Next-generation planetary tracking methods, such as interplanetary laser ranging (ILR) and same-beam interferometry (SBI) promise an orders-of-magnitude increase in the accuracy of measurements of solar system dynamics. This requires a reconsideration of modelling strategies for the translational and rotational dynamics of natural bodies, to ensure that model errors are well below the measurement uncertainties. The influence of the gravitational interaction of the full mass distributions of celestial bodies, the so-called figure-figure effects, will need to be included for selected future missions. The mathematical formulation of this problem to arbitrary degree is often provided in an elegant and compact manner that is not trivially relatable to the formulation used in space geodesy and ephemeris generation. This complicates the robust implementation of such a model in operational software packages. We formulate the problem in a manner that is directly compatible with the implementation used in typical dynamical modelling codes: in terms of spherical harmonic coefficients and Legendre polynomials. An analytical formulation for the associated variational equations for both translational and rotational motion is derived. We apply our methodology to both Phobos and the KW4 binary asteroid system, to analyze the influence of figure-figure effects during estimation from next-generation tracking data. For the case of Phobos, omitting these effects during estimation results in relative errors of 0.42% and 0.065% for the C¯ ₂₀ and C¯ ₂₂ spherical harmonic gravity field coefficients, respectively. These values are below current uncertainties, but orders of magnitude larger than those obtained from past simulations for accurate tracking of a future Phobos lander, showing the need to apply the methodology outlined in this manuscript for selected future missions.

@en

The mass sources that are responsible for the longwavelength gravity field of the Earth are not yet fully understood. The biggest candidate to describe these anomalies is mantle convection, yet models that can explain the mantle convection and the gravity field are not available. Another candidate is the core-mantle boundary with a density contrast of 4500 kg/m3 between the ferrous core and the silicate mantle. Due to the high density contrast variations in the core-mantle boundary can have a great impact on the gravity field. Also, density anomalies above the core-mantle boundary are large enough to be seen in the gravity field. The geometry of these anomalies amount to hundreds of km’s with density contrast up to 100 kg/m3. Sensitivity tests using similar dimensions and density contrasts show that gravity anomalies of 10-100 mGal can be computed, which is in the same as the longwavelength part of the gravity field. We argue that the gravity signal from structures at the deepest part of the lower mantle should be part of any global analysis. During the 3D Earth project, we will use seismic models of the core-mantle region to simulate more realistic geometries and density anomalies to study their effect on the global gravity field. This presentation is part of the ESA Support To Science Element - 3D Earth – which is presented by Jörg Ebbing @en

In the last decade, the gravity field of the Earth has been observed with increased coverage due to dedicated satellite missions, which resulted in higher resolution and more accurate global gravity field models than were previously available. These models make it possible to study large scale processes such as solid Earth deformation after large loading events such as retreat of ice sheets or to study lateral density variation in the lithospheric part of the upper mantle. However, to use the gravity data successfully, unwanted signal needs to be removed in order to extract the information of interest. For example, with lithosphere studies the gravity signal coming from the crust and the deep mantle needs to be removed. This is commonly done by filtering out long-wavelength signals from the solution to remove deep mantle effects, and by removing the crustal signal by forward modelling seismic-derived crustal models. With improved models of crustal structure and more accurate gravity data, new information about the upper mantle and lithosphere can be obtained. Adopting the increased resolution and accuracy of the global gravity field models, I have developed new approaches that focus on spectral analysis of the gravity field, which result in new insights of the upper mantle. The forward gravity field modelling method that I improve upon in this dissertation is mostly used for topographic/isostatic mass reduction of gravity data. The methodology is able to transform density-models into gravitational potential fields using a spherical harmonic representation. I show that this methodology in the existing form is not suited to be used for density layers in lower crustal and upper mantle regions. The binomial series inherent to this methodology do not converge when applied to deep mass structures, and therefore it is not possible to truncate the series at a low degree to approximate the mass. This approximation is crucial for the computational efficiency of the methodology. I propose a correction that mitigates this erroneous behaviour, which enables this methodology to efficiently compute the potential field of deep situated masses. I benchmark the improved methodology with a tesseroid-based gravity-field modelling software, and I show that my software is accurate within ±4 mGal, when modelling the Moho density interface (with a range in signal of ±500 mGal. The improved methodology is used in the studies described in this thesis. With an efficient and accurate forward modelling methodology, I am able to use global gravity field data in studies of the solid Earth. In the central part of Fennoscandia the crust is currently uplifting, because of the delayed response of the viscous mantle to melting of the regional Late Pleistocene ice sheet. This process, called glacial isostatic adjustment (GIA), causes a negative anomaly in the present-day static gravity field as isostatic equilibrium has not been reached yet. Several studies have used this anomaly as a constraint on models of GIA, but the uncertainty in crustal and upper mantle structures had not been properly taken into account. In revisiting this problem, I show that the GIA gravity signal overlaps with mantle convection signals, such that a simple spherical harmonic truncation is not sufficient to separate these two phenomena. Furthermore, I find that, in contrast to the other studies, the effect of crustal anomalies on the gravity field cannot be effectively removed, because of the relative large uncertainties in the crustal density models. Therefore, I propose to correct the observed gravity field for GIA with numerical modelling results when constructing geophysical models that assume isostatic equilibrium. I show that correcting for GIA results in a significant vertical readjustment of the geometry of structural layers in the modelled crust of 5-10 percent. Correcting the gravity field for GIA prior to assuming isostatic equilibrium might be relevant in other areas with ongoing post-glacial rebound such as North America and the polar regions. Uncertainty in lithospheric density models is still the limiting factors in solid Earth studies and needs to be improved. Lithospheric density anomalies can, among other methods, be estimated from seismic tomography, gravity studies, or joint studies using both datasets. I compare different gravity-based density models of the lithosphere to a tomographic-derived solution and characterise the sources that introduce large uncertainties in the density models of the lithosphere. To study the uncertainty between global and regional crustal models, I select a region where the crust is explored in great measure with seismic profiles, namely the British Isles and surrounding areas, where I use three crustal models to quantify the crustal uncertainty: CRUST1.0, EUCrust-07, and a high-resolution regional P-wave velocity model of the region. The crustal models contribute to the uncertainty of the density of the lithosphere with ±110 kg/m3. Furthermore, I study various P-wave velocity-to-density conversions to quantify the uncertainty introduced by these conversion methods (±10 kg/m3. All different crustal density models are forward modelled into gravity anomalies using the improved methodology of Chapter 2 and these gravity anomalies are subsequently removed from the gravity observations. The unmodelled long-wavelength signal in the gravity field representing mass anomalies in the deep mantle are removed from the observation by spherical harmonic truncation, introducing an uncertainty of ±5 kg/m3. Also, the choice of density background model (±20 kg/m3) and lithosphere-asthenosphere boundary uncertainty (±30 kg/m3) have a small but significant effect on the estimated lithosphere densities. However, the inhomogeneous spatial distribution of profiles of controlled-source seismic exploration of the crustal thickness and density distribution proves to be the largest source of uncertainty (±110 kg/m3). The gravity-based lithospheric density solutions with a variation of ±100 kg/m3 are completely different in magnitude and spatial signature to the densities (±35 kg/m3) derived from a shear wave velocity model. This demonstrates that the tomographic model has a limited resolution, which can be related to regularisation that is used in the construction of global tomographic models. To account for this spectral imbalance, I spatially filter the gravity-based density models, resulting in similarities in spatial correlation and magnitude between that of the gravity-based and the tomographic-derived density. With the filtered gravity-based density I am able to estimate lateral varying conversion values between shear wave velocity and density for the lithosphere, which shows a correlation with major tectonic regions. This correlation shows that the independent gravity-based solutions, despite being filtered, can help in identifying different compositional domains in the lithosphere. Satellite observations also provide global data on the temporal variations of the gravity field. In the last study, I show that global gravity-change observations from the GRACE satellite mission can be used to study GIA in the Barents Sea Region. The Barents Sea is subject to ongoing postglacial uplift since the melting of the Weichselian ice sheet that covered this region. The deglaciation history is not well known because there is only data from locations close to the boundary of the former ice sheet, in Franz Joseph Land, Svalbard, and Novaya Zemlya. At these locations the magnitude of the GIA uplift is limited, reducing the signal-to-noise of the data. The GRACE mission measures the gravity-change due to GIA at the center of the Barents Sea, where the maximum uplift and ongoing gravity-change is situated. I show that the linear trend in the gravity-change derived from a decade of observations from the GRACE satellite mission can constrain the volume of the ice sheet after correcting for current ice-melt, hydrology and far-field gravitational effects. Regional ice loading models based on new geologically-inferred ice margin chronologies show a significantly better fit to the GRACE data than the global ice models ICE-5G and ICE-6G_C. The regional ice models in this study contain less ice mass during LGM in the Barents Sea than ICE-5G (5-6.3 m equivalent sea level vs. 8.5 m). Also, I show that the GRACE gravity-change is sensitive to the upper mantle viscosity underneath the Barents sea, for which I found a minimum value of 4x1020 Pas, regardless of the ice loading history. The GRACE gravity-change should be used as a constraint in any future GIA modelling of the Barents Sea, because it is the only measurement that captures the signal of maximum GIA. The high resolution and accurate global gravity field models do give new insights in the structure and density distribution of the upper mantle. The presented studies in this dissertation demonstrate that analysing the spectral signature of gravity data is very useful. Medium-to-short-scale features, like lateral density variation in the lithosphere and GIA gravity-change in the Barents Sea can be separate from other gravity-change sources by applying spectral filters. For longer wavelength signals, such as the GIA static gravity signal in Fennoscandia, this proves to be more difficult due to the overlap in the long-wavelength region by mantle convection signals and other deep mantle signals. On the whole, the global gravity field models and their spectral signature play an important part in building a global density model of the Earth, in which lithosphere, GIA, but also mantle convection and core-mantle boundary effects need to be combined to explain the gravity field. @en

Lithospheric density structure can be constructed from seismic tomography, gravity modelling, or using both data sets. The different approaches have their own uncertainties and limitations. This study aims to characterize and quantify some of the uncertainties in gravity modelling of lithosphere densities. To evaluate the gravity modelling we compare gravity-based and seismic velocity-based approaches to estimating lithosphere densities. In this study, we use a crustal model together with lithospheric isostasy and gravity field observations to estimate lithosphere densities. To quantify the effect of uncertainty in the crustal model, three models are implemented in this study: CRUST1.0, EuCrust-07 and a high-resolution P-wave velocity model of the British Isles and surrounding areas. Different P-wave velocity-to-density conversions are used to study the uncertainty in these conversion methods. The crustal density models are forward modelled into gravity field quantities using a method that is able to produce spherical harmonic coefficients. Deep mantle signal is assumed to be removed by removing spherical harmonic coefficients of degree 0–10 in the observed gravity field. The uncertainty in the resulting lithosphere densities due to the different crustal models is ±110 kg m−3, which is the largest uncertainty in gravity modelling. Other sources of uncertainty, such as the VP to density conversion (±10 kg m−3), long-wavelength truncation (±5 kg m−3), choice of reference model (<±20 kg m−3) and Lithosphere Asthenosphere Boundary uncertainty (±30 kg m−3), proved to be of lesser importance. The resulting lithosphere density solutions are compared to density models based on a shear wave velocity model. The comparison shows that the gravity-based models have an increased lateral resolution compared to the tomographic solutions. However, the density anomalies of the gravity-based models are three times higher. This is mainly due to the high resolution in the gravity field. To account for this, the gravity-based density models are filtered with a spatial Gaussian filter with 200 km half-width, which results in similar density estimates (±35 kg m−3) with the tomographic approach. Lastly, the gravity-based density is used to estimate laterally varying conversion factors, which correlate with major tectonic regions. The independent gravity-based solutions could help in identifying different compositional domains in the lithosphere, when compared to the tomographic solutions.@en

This article reviews a spectral forward gravity field modelling method that was initially designed for topographic/isostatic mass reduction of gravity data. The method transforms 3D spherical density models into gravitational potential fields using a spherical harmonic representation. The binomial series approximation in the approach, which is crucial for its computational efficiency, is examined and an error analysis is performed. It is shown that, this method cannot be used for density layers in crustal and upper mantle regions, because it results in large errors in the modelled potential field. Here, a correction is proposed to mitigate this erroneous behaviour. The improved method is benchmarked with a tesseroid gravity field modelling method and is shown to be accurate within ±4 mGal for a layer representing the Moho density interface, which is below other errors in gravity field studies. After the proposed adjustment the method can be used for the global gravity modelling of the complete Earth's density structure.

@en

In November 1934, Den Helder, The Netherlands, the start of a remarkable voyage commenced. The Hr. Ms.K-XVIII, a Dutch submarine, was about to set sail to Soerabaya, Indonesia. Onboard was a Dutch professor, Felix Andries Vening Meinesz. He was able to measure the Earth’s gravity field with similar precision as on land for the first time in history using his innovative pendulum apparatus. His ground breaking data and systematic way of working changed the way of performing scientific expeditions.@en