Gravimetric Terrain Corrections Research Articles

Investigation of the Geoid Model Accuracy Improvement in Turkey

This study investigated the accuracy improvement of the gravimetric geoid model in Turkey using densified grids of the gravity anomalies with the least-squares modification of the Stokes integral with additive corrections (LSMSA) method. In LSMSA computations of the geoid models, the surface gravity anomalies, which were derived from the complete Bouguer anomaly grid data set by forward modeling the topographic masses, were used. In the computations, in addition to the original 5′-resolution (∼9 km) Bouguer anomaly data set, its densified versions to 1′-resolution (∼2 km) also were used. The densification of the gravity grids was carried out using both planar- and spherical-type complete Bouguer anomalies. Thus, the effects of both gravity grid densification and the applied approximation for the Bouguer reduction process in gravity gridding on the geoid model accuracies were examined in order. The Bouguer gravity reductions were carried out using classical formulas for the planar approximation, whereas the gravimetric terrain corrections from the high-resolution SRTM2gravity global model were employed for the spherical approximation in numerical tests. The calculated geoid models were validated at 100 homogeneously distributed GPS/leveling (GPS means Global Positioning System) benchmarks over the country. In the tests, the geoid undulations derived from the models also were compared with the observations at six tide gauge stations along the coastlines. The latter provided an additional check for the assessment of the models’ compatibility at the coastal boundaries. In conclusion, the accuracy of the best geoid model solution (8.6 cm by means of the standard deviation of geoid undulation differences at GPS/leveling benchmarks) was achieved using 1′-resolution gravity anomalies restored from the spherical complete Bouguer anomalies in Turkey.

Efficient spatial-spectral computation of local planar gravimetric terrain corrections from high-resolution digital elevation models

SUMMARY Computation of gravimetric terrain corrections (TCs) is a numerical challenge, especially when using very high-resolution (say, ∼30 m or less) digital elevation models (DEMs). TC computations can use spatial or/and spectral techniques: Spatial domain methods are more exact but can be very time-consuming; the discrete/fast Fourier transform (D/FFT) implementation of a binomial expansion is efficient, but fails to achieve a convergent solution for terrain slopes >45°. We show that this condition must be satisfied for each and every computation-roving point pair in the whole integration domain, not just at or near the computation points. A combination of spatial and spectral methods has been advocated by some through dividing the integration domain into inner and outer zones, where the TC is computed from the superposition of analytical mass-prism integration and the D/FFT. However, there remain two unresolved issues with this combined approach: (1) deciding upon a radius that best separates the inner and outer zones and (2) analytical mass-prism integration in the inner zone remains time-consuming, particularly for high-resolution DEMs. This paper provides a solution by proposing: (1) three methods to define the radius separating the inner and outer zones and (2) a numerical solution for near-zone TC computations based on the trapezoidal and Simpson's rules that is sufficiently accurate w.r.t. the exact analytical solution, but which can reduce the computation time by almost 50 per cent.

SRTM2gravity: An Ultrahigh Resolution Global Model of Gravimetric Terrain Corrections

AbstractWe present a new global model of spherical gravimetric terrain corrections that take into account the gravitational attraction of Earth's global topographic masses at 3″ (~90 m) spatial resolution. The conversion of Shuttle Radar Topography Mission‐based digital elevation data to implied gravity effects relies on the global evaluation of Newton's law of gravitation, which represents a computational challenge for 3″ global topography data. We tackled this task by combining spatial and spectral gravity forward modeling techniques at the 0.2‐mGal accuracy level and used advanced computational resources in parallel to complete the 1 million CPU‐hour‐long computation within ~2 months. Key outcome is a 3″ map of topographic gravity effects reflecting the total gravitational attraction of Earth's global topography at ~28 billion computation points. The data, freely available for use in science, teaching, and industry, are immediately applicable as new in situ terrain correction to reduce gravimetric surveys around the globe.

Fast-Fourier-based error propagation for the gravimetric terrain correction

We have identified a gap in the literature on error propagation in the gravimetric terrain correction. Therefore, we have derived a mathematical framework to model the propagation of spatially correlated digital elevation model errors into gravimetric terrain corrections. As an example, we have determined how such an error model can be formulated for the planar terrain correction and then be evaluated efficiently using the 2D Fourier transform. We have computed 18.3 billion linear terrain corrections and corresponding error estimates for a 1 arc-second ([Formula: see text]) digital elevation model covering the whole of the Australian continent.

New high‐resolution grid of gravimetric terrain corrections over Australia

A 9 arc‐second grid of gravimetric terrain corrections has been computed over Australia using ver.2 of the GEODATA 9 arc‐second digital elevation model (DEM) and the fast Fourier transform technique. This supersedes the 27 arc‐second grid previously reported in this journal, computed from ver.1 GEODATA DEM. The improved resolution is possible because of the removal of errors in ver.1 DEM.

Anomalously large gradients in Version 1 of the “GEODATA 9 SECOND” Digital Elevation Model of Australia, and their effects on gravimetric terrain corrections

Large gradients, when calculated by a first-difference method, have been detected in Version 1 of the 9 arc-second National Digital Elevation Model (DEMv1) of Australia released by the Australian Surveying and Land Information Group. Gradient values implied by the mean elevations in the DEMv1 between adjacent grid cells of up to 74° have been observed, most notably in Australia's more mountainous regions in the east. Comparisons with topographic maps indicate that these are anomalous gradients in the DEMv1 that are not present in the mapped topography. It is recommended that the first-difference method is used to test DEMs before they are used to compute terrain corrections.

Terrain correcting Australian gravity observations using the national digital elevation model and the fast Fourier transform

The release of a digital elevation model (DEM) for Australia on a 9″ (∼250 m) grid has enabled the computation of gravimetric terrain corrections thus allowing the computation of complete Bouguer anomalies across the continent. The terrain correction was calculated through a two‐dimensional fast Fourier transform algorithm applied to a linear, planar approximation of the terrain‐correction formula, and with a constant topographic density of 2670 kg.m‐3. The technique was applied to two datasets in order to test for instabilities in the terrain‐correction algorithm: the original 9″ DEM, and a 27″ DEM averaged from the 9″ data. The 27″ terrain corrections were compared with values supplied by the Australian Geological Survey Organisation in Tasmania: 86% of these data were found to agree within 3.91 μm.s‐2; 98% agreed to within 5.32 μm.s‐2 (1σ).

Improved Gravimetric Terrain Corrections

The objective of this paper is to improve gravimetric terrain corrections by: (1) investigating the effect of different topographic representations that are suitable for efficient processing of high volumes of data (e.g. The mass prism and the mass line models) on terrain corrections and on geiod computations; (2) accelerating the convergence of fast Fourier transform (FFT)-based terrain-correction formulae; and (3) developing a set of new formulae corresponding to the mass prism topographic model, which can be evaluated efficiently with the 2-D FFT. Terrain corrections were computed on a grid of 600 by 600 points with spacing 30″ by 60″ in the Rocky Mountains of British Columbia, Canada. The effect of using the mass line model instead of the mass prism model is 7.4 mGal (maximum) and 0.7mGal (rms) on the terrain corrections, and 24cm (maximum) and 16cm (rms) on the geoid undulations. The optimizations made on the FFT-based terrain-correction formulae effectively speed up the convergence. The newly developed mass prism terrain-correction formula significantly reduced the required computer time and provided identical results with those from the rigorous numerical integration. On an IBM/RISC machine running AIX, the computation of the 15 convolutions (the matrix size was expanded to 1200 by 1200 after 100 per cent zero-padding) via the new formulae only took 15 min (user time), while the numerical summation method required 83.5 days.

Gravimetric terrain corrections by triangular‐element method

Terrain corrections for gravity data are a critical concern in rugged topography, because the magnitude of the corrections may be large relative to the anomalies of interest. The terrain‐correction process, however, is very tedious. At present, rectangular prism and fan‐shaped prism methods are commonly used for the corrections; but these methods assume elements have horizontal tops, an assumption which does not reflect true topography, especially near the station. A triangular‐element method which allows dipping surfaces has been developed using a Gaussian formulation to improve the correction process. This method involves a simple formula employing a surface rather than a volume integral for computing the terrain corrections with high accuracy.

Optimization in gravimetric terrain corrections

Terrain corrections for gravimetric measurements require topographical data of appropriate accuracy and density. The situation is clearly dependent on the ruggedness of the topography and the accuracy specifications for the terrain corrections. From the optimization point of view, this computational problem has been investigated and computer software developed for the near, intermediate, and distant zones surrounding the gravity stations. Test results indicate that the general accuracy objective of 1.0–1.5 mGal (10–15 μm s−2) for the gravimetric terrain corrections is easily achievable in practice with appropriate topographical data. For geophysical applications, higher accuracies are certainly achievable with the method, even in rugged areas.

Gravimetric terrain corrections in western Canada

Terrain corrections for gravimetric measurements have been studied in terms of accuracy requirements and automated computations. Geodetic and geophysical applications in western Canada have been considered specifically because of complications arising from the rugged topography. Comparing the computation methods of Nagy and Mathisen in relation to the theoretical results, the former is shown to be more reliable with simulated accidented topography. Other approaches are also briefly discussed and general recommendations are made for an optimal technique to compute gravimetric terrain corrections in western Canada.

THE USE OF ELECTRONIC DIGITAL COMPUTORS FOR THE EVALUATION OF GRAVIMETRIC TERRAIN CORRECTIONS*

AbstractA method is described in which the major part of gravimetric terrain corrections may be estimated by digital computing methods.