Correction to: On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

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 2013, as given in the second paragraph on page 1370, and repeated here for convenience.

(1) A different initial guess on the regularization parameter alters the search space for the applied L-curve method, giving different final regularization parameters α. In this particular case, the value of α 0 that we implemented was larger than prescribed by Eq. (1), giving systematically too large final values of α.
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: Gravity anomalies in mGal (= 10 −5 ms −2 ) and geoid heights in cm The entries related to results with the SK in Table 4 should read: Gravity anomalies in mGal (= 10 −5 ms −2 ) and geoid heights in cm 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 ∼ 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 ∼ 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 5×10 −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 1×10 −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