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Isostatic rebound corrections are still on a squishy footing

September 18, 2019

This blog post covers the subject of whether or not you should account for isostatic elevation changes in production rate calculations, comes to the conclusion that we don’t really know the answer, highlights one cautionary tale that is probably relevant, and suggests some ways to fix the situation.

This subject is important because production rates vary with atmospheric pressure. Higher  elevation, less atmosphere, higher production rate. In exposure-dating, because it’s not possible to measure the mean atmospheric pressure over the entire exposure duration of a sample, what we actually do in nearly all cases is (i) measure the elevation, (ii) convert to atmospheric pressure using a relationship based on the modern atmosphere, and (iii) assume that the resulting pressure applies for the entire exposure duration. Unfortunately, for sites that are more than a few thousand years old, and especially for sites that are, or were, marginal to large ice sheets, this usual procedure has to be at least partly wrong on two counts. First, as ice sheets, sea level, and global temperature have changed in the past, the mass of the atmosphere has moved around: the atmosphere has to get out of the way of ice sheets, fill up space created by eustatic sea-level fall, and shrink and expand as global temperature cools and warms. The pressure-elevation relationship in many places was probably quite different in the past. Second, because of glacioisostatic depression and rebound associated with the advance and retreat of large ice sheets, many potential exposure-dating sites have not only had the atmosphere move around on them, but have themselves moved up and down in the atmosphere. These observations would tend to indicate that maybe we shouldn’t just compute the production rate at the present elevation — we should account for the fact that all these things are changing.

Several folks have tried to do this. Or, at least, part of it. Specifically, a number of exposure-dating papers have extracted elevation time series for their sample sites from glacioisostasy models (such as the ICE-XG series of global models generated by Richard Peltier and colleagues), computed production rates corresponding to those elevations, and   then, basically, used a time-averaged production rate instead of an instantaneous modern production rate to compute the exposure ages. In effect this approach assumes that the atmosphere is stationary and the sites are moving up and down in it, and in all cases where I can remember it being applied, it results in lower calculated production rates (because the sites were ice-marginal, so at lower elevation in the past than they are now) and therefore older calculated exposure ages.

OK, this seems like a good idea. And it is possible to do this for nearly any exposure-dating study that deals with LGM-to-present events, because there exist global models for isostatic depression and relative sea-level change for the last glacial cycle. Of course, it’s not easily possible for pre-LGM exposure-dating applications.

However, there are several problems with this otherwise good idea.

One, as noted above, it’s not just the sites that are moving up and down in the atmosphere, the atmosphere is moving around as well. This aspect of the problem is the subject of a very good paper from quite a long time ago by Jane Willenbring (Staiger). This paper looked mainly at the atmosphere effects and pointed out, among other things, that they may act to offset isostatic depression near ice sheet margins, resulting in less variation in production rates than one would expect from each process individually.

Two, the production rate calibration sites are moving up and down too. Suppose that your production rate calibration data experienced an increase in the surface production rate over time, as expected for glacioisostatic rebound in the absence of any offsetting atmospheric effects. If you don’t account for this — that is, you just use the modern elevations of calibration sites — when using these data for production rate calibration, you will underestimate the production rate. If you then use this calibrated production rate, but make an isostatic rebound correction when computing ages for unknown-age sites, you will make your production rate estimate at your site, which is already too low globally because you forgot to include isostasy in the production rate calibration, even more too low locally. Thus, the resulting exposure ages will be too old, twice. Instead, to do this correctly, you must include the isostatic rebound correction in both the production rate calibration phase and the exposure-age calculation phase.

There exist several papers that appear on a reasonably close reading to have made this mistake and doubly-compounded errors, although in many cases it is hard to tell because the details have disappeared into a methods supplement. Because this is a blog entry and not an actual journal article, I don’t have to name them all in detail. But the point is that ages calculated in this way are unambiguously wrong. If you want to use an isostatic rebound correction in your exposure-age calculations, you must also use it in your production rate calibration. Or you can do neither: if you have ice-marginal calibration sites and ice-marginal unknown-age sites with similar rebound histories, and you don’t make the correction at either one, you will get close to the right answer, because any errors you introduced by skipping the correction in part 1 will be canceled by the offsetting error in part 2. Thus, no matter what, it is unquestionably better to not do this correction at all than to do it halfway.

The third problem is that the existing calibration data set does not show any evidence that a correction based on elevation change alone is either necessary or desirable. This is kind of a complicated claim, and it’s important to clarify the difference between (i) testing a particular model, that is, asking “does a particular glacioisostasy model, ICE-5G or whatever, improve the performance of production rate scaling models when measured against calibration data” and (ii) asking whether the calibration data themselves provide any evidence that production rates are affected by glacioisostatic elevation change considered alone. I would argue that we have to do (ii) first. Basically, if we can’t establish (ii) from the calibration data, then it is very difficult to determine what the answer to (i) would mean, or if a “yes” answer was meaningful at all.

So, here I will consider item (ii). The basic principle here is that the existing set of calibration data includes both calibration sites that are marginal to large ice sheets, and those that are not. If glacioisostatic elevation change by itself is an important control on production rates, then ice-sheet-marginal calibration sites will yield lower production rates than far-field ones.

This seems like a pretty simple question, but it is not obvious that anyone has actually answered it. A recent paper by Richard Jones and others gets tantalizingly close to at least asking the question, but then disappointingly stops at the last minute. The bulk of this paper is a quite thorough global calculation of the effect on time-integrated production rates expected from glacioisostatic elevation change. In the course of discussing whether or not this should be routinely taken into account in exposure-dating studies, the authors point out that the four (or six, depending on how you group the Scottish data) calibration sites for beryllium-10 data in the widely used “CRONUS-Earth primary calibration data set” are all in areas that experienced very small glacioisostatic elevation changes, so if you use this calibration data set, including or not including a glacioisostatic elevation change correction in the production rate calibration phase doesn’t really make any difference. This is true and, of course, represents a loophole in the absolutely strict rule described above that if you want to use an isostatic correction in exposure-age calculations, you must also use it in production rate calibration. After this, however, the paper stops thinking about calibration data and goes on to other things. Stopping here is extremely disappointing in the context of the otherwise very clear and comprehensive paper, because there are not just four Be-10 production rate calibration sites. In fact, there are 38, seventeen of which are from ice-sheet-marginal features. Although this is the subject of a whole different discussion, there is no particular reason that most of these sites are more or less reliable than the “CRONUS primary” calibration data, which, IMHO, were somewhat arbitrarily selected. Thus, it is possible to test production rate predictions based on glacioisostatic elevation change against a fairly large data set of calibration measurements from both ice-sheet-marginal and far-field sites, and as I read this paper I was very surprised that the authors did not do it. Or, at least, I could not find it in the paper. Why not?

So, let’s try it, at least to the extent feasible without making this blog posting any longer than its current 14,000 words. Specifically, we are asking the question: are production rates inferred from ice-sheet-proximal calibration sites lower than those inferred from other calibration sites? If glacioisostatic elevation change by itself is an important control on production rates, we expect the answer to be yes. However…

…the answer is no. The above histograms show Be-10 production rate scaling parameters for LSDn scaling inferred from all Be-10 calibration sites in the ICE-D:CALIBRATION database (top panel) as well as ice-sheet-marginal and far-field subsets (lower two panels). The blue histograms are for all samples; the red histograms are for averages from each site. Remember, the LSDn scaling algorithm directly predicts production rates with units of atoms/g/yr instead of nondimensional scaling factors, so the calibrated parameter is accordingly a nondimensional correction factor rather than a “reference production rate” with physical units. Regardless, the mean and standard deviation of correction factors inferred from ice-sheet-marginal and far-field calibration sites are, basically, the same. Almost exactly the same.

This result, therefore, fails to provide any evidence that production rates are significantly affected by glacioisostatic elevation change alone. Instead, it tends to suggest that elevation-change effects at ice-marginal sites could very well be substantially offset by atmospheric effects. This, in turn, implies that one should probably not include isostatic effects in a production rate scaling model without also including atmospheric mass redistribution effects.

There is no point in giving up too soon, though, so let’s try to come up with some other first-principles predictions that we can test with the calibration data. The simplest prediction was just that production rates from ice-marginal calibration sites should be lower. A slightly more complex prediction is that we expect that this effect should be more pronounced for younger calibration sites. Suppose an ice sheet margin retreats between the LGM and the mid-Holocene, and we have ice-marginal calibration sites that deglaciated at a range of times during this retreat. In general terms, the sites that are closer to the center of the ice sheet deglaciated more recently and should be more affected by isostatic depression: not only is the magnitude of the depression typically larger at sites near the center of the ice sheet, but the period of time during which the sites are isostatically depressed makes up a larger fraction of their total exposure history. Therefore, we expect a correlation between the age of ice-marginal calibration sites and the inferred production rates: all sites should give lower production rates than non-ice-marginal sites, but the younger ice-marginal sites should yield lower production rates than the older ones. Is this true?

The answer: well, sort of. The plot above shows values of the LSDn calibration parameter inferred from far-field sites (blue) and ice-sheet-marginal sites (red). The solid and dotted black lines give the mean and standard deviation of the entire data set. Production rates inferred from non-ice-proximal calibration data are uncorrelated with site age, as they should be, but those inferred from the ice-marginal sites are strongly correlated with site age. This correlation would be a point in favor of an elevation-change effect being important…except for one thing. As expected from the previous observation that the two groups yield the same mean production rate, half the ice-marginal sites give production rate estimates that are HIGHER, not lower, than the mean of the non-ice-marginal sites. If glacioisostatic elevation change is the primary control on production rate, we do not expect this. Forebulge effects are not expected for sites that are well within the LGM boundaries of the ice sheets, so (at least to me) there appears no obvious way that elevation change alone can account for both (i) the similarity in the distribution of production rate estimates between production rates inferred from ice-marginal and far-field sites, and (ii) the strong age-production rate correlation for ice-marginal sites. Overall, this result is sort of ambiguous but, again, appears to indicate that correcting for glacioisostatic elevation changes by themselves is oversimplified.

The summary up to this point is that arguing that glacioisostatic elevation change should be an element in production rate scaling models is probably reasonable. However, both the calculations in Jane’s paper and an analysis of available calibration data indicate that it is probably oversimplified, and probably not a good idea, to correct only for the elevation changes without also considering simultaneous atmospheric mass redistribution.

On to the cautionary tale. The cautionary tale part of this posting is that this situation is very similar to a somewhat obscure, but analogous and pertinent, aspect of cosmogenic-nuclide applications to erosion-rate measurements. Specifically, this is the question of whether or not to apply topographic shielding corrections to production rate estimates for basin-scale erosion-rate calculations.

In the isostasy example, we know that there are two time-dependent effects on production rates in ice-sheet marginal areas, and we think they probably offset each other, at least to some extent. One of them (isostatic elevation change) is easy to calculate, because there exist a number of global, time-dependent, LGM-to-present glacioisostasy models that can be easily used to produce a time-dependent elevation change estimate for any site on Earth. The second one (atmospheric mass redistribution) is difficult to calculate, because there are no equivalent, easily accessible, LGM-to present models for atmospheric pressure distribution.

In the topographic shielding example, we know that the erosion rate we infer from a nuclide concentration in sediment leaving a basin is inversely proportional to the mean production rate in the basin, and directly proportional to the effective attenuation  length for subsurface production. In a steep basin where many parts of the basin are shielded by surrounding topography, shielding reduces both the mean production rate and the attenuation length. Because these have opposite proportionalities, therefore, corrections for these effects offset each other. Again, one of these corrections is easy: using digital elevation models to compute the reduction in the mean surface production rate in a basin due to topographic shielding is a satisfying and fun application of raster GIS. And one of them is hard: computing the shielding effect on the subsurface attenuation length is computationally painful, time-consuming, and really not fun at all. The result is that for quite a long time, many researchers just did the easy correction and ignored the hard correction, until a thoroughly helpful paper by Roman DiBiase did the hard part of the correction, and showed that doing only the easy correction results in an answer that is more wrong than it would have been if you had ignored both corrections. Oops.

The analogy is pretty clear. Both cases feature two offsetting corrections, one easy and one hard, and a seductive heuristic trap created by the availability of code to do the easy correction but not the hard one. It seems like it might be a good idea to learn from history, and try not to shove our left foot in a mouth still filled with the taste of our right shoe.

So, to conclude, what should we do or not do about this? Well, for one thing, we should probably not apply a time-dependent isostatic rebound model to production rate calculations without an atmosphere model as well. The evidence seems to indicate that this is probably oversimplified and may not improve the accuracy of exposure ages. However, it is not obvious that there is a good candidate atmosphere model at the moment; this may require some work. And then, if we use both models, it is necessary to apply them to both calibration data and unknown-age sites, which requires hard-coding them into everyone’s  exposure-age calculation code. We should complete the Jones paper by doing a better job than I have done above in examining more carefully whether scatter in fitting calibration data to production rate scaling models can be explained by glacioisostatic or atmospheric-mass-redistribution effects. And, finally, we will probably need to collect more calibration data. This is because applying these corrections will very likely have the largest effect on inferred exposure ages for times and places where we do not have calibration data. It is probably not a good idea to extrapolate production rate corrections into places outside the range of calibration data where being wrong could incur large and hard-to-identify errors. Targeting places and times where we expect the largest corrections, and also places and times where elevation-change and mass-redistribution effects are likely to have offsetting or compounding effects, to collect new calibration data would add a lot of credibility to the argument that these effects are important.

 

 

 

One Comment leave one →
  1. Richard Selwyn Jones permalink
    November 20, 2019 12:39

    Thanks for continuing the discussion on this topic.

    We agree that there are multiple ways to assess whether glacial isostatic adjustment (GIA) corrections are necessary, and that a correction cannot be accurate if it ignores possible GIA effects on the production rate calibration data. As mentioned in the original blog post, there is no obvious reason why the “CRONUS primary” calibration data are more reliable than that of the rest of the Be-10 calibration sites, and therefore all sites should be included in an assessment of the production rate.

    Similar to our original analysis for the “primary” sites (Fig 3 in our paper; https://www.sciencedirect.com/science/article/pii/S0277379118308825), we have now evaluated the potential GIA impact at all calibration sites (taken from ICE-D; http://calibration.ice-d.org/). This essentially predicts what the production rate would be at the sites based on modelled GIA (using ICE-6G combined with an approximation of the VM5a Earth model).

    Figure 1 – Site-specific production rate difference (corrected-uncorrected for GIA) vs. time since exposure: https://drive.google.com/file/d/1qLQp8WvWqK7NBYzcoEPQitbF8sMJojyl/view.

    In this first figure, you can see that the production rate at the sites is effected by GIA. At several sites, production was slightly lower at the time of exposure due to isostatic depression (i.e. those proximal to the ice sheets), and at a few sites, production was higher in the past as they were located on a peripheral bulge (relatively distal to the ice sheets). However, the magnitude of the effect at each site needs to be substantial enough to have a notable impact the production rate.

    Figure 2 – Ratios of nuclide production from GIA-corrected versus uncorrected sample elevations for all Be-10 calibration sites: https://drive.google.com/file/d/1gwUxoT0R60g3gcELhf8AERXyldXisYQq/view.

    Figure 2 shows that the GIA-corrected production rates (time- and site-averaged) at the sites do not significantly differ from the uncorrected rates – both the site uncertainties (vertical red lines) and the mean of the sites (dashed line) with standard deviation (dotted lines), overlap with the 1:1 line (solid line). The largest potential effect is recorded at the Swedish sites (Billingen; no. 32-37), which is not surprising considering the large ice loading there relative to the other sites, but this is still not significantly different from the 1:1 line or the mean of the sites.

    Based on this assessment, modelled GIA predicts no significant impact on the production rate at these sites, which explains why the assessment of ‘ice-proximal’ vs ‘far-field’ calibration sites in the original blog post did not identify a notable GIA signal. Perhaps this means that an uncorrected calibration dataset can be used with a GIA model to correct exposure ages, but even if we assume that this is the case, there is uncertainty in the GIA values used to make the correction that needs to be considered.

    Importantly, atmospheric effects may have, at least partly, cancelled out a GIA effect in some rebounding regions; although this is spatially and temporally variable, and may have enhanced the GIA effect in some other regions. More calibration sites and combined time-dependent atmosphere/GIA modelling should hopefully get to the bottom of this.

    Figure 3 – Mean age correction for Antarctic samples exposed over the last 25 kyr based on W12 and ICE-6G GIA models (the number of samples in each time interval is the same for the two model corrections): https://drive.google.com/file/d/1mI4g4kS37ti_ZlKCl3Vu-ndX97sxPejp/view.

    A final word of caution – uncertainty in GIA models may be larger than the uncertainty on potential atmospheric effects, and not least in Antarctica. Figure 3 shows the difference through time between Antarctic exposure ages that are corrected (assuming an unaffected calibration dataset) and uncorrected for GIA, using two GIA models with very different ice loading histories (we initially only looked at a 20 ka time slice in the supplementary info of our paper). In all cases, the correction acts to make the exposure age younger, with largest corrections applied to samples exposed near to the time of greatest ice volume and, thus, isostatic depression. But the correction differs by up to several thousand years depending on the GIA model used.

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