Analyzing the Diurnal Cycle by Bayesian Interpolation on a Sphere for Mapping GNSS Radio Occultation Data

Bayesian interpolation has previously been proposed as a strategy to construct maps of radio occultation (RO) data, but that proposition did not consider the diurnal dimension of RO data. In this work, the basis functions of Bayesian interpolation are extended into the domain of the diurnal cycle, thus enabling monthly mapping of radio occultation data in synoptic time and analysis of the atmospheric tides. The basis functions are spherical harmonics multiplied by sinusoids in the diurnal cycle up to arbitrary spherical harmonic degree and diurnal cycle harmonic. Bayesian interpolation requires a regularizer to impose smoothness on the fits it produces, thereby preventing the overfitting of data. In this work, a formulation for the regularizer is proposed and the most probable values of the parameters of the regularizer determined. Special care is required when obvious gaps in the sampling of the diurnal cycle are known to occur in order to prevent the false detection of statistically significant high-degree harmonics of the diurnal cycle in the atmosphere. Finally, this work probes the ability of Bayesian interpolation to generate a valid uncertainty analysis of the fit. The postfit residuals of Bayesian interpolation are dominated not by measurement noise but by unresolved variability in the atmosphere, which is statistically nonuniform across the globe, thus violating the central assumption of Bayesian interpolation. The problem is ameliorated by constructing maps of RO data using Bayesian interpolation that partially resolve the temporal variability of the atmosphere, constructing maps for approximately every 3 days of RO data.

Applications for CryoSat-2 satellite magnetic data in studies of Earth’s core field variations

We use 20 years of continuous magnetic field measurements from the Ørsted, CHAMP and Swarm satellite missions, supplemented by calibrated platform magnetometer data from the CryoSat-2 satellite, to study time variations of the Earth’s core field at satellite altitude and at the core–mantle boundary (CMB). From the satellite data we derive composite time series of the core field secular variation (SV) with 4-month cadence, at 300 globally distributed Geomagnetic Virtual Observatories (GVO). A previous gap in the GVO series between 2010 and 2014 is successfully filled using CryoSat-2, and sub-decadal variations are identified during this period. Tests showed that similar sub-decadal SV patterns were obtained from the CryoSat-2 data regardless of whether IGRF-13 or CHAOS-6x9 was used in their calibration. Cryosat-2 radial field SV series at non-polar latitudes have a mean standard deviation level compared to smoothing spline fits of 3.5 nT/yr compared to 1.8 nT/yr for CHAMP and 0.9 nT/yr for Swarm. GVO radial SV series display regional fluctuations with 5–10 years duration and amplitudes reaching 20 nT/yr, most notably at low latitudes over Indonesia (2014), over South America and the South Atlantic (2007, 2011 and 2014), and over the central Pacific (2017). Applying the Subtractive Optimally Localized Averages (SOLA) method, we also map the radial SV at the CMB as a collection of locally averaged SV estimates. We demonstrate that using 2-year windows of CryoSat-2 data, it is possible to reliably estimate the SV and its time derivative, the secular acceleration (SA), at the CMB, with a spatial resolution, corresponding to spherical harmonic degree 10. Along the CMB geographic equator, we find strong SA features with amplitude $$\pm 2.5\mu \mathrm{T}/\mathrm{yr}^2$$ ± 2.5 μ T / yr 2 under Indonesia from 2011–2014, under central America from 2015 to 2019, and sequences of SA with alternating sign under the Atlantic during 2004–2019. We find that platform magnetometer data from CryoSat-2 make a valuable contribution to the emerging picture of sub-decadal core field variations. Using 1-year windows of data from the Swarm satellites, we show that it is possible to study SA changes at low latitudes on timescales down to 1 year, with spatial resolution corresponding to spherical harmonic degree 10. We find strong positive and negative SA features appearing side-by-side in the Pacific in 2017, and thereafter drift westward.

The CHAOS-7 geomagnetic field model and observed changes in the South Atlantic Anomaly

We present the CHAOS-7 model of the time-dependent near-Earth geomagnetic field between 1999 and 2020 based on magnetic field observations collected by the low-Earth orbit satellites Swarm, CryoSat-2, CHAMP, SAC-C and Ørsted, and on annual differences of monthly means of ground observatory measurements. The CHAOS-7 model consists of a time-dependent internal field up to spherical harmonic degree 20, a static internal field which merges to the LCS-1 lithospheric field model above degree 25, a model of the magnetospheric field and its induced counterpart, estimates of Euler angles describing the alignment of satellite vector magnetometers, and magnetometer calibration parameters for CryoSat-2. Only data from dark regions satisfying strict geomagnetic quiet-time criteria (including conditions on IMF $$B_z$$ B z and $$B_y$$ B y at all latitudes) were used in the field estimation. Model parameters were estimated using an iteratively reweighted regularized least-squares procedure; regularization of the time-dependent internal field was relaxed at high spherical harmonic degree compared with previous versions of the CHAOS model. We use CHAOS-7 to investigate recent changes in the geomagnetic field, studying the evolution of the South Atlantic weak field anomaly and rapid field changes in the Pacific region since 2014. At Earth’s surface a secondary minimum of the South Atlantic Anomaly is now evident to the south west of Africa. Green’s functions relating the core–mantle boundary radial field to the surface intensity show this feature is connected with the movement and evolution of a reversed flux feature under South Africa. The continuing growth in size and weakening of the main anomaly is linked to the westward motion and gathering of reversed flux under South America. In the Pacific region at Earth’s surface between 2015 and 2018 a sign change has occurred in the second time derivative (acceleration) of the radial component of the field. This acceleration change took the form of a localized, east–west oriented, dipole. It was clearly recorded on ground, for example at the magnetic observatory at Honolulu, and was seen in Swarm observations over an extended region in the central and western Pacific. Downward continuing to the core–mantle boundary, we find this event originated in field acceleration changes at low latitudes beneath the central and western Pacific in 2017.

Long-wavelength lithospheric magnetic field of China

SUMMARY We present new regional models, denoted CLAS, of the Chinese lithospheric field, combining the long-wavelength information provided by satellite-derived models: CHAOS-6, MF7, LCS-1 and NGDC720, and an extremely high-quality compilation of 97 994 aeromagnetic survey data with 10 km × 10 km resolution for shorter wavelength. The models are estimated using a depleted basis of global spherical harmonic functions centred on China. CLAS models are determined include harmonic degrees up to 400. Although some accuracy of aeromagnetic data is lost in order to balance the consistent of two data sets, the results show that CLAS models have a high correlation with the satellite models at low-degree terms (degree correlation > 0.9) but with more power at high-degree terms, reflecting more features of the lithospheric field in continental China. Examples of improvement include Changbai mountains, Sichuan Basin and Qinghai–Tibet Plateau. CLAS models have good agreement (coherence > 0.9) with Chinese aeromagnetic data at wavelength down to about 100 km (corresponding to spherical harmonic degree n = 400), filling the usual gap between satellite models and aeromagnetic data. Comparison with aeromagnetic data filtered at 100 km gives good agreement (correlation > 0.95). The residuals between CLAS models and aeromagnetic data are still large (rms > 70 nT), but with most of misfits arising from shorter wavelength fields that the model cannot fit at degree up to 400; such misfit could be reduced by increasing the model degree. We provide a geological example of how the inclusion of satellite data can change the geological conclusions that can be drawn from the magnetic information. However, the two data sets are not completely consistent, future models should start from a reanalysis of the aeromagnetic data and its line levelling to ensure consistency with the satellite model.

Inferences of the lunar interior from a probabilistic gravity inversion approach

<p>The shape and location of density anomalies inside the Moon provide insights into processes that produced them and their subsequent evolution. Gravity measurements provide the most complete data set to infer these anomalies on the Moon [1]. However, gravity inversions suffer from inherent non-uniqueness. To circumvent this issue, it is often assumed that the Bouguer gravity anomalies are produced by the relief of the crust-mantle or other internal interface [2]. This approach limits the recovery of 3D density anomalies or any anomaly at different depths. In this work, we develop an algorithm that provides a set of likely three-dimensional models consistent with the observed gravity data with no need to constrain the depth of anomalies a priori.</p><p>The volume of a sphere is divided in 6480 tesseroids and n Voronoi regions. The algorithm first assigns a density value to each Voronoi region, which can encompass one or more tesseroids. At each iteration, it can add or delete a region, or change its location [2, 3]. The optimal density of each region is then obtained by linear inversion of the gravity field and the likelihood of the solution is calculated using Bayes’ theorem. After convergence, the algorithm then outputs an ensemble of models with good fit to the observed data and high posterior probability. The ensemble might contain essentially similar interior density distribution models or many different ones, providing a view of the non-uniqueness of the inversion results.</p><p>We use the lunar radial gravity acceleration obtained by the GRAIL mission [4] up to spherical harmonic degree 400 as input data in the algorithm. The gravity acceleration data of the resulting models match the input gravity very well, only missing the gravity signature of smaller craters. A group of models show a deep positive density anomaly in the general area of the Clavius basin. The anomaly is centered at approximately 50°S and 10°E, at about 800 km depth. Density anomalies in this group of models remain relatively small and could be explained by mineralogical differences in the mantle. Major variations in crustal structure, such as the near side / far side dichotomy and the South Pole Aitken basin are also apparent, giving geological credence to these models. A different group of models points towards two high density regions with a much higher mass than the one described by the other group of models. It may be regarded as an unrealistic model. Our method embraces the non-uniqueness of gravity inversions and does not impose a single view of the interior although geological knowledge and geodynamic analyses are of course important to evaluate the realism of each solution.</p><p>References: [1] Wieczorek, M. A. (2006), Treatise on Geophysics 153-193. doi: 10.1016/B978-0-444-53802-4.00169-X. [2] Izquierdo, K et al. (2019) Geophys. J. Int. 220, 1687-1699, doi: 10.1093/gji/ggz544, [3]  Izquierdo, K. et al., (2019) LPSC 50, abstr. 2157. [4] Lemoine, F. G., et al. ( 2013), J. Geophys. Res. 118, 1676–1698 doi: 10.1002/jgre.20118.</p><p> </p>

Triple-Pair Constellation Configurations for Temporal Gravity Field Retrieval

The goal of next-generation gravity missions (NGGM) is to improve the monitoring of mass transport in the Earth system by an increased space-time sampling capability as well as higher accuracies of a new generation of instrumentation, but also to continue the monitoring time series obtained by past and current missions such as GRACE and GRACE Follow-On. As the likelihood of three satellite pairs being simultaneously in orbit in the mid-term future increased, we have performed a closed-loop simulation to investigate the impact of a third pair in either polar or inclined orbit as an addition to a Bender-type constellation with NGGM instrumentation. For the additional pair, GRACE-like as well as NGGM instrumentation was tested. The analysis showed that the third pair mainly increases the redundancy of the monitoring system but does not significantly improve de-aliasing capabilities. The best-performing triple-pair scenario comprises a third inclined pair with NGGM sensors. Starting with a Bender-type constellation of a polar and an inclined satellite pair, simulation results indicate an average improvement of 11% in case of adding the third pair in a near-polar orbit, and of 21% for the third pair placed in an inclined orbit. The most important advantage of a multi-pair constellation, however, is the possibility to recover daily gravity fields with higher spatial resolution. In the case of the investigated triple-pair scenarios, a meaningful daily resolution with a maximum spherical harmonic degree of 26 can be achieved, while a higher daily parametrization up to degree 40 results in spatial aliasing and thus would need additional constraints or prior information.

Magneto-gravity wave packet dynamics in strongly magnetized cores of evolved stars

ABSTRACT Magnetic fields are believed to be generated in the cores of massive main-sequence stars, and these may survive on to later stages of evolution. Observations of depressed dipole modes in red giant stars have been touted as evidence for these fields, but the predictions of existing magnetic theories have difficulty accommodating several aspects, including the need to return a fraction of wave energy from the core to the envelope, and the persistent gravity-like character of affected modes. In this work, we perform a Hamiltonian ray-tracing study investigating the dynamics of magneto-gravity waves in full spherical geometry, using realistic stellar models and magnetic field configurations. This technique applies in the limit where wavelengths are much shorter than scales of background variation. We conduct a comprehensive exploration of parameter space, examining the roles of wave frequency, spherical harmonic degree, wavevector polarization, incoming latitude, field strength, field radius, and evolutionary state. We demonstrate that even in the presence of a strong field, there exist trajectories where waves remain predominantly gravity-like in character, and these are able to undergo reflection out of the core, much like pure gravity waves. The remaining trajectories are ones where waves acquire significant Alfvén character, becoming trapped and eventually dissipated. Orientation effects, i.e. wavevector polarization and incoming latitude, are found to be crucial factors in determining the outcome (trapped versus reflected) of individual wave packets. The allowance for partial energy return from the core offers a solution to the conundrum faced by the magnetic hypothesis.

Linking zonal winds and gravity: the relative importance of dynamic self-gravity

ABSTRACT Recent precise measurements of Jupiter’s and Saturn’s gravity fields help constraining the properties of the zonal flows in the outer envelopes of these planets. The link is provided by a simplified dynamic equation, which connects zonal flows to related buoyancy perturbations. These can result from density perturbations but also from the gravity perturbations. Whether the latter effect, which we call dynamic self-gravity (DSG), must be included or is negligible has been a matter of intense debate. We show that the second-order differential equation for the gravity perturbations becomes an inhomogeneous Helmholtz equation when assuming a polytrope of index unity for density and pressure. This equation can be solved semi-analytically when using modified spherical Bessel functions for describing the radial dependence. The respective solutions allow us to quantify the impact of the DSG on each gravity harmonic, practically independent of the zonal flow or the details of the planetary interior model. We find that the impact decreases with growing spherical harmonic degree ℓ. For degrees ℓ = 2 to about ℓ = 4, the DSG is a first-order effect and should be taken into account in any attempt of inverting gravity measurements for zonal flow properties. For degrees of about ℓ = 5 to roughly ℓ = 10, the relative impact of DSG is about 10 per cent and thus seems worthwhile to include, in particular since this comes at little extra cost with the method presented here. For yet higher degrees, it seems questionable whether gravity measurements or interior models will ever reach the precision required for disentangling the small DSG effects, which amount to only a few per cent at best.

Finite-frequency imaging of the global 410- and 660-km discontinuities using SS precursors

SUMMARY We report finite-frequency imaging of the global 410- and 660-km discontinuities using boundary sensitivity kernels for traveltime measurements made on SS precursors. The application of finite-frequency sensitivity kernels overcomes resolution limits in previous studies associated with large Fresnel zones of SS precursors and their interferences with other seismic phases. In this study, we calculate the finite-frequency sensitivities of SS waves and their precursors based on a single-scattering (Born) approximation in the framework of travelling-wave mode summation. The global discontinuity surface is parametrized using a set of triangular gridpoints with a lateral spacing of about 4°, and we solve the linear finite-frequency inverse problem (2-D tomography) based on singular value decomposition (SVD). The new global models start to show a number of features that were absent (or weak) in ray-theoretical back-projection models at spherical harmonic degree l > 6. The thickness of the mantle transition zone correlates well with wave speed perturbations at a global scale, suggesting dominantly thermal origins for the lateral variations in the mantle transition zone. However, an anticorrelation between the topography of the 410-km discontinuity and wave speed variations is not observed at a global scale. Overall, the mantle transition zone is about 2–3 km thicker beneath the continents than in oceanic regions. The new models of the 410- and 660-km discontinuities show better agreement with the finite-frequency study by Lawrence & Shearer than other global models obtained using SS precursors. However, significant discrepancies between the two models exist in the Pacific Ocean and major subduction zones at spherical harmonic degree >6. This indicates the importance of accounting for wave interactions in the calculations of sensitivity kernels as well as the use of finite-frequency sensitivities in data quality control.