Correction to: Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

The article: “Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction”, written by Miao Lin, Heiner Denker and Jürgen Müller, was originally published electronically on the publisher’s internet portal on 26 March 2020 without open access.

Report of the D-A-CH geoid and height unification project and prospects for the extension to the European Alps and beyond

<p>The D-A-CH geoid project was initiated in 2017 between the national mapping agencies of Germany (BKG), Austria (BEV) and Switzerland (swisstopo), as well as the regional authorities of the German federal states of Bavaria (LDBV) and Baden-Württemberg (LGL), with the motivation to better harmonize the basis for height determination.</p><p>In these countries, the official national height reference systems that are still in use apply different definitions of the height and the zero levels refer to different tide gauges and epochs. Additionally, the treatment of the permanent tide is not fully consistent. This causes differences at the decimeter scale which also vary along the national borders. At the same time, Austria and Switzerland do compute and store also EVRS-compatible geopotential numbers that are valuable for height system unification.</p><p>The ambitions of the initiative therefore mirror the situation as described above ‒ to foster and to intensify the cooperation between the partners regarding regional gravity field modeling and to provide better information about the transformations between the national height systems.</p><p>It was agreed that the cooperation should first focus on a case study area around Lake Constance, with envisaged extension to the complete territories of the “D-A-CH countries” and/or, ideally, to the most of the European Alps. The following achievements have been reached for the focus area:</p><p>In view of these developments, and taking into account that these challenges are not unique for this specific area, it is planned to extend this initiative to the computation of the entire European Alps (and surrounding lowland areas) and rename the project to “European Alps Geoid (EAlpG)”.</p><p>We believe that this project can contribute to a better understanding of height differences across borders. Such height differences are for instance of great interest for ground water level investigations or flood protection. Other crucial applications for cross-border height unification are engineering projects such as tunnels, bridges, supply lines, etc.</p><p>What is more, these activities shall be embedded in a pan-European geoid initiative within EUREF. Contributing to the upcoming EUREF Working Group “European Height Reference Surface”, the European Alps Geoid will be one of many cornerstones to build an official EVRS height reference surface.</p><p>Potential cooperation partners have been contacted. Nevertheless, the initiative shall be open to interested parties. A virtual meeting is planned to be held shortly after the vEGU2021.</p>

Determination of the Regularization Parameter to Combine Heterogeneous Observations in Regional Gravity Field Modeling

Various types of heterogeneous observations can be combined within a parameter estimation process using spherical radial basis functions (SRBFs) for regional gravity field refinement. In this process, regularization is in most cases inevitable, and choosing an appropriate value for the regularization parameter is a crucial issue. This study discusses the drawbacks of two frequently used methods for choosing the regularization parameter, which are the L-curve method and the variance component estimation (VCE). To overcome their drawbacks, two approaches for the regularization parameter determination are proposed, which combine the L-curve method and VCE. The first approach, denoted as “VCE-Lc”, starts with the calculation of the relative weights between the observation techniques by means of VCE. Based on these weights, the L-curve method is applied to determine the regularization parameter. In the second approach, called “Lc-VCE”, the L-curve method determines first the regularization parameter, and it is set to be fixed during the calculation of the relative weights between the observation techniques from VCE. To evaluate and compare the performance of the two proposed methods with the L-curve method and VCE, all these four methods are applied in six study cases using four types of simulated observations in Europe, and their modeling results are compared with the validation data. The RMS errors (w.r.t the validation data) obtained by VCE-Lc and Lc-VCE are smaller than those obtained from the L-curve method and VCE in all the six cases. VCE-Lc performs the best among these four tested methods, no matter if using SRBFs with smoothing or non-smoothing features. These results prove the benefits of the two proposed methods for regularization parameter determination when different data sets are to be combined.

Precise LEO satellite orbit determination and Earth gravity field modeling with carrier-range method

<p>Precise LEO satellite orbit determination(OD) and Earth gravity field modeling are researched in this study.</p><p>Firstly, on the basis of Precise Point Positioning Ambiguity Resolution(PPPAR), a kinematic LEO satellite OD algorithm based on the epoch-difference and post-facto iteration is introduced, which plays a vital rule in the detection of the phase cycle slip to achieve the best orbit accuracy. The experiments of GRACE satellite OD with zero-difference IF combination observations spanning one year of 2010 show that, compared to the JPL reference orbits, the daily average 3D RMS is generally below 5.0cm for the float solution, while that is below 4.0cm for the fixed solution.</p><p>Secondly, to solve the problem that specific a-priori information like earth gravity field model must be involved in LEO’ reduced dynamic OD, the simultaneous solution method, which is specially on the relation with the kinematic OD and reduced dynamic OD, is used and the carrier-range, which can be recovered from phase observations once the kinematic OD process using Integer Ambiguity Resolution (IAR) technology is carried out, is naturally applied to this method. With the experiments based on the data over a period of the year of 2010, comes some evacuations, including the external checks on the accuracy of the orbits and the analysis on the earth gravity model. The numerical results show that, compared to the JPL reference orbits, the 3D RMS is below 3.0cm and the RMS is below 2.0cm for each component. As for the accuracy of gravity field model, compared to some contemporary significant earth gravity model, the model of the single month solution behaves very well below the 60 degree of the gravity field’s coefficients, while over the 60 degree, only the UTCSR model quite corresponds to the model computed by this method. Therefore, due to the promotion of the orbital accuracy and gravity field model, we suggest that the recovered carrier-range should be implemented in the simultaneous method for the better product solution of the LEO’s missions.</p>

Assessment of local covariance estimation through Least Squares Collocation over Iran

Covariance determination as the heart of Least Squares Collocation gravity field modeling is based on fitting an analytical covariance to the empirical covariance, which is stemmed from gravimetric data. The main objective of this study is to process different local covariance strategies over four regions with different topography and spatial data distribution in Iran. For this purpose, Least Squares Collocation based on Remove – Compute – Restore technique is implemented. In the Remove step, gravity reduction in regions with a denser distribution and a rougher topography is more effective. In the Compute step, the assessment of the Collocation estimates on the gravity anomaly control points illustrates that data density is more relevant than topography roughness to have a good covariance determination. Moreover, among the different attempts of localizing the covariance estimation, a recursive approach correcting the covariance parameters based on the agreement between Least Squares Collocation estimates and control points shows better performance. Furthermore, we could see that covariance localization in a region with sparse or bad distributed observations is a challenging task and may not necessarily improve the Collocation gravity modeling. Indeed, the geometrical fitness of the empirical and analytical covariances – which is usually a qualitative test to verify the precision of the covariance determination – is not always an adequate criterion.

A New Approach to Earth’s Gravity Field Modeling Using GPS-Derived Kinematic Orbits and Baselines

Earth’s gravity field recovery from GPS observations collected by low earth orbiting (LEO) satellites is a well-established technique, and kinematic orbits are commonly used for that purpose. Nowadays, more and more satellites are flying in close formations. The GPS-derived kinematic baselines between them can reach millimeter precision, which is more precise than the centimeter-level kinematic orbits. Thus, it has long been expected that the more precise kinematic baselines can deliver better gravity field solutions. However, this expectation has not been met yet in practice. In this study, we propose a new approach to gravity field modeling, in which kinematic orbits of the reference satellite and baseline vectors between the reference satellite and its accompanying satellite are jointly inverted. To validate the added value, data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission are used. We derive kinematic orbits and inter-satellite baselines of the twin GRACE satellites from the GPS data collected in the year of 2010. Then two sets of monthly gravity field solutions up to degree and order 60 are produced. One is derived from kinematic orbits of the twin GRACE satellites (‘orbit approach’). The other is derived from kinematic orbits of GRACE A and baseline vectors between GRACE A and B (‘baseline approach’). Analysis of observation postfit residuals shows that noise in the kinematic baselines is notably lower than the kinematic orbits by 50, 47 and 43% for the along-track, cross-track and radial components, respectively. Regarding the gravity field solutions, analysis in the spectral domain shows that noise of the gravity field solutions beyond degree 10 can be significantly reduced when the baseline approach is applied, with cumulative errors up to degree 60 being reduced by 34%, when compared to the orbit approach. In the spatial domain, the recovered mass changes with the baseline approach are more consistent with those inferred from the K-Band Ranging based solutions. Our results demonstrate that the proposed baseline approach is able to provide better gravity field solutions than the orbit approach. The findings may facilitate, among others, bridging the gap between GRACE and GRACE Follow-On satellite mission.