This post is mostly about production rate scaling models, but in addition is neat because it’s an actual useful application of the ICE-D Antarctic exposure age database. It’s a pretty simple one, but hey, mighty oaks from little acorns grow. This is another immensely long post, but the short summary is that a look at samples from Antarctica with very high Be-10 and Al-26 concentrations seems to indicate that the relatively recent, and extremely complicated, production rate scaling model of Nat Lifton and colleagues (the “LSD” scaling method) works better at high latitudes than the simpler and very widely used scaling method based on the work of Devendra Lal in the 1980’s. This is interesting because so far it has been quite difficult to determine whether or not there is any substantive difference in performance between these scaling methods.

We start with some recent Be-10 measurements by Gordon Bromley of the University of Maine on sandstone boulders, presumably glacially transported, at the same site in the southern Transantarctic Mountains where we recently observed what appears to be the highest concentration of cosmogenic He-3 ever observed in a terrestrial sample. These Be-10 measurements are interesting because, even after a reasonably thorough effort to make sure that no mistakes were made, they appear to show that these samples contain an impossible amount of Be-10. What does impossible mean? Basically, if a surface is left undisturbed and exposed to the cosmic-ray flux long enough, the concentration of cosmic-ray-produced Be-10 will eventually build up to a level high enough that the rate that Be-10 is lost by radioactive decay is equal to the production rate. At this point, the amount of Be-10 can’t increase any more. So this relationship provides a limit to the maximum possible concentration of Be-10 that can be present in quartz at a particular site. Generally this condition is referred to as either ‘production-decay equilibrium,’ or, more commonly, just ‘saturation.’ In math, accumulation of Be-10 in a non-eroding surface is governed by the differential equation:

$\frac{dN}{dt} = P - N\lambda$

where N is the Be-10 concentration (atoms/g), P is the production rate at the site (atoms/g/yr), and $\lambda$ is the decay constant for Be-10 (4.99e-7 /yr). So saturation occurs when $dN/dt = 0$, or when $N = P/\lambda$. Thus, if we know the production rate at some site, we equivalently know the saturation concentration.

At Gordon’s sample site, the production rate (calculated with the Antarctic atmosphere model and scaling scheme of Stone(2000) and the “primary” production rate calibration data set of Borchers and others (2015), the Be-10 production rate is 38.2 atoms/g/yr, which means the saturation concentration for Be-10 is 7.65 x 10^7 atoms/g. However, the sample actually contains 8.5 x 10^7 atoms/g. Inconceivable!  If we accept this production rate estimate, this is an impossible Be-10 concentration.

OK, that seems weird, but it’s not actually all that  unusual in Antarctica. To show this, look at nearly all known Be-10 and Al-26 measurements from Antarctica as they are represented in the ICE-D:ANTARCTICA database (making sure they are all correctly normalized to the ’07KNSTD’ and ‘KNSTD’ standardizations, respectively). This data set looks like this:

What I have done here is just plot nuclide concentration vs. elevation. Because production rates increase with elevation, the nuclide concentration associated with a particular exposure age also increases with elevation, so the envelope of possible nuclide concentrations that one could potentially observe widens as elevation increases. There are about 1200 Be-10 measurements and 500 Al-26 measurements represented here.

Now, because we are using an atmosphere model and scaling scheme that vary only with elevation (magnetic cutoff rigidity is pretty much effectively zero in Antarctica for all practical purposes), predicted saturation concentrations for Be-10 and Al-26 are also only a function of elevation, so we can plot them on these axes, as follows:

Here the black line shows predicted saturation concentrations. It is evident that there are lots of samples in Antarctica, mostly at high elevation above about 2000 m, whose Be-10 concentrations appear to be above saturation. Gordon’s samples are sitting off to the right of the line at about 2300 m elevation. Some measurements are up to 12% above saturation concentrations calculated in this way.

There are four possible reasons for this. One, the measurements are somehow messed up. This is unlikely, because the measurements represented here are from several different laboratories and/or AMS facilities, and it’s hard to come up with any reason why they would all be similarly spurious. Certainly it is suspicious that the maximum amount that some samples exceed predicted saturation concentrations (12%) is about the same as the difference between the KNSTD and 07KNSTD standardization (11%). However, I spot-checked most of these samples, most of the measurements are well documented, and I am pretty sure that this error has not been made.

The second reason is that we might be overestimating the atmospheric pressure at high-elevation sites, and therefore underestimating the production rate and the saturation concentration. That possibility we can evaluate by using a totally different atmosphere model, one derived from the ERA40 reanalysis and adapted for use in estimating production rates by Nat Lifton (you can read about it here). To do this, we need slightly different axes, because in the ERA40-derived atmosphere model, atmospheric pressure varies with location as well as elevation. So instead of plotting just raw nuclide concentrations, we will plot the ratio of the measured nuclide concentration at a particular site to the saturation concentration predicted for that site. For samples above saturation, this ratio will be greater than one.

The two left-hand panels use the Stone (2000) Antarctic atmosphere, and the two right-hand panels use the ERA40-derived atmosphere. Basically, there is no difference. Both atmosphere models with this scaling scheme indicate that there are a bunch of samples at high elevations that exceed saturation concentrations for both Be-10 and Al-26. The fact that two independent atmosphere models agree on this point would tend to indicate that our problem cannot be explained just by a bad atmosphere approximation.

The third reason is that the sites where we observe greater-than-saturated nuclide concentrations might have changed elevation. If the samples had spent a significant amount of their exposure history at an elevation higher than their present elevation, they could potentially have higher nuclide concentrations than they could ever achieve at their present elevations. Elevation changes could be caused by local effects (for example, all the samples are on the hanging wall of an active normal fault), or by regional effects (for example, changes in dynamic topography during the several-million-year exposure history of these samples; or glacial isostasy, if the ice sheet surrounding sample sites had continuously thickened during the period of exposure). The amount by which the offending samples exceed predicted saturation concentrations requires approximately 300-400 meters of elevation change in the last few million years. We can evaluate some of these possibilities by looking at where these samples are actually located in Antarctica. Here they are:

The white dots are Be-10 measurements exceeding saturation, and the gray ones are for Al-26. There are some more white dots hidden behind some of the gray dots. These samples are widely distributed around the high mountains of East Antarctica, which makes it unlikely that concentrations above saturation are due to local faulting or mass-movement effects at a particular location. The background image in this figure is a model for changes in dynamic topography in the last 3 Ma from a recent paper by Austermann and others; the color scale shows total uplift in the past 3 Ma in meters. Samples exhibiting Be-10 concentrations above saturation are not located in areas that have experienced signficant regional subsidence, at least due to this process, over long time periods. In fact a few of them are located in areas that are predicted to have experienced significant uplift. So we can probably exclude local faulting as a blanket explanation for the many samples in Antarctica that have Be-10 concentrations above saturation, and we can also exclude long-wavelength elevation change due to mantle dynamics. What about glacial isostasy? If the ice sheet in the vicinity of these samples had continuously thickened over the past several million years, then the elevation of the samples would have lowered during that time due to isostatic compensation, potentially leading to above-saturation nuclide concentrations. However, we need this change to be large — order 1000 m of ice to obtain the needed few hundred meters of isostatic response — and this is not consistent with sea level records or ice sheet modeling.

For example, the above figure shows changes in bed elevation from a 5-million-year-long Antarctic ice sheet simulation by David Pollard and others, at two locations that are representative of where we see Be-10 concentrations above saturation: the central Transantarctic Mountains (red), and the East Antarctic marginal highland in the area of the Sor Rondane mountains (blue). At both of these sites, this model run predicts a long-term decrease in land surface elevation over the last ~4 Ma, presumably driven by Antarctic ice sheet expansion due to lower sea levels in the Pleistocene. However, the magnitude of the decrease is only several tens of meters, much too small to account for the above-saturation Be-10 concentrations. Thus, long-term lowering of sample sites due to glacial isostasy does not appear to be a good explanation for above-saturation cosmogenic-nuclide concentrations either.

The fourth possibility, of course, is that our production rate scaling is incorrect. I’ve left it for last, of course, because it’s the most likely. We don’t have any Be-10 or Al-26 production rate calibration data at high elevation near the poles (see here), so we are relying on the scaling method of Stone (2000), which is of course just based on the work of Devendra Lal, to extrapolate from production rate calibration sites at low elevation to sample sites at high elevation. We can test whether inaccuracy in this scaling method explains the apparently-above-saturation samples, by trying a different scaling method. Specifically, let’s try the more recent (and much more complicated) scaling method of Lifton and others (2015), now commonly known as the ‘LSD’ scaling method. At low magnetic cutoff rigidity (i.e., in polar regions), this scaling method predicts a larger altitude dependence than Lal-based scaling methods. Because magnetic field variability is out of the picture at polar latitudes, this elevation dependence is the primary difference between the two scaling methods. Here are the results, calculated in the same way as for the four-panel figure above (compare with the first two panels of that figure):

Using the LSD scaling method completely fixes the problem (for both atmosphere models, although only one is shown here). Here is another view that copies the first figure that I started with above, showing raw concentration vs. elevation, compared with saturation concentrations. The solid lines are saturation concentrations predicted using the Stone/Lal scaling method as shown above, and the dashed lines are saturation concentrations predicted using the LSD scaling method. If we use the LSD scaling method, there are no measured Be-10 or Al-26 concentrations in Antarctica that are impossibly high.

So, overall, nuclide concentrations in many high-elevation Antarctic surfaces are inconsistent with production rates estimated using the Lal/Stone scaling method, but are consistent with production rates estimated using the LSD method. This would appear to imply that LSD scaling performs better for this data set, and, by implication, very likely for high-elevation sites in polar regions generally.

This is interesting for several reasons.

First, it would tend to indicate that one should probably use LSD rather than Lal-based scaling to compute exposure ages from old, high-elevation glacial deposits in  Antarctica. It makes a pretty big difference: almost 20% at 2500 m elevation. For example, you might find a moraine at this elevation that belonged to the mid-Pliocene warm period (ca. 3-3.3 Ma) if you used Lal-based scaling, but was coeval with the onset of Northern Hemisphere glaciation (ca. 2.7 Ma) if you used LSD scaling. Certainly some potential for confusion there.

Second, although the LSD scaling method is very much more complex than the Lal-based methods, and there are some physical arguments that indicate that LSD should be more accurate than Lal, so far it has been very difficult to show that there is any practical difference in performance between the two. This  issue is important mainly because of the complexity of the LSD scheme — it requires a lot more computation, so code to implement it is more complicated and very much slower. If we don’t have to use it, we’d rather not. The calibration exercise of Borchers and others (2015) showed that for available Be-10 and Al-26 calibration data, there was basically no detectable difference in performance, and that’s been taken (certainly by me) to indicate that for most practical purposes, it’s fine to use the simpler method. However, the Antarctic exposure-age data set would tend to indicate the opposite: that the Lal-based methods, when calibrated with existing calibration data sets that lack coverage at high altitudes in polar regions, significantly underestimate production rates at these sites.

Third, if we accept that the LSD scaling method is correct, we’ve gone from the conclusion that there are a lot of Be-10 and Al-26-saturated surfaces in Antarctica, to the conclusion that there aren’t any. No known measurements from Antarctica quite overlap with predicted saturation values. In one sense, this is not really a huge surprise, because for a site to reach true production-decay saturation, the surface erosion rate must be exactly zero for several million years. Really exactly zero erosion for millions of years is a big ask — this doesn’t even occur in outer space — and is probably unlikely to occur in reality, even at -60° C in the upper Transantarctic Mountains. Again, if we accept that the LSD scaling method is correct, we can estimate steady-state erosion rates (that is, maximum limits on the erosion rate) for the samples that are closest to saturation; for the Be-10 measurements this is in the range of 1-2 cm per million years; for the Al-26 measurements it is 3-4 cm per million years. It is not clear why Be-10 measurements are closer to saturation than Al-26 measurements, although because they are all very close to saturation, this calculation is quite sensitive to the calibrated reference production rates for both nuclides, so this could easily be explained by inaccuracies in those values. In any case, these are plausible values for the lowest erosion rates in Antarctica. Not zero, but extremely slow.

So the summary here is that it appears that the LSD scaling method is consistent with the overall data set of Be-10 and Al-26 concentrations in Antarctic surfaces, and the Lal-Stone scaling method isn’t. One final point, however, is that Stone (2000) also considered this issue (with fewer data; see Figure 4 in that paper) and concluded that the Lal scaling method did not, in fact, undershoot saturation concentrations. The difference, I am fairly sure, is that new calibration data imply lower production rates than the calibration data available at the time of that paper; if currently available production rate calibration data had been used in that paper they would have led to the same conclusion I come to here, that Lal-based scaling underestimates saturation concentrations at high elevations in Antarctica. However, there are a lot of moving parts involved in comparing the two calculations (Be-10 standardizations, production rates, decay constants, etc….) and that claim could probably use further investigation.

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