the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A Review of Different Mascon Approaches for Regional Gravity Field Modelling since 1968
- Institute of Geodesy, University of Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany
- Institute of Geodesy, University of Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany
Abstract. The geodetic and geophysical literature shows an abundance of mascon approaches for modelling the gravity field of Moon or Earth on global or regional scale. This article illustrates the differences and similarities of the methods, which are labelled as mascon approaches by their authors.
Point mass mascons and planar disc mascons were developed for modelling the Lunar gravity field from Doppler tracking data. These early models had to consider restrictions in observation geometry, computational resources or geographical pre-knowledge, which influenced the implementation. Mascon approaches were later adapted and applied for the analysis of GRACE-observations of the Earth’s gravity field with the most recent methods based on the simple layer potential.
Differences among the methods relate to the geometry of the mascon patches and in the implementation of the gradient and potential for the field analysis and synthesis. Most mascon approaches provide a direct link between observation and mascon parameter –usually the surface density or the mass of an element–, while some methods serve as a post processing tool of spherical harmonic solutions. This article provides an historical overview of the different mascon approaches and sketches their properties from theoretical perspective.
Markus Antoni
Status: final response (author comments only)
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RC1: 'Comment on hgss-2022-4', Anonymous Referee #1, 26 Apr 2022
The authors present a review of different mascon approaches for regional gravity field modelling over the last ca. 50 years. Since the mascon method, based on simplifications such as single-layer approximation etc., was originally developed for planetary geodesy and now represents a standard tool to model time-variable effects in the system Earth, the manuscript also focusses more on these aspects, in particular in context of spherical harmonic analysis. Typical applications to regional modeling of the static gravity field and geoid by means of localising base functions (e.g. pointmasses) in the space domain are mentioned only briefly at the end of Section 2.3.
The main contribution of this manuscript is that it categorizes and shortly characterizes the approaches and their properties from the relevant literature according to clearly defined criteria. In particular, formulae are presented in a unified notation and interpreted in context. Numerical aspects of the practical implementation are not discussed in detail but in the general terms of these criteria. Thus, the manuscript is well structured and easy to read, also for non-expert readers. English style is good and mostly correct.
I have only some minor proofreading corrections (grammar and interpunction), typesetting suggestions as well one recommendation for additional references.
Additional references:
The following references could be added to the sequence cited on line 127, the first one maybe also already in the Introduction.
Barthelmes (1986) already gives a comprehensive and instructive overview and characterization of mascons. However, he focusses on pointmasses, resembling zero-order multipoles in free space, and describes their conceptional and numerical relation with spherical harmonics, resembling an n-order multipole expansion in the Earth's center, and other representations that can be denoted as "mascons".
- Barthelmes, F. (1986): Untersuchungen zur Approximation des äußeren Gravitationsfeldes der Erde durch Punktmassen mit optimierten Positionen. PhD thesis, Veröffentlichungen des Zentralinstituts für Physik der Erde 92. https://gfzpublic.gfz-potsdam.de/pubman/item/item_236018_1/component/file_236017/barthelmes_diss1986.pdf (in German, last visited 26 April 2022).
- Claessens, S., Featherstone, W., Barthelmes, F. (2001): Experiences with Point-Mass Gravity Field Modelling in the Perth Region, Western Australia. Geomatics Research Australasia 75, pp. 53-86. http://hdl.handle.net/20.500.11937/31745 (last visited 26 April 2022).
Proofreading corrections:
1. Line 16: an irregular
2. Line 23: "where [...] modelling overcome [...]" -> Grammar; either it should read "overcomes", or a word is missing.
3. Line 94-95: Typesetting of the period in context with the brackets looks odd. I guess the brackets can be just left out.
4. Line 127: No comma is needed after "It should be pointed out".
5. Line 157: No comma is needed after "It also turns out".
6. Line 210: I guess past tense applies here: "The approach arised ... "
7. Line 220: No comma is needed after "Taking into account".
8. Line 221: within an area
9. Line 222: is re-written [...] as
10. Line 260: most methods require
11. Line 264: analogously
12. Line 293: radians
13. Line 299: comma missing as vector field separator in Eq. 16
14. Line 301: Sunseri
15. Line 310: leads to a class A mascon
16. Line 332: w.r.t.
17. Line 334: by the variational equation
18. Line 336: similarly
19. Line 338: period missing at the end
20. Line 344: period missing at the end
21. Line 346: GRACE mission; no comma needed after "Since the successful GRACE mission"
22. Line 349: no comma needed after "The question arises"
23. Line 351: temporally variable; or even better time-variable
24. Line 384: by by
25. Line 404: to the origin
26. Line 416: was first introduced
27. Line 419: were the point mass models
28. Line 431: no comma neededLaTeX typesetting and style:
It appears that some LaTeX package like "siunitx", or at least inline "math mode", was used to typeset numbers and units in the main text, e.g. "100 km" and "50 km" on line 106, or "48" and "50" on line 109. This is fine in general. However, although this may be just my personal taste, the Computer Modern font in the "unit mode" in combination with the Times New Roman font in "text mode" looks a bit weird and ugly. Furthermore, the spacing between number and unit appears bigger than the spacing to the surrounding text. Usually, a pair of number and unit is set as a half space (LaTeX code \,).
If at all, the style should at least be used consistently throughout the manuscript. For example, just a few lines later (line 112), the "216" in "asteroid 216" is set in text mode (Times New Roman"). The same holds e.g. for the "40962" in "40962 vertices" on line 242. I suggest to make a thoughtful and clear decision which font and mode should be used for actual inline formulas in the text (e.g. line 222), for number-unit pairs (e.g. as "X km") and for simple numbers (e.g. "X [objects]"). I personally would prefer to use math mode with the corresponding font for inline formulas but text mode (or at least text font) for number-unit and number-object pairs.
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AC1: 'Reply on RC1', Markus Antoni, 30 Apr 2022
Dear Reviewer,
thank you very much for your quick and positive feedback about concept and intention, and also the detailed proofreading corrections.
I read the suggested contributions and included them into my article in the section 1 and section 2.3 as follows (Barthelmes [1986] is not using the term mascon for his modelling as far as I could see):
Mass anomalies are sometimes called mascons without using the phrase for the mathematical modelling as well (c.g. Barthelmes1986, p. 35). The focus of this thesis is on the point mass modelling, but it also sketches simple layer potential and discretion of surface elements, and these 3 aspects will be identified as mascon approach in the sense of the following article.
It should be pointed out that the modelling by point masses is applied for example in (Baur 2011) or (Barthelmes 1986, Claessens et al 2001, Lin 2014} without being labelled as mascon approach by the authors, and that in the latter examples also the positions of the masses are estimated for regional studies of the Earth's gravity field. An iterative algorithm is developed and justified via quasi-orthogonality inthe sense of an inner product in (Barthelmes, 1986). To stabilize the optimization process, the possible movement per point mass shall be restricted in depth but also in radial or tangential direction w.r.t. to an initial position.
I applied your "proofreading corrections" 1 to 28 with the exception of No 3 into my new version. In No 3, a removing of the brackets would bring a "negative statement within my explaination" and "stop the reading flow". My new solution is another footnote, which is hopefully ok for you.
I used the LaTeX package "units". Accidetly there were some extra spacings in some brackets, which I corrected. Now the spacing between number and unit should be fine.
I agree, that the number of vertices should be in math mode for consistency and corrected it. In my personal understanding, this should not hold for "asteroid 216" and "Luna 2", as the number is here part of the name...?
Best Regards
Antoni
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AC1: 'Reply on RC1', Markus Antoni, 30 Apr 2022
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RC2: 'Comment on hgss-2022-4', Anonymous Referee #2, 16 Jun 2022
The author provides a review of different mascon technices, from a viewpoint of consistent mathematical representation of the mascons. That’s a laudable effort given that mascons enjoy increasing popularity with the GRAIL and GRACE/-FO missions. A systematic presentation of the approaches would be helpful not only for junior researchers in the field.
However I believe the author chose the wrong journal for publishing his findings. In HGSS the focus should be on the historical context, and that would need to include both the historical technological situation and consequently the spacecraft and ground instrumentation and the line-of-sight data limitations, but in particular also the geophysical problems that people tried to address through mascon approaches. Which was, and is to some extent, the quest for testing geophysical hypotheses on crater formation, impact estimates and thermal regimes, and strength of the underlying lunar crust. See papers by Arkani-Hamed (On the origin of lunar mascons) and others. Both aspects are nearly absent in the paper. Also little is said about more recent planetary applications. I think the paper is well-intended but for HGSS it misses its mark. It should be submissed to e.g. the Journal of Geodesy. If the author insists on HGSS, my suggestion would be to entirely remove the GRACE applications (all this is essentially known, including the authors conclusions in sec 5) and focus his review on the Moon (I can recommend reading the Phd theses by Rune Floberghagen and Sander Goossens), but definitely including an overview of how mascons helped to improve the lunar geophysics.
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AC2: 'Reply on RC2', Markus Antoni, 17 Jun 2022
Dear reviewer,
thank you very much for your your feedback, the assessment about possible users and benefits, and the basic considerations.
I selected the HGSS as possible journal, because I describe here a historical developement of the mathematical mascon concepts. A quest for geophysical hypotheses might be beyond my experience and knowledge. If these aspects are a fundamental pre-requirement, then indeed I missed the mark.
(To focus the paper on the Moon only, is not in my intention. I wanted to summarize also the mascons solutions of the last 2 decades due to the GRACE mission.)
Best Regards
Antoni
- EC1: 'Reply on AC2', Johannes Ihde, 01 Jul 2022
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AC2: 'Reply on RC2', Markus Antoni, 17 Jun 2022
Markus Antoni
Markus Antoni
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