Unified Land–Ocean Quasi-Geoid Computation from Heterogeneous Data Sets Based on Radial Basis Functions

Round 1

Reviewer 1 Report

An analysis of the remote-derived gravity and other data is a very complex and specific process. Any new step in this field gives us new useful information. The revised version of this MS is ready for publication.

Author Response

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Reviewer 2 Report

In the new version of the manuscript there are still some problems that should be addressed.

 

1. The comparison between gravimetric geoid undulations and GPS/levelling undulations requires the bias reduction to taken into account for the different datum. I don’t agree the authors that affirm that it is not necessary because it's a systematic error that it's removed in the comparison procedure. I think I think that any source of data (terrestrial, shipborne etc…) contribute to define a reference system, that can change solution by solution, so it's not a systematic error. The datum transformation requires a simple formula, where the  difference between the gravimetric geoid and the GPS/levelling geoid can be expressed in term of three parameters (i.e. Moritz)
2. I asked the authors to give the statistics of the comparison of a solution that don’t introduce the DTU15 data, because I think that the information coming by the satellite altimetry are already introduced by the GGM. So in Tables 3 and 6 I expect to see rows like “terrestrial + shipborne” or “terrestrial+shipborne+airborne”, without “+DTU15”
3. The comparison with the GEOID18 model for the ocean geoid it’s not significant because your solutions and the GEOID18 are not independent, but you are using a subset of the same observations, at least. I suggest to analysis the differences in the GPS/levelling points along the coast, because any improvement in the gravity field modeling over the ocean will be reflect in the geoid estimate especially along the coast.
4. I don’t agree this assumption “When the calculation area and data number become large, it will be quite difficult to reverse the high-order covariance matrix and the calculation efficiency will be seriously reduced” because the Fast Collocation approach (Bottoni&Barzaghi, 1993) has solve this problem
5. Check formula (6) because in Moritz & Hofmann-Wellenhof (pag. 327) is different
6. You didn’t compute the RTM for the shipborne and DTU15 gravity data, but in my opinion it’s necessary
7.  

Minor remarks:

• Line 176: “GPS and leveling surveys are difficult to be carried out offshore”. I suggest “impossibile” rather than “difficult”

Author Response

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Round 2

Reviewer 2 Report

Dear authors, 

thank you for taking my comments and suggestions into consideration. 

I think that the manuscript can be published. 

 

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The authors of the manuscript “Unified Land-Ocean Quasi-Geoid Computation from Heterogeneous Data Sets by Using Radial Basis Functions” focus their work in the computation of the quasi-geoid undulation using different sources of gravity data (terrestrial, shipborne, altimetry ). The proposed estimation method  is based on Radial Basis Function (RBF), because the authors hypothesis is that this method is very efficient in this type of computation, where heterogeneous data are used. Two testing area in USA are selected and the proposed method is compared with the one based on the Stokes integral.

In my opinion the manuscript requires a very deep revision because the results are not presented in a proper way (figures and figure captions should be improve), some theoretical points should be justified and some critical issues are present. In following the list of the most critical point

1. The altimetric data contribute to most the GGM models, as EIGEN-6c4, so I can not understand the necessity to introduce these data in the geoid computation, moreover the actual limit of the satellite altimetry is in the boundary between open sea and coasts because the altimetric signal is interrupted. 
2. In your estimates the DTU15 data are always present. Have you ever tested the geoid computation without this set of data? I suggest a solution based on terrestrial and shipborne observations. 
3. When you compare gravimetric geoid undulations with GPS/levelling derived undulation you often have to take into account for different datum, did you check this aspect? In the manuscript there isn’t any information about it. 
4. The previous problem could be the reason of the anomalous values reported in Table 2. A mean value of about 70 cm with a std of 5 cm suggest the presence of a bias in the data. If you estimate and remove this bias I aspect that conclusions about the Stokes approach will be different, and also the consideration about Gravsoft software at lines 579-580 that I find inappropiate. 
5. In my opinion in the east coast experiment the RTC (residual terrain correction) should be computed, although the authors affirm that “The terrain of the experiment area is quite flat, so the impact of terrain mass can be ignored”. Before ignoring this correction, its values should be quantified. 
6. I have some doubts concerning the selection of points from GEOID18, because I aspect that the control points and the external checkpoints are very correlated because they are extracted by the same model.
7. In the Introduction Section, I agree about the importance of covariance models in LSC, however the sentence about the difficulty in constructing an appropriate and accurate local model (Line 48-49) should be better justify because, in my opinion, it doesn't represent a limitation in the application of the technique. 
8. In the presentation of the method some information is missing:
   • Line 208: what is the correlation length parameter?
   • Line 210: what is the parameter q? How do you fix this value?
   • In Equation (5) the coefficients µi has not been defined. 

Minor remarks:

1. Figure 2, Figure 3, Figure 4: missing information about the color scale. The represented quantity and the measure unit (free-air and mGal) should be reported.
2. Figure 5: missing information about the color scale. Is it the topographic height?
3. Figure 6: missing information about the color scale. Is it the geoid undulation?
4. Analogous problem for the other figures (i.e. Figures 14, 15, 21, 22, 25): missing information about the color scale.
5. Figure 21, 22 and 25: please, use the color scale in figures a) and b)
6. Figure 20: the plot should be more described
7. Line 304: GVC —> GVC (general cross validation)
8. Line 304: does STD mean Standard Deviation?  

Author Response

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Reviewer 2 Report

The ms attempts to design a multi-scale RBF model to construct the unified land-ocean quasi-geoid

geoid fusing the measured terrestrial, shipborne, satellite altimetry and airborne gravity data in coastal areas.

 

It is my view that the ms is not well written as it misses many fundamental elements that make the development of the RBFs unclear. Perhaps there are some language issues or terminology that has not been either referenced or explained. It is very hard to check the validity of the RBFs and assess the final results.

 

Apparently, the authors follow the classical RCR method to calculate the quasi-geoid which they compare with geoid models and also with GPS on benchmarks (see for instance lines 177-184 and 192 and bottom part of flow chard in Fig. 7). On land, the quasi geoid is different from the geoid whereas in the oceans they are almost identical. It is not clear how these comparisons are done as I could not find any transformation of the quasi-geoid to geoid before the comparisons.

 

On Lines 208-209 the authors introduce χ as correlation length parameter. How id this parameter defined? The explanation given in line 211 does not really explain it. Is this parameter the same as Ψ in Fig. 8 or the same as ψ_m in Eq. (2)? And why is ψ_m  called Legendre coefficient, since it defines the shape factor of RBFs?  And what is q?

 

Fig. 9 is not clear. It is missing description

 

There are many points I could make on the ms but at this point of the review it will make little sense as the ms is not, by and large, clearly written. Since the main contribution of the ms is the development of the RBF I would strongly suggest that the authors rewrite Section 2.3 by providing clear step-by-step explanations. In this section, the authors introduce new variables namely x, and y (also with subscripts) which the call coordinates? Are they latitude and longitude? Or what is the coordinate system they consider? For instance, in Fig 10 they label the axes as degrees… are they latitude and longitude? And what is Reuter grid?

Downward continuation of the airborne data? 

Overall, the ms is not well written and offers very little to the reader. To review this ms properly and make helpful comments to the authors it must be re-written. I cannot suggest “reject” as the validity of the algorithms and by extension the validity of the results cannot be checked. For now, I suggest a major overhaul of the ms, that will probably take long time to achieve.

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

Indeed, it is an interesting and useful theoretical and practical investigation. 

The authors demonstrate a top knowledge of the studied problem.

However, there are some shortcomings.

In the Introduction, some additional references can be added:

Eppelbaum, L.V. and Katz, Yu.I., 2017. A New Regard on the Tectonic Map of the Arabian-African Region Inferred from the Satellite Gravity Analysis. Acta Geophysica, 65, 607-626.

Braitenberg, C. and Ebbing, J., 2009. New insights into the basement
structure of the West Siberian Basin from forward and inverse modeling of GRACE satellite gravity data. J Geophys Res, 114(B06402), 1–15.

Many Figures are too small and have a low quality:

Figures 1a, 1b, 2a, b, 3a, b, 4a, b, 5a, b, 14a, b, 19a, b, 20, 21a, b, 22a, b, 25 a, b, 26 a, b, 27 a, b, 28 a,b,c,d 

 

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I thank the authors for the  kindly replies of the authors, however your explanations do not solve all  my perplexities. 

The two main criticisms concern: 

1. In the comparison with the GPS/levelling geoid undulation values, the bias due to the difference in the reference systems should be removed. This bias is not a constant value but depends at least on three parameter, so in a local application it can be considered as a plane. So, the authors can not affirm that they take into account just for the standard deviation. With the SRBF you don't have this problem (look at Table 3 and 6, where the mean values are in the order of 1-2 cm) because in the proposed method some GPS/levelling and  GEOID18 values are already used as control points to constrain your solution. 

Please see the following reference: Heiskanen, W.A.; Moritz, H. Physical Geodesy, Institute of Physical Geodesy; Technical University of Graz: Graz, Austria, 1990.

 

 

2. The input data seem to be too much correlated each other. In particular the DTU database is used in the computation of the GGM so the data should be used taking into account for this. I asked the authors to provide a solution without introducing the DTU data, and they have replied that “the modeling results are not as good as the fusion of these three types of data all”, but this is not shown in the results. In Table 3 and 6 the DTU data are always added to the terrestrial data. As the authors have computed solutions like “terrestrial+DTU”, “terrestrial+DTU+shipborne” etc… I think that they should compute the solution “terrestrial+shipborne+airborne” without DTU.

Reviewer 2 Report

N/A