the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Jan 2022
Regional mantle viscosity constraints for North America reveal upper mantle strength differences across the continent
Abstract. We present regional constraints of mantle viscosity for North America using a local Bayesian joint inversion of mantle flow and glacial isostatic adjustment (GIA) models. Our localized mantle flow model uses new local geoid kernels created via spatio-spectral localization using Slepain basis functions, convolved with seismically derived mantle density to calculate and constrain against the regional free-air gravity field. The joint inversion with GIA uses two deglaciation of ice sheet models (GLAC1D-NA and ICE-6G-NA) and surface relative sea level data. We solve for the local 1D mantle viscosity structure for the entire North America (NA) region, the eastern region including Hudson Bay, and the western region of North America extending into the Pacific plate.
Our results for the entire NA region show one order of magnitude viscosity jump at the 670 km boundary using a high seismic density scaling parameter (e.g., δlnp/δlnvs = 0.3). Seismic scaling parameter demonstrates significant influence on the resulting viscosity profile. However, when the NA region is further localized into eastern and western parts, the scaling factor becomes much less important for dictating the resulting upper mantle viscosity characteristics. Rather the respective local mantle density heterogeneities provide the dominate control on the upper mantle viscosity. We infer local 1D viscosity profiles that reflect the respective tectonic settings of each region's upper mantle, including a weak and shallow asthenosphere layer in the west, and deep sharp viscosity jumps in the eastern transition zone, below the suggested/proposed depth range of the eastern continental root.
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CC1: 'Comment on se-2021-151', Tanghua Li, 22 Feb 2022
Dear Editor,
The manuscript “Regional mantle viscosity constraints for North America reveal upper mantle strength differences across the continent” by Osei Tutu & Harig investigates the regional mantle viscosity values for North America using a local Bayesian joint inversion of mantle flow and GIA models. They use the region free-air gravity data and RSL data. The topic is interesting and worth studying since North America is a key region to deduce the mantle viscosity structure and to better understand the solid Earth behavior due to the loading and uploading of the Laurentide ice sheet.
The figures are of good quality and easy to read. However, several aspects of this manuscript need to be improved/addressed before it can be considered for publication in Solid Earth.
In this study, the RSL database from Tushingham and Peltier (1991) is used, which is pretty old and outdated. There are more recent high-quality standardized RSL databases for eastern Canadian coast (Vacchi et al., 2018), including Hudson Bay, U.S. Atlantic coast (Englehart & Horton, 2012), Pacific coast (Engelhart et al., 2015), and they are all freely available. Why don’t you use the recent ones? The importance of the standardization of RSL data has been demonstrated in many studies, especially for the differentiation of sea-level index points (SLIPs) and limiting data, which are not considered in Tushingham and Peltier (1991). Moreover, your eastern region () includes U.S. Atlantic coast, but no RSL data from the Atlantic coast of NA, which have been shown to be vital to constrain the viscosity and may result in totally different upper mantle viscosity values as that from Hudson Bay (e.g., Engelhart et al., 2011).
Why restrict ice melting history and RSL data to < 10 kyr BP (line 142)? What’s the significance behind this? Does that mean you only have a deglaciation stage from 10 ka BP till present but no glaciation stage in your GIA ice model? You cannot only use ice melting < 10 kyr BP to even only study the RSL < 10 ka BP. Because the solid Earth response to loading/glaciation and unloading/deglaciation events is a delayed process, what happened before 10 kyr BP (e.g., the LGM) definitely significantly affect the RSL <10kyr BP.
Line 244-245, “differences in our inferred 1D regional …” and as you stated in the abstract, this means the uncertainty of inferred 1Dregional viscosity due to the uncertainty of scaling factor is significantly dramatic, then comes the question about the validity and accuracy of this approach to infer the 1D regional viscosity. In that case, you need to assess the uncertainty of the inferred viscosity.
Whole North America, eastern sub-region, western sub-region, you need to follow the same order through the whole paper for each section. Besides, suggest you consistently using “sub-region” when refer to western and eastern North America region, as you used in line 197. Please consistently use the same expression for the seismic velocity to density scaling parameter.
The usage of abbreviation and full name in this manuscript is really in chaos, like relative sea level (e.g., line 137, 163, 403) and RSL (e.g., line 44, 139, 299), North America (e.g., 83, 191, 256) and NA (e.g., line 18, 55, 98), glacial isostatic adjustment (e.g., line 189, 416) and GIA (e.g., line 23, 36) … Also misuse of italic for some common words, like localized in line 104, volume in line 106,
Citation formats in the main text need to be double checked.
Detailed comments:
1, Replace the word “strength” in the title, maybe by “viscosity”, as “strength” only appears once in the title and then never been used any more in the main text.
2, Line 6, replace “North America” with “NA”.
3, Line 26, I don’t see any paper you cited in lines 23-25 that investigates 3D GIA. There are some recent 3D GIA papers in North America, e.g., Clark et al. (2019), Kuchar et al. (2019), Li et al. (2018, 2020), Li & Wu (2019).
4, Line 42, for the refs, should be “James et al., 2009a, b” and “Yousefi et al., 2018, 2021”.
5, Line 44, change “constraints” to “constrain”. And “testing different RSL curves” doesn’t make sense, please rephrase.
6, Line 49-50, the sentence “using only …” doesn’t make sentence, please rephrase. It is the viscosity values that constrained by postglacial rebound data only with GIA modelling and viscosity values that revealed by mantle flow modelling have some differences.
7, Line 53, delete the “North America”, you have used the abbreviation “NA” in line 18 for the first time.
8, Line 56, why the two “local” here are italic? Any special indication?
9, Line 104, 106, why keep using italic “localized” “volume” for some usual words?
10, Line 126, delete “a” and add “model” after GIA.
11, Line 133, I doubt you can use “ice thickness datasets” here, better change to “models”; use “ICE-6G_C” to be consistent with the original paper Peltier et al. (2015).
12, Line 134, delete “North America”, as you used “Laurentide” already, use “component”.
13, Line 137, change to “and compare the modeled RSL predictions against RSL data using a misfit function”. Model outputs are predictions, not data. Besides, better to use RSL abbreviation as it is used dozens of times in the paper. Especially you are using NRSL data to calculate the misfit statistics, the old database doesn’t have a clear differentiation of sea level index point and marine/terrestrial limiting data, which will affect your misfit calculation.
14, Line 140, delete “as shown in”.
15, Line 150, move “to take advantage …” to the beginning of the sentence and change to “Due to increasing availability and declining expense of computing resources,”.
16, Line 153 delete “modeled-” if you are referring to observational data, or change “modeled-observation data” to “model prediction” if you are referring to prediction.
17, Fig. 2, there is no label for Fig. b-d.
18, Line 189, add “described” before “above”.
19, Line 192, change “east-west continental divide” to “east-west continental sub-regions”
20, Line 193, need refs after “3D finite element or finite volume GIA modeling”. What does “these authors” refer to?
21, Line 196, change to “We explore the lateral changes in upper mantle viscosity structures between the eastern cratonic and western region with our joint inversion.”
22, Line 222, add “of” after “a series”.
23, Line 223, change “based in” to “based on”.
24, Line 238, delete the “a” before “weaker”.
25, Line 239, change to “with the depth-dependent seismic-to-density parameter scaling profile from Simmons et al. (2010).”
26, Line 241, change “that in total results in” to “which is”.
27, Line 243, change “strong” to “high”, the viscosity values can be high or low, cannot use “strong”.
28, Line 256, change “east and west regions” to “east and west sub-regions”, as you used in line 197.
29, Fig. 4a-d, why the title is “Western shallow lithosphere” and “East cratonic lithosphere”? Where does the “lithosphere” come from? Shouldn’t it be something like “Western sub-region” and “Eastern sub-region”?
30, Line 265, change the sentence “This aligns …” to be more conservative (e.g., might be correspond wtih …), as they show notable differences (410 km, 250-350 km).
31, Line 267, change “gave” to “gives” to be consistent with the rest regarding the grammatical tense.
32, Fig. 5b-c, where is the “GLAC-ICE” from? You use “GLAC1D-NA” in the main text, do you mean the same one? Besides, for Fig. b, please use the same line color but different line style for the ice model results, just like Fig. c
33, Line 299, you refer to Fig. 7 before Fig. 6, please reorder the figures to match where they are referred in the main text.
34, Line 298-301, Fig. 7 shows two results that used the Simmons et al. (2010) seismic conversion profile, why here you compare the solid lines with the dashed lines (using constant conversion parameter of 0.3) in Fig. 5b? The whole sentence doesn’t support your point.
35, Line 310, change “a weighting the” to “a weighting of”, move the “(i.e., α = 1)” in line 314 to the end of “those have no weighting applied”. Line 310-313, the first two sentences tell the same thing, please combine and only keep one sentence. Moreover, add one sentence before “The most obvious impact …” to explain what it means by changing alpha from 1 to 2, double the weight of GIA data in constraining the viscosity? If yes, the viscosity values constrained by higher weight of GIA data should be closer to the results derived by GIA modelling.
36, Line 318, change “with (Simmons et al., 2010) seismic-” to “with Simmons et al. (2010) seismic-”.
37, Fig. 6a-b, why the title is “Western shallow lithosphere” and “East cratonic lithosphere”? Where does the “lithosphere” come from? Shouldn’t be something like “Western sub-region” and “Eastern sub-region”? In the text in line 318 and line 329, you say using Simmons et al. (2010) scaling, while in the figure caption, you say using constant scaling of 0.3, which is correct?
38, Line 322-323, the citation should be “James et al., 2000, 2009a, b”, there is no Fig. 6d but you refer to Fig. 6d. Line 324, change “larger” to “large”, change “regions” to “regional”.
39, Fig. 7, change the shades to be slightly transparent, now cannot distinguish the overlay zone of blue and dark grey shades. The error bars are for 1σ or 2σ? Please make it clear. The deep blue and deep black solid lines are referring to average viscosity profile results? Please add description.
40, Fig. 8, the ICE-6G-Na is not included in the caption. Alternatively, you could shorten the caption as “Similar as Fig. 7, but joint inversion is based on dlnρ/dlnvs = 0.3.” Fig. 8 is not mentioned in the text.
41, Line 338, the second “e.g.,” should be before the citation. Line 339, should be “geodynamic” not “geodynamics”.
42, Line 340-341, a channel needs the viscosity value to be lower or higher than both the upper and lower parts, which is true for Fig. 3a & d for the asthenosphere and transition zone, but not the case for Fig. 3b & e. Please revise the sentence.
43, Line 362-364, add a “,” before “we inferred”, add “, which” after “in the transition zone”. Line 367, for the “moderately weak channel”, do you indicate upper mantle? As upper mantle has a relatively weaker viscosity in Fig. 3b & e, Fig. 4a & b, Fig. 5, and Fig. 6a. If yes, change “weak channel” to “upper mantle”.
44, Line 391, add “sub-region” after “east-west”
45, Line 431, Fig. 9 is for the western NA sub-region, do you mean Fig. 7?
46, Line 432, the resulting upper mantle viscosity of ~1022 Pa s (Fig. 6b) might be related your thin lithosphere around the Hudson Bay. All your RSL data in the eastern NA sub-region are from Hudson Bay, while previous studies (e.g., Wang & Wu, 2006) showed that the lithospheric thickness around Hudson Bay is around 200 km. You can add some explanation here on this point.
47, Line 463-464, for the last sentence “Our results also show …”, I don’t know see the evidence in the main text to show this statement, you can rephase as “Our results imply the potential need to consider lateral variation of mantle seismic …”
48, Fig. A4, all the predictions and RSL data are compressed horizontally/temporally compared with other plots (e.g., Fig.7), please double check.
References
Clark, J., Mitrovica, J. X., & Latychev, K. (2019). Glacial isostatic adjustment in Central Cascadia: Insights from three‐dimensional Earth modeling. Geology, 47(4), 295–298.
Engelhart, S. E., Peltier, W. R., & Horton, B. P. (2011). Holocene relative sea-level changes and glacial isostatic adjustment of the US Atlantic coast. Geology, 39(8), 751-754.
Engelhart, S. E., & Horton, B. P. (2012). Holocene sea level database for the Atlantic coast of the United States. Quaternary Science Reviews, 54, 12–25.
Engelhart, S. E., Vacchi, M., Horton, B. P., Nelson, A. R., & Kopp, R. E. (2015). A sea‐level database for the Pacific coast of Central North America. Quaternary Science Reviews, 113, 78–92.
Kuchar, J., Milne, G., & Latychev, K. (2019). The importance of lateral Earth structure for North American glacial isostatic adjustment. Earth and Planetary Science Letters, 512, 236–245.
Li, T., & Wu, P. (2018). Laterally heterogeneous lithosphere, asthenosphere and sub‐lithospheric properties under Laurentia and Fennoscandia from glacial isostatic adjustment. Geophysical Journal International, 216(3), 1633–1647.
Li, T., Wu, P., Steffen, H., & Wang, H. (2018). In search of laterally heterogeneous viscosity models of glacial isostatic adjustment with the ICE‐6G_C global ice history model. Geophysical Journal International, 214(2), 1191–1205.Li, T., Wu, P., Wang, H., Steffen, H., Khan, N.S., Engelhart, S.E., Vacchi, M., Shaw, T.A., Peltier, W.R. and Horton, B.P., 2020. Uncertainties of glacial isostatic adjustment model predictions in North America associated with 3D structure. Geophysical Research Letters, 47(10), p.e2020GL087944.
Vacchi, M., Engelhart, S. E., Nikitina, D., Ashe, E. L., Peltier, W. R., Roy, K., et al. (2018). Postglacial relative sea‐level histories along the eastern Canadian coastline. Quaternary Science Reviews, 201, 124–146.
Wang, H., & Wu, P. (2006). Effects of lateral variations in lithospheric thickness and mantle viscosity on glacially induced relative sea levels and long wavelength gravity field in a spherical, self-gravitating Maxwell Earth. Earth and Planetary Science Letters, 249(3-4), 368-383.
Citation: https://doi.org/10.5194/se-2021-151-CC1 -
RC1: 'Comment on se-2021-151', Wouter van der Wal, 23 Mar 2022
The long-wavelength geoid or gravity anomaly in North America is thought to reflect mantle density anomalies and associated flow, and glacial isostatic adjustment. The paper uses the gravity anomaly together with RSL data in North America in a joint inversion for the radial viscosity profile, showing large sensitivity to the seismic to density scaling. New in the paper is a regional inversion based on representations of the kernels by Slepian functions. The inversion is done separately for a western region and eastern region in North America. The obtained viscosity profiles show a weak asthenosphere in the west and and large viscosity jump in the eastern region, in agreement with the expected first order difference in mantle structure. A constraint on lateral viscosity variation for North America is welcome and would have implications for geodynamic models including glacial isostatic adjustment and tectonics. The method and figures re clear. However there are a few main issues that should be addressed, the first of which is likely to impact the results and hence will require a major revision of the paper. In addition, the text requires additional discussion on the potential impact of assumptions, and references on some aspects of the paper are currently missing, see the specific comments. There are several incomplete sentences and typos, please see the annotated pdf where some of the textual issues are pointed out. The references mentioned in the comments can be found at the end of the review.
best regards,
Wouter van der WalMain issues
The long-wavelength gravity field is not only caused by GIA and mantle convection but also anomalies in the crustal thickness and density anomalies in the lithosphere. Correcting for a crustal or lithospheric signal is done in recent papers that fit gravity anomaly data in North America (Kaban et al. 2014; Metivier et al. 2016, section 3.2; Reusen et al. 2020) and it is standard in global studies also when long-wavelength signal is studied (e.g. Wen and Anderson 1997). In North America the crustal signal contributes tens of mGal up to spherical harmonic degree 15 (Reusen et al. 2020 figure 6). This is especially significant in the western region where the gravity anomaly itself is not as large. Therefore the gravity anomaly needs to be corrected for variations in crustal thickness and density anomalies in the lithosphere before fitting the GIA and mantle convection model. There is the additional complication that crustal thickness variations will contain part of the GIA signal and isostasy can not be assumed in the region (Reusen et al. 2020).
The paper inverts a long-wavelength signal with a regional model. However, most of the variance in the gravity field is coming from degree 2 and 3 which are caused by very deep sources (e.g. Liu and Zhong 2016), which means the gravity anomaly will also be sensitive to anomalies in a much wider region surrounding the region of interest (I could not immediately find references that shows kernels as a function of horizontal distance). It is not clear how accurate the regional inversion is when signal outside the region of interest is not included, but in my opinion this should be demonstrated in the paper which proposes regional inversion.This can be investigated for example, by fitting only the higher degree signal, or by varying the size of the region of interest .
Referencing: The non-uniqueness in the inversion is mentioned (line 396) but not discussed. It is investigated for GIA by Paulson et al. (2007) and for mantle convection by Thoroval and Richards (1997). The effect of lateral viscosity variations on dynamic topography or the geoid is discussed in e.g. Ghosh et al. (2010), Cadek and Fleitout 2003. Results of viscosity inferences can be compared with other inversions of viscosity profiles for North America (Wolf et al. 2006; Kuchar et al 2019; Metivier et al. 2016; Reusen et al. 2020; Mao and Zhong 2021), at least the ones that also use gravity data.
In section 3.1 the gravity anomaly is fit with only the geoid kernels, resulting in a variance reduction of around 40%. The results are very different from those of the joined inversion. Since the added value of the manuscript is in doing a joined inversion, it would help the flow of the paper to remove section 3.1 or place it in an appendix.
Specific comments
Title: strength of a material usually refers to yield stress. I suggest to change the title to something like the following: Regional gravity constraints for North America reveal upper mantle viscosity differences across the continent.
49: It is stated that Mitrovica and Forte showed considerable potential, but it is not clear what is the gap in the literature that you will address. The text from line 225 onwards is useful to add in the introduction.
82: subset: this seems important information that is not discussed further in the paper. How is the choice made which functions are included and how could that affect the results?
128: scaling with the 10^21 Pa s value. This effect of this choice should be discussed.
133: The paper of Tarasov et al. (2012) does not present a single ice model as far as I know.
233: The scaling is a crucial parameters and constraints on them have implications for other models. I suggest to better introduce the choices made for the scalings. Are the chosen values common in the literature? Is it expected that they hold for North America?
299: ICE-6G is created by fitting RSL data, therefore a good fit with RSL data is to be expected. Is the fit obtained here better than the fit of the original model?
325: The fit with RSL data is poor as you also note in line 434. Looking at figure 9 it is unlikely that that is due to missed tectonic signal. It is likely that the crustal signal plays a large role in explaining the gravity anomaly; this should be quantified.
334: That is surprising given that most of the gravity anomaly signal is in the eastern region. Can you speculate why the joined inversion is dominated by the solution for the western region?
References mentioned in the review comments
- Äadek, O. and Fleitout, L., 2006. Effect of lateral viscosity variations in the core-mantle boundary region on predictions of the long-wavelength geoid. Studia Geophysica et Geodaetica, 50(2), pp.217-232.
- Ghosh, A., Becker, T.W. and Zhong, S.J., 2010. Effects of lateral viscosity variations on the geoid. Geophysical Research Letters, 37(1).
- Kaban, M.K., Tesauro, M., Mooney, W.D. and Cloetingh, S.A., 2014. Density, temperature, and composition of the North American lithosphere—New insights from a joint analysis of seismic, gravity, and mineral physics data: 1. Density structure of the crust and upper mantle. Geochemistry, Geophysics, Geosystems, 15(12), pp.4781-4807.
- Kuchar, J., Milne, G. and Latychev, K., 2019. The importance of lateral Earth structure for North American glacial isostatic adjustment. Earth and Planetary Science Letters, 512, pp.236-245.
- Mao, W. and Zhong, S., 2021. Constraints on mantle viscosity from intermediateâwavelength geoid anomalies in mantle convection models with plate motion history. Journal of Geophysical Research: Solid Earth, 126(4), p.e2020JB021561.
- Métivier, L., Caron, L., Greff-Lefftz, M., Pajot-Métivier, G., Fleitout, L. and Rouby, H., 2016. Evidence for postglacial signatures in gravity gradients: A clue in lower mantle viscosity. Earth and Planetary Science Letters, 452, pp.146-156.
- Paulson, A., Zhong, S. and Wahr, J., 2007. Limitations on the inversion for mantle viscosity from postglacial rebound. Geophysical Journal International, 168(3), pp.1195-1209.
- Sella, G.F., Stein, S., Dixon, T.H., Craymer, M., James, T.S., Mazzotti, S. and Dokka, R.K., 2007. Observation of glacial isostatic adjustment in “stable” North America with GPS. Geophysical Research Letters, 34(2).
- Thoraval, C. and Richards, M.A., 1997. The geoid constraint in global geodynamics: viscosity structure, mantle heterogeneity models and boundary conditions. Geophysical Journal International, 131(1), pp.1-8.
- Wen, L. and Anderson, D.L., 1997. Layered mantle convection: A model for geoid and topography. Earth and Planetary Science Letters, 146(3-4), pp.367-377.
- Wolf, D., Klemann, V., Wünsch, J. and Zhang, F.P., 2006. A reanalysis and reinterpretation of geodetic and geological evidence of glacial-isostatic adjustment in the Churchill region, Hudson Bay. Surveys in Geophysics, 27(1), pp.19-61.
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RC2: 'Comment on se-2021-151', Anonymous Referee #2, 25 Mar 2022
In this manuscript, Osei Tutu and Harig present a series of calculations to constrain the local radial viscosity profiles beneath the North American (NA) continent, using localized free-air gravity kernels and GIA models for the inversion. Due to the strong tectonic contrast between the tectonically active western cordilleran regions and the eastern cratonic regions across the whole NA continent, the authors also zoom into the eastern and the western part of the continent separately using the same joint inversion technique. The inversion results over the whole NA continent show that the density-to-velocity scaling used (constant at 0.3 or depth-dependent following Simmons et al., 2010) has a strong influence on the resulting radial viscosity profiles. The focus on either the western continent plus Pacific margin or the eastern cratonic continent leads to more robust results. The western localized inversion shows a weak and shallow asthenosphere layer, while the eastern localized inversion shows a strong upper mantle and a weak transition zone. Beneath the western NA continent + Pacific margin, the results of the weak asthenosphere/upper mantle in this manuscript are different from the previous postglacial and mantle viscosity studies (e.g., James et al., 2000, 2009a, b) by ~10 orders of magnitude in light of the predicted uplift rate, as authors mention in the text. Beneath the eastern part of the NA continent, the authors attribute the strong upper mantle to the existence of the stiff cratonic root and the weak transition zone to the hot mantle material and volatiles/fluids associated with the underlying descending Farallon slab. Although this study seems systematic and robust, I have three significant concerns/suggestions.
First, after applying a windowing function to the global GRACE gravity data (Slepian function spectral localization technique presented by Wieczorek and Simons,
2005), the authors use two spherical harmonic filters from degree 2-10 and degree 2-15 respectively to perform the regional inversion using the localized geoid kernels. They found there is no significant difference of the results between the two filters in the regional mantle flow modeling below the NA continent. However, as agreed within the geodynamic community, the long-wavelength geoid (2-8 degrees) is dominated by broader mantle density anomalies including LLSVPs and global subducting slabs (e.g., Liu and Zhong, 2016; Ghosh et al., 2010; Mao and Zhong, 2021). Even if the GRACE gravity data is localized into the NA region using Slepian function, part of the local gravity/geoid in the long-wavelength window may still come from other broader-scale mantle density heterogeneities outside of the NA cap. Therefore, I think the author should filter the gravity signal over the NA continent into a shorter-wavelength window. Considering the scale of the NA continent, maybe spherical harmonic degree higher than 8 would be accepted (e.g., 8 – 20 or 8 - 25). The author can try different window in shorter wavelength. Current window starting from degree 2 and 3 is definitely in too long wavelength for the NA continent. The author could also try decomposing the spatial GRACE gravity field over the NA continent or Hudson Bay to find out the appropriate wavelength for the inversion. In addition, based on the findings in Wieczorek and Simons 2005 (Page 670), after windowing a global field by a taper of bandwidth L, the resultant coefficients are only reliable up to Lf – L (Lf is the maximun degree of the field data), and the first L coefficients exhibit large uncertainties. This study seems using a taper of bandwidth L =15 (from the caption of Fig. 1?), so only the GRACE data higher than spherical harmonic degree 15 can be robustly used for this kind of inversion. The authors should also explicitly denote in the manuscript on the bandwidth taper used in this study and perform the corresponding inversion consistent with that bandwidth. I strongly suggest that the authors perform the whole calculation again using the GRACE data in shorter-wavelengths, which can provide more robust constraint on the viscosity profiles below the NA continent. In addition, the variance reductions (in the longer-wavelengths considered here) of the results from only regional mantle modeling seems poorly fitting the observed GRACE gravity data (line #252 - #254).
Second, the NA plate includes a contrasting tectonic setting between the eastern and western half. The eastern cratonic root is reflected by strong high seismic velocity and the western asthenosphere is featured with strong slow seismic velocity. Thus, in the mantle flow modeling focusing on the North American plate, the lateral perturbations of the velocity-to-density scaling should be incorporated into the density/buoyancy field, as the authors also mention in the Discussion section. For example, Ghosh et al., 2013 use zero velocity-to-density scaling for the high velocity anomalies of the cratonic root. Forte et al., 2010 uses two different density-to-velocity scaling profiles for the cratonic root and the ambient upper mantle. The buoyancy of the cratonic keel is still in debate, but for the viscosity inversion, the authors at least should try a simple isopycnic hypothesis (Jordan, 1978) that assuming a neutrally-buoyant cratonic keel down to 250 or 300 km. In addition to the cratonic keel, the lateral perturbation of the velocity-to-density scaling for the slow seismic velocity below the western NA should also be considered. Below the western NA and the neighboring Pacific margin, significant fraction of the partial melt may be present, which explains the very slow seismic velocity in this region. Liu and King, 2022 tested the lateral perturbation of the velocity-to-density scaling in this local area associated with the existence of the partial melt and found that this considerably influences the mantle flow patterns beneath the NA continent. They found that the velocity-to-density scaling for the slow seismic velocity in this area should be as small as 0.05 or below. If one only considers a constant velocity-to-density scaling of 0.3 or a 1D velocity-to-density scaling from Simmons et al., 2010 as did in this study, the buoyancy structures below the NA continent cannot be correctly represented. This substantial inaccuracy in the density structures may significantly bias the recovered viscosity profiles as the definition of the geoid kernels (eq. 4b in the manuscript). I think for this regional study specifically focusing on the NA continent, considering the lateral perturbations of the density-to-velocity scaling from the chemically-depleted cratonic keel below the eastern NA and the present of the partial melt in the western NA is crucial to correctly estimate the buoyancy/density structures in the mantle flow modeling. The seismic tomography model from French and Romanowicz, 2015 is used in this study, so the authors should be able to define the geometry of these two local tectonic areas based on the seismic velocity magnitudes in the tomography model. Then the velocity-to-density scaling for each area could be adjusted. I guess the resulting viscosity profiles from this new inversion may be improved to fit GRACE gravity data and RSL data better. The variance reductions of the results from only regional mantle modeling seems poorly fitting the observed GRACE gravity data (line #252 - #254).
Third, due to the strong lateral viscosity variations (LVVs) across the NA continent between the weak western asthenosphere and the strong eastern cratonic keel, the effects of the mode-coupling (for example, see Stewart, 1992) on the geoid/free-air gravity between different spherical harmonic wavelengths may be considerably strong below the NA continent. For example, in this scenario, the geoid/gravity signal in the shorter-wavelength may include a part coming from the contribution of the buoyancy structures in the long-wavelengths through mode-coupling. I noticed that the current inversion based on the geoid kernel (eq. 4b in the manuscript) can only constrain 1D radial viscosity profile. The 3D viscosity structures cannot be incorporated into this kind of inversion. It seems that the author does not mention how they calculate regional mantle flow. I think the author should add another paragraph in Method section to describe the technical detail of their mantle flow modeling, including initial condition set-up, boundary condition, solver (spectral or spatial?), etc. If it’s a code that can handle LVVs, such as CitcomCU, CitcomS, or ASPECT, I suggest the author run another numerical model incorporating LVVs (strong cratonic keel and weak western asthenosphere, temperature-dependence, etc.) for the whole NA region, and then compare the predicted geoid/gravity in the appropriate wavelengths corresponding to the NA continent (see my previous point) with the original model in the manuscript that does not include LVVs. If the authors use a spectral code (propagator matrix technique) that cannot handle LVVs, I suggest they should add a paragraph in the manuscript to discuss the potential bias caused by the mode-coupling due to the strong LVVs. Previous global mantle flow modeling has incorporated the LVVs from strong cratonic root, weak plate boundaries, and temperature-dependent viscosity (e.g., Zhong 2001; Becker 2006; Miller and Becker, 2012; Becker 2017).
Some specific minor points that would help make the manuscript clearer and better for the readers:
- The authors should clarify whether self-gravity is included in the geoid calculation. As Zhong et al., 2008 clearly show, self-gravity has a considerable influence on the long-wavelength geoid. Further considering the effects of mode-coupling in LVVs, the effects of self-gravity may be even larger than expected for the Earth. The geoid predicted from the mantle flow models with the effects of self-gravity is also not comparable with the geoid results without the effects of self-gravity.
- Non-hydrostatic free-air gravity. This is mentioned at line #222. I think the authors should make this point more explicitly expressed. How the hydrostatic flattening is removed? Following which paper? Nakiboglu, 1982 and Chambat et al., 2010 are two references commonly used in the community to perform the correction on the gravity date associated with hydrostatic flattening (e.g., Ghosh et al., 2013).
- Slepian functions. In Method section, although the authors describe the utilization and advantages of Slepian functions as a localization technique in spectral field, the authors should more explicitly describe which kinds of Slepian functions are used in this study around line #105, such as which degree the bandlimited space-concentrated tapers up to and how they are linked to the geoid kernels. In addition, in Fig. 1, what does functions 1, 2, 3, 4, 9 mean? These should be explained in detail as some new texts and equations.
- Letter symbols of subfigures. I notice that the authors put letter symbols for some subfigures but the letter symbols for other subfigures are missing, such as Fig. 2, Fig. 7, and Fig. 8. The letter symbols (a, b, c, d, ….) for each subfigure should be appended for the convenience of the readers.
- Be careful of the explanation of the abbreviation, such as RSL (Relative Sea Level?). The full name should be mentioned first with the abbreviation in the parathesis, and then the abbreviation can be used elsewhere.
- There are some misused phrases in the manuscript. Line #29: Fee-air ---> Free-air;
- Line #66: patial sphere ---> partial sphere;
- Line #193: basin specific multiple 1D Earth models ---> basic specific multiple 1D Earth models;
- Line #238: a relatively a weaker ---> a relatively weaker;
- Line #243 and elsewhere: strong viscosity values ---> strong layer or large viscosity values;
- Line #264: strong viscosity interface. Be careful of the word “interface”. Interface means “boundaries” instead of “layers”. Strong interface is a vague expression. You should either use strong layer or an interface from a strong layer to a weak layer. I notice this problem appears in many places (e.g., line #294, line #347, line #451).
- Line #330: “both” should be deleted?
- Line #333 - #334: why does the western upper mantle dominates? Note that there is also no cratonic keel in the western NA continent.
- Line # 358 – Line # 360: This sentence needs more explanations on how it is more in line with the findings by Hager and Richards, 1989.
- Line #388: serving a a first order ---> serving as a first order.
- Line #398: what are clear geophysical arguments? This place needs a couple of citations.
- Line #424: missing parenthesis for the citation.
- Line #441: conclusion ---> Conclusion
- Line #442: based, created: this is a problematic vague expression.
- Line #450: inversion the entire ---> inversion of the entire.
- Line #458 - #459: This sentence needs to be rearranged.
- Line #460 - #463: This long sentence is vague and needs to be explained in detail.
- A1: “Alpha indicates the eigenfunction number and rank”. Can you explain this definition clearer? What does the eigenfunction number and rank mean?
- Line #323: There is no Fig. 6d.
- In Reference section, the reference papers should be listed in alphabetic order.
Overall, I think this manuscript could potentially become an appropriate paper including significant contributions to the exploration of the lateral strength variations below the NA continent if the authors consider following my three major suggestions above to improve the robustness and quality of this study. The manuscript also needs more careful and clear writing/organization. Considering that ~8 weeks turnaround may be not enough for re-performing the calculations, I suggest the paper should be returned for major revision to the authors. If the authors can address my concerns above, they should be encouraged and welcomed to submit the substantially-revised manuscript back to SE. I will be happy to see the revised manuscript again.
References:
- Liu, Xi, and Shijie Zhong. "Constraining mantle viscosity structure for a thermochemical mantle using the geoid observation." Geochemistry, Geophysics, Geosystems3 (2016): 895-913.
- Ghosh, A., T. W. Becker, and S. J. Zhong. "Effects of lateral viscosity variations on the geoid." Geophysical Research Letters1 (2010).
- Mao, Wei, and Shijie Zhong. "Constraints on mantle viscosity from intermediateâwavelength geoid anomalies in mantle convection models with plate motion history." Journal of Geophysical Research: Solid Earth4 (2021): e2020JB021561.
- Ghosh, A., T. W. Becker, and E. D. Humphreys. "Dynamics of the North American continent." Geophysical Journal International 194.2 (2013): 651-669.
- Forte, A. M., et al. "Deep-mantle contributions to the surface dynamics of the North American continent." Tectonophysics481.1-4 (2010): 3-15.
- Liu, Shangxin, and Scott D. King. "Dynamics of the North American plate: largeâscale driving mechanism from farâfield slabs and the interpretation of shallow negative seismic anomalies." Geochemistry, Geophysics, Geosystems (2022): e2021GC009808.
- Jordan, Thomas H. "Composition and development of the continental tectosphere." Nature 274.5671 (1978): 544-548.
- Stewart, Cheryl A. "Thermal convection in the Earth's mantle: Mode coupling induced by temperatureâdependent viscosity in a threeâdimensional spherical shell." Geophysical research letters 19.4 (1992): 337-340.
- Zhong, Shijie. "Role of oceanâcontinent contrast and continental keels on plate motion, net rotation of lithosphere, and the geoid." Journal of Geophysical Research: Solid EarthB1 (2001): 703-712.
- Becker, Thorsten W. "On the effect of temperature and strain-rate dependent viscosity on global mantle flow, net rotation, and plate-driving forces." Geophysical Journal International2 (2006): 943-957.
- Miller, Meghan S., and Thorsten W. Becker. "Mantle flow deflected by interactions between subducted slabs and cratonic keels." Nature Geoscience10 (2012): 726-730.
- Becker, Thorsten W. "Superweak asthenosphere in light of upper mantle seismic anisotropy." Geochemistry, Geophysics, Geosystems5 (2017): 1986-2003.
- Zhong, Shijie, et al. "A benchmark study on mantle convection in a 3âD spherical shell using CitcomS." Geochemistry, Geophysics, Geosystems 9.10 (2008).
- Nakiboglu, S. M. "Hydrostatic theory of the Earth and its mechanical implications." Physics of the Earth and Planetary Interiors 28.4 (1982): 302-311.
- Chambat, F., Y. Ricard, and Bernard Valette. "Flattening of the Earth: further from hydrostaticity than previously estimated." Geophysical Journal International 183.2 (2010): 727-732.
Citation: https://doi.org/10.5194/se-2021-151-RC2
Status: closed
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CC1: 'Comment on se-2021-151', Tanghua Li, 22 Feb 2022
Dear Editor,
The manuscript “Regional mantle viscosity constraints for North America reveal upper mantle strength differences across the continent” by Osei Tutu & Harig investigates the regional mantle viscosity values for North America using a local Bayesian joint inversion of mantle flow and GIA models. They use the region free-air gravity data and RSL data. The topic is interesting and worth studying since North America is a key region to deduce the mantle viscosity structure and to better understand the solid Earth behavior due to the loading and uploading of the Laurentide ice sheet.
The figures are of good quality and easy to read. However, several aspects of this manuscript need to be improved/addressed before it can be considered for publication in Solid Earth.
In this study, the RSL database from Tushingham and Peltier (1991) is used, which is pretty old and outdated. There are more recent high-quality standardized RSL databases for eastern Canadian coast (Vacchi et al., 2018), including Hudson Bay, U.S. Atlantic coast (Englehart & Horton, 2012), Pacific coast (Engelhart et al., 2015), and they are all freely available. Why don’t you use the recent ones? The importance of the standardization of RSL data has been demonstrated in many studies, especially for the differentiation of sea-level index points (SLIPs) and limiting data, which are not considered in Tushingham and Peltier (1991). Moreover, your eastern region () includes U.S. Atlantic coast, but no RSL data from the Atlantic coast of NA, which have been shown to be vital to constrain the viscosity and may result in totally different upper mantle viscosity values as that from Hudson Bay (e.g., Engelhart et al., 2011).
Why restrict ice melting history and RSL data to < 10 kyr BP (line 142)? What’s the significance behind this? Does that mean you only have a deglaciation stage from 10 ka BP till present but no glaciation stage in your GIA ice model? You cannot only use ice melting < 10 kyr BP to even only study the RSL < 10 ka BP. Because the solid Earth response to loading/glaciation and unloading/deglaciation events is a delayed process, what happened before 10 kyr BP (e.g., the LGM) definitely significantly affect the RSL <10kyr BP.
Line 244-245, “differences in our inferred 1D regional …” and as you stated in the abstract, this means the uncertainty of inferred 1Dregional viscosity due to the uncertainty of scaling factor is significantly dramatic, then comes the question about the validity and accuracy of this approach to infer the 1D regional viscosity. In that case, you need to assess the uncertainty of the inferred viscosity.
Whole North America, eastern sub-region, western sub-region, you need to follow the same order through the whole paper for each section. Besides, suggest you consistently using “sub-region” when refer to western and eastern North America region, as you used in line 197. Please consistently use the same expression for the seismic velocity to density scaling parameter.
The usage of abbreviation and full name in this manuscript is really in chaos, like relative sea level (e.g., line 137, 163, 403) and RSL (e.g., line 44, 139, 299), North America (e.g., 83, 191, 256) and NA (e.g., line 18, 55, 98), glacial isostatic adjustment (e.g., line 189, 416) and GIA (e.g., line 23, 36) … Also misuse of italic for some common words, like localized in line 104, volume in line 106,
Citation formats in the main text need to be double checked.
Detailed comments:
1, Replace the word “strength” in the title, maybe by “viscosity”, as “strength” only appears once in the title and then never been used any more in the main text.
2, Line 6, replace “North America” with “NA”.
3, Line 26, I don’t see any paper you cited in lines 23-25 that investigates 3D GIA. There are some recent 3D GIA papers in North America, e.g., Clark et al. (2019), Kuchar et al. (2019), Li et al. (2018, 2020), Li & Wu (2019).
4, Line 42, for the refs, should be “James et al., 2009a, b” and “Yousefi et al., 2018, 2021”.
5, Line 44, change “constraints” to “constrain”. And “testing different RSL curves” doesn’t make sense, please rephrase.
6, Line 49-50, the sentence “using only …” doesn’t make sentence, please rephrase. It is the viscosity values that constrained by postglacial rebound data only with GIA modelling and viscosity values that revealed by mantle flow modelling have some differences.
7, Line 53, delete the “North America”, you have used the abbreviation “NA” in line 18 for the first time.
8, Line 56, why the two “local” here are italic? Any special indication?
9, Line 104, 106, why keep using italic “localized” “volume” for some usual words?
10, Line 126, delete “a” and add “model” after GIA.
11, Line 133, I doubt you can use “ice thickness datasets” here, better change to “models”; use “ICE-6G_C” to be consistent with the original paper Peltier et al. (2015).
12, Line 134, delete “North America”, as you used “Laurentide” already, use “component”.
13, Line 137, change to “and compare the modeled RSL predictions against RSL data using a misfit function”. Model outputs are predictions, not data. Besides, better to use RSL abbreviation as it is used dozens of times in the paper. Especially you are using NRSL data to calculate the misfit statistics, the old database doesn’t have a clear differentiation of sea level index point and marine/terrestrial limiting data, which will affect your misfit calculation.
14, Line 140, delete “as shown in”.
15, Line 150, move “to take advantage …” to the beginning of the sentence and change to “Due to increasing availability and declining expense of computing resources,”.
16, Line 153 delete “modeled-” if you are referring to observational data, or change “modeled-observation data” to “model prediction” if you are referring to prediction.
17, Fig. 2, there is no label for Fig. b-d.
18, Line 189, add “described” before “above”.
19, Line 192, change “east-west continental divide” to “east-west continental sub-regions”
20, Line 193, need refs after “3D finite element or finite volume GIA modeling”. What does “these authors” refer to?
21, Line 196, change to “We explore the lateral changes in upper mantle viscosity structures between the eastern cratonic and western region with our joint inversion.”
22, Line 222, add “of” after “a series”.
23, Line 223, change “based in” to “based on”.
24, Line 238, delete the “a” before “weaker”.
25, Line 239, change to “with the depth-dependent seismic-to-density parameter scaling profile from Simmons et al. (2010).”
26, Line 241, change “that in total results in” to “which is”.
27, Line 243, change “strong” to “high”, the viscosity values can be high or low, cannot use “strong”.
28, Line 256, change “east and west regions” to “east and west sub-regions”, as you used in line 197.
29, Fig. 4a-d, why the title is “Western shallow lithosphere” and “East cratonic lithosphere”? Where does the “lithosphere” come from? Shouldn’t it be something like “Western sub-region” and “Eastern sub-region”?
30, Line 265, change the sentence “This aligns …” to be more conservative (e.g., might be correspond wtih …), as they show notable differences (410 km, 250-350 km).
31, Line 267, change “gave” to “gives” to be consistent with the rest regarding the grammatical tense.
32, Fig. 5b-c, where is the “GLAC-ICE” from? You use “GLAC1D-NA” in the main text, do you mean the same one? Besides, for Fig. b, please use the same line color but different line style for the ice model results, just like Fig. c
33, Line 299, you refer to Fig. 7 before Fig. 6, please reorder the figures to match where they are referred in the main text.
34, Line 298-301, Fig. 7 shows two results that used the Simmons et al. (2010) seismic conversion profile, why here you compare the solid lines with the dashed lines (using constant conversion parameter of 0.3) in Fig. 5b? The whole sentence doesn’t support your point.
35, Line 310, change “a weighting the” to “a weighting of”, move the “(i.e., α = 1)” in line 314 to the end of “those have no weighting applied”. Line 310-313, the first two sentences tell the same thing, please combine and only keep one sentence. Moreover, add one sentence before “The most obvious impact …” to explain what it means by changing alpha from 1 to 2, double the weight of GIA data in constraining the viscosity? If yes, the viscosity values constrained by higher weight of GIA data should be closer to the results derived by GIA modelling.
36, Line 318, change “with (Simmons et al., 2010) seismic-” to “with Simmons et al. (2010) seismic-”.
37, Fig. 6a-b, why the title is “Western shallow lithosphere” and “East cratonic lithosphere”? Where does the “lithosphere” come from? Shouldn’t be something like “Western sub-region” and “Eastern sub-region”? In the text in line 318 and line 329, you say using Simmons et al. (2010) scaling, while in the figure caption, you say using constant scaling of 0.3, which is correct?
38, Line 322-323, the citation should be “James et al., 2000, 2009a, b”, there is no Fig. 6d but you refer to Fig. 6d. Line 324, change “larger” to “large”, change “regions” to “regional”.
39, Fig. 7, change the shades to be slightly transparent, now cannot distinguish the overlay zone of blue and dark grey shades. The error bars are for 1σ or 2σ? Please make it clear. The deep blue and deep black solid lines are referring to average viscosity profile results? Please add description.
40, Fig. 8, the ICE-6G-Na is not included in the caption. Alternatively, you could shorten the caption as “Similar as Fig. 7, but joint inversion is based on dlnρ/dlnvs = 0.3.” Fig. 8 is not mentioned in the text.
41, Line 338, the second “e.g.,” should be before the citation. Line 339, should be “geodynamic” not “geodynamics”.
42, Line 340-341, a channel needs the viscosity value to be lower or higher than both the upper and lower parts, which is true for Fig. 3a & d for the asthenosphere and transition zone, but not the case for Fig. 3b & e. Please revise the sentence.
43, Line 362-364, add a “,” before “we inferred”, add “, which” after “in the transition zone”. Line 367, for the “moderately weak channel”, do you indicate upper mantle? As upper mantle has a relatively weaker viscosity in Fig. 3b & e, Fig. 4a & b, Fig. 5, and Fig. 6a. If yes, change “weak channel” to “upper mantle”.
44, Line 391, add “sub-region” after “east-west”
45, Line 431, Fig. 9 is for the western NA sub-region, do you mean Fig. 7?
46, Line 432, the resulting upper mantle viscosity of ~1022 Pa s (Fig. 6b) might be related your thin lithosphere around the Hudson Bay. All your RSL data in the eastern NA sub-region are from Hudson Bay, while previous studies (e.g., Wang & Wu, 2006) showed that the lithospheric thickness around Hudson Bay is around 200 km. You can add some explanation here on this point.
47, Line 463-464, for the last sentence “Our results also show …”, I don’t know see the evidence in the main text to show this statement, you can rephase as “Our results imply the potential need to consider lateral variation of mantle seismic …”
48, Fig. A4, all the predictions and RSL data are compressed horizontally/temporally compared with other plots (e.g., Fig.7), please double check.
References
Clark, J., Mitrovica, J. X., & Latychev, K. (2019). Glacial isostatic adjustment in Central Cascadia: Insights from three‐dimensional Earth modeling. Geology, 47(4), 295–298.
Engelhart, S. E., Peltier, W. R., & Horton, B. P. (2011). Holocene relative sea-level changes and glacial isostatic adjustment of the US Atlantic coast. Geology, 39(8), 751-754.
Engelhart, S. E., & Horton, B. P. (2012). Holocene sea level database for the Atlantic coast of the United States. Quaternary Science Reviews, 54, 12–25.
Engelhart, S. E., Vacchi, M., Horton, B. P., Nelson, A. R., & Kopp, R. E. (2015). A sea‐level database for the Pacific coast of Central North America. Quaternary Science Reviews, 113, 78–92.
Kuchar, J., Milne, G., & Latychev, K. (2019). The importance of lateral Earth structure for North American glacial isostatic adjustment. Earth and Planetary Science Letters, 512, 236–245.
Li, T., & Wu, P. (2018). Laterally heterogeneous lithosphere, asthenosphere and sub‐lithospheric properties under Laurentia and Fennoscandia from glacial isostatic adjustment. Geophysical Journal International, 216(3), 1633–1647.
Li, T., Wu, P., Steffen, H., & Wang, H. (2018). In search of laterally heterogeneous viscosity models of glacial isostatic adjustment with the ICE‐6G_C global ice history model. Geophysical Journal International, 214(2), 1191–1205.Li, T., Wu, P., Wang, H., Steffen, H., Khan, N.S., Engelhart, S.E., Vacchi, M., Shaw, T.A., Peltier, W.R. and Horton, B.P., 2020. Uncertainties of glacial isostatic adjustment model predictions in North America associated with 3D structure. Geophysical Research Letters, 47(10), p.e2020GL087944.
Vacchi, M., Engelhart, S. E., Nikitina, D., Ashe, E. L., Peltier, W. R., Roy, K., et al. (2018). Postglacial relative sea‐level histories along the eastern Canadian coastline. Quaternary Science Reviews, 201, 124–146.
Wang, H., & Wu, P. (2006). Effects of lateral variations in lithospheric thickness and mantle viscosity on glacially induced relative sea levels and long wavelength gravity field in a spherical, self-gravitating Maxwell Earth. Earth and Planetary Science Letters, 249(3-4), 368-383.
Citation: https://doi.org/10.5194/se-2021-151-CC1 -
RC1: 'Comment on se-2021-151', Wouter van der Wal, 23 Mar 2022
The long-wavelength geoid or gravity anomaly in North America is thought to reflect mantle density anomalies and associated flow, and glacial isostatic adjustment. The paper uses the gravity anomaly together with RSL data in North America in a joint inversion for the radial viscosity profile, showing large sensitivity to the seismic to density scaling. New in the paper is a regional inversion based on representations of the kernels by Slepian functions. The inversion is done separately for a western region and eastern region in North America. The obtained viscosity profiles show a weak asthenosphere in the west and and large viscosity jump in the eastern region, in agreement with the expected first order difference in mantle structure. A constraint on lateral viscosity variation for North America is welcome and would have implications for geodynamic models including glacial isostatic adjustment and tectonics. The method and figures re clear. However there are a few main issues that should be addressed, the first of which is likely to impact the results and hence will require a major revision of the paper. In addition, the text requires additional discussion on the potential impact of assumptions, and references on some aspects of the paper are currently missing, see the specific comments. There are several incomplete sentences and typos, please see the annotated pdf where some of the textual issues are pointed out. The references mentioned in the comments can be found at the end of the review.
best regards,
Wouter van der WalMain issues
The long-wavelength gravity field is not only caused by GIA and mantle convection but also anomalies in the crustal thickness and density anomalies in the lithosphere. Correcting for a crustal or lithospheric signal is done in recent papers that fit gravity anomaly data in North America (Kaban et al. 2014; Metivier et al. 2016, section 3.2; Reusen et al. 2020) and it is standard in global studies also when long-wavelength signal is studied (e.g. Wen and Anderson 1997). In North America the crustal signal contributes tens of mGal up to spherical harmonic degree 15 (Reusen et al. 2020 figure 6). This is especially significant in the western region where the gravity anomaly itself is not as large. Therefore the gravity anomaly needs to be corrected for variations in crustal thickness and density anomalies in the lithosphere before fitting the GIA and mantle convection model. There is the additional complication that crustal thickness variations will contain part of the GIA signal and isostasy can not be assumed in the region (Reusen et al. 2020).
The paper inverts a long-wavelength signal with a regional model. However, most of the variance in the gravity field is coming from degree 2 and 3 which are caused by very deep sources (e.g. Liu and Zhong 2016), which means the gravity anomaly will also be sensitive to anomalies in a much wider region surrounding the region of interest (I could not immediately find references that shows kernels as a function of horizontal distance). It is not clear how accurate the regional inversion is when signal outside the region of interest is not included, but in my opinion this should be demonstrated in the paper which proposes regional inversion.This can be investigated for example, by fitting only the higher degree signal, or by varying the size of the region of interest .
Referencing: The non-uniqueness in the inversion is mentioned (line 396) but not discussed. It is investigated for GIA by Paulson et al. (2007) and for mantle convection by Thoroval and Richards (1997). The effect of lateral viscosity variations on dynamic topography or the geoid is discussed in e.g. Ghosh et al. (2010), Cadek and Fleitout 2003. Results of viscosity inferences can be compared with other inversions of viscosity profiles for North America (Wolf et al. 2006; Kuchar et al 2019; Metivier et al. 2016; Reusen et al. 2020; Mao and Zhong 2021), at least the ones that also use gravity data.
In section 3.1 the gravity anomaly is fit with only the geoid kernels, resulting in a variance reduction of around 40%. The results are very different from those of the joined inversion. Since the added value of the manuscript is in doing a joined inversion, it would help the flow of the paper to remove section 3.1 or place it in an appendix.
Specific comments
Title: strength of a material usually refers to yield stress. I suggest to change the title to something like the following: Regional gravity constraints for North America reveal upper mantle viscosity differences across the continent.
49: It is stated that Mitrovica and Forte showed considerable potential, but it is not clear what is the gap in the literature that you will address. The text from line 225 onwards is useful to add in the introduction.
82: subset: this seems important information that is not discussed further in the paper. How is the choice made which functions are included and how could that affect the results?
128: scaling with the 10^21 Pa s value. This effect of this choice should be discussed.
133: The paper of Tarasov et al. (2012) does not present a single ice model as far as I know.
233: The scaling is a crucial parameters and constraints on them have implications for other models. I suggest to better introduce the choices made for the scalings. Are the chosen values common in the literature? Is it expected that they hold for North America?
299: ICE-6G is created by fitting RSL data, therefore a good fit with RSL data is to be expected. Is the fit obtained here better than the fit of the original model?
325: The fit with RSL data is poor as you also note in line 434. Looking at figure 9 it is unlikely that that is due to missed tectonic signal. It is likely that the crustal signal plays a large role in explaining the gravity anomaly; this should be quantified.
334: That is surprising given that most of the gravity anomaly signal is in the eastern region. Can you speculate why the joined inversion is dominated by the solution for the western region?
References mentioned in the review comments
- Äadek, O. and Fleitout, L., 2006. Effect of lateral viscosity variations in the core-mantle boundary region on predictions of the long-wavelength geoid. Studia Geophysica et Geodaetica, 50(2), pp.217-232.
- Ghosh, A., Becker, T.W. and Zhong, S.J., 2010. Effects of lateral viscosity variations on the geoid. Geophysical Research Letters, 37(1).
- Kaban, M.K., Tesauro, M., Mooney, W.D. and Cloetingh, S.A., 2014. Density, temperature, and composition of the North American lithosphere—New insights from a joint analysis of seismic, gravity, and mineral physics data: 1. Density structure of the crust and upper mantle. Geochemistry, Geophysics, Geosystems, 15(12), pp.4781-4807.
- Kuchar, J., Milne, G. and Latychev, K., 2019. The importance of lateral Earth structure for North American glacial isostatic adjustment. Earth and Planetary Science Letters, 512, pp.236-245.
- Mao, W. and Zhong, S., 2021. Constraints on mantle viscosity from intermediateâwavelength geoid anomalies in mantle convection models with plate motion history. Journal of Geophysical Research: Solid Earth, 126(4), p.e2020JB021561.
- Métivier, L., Caron, L., Greff-Lefftz, M., Pajot-Métivier, G., Fleitout, L. and Rouby, H., 2016. Evidence for postglacial signatures in gravity gradients: A clue in lower mantle viscosity. Earth and Planetary Science Letters, 452, pp.146-156.
- Paulson, A., Zhong, S. and Wahr, J., 2007. Limitations on the inversion for mantle viscosity from postglacial rebound. Geophysical Journal International, 168(3), pp.1195-1209.
- Sella, G.F., Stein, S., Dixon, T.H., Craymer, M., James, T.S., Mazzotti, S. and Dokka, R.K., 2007. Observation of glacial isostatic adjustment in “stable” North America with GPS. Geophysical Research Letters, 34(2).
- Thoraval, C. and Richards, M.A., 1997. The geoid constraint in global geodynamics: viscosity structure, mantle heterogeneity models and boundary conditions. Geophysical Journal International, 131(1), pp.1-8.
- Wen, L. and Anderson, D.L., 1997. Layered mantle convection: A model for geoid and topography. Earth and Planetary Science Letters, 146(3-4), pp.367-377.
- Wolf, D., Klemann, V., Wünsch, J. and Zhang, F.P., 2006. A reanalysis and reinterpretation of geodetic and geological evidence of glacial-isostatic adjustment in the Churchill region, Hudson Bay. Surveys in Geophysics, 27(1), pp.19-61.
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RC2: 'Comment on se-2021-151', Anonymous Referee #2, 25 Mar 2022
In this manuscript, Osei Tutu and Harig present a series of calculations to constrain the local radial viscosity profiles beneath the North American (NA) continent, using localized free-air gravity kernels and GIA models for the inversion. Due to the strong tectonic contrast between the tectonically active western cordilleran regions and the eastern cratonic regions across the whole NA continent, the authors also zoom into the eastern and the western part of the continent separately using the same joint inversion technique. The inversion results over the whole NA continent show that the density-to-velocity scaling used (constant at 0.3 or depth-dependent following Simmons et al., 2010) has a strong influence on the resulting radial viscosity profiles. The focus on either the western continent plus Pacific margin or the eastern cratonic continent leads to more robust results. The western localized inversion shows a weak and shallow asthenosphere layer, while the eastern localized inversion shows a strong upper mantle and a weak transition zone. Beneath the western NA continent + Pacific margin, the results of the weak asthenosphere/upper mantle in this manuscript are different from the previous postglacial and mantle viscosity studies (e.g., James et al., 2000, 2009a, b) by ~10 orders of magnitude in light of the predicted uplift rate, as authors mention in the text. Beneath the eastern part of the NA continent, the authors attribute the strong upper mantle to the existence of the stiff cratonic root and the weak transition zone to the hot mantle material and volatiles/fluids associated with the underlying descending Farallon slab. Although this study seems systematic and robust, I have three significant concerns/suggestions.
First, after applying a windowing function to the global GRACE gravity data (Slepian function spectral localization technique presented by Wieczorek and Simons,
2005), the authors use two spherical harmonic filters from degree 2-10 and degree 2-15 respectively to perform the regional inversion using the localized geoid kernels. They found there is no significant difference of the results between the two filters in the regional mantle flow modeling below the NA continent. However, as agreed within the geodynamic community, the long-wavelength geoid (2-8 degrees) is dominated by broader mantle density anomalies including LLSVPs and global subducting slabs (e.g., Liu and Zhong, 2016; Ghosh et al., 2010; Mao and Zhong, 2021). Even if the GRACE gravity data is localized into the NA region using Slepian function, part of the local gravity/geoid in the long-wavelength window may still come from other broader-scale mantle density heterogeneities outside of the NA cap. Therefore, I think the author should filter the gravity signal over the NA continent into a shorter-wavelength window. Considering the scale of the NA continent, maybe spherical harmonic degree higher than 8 would be accepted (e.g., 8 – 20 or 8 - 25). The author can try different window in shorter wavelength. Current window starting from degree 2 and 3 is definitely in too long wavelength for the NA continent. The author could also try decomposing the spatial GRACE gravity field over the NA continent or Hudson Bay to find out the appropriate wavelength for the inversion. In addition, based on the findings in Wieczorek and Simons 2005 (Page 670), after windowing a global field by a taper of bandwidth L, the resultant coefficients are only reliable up to Lf – L (Lf is the maximun degree of the field data), and the first L coefficients exhibit large uncertainties. This study seems using a taper of bandwidth L =15 (from the caption of Fig. 1?), so only the GRACE data higher than spherical harmonic degree 15 can be robustly used for this kind of inversion. The authors should also explicitly denote in the manuscript on the bandwidth taper used in this study and perform the corresponding inversion consistent with that bandwidth. I strongly suggest that the authors perform the whole calculation again using the GRACE data in shorter-wavelengths, which can provide more robust constraint on the viscosity profiles below the NA continent. In addition, the variance reductions (in the longer-wavelengths considered here) of the results from only regional mantle modeling seems poorly fitting the observed GRACE gravity data (line #252 - #254).
Second, the NA plate includes a contrasting tectonic setting between the eastern and western half. The eastern cratonic root is reflected by strong high seismic velocity and the western asthenosphere is featured with strong slow seismic velocity. Thus, in the mantle flow modeling focusing on the North American plate, the lateral perturbations of the velocity-to-density scaling should be incorporated into the density/buoyancy field, as the authors also mention in the Discussion section. For example, Ghosh et al., 2013 use zero velocity-to-density scaling for the high velocity anomalies of the cratonic root. Forte et al., 2010 uses two different density-to-velocity scaling profiles for the cratonic root and the ambient upper mantle. The buoyancy of the cratonic keel is still in debate, but for the viscosity inversion, the authors at least should try a simple isopycnic hypothesis (Jordan, 1978) that assuming a neutrally-buoyant cratonic keel down to 250 or 300 km. In addition to the cratonic keel, the lateral perturbation of the velocity-to-density scaling for the slow seismic velocity below the western NA should also be considered. Below the western NA and the neighboring Pacific margin, significant fraction of the partial melt may be present, which explains the very slow seismic velocity in this region. Liu and King, 2022 tested the lateral perturbation of the velocity-to-density scaling in this local area associated with the existence of the partial melt and found that this considerably influences the mantle flow patterns beneath the NA continent. They found that the velocity-to-density scaling for the slow seismic velocity in this area should be as small as 0.05 or below. If one only considers a constant velocity-to-density scaling of 0.3 or a 1D velocity-to-density scaling from Simmons et al., 2010 as did in this study, the buoyancy structures below the NA continent cannot be correctly represented. This substantial inaccuracy in the density structures may significantly bias the recovered viscosity profiles as the definition of the geoid kernels (eq. 4b in the manuscript). I think for this regional study specifically focusing on the NA continent, considering the lateral perturbations of the density-to-velocity scaling from the chemically-depleted cratonic keel below the eastern NA and the present of the partial melt in the western NA is crucial to correctly estimate the buoyancy/density structures in the mantle flow modeling. The seismic tomography model from French and Romanowicz, 2015 is used in this study, so the authors should be able to define the geometry of these two local tectonic areas based on the seismic velocity magnitudes in the tomography model. Then the velocity-to-density scaling for each area could be adjusted. I guess the resulting viscosity profiles from this new inversion may be improved to fit GRACE gravity data and RSL data better. The variance reductions of the results from only regional mantle modeling seems poorly fitting the observed GRACE gravity data (line #252 - #254).
Third, due to the strong lateral viscosity variations (LVVs) across the NA continent between the weak western asthenosphere and the strong eastern cratonic keel, the effects of the mode-coupling (for example, see Stewart, 1992) on the geoid/free-air gravity between different spherical harmonic wavelengths may be considerably strong below the NA continent. For example, in this scenario, the geoid/gravity signal in the shorter-wavelength may include a part coming from the contribution of the buoyancy structures in the long-wavelengths through mode-coupling. I noticed that the current inversion based on the geoid kernel (eq. 4b in the manuscript) can only constrain 1D radial viscosity profile. The 3D viscosity structures cannot be incorporated into this kind of inversion. It seems that the author does not mention how they calculate regional mantle flow. I think the author should add another paragraph in Method section to describe the technical detail of their mantle flow modeling, including initial condition set-up, boundary condition, solver (spectral or spatial?), etc. If it’s a code that can handle LVVs, such as CitcomCU, CitcomS, or ASPECT, I suggest the author run another numerical model incorporating LVVs (strong cratonic keel and weak western asthenosphere, temperature-dependence, etc.) for the whole NA region, and then compare the predicted geoid/gravity in the appropriate wavelengths corresponding to the NA continent (see my previous point) with the original model in the manuscript that does not include LVVs. If the authors use a spectral code (propagator matrix technique) that cannot handle LVVs, I suggest they should add a paragraph in the manuscript to discuss the potential bias caused by the mode-coupling due to the strong LVVs. Previous global mantle flow modeling has incorporated the LVVs from strong cratonic root, weak plate boundaries, and temperature-dependent viscosity (e.g., Zhong 2001; Becker 2006; Miller and Becker, 2012; Becker 2017).
Some specific minor points that would help make the manuscript clearer and better for the readers:
- The authors should clarify whether self-gravity is included in the geoid calculation. As Zhong et al., 2008 clearly show, self-gravity has a considerable influence on the long-wavelength geoid. Further considering the effects of mode-coupling in LVVs, the effects of self-gravity may be even larger than expected for the Earth. The geoid predicted from the mantle flow models with the effects of self-gravity is also not comparable with the geoid results without the effects of self-gravity.
- Non-hydrostatic free-air gravity. This is mentioned at line #222. I think the authors should make this point more explicitly expressed. How the hydrostatic flattening is removed? Following which paper? Nakiboglu, 1982 and Chambat et al., 2010 are two references commonly used in the community to perform the correction on the gravity date associated with hydrostatic flattening (e.g., Ghosh et al., 2013).
- Slepian functions. In Method section, although the authors describe the utilization and advantages of Slepian functions as a localization technique in spectral field, the authors should more explicitly describe which kinds of Slepian functions are used in this study around line #105, such as which degree the bandlimited space-concentrated tapers up to and how they are linked to the geoid kernels. In addition, in Fig. 1, what does functions 1, 2, 3, 4, 9 mean? These should be explained in detail as some new texts and equations.
- Letter symbols of subfigures. I notice that the authors put letter symbols for some subfigures but the letter symbols for other subfigures are missing, such as Fig. 2, Fig. 7, and Fig. 8. The letter symbols (a, b, c, d, ….) for each subfigure should be appended for the convenience of the readers.
- Be careful of the explanation of the abbreviation, such as RSL (Relative Sea Level?). The full name should be mentioned first with the abbreviation in the parathesis, and then the abbreviation can be used elsewhere.
- There are some misused phrases in the manuscript. Line #29: Fee-air ---> Free-air;
- Line #66: patial sphere ---> partial sphere;
- Line #193: basin specific multiple 1D Earth models ---> basic specific multiple 1D Earth models;
- Line #238: a relatively a weaker ---> a relatively weaker;
- Line #243 and elsewhere: strong viscosity values ---> strong layer or large viscosity values;
- Line #264: strong viscosity interface. Be careful of the word “interface”. Interface means “boundaries” instead of “layers”. Strong interface is a vague expression. You should either use strong layer or an interface from a strong layer to a weak layer. I notice this problem appears in many places (e.g., line #294, line #347, line #451).
- Line #330: “both” should be deleted?
- Line #333 - #334: why does the western upper mantle dominates? Note that there is also no cratonic keel in the western NA continent.
- Line # 358 – Line # 360: This sentence needs more explanations on how it is more in line with the findings by Hager and Richards, 1989.
- Line #388: serving a a first order ---> serving as a first order.
- Line #398: what are clear geophysical arguments? This place needs a couple of citations.
- Line #424: missing parenthesis for the citation.
- Line #441: conclusion ---> Conclusion
- Line #442: based, created: this is a problematic vague expression.
- Line #450: inversion the entire ---> inversion of the entire.
- Line #458 - #459: This sentence needs to be rearranged.
- Line #460 - #463: This long sentence is vague and needs to be explained in detail.
- A1: “Alpha indicates the eigenfunction number and rank”. Can you explain this definition clearer? What does the eigenfunction number and rank mean?
- Line #323: There is no Fig. 6d.
- In Reference section, the reference papers should be listed in alphabetic order.
Overall, I think this manuscript could potentially become an appropriate paper including significant contributions to the exploration of the lateral strength variations below the NA continent if the authors consider following my three major suggestions above to improve the robustness and quality of this study. The manuscript also needs more careful and clear writing/organization. Considering that ~8 weeks turnaround may be not enough for re-performing the calculations, I suggest the paper should be returned for major revision to the authors. If the authors can address my concerns above, they should be encouraged and welcomed to submit the substantially-revised manuscript back to SE. I will be happy to see the revised manuscript again.
References:
- Liu, Xi, and Shijie Zhong. "Constraining mantle viscosity structure for a thermochemical mantle using the geoid observation." Geochemistry, Geophysics, Geosystems3 (2016): 895-913.
- Ghosh, A., T. W. Becker, and S. J. Zhong. "Effects of lateral viscosity variations on the geoid." Geophysical Research Letters1 (2010).
- Mao, Wei, and Shijie Zhong. "Constraints on mantle viscosity from intermediateâwavelength geoid anomalies in mantle convection models with plate motion history." Journal of Geophysical Research: Solid Earth4 (2021): e2020JB021561.
- Ghosh, A., T. W. Becker, and E. D. Humphreys. "Dynamics of the North American continent." Geophysical Journal International 194.2 (2013): 651-669.
- Forte, A. M., et al. "Deep-mantle contributions to the surface dynamics of the North American continent." Tectonophysics481.1-4 (2010): 3-15.
- Liu, Shangxin, and Scott D. King. "Dynamics of the North American plate: largeâscale driving mechanism from farâfield slabs and the interpretation of shallow negative seismic anomalies." Geochemistry, Geophysics, Geosystems (2022): e2021GC009808.
- Jordan, Thomas H. "Composition and development of the continental tectosphere." Nature 274.5671 (1978): 544-548.
- Stewart, Cheryl A. "Thermal convection in the Earth's mantle: Mode coupling induced by temperatureâdependent viscosity in a threeâdimensional spherical shell." Geophysical research letters 19.4 (1992): 337-340.
- Zhong, Shijie. "Role of oceanâcontinent contrast and continental keels on plate motion, net rotation of lithosphere, and the geoid." Journal of Geophysical Research: Solid EarthB1 (2001): 703-712.
- Becker, Thorsten W. "On the effect of temperature and strain-rate dependent viscosity on global mantle flow, net rotation, and plate-driving forces." Geophysical Journal International2 (2006): 943-957.
- Miller, Meghan S., and Thorsten W. Becker. "Mantle flow deflected by interactions between subducted slabs and cratonic keels." Nature Geoscience10 (2012): 726-730.
- Becker, Thorsten W. "Superweak asthenosphere in light of upper mantle seismic anisotropy." Geochemistry, Geophysics, Geosystems5 (2017): 1986-2003.
- Zhong, Shijie, et al. "A benchmark study on mantle convection in a 3âD spherical shell using CitcomS." Geochemistry, Geophysics, Geosystems 9.10 (2008).
- Nakiboglu, S. M. "Hydrostatic theory of the Earth and its mechanical implications." Physics of the Earth and Planetary Interiors 28.4 (1982): 302-311.
- Chambat, F., Y. Ricard, and Bernard Valette. "Flattening of the Earth: further from hydrostaticity than previously estimated." Geophysical Journal International 183.2 (2010): 727-732.
Citation: https://doi.org/10.5194/se-2021-151-RC2
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