Regularization of the Gravity Field Inversion Process with High-Dimensional Vector Autoregressive Models

Earth’s gravitational field provides invaluable insights into the changing nature of our planet. It reflects mass change caused by geophysical processes like continental hydrology, changes in the cryosphere or mass flux in the ocean. Satellite missions such as the NASA/DLR operated Gravity Recovery and Climate Experiment (GRACE), and its successor GRACE Follow-On (GRACE-FO) continuously monitor these temporal variations of the gravitational attraction. In contrast to other satellite remote sensing datasets, gravity field recovery is based on geophysical inversion which requires a global, homogeneous data coverage. GRACE and GRACE-FO typically reach this global coverage after about 30 days, so short-lived events such as floods, which occur on time frames from hours to weeks, require additional information to be properly resolved. In this contribution we treat Earth’s gravitational field as a stationary random process and model its spatio-temporal correlations in the form of a vector autoregressive (VAR) model. The satellite measurements are combined with this prior information in a Kalman smoother framework to regularize the inversion process, which allows us to estimate daily, global gravity field snapshots. To derive the prior, we analyze geophysical model output which reflects the expected signal content and temporal evolution of the estimated gravity field solutions. The main challenges here are the high dimensionality of the process, with a state vector size in the order of 103 to 104, and the limited amount of model output from which to estimate such a high-dimensional VAR model. We introduce geophysically motivated constraints in the VAR model estimation process to ensure a positive-definite covariance function.

MULTI-SCALE MECHANICAL BEHAVIOR ANALYSIS ON FLUID–SOLID COUPLING FOR OSTEONS IN VARIOUS GRAVITATIONAL FIELDS

Background: Compact bone mainly consists of cylindrical osteon structures. In microgravity, the change in the mechanical microenvironment of osteocytes might be the root cause of astronauts’ bone loss during space flights. Methods: A multi-scale three-dimensional (3D) fluid–solid coupling finite element model of osteons with a two-stage pore structure was developed using COMSOL software based on the natural structure of osteocytes. Gradients in gravitational fields of [Formula: see text]1, 0, 1, 2.5, and 3.7[Formula: see text]g were used to investigate the changes in the mechanical microenvironment on osteocyte structure. The difference in arteriole pulsating pressure and static compression stress caused by each gravity gradient was investigated. Results: The mechanical response of osteocytes increased with the value of g, compared with the Earth’s gravitational field. For instance, the fluid pressure of osteocytes and the von Mises stress of bone matrix near lacunae decreased by 31.3% and 99.9%, respectively, in microgravity. Under static loading, only about 16.7% of osteocytes in microgravity and 58.3% of osteocytes in the Earth’s gravitational field could reach the fluid shear stress threshold of biological reactions in cell culture experiments. Compared with the Earth’s gravitational field, the pressure gradient inside osteocytes severely decreased in microgravity. Conclusion: The mechanical microenvironment of osteocytes in microgravity might cause significant changes in the mechanical microenvironment of osteocytes, which may lead to disuse osteoporosis in astronauts.

Quantum mechanical rotation of a photon polarization by Earth’s gravitational field

We describe the quantum mechanical rotation of a photon state, the Wigner rotation—a quantum effect that couples a transformation of a reference frame to a particle’s spin, to investigate geometric phases induced by Earth’s gravitational field for observers in various orbits. We find a potentially measurable quantum phase of the Wigner rotation angle in addition to the rotation of standard fame, the latter of which is computed and agrees well with the geodetic rotation. When an observer is in either a circular or a spiraling orbit containing non-zero angular momentum, the additional quantum phase contributes 10−6 degree to 10−4 degree respectively, depending on the altitude of the Earth orbit. In the former case, the additional quantum phase is dominant over the near-zero classical geodetic rotation. Our results show that the Wigner rotation represents a non-trivial semi-classical effect of quantum field theory on a background classical gravitational field.

Satellite Gravimetry: A Review of Its Realization

Since Kepler, Newton and Huygens in the seventeenth century, geodesy has been concerned with determining the figure, orientation and gravitational field of the Earth. With the beginning of the space age in 1957, a new branch of geodesy was created, satellite geodesy. Only with satellites did geodesy become truly global. Oceans were no longer obstacles and the Earth as a whole could be observed and measured in consistent series of measurements. Of particular interest is the determination of the spatial structures and finally the temporal changes of the Earth's gravitational field. The knowledge of the gravitational field represents the natural bridge to the study of the physics of the Earth's interior, the circulation of our oceans and, more recently, the climate. Today, key findings on climate change are derived from the temporal changes in the gravitational field: on ice mass loss in Greenland and Antarctica, sea level rise and generally on changes in the global water cycle. This has only become possible with dedicated gravity satellite missions opening a method known as satellite gravimetry. In the first forty years of space age, satellite gravimetry was based on the analysis of the orbital motion of satellites. Due to the uneven distribution of observatories over the globe, the initially inaccurate measuring methods and the inadequacies of the evaluation models, the reconstruction of global models of the Earth's gravitational field was a great challenge. The transition from passive satellites for gravity field determination to satellites equipped with special sensor technology, which was initiated in the last decade of the twentieth century, brought decisive progress. In the chronological sequence of the launch of such new satellites, the history, mission objectives and measuring principles of the missions CHAMP, GRACE and GOCE flown since 2000 are outlined and essential scientific results of the individual missions are highlighted. The special features of the GRACE Follow-On Mission, which was launched in 2018, and the plans for a next generation of gravity field missions are also discussed.

Technology of determining the Earth’s gravitational field parameters using gradiometric measurements Part 6. Calculation the components of the gravitational potential tensor in the earth’s spatial rectangular coordinate system

This is the sixth one in a series of articles describing the technology of determining the Earth’s gravitational field parameters through gradiometric measurements performed with an onboard satellite electrostatic gradiometer. It provides formulas for calculating the components of the gravitational potential tensor in a geocentric spatial rectangular earth coordinate system in order to convert them into a gradiometric one and obtain a free term for the equations of correcting gradiometric measurements when determining the parameters of the Earth’s gravitational field. The components of the gravitational gradient tensor are functions of test masses accelerations measured by accelerometers and relate to the gradiometer coordinate system, while the desired parameters of the Earth’s gravitational field model relate to the Earth’s coordinate one. The components of the gravitational gradient tensor are the second derivatives of the gravitational potential in rectangular coordinates. The calculated values of the gravitational potential tensor components in the earth’s spatial rectangular coordinate system are obtained through double differentiation of the gravitational potential formula. Basing on the obtained formulas, an algorithm and a program in the Fortran algorithmic language were developed. Using this program, experimental calculations were performed, the results of which were compared with the data of the EGG_TRF_2 product.

Formation of an expanded criteria base for managing the effectiveness of the most important elements of advanced space systems for monitoring the Earth's gravitational field

В статье рассматриваются подходы к формированию критериальной базы оценки эффективности при проектировании перспективных космических систем на примере космических систем мониторинга гравитационного поля Земли. Сформулированы основные требования к созданию критериальной базы. Дано определение критерия и обобщенного критерия эффективности применительно к элементам перспективных космических систем. Рассмотрены общие и частные критерии эффективности перспективных космических систем в зависимости от назначения конкретной системы (оборонного применения, гражданские, многоцелевые). The article considers approaches to the formation of a criteria base for evaluating the effectiveness of the design of advanced space systems on the example of space systems for monitoring the Earth's gravitational field. The main requirements for creating a criteria base are formulated. The definition of the criterion and the generalized criterion of efficiency in relation to the elements of advanced space systems is given. General and specific criteria for the effectiveness of advanced space systems are considered, depending on the purpose of a particular system (defense, civil, multi-purpose).

On the accuracy of (quasi) geoid models relatively UELN/EVRS2000 height systems

Nowadays the Baltic Height System 1977 operates in Ukraine, the starting point of which is the zero of the Kronstadt banchmark. However, the current height system in Ukraine is morally obsolete primarily due to the great distance from the zero-point of height (about 2 thousand km) and the difficulty of adapting to the use of satellite surveying methods. Therefore, today it does not correspond to the level of development of modern geospatial technologies and it needs to be modernized. The most optimal way to modernize the height network of Ukraine is its integration into the United European Leveling Network UELN, the zero point of which is the Amsterdam banchmark. Within the framework of such integration it is necessary to create a high-precision geoid model for the territory of Ukraine, connected to the UELN/EVRS2000 height system. Aim. The purpose of the work is to compare the accuracy of different geoid/quasigeoid models and the global gravitational field of the Earth on the Western Ukraine area (border region) relative to the heights of points in the height system UELN/EVRS2000, where GNSSleveling is performed, and to determine the most optimal model in relation to which a high-precision geoid model can be created, consistent with the UELN/EVRS2000 height system.Method. To obtain the heights of leveling points on the Ukraine area in the UELN/EVRS2000 height system we performed I class leveling on two lines from the fundamental benchmarks on the territory of Ukraine (heights are known in the Baltic height system 1977) to I class benchmarks on the Poland area (UELN/EVRS2000 height system). GNSS-leveling in static mode (1-second sampling observations for more than 6 hours) was performed on all fundamental and ground benchmarks, as well as horizontal marks. Results. The heights of the quasigeoid at 26 points are obtained from the performed measurements. The heights are compared with three global models of the Earth’s gravitational field: EGM2008, EIGEN-6C4 and XGM2019e_2159 (the maximum order of all these models is 2190), as well as with the European geoid EGG2015. It is established that the best accuracy (≈ 7 cm) allows to obtain the European geoid EGG2015. Scientific novelty and practical significance. For the first time the accuracy of the Earth’s gravitational field models and geoid models on the Ukraine area is investigated at points where the height in the UELN/EVRS2000 height system is known. We established that when constructing a high-precision quasigeoid using the “Remove–Restore” procedure, it is best to use the European geoid EGG2015 as a systematic component.