Regional Seafloor Topography by Extended Kalman Filtering of Marine Gravity Data without Ship-Track Information
, Guillaume Ramillien 1,3, Benjamin Beirens 1,2, José Darrozes 1,2, Didier Rouxel 4, Thierry Schmitt 4, Corinne Salaün 4 and Frédéric Frappart 5Round 1
Reviewer 1 Report
The authors applied the extended Kalman filter method to the inversion of seafloor topography, and obtained relatively accurate seafloor topography using only the relevant gravity data in the region, unconstrained by ship measurement data, which has a corresponding contribution to the topographic recovery of unknown sea areas. In my opinion current form of work is suitable for publication. I add some, comments and suggestions which may be useful for the authors.
- Title: Only gravity information is not rigorous, and the construction of the seafloor topography involves geophysical parameters (elastic thickness Te, etc.), which can be modified to without ship-tracks information.
- In the keyword list, I suggest changing those keywords that are already stated in the title. For example, "inversion method" is not precise enough, and "gravity field" can be considered to be changed.
- Formula 5: The Brun's formula is the relationship between the disturbing potential and the geoid height, the symbol of the disturbing potential is T.
- While we understand the meaning of your expression, please write out what these abbreviations means before using them for the first time (e.g., FAA).
- In Figure 3, it is recommended to add the bathymetric scale because the known bathymetric range is indicated in the figure legend, which can be obtained from existing models as an introduction to the study area.
- Line 453：The sentence “combined inversion unregularly sampled versus ship-tracks bathymetry (Figure 12a)”is incorrectly, should be changed to “combined inversion unregularly sampled versus ship-tracks bathymetry (Figure 12b)”.
- Line 349-354: It is mentioned that there are different improvements in the absolute error for different elastic thickness Te, but this is not shown in the figure and a part of the result support can be added, similar to Figure 7(a)
- The format of symbols is not customary, and there are more errors in the use of arithmetic symbols (e.g. Figure 1 names, Appendices A and B, etc.) Please, change them.
Comments for author File: 
Comments.docx
Author Response
- Title: Only gravity information is not rigorous, and the construction of the seafloor topography involves geophysical parameters (elastic thickness Te, etc.), which can be modified to without ship-tracks information.
Corrected. The new title is:
“Regional seafloor topography by Extended Kalman Filtering of marine gravity data without ship-track information”.
- In the keyword list, I suggest changing those keywords that are already stated in the title. For example, "inversion method" is not precise enough, and "gravity field" can be considered to be changed.
Corrected. We have changed the keywords “inversion method” and “gravity field” by “Kalman Filter” and “marine gravity anomalies”
- Formula 5: The Brun's formula is the relationship between the disturbing potential and the geoid height, the symbol of the disturbing potential is T.
Corrected. We changed the symbol of the disturbing potential by T and added the related reference:
Hofmann-Wellenhof, B.; Moritz, H. Physical Geodesy; 2nd ed.; Springer-Verlag: Wien, 2006; ISBN 978-3-211-33544-4
- While we understand the meaning of your expression, please write out what these abbreviations means before using them for the first time (e.g., FAA).
Corrected. We have defined the abbreviations before using the Generic mapping tools (GMT), Geoid height (GEOID), Free-Air Anomalies (FAA), combined inversion (GEOID + FAA), inversion (INV). In Table 1, we have changed COMB by GEOID+FAA to be consistent with the definitions and the abbreviations.
- In Figure 3, it is recommended to add the bathymetric scale because the known bathymetric range is indicated in the figure legend, which can be obtained from existing models as an introduction to the study area.
Corrected. The color scale has been added in Figure 3.
- Line 453：The sentence “combined inversion unregularly sampled versus ship-tracks bathymetry (Figure 12a)”is incorrectly, should be changed to “combined inversion unregularly sampled versus ship-tracks bathymetry (Figure 12b)”.
Corrected. The sentence has been changed.
- Line 349-354: It is mentioned that there are different improvements in the absolute error for different elastic thickness Te, but this is not shown in the figure and a part of the result support can be added, similar to Figure 7(a).
Corrected. Another figure about the sensitivity of Te has been added to show RMS values and the improvements in the absolute error for different elastic thickness.
- The format of symbols is not customary, and there are more errors in the use of arithmetic symbols (e.g. Figure 1 names, Appendices A and B, etc.) Please, change them.
Corrected. We uniformed the format of symbols and arithmetic symbols.
Reviewer 2 Report
Major considerations
This paper addresses the gravity-bathymetric inversion problem using an Extended Kalman Filter (EKF). The authors distinguish themself from prior work in a number of ways. The approach is validated using gravity data from the Great Meteor seamount in the Atlantic ocean using gravity data to predict depths from acoustic ship soundings. The model offers solutions with accuracies comparable to well known bathymetric grids, such as GEBCO2020 and ETOPO1.
This paper offers an exciting and innovative update to the bathymetric inversion problem and I highly recommend its inclusion in this journal. I am impressed with its efficient use of irregular grid spacing. The realization of the EKF is consistent with the textbook formulation and applicable to this problem. I am unable to find any major methodological errors and I therefore recommend publication after the “minor considerations” have been addressed.
Minor considerations
The results in Table 1 are impressive. As various 18th and 19th-century geodesists learned, compensation is critical to predicting the gravity effects of topographic features.
Line 133: The value of the gravitational constant is inconsistent with the value on line 518.
Line 233: A 20-30 km half-wavelength is consistent with what’s prescribed in textbooks, including Wellenhof & Moritz.
Figure 11: “Irregular grids” rather than “Unreguarly grids”
Equation A-2 and subsequent equations: Use a symbol other than d to express distance as this can easily be confused with the differential operator, which appears frequently in this manuscript.
The color palettes for Figures 3, 4, 8, 9, 10, and 12 should be updated. Authors should strive for modern, perceptually uniform color maps when presenting data. The color maps used in this manuscript are well known for accessibility issues and lack of uniformity. By sweeping through all possible hues, they mix colors that are challenging to distinguish for readers with common forms of color vision deficiency. Furthermore, both scales obscure and distort variation in the data by incrementing through color values in perceptually nonuniform steps. The images in the right side of Figure 8 obscure variation in the data by using a single distinct shade of green for a quarter of the data range, obscuring high-frequency features that should be more apparent in FAA grids. The color scale on the left side of the same figure is an improvement, but it also creates false contours around 1500 meters and 6000 meters because of the irregular treatment of those red and blue bands. The authors should switch to a color scale that is both accessible and more illustrative of the data such as Parula, Viridis, coolwarm, or similar color maps developed in the perceptually uniform CIELab color space. Figures 9, 10, and 12b should use a diverging color scale to aid the reader in identifying the sign and central tendency of the residuals, which Table 1 shows have a nonzero mean.
For Figures 5, 6, and 7, the authors are cautioned to avoid combinations of color that are difficult to distinguish for readers with common color deficiencies, including red and green; and cyan and magenta.
Further reading:
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0199239
https://www.osti.gov/biblio/1363736
https://doi.org/10.2312/eurorv3.20171107
https://www.kennethmoreland.com/color-advice/
Author Response
Line 133: The value of the gravitational constant is inconsistent with the value on line 518.
Corrected. We have changed the value on line 518 by the value defined in line 133 (this latter value is used in our computations)
Line 233: A 20-30 km half-wavelength is consistent with what’s prescribed in textbooks, including Wellenhof & Moritz.
Agree. In line 213 we have added the famous reference Wellenhof & Moritz.
Figure 11: “Irregular grids” rather than “Unreguarly grids”
Corrected. We have changed “Unreguarly grids” by “Irregular grids”
Equation A-2 and subsequent equations: Use a symbol other than d to express distance as this can easily be confused with the differential operator, which appears frequently in this manuscript.
Corrected. We have changed d by norm notation ||P-M||
The color palettes for Figures 3, 4, 8, 9, 10, and 12 should be updated. Authors should strive for modern, perceptually uniform color maps when presenting data. The color maps used in this manuscript are well known for accessibility issues and lack of uniformity. By sweeping through all possible hues, they mix colors that are challenging to distinguish for readers with common forms of color vision deficiency. Furthermore, both scales obscure and distort variation in the data by incrementing through color values in perceptually nonuniform steps. The images in the right side of Figure 8 obscure variation in the data by using a single distinct shade of green for a quarter of the data range, obscuring high-frequency features that should be more apparent in FAA grids. The color scale on the left side of the same figure is an improvement, but it also creates false contours around 1500 meters and 6000 meters because of the irregular treatment of those red and blue bands. The authors should switch to a color scale that is both accessible and more illustrative of the data such as Parula, Viridis, coolwarm, or similar color maps developed in the perceptually uniform CIELab color space. Figures 9, 10, and 12b should use a diverging color scale to aid the reader in identifying the sign and central tendency of the residuals, which Table 1 shows have a nonzero mean.
Corrected. We have used the scientific color maps provided by Fabio Crameri and supported by GMT.
For Figures 5, 6, and 7, the authors are cautioned to avoid combinations of color that are difficult to distinguish for readers with common color deficiencies, including red and green; and cyan and magenta.
Corrected. We have changed colors of Figures 5, 6, and 7.
Further reading:
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0199239
https://www.osti.gov/biblio/1363736
https://doi.org/10.2312/eurorv3.20171107
https://www.kennethmoreland.com/color-advice/
Reviewer 3 Report
This is a good article that could be improved with some of the specific suggestions below.
Basic editing needs to include singular/plural agreement.
L 58-60: Based on titles of references it is not clear that the statement about “global bathymetric maps” is quite correct. Perhaps “regional”, “large area”. It seems that Ref 6 onward did some sort of maps as noted in L 67.
L 103: The sentence does not make sense. What does proxy mean – constraint, prior?
L 107: A different word that “cumulating” (not really a word) is needed – using?
Fig 1: The model seems to have 4 layers (plus water), but the text and caption say 3. The figure seems to show that a significant layer of sediment only exists under the topography or that the topography spreads/deepens the sediment layer. If this is really the intention then some discussion is needed. If the sediment layer is basically uniform, it is hard to see why it needs to be called out separately from the crust.
This also perhaps affects Appendix B, which has no reference to rho_s.
L 556: “Spherical cap of constant thickness” does not make sense. Does it mean circular cap, so a cylinder?
L 169: While it is obviously statistically/mathematically convenient, it is very unlikely to be true for real altimeter data that the noise is white and uncorrelated. If one is using the difference of along-track sea surface heights, it is certainly much truer than if using the total height (or residual from a fixed mean sea surface). The main height errors in modern altimetry are from propagation media or sea state bias, all of which have correlation scales of 10s-100 km.
Also, the errors on the initial topographic state are likely to be correlated over significant spatial scales.
A comment on this (or a pointer to the discussion section) should be included.
Fig 2: Appendix C is not very long and should be moved here for easier reading, understanding of the figure. This will also help with lines 182-187 and/or that in App C many of the terms are undefined. If App C is left as standalone at the end all terms should be redefined.
Eq 7 indicates that the matrices are the size of the topo to be estimated. How does that help the “very large matrix” issue raised at the beginning?
L 221: Define (and perhaps briefly describe) GMT (ref 46, 52) at first use.
L 233: Typo? Resolution = 1 arc minute?
Fig 9 would be much more informative if the color scale were limited to <~1000m and areas simply allowed to saturate in the maps. The good quality in the FAA and combined maps would be much more obvious.
Fig 10: Similar comment to Fig 9 except maximum scale = +/-500m.
Author Response
Basic editing needs to include singular/plural agreement.
Agree. The manuscript has been re-read several times to track singular/plural inconsistencies.
L 58-60: Based on titles of references it is not clear that the statement about “global bathymetric maps” is quite correct. Perhaps “regional”, “large area”. It seems that Ref 6 onward did some sort of maps as noted in L 67.
Corrected. We have changed “global bathymetric maps” to “bathymetric maps over large areas”
L 103: The sentence does not make sense. What does proxy mean – constraint, prior?
Corrected. We have changed proxy by “reference for validation” to be clear.
L 107: A different word that “cumulating” (not really a word) is needed – using?
Corrected. The word “cumulating” has been removed.
Fig 1: The model seems to have 4 layers (plus water), but the text and caption say 3.
Corrected. We have changed 3 layers by 4 layers
The figure seems to show that a significant layer of sediment only exists under the topography or that the topography spreads/deepens the sediment layer. If this is really the intention then some discussion is needed. If the sediment layer is basically uniform, it is hard to see why it needs to be called out separately from the crust.
We used a simplified model based on Watts (2001) which considers different assumptions:
- the sediment layer has the same density as the surrounding load but a different density of crust.
-the sediment layer, load and crust have the same density.
In others words, we assume that the eroded material comes from the volcanic edifice itself. Of course, in some cases, the density could be significantly different at the edge of the seamount. The evaluation of sediment layer impact in deflection modelling is an entire research subject that could be developed in a specific study.
We added in Line 119: “This is a simplified model which allows considering different assumptions ρs=ρh or ρs=ρh=ρc.”
This also perhaps affects Appendix B, which has no reference to rho_s.
In Appendix B ρs is referenced in Equation B3 as proposed by Calmant 1994. The reference has been added in Appendix B.
L 556: “Spherical cap of constant thickness” does not make sense. Does it mean circular cap, so a cylinder?
The “Spherical cap of constant thickness” is already used by several authors please see Heck and Seitz 2007, Nozaki 2004 and is defined as a portion of the spherical shell of constant thickness.
L 169: While it is obviously statistically/mathematically convenient, it is very unlikely to be true for real altimeter data that the noise is white and uncorrelated. If one is using the difference of along-track sea surface heights, it is certainly much truer than if using the total height (or residual from a fixed mean sea surface). The main height errors in modern altimetry are from propagation media or sea state bias, all of which have correlation scales of 10s-100 km.
Agree. Besides Previous studies (Calmant 1994) on seafloor topography inversion used the assumption of white noise for altimeter data errors, and specific would be driven to quantify the statistics of altimeter measurement errors or the structure of matrix Rk) at short wavelengths.
Also, the errors on the initial topographic state are likely to be correlated over significant spatial scales.
Agree. The starting a priori covariance of the topographic heights is of Hirvonen-type (Equation 1) which takes into account spatial correlation and the corresponding covariance matrix Pk evolves during the KF process, as well as the length of correlation (or spatial scale) between topographic errors.
A comment on this (or a pointer to the discussion section) should be included.
We have added a comment in section 3.1.2 Line 310.
“Obviously, the real altimeter data are probably spatially correlated with no white noise. Further studies would be needed to determine a realistic structure of matrix Rk-1.”
Fig 2: Appendix C is not very long and should be moved here for easier reading, understanding of the figure. This will also help with lines 182-187 and/or that in App C many of the terms are undefined. If App C is left as standalone at the end all terms should be redefined.
We moved Appendix C to section 2.1
Eq 7 indicates that the matrices are the size of the topo to be estimated. How does that help the “very large matrix” issue raised at the beginning?
Fik dimensions in Equation 7 depends on the size of topography M to be estimated but also on the size of input observations N. The number of observations can be huge. The Kalman filter method allows to combine progressively different types of observations and to update the solution at every iteration reducing the problem dimensions. If we use classical inversion methods based on non-linear least squares, for example, all observations are included in the Jacobian matrix then it becomes a very large matrix.
L 221: Define (and perhaps briefly describe) GMT (ref 46, 52) at first use.
Corrected. We have defined GMT (Generic mapping tools) at line 259
L 233: Typo? Resolution = 1 arc minute?
Agree. Yes, this is a typo. We changed by: “ETOPO 1 provides land and seafloor topography with a spatial resolution of 1 arc-minute”
Fig 9 would be much more informative if the color scale were limited to <~1000m and areas simply allowed to saturate in the maps. The good quality in the FAA and combined maps would be much more obvious.
As required by the other reviewer, we have changed the colour palettes in order to better represent the results.
Fig 10: Similar comment to Fig 9 except maximum scale = +/-500m.
We have also changed the colour palettes of Figure 10 in order to better represent the results.
