COMPUTATION OF GRAVITY FIELD FUNCTIONALS WITH A LOCALIZED LEVEL ELLIPSOID

The description of the earth’s gravity field is usually expressed in terms of spherical harmonic coefficients, derived from global geopotential models. These coefficients may be used to evaluate such quantities as geoid undulations, gravity anomalies, gravity disturbances, deflection of the vertical, etc. To accomplish this, a global reference normal ellipsoid, such as WGS84 and GRS80, is required to provide the computing reference surface. These global ellipsoids, however, may not always provide the best fit of the local geoid and may provide results that are aliased. In this study, a regional or localized geocentric level ellipsoid is used alongside the EGM2008 to compute gravity field functionals in the state of Johor. Residual gravity field quantities are then computed using GNSS-levelled and raw gravity data, and the results are compared with both the WGS84 and the GRS80 equipotential surfaces. It is demonstrated that regional level ellipsoids may be used to compute gravity field functionals with a better fit, provided the zero-degree spherical harmonic is considered. The resulting residual quantities are smaller when compared with those obtained with global ellipsoids. It is expected that when the remove-compute-restore method is employed with such residuals, the numerical quadrature of the Stoke’s integral may be evaluated on reduced gravity anomalies that are smoother compared to when global equipotential surfaces are used

Evaluation of GGMs Based on the Terrestrial Gravity Data of Gravity Disturbance and Moho Depthin Afar, Ethiopia

To estimate Moho depth, geoid, gravity anomaly, and other geopotential functionals, gravity data is needed. But, gravity survey was not collected in equal distribution in Ethiopia, as the data forming part of the survey were mainly collected on accessible roads. To determine accurate Moho depth using Global Gravity Models (GGMs) for the study area, evaluation of GGMs is needed based on the available terrestrial gravity data. Moho depth lies between 28 km and 32 km in Afar. Gravity disturbances (GDs) were calculated for the terrestrial gravity data and the recent GGMs for the study area. The model-based GDs were compared with the corresponding GD obtained from the terrestrial gravity data and their differences in terms of statistical comparison parameters for determining the best fit GGM at a local scale in Afar. The largest standard deviation (SD) (36.10 mGal) and root mean square error (RMSE) (39.00 mGal) for residual GD and the lowest correlation with the terrestrial gravity (0.61 mGal) were obtained by the satellite-only model (GO_CONS_GCF_2_DIR_R6). The next largest SD (21.27 mGal) and RMSE (25.65 mGal) for residual GD were obtained by the combined gravity model (XGM2019e_2159), which indicates that it is not the best fit model for the study area as compared with the other two GGMs. In general, the result showed that the combined models are more useful tools for modeling the gravity field in Afar than the satellite-only GGMs. But, the study clearly revealed that for the study area, the best model in comparison with the others is the EGM2008, while the second best model is the EIGEN6C4.

Towards an updated, enhanced regional gravity field solution for Antarctica

<p>In the frame of the IAG Subcommission 2.4f “Gravity and Geoid in Antarctica” (AntGG) a first Antarctic-wide grid of ground-based gravity anomalies was released in 2016 (Scheinert et al. 2016). That data set was provided with a grid space of 10 km and covered about 73% of the Antarctic continent. Since then a considerably amount of new data has been made available, mainly collected by means of airborne gravimetry. Regions which were formerly void of any terrestrial gravity observations and have now been surveyed include especially the polar data gap originating from GOCE satellite gravimetry. Thus, it is timely to come up with an updated and enhanced regional gravity field solution for Antarctica. For this, we aim to improve further aspects in comparison to the AntGG 2016 solution: The grid spacing will be enhanced to 5 km. Instead of providing gravity anomalies only for parts of Antarctica, now the entire continent should be covered. In addition to the gravity anomaly also a regional geoid solution should be provided along with further desirable functionals (e.g. gravity anomaly vs. disturbance, different height levels).</p><p>We will discuss the expanded AntGG data base which now includes terrestrial gravity data from Antarctic surveys conducted over the past 40 years. The methodology applied in the analysis is based on the remove-compute-restore technique. Here we utilize the newly developed combined spherical-harmonic gravity field model SATOP1 (Zingerle et al. 2019) which is based on the global satellite-only model GOCO05s and the high-resolution topographic model EARTH2014. We will demonstrate the feasibility to adequately reduce the original gravity data and, thus, to also cross-validate and evaluate the accuracy of the data especially where different data set overlap. For the compute step the recently developed partition-enhanced least-squares collocation (PE-LSC) has been used (Zingerle et al. 2021, in review; cf. the contribution of Zingerle et al. in the same session). This method allows to treat all data available in Antarctica in one single computation step in an efficient and fast way. Thus, it becomes feasible to iterate the computations within short time once any input data or parameters are changed, and to easily predict the desirable functionals also in regions void of terrestrial measurements as well as at any height level (e.g. gravity anomalies at the surface or gravity disturbances at constant height).</p><p>We will discuss the results and give an outlook on the data products which shall be finally provided to present the new regional gravity field solution for Antarctica. Furthermore, implications for further applications will be discussed e.g. with respect to geophysical modelling of the Earth’s interior (cf. the contribution of Schaller et al. in session G4.3).</p>

Harmonica and Boule: Modern Python tools for geophysical gravimetry

<p>Gravimetry is a routine part of the geophysicists toolset, historically used in geophysics following the geodetic definitions of gravity anomalies and their related “reductions”. Several authors have shown that the geodetic concept of a gravity anomaly does not align with goals of gravimetry in geophysics (the investigation of anomalous density distributions). Much of this confusion likely stems from the lack of widely available tools for performing the corrections needed to arrive at a geophysically meaningful gravity disturbance. For example, free-air corrections are completely unnecessary since analytical expressions for theoretical gravity at any point have existed for over a decade. Since this is not easily done in a spreadsheet or short script, modern tools for processing and modelling gravity data for geophysics are needed. These tools must be trustworthy (i.e., extensively tested) and designed with software development and geophysical best practices in mind.</p><p>We present the Python libraries Harmonica and Boule, which are part of the Fatiando a Terra project (https://www.fatiando.org). Both tools are open-source under the permissive BSD license and are developed in the open by a community of geoscientists and programmers.</p><p>Harmonica provides tools for processing, forward modelling, and inversion of gravity and magnetic data. The first release of Harmonica was focused on implementing methods for processing and interpolation with the equivalent source technique, as well as forward modelling with right-rectangular prisms, point sources, and tesseroids. Current work is directed towards implementing a processing pipeline for gravity data, including topographic corrections in Cartesian and spherical coordinates, atmospheric corrections, and more. The software is still in early stages of development and design and would benefit greatly from community involvement and feedback.</p><p>Boule implements reference ellipsoids (including oblate ellipsoids, spheres, and soon triaxial ellipsoids), conversions between ellipsoidal and geocentric spherical coordinates, and normal gravity calculations using analytical solutions for gravity fields at any point outside of the ellipsoid. It includes ellipsoids for the Earth as well as other planetary bodies in the solar system, like Mars, the Moon, Venus, and Mercury. This enables the calculation of gravity disturbances for Earth and planetary data without the need for free-air corrections. Boule was created out of the shared needs of Harmonica, SHTools (https://github.com/SHTOOLS), and pygeoid (https://github.com/ioshchepkov/pygeoid) and is developed with input from developers of these projects.</p><p>We welcome participation from the wider geophysical community, irrespective of programming skill level and experience, and are actively searching for interested developers and users to get involved in shaping the future of these projects.</p>

Geophysical subsurface modelling based on the updated, enhanced regional gravity field solution in Antarctica

<p>The gravity field reflects mass inhomogeneities (mostly) inside the Earth. Therefore, gravity inversion and geophysical gravity field modelling are important tools to study the Earth's inner structure and tectonic evolution. In Antarctica, it is extremely challenging to carry out geoscientific studies due to its harsh environment and difficult logistics. Additionally, the continent is covered by an up to 5 km thick ice sheet. However, in the framework of IAG Subcommission 2.4f “Gravity and Geoid in Antarctica” (AntGG) a large database of airborne, shipborne and ground based gravity data has been compiled. Especially airborne data have been acquired during recent years, among others in the area of the polar gap of satellite gravity data. Now, in a joint project funded by the German Research Foundation (DFG) all existing and new gravity data were processed to infer an enhanced gravity field solution for Antarctica (see contribution by Scheinert et al., session G1.5). Processed data e.g. gravity disturbances on the ground or a constant height and other functionals will be provided on a regular grid with 5 km grid spacing. Subsequently, the new Antarctic gravity field solution can now be used for further geophysical and tectonic studies. We use the newly calculated gravity disturbances to study subglacial topography, sediment thickness and Moho depth and to improve respective existing models in Antarctica. For this, we apply 2D Parker-Oldenburg inversion in combination with results from other gravity based studies and further constraining data (e.g. seismic data and ice penetrating radar). We investigate how the higher resolution (5 km) of the new Antarctic gravity field solution facilitates the study of smaller regions in more detail, specifically parts of Wilkes Land, Dronning Maud Land and the Weddell Sea. Additionally, we will infer accuracy estimates for the resulting boundaries in terms of the used inversion parameters (density contrast, average density and filter wavelengths) and their respective gravity signal. Thus, the challenges of gravity field inversion in Antarctica will be discussed in detail and first results of the subsurface modelling will be presented.</p>

Validation of calibrated GOCE gravity gradients GRD_SPW_2 by least-squares spectral weighting

<p>The Gravity field and steady-state Ocean Circulation Explorer (GOCE) was the first mission which carried a novel instrument, gradiometer, which allowed to measure the second-order directional derivatives of the gravitational potential or gravitational gradients with uniform quality and a near-global coverage. More than three years of the outstanding measurements resulted in two levels of data products (Level 1b and Level 2), six releases of global gravitational models (GGMs), and several grids of gravitational gradients (see, e.g., ESA-funded GOCE+ GeoExplore project or Space-wise GOCE products). The grids of gravitational gradients represent a step between gravitational gradients measured directly along the GOCE orbit and data directly from GGMs. One could use grids of gravitational gradients for geodetic as well as for geophysical applications. In this contribution, we are going to validate the official Level 2 product GRD_SPW_2 by terrestrial gravity disturbances and GNSS/levelling over two test areas located in Europe, namely in Norway and former Czechoslovakia (now Czechia and Slovakia). GRD_SPW_2 product contains all six gravity gradients at satellite altitude from the space-wise approach computed only from GOCE data for the available time span (r-2, r-4, and r-5) and provided on a 0.2 degree grid. A mathematical model based on a least-squares spectral weighting will be developed and the corresponding spectral weights will be presented for the validation of gravitational gradients grids. This model allows us to continue downward gravitational gradients grids to an irregular topographic surface (not to a mean sphere) and transform them into gravity disturbances and/or geoidal heights in one step. Before we compared results obtained by spectral downward continuation, we had to remove the high-frequency part of the gravitational signal from terrestrial data because in gravitational gradients measured at GOCE satellite altitude is attenuated. To do so we employ EGM2008 up to d/o 2160 and the residual terrain model correction (RTC) has been a) interpolated from ERTM2160 gravity model, b) synthesised from dV_ELL_Earth2014_5480_plusGRS80, c) calculated from a residual topographic model by forward modelling in the space domain.  </p>

The nearby spiral density-wave structure of the Galaxy: line-of-sight velocities of the Gaia DR2 OB stars

ABSTRACT Distances and line-of-sight velocities of 964 Gaia Data Release 2 (DR2) OB stars of Xu et al. within 3 kpc from the Sun and 500 pc from the Galactic mid-plane with accuracies of <50 per cent are selected. The data are used to find small systematic departures of velocities from the mean circular motion for the stars in the solar neighborhood due to the spiral compression-type (Lin–Shu-type) waves, or spiral density waves, e.g. those produced by real instabilities of spontaneous gravity disturbances, a central bar or a companion system. A key point of the study is that our results are consistent with the ones extracted from the asymptotic density-wave theory. Revised parameters of density waves in the solar vicinity of the Galaxy are also provided.

Vertical Deflections and Gravity Disturbances Derived from HY-2A Data

The first Chinese altimetry satellite, Haiyang-2A (HY-2A), which was launched in 2011, has provided a large amount of sea surface heights which can be used to derive marine gravity field. This paper derived the vertical deflections and gravity disturbances using HY-2A observations for the major area of the whole Earth’s ocean from 60°S and 60°N. The results showed that the standard deviations (STD) of vertical deflections differences were 1.1 s and 3.5 s for the north component and the east component between HY-2A’s observations and those from EGM2008 and EIGEN-6C4, respectively. This indicates the accuracy of the east component was poorer than that of the north component. In order to clearly demonstrate contribution of HY-2A’s observations to gravity disturbances, reference models and the commonly used remove-restore method were not adopted in this study. Therefore, the results can be seen as ‘pure’ signals from HY-2A. Assuming the values from EGM2008 were the true values, the accuracy of the gravity disturbances was about −1.1 mGal in terms of mean value of the errors and 8.0 mGal in terms of the STD. This shows systematic errors if only HY-2A observations were used. An index of STD showed that the accuracy of HY-2A was close to the theoretical accuracy according to the vertical deflection products. To verify whether the systematic errors of gravity field were from the long wavelengths, the long-wavelength parts of HY-2A’s gravity disturbance with wavelengths larger than 500 km were replaced by those from EGM2008. By comparing with ‘pure’ HY-2A version of gravity disturbance, the accuracy of the new version products was improved largely. The systematic errors no longer existed and the error STD was reduced to 6.1 mGal.

High-resolution numerical modelling of the altimetry-derived gravity disturbances and disturbing gradients

<p>Recent high-resolution mean sea surface models obtained from satellite altimetry in a combination with the GRACE/GOCE-based global geopotential models provide valuable information for detailed modelling of the altimetry-derived gravity data. Our approach is based on a numerical solution of the altimetry-gravimetry boundary-value problem using the finite volume method (FVM). FVM discretizes the 3D computational domain between an ellipsoidal approximation of the Earth's surface and an upper boundary chosen at a mean altitude of the GOCE satellite orbits. A parallel implementation of the finite volume numerical scheme and large-scale parallel computations on clusters with distributed memory allow to get a high-resolution numerical solution in the whole 3D computational domain. Our numerical experiment presents the altimetry-derived gravity disturbances and disturbing gradients determined with the high-resolution 1 x 1 arc min at two altitude levels; on the reference ellipsoid and at the altitude of 10 km above the ellipsoid. As input data, the Dirichlet boundary conditions over oceans/seas are considered in the form of the disturbing potential. It is obtained from the geopotential evaluated on the DTU18 mean sea surface model from the GO_CONS_GCF_2_TIM_R5 geopotential model and then filtered using the nonlinear diffusion filtering. On the upper boundary, the FVM solution is fixed to the disturbing potential generated from the GO_CONS_GCF_2_DIR_R5 model while exploiting information from the GRACE and GOCE satellite missions.</p>

Terrestrial strapdown inertial gravimetry in the Bavarian Alps

<p>Around 100km south of Munich, the Institute of Astronomical and Physical Geodesy of the Technical University of Munich established a gravimetric-astrogeodetic testing ground over the last 20 years. Precise gravity values as well as vertical deflections exist for hundreds of points. End of 2019, a car-based strapdown inertial gravimetry survey was realized in this area along a ~25km track. For this track, a few gravity values and several vertical deflections (spacing around 200m) are available (Hirt and Flury 2008). Navigation-grade IMU (inertial measurement unit), GNSS (global navigation satellite systems) and additional relative gravimeter observations were recorded during the survey. With this setup, it is possible to evaluate the capabilities of terrestrial scalar and vector strapdown inertial gravimetry.</p><p>This contribution gives an overview about the testing ground, the recently conducted survey and the data processing. The main part treats the analyses regarding the accuracy of 1D- and 3D-strapdown inertial gravimetry. Furthermore, attention is payed to the kinematic IMU performance (noise behavior), the benefit of special IMU calibrations (Becker 2016) and a comparison of the results with pure model based gravity disturbances.</p><p><strong>Literature</strong></p><ul><li>Becker, D. (2016). Advanced Calibration Methods for Strapdown Airborne Gravimetry. PhD thesis, Technische Universität Darmstadt, Fachbereich Bau- und Umweltingenieurwissenschaften, Schriftenreihe der Fachrichtung Geodäsie Heft 51. ISBN 978-3-935631-40-2.</li> <li>Hirt, C. and Flury J. (2008). Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data. J Geod (2008) 82:231–248, Springer-Verlag. DOI 10.1007/s00190-007-0173-x.</li> </ul>