Differential geometry and curvatures of equipotential surfaces in the realization of the World Height System

2020 ◽  
Author(s):  
Petr Holota ◽  
Otakar Nesvadba
Keyword(s):  
Differential Geometry ◽  
Gravity Field ◽  
Physical Geodesy ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
The Earth ◽  
The World ◽  
Height System ◽  
Equipotential Surfaces ◽  
Gravity Field Models

<p>The notion of an equipotential surface of the Earth’s gravity potential is of key importance for vertical datum definition. The aim of this contribution is to focus on differential geometry properties of equipotential surfaces and their relation to parameters of Earth’s gravity field models. The discussion mainly rests on the use of Weingarten’s theorem that has an important role in the theory of surfaces and in parallel an essential tie to Brun’s equation (for gravity gradient) well known in physical geodesy. Also Christoffel’s theorem and its use will be mentioned. These considerations are of constructive nature and their content will be demonstrated for high degree and order gravity field models. The results will be interpreted globally and also in merging segments expressing regional and local features of the gravity field of the Earth. They may contribute to the knowledge important for the realization of the World Height System.</p>

Subtleties in spherical harmonic synthesis of the gravity field

2020 ◽  
Author(s):  
Ropesh Goyal ◽  
Sten Claessens ◽  
Will Featherstone ◽  
Onkar Dikshit
Keyword(s):  
Gravity Field ◽  
Spherical Harmonic ◽  
Gravitational Constant ◽  
Angular Rate ◽  
Physical Geodesy ◽  
Major Axis ◽  
Gravity Potential ◽  
Semi Major Axis ◽  
The Earth ◽  
Spherical Harmonic Synthesis

<p>Spherical harmonic synthesis (SHS) can be used to compute various gravity functions (e.g., geoid undulations, height anomalies, deflections of vertical, gravity disturbances, gravity anomalies, etc.) using the 4pi fully normalised Stokes coefficients from the many freely available Global Geopotential Models (GGMs).  This requires a normal ellipsoid and its gravity field, which are defined by four parameters comprising (i) the second-degree even zonal Stokes coefficient (J2) (aka dynamic form factor), (ii) the product of the mass of the Earth and universal gravitational constant (GM) (aka geocentric gravitational constant), (iii) the Earth’s angular rate of rotation (ω), and (iv) the length of the semi-major axis (a). GGMs are also accompanied by numerical values for GM and a, which are not necessarily identical to those of the normal ellipsoid.  In addition, the value of W<sub>0,</sub> the potential of the geoid from a GGM, needs to be defined for the SHS of many gravity functions. W<sub>0</sub> may not be identical to U<sub>0</sub>, the potential on the surface of the normal ellipsoid, which follows from the four defining parameters of the normal ellipsoid.  If W<sub>0</sub> and U<sub>0</sub> are equal and if the normal ellipsoid and GGM use the same value for GM, then some terms cancel when computing the disturbing gravity potential.  However, this is not always the case, which results in a zero-degree term (bias) when the masses and potentials are different.  There is also a latitude-dependent term when the geometries of the GGM and normal ellipsoids differ.  We demonstrate these effects for some GGMs, some values of W<sub>0</sub>, and the GRS80, WGS84 and TOPEX/Poseidon ellipsoids and comment on its omission from some public domain codes and services (isGraflab.m, harmonic_synth.f and ICGEM).  In terms of geoid heights, the effect of neglecting these parameters can reach nearly one metre, which is significant when one goal of modern physical geodesy is to compute the geoid with centimetric accuracy.  It is also important to clarify these effects for all (non-specialist) users of GGMs.</p>

Preliminary monthly gravity field models from kinematic orbits of SWARM Satellites

2021 ◽  
Author(s):  
Xingfu Zhang ◽  
Qiujie Chen ◽  
Yunzhong Shen
Keyword(s):  
Gravity Field ◽  
Large Scale ◽  
Earth System ◽  
The Earth ◽  
Swarm Satellites ◽  
Variable Gravity ◽  
Data Gap ◽  
One Year ◽  
Gravity Recovery ◽  
Gravity Field Models

<p>      Although the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE FO) satellite missions play an important role in monitoring global mass changes within the Earth system, there is a data gap of about one year spanning July 2017 to May 2018, which leads to discontinuous gravity observations for monitoring global mass changes. As an alternative mission, the SWARM satellites can provide gravity observations to close this data gap. In this paper, we are dedicated to developing alternative monthly time-variable gravity field solutions from SWARM data. Using kinematic orbits of SWARM from ITSG for the period January 2015 to September 2020, we have generated a preliminary time series of monthly gravity field models named Tongji-Swarm2019 up to degree and order 60. The comparisons between Tongji-Swarm2019 and GRACE/GRACE-FO monthly solutions show that Tongji-Swarm2019 solutions agree with GRACE/GRACE-FO models in terms of large-scale mass change signals over amazon, Greenland and other regions. We can conclude that Tongji-Swarm2019 monthly gravity field models are able to close the gap between GRACE and GRACE FO.</p>

scholarly journals Evaluation of Gravity Data Derived from Global Gravity Field Models Using Terrestrial Gravity Data in Enugu State, Nigeria

2018 ◽  
Vol 8 (1) ◽  
pp. 145-153 ◽  
Author(s):  
O.I. Apeh ◽  
E.C. Moka ◽  
V.N. Uzodinma
Keyword(s):  
Gravity Field ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Mean Square ◽  
Bouguer Gravity ◽  
Bouguer Anomalies ◽  
The Earth ◽  
Global Gravity ◽  
Enugu State ◽  
Gravity Field Models

Abstract Spherical harmonic expansion is a commonly applied mathematical representation of the earth’s gravity field. This representation is implied by the potential coeffcients determined by using elements/parameters of the field observed on the surface of the earth and/or in space outside the earth in the spherical harmonic expansion of the field. International Centre for Gravity Earth Models (ICGEM) publishes, from time to time, Global Gravity Field Models (GGMs) that have been developed. These GGMs need evaluation with terrestrial data of different locations to ascertain their accuracy for application in those locations. In this study, Bouguer gravity anomalies derived from a total of eleven (11) recent GGMs, using sixty sample points, were evaluated by means of Root-Mean-Square difference and correlation coeficient. The Root-Mean-Square differences of the computed Bouguer anomalies from ICGEMwebsite compared to their positionally corresponding terrestrial Bouguer anomalies range from 9.530mgal to 37.113mgal. Additionally, the correlation coe_cients of the structure of the signal of the terrestrial and GGM-derived Bouguer anomalies range from 0.480 to 0.879. It was observed that GECO derived Bouguer gravity anomalies have the best signal structure relationship with the terrestrial data than the other ten GGMs. We also discovered that EIGEN-6C4 and GECO derived Bouguer anomalies have enormous potential to be used as supplements to the terrestrial Bouguer anomalies for Enugu State, Nigeria.

GRACE-FO accelerometer data: An alternative approach using Least Squares Spectral Analysis

2021 ◽  
Author(s):  
Myrto Tzamali ◽  
Spiros Pagiatakis
Keyword(s):  
Magnetic Field ◽  
Least Squares ◽  
Frequency Domain ◽  
Gravity Field ◽  
Thermal Sensitivity ◽  
Accelerometer Data ◽  
Data Set ◽  
The Earth ◽  
Gravity Field Models ◽  
The Magnetic Field

<p>Technological advances in satellite geodesy have been demanding more and more accurate gravity field models but also precise measurements of the movement of water along the Earth system. GRACE-FO (GFO) mission is dedicated to monitor the Earth with a purpose of estimating the gravity field and the hydrological cycles. For the extraction of monthly gravity field models the non-gravitational accelerations are essential. The performance of GFO accelerometers (ACC) is not the optimal.  The ACC measurements present immense spikes, spurious signals and bias jumps on all three axes affecting the validity of the measurements. The bias jumps are similar to those presented at GRACE measurements and they have been related to the satellites’ entrance to and exit from the Earth’s shadow. The dominant spikes, mainly appearing in the equatorial region, have been connected to the thermal sensitivity of the instrument or the orientation of the magnetic field lines. We propose an alternative dataset generated from Level 1A of GFO C with corresponding Gaussian weights and an optimal correction of the bias jumps, along with the estimation of linear and quadratic trends using the Least Squares methodology in the frequency domain and in all three axes. The method does not remove spikes, nor does it interpolate missing values. The new 1B dataset with estimated variances shows no spike effects in the frequency domain contrastingly to the existing ACT Level 1B data. Also, a preliminary analysis of the daily amplitudes of the orbital period and semi-period components of the ACT Level 1B data set spanning one year, reveals a strong periodic signal of ~ 153 days. This signal vanishes when the proposed weighted data set is used. This signal could be related to calibration deficiencies or a systematic error in the ACC data that requires further study. The same weighted filtering approach is proposed for the ACC measurements of Swarm C satellite, a LEO constellation that measures the magnetic field of the Earth. The ACC measurements of Swarm display low signal to noise ratio due to an increased thermal sensitivity of the instrument. A weighted Gaussian filter applied on the Swarm ACC measurements reduces the contribution of the dominant spikes in the frequency domain and displays the non-gravitational signals more clearly leading to a more extended use of Swarm non-gravitational accelerations measurements.</p>

Comparison of Satellite-Only Gravity Field Models Constructed with All and Parts of the GOCE Gravity Gradient Dataset

2016 ◽  
Vol 39 (3-4) ◽  
pp. 238-255
Author(s):  
Sean L. Bruinsma ◽  
Christoph Förste ◽  
Sandrine Mulet ◽  
Marie-Hélène Rio ◽  
Oleg Abrikosov ◽  
...  
Keyword(s):  
Gravity Field ◽  
Gravity Gradient ◽  
Gravity Field Models

scholarly journals GOCE-Derived Coseismic Gravity Gradient Changes Caused by the 2011 Tohoku-Oki Earthquake

2019 ◽  
Vol 11 (11) ◽  
pp. 1295
Author(s):  
Xinyu Xu ◽  
Hao Ding ◽  
Yongqi Zhao ◽  
Jin Li ◽  
Minzhang Hu
Keyword(s):  
Gravity Field ◽  
Ocean Circulation ◽  
Spherical Harmonic ◽  
Least Squares Method ◽  
Time Variable ◽  
Gravity Models ◽  
Gravity Gradient ◽  
Gravity Field Recovery ◽  
Before And After ◽  
Gravity Field Models

In contrast to most of the coseismic gravity change studies, which are generally based on data from the Gravity field Recovery and Climate Experiment (GRACE) satellite mission, we use observations from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) Satellite Gravity Gradient (SGG) mission to estimate the coseismic gravity and gravity gradient changes caused by the 2011 Tohoku-Oki Mw 9.0 earthquake. We first construct two global gravity field models up to degree and order 220, before and after the earthquake, based on the least-squares method, with a bandpass Auto Regression Moving Average (ARMA) filter applied to the SGG data along the orbit. In addition, to reduce the influences of colored noise in the SGG data and the polar gap problem on the recovered model, we propose a tailored spherical harmonic (TSH) approach, which only uses the spherical harmonic (SH) coefficients with the degree range 30–95 to compute the coseismic gravity changes in the spatial domain. Then, both the results from the GOCE observations and the GRACE temporal gravity field models (with the same TSH degrees and orders) are simultaneously compared with the forward-modeled signals that are estimated based on the fault slip model of the earthquake event. Although there are considerable misfits between GOCE-derived and modeled gravity gradient changes (ΔVxx, ΔVyy, ΔVzz, and ΔVxz), we find analogous spatial patterns and a significant change (greater than 3σ) in gravity gradients before and after the earthquake. Moreover, we estimate the radial gravity gradient changes from the GOCE-derived monthly time-variable gravity field models before and after the earthquake, whose amplitudes are at a level over three times that of their corresponding uncertainties, and are thus significant. Additionally, the results show that the recovered coseismic gravity signals in the west-to-east direction from GOCE are closer to the modeled signals than those from GRACE in the TSH degree range 30–95. This indicates that the GOCE-derived gravity models might be used as additional observations to infer/explain some time-variable geophysical signals of interest.

Optical clocks for gravity field observation and further geodetic applications

2020 ◽  
Author(s):  
Hu Wu ◽  
Jürgen Müller ◽  
Annike Knabe
Keyword(s):  
Gravity Field ◽  
Rapid Development ◽  
Satellite Orbit ◽  
Equipotential Surface ◽  
Gravity Potential ◽  
Height Difference ◽  
The Earth ◽  
Temporal Gravity ◽  
Reference Clock

<p>In the past three decades, optical clocks and frequency transfer techniques have experienced a rapid development. They are approaching a fractional frequency uncertainty of 1.0x10<sup>-18</sup>, corresponding to about 1.0 cm in height. This makes them promising to realize “relativistic geodesy”, and it opens a new door to directly obtain gravity potential values by the comparison of clock frequencies. Clocks are thus considered as a novel candidate for determining the Earth’s gravity field. We propose to use a spaceborne clock to obtain gravity potential values along a satellite orbit through its comparison with reference clocks on ground or with a co-orbital clock. The sensitivity of clock measurements is mapped to gravity field coefficients through closed-loop simulations.</p><p>In addition, clocks are investigated for other geodetic applications. Since they are powerful in providing the height difference between distant sites, clocks can be applied for the unification of local/regional height systems, by estimating the offsets between different height datums and the systematic errors within levelling networks. In some regions like Greenland, clocks might be a complementary tool to GRACE(-FO) for detecting temporal gravity signals. They can be operated at locations of interest and continuously track changes w.r.t. reference clock stations. The resulting time-series of gravity potential values reveal the temporal gravity signals at these points. Moreover, as the equipotential surface at a high satellite altitude is more regular than that on the Earth’s surface, a couple of clocks in geostationary orbits can realize a space-based reference for the determination of physical heights at any point on the Earth through clock comparisons.</p><p>We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2123/1 (Project-ID: 390837967).</p>

Fundamental Notions in Relativistic Geodesy - physics of a timelike Killing vector field

2020 ◽  
Author(s):  
Dennis Philipp ◽  
Claus Laemmerzahl ◽  
Eva Hackmann ◽  
Volker Perlick ◽  
Dirk Puetzfeld ◽  
...  
Keyword(s):  
Gravity Field ◽  
Vector Fields ◽  
Dimensional Space ◽  
Killing Vector ◽  
Three Dimensional ◽  
Scalar Function ◽  
Gravity Potential ◽  
Static Case ◽  
The Earth ◽  
Timelike Killing Vector

<p>The Earth’s geoid is one of the most important fundamental concepts to provide a gravity field- related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions within Einstein’s theory of General Relativity is inevitable.<span> </span></p><p>Building on the theoretical construction of isochronometric surfaces we define a relativistic gravity potential as a generalization of known (post-)Newtonian notions. It exists for any stationary configuration and rigidly co-rotating observers; it is the same as realized by local plumb lines and determined by the norm of a timelike Killing vector. In a second step, we define the relativistic geoid in terms of this gravity potential in direct analogy to the Newtonian understanding. In the respective limits, it allows to recover well-known results. Comparing the Earth’s Newtonian geoid to its relativistic generalization is a very subtle problem. However, an isometric embedding into Euclidean three-dimensional space can solve it and allows an intrinsic comparison. We show that the leading-order differences are at the mm-level.<span> </span>In the next step, the framework is extended to generalize the normal gravity field as well. We argue that an exact spacetime can be constructed, which allows to recover the Newtonian result in the weak-field limit. Moreover, we comment on the relativistic definition of chronometric height and related concepts.</p><p>In a stationary spacetime related to the rotating Earth, the aforementioned gravity potential is of course not enough to cover all information on the gravitational field. To obtain more insight, a second scalar function can be constructed, which is genuinely related to gravitomagnetic contributions and vanishes in the static case. Using the kinematic decomposition of an isometric observer congruence, we suggest a potential related to the twist of the worldlines therein. Whilst the first potential is related to clock comparison and the acceleration of freely falling corner cubes, the twist potential is related to the outcome of Sagnac interferometric measurements. The combination of both potentials allows to determine the Earth’s geoid and equip this surface with coordinates in an operational way. Therefore, relativistic geodesy is intimately related to the physics of timelike Killing vector fields.</p>

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