How Dynamic is Topography in the Western U.S.?

We have the unusual presence of two papers published in the same year essentially trying to do the same thing: determine how much topography is caused by isostasy in the crust and mantle and how much might be non-isostatic (or dynamic). And the two disagree strongly on the amount of dynamic topography. So let’s break this down and see where these papers really differ. The two are Levandowski et al. in JGR in early 2014 and Becker et al. in EPSL in late 2013 (well, the publication date is 2014; it came online back in October).

Advisory: GG is a coauthor on the Levandowski et al. paper.  While this might be reflected in preferences expressed below, hopefully the analysis is still helpful to those with different preferences.

Outline: Basically both papers include the same routine: take a seismic model, convert it to density, subtract the effects of that density model and see what is left.  The Becker et al. paper has a number of other analyses going on that make it harder to follow the most direct route. There are some wrinkles, but we’ll look at the seismic models first, then their conversion to density, and then how that is used. This is a bit more technical than many other posts here…

Crust and Moho Seismology: Becker et al. consider the depth of the Moho from a couple of different receiver function studies, which means that there is some conflation of crustal velocity and Moho depth.  The Levander and Miller model uses CRUST2.0 velocities, which are based on interpolation and extrapolation of P-wave models across the west; the Lowry and Pérez-Gussinyé paper isn’t terribly clear about how (or even if) they determine absolute wave speeds in the crust (the paper is focused vp/vs ratios, which are kind of ignored in the Levander and Miller paper); they use the H-k stacking approach of Liu and Kanamori (2000), which relies on reverberations to remove ambiguity with uncertain in vp/vs (but still requires something to be assumed for a mean crustal vp; Liu & Kanamori basically argued that uncertainty in vp was not significant in the error on depth of Moho). Each of the models has a weakness when used for determining density as neither is fully self-consistent between crustal velocities and Moho depths; the Levander and Miller model relies on CRUST2.0, which is not always based on observation and which conflicts with receiver function Moho depths in several places; Lowry and Pérez-Gussinyé don’t have any obvious control on variations in mean wavespeed, which, while second order for Moho depth, is important in determining mean crustal density. Levandowski et al. relied on the SV-wave model of Shen et al. (JGR, 2013) constructed from ambient noise and ballistic surface waves combined with azimuthally filtered receiver functions. The crust includes a sediment layer, which can be a major contributor to Ps and Sp delay times for stations on basins. This is fully self-consistent but as it relies on SV, any variations in radial anisotropy will map into density unreliably. Moho depths are specifically at stations but the surface wave dispersion curves effectively cover a volume, so errors are to be expected in places where there are strong slopes on the Moho. The parameterization of the inversion allows for only a single Moho (this is true as well of the papers used by Becker et al.); areas that might have a “double Moho” will probably be aliased unpredictably.

Mantle Seismology: Becker et al. seem to rely on Ps (mainly) and Sp receiver function estimates of the lithosphere-asthenosphere boundary from Levander and Miller (2012): “Here, we will simply test if the LAB depths from Levander and Miller (2012) (P receiver function based as in Fig. 2) exert some control on the isostatic adjustment of a variable thickness, and constant density, lithosphere, assuming that a thermal boundary layer of variable thickness should approximately scale with the LAB.” GG thinks this means that some contribution to isostasy is considered to be proportional to mantle lithosphere thickness, though the constant of proportionality is either to be determined or is simply left unclear. There are many weaknesses here: first, there is considerable controversy as to the significance of the Ps and Sp “LAB” converters as they appear to be well within the lithosphere in the eastern U.S., which suggests they may be inappropriate markers in much of the western U.S. as well and are almost certainly inappropriate in the easternmost areas covered in the paper. Second, the Ps converters are frequently tangled with reverberations from the crust and are often unreliable; Sp is a difficult phase both because signal to noise is often weak but also  because the converted phase travels at a high incidence angle through the lithosphere, so the LAB piercing point of the Sp wave is far from that of the direct S.  They also derive the thickness of a constant-density lithosphere that would add to crustal models to produce observed topography. Later on, density variations in the mantle lithosphere are derived from the S-wave tomography of Schmandt and Humphreys and from a thermal estimate from Lowry and Perez-Gussinye (these are both found to have little impact as discussed further below). Levandowski et al. uses Shen et al’s model, which is primarily ballistic surface waves with some control from the amplitude of the Psconversion at the Moho to construct a velocity model to depths of 150 km. No attempt to identify the base of the lithosphere is made. This can mean that density variations in the asthenosphere will be improperly connected to the lithosphere and exaggerate their influence [in practice, though, the places with thinner lithosphere have such low wave speeds that there are negligible density variations at greater depth]. As before, radial anisotropy variations will map into variations improperly. Lithosphere thicker than 150 km will not be fully represented within the calculation [this should be relatively minor as the bottom of a thermal boundary layer will be close to asthenospheric temperatures and in general highly depleted mantle is generally above 150 km depth].

Density: Becker et al. hold crustal and mantle lithospheric densities constant in the initial analysis. They derive possible deviations of density from the difference between constant-density models and topography, mainly for illustration and not analysis.  They apply a crustal density correction from Lowry and Perez-Gussinye. They consider a thermal density variation from S-wave tomography of Schmandt and Humphreys. When mixing and matching components of other models, there is always a risk of any covariance between model components really messing things up.  For instance, the S-wave tomography depends on the crustal model used, especially at shallow depths in the mantle; if you pair this tomography with a different crustal model, you have a hybrid that might not satisfy the tomographic dataset. Levandowski et al. use a vs to density relation in the crust derived empirically by Brocher (2005) with a correction for thermal variations derived from surface heat flow, assuming a steady-state thermal profile. Disequilibrium thermal states will have a minor impact.  A single-valued velocity-density relationship certainly has limitations that depend on the scale length of deviations from such a relationship; if large areas systematically lie off such a curve (e.g., crust largely made of quartzite), significant errors are possible.  In the mantle a simple thermal relation for velocity to density is used, accounting for anelastic effects near the solidus and assuming no density variations below a threshold velocity assumed to represent the solidus. Variations in composition are ignored [but discussed when interpreted] and variations in solidus due to presence of water or variations in composition could produce deceptive density estimates.

Analysis: Becker et al. use a frequency-domain coherence between the two Moho maps (with and without the mantle lithospheric thickness) and topography. This has the advantage of relative insensitivity to the absolute density contrast across the Moho or between the mantle lithosphere and asthenosphere; however, it fails to account for varying density contrasts across the Moho or lateral variations in crustal or mantle lithospheric density, any of which would reduce the coherence of the compositional proxy (Moho depth, LAB depth) to topography and so appear to show a failure of isostasy. Spatially, the paper subtracts out the constant density crust and considers a few different flavors of lateral variations that could reconcile predicted to observed topography. Most relevant for comparison, they take the constant-density crust and constant thickness mantle lithosphere and add in density variations in the crust from one paper and in the mantle from another to obtain something in Fig. 6a that should be most like the residual topography (Fig. 4g) of Levandowski et al.  Levandowski et al. works in the spatial domain, calculating topography from the crustal and mantle density structures. The initial calculation ignores crustal thermal variations, but these are then added in (based, as noted above, on heat flow). Areas where predicted topography are outside the uncertainty from observed topography are considered; areas where predicted topography is too high and heat flow is high are suggested to contain crustal melt; areas where predicted topography is too low are suggested to contain unusual amounts of quartz; areas where predicted topography is too high but cold are suggested to be affected by dynamic effects [basically, coupling to the slab, which exerts a greater downward force than is evident within the seismic model]. These effects are limited in extent and have a maximum contribution locally of 1 km of elevation. These posited possible effects are then reincorporated into the overall estimates of crustal and mantle contributions to topography.  Obviously the assignments of dynamic effects, melt and quartz variations are arbitrary and not based on independent evidence [though the text argues that they are broadly consistent with other observations].

Comparison of some chief results.  Both papers present topographic residuals:

Residual topography from Levandowski et al. (2014) before inferring crustal thermal variations, crustal melt, and quartz composition variations.

Residual topography (Fig. 4g) from Levandowski et al. (2014) before inferring crustal thermal variations, crustal melt, and quartz composition variations.

Preferred residual topography of Becker et al. (2014) includes Moho variations, crustal density variations from Lowry and Perez-Gussinye

Preferred residual topography (Fig. 5c) of Becker et al. (2014) includes Moho variations, crustal density variations from Lowry and Perez-Gussinye. No variation in the mantle (constant density and thickness mantle lithosphere used), but even using S-wave tomo correction (their Fig. 6A) looks nearly the same.

(Sorry, cannot get on one line). Both show observed topography – predicted, but use opposite senses for the color bar. Both have an anomaly in the northern Nevada where elevations are higher than predicted, but otherwise the large residuals seen in the Becker et al. paper are absent in the Levandowski et al. paper.  Why? It seems that the big difference is the mantle contribution.  Consider below the contribution from the mantle in Levandowski et al.

Mantle topography from Vs model of Shen et al, from Levandowski et al. 2014.

Mantle topography from Vs model of Shen et al, from Levandowski et al. 2014.

Vs at 120 km depth from Shen et al. (2013)

Vs at 120 km depth from Shen et al. (2013)

What we see is that the Levandowski et el. formulation suggests that about 1.5 km of topography should be coming from variations within the upper mantle imaged by Shen et al. Although the Vs tomography of Schmandt and Humphreys differs some from Shen et al., in Becker et al., its effects on the residual topography are nearly negligible (basically, Fig. 5c and Fig. 6a look almost the same).  It suggests that one of the main differences between these papers is buried in the way that mantle seismic velocities are mapped into density.  It would seem that if the mantle contribution of Levandowski were added to the Becker et al. residual topography that the topographic residual would be pretty small.  While there are additional terms in the Levandowski et al. formulation that are significant (e.g., a greater lateral variation in crustal buoyancy), this appears to be the big difference.  Either Levandowski et al. overestimated the density change for a given change in vs, or Becker et al. underestimated it. Both papers assume (away from the solidus) that a 5% change in vs will produce a 1% change in density, so for a 70 km thick mantle lithosphere (about as used in Becker et al.), a 5% change should produce a topographic change of 70 km x 3250 kg/m^3 x 1% / 3250 kg/m^3 = 700m [changed 1/8/2015 as Moho moves up with crust if only difference in mantle–this was error T. Becker complained about in a comment].  Comparing figures 5c and 6a in Becker et al., it doesn’t seem that there is hundreds to over 1000 m of difference between the two, while the Hm plot from Levandowski et al. (above) has that order of signal.

So it appears that there was some kind of blunder in calculating the topographic effect of mantle density variations in Becker et al.  With this corrected, it appears that most of the topographic anomaly will go away, much as was seen in Levandowski et al. This would mean that the need for the dynamic calculations (the remainder of the Becker et al. paper) would be irrelevant.

Is this right or did GG just add a new error in?

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6 responses to “How Dynamic is Topography in the Western U.S.?”

  1. Thorsten Becker says :

    This blog was pointed out to me. I read this quickly and welcome the discussion of what is basically a long standing issue, how to assign density and mantle flow anomalies in the context of topography, and how to call them, “static” or “dynamic”. There has been lots of good debate over the decade, and it continues to this day.

    It would appear that the presentation and discussion of GG has a few misinterpretations of the paper by Becker et al. (2014), for example when it comes to using receiver-function based lithospheric thickness estimates. We are well aware of the inherent uncertainties (in terms of measurement and interpretation), and actually agree with many of the statements of GG, and tried to say as much in our paper. For example, we suggest that using an “LAB” from the receiver function estimates does not improve the isostatic model precisely because the LAB might be seeing something different from a buoyant lithosphere.

    There appear to be some differences in how some of the corrections are applied in the two studies. However, the core of the disagreement seem to lie in the treatment of GG & Co. of all velocity anomalies in the mantle above some depth as lithospheric mass anomalies, and the subsequent removal from their “dynamic elevation” estimate. This is different from our definition of dynamic elevation as “everything in the flow regime” and hence the only non-dynamic contribution in the mantle coming from variations in thickness of the thermal boundary boundary layer. This is a choice and an assumption, and one that is motivated partly by what are perceived to be large uncertainties in inferring effective density anomalies within continental lithosphere, by ongoing imaging work and inferred moderate sub-Moho temperature variations, and by a general style of interpretation. We actually tried to provide some alternative explanations for topography as being of a “static” origin (e.g. Figure 5a and discussion of lithospheric thickness variations required), and recognize that there are uncertainties in residual topography maps.

    As is the case in other settings such as the Atlas or the Apennines, one is often faced with alternative models where all topography could be called static, and would be controlled by highly heterogeneous lithospheric structure, or some component might be inferred to be of the “dynamic”, mantle flow induced type. The western US is no different, and the two papers discussed by GG are just the latest in a series of studies making either claim. One of the problems with the static, heterogeneous lithosphere interpretation is that it can fail to make the connection to the dynamic, mantle convection related processes that actually led to the modified lithosphere in the first place.

    Lastly, the dynamic flow computations are not “irrelevant”, either way, since they stem from global mantle circulation models which are consistent with a large number of other constraints, such as plate motions, seismic anisotropy, and the geoid, and as such useful information. Should there be no residual topography of “dynamic” origin, as GG would like to have it, then we have a problem, and need to alter a range of independently established models, invoke widespread compositional anomalies, petrological or mineral physics complications, or all of the above. This may well be the case, but it would seem quite relevant for our understanding of mantle convection.

    I think future analysis, refined structural models, and joint inversions will show which of the interpretations will lead to a more parsimonious model of the interactions between the surface and the deep mantle.


    • cjonescu says :

      Corrected 1/8/15 for the error in calculating elevation change as noted in comment on this comment.

      How unexpected! A scientific discussion on a blog….Thank you very much for the comments on this blog post. In shooting from the hip, GG sometimes muddles things a bit and appreciates comments to clarify mistakes.

      Yes, certainly the Levandowski et al paper could fold some asthenospheric density anomalies into a “static” (isostatic) calculation (though not much as the velocity-density relation used basically results in no density decrease for velocities below a certain value, so areas with a thin lithosphere and thus low upper mantle wavespeeds will not contribute excess density), but GG disagrees that this is the core of the disagreement. There isn’t a huge difference in the overall crustal calculation, but where things seem to differ a lot is in the upper mantle. Perhaps this is due to some misunderstanding, so let’s review why this seems (from this perspective) to be an issue.

      The simplest observation is that the crustal residual topography (Fig. 5c) of Becker et al. and the initial mantle-generated topography (Hm, Fig. 4a) of Levandowski et al. look awfully similar, suggesting that combining the two would yield little residual topography, yet Becker et al. don’t see upper mantle really solving the topographic variations in Fig. 5(c) but Levandowski et al. do. It would seem that the difference lies in Hm from Levandowski et al. vs. the lithospheric mantle calculation made in Becker et al. to produce their Fig. 6(a). And so now the gory details of an attempt to see if this might be so….

      Figure 5(c) of Becker et al. shows the residual topography after removing an estimate of crustal density variations. We can quibble over differences in methodology to this point, but the differences between the two papers to this point are not huge. This figure is nearly identical to Figure 6(a), which includes an estimate of a contribution from density variations within a constant thickness mantle lithosphere. This suggests that variations in the mantle lithosphere have very little impact on topography. Is GG reading that right? This is a huge difference from Levandowski et al., hence the assertion that this is the core difference.

      Looking more closely, consider the extremes in the mantle tomography in Fig. 5(d). In the Snake River Plain we have -5% and near 40N in California we have +5%. If GG is reading the velocity density relation right, then, as in the original post, that 5% change in vs will produce a 1% change in density, so for a 70 km thick mantle lithosphere (about as used in Becker et al.), a 5% change should produce a topographic change of 70 km x 3250 kg/m^3 x 1% / 3250 kg/m^3 = 700m. Right? [maybe right now 🙂 ] Or is there a mistake? So we should see in Fig. 6(a) that the residual elevation of the northern California spot relative to the Snake River Plain (SRP) should rise up by 1400 m. In Figure 5(c) the No. California spot residual is -500m, SRP is near 0; in figure 6(a) No California spot is slightly higher (-250 to -500m) and SRP is now about -250m, so no. California rose by about 250-500m relative to SRP, much less than the 1400m expected. Maybe flexure is preventing these density variations from being fully realized? NE Wyoming is 4% fast but much broader than the Northern California anomaly (so less if any flexural effect) and so residual should increase 560m; it was between 0 and -750m in 5(c) and is between +250 and -500m in 6(a), an increase of 250m, similar to the northern California change and again well below that expected from the simple 1-D calculation.

      In comparison, the Levandowski et al. paper has about a 300m difference in support from the mantle between SRP and No. California; this is so small largely because we used a velocity-density curve that flattens as velocities get quite low and there isn’t as big an anomaly in Shen et al’s tomography. Between SRP and NE Wyoming, there is something like a >1 km difference in contribution from the mantle. Part of this is, perhaps, that Levandowski et al. go from Moho to 150 km depth (so a bit more than the 70 km thickness used in Becker et al), but this is mitigated again by the shape of the velocity-density curve used. So a fair bit of this greater value than in Becker et al. appears to be a different conversion of seismic velocity to density: Shen et al. velocities go from 4.2 to about 4.7 km/s (about -5% to +7%) (GG is being lazy in not similarly averaging Shen et al.’s model from Moho to 150 km and just using the 120 km map as a proxy for that average, but this should be close). From our simple calculation, there should be something over 1400m, but again Levandowski et al. have no density change below a certain vs value.

      Now there is an additional difference in that the vs tomography in fig 5(d) differs from that of Shen et al. used by Levandowski et al.; this is most pronounced in the central Great Basin. This contributes to the different findings of the two papers and may be a reflection of variation in radial anisotropy, which the original post noted was a weakness of relying on the Shen et al. model.

      Now another possibility that GG thinks is being brought up in this comment is that Levandowski et al. are incorporating stuff that is in the “flow regime” that properly should be considered dynamic. GG doesn’t think this is a big effect away from subduction zones is because the only parts of the model where asthenosphere is within the model are places where the wavespeeds are very low; variations below a certain level have no contribution.

      Also, it was not our intent to say there is no dynamic topography, merely that the magnitude of any signal was below our uncertainty and that variations in dynamic topography (as we defined it, BTW–by other definitions our mantle contributions are ‘dynamic’) are not needed outside of the Pacific Northwest. The reason for saying that the dynamic calculations were “irrelevant” [now there was flame bait, no?] was that if indeed there was some problem in the mantle lithosphere contribution such that the Levandowski et al. approach was correct, there might be no need to invoke dynamic topography. It is important to try and obtain observational constraints on dynamic (here, sublithospheric) topography and hence the origin of this discussion. In this is appears we agree: if there is little to no dynamic topographic variation in this region, then models predicting such are not valid. GG here makes no attempt to explore the validity or invalidity of the models shown per se.

      It may be worth asking if the density structure envisioned by Becker et al. will fit the gravity measurements in the WUS. The Levandowski et al. calculations do, to a surprisingly large degree (well, truth be told, it surprised us). Generally, nearly any isostatically balanced lithospheric model will closely approximate gravity; generating more than a kilometer of topography from mantle flow will probably violate the gravity observations (but in this case, GG cannot say for sure).

      P.S. One thing clearly better in Becker et al. than in Levandowski et al is the use of the color scheme: Becker et al. show nice clear contours so it is straightforward to estimate values on the not-quite-a-map figures. [There is some reason for this not being the case in Levandowski et al., but, well, we should have tried harder. At least our maps are real maps].


      • Thorsten Becker says :

        I agree that different choices in the lithospheric density scaling and how it is applied may be at the core of those specific differences in the residual computations in both papers.

        Both Becker et al. and GG use their respective equations 1 and 2 to correct topography for lithospheric density variations, which leads me to compute change in topography due to lithospheric density variation as
        -Drho_l l_l /rho_a

        where Drho_l is the lithospheric anomaly (3250 kg/m^3 x 1% in GG’s example), l_l is lithosheric mantle thickness (70 km the example), but rho_a is asthenospheric, rather than crustal, density, giving slightly (~86%) different numbers. Combine this with mantle thicknesses and density anomalies that are both somewhat smaller than what GG uses in his examples, the correction in Becker et al. has apparently smaller amplitude than in GG’s work, which appears to explain part of the difference in results.

        Those corrections, ambiguities, and uncertainties aside, I agree that it would be good to attempt a joint inversion, or check against gravity, something we have not done yet. However, we did look at the geoid in flow computations similar to those used in Becker et al. previously (Ghosh et al., GJI, 194, 651-669, 2013) and found the geoid anomalies to be consistent with the dynamic topography model, leaving the question of overall consistency and how the anomalies may balance with depth open in my opinion.

        I am happy to further continue this discussion offline.


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