Avoid common mistakes on your manuscript.
1 Correction to: J Geod (2017) 91:1367–1382 https://doi.org/10.1007/s00190-017-1030-1
The authors would like to apologize to the editor and readers for a bug in the numerical implementation of the formula for the initial guess on the regularization parameter \(\alpha _0\) (Naeimi 2013), as given in the second paragraph on page 1370, and repeated here for convenience,
A different initial guess on the regularization parameter alters the search space for the applied L-curve method, giving different final regularization parameters \(\alpha \). In this particular case, the value of \(\alpha _0\) that we implemented was larger than prescribed by Eq. (1), giving systematically too large final values of \(\alpha \).
Initial regularization parameters proposed according to Eq. (1) slightly affect the numerical results in Sect. 5, but do not invalidate, change or undermine the methods. The numerical results are positively affected in that the spline kernel (SK) results improve. Finally, the discussion and summary of the original paper are not affected.
In the following, we list the resulting changes.
Table 2 should read as follows:
The entries related to results with the SK in Table 3 should read:
The entries related to results with the SK in Table 4 should read:
In the sixth paragraph of Sect. 5 on page 1375,
“In East Frisia, using unmodified LSC (SKs), we get RMS differences of 1.75 cm and 1.83 cm (1.76 cm and 1.83 cm) in the data and target areas, respectively. In the Alpine region, using unmodified LSC (SKs), we get RMS differences of 5.73 cm and 5.18 cm (5.74 cm and 5.18 cm) in the data and target areas, respectively”,
should read
“In East Frisia, using unmodified LSC (SKs), we get RMS differences of 1.75 cm and 1.83 cm (1.89 cm and 1.83 cm) in the data and target areas, respectively. In the Alpine region, using unmodified LSC (SKs), we get RMS differences of 5.73 cm and 5.18 cm (5.91 cm and 5.18 cm) in the data and target areas, respectively”.
In the ninth paragraph of Sect. 5 on page 1379,
“Increasing the grid resolution to 2.5 arcmin leaves the LSC and SK results largely unaffected (with maximum improvements in the target areas of \(\sim \) 0.5 mm, and even a slight degradation of the LSC solution in East Frisia), while results by Stokes’s formula improve by 0.85 mm and 3.5 mm in East Frisia and Alpine region, respectively.”,
should read
“Increasing the grid resolution to 2.5 arcmin leaves the LSC and SK results largely unaffected (with maximum improvements in the target areas of \(\sim \) 1.1 mm, a slight degradation of the LSC solution in East Frisia, and SKs slightly outperforming LSC on the sub-mm level in both target areas), while results by Stokes’s formula improve by 0.85 mm and 3.5 mm in East Frisia and Alpine region, respectively.”.
In the eleventh paragraph of Sect. 5 on page 1380,
“Both cases give practically equal results (RMS differences between SK geoid solutions of 1 mm and 5\(\times \)10\(^{-2}\) mm in the data and target zones, respectively), suggesting that the SKs converge towards LSC depending on the signal resolution rather than the number of observations.”,
should read
“Both cases give practically equal results (RMS differences between SK geoid solutions of 3 mm and 1\(\times \)10\(^{-3}\) mm in the data and target zones, respectively), suggesting that the SKs converge towards LSC depending on the signal resolution rather than the number of observations.”,
In the second paragraph of Sect. 6 on page 1380
“At the 2.5 arcmin resolution, all methods agree within \(6\times 10^{-2}\) mm to 2.4 mm.”,
should read
“At the 2.5 arcmin resolution, all methods agree within \(1.1\times 10^{-3}\) mm to 2.4 mm.”.
In the third paragraph of Sect. 6 on page 1380
“Indeed, this is confirmed in our numerical examples, where LSC generally gives the smallest error. SKs perform very similar to LSC.”,
should read
“In our numerical examples, LSC gives the smallest error for the 5 arcmin resolution. The SKs perform very similar to LSC, and even show a sub-mm improvement over LSC on the 2.5 arcmin resolution.”.
Reference
Naeimi M (2013) Inversion of satellite gravity data using spherical radial base functions. Ph.D. thesis, Deutsche Geodätische Kommission Reihe C, Nr. 711
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ophaug, V., Gerlach, C. Correction to: On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation. J Geod 94, 60 (2020). https://doi.org/10.1007/s00190-020-01375-7
Published:
DOI: https://doi.org/10.1007/s00190-020-01375-7