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Compared to the SH solution, the added value of M RBF approaches is their convenience to implementing tailored regularization at areas of interest, as M RBFs are more regionally specified. To this end, we classify the M RBF gridding into the ocean and land areas, which yields 12 In the majority of cases, a valid assumption is that the geophysical signals over the oceans are far less rough than that over the continents. In particular, we zoom in to the region of South America in Figure 5 for a better comparison.

However, the physical constraint that is satisfied by the MRBFs ensures an efficient separation of ocean and land signals across the coastlines. As a result, this yields a more robust continental mass estimate that is less subjective to the ocean smoothing. This experiment demonstrates that the regularization of RBF solution has to be treated very carefully, while MRBF is robust to regularization in the sense of a flexible ocean smoothing without increasing instability.

However, one can observe from the above experiments that the side effects such as the spurious continental noise brought by the regularization into MRBF and RBF gravity fields are unequal. In general, the regularized gravity field from GRACE does not require to be spatially filtered as this has been often considered in the regularization.

Yet in this study, the simple standard Tikhonov regularization was found not to be sufficient to suppress the striping errors see Figure 5 , the magnitude of the noise is still nonnegligible for both RBF and MRBF solutions. In this way, the global mass anomaly in terms of EWH on January, May, and September is exemplarily shown in Figure 8 from the top to the bottom. Furthermore, as illustrated by the maps of differences in Figures 8 c, 8 f, and 8 i, discrepancies between the RBF and MRBF solution exist mostly at coastal areas, such as the coast of Greenland and Antarctica that have the most significant spatial leakage in ordinary solutions.

This finding indicates a possible reduction of ocean leakage as well as the improvement of the resolution at coastal areas for the MRBF solutions. In what follows we particularly investigate the seasonal and secular mass flux signals for gravity solutions of different types.

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Figure 9 provides insight into the yearly trend maps covering — To address whether the physical constraint embedded in MRBF affects the recovered gravity signals, we assess EWH trends from Figure 9 over a coastal area, i. In these regions, the proposed MRBF solutions appear in visual inspection to have better localized continental signals as well as less oceanic leakage than the RBF and SH solutions.

But more tests with independent data such as, e.


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This would, however, go beyond the scope of the paper. In particular, compared to the corresponding SH solutions, Mascon [ Watkins et al. This is not a coincidence but probably due to the particular treatment of the spatial leakage embedded in the inversion. Therefore, a set of scale factors [ Baur et al. The red and the brown curves are obviously more steep than the green and the blue, and this indicates that the M RBF might have a larger yearly trend and therefore lower the dependence on using scale factors. The magnitude of the differences might seem small, but it should not be ignored.

However, quantifying the MRBF's leakage reduction with in situ or other independent measurements will be more reliable, and this will be subject to future investigations. This case study also suggests that the MRBF solutions indicate stronger numerical stability during the regularization, due to the lower dependence between the oceanic and continental signals section 4.

In section 5 , after analyzing the annual amplitude and trend maps derived from the time series of M RBF and SH gravity fields, it was shown that the MRBF solution improves the gravity recovery at coastal regions in terms of both spatial resolution and magnitude, hinting that a more accurate modeling of coastal gravity signals could be expected.

Spherical Radial Basis Functions Theory And Applications

This will well minimize the striping error without the necessity of applying additional spatial averaging in a postprocessing step. This might be replaced by another type of RBF e. The authors thank Wang ChangQin from the Chinese Academy of Sciences for his great contribution to the development of gravity solver.

The authors would also like to thank Editor Paul Tregoning and the reviewers for their helpful comments. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors.

Any queries other than missing content should be directed to the corresponding author for the article. Volume , Issue 8. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

Spherical radial basis functions, theory and applications

If the address matches an existing account you will receive an email with instructions to retrieve your username. Open access. Research Article Free Access. Fan Yang Corresponding Author E-mail address: mrfanyang90 gmail. Yang, E-mail address: mrfanyang90 gmail. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access.


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Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Figure 1 Open in figure viewer PowerPoint. Figure 2 Open in figure viewer PowerPoint.

Two types of locations near the coastline and over the inner land far away from the coastline are investigated. Figure 3 Open in figure viewer PowerPoint. Figure 4 Open in figure viewer PowerPoint. Figure 5 Open in figure viewer PowerPoint. Figure 5 f presents the differences between Figures 5 e and 5 a with ocean mask. Figure 5 l presents the differences between Figures 5 k and 5 g with ocean mask. Figure 6 Open in figure viewer PowerPoint. Figure 7 Open in figure viewer PowerPoint. Figure 8 Open in figure viewer PowerPoint. Figure 9 Open in figure viewer PowerPoint.

Figure 10 Open in figure viewer PowerPoint. Figure 11 Open in figure viewer PowerPoint. Figure 12 Open in figure viewer PowerPoint. Baur, O.

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