Christian Hirt

Comparison and validation of recent freely-available ASTER-GDEM ver1, SRTM ver4.1 and GEODATA DEM-9S ver3 digital elevation models over Australia

This study investigates the quality (in terms of elevation accuracy and systematic errors) of three recent publicly available elevation model datasets over Australia: (i) the 9 arc second national GEODATA DEM-9S ver3 from Geoscience Australia and the Australian National University; (ii) the 3 arc second SRTM ver4.1 from CGIAR-CSI; and (iii) the 1 arc second ASTER-GDEM ver1 from NASA/METI. The main features of these datasets are reported from a geodetic point of view. Comparison at about 1 billion locations identifies artefacts (e.g. residual cloud patterns and stripe effects) in ASTER. For DEM-9S, the comparisons against the space-collected SRTM and ASTER models demonstrate that signal omission (due to the ∼270 m spacing) may cause errors of the order of 100–200 m in some rugged areas of Australia. Based on a set of geodetic ground control points over Western Australia, the vertical accuracy of DEM-9S is ∼9 m, SRTM ∼6 m and ASTER ∼15 m. However, these values vary as a function of the terrain type and shape. Thus, CGIAR-CSI SRTM ver4.1 may represent a viable alternative to DEM-9S for some applications. While ASTER GDEM has an unprecedented horizontal resolution of ∼30 m, systematic errors present in this research-grade version of the ASTER GDEM ver1 will impede its immediate use for some applications.

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.

Modern Determination of Vertical Deflections Using Digital Zenith Cameras

At the beginning of the 21st century, a significant technological change took place in geodetic astronomy. In Zurich and Hannover, digital zenith camera systems were developed based on digital imaging sensors (charge-coupled device) that strongly improved the degree of automation, efficiency, and accuracy of the observation of the direction of the plumb line and its vertical deflection. This paper describes the instrumental design of the new digital zenith camera systems and gives an overview of the data processing with focus on the models used for astrometric data reduction and tilt correction. Results of frequently repeated observations of vertical deflections and comparison measurements show an accuracy of vertical deflection measurements of better than 0.1 arc sec. Application examples for vertical deflection data from zenith camera observations, such as the high-precision local gravity field determination in engineering projects and gravity field validation are summarized.

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.

Earth2014: 1 arc-min shape, topography, bedrock and ice-sheet models – Available as gridded data and degree-10,800 spherical harmonics

Abstract 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/ .

Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

Forward gravity modeling in the spectral domain traditionally relies on spherical approximation. However, this level of approximation is insufficient for some present day high-accuracy applications. Here we present two solutions that avoid the traditional spherical approximation in spectral forward gravity modeling. The first solution (the extended integration method) applies integration over masses from a reference sphere to the topography and applies a correction for the masses between ellipsoid and sphere. The second solution (the harmonic combination method) computes topographic potential coefficients from a combination of surface spherical harmonic coefficients of topographic heights above the ellipsoid, based on a relation among spherical harmonic functions introduced by Claessens (2005). Using a degree 2160 spherical harmonic model of the topographic masses, both methods are applied to derive the Earths ellipsoidal topographic potential in spherical harmonics. The harmonic combination method converges fastest and—akin to the EGM2008 geopotential model—generates additional spherical harmonic coefficients in spectral band 2161 to 2190 which are found crucial for accurate evaluation of the ellipsoidal topographic potential at high degrees. Therefore, we recommend use of the harmonic combination method to model ellipticity in spectral-domain forward modeling. The method yields ellipsoidal topographic potential coefficients which are “compatible” with global Earth geopotential models constructed in ellipsoidal approximation, such as EGM2008. It shows that the spherical approximation significantly underestimates degree correlation coefficients among geopotential and topographic potential. The topographic potential model is, for example, of immediate value for the calculation of Bouguer gravity anomalies in fully ellipsoidal approximation.

Band‐limited topographic mass distribution generates full‐spectrum gravity field: Gravity forward modeling in the spectral and spatial domains revisited

Most studies on gravity forward modeling in the spectral domain truncate the gravitational potential spectra at a resolution commensurate with the input topographic mass model. This implicitly assumes spectral consistency between topography and implied topographic potential. Here we demonstrate that a band-limited topographic mass distribution generates gravity signals with spectral energy at spatial scales far beyond the input topographys resolution. The spectral energy at scales shorter than the resolution of the input topography is associated with the contributions made by higher-order integer powers of the topography to the topographic potential. The pth integer power of a topography expanded to spherical harmonic degree n is found to make contributions to the topographic potential up to harmonic degree p times n. New numerical comparisons between Newtons integral evaluated in the spatial and spectral domain show that this previously little addressed truncation effect reaches amplitudes of several mGal for topography-implied gravity signals. Modeling the short-scale gravity signal in the spectral domain improves the agreement between spatial and spectral domain techniques to the μGal level, or below 10−5 in terms of relative errors. Our findings have important implications for the use of gravity forward modeling in geophysics and geodesy: The topographic potential in spherical harmonics must be calculated to a much higher harmonic degree than resolved by the input topography if consistency between topography and implied potential is sought. With the improved understanding of the spectral modeling technique in this paper, theories, and computer implementations for both techniques can now be significantly better mutually validated.

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.

High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System

During the last years the observation of vertical deflections experienced a revival due to the development of state-of-the-art Digital Zenith Camera Systems in Zurich and Hanover. Other than analogue instruments of geodetic astronomy, the new digital observation Systems provide vertical deflection data very fast and highly-accurate. One main application for these instruments is the precise gravity field determination in local areas applying the classical method of astronomical leveling.

First Results from new High-precision Measurements of Deflections of the Vertical in Switzerland

In October 2003 two modernized digital zenith camera systems have been deployed in Switzerland during the campaign CHGeo2003. The mission was carried out under the auspices of the Swiss Geodetic Commission of the Swiss Academy of Sciences (SAS) and coordinated by the Swiss Federal Office of Topography (swisstopo). The goal of the campaign was to provide additional highly accurate deflections of the vertical in order to contribute to an improvement of the presently used Swiss geoid CHGeo98. The observations were carried out at 68 selected stations covering regions with old or inadequate data. Further reasons of the campaign were the proof of the field capability and the comparison of both systems concerning their accuracy by observing at several stations simultaneously. The paper describes the realization of the campaign CHGeo2003 and the results obtained with the digital systems.