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Using radial basis functions in airborne gravimetry for local geoid improvement

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Abstract

Radial basis functions (RBFs) have been used extensively in satellite geodetic applications. However, to the author’s knowledge, their role in processing and modeling airborne gravity data has not yet been fully advocated or extensively investigated in detail. Compared with satellite missions, the airborne data are more suitable for these kinds of localized basis functions especially considering the following facts: (1) Unlike the satellite missions that can provide global or near global data coverage, airborne gravity data are usually geographically limited. (2) It is also band limited in the frequency domain. (3) It is straightforward to formulate the RBF observation equations from an airborne gravimetric system. In this study, a set of band-limited RBF is developed to model and downward continue the airborne gravity data for local geoid improvement. First, EIGEN6c4 coefficients are used to simulate a harmonic field to test the performances of RBF on various sampling, noise, and flight height levels, in order to gain certain guidelines for processing the real data. Here, the RBF method not only successfully recovers the harmonic field but also presents filtering properties due to its particular design in the frequency domain. Next, the software was tested for the GSVS14 (Geoid Slope Validation Survey 2014) area in Iowa as well as for the area around Puerto Rico and the US Virgin Islands by use of the real airborne gravity data from the Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project. By fully utilizing the three-dimensional correlation information among the flight tracks, the RBF can also be used as a data cleaning tool for airborne gravity data adjustment and cleaning. This property is further extended to surface gravity data cleaning, where conventional approaches have various limitations. All the related numerical results clearly show the importance and contribution of the use of the RBF for high- resolution local gravity field modeling.

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Acknowledgements

The author sincerely appreciates all of the useful comments and suggestions from the Editors and all the anonymous reviewers, especially reviewer 3 who gave very useful insights to further discuss the numerical results and very practical instructions to present them in a much better way. The author also thanks Mr. Heck, and Mr. Hilla at NGS for their help on improving the language usage and writing styles. Without all of these above mentioned help, the paper cannot be in the current shape that can be considered here.

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    Xiaopeng Li

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Appendices

A: Using RBF to clean surface gravity data in the band of d/o 2-2160

Fig. 15
figure 15

Simulated gravity anomalies at real NGS surface gravity points

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Fig. 16
figure 16

Estimated survey biases versus the original artifacts that were added to the data

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Goce and Egm2008 Combination (GECO Gilardoni et al. 2016), published by the International Centre for Global Earth Models (ICGEM, http://icgem.gfz-potsdam.de/ICGEM/) is used to generate gravity anomalies at real NGS surface gravity points in the state of Iowa, as shown in Fig. 15.

Fig. 17
figure 17

The estimated survey biases in the historical NGS terrestrial gravity survey

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Fig. 18
figure 18

The geoid model errors due to the estimated survey biases in the historical NGS terrestrial gravity survey

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According to the metadata that is available to the author, the data as shown in Fig. 15 are separated into 13 subsets. Random biases are assigned as survey biases of each of the 13 subsets. 0.5 mGal noise is added to all of the data as the observation error. Figure 16 shows the estimated survey biases and the original added artifacts. It shows that the estimation is fairly close to the true values.

After verifying the program by use of simulated data, the program is applied to the real NGS surface gravity, where the residual terrain effects (from 5 arcminutes to 3 arcseconds) have been removed. The estimated survey biases of the real NGS terrestrial gravity survey in the target area are shown in Fig. 17. If we simply neglect this problem in the case of using Stokes’s integral, the kernel has to be truncated around degree 800 to avoid 1 cm extreme errors in the geoid model, please see Fig. 18. This means that the global reference field has to be reliable at degree 800 that is kind of a challenging thing if only the satellite information is used during building the global coefficient models for the purpose of remove, compute, and restore in traditional geoid modeling method.

B: The actual number of the line biases in Fig. 10

Table 3 The estimated line biases in the NGS GRAV-D flights over Iowa (units in mGal)
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Li, X. Using radial basis functions in airborne gravimetry for local geoid improvement. J Geod 92, 471–485 (2018). https://doi.org/10.1007/s00190-017-1074-2

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