Application of the nonlinear optimisation in regional gravity field modelling using spherical radial base functions

The gravity field is a signature of the mass distribution and interior structure of the Earth, in addition to all its geodetic applications especially geoid determination and vertical datum unification. Determination of a regional gravity field model is an important subject and needs to be investigated and developed. Here, the spherical radial basis functions (SBFs) are applied in two scenarios for this purpose: interpolating the gravity anomalies and solving the fundamental equation of physical geodesy for geoid or disturbing potential determination, which has the possibility of being verified by the Global Navigation Satellite Systems (GNSS)/levelling data. Proper selections of the number of SBFs and optimal location of the applied SBFs are important factors to increase the accuracy of estimation. In this study, the gravity anomaly interpolation based on the SBFs is performed by Gauss-Newton optimisation with truncated singular value decomposition, and a Quasi-Newton method based on line search to solve the minimisation problems with a small number of iterations is developed. In order to solve the fundamental equation of physical geodesy by the SBFs, the truncated Newton optimisation is applied as the Hessian matrix of the objective function is not always positive definite. These two scenarios are applied on the terrestrial free-air gravity anomalies over the topographically rough area of Auvergne. The obtained accuracy for the interpolated gravity anomaly model is 1.7 mGal with the number of point-masses about 30% of the number of observations, and 1.5 mGal in the second scenario where the number of used kernels is also 30%. These accuracies are root mean square errors (RMSE) of the differences between predicted and observed gravity anomalies at check points. Moreover, utilising the optimal constructed model from the second scenario, the RMSE of 9 cm is achieved for the differences between the gravimetric height anomalies derived from the model and the geometric height anomalies from GNSS/levelling points.

DETERMINATION AND MAPPING OF THE TERRESTRIAL GRAVITY ANOMALIES IN THE MOUNTAINOUS AREAS OF IRAQ

Gravity data and computing gravity anomalies are regarded as vital for both geophysics and physical geodesy fields. The mountainous areas of Iraq are characterized by the lack of regional gravity data because gravity surveys are rarely performed in the past four decades due to the Iraq-Iran war and the internal unstable political situation of this particular region. In addition, the formal map of the available terrestrial gravity which was published by the French Database of Bureau Gravimetrique International (International Gravimetric Bureau-in English) (BGI), introduces Iraq and the study area as a remote area and in white color because of the unavailability of gravity data. However, a dense and local (not regional) gravity data is available which was conducted by geophysics researchers 13 years ago. Therefore, the regional gravity survey of 160 gravity points was performed by the authors at an average 11 km apart, which was covers the whole area of Sulaymaniyah Governorate (part of the mountainous areas of Iraq). In spite of Although the risk of mine fields within the study area, suitable safe routes as well as a helicopter was used for the gravity survey of several points on the top of mountains. The survey was conducted via Lacoste and Romberg geodetic gravimeter and GPS handheld. The objective of the study is to determine and map the gravity anomalies for the entire study area, the data of which would assist different geosciences applications.

Harmonic Correction for Residual Terrain Modelling (RTM) Technique in Physical Geodesy Applications

<p>       In physical geodesy, the harmonic correction (HC), as one of the main problems when using residual terrain modelling (RTM), has become a research focus of high-frequency gravity field modelling. Over past decades, though various methods have been proposed to handle the HC issues for RTM technique, most of them focused on the HC for RTM gravity anomaly rather than other gravity functionals, such as RTM geoid height and gravity gradient. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a quantification is yet studied. In this study, besides the highlighted HC for gravity anomaly in previous studies, the expressions of HC terms for RTM geoid height are provided in the framework of the classical condensation method under infinite Bouguer plate approximation. The errors involved by various assumption of the classical condensation method, e.g., mass inconsistency between infinite masses in the HC and limited masses in the RTM, and planar assumption of the Earth’s surface, are further studied. Based on the derived formulas, the quantification of HC for RTM geoid height when reference surface is expanded to degree and order of 2,159 is given. Our results showed the significance of HC for RTM geoid height, with values up to ~10 cm, in cm-level and mm-level geoid determination. With integration masses extending up to a sufficient distance, such as 1° from calculation point for the determination of RTM geoid height, the errors due to an infinite Bouguer plate approximation are neglectable small. The validation through comparison with terrestrial measurements proved that the HC terms provided in this study can improve the accuracy of RTM derived geoid height and are expected to be useful for applications of RTM technique in regional and global gravity field modelling.</p>

Arne Bjerhammar- a personal summary of his academic deeds

Arne Bjerhammar is well known worldwide mainly for his research in physical geodesy but also for introducing a new matrix algebra with generalized inverses applied in geodetic adjustment. Less known are his developments in geodetic engineering and contributions to satellite and relativistic geodesy as well as studies on the relation between the Fennoscandia land uplift and the regional gravity low. Most likely part of his research has contributed to worldwide political relaxation during the cold war, which deed was honored by a certificate of achievement awarded by the Department of Research of the US army as well as the North Star Order by the King of Sweden. Arne Bjerhammar’s pioneer scientific production, in particular on a world geodetic system, towards what would become GPS, as well as relativistic geodesy, is still of great interest among the worldwide geodetic community, while the memories and spirit along his outstanding academic deeds have more or less fainted away from his home university (KTH) only a decade after he passed away.

Subtleties in spherical harmonic synthesis of the gravity field

<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>

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>

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

<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>

PRELIMINARY DETECTION OF GEOTHERMAL MANIFESTATION POTENTIAL USING MICROWAVE SATELLITE REMOTE SENSING

The satellite technology has developed significantly. The sensors of remote sensing satellites are in the form of optical, Microwave, and LIDAR. These sensors can be used for energy and mineral resources applications. The example of those applications are height model and the potential of geothermal manifestation detection. This study aims to detect the potential of geothermal manifestation using remote sensing. The study area is the Northern of the Inverse Arc of Sulawesi. The method used is remote sensing approach for its preliminary detection with 4 steps as follow (a) mining land identification, (b) geological parameter extraction, (c) preparation of standardized spatial data, and (d) geothermal manifestation. Mining lands identification is using Vegetation Index Differencing method. Geological parameters include structural geology, height model, and gravity model. The integration method is used for height model. The height model integration use ALOS PALSAR data, Icesat/GLAS, SRTM, and X SAR. Structural geology use dip and strike method. Gravity model use physical geodesy approach. Preparation of standardized spatial data with re-classed and analyzed using Geographic Information System between each geological parameter, whereas physical geodesy methods are used for geothermal manifestation detection. Geothermal manifestation using physical geodesy approach in Barthelmes method. Grace and GOCE data are used for gravity model. The geothermal manifestation detected from any parameter is analyzed by using geographic information system method. The result of this study is 10 area of geothermal manifestation potential. The accuracy test of this research is 87.5 % in 1.96 σ. This research can be done efficiently and cost-effectively in the process. The results can be used for various geological and mining applications.

A numerical test of the topographic bias

In 1962 A. Bjerhammar introduced the method of analytical continuation in physical geodesy, implying that surface gravity anomalies are downward continued into the topographic masses down to an internal sphere (the Bjerhammar sphere). The method also includes analytical upward continuation of the potential to the surface of the Earth to obtain the quasigeoid. One can show that also the common remove-compute-restore technique for geoid determination includes an analytical continuation as long as the complete density distribution of the topography is not known. The analytical continuation implies that the downward continued gravity anomaly and/or potential are/is in error by the so-called topographic bias, which was postulated by a simple formula of L E Sjöberg in 2007. Here we will numerically test the postulated formula by comparing it with the bias obtained by analytical downward continuation of the external potential of a homogeneous ellipsoid to an inner sphere. The result shows that the postulated formula holds: At the equator of the ellipsoid, where the external potential is downward continued 21 km, the computed and postulated topographic biases agree to less than a millimetre (when the potential is scaled to the unit of metre).