Bernhard Heck

Strategies for Solving the Vertical Datum Problem Using Terrestrial and Satellite Geodetic Data

The classical procedure of establishing precise height networks is based upon geodetic levelling, potentially including gravity information along the levelling lines. Since levelling is a relative operation some vertical datum must be fixed in order to determine “absolute” heights of benchmarks. In most cases the vertical datum of a height network has been defined by assigning zero height to the long-term mean value of local sea level observed at a fundamental tide gauge station. The vertical datum of largely extended height networks has often been fixed by employing several tide gauge stations situated along the coastline. In any case the definition of datum of classical vertical networks is connected with the concept of local mean sea level; the equipotential surface of the earth’s gravity field passing through the fundamental tide gauge mark is the reference surface of heights derived from levelling.

An evaluation of some systematic error sources affecting terrestrial gravity anomalies

Terrestrial free-air gravity anomalies form a most essential data source in the framework of gravity field determination. Gravity anomalies depend on the datums of the gravity, vertical, and horizontal networks as well as on the definition of a normal gravity field; thus gravity anomaly data are affected in a systematic way by inconsistencies of the local datums with respect to a global datum, by the use of a simplified free-air reduction procedure and of different kinds of height system. These systematic errors in free-air gravity anomaly data cause systematic effects in gravity field related quantities like e.g. absolute and relative geoidal heights or height anomalies calculated from gravity anomaly data.In detail it is shown that the effects of horizontal datum inconsistencies have been underestimated in the past. The corresponding systematic errors in gravity anomalies are maximum in mid-latitudes and can be as large as the errors induced by gravity and vertical datum and height system inconsistencies. As an example the situation in Australia is evaluated in more detail: The deviations between the national Australian horizontal datum and a global datum produce a systematic error in the free-air gravity anomalies of about −0.10 mgal which value is nearly constant over the continent

A comparison of different isostatic models applied to satellite gravity gradiometry

In satellite gradiometry, the gravitational signals originating from the Earth’s topography and its isostatic compensation can be recognized in the gravity gradients observed along the satellite orbit. One general task should be the reduction of these effects to produce a smooth gravity field suitable for downward continuation. Based on different isostatic models such as the Airy-Heiskanen model, the Pratt-Hayford model, the combination of the Airy-Heiskanen model (land area) and the Pratt-Hayford model (ocean area), and the generalized Helmert model, the topographic-isostatic effects are calculated for a GOCE-like satellite orbit. For the second vertical (radial) derivative of the gravitational potential the order of magnitude of both topographic and isostatic components amounts to about 10 E.U. while the combined topographic-isostatic effect reduces to about 1 E.U.. In this paper, the focus lies on the comparison between the classical isostatic models and the generalized Helmert model, consistently using a rigorous spherical formulation for all models. By variation of the depth of the condensation layer, it is possible to demonstrate that the classical isostatic models become equivalent to the Helmert model related to a specific condensation depth d. E.g., the standard Airy-Heiskanen model related to a normal crustal thickness T = 25 km is best approximated using the compensation depth d = 24 km in the generalized Helmert model. Instead of the conventional remove-restore techniques which lead to high numerical efforts, the use of the generalized Helmert model is recommended.

A Wavelet-Based Assessment of Topographic-Isostatic Reductions for GOCE Gravity Gradients

Gravity gradient measurements from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE) contain significant high- and mid-frequency signal components, which are primarily caused by the attraction of the Earth’s topographic and isostatic masses. In order to mitigate the resulting numerical instability of a harmonic downward continuation, the observed gradients can be smoothed with respect to topographic-isostatic effects using a remove–compute–restore technique. For this reason, topographic-isostatic reductions are calculated by forward modeling that employs the advanced Rock–Water–Ice methodology. The basis of this approach is a three-layer decomposition of the topography with variable density values and a modified Airy–Heiskanen isostatic concept incorporating a depth model of the Mohorovičić discontinuity. Moreover, tesseroid bodies are utilized for mass discretization and arranged on an ellipsoidal reference surface. To evaluate the degree of smoothing via topographic-isostatic reduction of GOCE gravity gradients, a wavelet-based assessment is presented in this paper and compared with statistical inferences in the space domain. Using the Morlet wavelet, continuous wavelet transforms are applied to measured GOCE gravity gradients before and after reducing topographic-isostatic signals. By analyzing a representative data set in the Himalayan region, an employment of the reductions leads to significantly smoothed gradients. In addition, smoothing effects that are invisible in the space domain can be detected in wavelet scalograms, making a wavelet-based spectral analysis a powerful tool.

Problems in the Definition of Vertical Reference Frames

Present and future satellite-borne gravity field missions will have a major impact on the unification of regional height systems into a consistent global vertical reference frame. On the background of the high precision and quality of the expected data the scientific concepts of vertical reference frames have to be discussed, before a global frame will be implemented or before international standards and conventions can be fixed.

On the non-linear geodetic boundary value problem for a fixed boundary surface

SummaryThe fixed gravimetric boundary value problem of Physical Geodesy is a nonlinear, oblique derivative problem. Expanding the non-linear boundary condition into a Taylor series—based upon some reference potential field approximating the geopotential—it is shown that the numerical convergence of this series is very rapid; only the quadratic term may have some practical impact on the solution. The secondorder solution term can be described by a spherical integral formula involving the deflections of the vertical with respect to the reference field. The influence of nonlinear terms on the figure of the level surfaces (e.g. the geoid) is roughly estimated to have an order of magnitude of some few centimetres, based upon a Somigliana-Pizzetti reference field; if on the other hand some high-degree geopotential model is used as reference then the effects by non-linearity are negligible from a practical point of view.

The free versus fixed geodetic boundary value problem for different combinations of geodetic observables

Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary.The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical harmonics with Wigner 3j-coefficients.

Improving the Stochastic Model of GNSS Observations by Means of SNR-based Weighting

In many GNSS software packages a sim- plified observation weighting model is used which is merely based on the satellite elevation angle and valid under the assumption of azimuthal symmetry. This elevation-dependent weighting model is only suitable for undisturbed GNSS signals based on the existing strong correlation between signal quality and satellite elevation angle. However, for high-precision geodetic applications this geometry-related weighting model becomes obsolete if observations are strongly affected by multipath effects, signal diffraction as well as receiver characteristic under non-ideal observation conditions. An improved observation weighting model based on signal-to noise power ratio measurements has been developed and experimentally implemented in the Bernese GPS software 5.0. Tests indicate that when this weighting model is used for low elevation data additional 10% ambiguities can be resolved and the accuracy of the estimated site-specific neutrosphere parameters can be improved by nearly 25% compared with the standard elevation-dependent weighting model.

Topographic–Isostatic Reduction of GOCE Gravity Gradients

Gravity gradients measured by ESA’s satellite mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) are highly sensitive to mass anomalies and mass transports in the Earth system. The high and mid-frequency gradient components are mainly affected by the attraction of the Earth’s topographic and isostatic masses. Due to these signal components, interpolation and prediction tasks, such as a harmonic downward continuation of the gradients, can be considered as ill-conditioned processes. One approach to mitigate the resulting instability is to smooth the observed gradients by applying topographic–isostatic reductions using a Remove–Compute–Restore technique. This paper presents a reduction concept based on the developed Rock–Water–Ice decomposition in which the topography is represented by a vertical three-layer model with variable density values. Geometry and density information is derived from the topographic data base DTM2006.0. Furthermore, the Airy–Heiskanen isostatic model is adapted to the Rock–Water–Ice approach and extended by including a depth model for the Mohorovicic discontinuity obtained from the global crust model CRUST 2.0. Since these data are provided in geographical coordinates, tesseroid bodies that are arranged on an ellipsoidal reference surface are used for mass discretization. The topographic–isostatic reduction values calculated along the orbit of the GOCE satellite reach a maximum of about 1 E (Eotvos unit, 1 E = 10−9 s−2) and lead to significant smoothing effects on gradient measurements, particularly in regions with highly variable topography. Taking one week of real GOCE measurements as example, the degree of smoothing is analyzed, showing a significant reduction of the standard deviation (about 30 %) and the range (about 20–40 %).

Determination of vertical recent crustal movements by levelling and gravity data

Abstract Recent vertical crustal movements are generally studied by comparing either repeated levellings or repeated gravity measurements; neither of these procedures gives strictly true vertical displacements of points. A theoretically more satisfactory approach can be based on a Stokes-like integral formula, requiring both types of data. Assuming a constant linear relationship between temporal changes of gravity and temporal changes of levelled “heights” in an area of interest, two variants of the original integral formula are derived, either variant requiring essentially only one type of data. The first variant can be used if, besides extended repeated levellings, the constants of the linear relationship are available (by using both types of data on some common points). Equivalently the second variant is based on repeated gravity measurements. Both variants contain a small surface integral term which is numerically estimated by model computations. Finally the theory is applied to the determination of real crustal movements in western Hokkaido, Japan. The temporal changes of gravity and of height at twenty points gave a correlation coefficient−0.66 which is statistically significant at the 1 per cent level. Neglecting the temporal changes of geoid undulations could cause systematic errors in indicated recent crustal movements in the area reaching the magnitude of one standard deviation of the relative height of the extreme points in the area. Further numerical investigations require the collection of precise data at some geodynamic test areas.