Automated Dynamic Mascon Generation for GRACE and GRACE-FO Harmonic Processing

Round 1

Reviewer 1 Report

Major considerations

This manuscript investigates the benefits of using a randomly generated Voronoi tessellation scheme to produce locally tailored mascons for better estimates of regional mass change time series. This is especially important for mountain glaciers, where mass change signals are highly concentrated. This approach stands in contrast to the globally optimized JPL and Goddard mascon grids, which are equal-area grids with mascons equally spaced along parallels. 

As Longueverne et al. (2010) and others established, leakage can be minimized by increasing the convexity of the integration kernel. Traditional mascon grids have ~90 degree corners, which naturally lead to ringing because corners are high-frequency features. The Voronoi tessellation scheme has an advantage in that its mascons are more convex than traditional sinusoidal grids. Additionally, because the JPL mascons are spherical caps stretched into quadrilaterals, some of their error comes from the unavoidable omission of signal between the caps. The CSR mascons use a hexagonal tessellation, which would minimize leakage globally because its mascons are maximally convex. The Voronoi tessellation guarantees that each cell is, on average, hexagonal and should take advantage of this intuition while guaranteeing variation in scale appropriate to the nature of the gravity signal.

One question I have for the authors: Is azimuthal symmetry in the Voronoi scheme desirable? The distribution of these Voronoi cells, particularly in Figure 1, reminds me of numerical integration templates, like the “Hammer net” used for terrain correction in terrestrial gravity studies. Like with the Hammer net, these schemes assign more area to less important mass variation outside of the region of interest.  However, the final distribution of cells seems unnecessarily stochastic, as if they’re converging on something that could be much more regular in distribution.

What happens if, instead of using a pole-centered traditional grid as a starting point for the tessellation scheme, the authors rotate the pole of the traditional grid to the point of interest? Instead of distributing the initial points in longitude at selected parallels, the starting points are evenly distributed in azimuth about the point of interest. My expectation is that the resultant tessellation will have higher radial symmetry and that the inner cells will be more regularly hexagonal. This could minimize leakage to a marginally greater degree than the pole-centered generator points.

My goal in asking these questions is to take the randomness out of the tessellation scheme and to help the authors arrive at generalized principles for generating the mascon distribution. This can help future scientists understand the guiding principles for optimality when generating mascon schemes. I don’t expect that rotating the distribution of points in Figure 1A will take much additional time for the authors.

I’m unsure if comparing mascon solutions generated from spherical harmonics to solutions generated from Level 1 range-rate data is appropriate. I understand the authors’ motivation for this approach. Modeling range-rate data is cumbersome and has a number of time-consuming barriers to entry. I also think making the mascons too large outside of the region of interest can lead to subtle aliasing and orbit errors and should be avoided with range-rate data. The reader is also interested in comparing this technique to established mascon time series, as the authors do. Nonetheless, I do think the authors should compare mascon time series derived from spherical harmonics projected onto their own scheme to the schemes of traditional GSFC and JPL mascons with the same spherical harmonics. Because the regularization matrices for these mascons are not necessarily available to the authors, the authors can expect to get entirely different results from what’s in Figure 5. The data going into Figure 5 should remain the same for this reason. However, I think an apple-to-apples comparison will properly illustrate the nature of leakage with traditional grids and better illustrate the problems the new approach avoids. I grant the authors wide latitude on how to address this.

The differences in seasonal amplitude in the Voronoi solutions are remarkable and somewhat counterintuitive. I suppose this is because the improved sampling suppresses the less concentrated hydrological background signals surrounding glaciers. I’m curious what future examination of this outcome will reveal. 

This manuscript is in a well-polished state and requires few, if any, revisions for grammar, spelling, and structure. The authors have put together an engaging reading experience and their techniques will be of benefit to the community. I recommend speedy acceptance after my questions about azimuthally symmetric starting points and consistency between grids have been addressed.

 

Minor considerations

Figure 1: This figure is difficult to interpret because both sides of the sphere are visible, resulting in a sort of Necker cube illusion. I suggest using an azimuthal, equal-area projection centered on the red dot to show the relative areas of the Voronoi cells. If the authors commit to an orthographic or perspective projection, it would be best to occlude the far side of the planet. An azimuthal projection would be especially important if the authors attempt my suggestion of rotating the initial starting point in A to be centered on the fixed point rather than the poles.

 

Figure 5: The green and red lines are difficult to distinguish for someone with the most common form of color vision deficiency. The authors could help this by making one of the lines dashed or changing the color scheme to something closer to Figure 4.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Authors:

The proposed paper presents a method for partitioning the Earth’s surface (given as a spherical model, i.e., constant radius) into Voronoi regions to represent the surface densities, called mascons. The method has the advantage to properly adjust the size of the areas according to an interest target region and thereby being more efficient for regional studies compared to the portioning scheme used by CSR, JPL, and GSFC for representing the mascons “blocks.” The study has the potential to be published in remote sensing. However, there are many problems in the introduction section that needs to be addressed before any potential publication. Comments and critics are provided below to improve the text further. The most important aspect is that previous studies on this topic have not been cited.

 

The best of luck,

 

Comments:

1. Lines 1-3: That statement is not valid; KBR data can be used as observations to any mascon parametrization, which can take into account regional and or basin characteristics. The authors probably had in mind the approaches used by the University of Texas’s CSR, JPL, and GSFC. Furthermore, a regularization scheme is necessary to control the inversion and the amount of noise in the solution. Please, revise that.
2. Lines 3-6: The approaches proposed by Ran et al. (2018b, a) and Baur and Sneeuw (2011), among others, are pretty efficient in reducing leakage and not challenging to be implemented. It is a surprise that the authors haven’t referenced those studies.
3. Line 21, the geoid, is just one of the infinity levels surfaces of the Earth’s gravity field, the one close to the mean sea level. The measurements acquired by GRACE satellites are used to estimate the Earth’s gravity field and its temporal variations.
4. Lines 31-33: Is leakage within the coefficients, or is it due to the post-processing filters? For example, the spherical harmonic coefficients of the EGM2008 contain leakage? That sentence might need some improvement since spherical harmonic coefficients are just used to represent the field (also has a solution of Laplace’s equation).
5. Line 40: Many studies have not been cited. They provide different approaches regarding the ones in Refs. [10-14]. For instance, the approach used by Jacob et al. [11,15] has been proposed by Tiwari et al. (2009), the so-called point-mass approach has been investigated by Baur and Sneeuw (2011), which was originally proposed by Forsberg and Reeh (2007) and improvements of the point-mass solutions presented by Ferreira et al. (2020). Furthermore, the method forwarded by Ran et al. (2018b, a) is quite efficient in the context discussed in the present study. So, it is important to recognize the previous studies and cite them accordingly.
6. In line 34 (abstract as well), the “word” mascon needs to be defined. Please, refer to the original text by Muller and Sjogren (1968).
7. Lines 41-43: This is misleading; spherical harmonics can also include regularization during the inversion (see, e.g., Bruinsma et al. 2010). The French colleagues (CNES/GRGS) have provided regularized harmonic solutions for the public domain for many years.
8. Lines 43-45: That is not true; it is possible to obtain the same results depending on the procedure. If mascon solutions are so good and superior to SHs, why do CSR, JPL, and GSFC provide different results? Besides, mascon solutions using Level-1b data need intermediate SH expansion to account for the contribution of the solid earth deformation. Otherwise, how are the Love numbers considered?
9. Lines 49-51: That is right, they used GLDAS as an a-prior guess to make things worse.
10. Lines 57-59: This contradicts what has been said in lines 41-43.
11. What is a spherical geodetic grid? The word geodetic might implies that there is an ellipsoid, so the use of spherical makes it confusing; this jargon perhaps comes from the Ref. [7]. Just say that the spherical model approximates the Earth…
12. So, in line 104, the geodetic grid means the field is represented on an ellipsoidal model such as WGS84. Is it right? See comment 11.
13. The method is implemented on an Earth spherical model of radius R as defined in Eq. (3). How could one account for the Earth’s ellipticity in such formulation? Mascon solutions such as those from CSR are finely given in WGS84. Further details on this have been provided by Ditmar (2018).
14. Line 224, JPL is based on equal-area spherical caps. The size of the caps changes w.r.t. the latitudes to keep the constant area, which is defined at the equator. GSFC is based on “equal-area” cells equivalent to an area of 1 deg by 1deg at the equator.
15. Not sure if the method would work given a spatial domain approach. For example, how would one solve Newton’s integral for a Voronoi region to properly fit the potential differences between the two space crafts to the surface densities over a target region like the method used by Ramillien et al. (2011)? The point is, Newton’s integral can be solved analytically for a spherical cap. It is also approximated by the so-called point mass solution for a spherical panel, whereas a proper solution based on the Taylor series has been forwarded. If one would like to use the method proposed in this study, how would Newton’s integral be analytically evaluated?
16. Figure 1 is too busy, difficult to extract information from it.

References:

Baur O, Sneeuw N (2011) Assessing Greenland ice mass loss by means of point-mass modeling: a viable methodology. J Geod 85:607–615. https://doi.org/10.1007/s00190-011-0463-1

Bruinsma S, Lemoine J-M, Biancale R, Valès N (2010) CNES/GRGS 10-day gravity field models (release 2) and their evaluation. Adv Sp Res 45:587–601. https://doi.org/10.1016/j.asr.2009.10.012

Ditmar P (2018) Conversion of time-varying Stokes coefficients into mass anomalies at the Earth’s surface considering the Earth’s oblateness. J Geod 92:1401–1412. https://doi.org/10.1007/s00190-018-1128-0

Ferreira VG, Yong B, Seitz K, et al (2020) Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica. Remote Sens 12:3197. https://doi.org/10.3390/rs12193197

Forsberg R, Reeh N (2007) Mass change of the Greenland Ice Sheet from GRACE. In: Dergisi H (ed) Gravity Field of the Earth – 1st meeting of the International Gravity Field Service. Ankara, pp 1–5

Muller PM, Sjogren WL (1968) Mascons: lunar mass concentrations. Science 161:680–684. https://doi.org/10.1126/science.161.3842.680

Ramillien G, Biancale R, Gratton S, et al (2011) GRACE-derived surface water mass anomalies by energy integral approach: application to continental hydrology. J Geod 85:313–328. https://doi.org/10.1007/s00190-010-0438-7

Ran J, Ditmar P, Klees R (2018a) Optimal mascon geometry in estimating mass anomalies within Greenland from GRACE. Geophys J Int 214:2133–2150. https://doi.org/10.1093/gji/ggy242

Ran J, Ditmar P, Klees R, Farahani HH (2018b) Statistically optimal estimation of Greenland Ice Sheet mass variations from GRACE monthly solutions using an improved mascon approach. J Geod 92:299–319. https://doi.org/10.1007/s00190-017-1063-5

Tiwari VM, Wahr J, Swenson S (2009) Dwindling groundwater resources in northern India, from satellite gravity observations. Geophys Res Lett 36:L18401. https://doi.org/10.1029/2009GL039401

 

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I am pleased that the pole rotation scheme worked out well and also produces a solution that minimizes leakage. The tessellation in Figure 1 could use a lighter, less saturated palette to not wash out the coastlines and national boundaries, but this is entirely up to the authors.

Reviewer 2 Report

Dear Authors,

Thanks for properly addressing my comments. I do support the publication of your study in the present form.