RTM gravity forward-modeling using topography/bathymetry data to improve high-degree global geopotential models in the coastal zone

We apply the residual terrain modeling (RTM) technique for gravity forward-modeling to successfully improve high-resolution global gravity fields at short spatial scales in coastal zones. The RTM scheme is combined with the concept of rock-equivalent topography, allowing to use a single uniform constant mass-density in the RTM forward-modeling, both at land and sea. SRTM30_PLUS bathymetry is merged with higher-resolution SRTM V4.1 land topography, and expanded into spherical harmonics to degree 2160, yielding a new and consistent high-degree RTM reference surface. The forward-modeling performance is demonstrated in coastal zones of Greece and Canada using ground-truth vertical deflections, gravity from land and shipborne gravimetry, and geoid heights from GPS/leveling, with improvements originating from bathymetry clearly identified. We demonstrate that the SRTM30_PLUS bathymetry carries information on gravity field structures at spatial scales less than 5 arc minutes, which can be used to augment EGM2008 in (rugged) coastal zones, both over land and marine areas. This may be of value (i) to partially reduce the signal omission error in EGM2008/GOCE-based height transfer in areas devoid of dense gravity data, (ii) to fill the gap between land gravity and shipborne gravity along rugged coastlines and (iii) for the development of next-generation altimetric gravity fields

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth's surface using the gradient approach.

Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth’s surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5):282–289, 1997) for degree 360 models, produce cm- to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultra-high resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points.Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth’s surface

Mean kernels to improve gravimetric geoid determination based on modified Stokes's integration

Gravimetric geoid computation is often based on modified Stokes's integration, where Stokes's integral is evaluated with some stochastic or deterministic kernel modification. Accurate numerical evaluation of Stokes's integral requires the modified kernel to be integrated across the area of each discretised grid cell (mean kernel). Evaluating the modified kernel at the centre of the cell (point kernel) is an approximation which may result in larger numerical integration errors near the computation point, where the modified kernel exhibits a strongly nonlinear behaviour. The present study deals with the computation of whole-of-thecell mean values of modified kernels, exemplified here with the Featherstone-Evans-Olliver (1998) kernel modification (Featherstone, W.E., Evans, J.D., Olliver, J.G., 1998. A Meissl modified Vancek and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. Journal of Geodesy 72(3), 154-160). We investigate two approaches (analytical and numerical integration) which are capable of providing accurate mean kernels. The analytical integration approach is based on kernel weighting factors which are used for the conversion of point to mean kernels. For the efficient numerical integration, Gauss-Legendre Quadrature is applied. The comparison of mean kernels from both approaches shows a satisfactory mutual agreement at the level of 10-4 and better, which is considered to be sufficient for practical geoid computation requirements. Closed-loop tests based on the EGM2008 geopotential model demonstrate that using mean instead of point kernels reduces numerical integration errors by ~65%. The use of mean kernels is recommended in remove-compute-restore geoid determination with the Featherstone-Evans-Olliver (1998) kernel or any other kernel modification under the condition that the kernel changes rapidly across the cells in the neighbourhood of the computation point

The Digital Zenith Camera - A New High-Precision and Economic Astrogeodetic Observation System for Real-Time Measurement of Deflections of the Vertical

During the last few years, new developments in the field of geodetic astronomy have been sparsely published. This might be due to the fact that the determination of deflections of the vertical still required relatively large efforts, both in time and in manpower, thus keeping the costs per point at a high level. Recently, the development of new high performance image sensors (CCD) at a reasonable price level enabled and initiated fundamental improvements in astrogeodetic observation instrumentations in terms of efficiency, automation, accuracy, and real-time capability. This promising development leads to a revitalization of astrogeodetic methods and offers very encouraging prospects for local high-precision astrogeodetic gravity field and geoid determinations.In this paper, two slightly different versions of the digital zenith camera, initially developed at the Institutfur Erdmessung, University of Hannover, are presented as high-precision state-of-the-art instruments. Using modern CCD technology for imaging stars and a GPS receiver, these systems allow the direct determination of the direction of the plumb line and thus its deflection from the ellipsoidal normal within a fully automated procedure in real-time. In addition to a description of the system’s design and performance, the processing steps are presented: image data acquisition, data transfer and processing giving deflections of the vertical immediately after measurement

Status of Geodetic Astronomy at the Beginning of the 21st Century

At the beginning of the 21st century, a significant technological change took place in geodetic astronomy. The use of digital imaging sensors strongly improved the degree of automation, efficiency and accuracy of methods for the observation of the direction of the plumb line and its vertical deflection. This paper outlines the transition of astrogeodetic techniques and applications from the analogue to the digital era and addresses instrumental developments and recently completed projects. Particular attention is given to Digital Zenith Camera Systemsrepresenting astrogeodetic state-of-the-art instrumentation. Moreover, accuracy issues, present application examples for highly-precise astrogeodetic gravity field determinations and some future applications are described

Comparison of free high resolution digital elevation data sets (ASTER GDEM2, SRTM v2.1/v4.1) and validation against accurate heights from the Australian National Gravity Database

Today, several global digital elevation models (DEMs) are freely available on the web. This study compares and evaluates the latest release of the Advanced Spaceborne Thermal Emission Reflectometer DEM (ASTER GDEM2) and two DEMs based on the Shuttle Radar Topography Mission (SRTM) as released by the United States Geological Survey (SRTM3 USGS version 2.1) and by the Consortium for Spatial Information (SRTM CGIAR-CSI version 4.1) over the Australian continent. The comparison generally shows a very good agreement between both SRTM DEMs; however, data voids contained in the USGS model over steep topographic relief are filled in the CGIAR-CSI model. ASTER GDEM2 has a northeast- to southwest-aligned striping error at the 10 m level and shows an average height bias of –5 m relative to SRTM models. The root-mean square (RMS) height error obtained from the differences between ASTER GDEM2 and SRTM over Australia is found to be around 9.5 m. An external validation of the models with over 228 000 accurate station heights from the Australian National Gravity Database allows estimating each models’ elevation accuracies over Australia: ASTER GDEM2 ~ 8.5 m, SRTM3 USGS ~ 6 m, SRTM CGIAR-CSI ~ 4.5 m (RMS). In addition, the dependence of the DEM accuracy on terrain type and land cover is analysed. Applying a cross-correlation image co-registration technique to 529 1 × 1 degree tiles and 138 2 × 2 degree tiles reveals a mean relative shift of ASTER GDEM2 compared with SRTM of –0.007 and –0.042 arc-seconds in north–south and –0.100 and –0.136 arc-seconds in east–west direction over Australia, respectively

Earth2014: 1 arc-min shape, topography, bedrock and ice-sheetmodels – Available as gridded data and degree-10,800 sphericalharmonics

Since the release of the ETOPO1 global Earth topography model through the US NOAA in 2009, new or significantly improved topographic data sets have become available over Antarctica, Greenland and parts of the oceans. Here, we present a suite of new 1(arc-min) models of Earth’s topography, bedrock and ice-sheets constructed as a composite from up-to-date topography models: Earth2014. Our model suite relies on SRTM30 PLUS v9 bathymetry for the base layer, merged with SRTM v4.1 topography over the continents, Bedmap2 over Antarctica and the new Greenland bedrock topography (GBT v3). As such, Earth2014 provides substantially improved information of bedrock and topography over Earth’s major ice sheets, and more recent bathymetric depth data over the oceans, all merged into readily usable global grids. To satisfy multiple applications of global elevation data, Earth2014 provides different representations of Earth’s relief. These are grids of (1) the physical surface, (2) bedrock (Earth’s relief without water and ice masses), (3) bedrock and ice (Earth without water masses), (4) ice sheet thicknesses, (5) rock-equivalent topography (ice and water masses condensed to layers of rock) as mass representation. These models have been transformed into ultra-high degree spherical harmonics, yielding degree 10,800 series expansions of the Earth2014 grids as input for spectral modelling techniques. As further variants, planetary shape models were constructed, providing distances between relief points and the geocenter. The paper describes the input data sets, the development procedures applied, the resulting gridded and spectral representations of Earth2014, external validation results and possible applications. The Earth2014 model suite is freely available via http://ddfe.curtin.edu.au/models/Earth2014/

Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data

At the beginning of the twenty-first century, a technological change took place in geodetic astronomy by the development of Digital Zenith Camera Systems (DZCS). Such instruments provide vertical deflection data at an angular accuracy level of 0".1 and better. Recently, DZCS have been employed for the collection of dense sets of astrogeodetic vertical deflection data in several test areas in Germany with high-resolution digital terrain model (DTM) data (10-50 m resolution) available. These considerable advancements motivate a new analysis of the method of astronomical-topographic levelling, which uses DTM data for the interpolation between the astrogeodetic stations. We present and analyse a least-squares collocation technique that uses DTM data for the accurate interpolation of vertical deflection data. The combination of both data sets allows a precise determination of the gravity field along profiles, even in regions with a rugged topography. The accuracy of the method is studied with particular attention on the density of astrogeodetic stations. The error propagation rule of astronomical levelling is empirically derived. It accounts for the signal omission that increases with the station spacing. In a test area located in the German Alps, the method was successfully applied to the determination of a quasigeoid profile of 23 km length. For a station spacing from a few 100 m to about 2 km, the accuracy of the quasigeoid was found to be about 1-2 mm, which corresponds to a relative accuracy of about 0.05-0.1 ppm. Application examples are given, such as the local and regional validation of gravity field models computed from gravimetric data and the economic gravity field determination in geodetically less covered regions