Spherical Harmonic Components Research Articles

Two-dimensional Eclipse Mapping of the Hot-Jupiter WASP-43b with JWST MIRI/LRS

We present eclipse maps of the two-dimensional thermal emission from the dayside of the hot-Jupiter WASP-43b, derived from an observation of a phase curve with the JWST MIRI/LRS instrument. The observed eclipse shapes deviate significantly from those expected for a planet emitting uniformly over its surface. We fit a map to this deviation, constructed from spherical harmonics up to order ℓmax=2 , alongside the planetary, orbital, stellar, and systematic parameters. This yields a map with a meridionally averaged eastward hot-spot shift of (7.75 ± 0.36)°, with no significant degeneracy between the map and the additional parameters. We show the latitudinal and longitudinal contributions of the dayside emission structure to the eclipse shape, finding a latitudinal signal of ∼200 ppm and a longitudinal signal of ∼250 ppm. To investigate the sensitivity of the map to the method, we fix the parameters not used for mapping and derive an “eigenmap” fitted with an optimized number of orthogonal phase curves, which yields a similar map to the ℓmax=2 map. We also fit a map up to ℓmax=3 , which shows a smaller hot-spot shift, with a larger uncertainty. These maps are similar to those produced by atmospheric simulations. We conclude that there is a significant mapping signal which constrains the spherical harmonic components of our model up to ℓmax=2 . Alternative mapping models may derive different structures with smaller-scale features; we suggest that further observations of WASP-43b and other planets will drive the development of more robust methods and more accurate maps.

Quantification of Global Cloud Properties With Use of Spherical Harmonic Functions

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

Effect of the low-order coefficients of the Earth gravity model in calculating the satellite orbit

The Earth's gravity model is a crucial factor in determining satellite orbits. Scientific organizations such as GFZ Potsdam in Germany, GRGS Toulouse in France, and AIUB in Switzerland have established Earth gravity models with increasing precision of spherical harmonic coefficients. Low-order coefficients, including C ̅_21,( S) ̅_21, C ̅_10,C ̅_11,( S) ̅_11and C ̅_20, play a vital role in describing changes in the Earth's poles, geometric center, and flattening. To evaluate the impact of these coefficients and understand altimeter satellite orbital error, the Propagerror program was developed. This program calculates satellite orbital error from the differential components of Earth gravity model spherical harmonic coefficients (dC/Slm), which can be obtained from the difference between two gravity models or from seasonal and annual components of spherical harmonics. By separating appropriate low-order components in the Earth gravity models, the Propagerror program enables the estimation of satellite orbital error. In this study, we isolate C ̅_10,C ̅_11,( S) ̅_11, and C ̅_21,( S) ̅_21 coefficients in the EIGEN-GRGS.RL02bis.MF and EIGEN-6S gravity models to assess the geophysical impact on satellite orbits. The influence of the geometry center elements results in a 2 cm error in the Jason-2 satellite, while the rotational axis elements have no effect. The C ̅_31,( S) ̅_31 coefficient has a 6-7 mm impact on the accuracy of the Jason-2 satellite, as demonstrated by the satellite error map in two situations with and without the C ̅_31,( S) ̅_31 harmonic coefficient. This study highlights the significance of regulating function coefficients in satellite orbit determination, particularly the low-level harmonic parameters. The Propagerror program provides insights into the impact of each spherical harmonic parameter on satellite orbits, contributing to the improvement of orbit accuracy and the understanding of the Earth's gravity model.

The Sun’s Mean Line-of-sight Field

In this paper, I regard the Sun-as-a-star magnetic field (i.e., the mean field) as a filter for the spherical harmonic components of the photospheric field, and then calculate the transmission coefficients of this filter. The coefficients for each harmonic, Y l m , are listed in three tables according to their dependence on B 0, the observer’s latitude in the star’s polar coordinate system. These coefficients are used to interpret the 46 yr sequence of daily mean-field measurements at the Wilcox Solar Observatory. I find that the nonaxisymmetric part of the field originates in the , , and a combination of the and harmonic components. The axisymmetric part of the field originates in plus a B 0-dependent combination of the and components. The power spectrum of the field has peaks at frequencies corresponding to the ∼27 day synodic equatorial rotation period and its second and third harmonics. Each of these peaks has fine structure on its low-frequency side, indicating magnetic patterns that rotate slowly under the influence of differential rotation and meridional flow. The sidebands of the fundamental mode resolve into peaks corresponding to periods of ∼28.5 and ∼30 days, which tend to occur at the start of sunspot maximum, whereas the ∼27 day period tends to occur toward the end of sunspot maximum. We might expect similar rotational sidebands to occur in magnetic observations of other Sun-like stars and to be a useful complement to asteroseismology studies of convection and magnetic fields in those stars.

Imaging the southern sky at 159 MHz using spherical harmonics with the engineering development array 2

Abstract One of the major priorities of international radio astronomy is to study the early universe through the detection of the 21 cm HI line from the epoch of reionisation (EoR). Due to the weak nature of the 21 cm signal, an important part in the detection of the EoR is removing contaminating foregrounds from our observations as they are multiple orders of magnitude brighter. In order to achieve this, sky maps spanning a wide range of frequencies and angular scales are required for calibration and foreground subtraction. Complementing the existing low-frequency sky maps, we have constructed a Southern Sky map through spherical harmonic transit interferometry utilising the Engineering Development Array 2 (EDA2), a Square Kilometre Array (SKA) low-frequency array prototype system. We use the m-mode formalism to create an all-sky map at 159 MHz with an angular resolution of 3 degrees, with data from the EDA2 providing information over +60 degrees to –90 degrees in declination. We also introduce a new method for visualising and quantifying how the baseline distribution of an interferometer maps to the spherical harmonics and discuss how prior information can be used to constrain spherical harmonic components that the interferometer is not sensitive to.

Probing the Astrometric Properties of Gaia EDR3 Quasars at the Faintest Magnitudes

Gaia early data release 3 (EDR3) provides about 1.6 million QSO-like objects. Measuring the systematic residuals of the quasars convincingly in Gaia EDR3 is paramount for further fundamental astronomy use. We aim to conduct an independent assessment of the astrometric quality of Gaia EDR3 utilizing a sizable and confirmed extragalactic sources catalog. By positional cross-matching the known quasars to EDR3, we obtain a list of 299,004 confirmed QSOs. Together with the quasars identified in EDR3, we calculated the generalized moving mean of proper motions and parallaxes. Using spherical harmonics, the parallaxes and the proper motions are expanded to investigate their astrometric features. The global parallax bias is ∼−21 μas and ∼−27 μas for the five-parameter solution and the six-parameter solution samples, respectively. The mean proper motion of the five-parameter solution QSO-like subset is consistent with zero, while the mean proper motion of the six-parameter solution subset is a few μas yr−1 biased. The typical rotation and glide vector for the five-parameter solution quasars are (+3, +2, +1) ± 0.5 μas yr−1 and (0, −4, −2) ± 0.5 μas yr−1, respectively. Small systematic dependence of the rotation on the Gaia color and the G magnitude exists. The parallaxes of EDR3 quasars show an rms amplitude of 9.9 μas spatial correlation with the angular of ≥18°. At the level of Gaia EDR3's uncertainties, the quadrupole components of the vectorial spherical harmonics show that the mass-energy density of the gravitational waves ΩGW is about 0.0073 h −2.

Efficient eddy current characterization using a 2D image-based sampling scheme and a model-based fitting approach.

To propose two innovations to existing eddy current characterization techniques, which include (1) an efficient spatio-temporal sampling scheme and (2) a model-based fitting of spherical harmonic eddy current components. This work introduces a three-plane 2D image-based acquisition scheme to efficiently sample eddy current fields. Additionally, a model-based spherical harmonic decomposition is presented, which reduces fitting noise using a rank minimization to impose an exponential decay on the eddy current amplitude evolution. Both techniques are applied in combination and analyzed in simulations for their applicability in reconstructing suitable pre-emphasis parameters. In a proof-of-concept measurement, the routine is tested for its propriety to correctly quantify user-defined field dynamics. Furthermore, based on acquired precompensation and postcompensation eddy current data, the suitability of pre-emphasis parameters calculated based on the proposed technique is evaluated. Simulation results derived from 500 data sets demonstrate the applicability of the acquisition scheme for the spatio-temporal sampling of eddy current fields. Compared with a conventional data processing strategy, the proposed model-based approach yields pre-emphasis parameters that reduce the average maximum residual field offset within a 10-cm-diameter spherical volume from 3.17 Hz to 0.58 Hz. Experimental data prove the proposed routine to be suitable to measure and effectively compensate for eddy currents within 10 minutes of acquisition time. The proposed framework was found to be well-suited to efficiently characterize and compensate for eddy current fields in a one-time calibration effort. It can be applied to facilitate pre-emphasis implementations, such as for dynamic B0 shimming applications.

Synthesis Strategy for Stator Magnetic Field of Permanent Magnet Spherical Motor

Considering the basic role of the stator magnetic field (MF), this paper presents a synthesis strategy for stator MF of a permanent magnet spherical motor (PMSM). Given the structure of the PMSM, the MF of a stator coil group with unit current is obtained, and the spherical harmonic components are decomposed, so the relationship between the whole stator MF and coils currents can be deduced by the rotary transformation and superposition. Referring to the rotor MF, the expected stator MF is determined; then, the stator coil currents are calculated by solving a constrained optimization. Finally, the current excitation strategy of stator coils is verified by finite-element analysis and experiments.

Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz–Mie theory: scattering of light coming from two pinholes

In the scattering theory of light by a spherical particle, the crucial step for solving the scattering problem is to determine the so-called beam-shape coefficients (weighting coefficients of multipolar expansions) for an arbitrary incident electromagnetic field. We present an alternative analytical derivation of such coefficients using the angular spectrum method of field propagation. Different from previous approaches, our derivation requires only recurrence relations and orthogonality properties of associated Legendre functions. We provide closed-form expressions to obtain these coefficients in terms of the distribution of either electric or magnetic field of the secondary source. Our general expressions for obtaining the beam-shape coefficients are applied to the scattering problem in which a sphere is illuminated with light coming from two pinholes. For this problem, the beam-shape coefficients can be analytically obtained and they are merely related to the Cartesian components of the vector spherical harmonics whose arguments are the vector positions from the pinholes to the center of the sphere. Specifically, we simulate the scattering fields when the scatterer is a dielectric sphere (size larger than the wavelength of the incident field) for on- and off-resonance conditions; subwavelength confinement of the electromagnetic field is attained.

Magnetic field topology and chemical spot distributions of the Ap star HD 119419

Context.Analysis of high-resolution spectropolarimetric time-series observations of early-type magnetic stars is currently the most advanced method of obtaining detailed information on their surface magnetic field topologies and horizontal spot distributions.Aims.In this study we analyse a new set of high-quality full Stokes vector observations of the magnetic Ap star HD 119419 – a member of the 14 Myr old Lower Cen-Cru association – for the purpose of studying the surface field topology and mapping the chemical abundance spots.Methods.We made use of the circular and linear polarisation data collected for HD 119419 with the HARPSpol instrument at the ESO 3.6-m telescope. These observations were analysed with a multi-line magnetic diagnostic technique and modelled in detail with a Magnetic Doppler imaging (MDI) code.Results.We present a new set of high-precision mean longitudinal magnetic field measurements and derive a revised stellar rotational period by comparing our measurements with the literature data. We also redetermine the basic stellar atmospheric parameters. Our four Stokes parameter magnetic inversions reveal a moderately complex surface field topology with a mean field strength of 18 kG and a maximum local strength of 24 kG. A poloidal dipolar component dominates the magnetic energy spectrum of the surface field in HD 119419. However, significant contributions of the higher-order spherical harmonic components are also present. We show that the dipole plus quadrupole part of the reconstructed field geometry is incapable of reproducing the observed amplitudes and shapes of the StokesQandUprofiles. The chemical abundance distributions of Fe, Cr, Ti, and Nd, derived self-consistently with the magnetic field geometry, are characterised by large abundance gradients and a lack of clear correlation with the magnetic field structure.Conclusions.This full Stokes vector analysis of HD 119419 extends the modern hot-star magnetic mapping investigations to an open cluster Ap star with a well-determined age. Further, MDI studies of cluster members will allow us to study the field topologies and chemical abundance spots as a function of stellar age.

Real-time localization and tracking of musical instruments with spherical microphone arrays

This research describes a method for dynamic beamforming with a particle filter to localize and track musical instruments in a real-time context. Using a spherical harmonic framework, spherical microphone arrays are able to decompose three-dimensional sound fields into their basic components, which enables detailed analysis and efficient spatial filtering algorithms. In recent years, methods for determining relative source positions around an array using steerable beams have been studied. By creating multiple weighting functions based on spherical harmonic components, many beams can be generated simultaneously and can be used to dynamically track instruments via an iterative process.

Modeling study of seated reach envelopes based on spherical harmonics with consideration of the difficulty ratings

Reach envelopes are very useful for the design and layout of controls. In building reach envelopes, one of the key problems is to represent the reach limits accurately and conveniently. Spherical harmonics are proved to be accurate and convenient method for fitting of the reach capability envelopes. However, extensive study are required on what components of spherical harmonics are needed in fitting the envelope surfaces. For applications in the vehicle industry, an inevitable issue is to construct reach limit surfaces with consideration of the seating positions of the drivers, and it is desirable to use population envelopes rather than individual envelopes. However, it is relatively inconvenient to acquire reach envelopes via a test considering the seating positions of the drivers. In addition, the acquired envelopes are usually unsuitable for use with other vehicle models because they are dependent on the current cab packaging parameters. Therefore, it is of great significance to construct reach envelopes for real vehicle conditions based on individual capability data considering seating positions. Moreover, traditional reach envelopes provide little information regarding the assessment of reach difficulty. The application of reach envelopes will improve design quality by providing difficulty-rating information about reach operations. In this paper, using the laboratory data of seated reach with consideration of the subjective difficulty ratings, the method of modeling reach envelopes is studied based on spherical harmonics. The surface fitting using spherical harmonics is conducted for circumstances both with and without seat adjustments. For use with adjustable seat, the seating position model is introduced to re-locate the test data. The surface fitting is conducted for both population and individual reach envelopes, as well as for boundary envelopes. Comparison of the envelopes of adjustable seat and the SAE J287 control reach envelope shows that the latter is nearly at the middle difficulty level. It is also found that the abilities of reach envelope models in expressing the shape of the reach limits based on spherical harmonics depends both on the terms in the model expression and on the data used to fit the envelope surfaces.

Using hybrid GPU/CPU kernel splitting to accelerate spherical convolutions

We present a general method for accelerating by more than an order of magnitude the convolution of pixelated functions on the sphere with a radially-symmetric kernel. Our method splits the kernel into a compact real-space component and a compact spherical harmonic space component. These components can then be convolved in parallel using an inexpensive commodity GPU and a CPU. We provide models for the computational cost of both real-space and Fourier space convolutions and an estimate for the approximation error. Using these models we can determine the optimum split that minimizes the wall clock time for the convolution while satisfying the desired error bounds. We apply this technique to the problem of simulating a cosmic microwave background (CMB) anisotropy sky map at the resolution typical of the high resolution maps produced by the Planck mission. For the main Planck CMB science channels we achieve a speedup of over a factor of ten, assuming an acceptable fractional rms error of order 10−5 in the power spectrum of the output map.

Multiple 3D Far-Field/Near-Field Moving Target Localization Using Wideband Echo Chirp Signals

In this paper, we propose a technique for active detection and estimation of the range, radial velocity and direction-of-arrival (DOA) of multiple moving targets in the three-dimensional space for both far-field and near-field cases using wideband chirp signals. The proposed technique is an extension of the method recently proposed by Leong in [“Multiple target localization using wideband echo chirp signals,” IEEE Trans. Signal Process., vol. 61, no. 16, pp. 4077-4089, Aug. 2013] for the localization of multiple far-field targets in the two-dimensional space. In our approach, the received signals at a sensor array are first decomposed into the spherical harmonic components and a modal preprocessing step is then carried out on the received wideband signals so that they can be further processed using narrowband techniques for parameter estimation. The application of fractional Fourier transform on the modal preprocessed signals allows the proposed method to identify closely spaced targets. This is demonstrated via computer simulations where the proposed technique achieved low root-mean-square errors in the parameters estimated in a scenario with closely spaced targets.

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

The influence of changes in surface ice-mass redistribution and associated viscoelastic response of the Earth, known as glacial isostatic adjustment (GIA), on the Earth's rotational dynamics has long been known. Equally important is the effect of the changes in the rotational dynamics on the viscoelastic deformation of the Earth. This signal, known as the rotational feedback, or more precisely, the rotational feedback on the sea level equation, has been mathematically described by the sea level equation extended for the term that is proportional to perturbation in the centrifugal potential and the second-degree tidal Love number. The perturbation in the centrifugal force due to changes in the Earth's rotational dynamics enters not only into the sea level equation, but also into the conservation law of linear momentum such that the internal viscoelastic force, the perturbation in the gravitational force and the perturbation in the centrifugal force are in balance. Adding the centrifugal-force perturbation to the linear-momentum balance creates an additional rotational feedback on the viscoelastic deformations of the Earth. We term this feedback mechanism, which is studied in this paper, as the rotational feedback on the linear-momentum balance. We extend both the time-domain method for modelling the GIA response of laterally heterogeneous earth models developed by Martinec and the traditional Laplace-domain method for modelling the GIA-induced rotational response to surface loading by considering the rotational feedback on linear-momentum balance. The correctness of the mathematical extensions of the methods is validated numerically by comparing the polar-motion response to the GIA process and the rotationally induced degree 2 and order 1 spherical harmonic component of the surface vertical displacement and gravity field. We present the difference between the case where the rotational feedback on linear-momentum balance is considered against that where it is not. Numerical simulations show that the resulting difference in radial displacement and sea level change between these situations since the Last Glacial Maximum reaches values of ±25 and ±1.8 m, respectively. Furthermore, the surface deformation pattern is modified by up to 10 per cent in areas of former or ongoing glaciation, but by up to 50 per cent at the bottom of the southern Indian ocean. This also results in the movement of coastlines during the last deglaciation to differ between the two cases due to the difference in the ocean loading, which is seen for instance in the area around Hudson Bay, Canada and along the Chinese, Australian or Argentinian coastlines.

Perceptually evaluating ambisonic reproduction of room acoustics

Spherical microphone array technology allows for the recording of auditory scenes in three dimensions by decomposing a soundfield into its spherical harmonic components. The performance of the array is affected by certain factors in its design, including the size of the array, whether an open or a rigid sphere configuration is used, and the number of sensors and their placement scheme. Ambisonics is a system designed to reconstruct a soundfield from its spherical harmonic components. The process of ambisonic decoding determines the signals that are fed to each loudspeaker in the array to simulate a given soundfield. Such systems have certain accuracy constraints, particularly at higher frequencies, and different decoding methods can be used at those frequencies to recreate more accurate spatial cues, particularly ILD cues. This paper examines how to develop a decoding scheme that addresses the constraints in both the recording and playback phases, and uses binaural modeling to determine its efficacy. Reconstruction of both simulated and real rooms is examined, with respect to the accurate reproduction of important room acoustic metrics.

Surface atmospheric pressure excitation of the translational mode of the inner core

Using hourly atmospheric surface pressure field from ECMWF (European Centre for Medium-Range Weather Forecasts) and from NCEP (National Centers for Environmental Prediction) Climate Forecast System Reanalysis (CFSR) models, we show that atmospheric pressure fluctuations excite the translational oscillation of the inner core, the so-called Slichter mode, to the sub-nanogal level at the Earth surface. The computation is performed using a normal-mode formalism for a spherical, self-gravitating anelastic PREM-like Earth model. We determine the statistical response in the form of power spectral densities of the degree-one spherical harmonic components of the observed pressure field. Both hypotheses of inverted and non-inverted barometer for the ocean response to pressure forcing are considered. Based on previously computed noise levels, we show that the surface excitation amplitude is below the limit of detection of the superconducting gravimeters, making the Slichter mode detection a challenging instrumental task for the near future.

Geometric distortion and shimming considerations in a rotating MR‐linac design due to the influence of low‐level external magnetic fields

This work investigates with simulation the effect of external stray magnetic fields on a recently reported MRI-linac hybrid, which by design will rotate about the patient axis during therapy. During rotation, interactions with magnetic fields from the earth or nearby ferromagnetic structures may cause unacceptable field distortions in the imaging field of view. Optimal approaches for passive shimming implementation, the degree and significance of residual distortion, and an analysis of the active shimming requirements for further correction are examined. Finite element simulations were implemented on two representative types of biplanar magnet designs. Each of these magnet designs, consisting of a 0.2 T four-post and a 0.5 T C-type unit, was simulated with and without an external field on the order of the earth's field (0.5 G) over a range of rotated positions. Through subtraction, the field distribution resulting from the external field alone could be determined. These measured distributions were decomposed into spherical harmonic components, which were then used to investigate the effect of their selective removal to simulate the effects of passive and active shimming. Residual fields after different levels of shim treatment were measured and assessed in terms of their imaging consequence. For both magnet types, the overall success of a passive shim implementation was highly dependent on the orientation for which it was based. If this orientation was chosen incorrectly, the passive shim would correct for the induced fields at that location, but the overall maximal distortion at other locations was exacerbated by up to a factor of two. The choice of passive shim orientation with the least negative consequence was found to be that where the magnet B(0) axis and transaxial component of the external field were aligned. Residual fields after passive shimming and frequency offset were found to be low in the simulated scenarios, contributing to <1 mm of distortion for most standard imaging sequences (based on a 0.5 G external field). However, extremely rapid single-shot sequences could be distorted by these residual fields to well over 5 mm. These residuals when analyzed were found to correspond primarily to second-order spherical harmonic terms. One term in particular was found to account for the vast majority of these residual fields, defined by the product of the two axes perpendicular to the axis of rotation. The implementation of this term would allow the resulting geometric distortion to fall to the order of 1 mm, even for single-shot sequences. After appropriate passive shimming, the imaging distortion due to an external field of 0.5 G was found to be important only in rapid single-shot sequences, which are especially susceptible to field inhomogeneity. Should it be desirable to use these sequences for real-time tracking, made conceivable due to the lower susceptibility concerns at low field, these residual fields should be addressed. The ability to use only one second-order term for this correction will reduce the cost impact of this decision.

Exploring spherical microphone arrays for room acoustic analysis

Spherical microphone arrays offer several advantages over linear microphone arrays and single sensor microphones for room acoustic analysis. Some advantages are the ability to: a) steer the directional response in 3D space, b) change the beampattern shape (independent of the look direction) and c) spatial decomposition of the sound field into spherical harmonic orthonormal components. All of these features are available online and offline meaning that the analysis can be performed after the measurement has been done. We will present standard measurements such as spatially dependent reverberation time, diffuseness, etc. that take advantage of the spherical array decomposition of the soundfield. We will also revisit the spatial correlation function, a measure very suitable for sphercial array based room analysis. Results for various setups will be presented.

Ongoing glacial isostatic contributions to observations of sea level change

Studies determining the contribution of water fluxes to sea level rise typically remove the ongoing effects of glacial isostatic adjustment (GIA). Unfortunately, use of inconsistent terminology between various disciplines has caused confusion as to how contributions from GIA should be removed from altimetry and GRACE measurements. In this paper, we review the physics of the GIA corrections applicable to these measurements and discuss the differing nomenclature between the GIA literature and other studies of sea level change. We then examine a range of estimates for the GIA contribution derived by varying the Earth and ice models employed in the prediction. We find, similar to early studies, that GIA produces a small (compared to the observed value) but systematic contribution to the altimetry estimates, with a maximum range of -0.15 to -0.5 mm yr-1. Moreover, we also find that the GIA contribution to the mass change measured by GRACE over the ocean is significant. In this regard, we demonstrate that confusion in nomenclature between the terms 'absolute sea level' and 'geoid' has led to an overestimation of this contribution in some previous studies. A component of this overestimation is the incorrect inclusion of the direct effect of the contemporaneous perturbations of the rotation vector, which leads to a factor of ˜two larger value of the degree two, order one spherical harmonic component of the model results. Aside from this confusion, uncertainties in Earth model structure and ice sheet history yield a spread of up to 1.4 mm yr-1 in the estimates of this contribution. However, even if the ice and Earth models were perfectly known, the processing techniques used in GRACE data analysis can introduce variations of up to 0.4 mm yr-1. Thus, we conclude that a single-valued 'GIA correction' is not appropriate for sea level studies based on gravity data; each study must estimate a bound on the GIA correction consistent with the adopted data-analysis scheme