What is the weight: how and why it occursfor gaseous matter

The ability of any satellite gravity mission concept to monitor mass transport processes in the Earth system is typically tested well ahead of its implementation by means of various simulation studies. Those studies often extend from the simulation of realistic orbits and instrumental data all the way down to the retrieval of global gravity field solution time-series. Basic requirement for all these simulations are realistic representations of the spatio-temporal mass variability in the different sub-systems of the Earth, as a source model for the orbit computations. For such simulations, a suitable source model is required to represent (i) high-frequency (i.e., sub-daily to weekly) mass variability in the atmosphere and oceans, in order to realistically include the effects of temporal aliasing due to non-tidal high-frequency mass variability into the retrieved gravity fields. In parallel, (ii) low-frequency (i.e., monthly to interannual) variability needs to be modelled with realistic amplitudes, particularly at small spatial scales, in order to assess to what extent a new mission concept might provide further insight into physical processes currently not observable. The new source model documented here attempts to fulfil both requirements: Based on ECMWF’s recent atmospheric reanalysis ERA-Interim and corresponding simulations from numerical models of the other Earth system components, it offers spherical harmonic coefficients of the time-variable global gravity field due to mass variability in atmosphere, oceans, the terrestrial hydrosphere including the ice-sheets and glaciers, as well as the solid Earth. Simulated features range from sub-daily to multiyear periods with a spatial resolution of spherical harmonics degree and order 180 over a period of 12 years. In addition to the source model, a de-aliasing model for atmospheric and oceanic high-frequency variability with augmented systematic and random noise is required for a realistic simulation of the gravity field retrieval process, whose necessary error characteristics are discussed. The documentation is organized as follows: The characteristics of the updated ESM along with some basic validation are presented in Volume 1 of this report (Dobslaw et al., 2014). A detailed comparison to the original ESA ESM (Gruber et al., 2011) is provided in Volume 2 (Bergmann-Wolf et al., 2014), while Volume 3 (Forootan et al., 2014) contains a description of the strategy to derive a realistically noisy de-aliasing model for the high-frequency mass variability in atmosphere and oceans. The files of the updated ESA Earth System Model for gravity mission simulation studies are accessible at DOI:10.5880/GFZ.1.3.2014.001.

The ability of any satellite gravity mission concept to monitor mass transport processes in the Earth system is typically tested well ahead of its implementation by means of various simulation studies. Those studies often extend from the simulation of realistic orbits and instrumental data all the way down to the retrieval of global gravity field solution time-series. Basic requirement for all these simulations are realistic representations of the spatio-temporal mass variability in the different sub-systems of the Earth, as a source model for the orbit computations. For such simulations, a suitable source model is required to represent (i) high-frequency (i.e., subdaily to weekly) mass variability in the atmosphere and oceans, in order to realistically include the effects of temporal aliasing due to non-tidal high-frequency mass variability into the retrieved gravity fields. In parallel, (ii) low-frequency (i.e., monthly to interannual) variability needs to be modelled with realistic amplitudes, particularly at small spatial scales, in order to assess to what extent a new mission concept might provide further insight into physical processes currently not observable. The new source model documented here attempts to fulfil both requirements: Based on ECMWF’s recent atmospheric reanalysis ERA-Interim and corresponding simulations from numerical models of the other Earth system components, it offers spherical harmonic coefficients of the time-variable global gravity field due to mass variability in atmosphere, oceans, the terrestrial hydrosphere including the ice-sheets and glaciers, as well as the solid Earth. Simulated features range from sub-daily to multiyear periods with a spatial resolution of spherical harmonics degree and order 180 over a period of 12 years. In addition to the source model, a de-aliasing model for atmospheric and oceanic high-frequency variability with augmented systematic and random noise is required for a realistic simulation of the gravity field retrieval process, whose necessary error characteristics are discussed. The documentation of the updated ESA Earth System Model (updated ESM) for gravity mission simulation studies is organized as follows: The characteristics of the updated ESM along with some basic validation is presented in Volume 1. A detailed comparison to the original ESA ESM (Gruber et al., 2011) is provided in Volume 2, while Volume 3 contains the description of a strategy to derive realistic errors for the de-aliasing model of high-frequency mass variability in atmosphere and ocean.

Next-generation satellite gravimetry for measuring mass transport in the Earth system

The Swarm satellite constellation?s GPS receivers provide valuable gravimetric data, with which it is possible to observe Earth?s large-scale mass transport process. These data have become increasingly relevant given the on-going GRACE/GRACE-FO gap, and are thus needed to provide continuous observations of the Earth system. In this context, the overall accuracy and maximum resolution of the Swarm temporal gravity field models are parameters with interest to the wider geophysical community. We assess the signal contents of the gravity field model resulting from the combination at the solution level of four individual solutions produced considering different gravity field estimation approaches. The combination considers Variance Component Estimation (VCE) and is a service kindly provided by the European Gravity Service for Improved Emergency Management (EGSIEM) initiative. We assume that past GRACE solutions provide an accurate measure of the signal at the spatial lengths captured by the Swarm solutions. On the basis of this, we derive per-degree correlation coefficients and spatial correlation maps for a selection of monthly solutions that were obtained under diverse conditions of geomagnetic and ionospheric activities, as well as variability of non-gravitational accelerations.

Gravity Field Constraints on the Upper Mantle of Northwestern Europe

Earth gravitational model 2008

Some unified theories on the gravitational and electromagnetic fields are researched. We investigate mainly two new geometric unified theories. A method is that the gravitational field and the source-free electromagnetic field can be unified by the equations Rklmi = κTklmi* in the Riemannian geometry, both are contractions of im and ik, respectively. If Rklmi = κTklmi* =constant, it will be equivalent to the Yang’s gravitational equations Rkm;l –Rkl;m = 0, which include Rlm= 0. From Rlm= 0 we can obtain the Lorentz equations of motion, the first system and second source-free system of Maxwell’s field equations. This unification can be included in the gauge theory, and the unified gauge group is SL(2,C) × U(1)=GL(2,C), which is just the same as the gauge group of the Riemannian manifold. Another unified method on the general nonsymmetric metric field with high-dimensional space-time and its matrix representations are analyzed mathematically. Further, the general unified theory of five-dimensional space-time combined quantum theory and four interactions is researched. Some possible unification ways on the gravitational and electromagnetic fields are discussed. The general matrix and various corresponding theories may decompose to a sum of symmetry and antisymmetry. Moreover, we proposed an imaginative representation on the ten dimensional space-time.

In this thesis we have implemented and studied on detail three different potential field inversion algorithms proposed by Li and Oldenburg (2003), Portniaguine and Zhdanov (2002) and Pilkington (2009). We focused our attention on the dependency of the solution with respect to external constraints and particularly with respect to the depth weighting function. This function is necessary to counteract the natural decay of the data kernels with depth, so providing depth resolution to the inverse solution. We derived invariance rules for either the minimum-length solution and for the regularized inversion with depth weighting and positivity constraints. For a given source class, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its kth order vertical derivatives. A further invariance rule regards the inversion of homogeneous fields: the homogeneity degree of the magnetization distribution obtained inverting any of the k-order vertical derivatives of the magnetic field is the same as that of the magnetic field, and does not depend on k. Similarly, the homogeneity degree of the density distribution obtained inverting any of the k-order vertical derivatives of the gravity field is the same as that of the 1st order vertical derivative of the gravity field, and does not depend on k. This last invariance rule allowed us using the exponent β of the depth weighting function corresponding to the structural index of the magnetic case, no matter the order of differentiation of the magnetic field. We also illustrated how the combined effect of regularization and depth weighting could influence the estimated source model depth, in the regularized inversion with depth weighting and positivity constraints. We found that too high regularization parameter will deepen the inverted source-density distribution, so that a lower value for the exponent of the depth weighting function should be used, with respect to the structural index N of the magnetic field (or of the 1st vertical derivative of the gravity field). In the attempt to keep the regularization parameter as low as possible, the GCV method yielded better results than the χ2 criterion. Furthermore we introduced a new approach to improve the resolution of the model, based on inversion of data with a differentiation order greater than that of the kernel. We analyzed also the case of a field generated by sources with different structural indices. This is a very important case, because it is the most common situation in real data. In this case, there isn’t a unique value for β allowing to obtain accurate estimations of depth to all the sources. Thus the depth weighting exponent β must be varied according to the structural index estimated for each source and according to the invariance rules. Furthermore we studied the dependency of the model obtained by inversion on the depth weighting function when a priori information is included in the inversion. We presented a self-constrained inversion procedure based only on the constraints retrieved by previous potential field anomaly interpretation steps. We showed that adding, as inversion constraints, information retrieved by a previous analysis of the data has a great potential to lead to well-constrained solutions with respect to the source depth and to the horizontal variations of the source-density distribution. Our analysis on both synthetic and real data demonstrated that the more self-constraints are included in the inversion, the less important is the role of the tuning of the depth-weighting function through the actual value of the source structural index. Another type of a priori information regards the compactness of solution. This constraint can be imposed using the focusing inversion algorithm (Portniaguine and Zhadanov, 2002) or using sparseness constraints (Pilkington, 2009). In this case, imposing this type of constraint tends to decrease the importance of the depth weighting function.

SUMMARY Satellite gravity missions so far are medium size satellites consisting of one or a pair of satellites flying in near-polar or sun-synchronous orbital planes. Due to the limited observation geometry and the related space–time sampling, high-frequency non-tidal mass variation signals from atmosphere and ocean cannot be observed and cause temporal aliasing. For current single-pair satellite gravimetry missions as GRACE and GRACE Follow-On (GRACE-FO) temporal aliasing is the limiting factor and represents the major error source in the gravity field time-series. Adding a second inclined satellite pair to a GRACE-like polar pair (Bender constellation) currently is the most promising solution to increase the spatio-temporal resolution and to significantly reduce the temporal aliasing error. This shall be implemented with the MAGIC mission in future. With the ongoing developments in miniaturization of satellites and gravity-relevant instruments (accelerometers and intersatellite ranging), in future constellations of multiple small satellite pairs may solve this problem even beyond the capabilities of a Bender constellation. Therefore, in this study the capabilities of such constellations flying in specific formations are investigated in order to enable a retrieval of the temporal gravity field on short time scales. We assess the performance of up to 18 satellite pairs. The satellite configurations cover satellite pairs in polar and inclined orbits flying in pair-wise or pearl-string formation with varying mean anomalies and right ascensions of the ascending node (RAAN). As future potential miniaturized instruments optomechanical accelerometers with similar performance as those flying on GRACE-FO are a candidate, while for the intersatellite ranging instrument still some technological development is required. Therefore, in this study a microwave ranging system equivalent to the GRACE and GRACE-FO instruments performance is taken as baseline assuming that such instruments can be miniaturized in future as well. In numerical closed-loop simulations, up to nine different satellite configurations with up to 18 satellite pairs are evaluated based on the retrieval of the non-tidal temporal gravity field on a monthly basis. From the simulation results it is concluded that the best-performing satellite constellation of 18 polar satellite pairs already is outperformed by a typical Bender-like constellation of 1 polar and 1 inclined pair. In general, we identify that increasing the number of satellite pairs leads to an improved gravity field retrieval, either at low spherical harmonic degree and order (d/o) by the shift in RAAN or at high d/o by the shift in mean anomaly. By a two-step simulation approach, co-estimating also (sub-)daily gravity fields for selected configurations with a large number of satellite pairs distributed equally over the globe, it is possible to retrieve stand-alone gravity fields at 24, 12 and 6 hr temporal resolution. Ultimately it is concluded that a network of miniaturized satellites with instrument performances similar to GRACE-FO and flying in a well-defined constellation has the potential to observe (sub-)daily mass variations and therefore could drastically reduce the problem of temporal aliasing due to high frequency mass variations in the Earth system.

<p>In the frame of the CubeGrav project, funded by the German Research Foundation, Cube-satellite networks for geodetic Earth observation are investigated on the example of the monitoring of Earth’s gravity field. Satellite gravity missions are an important element of Earth observation from space, because geodynamic processes are frequently related to mass variations and mass transport in the Earth system. As changes in gravity are directly related to mass variability, satellite missions observing the Earth’s time-varying gravity field are a unique tool for observing mass redistribution among the Earth’s system components, including global changes in the water cycle, the cryosphere, and the oceans. The basis for next generation gravity missions (NGGMs) is based on the success of the single satellite missions CHAMP and GOCE as well as the dual-satellite missions GRACE and GRACE-FO launched so far, which are all conventional satellites.    <br>In particular, feasibility as well as economic efficiency play a significant role for future missions, with a focus on increasing spatio-temporal resolution while reducing error effects. The latter include the aliasing of the time-varying gravity fields due to the under-sampling of the geophysical signals and the uncertainties in geophysical background models. The most promising concept for a future gravity field mission from the studies investigated is a dual-pair mission consisting of a polar satellite pair and an inclined (approx. 70°) satellite pair. Since the costs for a realization of the Bender constellation are very high, this contribution presents results of the CubeGrav project and focuses on alternative concepts in the form of different constellations and formations of small satellites. The latter includes both satellite pairs and chains consisting of trailing satellites. The aim is to provide a cost-effective alternative to the previous gravity field satellites while simultaneously increasing the spatiotemporal resolution and minimizing the above-mentioned error effects.</p><p>In numerical closed-loop simulations, the impact of different satellite formations and constellations will be investigated for the retrieval of monthly gravity fields. The configurations differ in the orbital setup including the number of orbital planes and key orbit parameters like altitude and inclination. The ground track coverage of the selected orbits will be analysed since an improved spatial sampling with specific sub-cycles is beneficial for estimating short-temporal gravity fields which will be co-parametrized in the overall solution approach. Due to the large number of observations, it is possible to retrieve sub-daily gravity fields down to quarter-day resolution, which exceeds the capabilities of the existing gravity mission like GRACE or GRACE-FO by far. These (sub-)daily gravity field solutions can also improve the overall monthly gravity product, which will be proven for several satellite constellations and formations. All in all, the opportunities and limits of multiple satellites pairs and chains of trailing satellites for achieving the highest possible spatial and temporal resolution shall be analysed in detail.</p>

Mass distribution and mass transport in the Earth system

GNSS-based low-degree spherical harmonic coefficients are an independent source of information for describing mass changes in the Earth system. In this study, we employ the mass load theory by an inverse GNSS approach to determine the changes in the Earth's gravity field with the spherical harmonic expansion up to degree and order 8, using daily station coordinate estimates from the third data reprocessing campaign of the International GNSS Service. Deriving a reliable series of variations in the Earth's dynamic oblateness terms (C20) and C30 is essential for supporting GRACE-based time-variable gravity field models. Consequently, our study focused on the comprehensive alternative and validation tool for the widely used Satellite Laser Ranging (SLR) series of C20 and C30 coefficients.The global mean sea level has risen significantly since the 1990s, largely due to mass loss from the Greenland and Antarctic ice sheets. This underscores the importance of continued monitoring of the global changes. Therefore, we conduct a detailed analysis of the impact of incorporating GNSS-derived coefficients into the official gravity field products provided by the GRACE and GRACE-FO missions on changes in the ice sheets of these regions. The findings highlight the benefits of the GNSS-GRACE integration as a crucial element in enhancing gravity models and improving the representation of mass changes within the Earth system. The combination of GRACE/GRACE-FO with the GNSS results in a linear trend in Antarctic ice sheets with a rate of -152 Gt/year between January 2007 and December 2020.Furthermore, we transform GNSS-based gravity field solutions into equivalent water heights and estimate annual terrestrial water storage (TWS) fluctuations in regions that are crucial for understanding large-scale hydrological dynamics, e.g., the Amazon and Brahmaputra river basins. Our solution is validated with GRACE/GRACE-FO data and global hydrological models, i.e., the Land Surface Discharge Model. The results show that the spatial and seasonal patterns of TWS changes derived from GNSS are consistent with GRACE/GRACE-FO and hydrological model estimates at the single-millimeter level within the range of the GNSS-based TVG model spherical harmonic expansion up to degree and order 5.

Dramatic alterations to the natural environment due to human activity have produced a permanent rupture in the Earth system; the relative stable epoch of the Holocene has given way to a volatile Anthropocene. Acceptance of these claims means that we now live in this altered physical reality, inviting us to rethink how we conceptualise disasters. Yet, disaster scholars have been hesitant to apply the Anthropocene label and to acknowledge the profound changes that it can bring to the study of disasters. This paper queries whether this label is a necessary adage or unnecessary baggage for disaster studies by examining the possibilities and the challenges associated with engaging with the Anthropocene. An analysis of the concepts, causes, and consequences of disasters reveals how the Anthropocene provides, as the very least, a theoretical heuristic for challenging linear temporal assumptions, the epistemological status of uncertainty, and the location of agency in disaster studies.

Abstract. Time variable gravity fields, reflecting variations of mass distribution in the system Earth is one of the key parameters to understand the changing Earth. Mass variations are caused either by redistribution of mass in, on or above the Earth's surface or by geophysical processes in the Earth's interior. The first set of observations of monthly variations of the Earth gravity field was provided by the US/German GRACE satellite mission beginning in 2002. This mission is still providing valuable information to the science community. However, as GRACE has outlived its expected lifetime, the geoscience community is currently seeking successor missions in order to maintain the long time series of climate change that was begun by GRACE. Several studies on science requirements and technical feasibility have been conducted in the recent years. These studies required a realistic model of the time variable gravity field in order to perform simulation studies on sensitivity of satellites and their instrumentation. This was the primary reason for the European Space Agency (ESA) to initiate a study on ''Monitoring and Modelling individual Sources of Mass Distribution and Transport in the Earth System by Means of Satellites''. The goal of this interdisciplinary study was to create as realistic as possible simulated time variable gravity fields based on coupled geophysical models, which could be used in the simulation processes in a controlled environment. For this purpose global atmosphere, ocean, continental hydrology and ice models were used. The coupling was performed by using consistent forcing throughout the models and by including water flow between the different domains of the Earth system. In addition gravity field changes due to solid Earth processes like continuous glacial isostatic adjustment (GIA) and a sudden earthquake with co-seismic and post-seismic signals were modelled. All individual model results were combined and converted to gravity field spherical harmonic series, which is the quantity commonly used to describe the Earth's global gravity field. The result of this study is a twelve-year time-series of 6-hourly time variable gravity field spherical harmonics up to degree and order 180 corresponding to a global spatial resolution of 1 degree in latitude and longitude. In this paper, we outline the input data sets and the process of combining these data sets into a coherent model of temporal gravity field changes. The resulting time series was used in some follow-on studies and is available to anybody interested.

The Earth's gravity field is a basic physical parameter, which reflects mass transport and re‐ distribution in the Earth System. It not only contributes to study the Earth's interior physical state and the dynamic mechanism in geophysics, but also provides an important way to re‐ search the Earth's interior mass distribution and characteristics. The gravity field and its changes with time is of great significance for studying various geodynamics and physical processes, especially for the dynamic mechanism of the lithosphere, mantle convection and lithospheric drift, glacial isostatic adjustment (GIA), sea level change, hydrologic cycle, mass balance of ice sheets and glaciers, rotation of the Earth and mass displacement [33; 37; 7; 39; 17 and 18]. For Geodesy, the gravity field is an important parameter to study the size and shape of the Earth. Meanwhile the Earth’s gravity field is very important to determine the trajectory of carrier rocket, long-range weapons, artificial Earth’s satellites and spacecrafts. In addition, the gravity field could provide some signals of pre-, co-, and post-earthquake with mass transport following earthquakes [25; 14]. Therefore, precisely determining Earth’s gravity field and its time-varying information are very important in geodesy, seismology, oceanography, space science and national defense as well as geohazards.