Verification of the Jakarta Geoid Model from the Gravity Data of 2.5 km Resolution with Gravimetric Geoid

To use the Global Navigation Satellite System (GNSS) correctly, the height information should be transformed into orthometric height by subtracting geoid undulation from it. This orthometric height is commonly used for practical purposes. In 2015 geoid of Jakarta has been produced, and it has an accuracy of 0.076 m. In the year 2019, airborne gravimetry has been done for the entire Java Island. The area of DKI Province cannot be measured because there is inhibition from Airnav. For this reason, terrestrial gravimetric measurements are carried out in this region by adding points outside the previously measured area. To compute the geoid in the Jakarta region is needed the Global Geopotential Model (GGM). In this paper, the GMM used is gif48. The “remove and restore” method will be used in calculating the geoid in this Jakarta region. Besides that in this geoid calculation also uses Stokes kernel and FFT to speed up the calculation. The verification of the resulting geoid is carried with 11 points in DKI Jakarta Province. This verification produces a standard deviation of 0.116 m and a root mean square of 0.411 m.

Comparison among Gravimetric, Astrogravimetric and Astrogeodetic Geoids: Case Study for Austria

<p>It is used to state that all geoid determination techniques should yield to the same geoid if the indirect effect is properly taken into account (Heiskanen and Moritz, 1967). The current study compares different geoid determination techniques for Austria. The used techniques are the gravimetric, astrogravimetric and astrogeodetic geoid determination techniques. The available data sets (gravity, deflections of the vertical, height, GPS) are described. The window remove-restore technique (Abd-Elmotaal and Kuehtreiber, 2003) has been used. The available gravity anomalies and the deflections of the vertical have been topographically-isostatically reduced using the Airy isostatic hypothesis. The reduced deflections have been used to interpolate deflections on a relatively dense grid covering the data window. These gridded reduced deflections have been used to compute an astrogeodetic geoid for Austria using least-squares collocation technique within the remove-restore scheme. The Vening Meinesz formula has been used to compute an astrogravimetric geoid for Austria. Another gravimetric geoid for Austria has been determined in the framework of the window remove-restore technique using Stokes integral with modified Stokes kernel. All computed geoids have been validated using GNSS/levelling derived geoid. A wide comparison among the derived geoids computed within the current investigation has been carried out.</p>

The modified integral method for the determination of gravity disturbance near the Earth’s surface

For the calculation of gravity disturbance in the Earth’s external gravity field, the Stokes-Pizzetti integral is a commonly used method. However, when the target point approaches the Earth’s surface, such problems as singularity and discontinuity arise due to the Stokes kernel structure itself. To settle the problems, firstly the reason for singularity and discontinuity was discussed, and then modification was made to the integral formula, by which the singularity at the surface point is eliminated. Finally the non-singular integral formulas for the calculation of disturbing gravity were derived. In numerical experiments, an area in China was selected to test the modified formula. Numerical results show that the modified formula performs much better than classical Stokes-Pizzetti integral formula when dealing with the calculation of the radial component of gravity disturbance near the Earth’s surface.

Accuracy of unmodified Stokes’ integration in the R-C-R procedure for geoid computation

Geoid determinations by the Remove-Compute-­Restore (R-C-R) technique involves the application of Stokes’ integral on reduced gravity anomalies. Numerical Stokes’ integration produces an error depending on the choice of the integration radius, grid resolution and Stokes’ kernel function.In this work, we aim to evaluate the accuracy of Stokes’ integral through a study on synthetic gravitational signals derived from EGM2008 on three different landscape areas with respect to the size of the integration domain and the resolution of the anomaly grid. The influence of the integration radius was studied earlier by several authors. Using real data, they found that the choice of relatively small radii (less than 1°) enables to reach an optimal accuracy. We observe a general behaviour coherent with these earlier studies. On the other hand, we notice that increasing the integration radius up to 2° or 2.5° might bring significantly better results. We note that, unlike the smallest radius corresponding to a local minimum of the error curve, the optimal radius in the range 0° to 6° depends on the terrain characteristics. We also find that the high frequencies, from degree 600, improve continuously with the integration radius in both semi-­mountainous and mountain areas.Finally, we note that the relative error of the computed geoid heights depends weakly on the anomaly spherical harmonic degree in the range from degree 200 to 2000. It remains greater than 10 % for any integration radii up to 6°. This result tends to prove that a one centimetre accuracy cannot be reached in semi-mountainous and mountainous regions with the unmodified Stokes’ kernel.

Reply to Comments to X. Li and Y. M. Wang (2011) Comparisons of geoid models over Alaska computed with different Stokes' kernel modifications, JGS 1(2): 136-142 by L. E. Sjöberg

Reply to Comments to X. Li and Y. M. Wang (2011) Comparisons of geoid models over Alaska computed with different Stokes' kernel modifications, JGS 1(2): 136-142 by L. E. SjöbergThe authors thank professor Sjöberg for having interest in our paper. The main goal of the paper is to test kernel modification methods used in geoid computations. Our tests found that Vanicek/Kleusberg's and Featherstone's methods fit the GPS/leveling data the best in the relative sense at various cap sizes. At the same time, we also pointed out that their methods are unstable and the mean values change from dm to meters by just changing the cap size. By contrast, the modification of the Wong and Gore type (including the spectral combination, method of Heck and Grüninger) is stable and insensitive to the truncation degree and cap size. This feature is especially useful when we know the accuracy of the gravity field at different frequency bands. For instance, it is advisable to truncate Stokes' kernel at a degree to which the satellite model is believed to be more accurate than surface data. The method of the Wong and Goretype does this job quite well. In contrast, the low degrees of Stokes' kernel are modified by Molodensky's coefficients tn in Vanicek/Kleusberg's and Featherstone's methods (cf. Eq. (6) in Li and Wang (2011)). It implies that the low degree gravity field of the reference model will be altered by less accurate surface data in the final geoid. This is also the cause of the larger variation in mean values of the geoid.

Comments to X. Li and Y. M. Wang (2011) Comparisons of geoid models over Alaska computed with different Stokes' kernel modifications, JGS 1(2): 136-142

Comments to X. Li and Y. M. Wang (2011) Comparisons of geoid models over Alaska computed with different Stokes' kernel modifications, JGS 1(2): 136-142Li and Wang recently compared geoid determination by various gravimetric methods for modifying Stokes' formula vs. using GPS/levelling geoid heights as a reference model. Possible large systematic errors in the differences of gravimetric and GPS/levelling geoid models deteriorate the results and conclusions. Moreover, spectral combination, the only stochastic method in the study, was applied in an unrealistic way.