Models

In the context of regional gravity field modeling, it is reasonable to relate the approach and the output models to a global reference.

• The earlier introduced normal potential can be well determined and easily modeled. Consequently, long wavelengths parts are described by normal potential models, which serve as reference for all regional measurements.

• Further, global satellite observations can be captured very well by global SH gravitational potential models. The according low- and medium-frequency parts cannot be resolved by spatially limited observations. Hence, referring to Fig. 1.1, the regional gravity modeling approach of this thesis incorporates global SH gravitational potential models (briefly denoted as SH models in the following) as so-called ”background models“. They also relate to a specific normal potential.

The underlying geometric reference ellipsoids, the global geophysical normal and gravitational potential models used in this work, as well as an available regional model, are introduced now.

3.2.1 Reference ellipsoids and normal potential models

As mentioned in Sec. 2.3.4, the normal potential U of a spheroid can be described by four parameters.

Various reference ellipsoids have been developed approximating either globally (e. g. GRS80, WGS84) or regionally (e. g. Bessel in Europe, Hayford in USA, Krassowsky in Russia) the Earth’s body. For gravity field modeling in terms of SBFs with global support, a global reference ellipsoid is reasonable. Table 3.6 lists the most established reference ellipsoids and their parameters describing the related normal gravity field. The referring normal potential models GRS80 and WGS84 can be developed as series expansions in terms of SHs according to Eq. (2.44). Herein, the SH coefficientsC_l,0are computed from the four parameters as described by Eqs. (2.47). Note, the TOPEX/Poseidon (TOPEX) ellipsoid is a pure geometric ellipsoid, specified by the parametersaand f^′. It is derived from altimetry measurements of the TOPEX/Poseidon mission and mainly serves as reference for the different altimetry missions.

Table 3.6: Reference ellipsoids and according parameters.

Reference ellipsoid GM semi major axisa inverse flattening¹/f’ angular velocityω GRS80 3 986 005×10⁸km³/s² 6 378 137.0 m 298.257 222 101 7.292 115×10⁻⁵rad/s WGS84 3 986 004.418×10⁸km³/s² 6 378 137.0 m 298.257 223 563 7.292 115×10⁻⁵rad/s

TOPEX 6 378 136.3 m 298.257

The Geodetic Reference System 1980 (GRS80) (Moritz, 2000) is the official reference system of the Interna-tional Union of Geodesy and Geophysics (IUGG). It replaced the previous Geodetic Reference System 1967 by more accurate values in 1979, and served as foundation for the more recent World Geodetic System 1984 (WGS84) (Hofmann-Wellenhof and Moritz, 2005, pp. 84). The latter is a conventional terrestrial reference system and delivers the basis, e. g., for GPS. The referring ellipsoids of revolution have an about 70 cm larger semi major axis than the TOPEX ellipsoid.

As different geometric or gravitational data sets relate to different reference ellipsoids or normal potential models, transformations are indispensable. Height transformations relate to Eq. (2.22), while the rescaling of the series expansion of a potential is performed by an according rescaling of the SH coefficients, cf. Eq. (2.42).

The formulas are applied in the pre-processing of the data in next sections.

In this work, GRS80 is chosen as reference for all regional gravity fields to be modeled. Consequently, all measurements have to be transformed in this system and all numerical implementations are restricted to the according values.

3.2.2 Global SH gravity field models

The global gravity field models addressed in this work relate on series expansions in terms of SHs. In literature, they are usually described by expanding the gravitational potential V in the series according to Eq. (2.40) starting from l = 0. The zero-degree term is set to one; setting the first-degree terms equal to zero, i. e.

assuming the origin of the global SH models coinciding with the geocenter, can be justified by deriving the low-degree terms (l = 2,3,4, ...up to aroundl = 20) from analysis of satellite orbit perturbations (Vermeer M., 2016, p. 46). A comprehensive set of models can be accessed from the International Centre for Global Earth Models (ICGEM), GFZ Potsdam, throughhttp://icgem.gfz-potsdam.de/ICGEM/.

EGM96 and EGM2008

With the Earth Gravitational Model 1996 (EGM96) researchers from The Ohio State University developed one of the first well-approved global gravity field models. This combination model was published byLemoine et al.(1998) and is mainly based on gravimetric data collected by the American NIMA (National Imagery and Mapping Agency) (Vermeer M., 2016, p. 46). The intensive investigations in global data collection enabled to expand a series in terms of SHs, cf. Eq. (2.34), up to degreeL =360. It served for many decades as ref-erence in various applications, such as referencing height systems on land, or determining the DOT over the sea.

At the beginning of this millennium, the CHAMP and especially the GRACE satellite missions revolutionized global gravity field determination by the ability of global gravity data collection. An enormous progress was achieved in precise long wavelength gravitational modeling. The National Geospatial-Intelligence Agency (NGA) developed a new release of EGM: EGM2008. This global SH model combines

• low- and medium-resolution GRACE data in terms of

• the global ITG-GRACE03s (static solution of ITG-GRACE2010, developed up to degree and order 180, Mayer-Gürr et al.(2010)), with

• high-resolution terrestrial, airborne and altimetry data in terms of a global 5 arc-minute grid of gravity anomalies, filled up with topography information in case of data gaps, as described by Pavlis et al.

(2012).

The SH series of EGM2008 is expanded up to degree 2190; complete gravitational information is contained up to degree and order 2159, corresponding to a spatial resolution down to less than 10 km – in case of data coverage. Up to present, this global gravity field model is unique in its high spectral and spatial resolution.

EIGEN-6C3stat

Based on the 4th Release of the GOCE direct approach (Pail et al., 2011a), the EIGEN-6C3stat was published by Foerste et al.(2014) in 2014 as a new release of EIGEN-6C (Foerste et al., 2012). It is a static high resolution global combined gravity field model, developed in SH expansion up to degree and order 1949. The data which are used for the SH series expansion stem from

• SLR (LAGEOS-1/2, 25 years),

• GRACE (GRGS RL02 from degree 2 to 100, including 8 years, and GFZ RL05 from degree 55 to 180, including 9 years from GPS-SST and K-band range-rate observations),

• GOCE (contributing up to degree 235 from 19 months SGG data)

• ground data (gridded global gravity anomalies from the DTU12 ocean geoid and the EGM2008 geoid).

Consequently, beyond degree 235, EIGEN-6C3stat is a reconstruction of the EGM2008 model.

GOCO03s and GOCO05s

The GOCO series presents global SH gravity models mainly from a combination of GRACE and GOCE data.

The satellite-only GOCO03s model (Mayer-Gürr et al., 2012) incorporates

• 7.5 years of GRACE and

• 18 months of GOCE data. Further data stem from

• CHAMP (8 years) and

• SLR (5 years).

The series expansion is developed up to degree and order 250. GRACE information is contained in the long wavelengths from the SH model ITG-Grace2010s. The content in the medium wavelengths stems from GOCE gradiometry applying the processing strategy ofPail et al.(2011b). Due to regularization, full signal content is only ensured up to approximately degree 200.

The latest release GOGO05s (Mayer-Gürr et al., 2015) is given with a spectral resolution up to degree and order 280 and contains significant information at least up to degree 200, i. e. it delivers a spatial resolution of around 100 km. The combined model consists of

• GOCE gradiometry data from the complete mission (48 months),

• the ITSG-Grace2014s model from 10.5 months GRACE observations (Mayer-Gürr et al., 2014),

• SLR measurements, as well as data from

• CHAMP, Swarm A+B+C, and the Earth observation satellites TerraSAR-X¹⁸and TanDEM-X¹⁹. GRACE contributes to low- and medium-resolution spectral content approximately up to degree 150, whereas the significant GOCE information predominates from degree 120 on.

3.2.3 Regional model: GCG2011

The German Combined QuasiGeoid 2011 (GCG2011) is the official German height reference of the ”Arbeits-gemeinschaft der Vermessungsverwaltungen der Länder der Bundesrepublik Deutschland (AdV)“ describing quasigeoid heights, according to Eq. (2.63) w.r.t. the reference ellipsoid GRS80.

It is provided in terms of a 1^′ longitude times 1.5^′latitude geographic grid byBKG (2011). The corresponding spatial reso-lution yields about 1.8 km × 1.7 km; the position coordinates refer to European Ter-restrial Reference System 1989 (ETRS89) (Boucher and Altamimi, 1992). The quasi-geoid heights in Fig. 3.8 are given in the ”Deutsches Haupthöhennetz (DHHN) 1992“ (Weber, 1994). The DHHN is the pri-mary leveling network of Germany. The res-olution counts 1 mm, the accuracy reaches 1 cm to 2 cm and varies depending on (1) geographical structures and (2) data avail-ability. Over the Alps, the values are less accurate with 3 cm to 4 cm, and over the North and Baltic Sea even worse with 4 cm to 10 cm. The GCG2011 is obtained by aver-aging two independent solutions from BKG and IfE (Institut für Erdmessung der Leib-nitz Universität Hannover). Whereas BKG uses an adjustment approach based on point masses, IfE applies integration and colloca-tion. Based on a new release DHHN2016, a new release GCG2016 is planned, as well

(AdV, 2014). Figure 3.8: German combined quasigeoid 2011; source:BKG(2011).