the third body. The secular increase in the orbital period of the system can be interpreted as being the result of mass transfer from the secondary component to the primary one at a rate of 2.1 × 10–7 M ☉ yr–1. Results of a time-series analysis performed after removing binary effects indicated that the hot component shows δ Scuti light variations with pulsational periods of 63 and 73 minutes. Spherical harmonic degrees (l) were determined to be 3 for the first frequency and 1-2 for the second frequency.

Spectral and spatial decomposition of lithospheric magnetic field models using spherical Slepian functions

Global magnetic field models are typically expressed as spherical-harmonic expansion coefficients. Slepian functions are linear combinations of spherical harmonics that produce new basis functions, which vanish approximately outside chosen geographical boundaries but also remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing spherical-harmonic models into portions that have significant magnetic field strength only in selected areas. Slepian functions are spatio-spectrally concentrated, balancing spatial bias and spectral leakage. Here, we employ them as a basis to decompose the global lithospheric magnetic field model MF7 up to degree and order 72, into two distinct regions. One of the resultant fields is concentrated within the ensemble of continental domains, and the other is localized over its complement, the oceans. Our procedure neatly divides the spectral power at each harmonic degree into two parts. The field over the continents dominates the overall crustal magnetic field, and each region has a distinct power-spectral signature. The oceanic power spectrum is approximately flat, while that of the continental region shows increasing power as the spherical-harmonic degree increases.We provide a further breakdown of the field into smaller, non-overlapping continental and oceanic regions, and speculate on the source of the variability in their spectral signatures.

An asteroseismic study of the Southern [formula omitted] Cephei star ALS 3721

We present the results of a new investigation aimed to identify the pulsational characteristics of the Southern β Cephei star ALS 3721. Spectroscopic and multicolour photometric data were acquired at the South African Astronomical Observatory (SAAO), South Africa in 2011. Frequency analysis showed that the oscillations of ALS 3721 could be attributed to the two main frequencies with higher significance. Stellar parameters and projected rotational velocity obtained by the spectra were also used to perform photometric mode identification. In order to determine spherical harmonic degrees (l), a principal method was followed by comparing the observed light amplitude ratios in different passbands with those computed from non-adiabatic pulsation models. In general case, therefore, the spherical harmonic degrees corresponding of the frequencies were found in the expected β Cephei range.

Spatiospectral concentration of vector fields on a sphere

We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.

Implication of the lopsided growth for the viscosity of Earth's inner core

A particular mode of convection, called translation, has recently been put forward as an important mode of inner core dynamics because this mechanism is able to explain the observed east–west asymmetry of P-wave velocity and attenuation (Monnereau et al., 2010). Translation is a particular solution to Navier–Stokes equation with permeable boundary conditions, but depending on the viscosity of the solid core, modes with higher spherical harmonics degree can develop. At low viscosity, these modes can be dominant and dissipate the degree l=1 of thermal heterogeneities. Hence, a viscosity threshold may be expected below which translation cannot take place, thereby constraining the viscosity of iron at inner core conditions.Using a hybrid finite-difference spherical harmonics Navier–Stokes solver, we investigate here the interplay between translation and convection in a 3D spherical model with permeable boundary conditions. Our numerical simulations show the dominance of pure translation for viscosities of the inner core higher than 4×1018Pas. Translation is almost completely hampered by convective motions for viscosities lower than 1017Pas and the phase change becomes an almost impermeable boundary. Between these values, a well developed circulation at the harmonic degree l=1 persists, but composed of localized cold downwellings, a passive upward flow taking place on the opposite side (the melting side). Such a convective structure remains compatible with the seismic asymmetry.

Mode identification in the high-amplitude δ Scuti star V2367 Cyg

We report on a multi-site photometric campaign on the high-amplitude $\delta$ Scuti star V2367 Cyg in order to determine the pulsation modes. We also used high-dispersion spectroscopy to estimate the stellar parameters and projected rotational velocity. Time series multicolour photometry was obtained during a 98-d interval from five different sites. These data were used together with model atmospheres and non-adiabatic pulsation models to identify the spherical harmonic degree of the three independent frequencies of highest amplitude as well as the first two harmonics of the dominant mode. This was accomplished by matching the observed relative light amplitudes and phases in different wavebands with those computed by the models. In general, our results support the assumed mode identifications in a previous analysis of Kepler data.

Subduction to the lower mantle – a comparison between geodynamic and tomographic models

Abstract. It is generally believed that subduction of lithospheric slabs is a major contribution to thermal heterogeneity in Earth's entire mantle and provides a main driving force for mantle flow. Mantle structure can, on the one hand, be inferred from plate tectonic models of subduction history and geodynamic models of mantle flow. On the other hand, seismic tomography models provide important information on mantle heterogeneity. Yet, the two kinds of models are only similar on the largest (1000 s of km) scales and are quite different in their detailed structure. Here, we provide a quantitative assessment how good a fit can be currently achieved with a simple viscous flow geodynamic model. The discrepancy between geodynamic and tomography models can indicate where further model refinement could possibly yield an improved fit. Our geodynamical model is based on 300 Myr of subduction history inferred from a global plate reconstruction. Density anomalies are inserted into the upper mantle beneath subduction zones, and flow and advection of these anomalies is calculated with a spherical harmonic code for a radial viscosity structure constrained by mineral physics and surface observations. Model viscosities in the upper mantle beneath the lithosphere are ~1020 Pas, and viscosity increases to ~1023 Pas in the lower mantle above D". Comparison with tomography models is assessed in terms of correlation, both overall and as a function of depth and spherical harmonic degree. We find that, compared to previous geodynamic and tomography models, correlation is improved, presumably because of advances in both plate reconstructions and mantle flow computations. However, high correlation is still limited to lowest spherical harmonic degrees. An important ingredient to achieve high correlation – in particular at spherical harmonic degree two – is a basal chemical layer. Subduction shapes this layer into two rather stable hot but chemically dense "piles", corresponding to the Pacific and African Large Low Shear Velocity Provinces. Visual comparison along cross sections indicates that sinking speeds in the geodynamic model are somewhat too fast, and should be 2 ± 0.8 cm yr−1 to achieve a better fit.

Lava flow recordings of the Blake and post-Blake excursions: refining the Quaternary Geomagnetic Instability Time Scale (GITS)

Complex polarity reversals in numerical dynamos driven by thermo-chemical convection are analyzed in terms of magnetic field intensity variations, transitional field structure, and other observable characteristics. Our most Earth-like dynamos are characterized by long stable polarity chrons with dipole-dominant magnetic fields punctuated by occasional polarity reversals, and are found within a transition region of parameter space between non-reversing strongly dipolar dynamos and chaotic multi-polar-type dynamos. Dynamos in the transition region have elevated dipole terms, reduced quadrupole terms, magnetic energy that decreases slowly with spherical harmonic degree, and broadband dipole frequency spectra. Their axial dipole intensity histograms are trimodal, with large modes representing the stable polarity states and an intermediate mode representing the transitional multi-polar state. The dipole family intensity exceeds the quadrupole family intensity on the core–mantle boundary during stable polarity times, but during transitions the two families are similar. A complex dynamo model reversal is compared with paleomagnetic reconstructions around the Matuyama–Brunhes polarity transition. Both start with a gradual decrease of the dipole intensity, followed by a precursor reversal and transient polarity recovery, then a rapid dipole collapse and a final reversal that initiates with reverse flux generation in one hemisphere. Virtual geomagnetic poles (VGPs) from sites near the reverse flux trace complex paths and cross the equator several thousand years before the simpler VGP paths from more distant sites, and magnetic intensity variations during the dynamo model reversal correlate with intensity variations inferred for the Matuyama–Brunhes transition.

On the influence of a translating inner core in models of outer core convection

It has recently been proposed that the hemispheric seismic structure of the inner core can be explained by a self-sustained rigid-body translation of the inner core material, resulting in melting of the solid at the leading face and a compensating crystallisation at the trailing face. This process induces a hemispherical variation in the release of light elements and latent heat at the inner-core boundary, the two main sources of thermochemical buoyancy thought to drive convection in the outer core. However, the effect of a translating inner core on outer core convection is presently unknown. In this paper we model convection in the outer core with a nonmagnetic Boussinesq fluid in a rotating spherical shell driven by purely thermal buoyancy, incorporating the effect of a translating inner core by a time-independent spherical harmonic degree and order 1 (Y11) pattern of heat-flux imposed at the inner boundary. The analysis considers Rayleigh numbers up to 10 times the critical value for onset of nonmagnetic convection, a parameter regime where the effects of the inhomogeneous boundary condition are expected to be most pronounced, and focuses on varying q∗, the amplitude of the imposed boundary anomalies. The presence of inner boundary anomalies significantly affects the behaviour of the model system. Increasing q∗ leads to flow patterns dominated by azimuthal jets that span large regions of the shell where radial motion is significantly inhibited. Vigorous convection becomes increasingly confined to isolated regions as q∗ increases; these regions do not drift and always occur in the hemisphere subjected to a higher than average boundary heat-flux. Effects of the inner boundary anomalies are visible at the outer boundary in all inhomogeneous models considered. At low q∗ the expression of inner boundary effects at the core surface is a difference in the flow speed between the two hemispheres. As q∗ increases the spiralling azimuthal jets driven from the inner boundary are clearly visible at the outer boundary. Finally, our results suggest that, when the system is heated from below, a Y11 heat-flux pattern imposed on the inner boundary has a greater overall influence on the spatio-temporal behaviour of the flow than the same pattern imposed at the outer boundary.

Short guide to direct gravitational field modelling with Hotine’s equations

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eotvos tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine’s equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closed-loop tests in which the simulated input is inverted in the least square sense using the rotated Hotine’s equations. The precision achieved is at the level of rounding error with RMS about $$10^{-12}{-}10^{-14}$$ m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.

Saturn’s high degree magnetic moments: Evidence for a unique planetary dynamo

Saturn’s intrinsic magnetic field has long been known to be highly axisymmetric. None of the existing field models goes beyond spherical harmonic degree 3. In this paper, we assume that the axisymmetry extends to higher degrees. This allows us to derive the degree 4 and 5 axial moments by searching their parameter space based on the Cassini Saturn Orbital Insertion (SOI) measurements and on our apriori knowledge of the first three zonal moments for Saturn. Gauss coefficients g40 and g50 are found to be 65±70nT and 185±100nT, respectively. The Lowes spectrum of Saturn’s field takes a zig-zag shape up to degree 5 with pronounced odd degree moments. All first three odd axial moments share the same sign. This reveals that the magnetic flux inside Saturn is strongly concentrated near the spin-poles, in contrast to the field geometry inside the Earth and convection-driven geodynamo models. The mechanically driven spherical Couette dynamo is considered as an alternative which shows intriguingly similar field properties to those observed at Saturn.

HIGH-PRECISION MACLAURIN-BASED MODELS OF ROTATING LIQUID PLANETS

We present an efficient numerical self-consistent field method for calculating a gravitational model of a rotating liquid planet to spherical harmonic degree ~30 and a precision ~10–12 in the external gravity field. The method's accuracy is validated by comparing results, for Jupiter rotation parameters, with the exact Maclaurin constant-density solution. The method can be generalized to non-constant density.

Time-variable gravity field from satellite constellations using the energy integral

The magnetic field mission Swarm, expected to be launched in 2012, comprises a constellation of three satellites. As all of them are equipped with GPS receivers and accelerometers, they can be used for gravity field recovery. We study the capability of a Swarm-like constellation for (time-variable) gravity field recovery and compare it with a gravity field tandem mission of GRACE-type. Due to the lower accuracy of the GPS measurements compared with GRACE low—low satellite-to-satellite tracking (SST), the accuracy of a Swarm derived gravity field cannot compete with the state-of-the-art GRACE models. However, unlike the GRACE mission, Swarm allows for the derivation of GPS-baselines between the satellites in directions other than purely along-track. This makes Swarm an interesting case for studying the error structure of gravity field models derived from various constellation geometries. Therefore, one focus of this study is the general analysis of different baseline constellations independent of the observation accuracy, that is, we do not restrict ourselves to just the actual Swarm case of pure GPS-baselines. To make the results comparable to GRACE, we explicitly study the error behaviour of the different Swarm baseline geometries assuming GRACE-type K-band ranging (KBR) links. This gives an indication of candidate constellations for future missions; at least in regards to the general error structure. To study the potential of Swarm for recovering time-variable components of the gravity field, we have set up a 2-yr simulation to recover annual and semi-annual components of continental hydrology. We show that Swarm has the potential to recover the annual signal up to spherical harmonic degree 6. This is of interest should there be a gap between the end of the GRACE mission and the launch of a follow-on mission. All our simulations make use of the energy integral method. This method is usually (implicitly) formulated for static potential fields. Therefore, it is necessary for our study to investigate the properties of this method when applied to the analysis of time-variable fields.

Three ways to solve the orbit of KIC 11 558 725: a 10-day beaming sdB+WD binary with a pulsating subdwarf

The recently discovered subdwarf B (sdB) pulsator KIC11558725 features a rich g-mode frequency spectrum, with a few low-amplitude p-modes at short periods, and is a promising target for a seismic study aiming to constrain the internal structure of this star, and of sdB stars in general. We have obtained ground-based spectroscopic Balmer-line radial-velocity measurements of KIC11558725, spanning the 2010 and 2011 observing seasons. From these data we have discovered that KIC11558725 is a binary with period P=10.05 d, and that the radial-velocity amplitude of the sdB star is 58 km/s. Consequently the companion of the sdB star has a minimum mass of 0.63 M\odot, and is therefore most likely an unseen white dwarf. We analyse the near-continuous 2010-2011 Kepler light curve to reveal orbital Doppler-beaming light variations at the 238 ppm level, which is consistent with the observed spectroscopic orbital radial-velocity amplitude of the subdwarf. We use the strongest 70 pulsation frequencies in the Kepler light curve of the subdwarf as clocks to derive a third consistent measurement of the orbital radial-velocity amplitude, from the orbital light-travel delay. We use our high signal-to-noise average spectra to study the atmospheric parameters of the sdB star, deriving Teff = 27 910K and log g = 5.41 dex, and find that carbon, nitrogen and oxygen are underabundant relative to the solar mixture. Furthermore, we extract more than 160 significant frequencies from the Kepler light curve. We investigate the pulsation frequencies for expected period spacings and rotational splittings. We find period-spacing sequences of spherical-harmonic degrees \ell=1 and \ell=2, and we associate a large fraction of the g-modes in KIC11558725 with these sequences. From frequency splittings we conclude that the subdwarf is rotating subsynchronously with respect to the orbit.

Global height system unification with GOCE: a simulation study on the indirect bias term in the GBVP approach

One of the main objectives of ESA’s Gravity Field and Steady-State Ocean Circulation mission GOCE (Gravity field and steady-state ocean circulation mission, 1999) is to allow global unification of height systems by directly providing potential differences between benchmarks in different height datum zones. In other words, GOCE provides a globally consistent and unbiased geoid. If this information is combined with ellipsoidal (derived from geodetic space techniques) and physical heights (derived from leveling/gravimetry) at the same benchmarks, datum offsets between the datum zones can be determined and all zones unified. The expected accuracy of GOCE is around 2–3 cm up to spherical harmonic degree nmax ≈ 200. The omission error above this degree amounts to about 30 cm which cannot be neglected. Therefore, terrestrial residual gravity anomalies are necessary to evaluate the medium and short wavelengths of the geoid, i.e. one has to solve the Geodetic Boundary Value Problem (GBVP). The theory of height unification by the GBVP approach is well developed, see e.g. Colombo (A World Vertical Network. Report 296, Department of Geodetic Science and Surveying, 1980) or Rummel and Teunissen (Bull Geod 62:477–498, 1988). Thereby, it must be considered that terrestrial gravity anomalies referring to different datum zones are biased due to the respective datum offsets. Consequently, the height reference surface of a specific datum zone deviates from the unbiased geoid not only due to its own datum offset (direct bias term) but is also indirectly affected by the integration of biased gravity anomalies. The latter effect is called the indirect bias term and it considerably complicates the adjustment model for global height unification. If no satellite based gravity model is employed, this error amounts to about the same size as the datum offsets, i.e. 1–2 m globally. We show that this value decreases if a satellite-only gravity model is used. Specifically for GOCE with nmax ≈ 200, the error can be expected not to exceed the level of 1 cm, allowing the effect to be neglected in practical height unification. The results are supported by recent findings by Gatti et al. (J Geod, 2012).

The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients $a_{l}^{m}$ of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form θ p or |θ−θ 0| p respectively, where θ is the co-latitude and p>−1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, $E(l) = \sqrt{\sum_{m} (a_{l}^{m})^{2}}$ where l and m are the spherical harmonic degree and order, of l −(p+3/2) or l −(p+1) respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of E(l) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface.

Time-dependent convection models of mantle thermal structure constrained by seismic tomography and geodynamics: implications for mantle plume dynamics and CMB heat flux

SUMMARY One of the outstanding problems in modern geodynamics is the development of thermal convection models that are consistent with the present-day flow dynamics in the Earth’s mantle, in accord with seismic tomographic images of 3-D Earth structure, and that are also capable of providing a time-dependent evolution of the mantle thermal structure that is as ‘realistic’ (Earth-like) as possible. A successful realization of this objective would provide a realistic model of 3-D mantle convection that has optimal consistency with a wide suite of seismic, geodynamic and mineral physical constraints on mantle structure and thermodynamic properties. To address this challenge, we have constructed a time-dependent, compressible convection model in 3-D spherical geometry that is consistent with tomography-based instantaneous flow dynamics, using an updated and revised pseudo-spectral numerical method. The novel feature of our numerical solutions is that the equations of conservation of mass and momentum are solved only once in terms of spectral Green’s functions. We initially focus on the theory and numerical methods employed to solve the equation of thermal energy conservation using the Green’s function solutions for the equation of motion, with special attention placed on the numerical accuracy and stability of the convection solutions. A particular concern is the verification of the global energy balance in the dissipative, compressible-mantle formulation we adopt. Such validation is essential because we then present geodynamically constrained convection solutions over billion-year timescales, starting from present-day seismically constrained thermal images of the mantle. The use of geodynamically constrained spectral Green’s functions facilitates the modelling of the dynamic impact on the mantle evolution of: (1) depth-dependent thermal conductivity profiles, (2) extreme variations of viscosity over depth and (3) different surface boundary conditions, in this case mobile surface plates and a rigid surface. The thermal interpretation of seismic tomography models does not provide a radial profile of the horizontally averaged temperature (i.e. the geotherm) in the mantle. One important goal of this study is to obtain a steady-state geotherm with boundary layers which satisfies energy balance of the system and provides the starting point for more realistic numerical simulations of the Earth’s evolution. We obtain surface heat flux in the range of Earth-like values : 37 TW for a rigid surface and 44 TW for a surface with tectonic plates coupled to the mantle flow. Also, our convection simulations deliver CMB heat flux that is on the high end of previously estimated values, namely 13 TW and 20 TW, for rigid and plate-like surface boundary conditions, respectively. We finally employ these two end-member surface boundary conditions to explore the very-long-time scale evolution of convection over billion-year time windows. These billion-year-scale simulations will allow us to determine the extent to which a ‘memory’ of the starting tomography-based thermal structure is preserved and hence to explore the longevity of the structures in the present-day mantle. The two surface boundary conditions, along with the geodynamically inferred radial viscosity profiles, yield steady-state convective flows that are dominated by long wavelengths throughout the lower mantle. The rigid-surface condition yields a spectrum of mantle heterogeneity dominated by spherical harmonic degree 3 and 4, and the plate-like surface condition yields a pattern dominated by degree 1. Our exploration of the time-dependence of the spatial heterogeneity shows that, for both types of surface boundary condition, deep-mantle hot upwellings resolved in the present-day tomography model are durable and stable features. These deeply rooted mantle plumes show remarkable longevity over very long geological time spans, mainly owing to the geodynamically inferred high viscosity in the lower mantle.

Anomalously weak solar convection.

Convection in the solar interior is thought to comprise structures on a spectrum of scales. This conclusion emerges from phenomenological studies and numerical simulations, though neither covers the proper range of dynamical parameters of solar convection. Here, we analyze observations of the wavefield in the solar photosphere using techniques of time-distance helioseismology to image flows in the solar interior. We downsample and synthesize 900 billion wavefield observations to produce 3 billion cross-correlations, which we average and fit, measuring 5 million wave travel times. Using these travel times, we deduce the underlying flow systems and study their statistics to bound convective velocity magnitudes in the solar interior, as a function of depth and spherical-harmonic degree ℓ. Within the wavenumber band ℓ < 60, convective velocities are 20-100 times weaker than current theoretical estimates. This constraint suggests the prevalence of a different paradigm of turbulence from that predicted by existing models, prompting the question: what mechanism transports the heat flux of a solar luminosity outwards? Advection is dominated by Coriolis forces for wavenumbers ℓ < 60, with Rossby numbers smaller than approximately 10(-2) at r/R([symbol: see text]) = 0.96, suggesting that the Sun may be a much faster rotator than previously thought, and that large-scale convection may be quasi-geostrophic. The fact that isorotation contours in the Sun are not coaligned with the axis of rotation suggests the presence of a latitudinal entropy gradient.

Assessment of the GOCE-Based Global Gravity Models in Canada

The aim of this study is to test the first, second and third generation GOCE geoid solutions, obtained from the first 2, 8 and 18-month observations, respectively. These solutions are assessed over Canada and for two sub-regions (the Great Lakes and Rocky Mountains). The Canadian GPS/leveling-derived geoid heights are used as independent control values in the assessment of the GOCE geoid models. The study is conducted in two steps. First, the geoid models are computed from satellite-only models and truncated to different spherical harmonic degrees. These models are compared with the GPS/leveling geoid heights which are reduced to the same spectral band as the satellite models by EGM2008 predicted frequency components higher than the truncation degrees. The results suggest that the GOCE models show a full power of signal up to about spherical harmonic degree 180. Moreover, the second and third generation GOCE models (with the exception of the direct approach models) provide better agreement with the GPS/leveling-derived geoid undulations than the first generation models due to the longer observation period. The second step involves the combination of the two third generation GOCE models with terrestrial data. These models are tested against to the GPS/leveling-derived geoid undulations in full spectrum. EGM2008 global geopotential model and Canadian gravimetric geoid model CGG2005 are also included in the comparisons to measure improvement provided by the GOCE models. The GOCE-combined models yielded GPS/leveling results that are comparable with those obtained from EGM2008 and CGG2005 models. The best comparative results with the combined models give standard deviations of 4.8 cm, 6.0 cm and 12.2 cm for the Great Lakes, Rocky Mountains and Canada, respectively. These results indicate that the third generation GOCE models conform to the Canadian terrestrial gravity data from degrees 90 to 180. The new generation models show evident improvement over the first and second generation models.

WE-G-217A-07: Increased MR Spatial Accuracy with Improved Gradient Nonlinearity and Magnet Inhomogeneity Correction.

To develop improved distortion correction of MR images based on higher degree spherical harmonic models of the gradient system and the main magnetic field. The induced field gradient along all three axes can be modeled by first order spherical harmonics. These models provide a true encoding of the physical location of a spin to the frequency at which it is detected. Currently on many commercial systems, only the lower 5 degrees of the model are used for gradient nonlinearity correction. While this provides sufficient accuracy for diagnostic imaging, the gradient nonlinearity correction was extended to include all first order harmonics up to the 9th degree to improve the spatial accuracy of the images. Using zeroth degree spherical harmonics up to the 20th order, a model of the main magnetic field was also incorporated into the correction algorithm. Shifts caused by field inhomogeneity were calculated using knowledge of the receiver bandwidth, frequency encode direction, and the magnetic field at any given point. These corrections were applied to images of a 50 cm diameter phantom, acquired with an extended FOV 3D FGRE sequence. Any improvements in spatial accuracy were measured in the resulting images. Visual improvements in spatial accuracy were observed with both corrections. With standard gradient nonlinearity correction, edges of the phantom were distorted in a wave-like fashion. With accurate models, almost all of the errors at the edges of the phantom were corrected when both gradient and field homogeneity corrections were applied. With accurate models of the gradient and magnetic field, the two greatest system-induced spatial errors can be corrected. This improved spatial accuracy enables the use of widebore MR scanners for therapy planning on large FOV images and guidance of percutaneous devices. Further applications include extended FOV imaging for combined PET-MR systems. All authors are employed by General Electric Healthcare.