A paper by Joerg Schaefer and numerous others, about exposure ages on Holocene moraines in New Zealand, came out in Science a couple of weeks ago. The overall point of this paper is that they exposure-dated a large number of Holocene moraines and discovered that Holocene glacier advances in New Zealand were neither synchronous or asychronous with Northern Hemisphere glacier advances. This is interesting from the paleoclimate perspective, and I’ve talked about why at some length in a commentary article in the same issue (you can get this article by clicking through this page).

From the perspective of cosmogenic-nuclide exposure-dating nerds, there are some other interesting features to this article that are much too obscure for the general Science readership. Thus, commentary here. Mainly, the Schaefer paper is notable because i) they measured a very large number of exposure ages, ii) they were very young exposure ages, and iii) they are very precisely measured. Items ii) and iii) mainly highlight hard work in the Lamont chem lab and impressive AMS skills at LLNL-CAMS, but i) is interesting because the size of the data set makes it possible to take a look at some of the beliefs that come into relating exposure ages of moraine boulders to the age of the glacier advance that formed the moraine.

Basically, when one measures the exposure age of a bunch of boulders on a particular moraine, they aren’t all the same. They differ, for three reasons. First, all boulders could have the same true exposure age but their measured exposure ages would still differ due to measurement uncertainty. Second, the boulders could have different true exposure ages because of post-emplacement moraine disturbance, that is, some boulders may have been uncovered, or had their surfaces eroded, after the moraine was abandoned. This is most likely not the primary issue for the very young moraines in the Schaefer paper. Third, some or all boulders could have had an “inherited” exposure age when they were initially emplaced on the moraine, that was left over from some past period of exposure that the boulder experienced.

Mostly people deal with this issue by using some sort of a statistical test to determine whether or not the distribution of measured ages could be accounted for by measurement error alone. If their observed ages pass this test, then they can reasonably average the measured ages and argue that this number is really the true age of the moraine. The reduced chi-squared statistic is commonly used for this; this statistic compares the deviations of the measurements from their mean with the uncertainty of all the measurements and comes up with a summary statistic. If the value is approximately equal to 1, then the scatter in the measurements is about as expected from the measurement uncertainties. If it’s significantly larger (what “significantly” means depends on the number of data), then some other source of scatter is present, which implies that averaging the data probably does not give the true moraine age.

The interesting thing about the Schaefer data is that the reduced chi-squared value for exposure ages from a particular moraine increases — a lot — as the age of the moraine becomes smaller. That is, for the older moraines, the exposure ages are scattered about as much as one expects from measurement uncertainty — reduced chi-squared values are near one — but for the younger moraines, the scatter is much more than expected from measurement uncertainty alone. The following plot shows the relationship between mean moraine age and reduced chi-squared for the Schaefer data set.

So the question is, what source of scatter becomes more important as exposure ages get younger? Boulder surface erosion and moraine degradation cannot explain this effect: both of these processes would result in greater excess scatter in *older*, rather than *younger*, moraines. The obvious answer to this is cosmogenic-nuclide inheritance. Many if not most of the boulders on these moraines must have originated as supraglacial debris shed from cliffs above the glaciers, so they were exposed to the cosmic-ray flux for at least some time prior to transport and emplacement in the moraines. Thus, they must have contained some inherited Be-10 at the time they were delivered to the moraines. Of course, we don’t know how much. The advantage of this data set is that it enables us to estimate how much. So make two assumptions: first, all moraine boulders were emplaced at the same time and experienced simple exposure thereafter; second, all boulders contain inherited Be-10 whose concentration is variable, but obeys a uniform distribution between 0 and a maximum value. With these assumptions, the exposure age of a moraine boulder is:

Where *t* is the apparent Be-10 exposure age of a boulder (yrs; this is what we observe and is just the measured Be-10 concentration divided by the production rate); *t_true *is the true depositional age of the moraine (this is what we want to know); *e(t*) is a measurement error term (yr) which is normally distributed around zero and has a standard deviation equal to the 1-sigma measurement error of the apparent age; and *t_i *is an inherited Be-10 concentration, expressed as an age, which is uniformly distributed between 0 and a maximum inherited age *t_i,max* (alternatively, *t_i *is the inherited Be-10 concentration divided by the Be-10 production rate). By generating random values of *t_i *and *e(t) *that follow the specified probability distributions, we can simulate the apparent exposure age distributions we would observe on a moraine of a certain age. We can then repeat this experiment multiple times and look at the distribution of reduced chi-squared values we would expect for moraines of various ages and uniformly distributed inheritance.

One important thing is that the measurement uncertainty is a function of the age — older boulders have more atoms and hence lower measurement uncertainties. We need to account for this, which we can accomplish by using the relationship between age and uncertainty in the Schaefer data to estimate measurement uncertainties in our simulation. Here is the relationship:

The red dots are the actual measurements; the blue line is a model relationship that I will use in the simulation.

To get to the point, here is the result of a Monte Carlo simulation that predicts the relationship between moraine age and reduced chi-squared — the relationship that caught my attention in the first place — by repeatedly evaluating the equation above to generate synthetic apparent age distributions, and then computing their reduced chi-squared values. The only free parameter left in our model is the maximum value of inheritance. It turns out that the value of this parameter that best reproduces the observed relationships is 200 years, that is,boulder inheritance is uniformly distributed between 0 and 200 yrs. The red dots are the data shown above; the solid blue line is the average value of the reduced chi-squared predicted in the simulation for a moraine of a particular age, and the dashed blue lines are the 1-sigma confidence bounds on that value.

Clearly this does a pretty good job of reproducing the observed striking relationship between reduced chi-squared and moraine age. For comparison, the next figure shows the result for 0-200 years (blue) compared with the results for 0-100 (red) and 0-300 (green) years:

So the summary of this all is that an extremely simple model for inheritance enables us to i) very precisely reproduce the striking relationship between moraine age and excess age scatter evident in the Schaefer data set, and ii) quantitatively estimate the mean inheritance in the boulders. To summarize, the average value of inheritance in these moraine boulders is most likely near 100 yrs.

The authors of the paper actually tried to estimate inheritance in a different way by dating several boulders on a historically observed moraine that was known to have formed ca. 1860-1895, i.e. 100 years ago. Several boulders from this moraine had apparent ages scattered between 150-200 yr, indicating inheritance of 50-100 yr. Which estimate is better? Both are consistent; a few values drawn from a uniform distribution between 0-200 yr could easily fall between 50-100 yr. For the older moraines, it seems most sensible to subtract 100 +/- 60 (the mean standard deviation of a uniform distribution between 0-200) years from apparent exposure ages to arrive at a better estimate for the true age of the moraine.

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## Be-10 inheritance in young New Zealand boulders

A paper by Joerg Schaefer and numerous others, about exposure ages on Holocene moraines in New Zealand, came out in Science a couple of weeks ago. The overall point of this paper is that they exposure-dated a large number of Holocene moraines and discovered that Holocene glacier advances in New Zealand were neither synchronous or asychronous with Northern Hemisphere glacier advances. This is interesting from the paleoclimate perspective, and I’ve talked about why at some length in a commentary article in the same issue (you can get this article by clicking through this page).

From the perspective of cosmogenic-nuclide exposure-dating nerds, there are some other interesting features to this article that are much too obscure for the general Science readership. Thus, commentary here. Mainly, the Schaefer paper is notable because i) they measured a very large number of exposure ages, ii) they were very young exposure ages, and iii) they are very precisely measured. Items ii) and iii) mainly highlight hard work in the Lamont chem lab and impressive AMS skills at LLNL-CAMS, but i) is interesting because the size of the data set makes it possible to take a look at some of the beliefs that come into relating exposure ages of moraine boulders to the age of the glacier advance that formed the moraine.

Basically, when one measures the exposure age of a bunch of boulders on a particular moraine, they aren’t all the same. They differ, for three reasons. First, all boulders could have the same true exposure age but their measured exposure ages would still differ due to measurement uncertainty. Second, the boulders could have different true exposure ages because of post-emplacement moraine disturbance, that is, some boulders may have been uncovered, or had their surfaces eroded, after the moraine was abandoned. This is most likely not the primary issue for the very young moraines in the Schaefer paper. Third, some or all boulders could have had an “inherited” exposure age when they were initially emplaced on the moraine, that was left over from some past period of exposure that the boulder experienced.

Mostly people deal with this issue by using some sort of a statistical test to determine whether or not the distribution of measured ages could be accounted for by measurement error alone. If their observed ages pass this test, then they can reasonably average the measured ages and argue that this number is really the true age of the moraine. The reduced chi-squared statistic is commonly used for this; this statistic compares the deviations of the measurements from their mean with the uncertainty of all the measurements and comes up with a summary statistic. If the value is approximately equal to 1, then the scatter in the measurements is about as expected from the measurement uncertainties. If it’s significantly larger (what “significantly” means depends on the number of data), then some other source of scatter is present, which implies that averaging the data probably does not give the true moraine age.

The interesting thing about the Schaefer data is that the reduced chi-squared value for exposure ages from a particular moraine increases — a lot — as the age of the moraine becomes smaller. That is, for the older moraines, the exposure ages are scattered about as much as one expects from measurement uncertainty — reduced chi-squared values are near one — but for the younger moraines, the scatter is much more than expected from measurement uncertainty alone. The following plot shows the relationship between mean moraine age and reduced chi-squared for the Schaefer data set.

So the question is, what source of scatter becomes more important as exposure ages get younger? Boulder surface erosion and moraine degradation cannot explain this effect: both of these processes would result in greater excess scatter in

older, rather thanyounger, moraines. The obvious answer to this is cosmogenic-nuclide inheritance. Many if not most of the boulders on these moraines must have originated as supraglacial debris shed from cliffs above the glaciers, so they were exposed to the cosmic-ray flux for at least some time prior to transport and emplacement in the moraines. Thus, they must have contained some inherited Be-10 at the time they were delivered to the moraines. Of course, we don’t know how much. The advantage of this data set is that it enables us to estimate how much. So make two assumptions: first, all moraine boulders were emplaced at the same time and experienced simple exposure thereafter; second, all boulders contain inherited Be-10 whose concentration is variable, but obeys a uniform distribution between 0 and a maximum value. With these assumptions, the exposure age of a moraine boulder is:Where

tis the apparent Be-10 exposure age of a boulder (yrs; this is what we observe and is just the measured Be-10 concentration divided by the production rate);t_trueis the true depositional age of the moraine (this is what we want to know);e(t) is a measurement error term (yr) which is normally distributed around zero and has a standard deviation equal to the 1-sigma measurement error of the apparent age; andt_iis an inherited Be-10 concentration, expressed as an age, which is uniformly distributed between 0 and a maximum inherited aget_i,max(alternatively,t_iis the inherited Be-10 concentration divided by the Be-10 production rate). By generating random values oft_iande(t)that follow the specified probability distributions, we can simulate the apparent exposure age distributions we would observe on a moraine of a certain age. We can then repeat this experiment multiple times and look at the distribution of reduced chi-squared values we would expect for moraines of various ages and uniformly distributed inheritance.One important thing is that the measurement uncertainty is a function of the age — older boulders have more atoms and hence lower measurement uncertainties. We need to account for this, which we can accomplish by using the relationship between age and uncertainty in the Schaefer data to estimate measurement uncertainties in our simulation. Here is the relationship:

The red dots are the actual measurements; the blue line is a model relationship that I will use in the simulation.

To get to the point, here is the result of a Monte Carlo simulation that predicts the relationship between moraine age and reduced chi-squared — the relationship that caught my attention in the first place — by repeatedly evaluating the equation above to generate synthetic apparent age distributions, and then computing their reduced chi-squared values. The only free parameter left in our model is the maximum value of inheritance. It turns out that the value of this parameter that best reproduces the observed relationships is 200 years, that is,boulder inheritance is uniformly distributed between 0 and 200 yrs. The red dots are the data shown above; the solid blue line is the average value of the reduced chi-squared predicted in the simulation for a moraine of a particular age, and the dashed blue lines are the 1-sigma confidence bounds on that value.

Clearly this does a pretty good job of reproducing the observed striking relationship between reduced chi-squared and moraine age. For comparison, the next figure shows the result for 0-200 years (blue) compared with the results for 0-100 (red) and 0-300 (green) years:

So the summary of this all is that an extremely simple model for inheritance enables us to i) very precisely reproduce the striking relationship between moraine age and excess age scatter evident in the Schaefer data set, and ii) quantitatively estimate the mean inheritance in the boulders. To summarize, the average value of inheritance in these moraine boulders is most likely near 100 yrs.

The authors of the paper actually tried to estimate inheritance in a different way by dating several boulders on a historically observed moraine that was known to have formed ca. 1860-1895, i.e. 100 years ago. Several boulders from this moraine had apparent ages scattered between 150-200 yr, indicating inheritance of 50-100 yr. Which estimate is better? Both are consistent; a few values drawn from a uniform distribution between 0-200 yr could easily fall between 50-100 yr. For the older moraines, it seems most sensible to subtract 100 +/- 60 (the mean standard deviation of a uniform distribution between 0-200) years from apparent exposure ages to arrive at a better estimate for the true age of the moraine.

## Like this:

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