Harmonic Degree Research Articles

A statistical assessment of seismic models of the U.S. continental crust using Bayesian inversion of ambient noise surface wave dispersion data

AbstractWe present a new approach for evaluating existing crustal models using ambient noise data sets and its associated uncertainties. We use a transdimensional hierarchical Bayesian inversion approach to invert ambient noise surface wave phase dispersion maps for Love and Rayleigh waves using measurements obtained from Ekström (2014). Spatiospectral analysis shows that our results are comparable to a linear least squares inverse approach (except at higher harmonic degrees), but the procedure has additional advantages: (1) it yields an autoadaptive parameterization that follows Earth structure without making restricting assumptions on model resolution (regularization or damping) and data errors; (2) it can recover non‐Gaussian phase velocity probability distributions while quantifying the sources of uncertainties in the data measurements and modeling procedure; and (3) it enables statistical assessments of different crustal models (e.g., CRUST1.0, LITHO1.0, and NACr14) using variable resolution residual and standard deviation maps estimated from the ensemble. These assessments show that in the stable old crust of the Archean, the misfits are statistically negligible, requiring no significant update to crustal models from the ambient noise data set. In other regions of the U.S., significant updates to regionalization and crustal structure are expected especially in the shallow sedimentary basins and the tectonically active regions, where the differences between model predictions and data are statistically significant.

The reversed and normal flux contributions to axial dipole decay for 1880–2015

The axial dipole component of Earth’s internal magnetic field has been weakening since at least 1840, an effect widely believed to be attributed to the evolution of reversed flux patches (RFPs). These are regions on the core-mantle boundary (CMB) where the sign of radial flux deviates from that of the dominant sign of hemispheric radial flux. We study dipole change over the past 135years using the field models gufm1, COV-OBS.x1 and CHAOS-6; we examine the impact of the choice of magnetic equator on the identification of reversed flux, the contribution of reversed and normal flux to axial dipole decay, and how reversed and normal field evolution has influenced the axial dipole. We show that a magnetic equator defined as a null-flux curve of the magnetic field truncated at spherical harmonic degree 3 allows us to robustly identify reversed flux, which we demonstrate is a feature of at least degree 4 or 5. Additionally, our results indicate that the evolution of reversed flux accounts for approximately two-thirds of the decay of the axial dipole, while one third of the decay is attributed to the evolution of the normal field. We find that the decay of the axial dipole over the 20th century is associated with both the expansion and poleward migration of reversed flux patches. In contrast to this centennial evolution, changes in the structure of secular variation since epoch 2000 indicate that poleward migration currently plays a much reduced role in the ongoing dipole decay.

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $l_c$, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $l_c=1$ and $l_c=2$ cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $Le<1$ but only continuous transition for $Le>1$. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2$l_c$ dimensional sphere and consists entirely of degenerate steady state solutions.

Regional spatiotemporal variations of a nondipole magnetic field over the Chinese mainland and neighboring regions in millennial scale

Spatiotemporal variations of the nondipole (ND) magnetic field over the Chinese mainland and neighboring regions from 10000 BC to 1990 AD were analyzed using the latest global geomagnetic models CALS10K.1b, CALS3K.4, and IGRF11. Moreover, for field sources, we investigated 2 n (n = 2–10) pole ND fields and their energies. The results suggest that the study period can be divided into three. The intensity of the ND field has been mainly positive since 10000 BC and lasted almost 7500 years, then gradually decreased to negative in 2500 BC to 1500 AD, and finally sharply increased to positive. The anomaly areas of the ND field in East Asia took shape for n = 3, when the anomaly areas in East Asia were shaped into closed circles in the mainland. This suggests that the first three harmonic degrees account for most of the ND field. The energy of the ND field rapidly attenuates at the core–mantle boundary and is stable at the surface.

On the frequency spectra of the core magnetic field Gauss coefficients

From monthly mean observatory data spanning 1957–2014, geomagnetic field secular variation values were calculated by annual differences. Estimates of the spherical harmonic Gauss coefficients of the core field secular variation were then derived by applying a correlation based modelling. Finally, a Fourier transform was applied to the time series of the Gauss coefficients. This process led to reliable temporal spectra of the Gauss coefficients up to spherical harmonic degree 5 or 6, and down to periods as short as 1 or 2years depending on the coefficient. We observed that a k-2 slope, where k is the frequency, is an acceptable approximation for these spectra, with possibly an exception for the dipole field. The monthly estimates of the core field secular variation at the observatory sites also show that large and rapid variations of the latter happen. This is an indication that geomagnetic jerks are frequent phenomena and that significant secular variation signals at short time scales – i.e. less than 2years, could still be extracted from data to reveal an unexplored part of the core dynamics.

Domains of pulsational instability of low-frequency modes in rotating upper main sequence stars

We determine instability domains on the Hertzsprung–Russell diagram for rotating main sequence stars with masses of 2–20 M⊙. The effects of the Coriolis force are treated wihin the traditional approximation. High-order g modes with harmonic degrees ℓ up to 4 and mixed gravity–Rossby modes with |m| up to 4 are considered. We include the effects of rotation in wider instability strips for a given ℓ compared to the non-rotating case and in an extension of the pulsational instability to hotter and more massive models. We present results for a fixed value of the initial rotation velocity as well as for a fixed ratio of the angular rotation frequency to its critical value. Moreover, we check how the initial hydrogen abundance, metallicity, overshooting from the convective core and opacity affect the pulsational instability domains. The effect of rotation on the period spacing is also discussed.

On equatorially symmetric and antisymmetric geomagnetic secular variation timescales

It has been suggested that the secular variation (SV) timescales of the geomagnetic field vary as 1/ℓ (where ℓ is the spherical harmonic degree), except for the dipole. Here we propose that the same scaling law applies for SV timescales defined for different symmetry classes of the geomagnetic field and SV. We decompose the field and its SV into symmetric and antisymmetric parts and show in geomagnetic field models and numerical dynamo simulations that the corresponding SV timescales also vary as 1/ℓ, again except for the dipole. The time-average antisymmetric/symmetric SV timescales are larger/smaller than the total, respectively. The difference in SV timescales between these two symmetry classes is probably due to different degrees of alignment of the core flow with different magnetic field structures at the core-mantle boundary. The symmetric dipole SV timescale in the recent geomagnetic field and in long-term time-averages from numerical dynamos is below the extrapolated 1/ℓ curve, whereas before ∼1965 the geomagnetic dipole tilt was rather steady and the symmetric dipole SV timescale exceeded the extrapolated 1/ℓ curve. We hypothesize that the period of nearly steady geomagnetic dipole tilt between 1810–1965 was anomalous for the geodynamo. Overall, the deviation of the dipole SV timescales from the 1/ℓ curves may indicate that magnetic diffusion contributes to the dipole SV more than it does for higher degrees.

A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models

We compile the GOCE-only satellite model GOSG01S complete to spherical harmonic degree of 220 using Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations along the GOCE orbit based on applying a least-squares analysis. The diagonal components (Vxx, Vyy, Vzz) of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations (ax, ay, az) along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. The analysis of spectral accuracy characteristics of the newly derived gravitational model GOSG01S and the existing models GOTIM04S, GODIR04S, GOSPW04S and JYY_GOCE02S based on their comparison with the ultra-high degree model EIGEN-6C2 reveals a significant consistency at the spectral window approximately between 80 and 190 due to the same period SGG data used to compile these models. The GOCE related satellite gravity models GOSG01S, GOTIM05S, GODIR05S, GOTIM04S, GODIR04S, GOSPW04S, JYY_GOCE02S, EIGEN-6C2 and EGM2008 are also validated by using GPS-leveling data in China and USA. According to the truncation at degree 200, the statistic results show that all GGMs have very similar differences at GPS-leveling points in USA, and all GOCE related gravity models have better performance than EGM2008 in China. This suggests that all these models provide much more information on the gravity field than EGM2008 in areas with low terrestrial gravity coverage. And STDs of height anomaly differences in China for the selected truncation degrees show that GOCE has improved the accuracy of the global models beyond degree 90 and the accuracies of the models improve from 24 cm to 16 cm. STDs of geoid height differences in USA show that GOSG01S model has best consistency comparing with GPS-leveling data for the frequency band of the degree between 20 and 160.

Gravity and zonal flows of giant planets: From the Euler equation to the thermal wind equation

AbstractAny nonspherical distribution of density inside planets and stars gives rise to a nonspherical external gravity and change of shape. If part or all of the observed zonal flows at the cloud deck of Jupiter and Saturn represent deep interior dynamics, then the density perturbations associated with the deep zonal flows could generate gravitational signals detectable by the Juno mission and the Cassini Grand Finale. Here we present a critical examination of the applicability of the thermal wind equation to calculate the wind‐induced gravity moments. Our analysis shows that wind‐induced gravity moments calculated from the thermal wind equation (TWE) are in overall agreement with the full solution to the Euler equation. However, the accuracy of individual high‐degree moments calculated from TWE depends crucially on retaining the nonsphericity of the background density and gravity. Only when the background nonsphericity of the planet is taken into account does the TWE make accurate enough prediction (with a few tens of percent errors) for individual high‐degree gravity moments associated with deep zonal flows. Since the TWE is derived from the curl of the Euler equation and is a local relation, it necessarily says nothing about any density perturbations that contribute irrotational terms to the Euler equation and that have a nonlocal origin. However, the predicted corrections from these density contributions to the low harmonic degree gravity moments are not discernible from insignificant changes in interior models, while the corrections at high harmonic degree are very small, a few percent or less.

Small-scale density variations in the lunar crust revealed by GRAIL

Data from the Gravity Recovery and Interior Laboratory (GRAIL) mission have revealed that ∼98% of the power of the gravity signal of the Moon at high spherical harmonic degrees correlates with the topography. The remaining 2% of the signal, which cannot be explained by topography, contains information about density variations within the crust. These high-degree Bouguer gravity anomalies are likely caused by small-scale (10′s of km) shallow density variations. Here we use gravity inversions to model the small-scale three-dimensional variations in the density of the lunar crust. Inversion results from three non-descript areas yield shallow density variations in the range of 100–200kg/m3. Three end-member scenarios of variations in porosity, intrusions into the crust, and variations in bulk crustal composition were tested as possible sources of the density variations. We find that the density anomalies can be caused entirely by changes in porosity. Characteristics of density anomalies in the South Pole-Aitken basin also support porosity as a primary source of these variations. Mafic intrusions into the crust could explain many, but not all of the anomalies. Additionally, variations in crustal composition revealed by spectral data could only explain a small fraction of the density anomalies. Nevertheless, all three sources of density variations likely contribute. Collectively, results from this study of GRAIL gravity data, combined with other studies of remote sensing data and lunar samples, show that the lunar crust exhibits variations in density by ± 10% over scales ranging fromcentimeters to100′s of kilometers.

Generation of MAC waves by convection in Earth's core

Author(s): Jaupart, E; Buffett, B | Abstract: Convection in Earth's core is a viable mechanism for generating MAC waves when the top of the core is stably stratified. We quantify the generation mechanism by extending the physical description of MAC waves to include a source term due to buoyancy forces in the convecting part of the core. Solutions for the forced motion are obtained using a Green's function, which is constructed from the eigenfunctions for the unforced motion. When the source term is evaluated using the output of a numerical geodynamo model, the largest excitation occurs at even spherical harmonic degrees, corresponding to waves with symmetric azimuthal flow about the equator. We also find that the magnitude of the source term decreases at periods shorter than about 60 yr. As a result most of the wave generation is confined to waves with periods of 60 yr or longer. Quantitative predictions for the wave amplitudes depend on the projection of the source term into the eigenfunction of the waves. Strong stratification limits the penetration of density anomalies into the stratified layer, which means that the source term is confined to the lowermost part of the layer. Overtones of MAC waves with large amplitudes in the lower part of the stratified layer are more effectively generated by convection, even though these waves are heavily damped by magnetic diffusion. Generation of MAC waves by convection establishes a physical link between observable wave motion and deeper convective processes. Detection of changes in the amplitude and phase of MAC waves would constrain the generation processes and offer insights into the nature of the convection.

Airborne geoid mapping of land and sea areas of East Malaysia

Abstract This paper describes the development of a new geoid-based vertical datum from airborne gravity data, by the Department of Survey and Mapping Malaysia, on land and in the South China Sea out of the coast of East Malaysia region, covering an area of about 610,000 square kilometres. More than 107,000 km flight line of airborne gravity data over land and marine areas of East Malaysia has been combined to provide a seamless land-to-sea gravity field coverage; with an estimated accuracy of better than 2.0 mGal. The iMAR-IMU processed gravity anomaly data has been used during a 2014-2016 airborne survey to extend a composite gravity solution across a number of minor gaps on selected lines, using a draping technique. The geoid computations were all done with the GRAVSOFT suite of programs from DTU-Space. EGM2008 augmented with GOCE spherical harmonic model has been used to spherical harmonic degree N = 720. The gravimetric geoid first was tied at one tide-gauge (in Kota Kinabalu, KK2019) to produce a fitted geoid, my_geoid2017_fit_kk. The fitted geoid was offset from the gravimetric geoid by +0.852 m, based on the comparison at the tide-gauge benchmark KK2019. Consequently, orthometric height at the six other tide gauge stations was computed from HGPS Lev = hGPS - Nmy_geoid2017_.t_kk. Comparison of the conventional (HLev) and GPS-levelling heights (HGPS Lev) at the six tide gauge locations indicate RMS height difference of 2.6 cm. The final gravimetric geoidwas fitted to the seven tide gauge stations and is known as my_geoid2017_fit_east. The accuracy of the gravimetric geoid is estimated to be better than 5 cm across most of East Malaysia land and marine areas

Sub-basin-scale sea level budgets from satellite altimetry, Argo floats and satellite gravimetry: a case study in the North Atlantic Ocean

Abstract. In this study, for the first time, an attempt is made to close the sea level budget on a sub-basin scale in terms of trend and amplitude of the annual cycle. We also compare the residual time series after removing the trend, the semiannual and the annual signals. To obtain errors for altimetry and Argo, full variance–covariance matrices are computed using correlation functions and their errors are fully propagated. For altimetry, we apply a geographically dependent intermission bias [Ablain et al.(2015)], which leads to differences in trends up to 0.8 mm yr−1. Since Argo float measurements are non-homogeneously spaced, steric sea levels are first objectively interpolated onto a grid before averaging. For the Gravity Recovery And Climate Experiment (GRACE), gravity fields full variance–covariance matrices are used to propagate errors and statistically filter the gravity fields. We use four different filtered gravity field solutions and determine which post-processing strategy is best for budget closure. As a reference, the standard 96 degree Dense Decorrelation Kernel-5 (DDK5)-filtered Center for Space Research (CSR) solution is used to compute the mass component (MC). A comparison is made with two anisotropic Wiener-filtered CSR solutions up to degree and order 60 and 96 and a Wiener-filtered 90 degree ITSG solution. Budgets are computed for 10 polygons in the North Atlantic Ocean, defined in a way that the error on the trend of the MC plus steric sea level remains within 1 mm yr−1. Using the anisotropic Wiener filter on CSR gravity fields expanded up to spherical harmonic degree 96, it is possible to close the sea level budget in 9 of 10 sub-basins in terms of trend. Wiener-filtered Institute of Theoretical geodesy and Satellite Geodesy (ITSG) and the standard DDK5-filtered CSR solutions also close the trend budget if a glacial isostatic adjustment (GIA) correction error of 10–20 % is applied; however, the performance of the DDK5-filtered solution strongly depends on the orientation of the polygon due to residual striping. In 7 of 10 sub-basins, the budget of the annual cycle is closed, using the DDK5-filtered CSR or the Wiener-filtered ITSG solutions. The Wiener-filtered 60 and 96 degree CSR solutions, in combination with Argo, lack amplitude and suffer from what appears to be hydrological leakage in the Amazon and Sahel regions. After reducing the trend, the semiannual and the annual signals, 24–53 % of the residual variance in altimetry-derived sea level time series is explained by the combination of Argo steric sea levels and the Wiener-filtered ITSG MC. Based on this, we believe that the best overall solution for the MC of the sub-basin-scale budgets is the Wiener-filtered ITSG gravity fields. The interannual variability is primarily a steric signal in the North Atlantic Ocean, so for this the choice of filter and gravity field solution is not really significant.

A high resolution model of linear trend in mass variations from DMT-2: Added value of accounting for coloured noise in GRACE data

We present a high resolution model of the linear trend in the Earth's mass variations based on DMT-2 (Delft Mass Transport model, release 2). DMT-2 was produced primarily from K-Band Ranging (KBR) data of the Gravity Recovery And Climate Experiment (GRACE). It comprises a time series of monthly solutions complete to spherical harmonic degree 120. A novel feature in its production was the accurate computation and incorporation of stochastic properties of coloured noise when processing KBR data. The unconstrained DMT-2 monthly solutions are used to estimate the linear trend together with a bias, as well as annual and semi-annual sinusoidal terms. The linear term is further processed with an anisotropic Wiener filter, which uses full noise and signal covariance matrices. Given the fact that noise in an unconstrained model of the trend is reduced substantially as compared to monthly solutions, the Wiener filter associated with the trend is much less aggressive compared to a Wiener filter applied to monthly solutions. Consequently, the trend estimate shows an enhanced spatial resolution. It allows signals in relatively small water bodies, such as Aral sea and Ladoga lake, to be detected. Over the ice sheets, it allows for a clear identification of signals associated with some outlet glaciers or their groups. We compare the obtained trend estimate with the ones from the CSR-RL05 model using (i) the same approach based on monthly noise covariance matrices and (ii) a commonly-used approach based on the DDK-filtered monthly solutions. We use satellite altimetry data as independent control data. The comparison demonstrates a high spatial resolution of the DMT-2 linear trend. We link this to the usage of high-accuracy monthly noise covariance matrices, which is due to an accurate computation and incorporation of coloured noise when processing KBR data. A preliminary comparison of the linear trend based on DMT-2 with that computed from GSFC_global_mascons_v01 reveals, among other, a high concentration of the signal along the coast for both models in areas like the ice sheets, Gulf of Alaska, and Iceland.

Can the comprehensive model phase 4 (CM4) predict the geomagnetic diurnal field for days away from quiet time?

Abstract. The most recent comprehensive model (CM4) of the geomagnetic field (Sabaka et al., 2004) has been used in conjunction with geomagnetic ground observatory station data to analyse and study the geomagnetic diurnal variation field for days away from quiet time and the CM4 prediction for these times. Even though much has been learnt about many components of the geomagnetic field, the diurnal variation field behaviour for days away from quiet time (moderately disturbed time) has not been intensively studied. Consequently, we analyse these, and the predictive ability of the CM4 for ground variations, and whether the CM4 prediction of the diurnal variation (whether at quiet time or away from quiet time) is valid outside the period of reference that from which the data were used in modelling. In carrying out the study, we compared the observatory station data and the CM4 prediction directly. Using the CM4 code, well-characterised internal and magnetospheric components were subtracted from the data, plots and global maps of the residual field generated and then compared with the CM4 to see how well the model performed in predicting the data at moderately disturbed time (Kp ≤ 5). The results show that the CM4 is valid and produces useful predictions outside the period covering the timespan of the model and during moderately disturbed time, despite the lack of active data in the original model dataset. The model predictability of the data increases as we move to higher spherical harmonic degree truncation, as the model–data misfit is reduced, but with increased roughness as a result of small-scale features incorporated. The observed results show that this relationship between the increase in spherical harmonic degree truncation and reduction in misfit can be restricted by data quality or quantity and global coverage or spread.

Asteroseismology of theδScuti star HD 50844

Aims. We aim to probe the internal structure and investigate more detailed information of the $\delta$ Scuti star HD 50844 with asteroseismology. Methods. We analyse the observed frequencies of the $\delta$ Scuti star HD 50844 obtained by Balona (2014), and search for possible multiplets based on the rotational splitting law of g-mode. We tried to disentangle the frequency spectra of HD 50844 by means of the rotational splitting only. We then compare them with theoretical pulsation modes, which correspond to stellar evolutionary models with various sets of initial metallicity and stellar mass, to find the best-fitting model. Results. There are three multiplets including two complete triplets and one incomplete quintuplet, in which mode identifications for spherical harmonic degree $l$ and azimuthal number $m$ are unique. The corresponding rotational period of HD 50844 is found to be 2.44$^{+0.13}_{-0.08}$ days. The physical parameters of HD 50844 are well limited in a small region by three modes identified as nonradial ones ($f_{11}$, $f_{22}$, and $f_{29}$) and by the fundamental radial mode ($f_{4}$). Our results show that the three nonradial modes ($f_{11}$, $f_{22}$, and $f_{29}$) are all mixed modes, which mainly represent the property of the helium core. The fundamental radial mode ($f_{4}$) mainly represents the property of the stellar envelope. In order to fit these four pulsation modes, both the helium core and the stellar envelope must be matched to the actual structure of HD 50844. Finally, the mass of the helium core of HD 50844 is estimated to be 0.173 $\pm$ 0.004 $M_{\odot}$ for the first time. The physical parameters of HD 50844 are determined to be $M=$ 1.81 $\pm$ 0.01 $M_{\odot}$, $Z=$ 0.008 $\pm$ 0.001. $T_{\rm eff}=$ 7508 $\pm$ 125 K, log$g=$ 3.658 $\pm$ 0.004, $R=$ 3.300 $\pm$ 0.023 $R_{\odot}$, $L=$ 30.98 $\pm$ 2.39 $L_{\odot}$.

Seismic evidence for a change in the large‐scale tomographic pattern across the D′′ layer

AbstractWe present SEISGLOB1, a pure SV tomographic model of Earth's mantle based on Rayleigh phase velocities and normal mode self‐ and cross‐coupling data. SEISGLOB1 is the first model that incorporates the cross‐coupling of normal modes since the pioneering work of Resovsky and Ritzwoller (1999). The simultaneous inversion of new cross‐coupling normal modes and self‐coupling of high‐order normal modes measured by Deuss et al. (2013) and Stoneley modes measured by Koelemeijer et al. (2013) allows us to show that the velocity structure at the base of the mantle is more complex than that expected from a dominant spherical harmonic degree 2 and that the relative strength of odd degrees has previously been underestimated. Near the core‐mantle boundary, the large low‐shear‐velocity provinces are less homogeneous than in previous studies, and various local maxima, often potentially associated with hot spot sources, are observed.

A model of Earth’s magnetic field derived from 2 years of Swarm satellite constellation data

More than 2 years of magnetic field data taken by the three-satellite constellation mission Swarm are used to derive a model of Earth’s magnetic field and its time variation. This model is called SIFMplus. In addition to the magnetic field observations provided by each of the three Swarm satellites, explicit advantage is taken of the constellation aspect of Swarm by including East–West magnetic intensity and vector field gradient information from the lower satellite pair. Along-track differences of the magnetic intensity as well as of the vector components provide further information concerning the North–South gradient. The SIFMplus model provides a description of the static lithospheric field that is very similar to models determined from CHAMP data, up to at least spherical harmonic degree $$n=75$$ . Also the core field part of SIFMplus, with a quadratic time dependence for $$n \le 6$$ and a linear time dependence for $$n=7$$ –15, demonstrates the possibility to determine high-quality field models from only 2 years of Swarm data, thanks to the unique constellation aspect of Swarm. To account for the magnetic signature caused by ionospheric electric currents at polar latitudes we co-estimate, together with the model of the core, lithospheric and large-scale magnetospheric fields, a magnetic potential that depends on quasi-dipole latitude and magnetic local time.

SENSITIVITY KERNELS FOR FLOWS IN TIME–DISTANCE HELIOSEISMOLOGY: EXTENSION TO SPHERICAL GEOMETRY

ABSTRACT We extend an existing Born approximation method for calculating the linear sensitivity of helioseismic travel times to flows from Cartesian to spherical geometry. This development is necessary for using the Born approximation for inferring large-scale flows in the deep solar interior. As first sanity check, we compare two f-mode kernels from our spherical method and from an existing Cartesian method. The horizontal and total integrals agree to within 0.3%. As a second consistency test, we consider a uniformly rotating Sun and a travel distance of 42°. The analytical travel-time difference agrees with the forward-modeled travel-time difference to within 2%. In addition, we evaluate the impact of different choices of filter functions on the kernels for a meridional travel distance of 42°. For all filters, the sensitivity is found to be distributed over a large fraction of the convection zone. We show that the kernels depend on the filter function employed in the data analysis process. If modes of higher harmonic degree (90 ≲ l ≲ 170) are permitted, a noisy pattern of a spatial scale corresponding to l ≈ 260 appears near the surface. When mainly low-degree modes are used (l ≲ 70), the sensitivity is concentrated in the deepest regions and it visually resembles a ray-path-like structure. Among the different low-degree filters used, we find the kernel for phase-speed-filtered measurements to be best localized in depth.

Spectral analysis of sea level during the altimetry era, and evidence for GIA and glacial melting fingerprints

We study the spatial patterns of the mass and steric components of sea-level change during the “altimetry era” (1992–today), and we characterize them at different scales by the orthonormal functions method. The spectrum of the altimetry-derived rate of sea-level rise is red and decays with increasing wavenumber nearly following a power law with exponent ≈2. By analyzing the degree correlation and the admittance function, we find that the altimetric rate of sea-level change is coherent with the total steric field in the whole range of wavelengths considered (down to ≈1000km), but particularly for wavelengths exceeding ≈2000km. Thermosteric and halosteric components are moderately anti-correlated within the range of wavelengths 1000–4000km. Their power spectrum varies significantly with the wavelength and, for ≈2000km, it is equally partitioned between the two components. The power of regional sea-level variations driven by Glacial Isostatic Adjustment and the melting of continental ice sheets is small compared to that held by the steric component, which explains most of the regional variability shown by the altimetry record. This causes the elusiveness of the “static” sea-level fingerprints, which at present are hidden in the pattern of the residual sea-level (i.e., the altimetry-derived sea-level minus the steric component). However, we find that at harmonic degree 2, mainly associated with rotational variations, the power of glacial melting is significant and it will progressively increase during next century in response to global warming. We also estimate that at the end of the Mid-Holocene the strength of the glacial isostatic readjustment fingerprints was ≈10 times larger than today, well above the long-wavelength component of residual sea-level.