Here is a screen grab of the header section of a SUERC results spreadsheet (with sample names blurred to protect the innocent).

After the column for the sample name, this shows the following data.

First, two columns headed “% of standard” and “ (% of standard)”. These columns are the most basic description of the actual AMS measurement: remember, what the AMS measurement actually does is compare the Be-10/Be-9 ratio in a sample to the Be-10/Be-9 ratio in a standard. The first line of the file tells us that a sample called “NIST,” which is presumably the NIST “SRM 4325″ Be standard material, has a Be-10/Be-9 ratio that is 100% of the standard, without uncertainty. The NIST standard material, as already discussed, is a stock of Be whose Be-10/Be-9 ratio is independently known. What this means, therefore, is that the NIST standard is the standard to which all the measurements are referenced. Subsequent lines then describe the relationship between the 10/9 ratio in each unknown sample and that in the standard. The 10/9 ratio in the first sample, for example, is 0.465% of the 10/9 ratio in the NIST standard. These lines have uncertainties in this relationship; the size of the uncertainty mostly depends on how many Be-10 atoms were actually counted. So these two columns, once again, are the actual data that was collected by the AMS — the relationship between the 10/9 ratio in  a sample and that in a standard.

The overall goal of this exercise, of course, is to determine the actual Be-10/Be-9 ratio in your sample. So if we define to be the 10/9 ratio in an unknown sample, to be the 10/9 ratio in the NIST standard, and to be the ratio of 10/9 ratios that we have measured, then: i) we want to know ,  ii) , and iii) we have measured , so iv) to compute the answer we want, we need a value for .

As discussed in the previous post, the absolute isotope ratio of the NIST Be standard is based on two measurements: a measurement of the total amount of Be present, and a measurement of the activity, that is, the rate of radioactive decay, of the Be-10 present. Thus, a value for the Be-10 decay constant, or equivalently the Be-10 half-life, is required to compute this ratio. If we assume a value for the Be-10 half-life, we can compute the amount of Be-10 present in the standard material, we can then compute the absolute 10/9 ratio of the standard, and we can apply the measurement described above to compute the absolute 10/9 ratio for an unknown sample. This is what happens in the red, blue, and green columns above.

The red columns — columns 4 and 5 if in the screen grab above — have the header line “10Be/9Be t(1/2)=1.53 Ma.” What this means is that a value of 1.53 Ma for the half-life of Be-10 was used to compute the amount of Be-10 present in the NIST standard, and thus its absolute 10/9 ratio. Given the Be-10 activity actually measured and the equations in the previous post, this yields an absolute 10/9 ratio of 3.06 x 10^-11 for the standard. Then to compute the absolute 10/9 ratio in the first sample, we apply the relationship described above: 3.06 x 10^-11 x 0.00465 = 1.42 x 10^-13. Again, if we assume that the NIST standard material has an absolute isotope ratio of 3.06 x 10^-11, which follows from its measured activity and the assumption that the Be-10 half-life is 1.53 Ma, then this sample has a 10/9 ratio of 1.42 x 10^-13. One would describe this measurement as being “normalized to the NIST standard with an assumed isotope ratio of 3.06 x 10^-11.” To use an atoms/g concentration calculated from this measured ratio in the online exposure age calculators, one would refer to the table of standardizations here and use the “NIST_30600″ standardization.

The next two columns, colored green, are headed “10Be/9Be t(1/2) = 1.34 Ma.” If we assume that the Be-10 half-life is 1.34 Ma, which was the value originally assumed in preparation of the NIST standard, then the activity measurement implies a true 10/9 ratio for the standard of 2.68 x 10^-11. This is the “certified” ratio for the NIST standard. In this case, we would compute the true 10/9 ratio of the sample by 2.68 x 10^-11 x 0.00465 = 1.25 x 10^-13. So we started with the same AMS measurement, but by assuming a different value for the Be-10 half-life, we obtained a different true isotope ratio for the NIST standard, and thus a different true isotope ratio for the sample. This value for the true 10/9 ratio in the sample would be described as “normalized to the NIST standard with an assumed isotope ratio of 2.68 x 10^-11,” and the standardization code for the online calculators is “NIST_Certified.”

The final two columns, colored blue, are headed “10Be/9Be t(1/2) = 1.36 Ma.” If we assume that the Be-10 half-life is 1.36 Ma, which was a value estimated by Kuni Nishiizumi as a byproduct of creating the implantation standards described in this previous post, then the activity measurement implies a true 10/9 ratio for the standard of 2.79 x 10^-11. In this case, we would compute the true 10/9 ratio of the sample by 2.79x 10^-11 x 0.00465 = 1.30 x 10^-13.  This value for the true 10/9 ratio in the sample would be described as “normalized to the NIST standard with an assumed isotope ratio of 2.79 x 10^-11,” and the standardization code for the online calculators is “NIST_27900.”

To summarize, there is a lot of redundant information in this spreadsheet. The actual measurement that was made — a comparison of the 10/9 ratio in the sample with that of the NIST standard — is presented four different ways. Personally, I find this confusing. The important thing, however, is that these all describe the same measurement. Calculating atoms/g in your sample using the ratio reported in the red columns and, in the online exposure age calculator, identifying it with the “NIST_30600″ standardization, will yield the same exposure age as calculating atoms/g using the ratio in the blue columns and identifying it with the “NIST_27900″ standardization.

Basically, the caption above says what this is: it’s a way of representing a lot of measurements of the same thing that have Gaussian uncertainties. You draw a Gaussian with mean and standard deviation corresponding to each of your individual measurements, and then add them all up to obtain a summary curve. The use of this type of diagram in geochronology dates back to the ’80′s, mostly in the fission-track and argon-argon dating literature — here is an interesting example from a paper on the ages of spherules in the lunar regolith from Tim Culler (Science 287 pp. 1785-1788):

More relevant to glacial chronology might be another example in a paper by Tom Lowell about radiocarbon dating of LGM moraines in Ohio (Lowell, T.V., 1995. The application of radiocarbon age estimates to the dating of glacial sequences: an example from the Miami Sublobe, Ohio, USA. Quaternary Science Reviews 14, 85–99.). The point of this post is to explain what the point of this diagram is, why and when you should use it, and how to apply it both rightly and wrongly to exposure-age data. As with all good statistical constructs, it can be useful for misleading readers into thinking what you want them to think.

First, what is the point of this diagram? Basically, we are using this diagram to describe the frequency distribution of observations. We have made a set of measurements of what we believe to be the same thing, and we want to represent the distribution of those measurements. Normally to carry out this task, we would use a histogram, which is a fairly basic sort of a diagram in which we divide the observation space into bins, determine how many observations fall into each bin, and then fill each bin with a bar whose height is proportional to the number of measurements. Let’s say we measured a bunch of exposure ages, in thousands of years BP (i.e., ka), on a moraine, and got the following results:

[23.1 24.1 16.3 24.1 21.3 15.9 17.8 20.5 24.6 24.6 16.6 24.7 24.6 19.9 23.0 16.4 19.2 24.2 22.9 24.6 ]

We could create a histogram of these data by defining bins, let’s say with a width of 2000 years and starting at 0, and assigning these data to bins. An exposure age of 19.2 ka goes in the 18-20 ka bin, an exposure age of 22.9 ka goes in the 22-24 ka bin, etc. and so on. This yields the following table of how many samples fall in each bin:

Which in turn produces the following histogram:

The x-axis is the exposure age, each bar is a bin, and the y-axis is the number of samples that fall into each bin. Three important points about histograms. First, they represent an observed frequency distribution of measurements. They’re not necessarily a probability distribution function for the ages of boulders on the moraine. If you i) made the additional assumption that the probability of observing a certain exposure age is exactly equal to the frequency distribution of exposure ages we have already observed (which is highly restrictive, but might be true if you had analysed all the boulders on the moraine), and then ii) renormalized the y-axis so that the sum of all bar heights was equal to 1, then you would arguably have a probability density function for boulder age. Second, you need to make two arbitrary decisions when you create a histogram: how wide are the bins, and where are they located? If you change these things, the histogram changes. Third, there is no uncertainty in histograms. Each measurement goes in one and only one bin.

Whether a histogram is or is not a probability density function is largely semantic and depends on your definition of terms, but the second and third points above mean that histograms are a lousy way to represent data when either one of two things are true: i) there are only a few measurements, and ii) the measurements have uncertainty associated with them. Obviously, these two things describe most geochronological data, cosmogenic-nuclide exposure ages in particular. We don’t collect very many because they’re expensive, and they have measurement uncertainty. So here is an example of how wrong you can go with a histogram representation of exposure-age data. Let’s say you analysed two boulders and found them to have apparent exposure ages of 16.9 +/- 2.1 ka and 18.2 +/- 1.5 ka. There are two important things about these results. First, the two ages are different. Second, they agree when their uncertainties are taken into account. However, it’s impossible to communicate both of these important observations at the same time using a histogram. Here’s one possible histogram for these ages:

This one gives the impression that the two ages are irreconcilably different. Wrong. So that’s misleading. How about this one:

That one indicates that the two ages are the same. Also wrong and misleading. The point is that when data are sparse and have measurement uncertainties, representing their distribution with histograms fails to communicate the information we are trying to communicate. This is the problem that “camel diagrams” are intended to solve. In constructing a histogram, we are basically representing each measurement by a rectangle with width equal to the bin width, and then adding the representations of all the samples together to get a summary histogram. Now what we will do instead is represent each measurement by something other than a rectangle. Usually, because we are generally working with cosmogenic-nuclide measurements that have normal, i.e. Gaussian, uncertainties, we represent each sample by a Gaussian-shaped curve. This is just a curve generated by the formula for a normal probability distribution:

where is the mean and is the standard deviation of the probability distribution. To do this for a single exposure age, we take the age we measured to be and the 1-standard-error uncertainty in the age to be . Doing this for the two data just mentioned above gives:
Representing the measurements as Gaussian curves visually communicates a lot of important things that we couldn’t communicate with the histogram. First, although the measurements are different, they are similar in light of their measurement uncertainties — if we envision each curve as something like a probability density function for the actual age of each of the samples, then the fact that there is a lot of overlap between the curves indicates that there is a high likelihood that they are both measurements of the same thing, and are different only because of measurement error. Second, we can compare the difference between the best estimate of each measurement — the location of each peak — and the size of the uncertainty on each measurement. In this case the measurements are more similar than their uncertainties, which also communicates the high likelihood that they are both measurements of the same thing. Third, because the formula for a Gaussian curve is defined such that the area under each curve is always the same, the height of each curve is inversely proportional to measurement uncertainty. This feature draws the eye immediately toward the most precise, i.e. tallest, measurements, and the viewer naturally tends to give those more weight. So representing data by continuous Gaussians instead of rectangles clears up a lot of the visual misrepresentation that histograms incur with small and uncertain data sets.

Typically one then adds the Gaussian curves corresponding to the single measurements together to come up with a summary plot, as follows:

The black line is the sum of the two individual Gaussians. The fact that it has only one peak correctly visually communicates the idea that the two measurements are both inaccurate measurements of the same thing — the true age of whatever we are dating — and considering them together tells us that this true age is likely to be somewhere between the two measurements, slightly closer to the more precise of the two. Basically we are performing a sort of a visual maximum likelihood estimate.

If we had data that didn’t agree even considering their uncertainties, we’d get something like this:

Because there’s not much overlap between the two Gaussian curves, they don’t add much to each other and we have a two-hump rather than one-hump summary plot. Hence the name “camel plot.” One hump good, two humps bad.

So the summary is that this type of a presentation, in which we represent observations by continuous functions rather than a histogram, solves the fact that a histogram fails to communicate the important information about small data sets with measurement uncertainty.

The next question is what to call it. The term “camel diagram,” while easy to remember, is pretty dumb. It’s not a histogram. It’s not really a probability density function (as suggested in the caption from Meredith Kelly’s paper given as an example above) because it’s not intended to represent the probability of observing a particular outcome — it’s intended to represent the frequency distribution of measurements already collected. This fact led Culler and others (other example above) to call it an “ideogram created by summing the Gaussians.” As the word “ideogram” is more commonly used to describe written characters in Chinese and other “ideographic” languages that communicate an entire idea or concept by a single character, using “ideogram” to describe this sort of a plot is, at the very least, confusing. Really what it is is a sort of a smoothed frequency distribution, and the proper statistical term for it is a “normal kernel density estimate.” This term communicates the fact that we are trying to estimate the frequency density of actual observations. The “kernel” is just what sort of shape is used to represent each datum. In a standard histogram, the kernel is a rectangle. Here it is the equation of a normal, i.e. Gaussian, PDF, so it is a “normal kernel.” In principle one could have any sort of kernel — triangular, Poisson, sinusoidal, anything you want. There is a lot of statistical research devoted to the proper way to construct a kernel density estimate.

When and why to use it? As noted above the value of this type of plot is in overcoming the fact that histograms are visually misleading for sparse and uncertain data. If you have sparse and uncertain data, the camel diagram is a very good way to visually communicate a lot of the important conclusions that should be drawn from the data. For this reason, it’s a very good way of presenting geochronological data. It doesn’t make a whole lot of sense to use it when the opposite things are true of your data set — data that are numerous and whose uncertainties are very small compared to the spread in their values are easily and honestly presented in a histogram.

Finally, several marginally related things to note. First, the fact that one adds the Gaussian kernels together is also related to the question of whether this diagram is a density estimate and not a probability estimate. If, as in the example above, we had two age measurements with Gaussian uncertainties, and we took each of those to be a probability density function for the age of the landform, then if we wanted to combine them into a single probability distribution, one could argue that we would instead want to multiply them to obtain the joint probability of both things being true at once.

Second, one potential serious error in the use of this diagram for geochronological data occurs in the situation where an age measurement is not distinguishable from zero. Let’s say you have an exposure age of 300 +/- 200 years on a Little Ice Age moraine. If you take this to be a Gaussian uncertainty, then you are saying there is a finite probability that the age is less than zero. Of course, it is not possible that the age of the boulder is less than zero, so taking this uncertainty to be Gaussian is wrong. Hence, a normal kernel density estimate representation of such data would also be misleading. In principle, you could overcome this by using a different type of kernel — Poisson for example — that always goes to zero at t = 0. Again, there is lots of statistical literature describing improved kernels that correct this problem.

Third, one question specifically about applying camel diagrams to exposure-age data: which uncertainty to use? Commonly in exposure dating we talk about two different values for the uncertainty. The so-called  ”internal” uncertainty includes only measurement uncertainty on the cosmogenic-nuclide concentration. So if we make a Be-10 measurement with 5% precision, then the internal error on the exposure age calculated from that Be-10 concentration is also 5%. The so-called “external” uncertainty adds uncertainty in the nuclide production rate that we use to compute the exposure age from the Be-10 concentration. For example, if the production rate uncertainty is 10%, then the same Be-10 measurement will yield an external uncertainty on the exposure age of about 12%. The important difference between these two is that the internal uncertainties are independent between exposure ages for samples from the same location, whereas the external uncertainties are not — they are all subject to a shared production rate uncertainty. This means that when comparing two samples at the same site to each other, one needs to use the internal uncertainty, not the external uncertainty: if we used the external uncertainty, we would often conclude that two samples agreed within their respective uncertainties when in fact this was not true. Because constructing a “camel diagram” for exposure ages from a particular landform is basically an exercise in comparing a set of samples to each other — you want to come to a conclusion about whether exposure ages are scattered due  to postdepositional disturbance, for example — in most cases you should use the internal uncertainty along in constructing the diagram.

Lastly, here is some MATLAB code for actually constructing camel diagrams.

1. The measurement we need to compute an exposure age is the amount of Be-1o in a sample. We determine this by measuring the amount of Be in our sample, determining the Be-10/Be-9 ratio by AMS, and  multiplying.

2. AMS measurement of the 10/9 ratio is done by comparing the 10/9 ratio in the sample to the 10/9 ratio of a standard  whose absolute isotope ratio is already known.

3. The absolute isotope ratio of the standard is usually defined by a decay-counting measurement to determine how much Be-10 is present. This requires knowing the half-life of Be-10. If you use a different value of the half-life,this implies a different absolute isotope ratio for the standard, a different isotope ratio for your sample, and, eventually, a different exposure age. So from this perspective, an exposure age can be said to be “normalized to a particular value of the half-life.” Unfortunately, this description is extremely confusing if you are not completely familiar with the preparation of AMS standards.

The point of the previous post was to make all readers completely familiar with the preparation of AMS isotope ratio standards. In case that failed, the point of this post is to explain how to reduce the confusion caused by the semi-equivalency of the value of the Be-10 half-life and the number of Be-10 atoms in your sample. I summarize a couple of steps that have been taken in the past few years to alleviate this, as well as recommendations for how to keep things simple and reduce confusion as much as possible.

Step 1. Make an AMS standard whose absolute isotope ratio is determined independently of the Be-10 half-life. Kuni Nishiizumi and a number of co-authors accomplished this in a 2007 paper:

K. Nishiizumi, M. Imamura, M. Caffee, J. Southon, R. Finkel, and J. McAnich. Absolute calibration of Be-10 AMS standards. Nuclear Instruments and Methods in Physics Research B, 258:403–413, 2007.

These authors made the observation that the whole point of an accelerator mass spectrometer is to detect and count the number of atoms of Be-10 that enter a detector. If you could hang onto all those atoms, you would have a precisely determined number of Be-1o atoms that could be used to mix up an AMS standard with known 10/9 ratio. And you would not have determined the number of atoms of Be-10 by decay counting, so this would be independent of the Be-10 half-life. They implemented this very smart idea by placing a silicon wafer in the LLNL AMS detector array that would trap Be-10 atoms entering the detector. They then put a Be-10-rich cathode in the accelerator source and injected a large number of Be-10 atoms into the detector. These were detected, counted, and came to rest in the silicon wafer. They then dissolved the wafer and added a gravimetrically determined amount of Be-9. The result: a stock of Be (I will call this the “implantation standard”) whose absolute 10/9 ratio was known independently of the half-life. They then used this stock of Be to prepare several samples for AMS analysis, and used the AMS to compare the 10/9 ratio of this material to that in commonly used AMS standards (whose isotope ratios were previously only known by a decay-counting measurement). The result: absolute measurements of the 10/9 ratios in these standards that were independent of the value of the Be-10 half-life.

This is an enormously valuable contribution because it decoupled the question of “how many atoms of Be-10 are there in my sample” from the question of “what is the Be-1o half-life.” In addition, this paper includes a long list of absolute isotope ratios determined for various commonly used AMS standards, all referenced to the implantation standard. This list is important because it allows interrelation of AMS measurements normalized to different standards. For example, these measurements showed that the “07KNSTD3110″ standard  material has an absolute 10/9 ratio of 2.85 x 10^-12. If we want to compare that with measurements made against the “LLNL3000″ standard material with an assumed isotope ratio of 3 x 10^-12, the intercomparison measurements in this paper reveal that we have to multiply the latter measurements by 0.8644. Some of these intercomparison measurements existed before, but this paper put a large set of internally consistent intercomparison measurements in one place.

Step 2. Strongly encourage anyone calculating exposure ages and/or erosion rates to determine and specify the Be measurement standard used for their AMS measurements. This is critically important for the usefulness and scientific longevity of AMS results — if there’s no information about what actual AMS standard measurements are linked to, then it’s impossible to determine how many atoms of Be-10 are actually in the samples. If you can’t determine how many atoms of Be-10 were observed in a study, then obviously the study is totally useless to future researchers.

This is easy for some measurements — because the AMS lab that made the measurements places this information at the top of all of their results spreadsheets supplied to users. Some labs (i.e., LLNL) clearly state the standard material that they used along with the absolute isotope ratio assumed for that material. This is the simplest and easiest-to-understand approach. Other labs  (i.e., SUERC) state the standard material used and the half-life of Be-10 that they used to define its absolute isotope ratio. This approach provides the information in a somewhat less accessible form: the user must obtain more information (i.e. the activity of the standard) to determine what absolute isotope ratio was assumed for the standard. As discussed in the last post, stating the half-life assumed in interpreting the activity of the standard material does fully define the standardization, but experience shows that it is much more confusing for users who are not familiar with how AMS standards are produced. Other labs (e.g., PRIME) do not by default supply standardization information with their results, so users must look on a lab website (e.g., here for PRIME) or fish around in the AMS literature. This approach is the least optimal.

In 2009, I decided that the best way to address this situation was not simply to hector scientists about proper data reporting (although this is also a popular strategy, as documented in this post) but to give them an incentive to do it properly. The online exposure age calculator provided this opportunity. The calculator had started to become very commonly used by Earth scientists who wished to use cosmogenic-nuclide exposure ages and erosion rates in their research, but were not specialists in the method and were not completely familiar with all details of Be-10 measurement and production rate estimation. I modified the online calculators to require as input a description of the Be-1o AMS standard used for the measurements. Basically, I trolled through the tables in the Nishiizumi et al. (2007) paper as well as some other published intercomparisons, and defined a number of possible combinations of actual standard material and assumed isotope ratio. These possible standardizations are tabulated here. This meant that users had to figure out which of these standardizations applied to their measurements, and enter it. Because the online calculator was supplying users a valuable and time-saving service — computing exposure ages according to commonly accepted practice without requiring the user to understand all details — users were willing to incur a small amount of extra work, that is, figuring out the AMS standard used for their measurements, to gain this benefit. The way this happened in practice was that users would ask their AMS lab which standardization applied — the AMS folks would either tell them (if the relevant standard was tabulated) or contact me to add their particular standard/ratio comparison to the approved list. Basically, this worked — by providing users a benefit for properly assembling the information, they became motivated to obtain this information from those responsible for their AMS measurements. In addition, the fact that all users of the online calculator knew that this information was required, motivated them to demand it from others in the course of paper or proposal review. Essentially, adding a standardization requirement to the online calculator harnessed worthwhile incentives and peer pressure to improve data reporting. At present, nearly all publications about cosmogenic-nuclide exposure dating properly describe the standardization of their Be-10 measurements. This is a huge improvement over the past situation.

Recommendations. Finally, I have some recommendations for the best way to make Be-10 measurement standardization as clear as possible for all users.

The first is to reduce confusion about the link between the half-life of Be-10 and the assumed isotope ratio of a standard by not describing AMS standards in terms of a half-life. This means saying “the NIST SRM 4329 standard with an assumed 10/9 ratio of 2.68 x 10^-11″ and not saying “the NIST SRM 4329 standard with a Be-10 half-life of 1.34 Ma.” It also means not making statements such as “these Be-10 measurements are normalized to a Be-10 half-life of 1.5 Myr.” The reasoning here is as follows: what we are really trying to do is determine the number of atoms of Be-10 in a sample. Thus, we should describe how these measurements are standardized by likewise stating how many atoms of Be-10 are in the standard. In addition, all AMS standards must by definition have an absolute 10/9 ratio, but this is not always determined by decay counting. Thus, the absolute isotope ratio is a common descriptor that can be applied to any AMS standard. Yes, it is true that stating a standard material and an assumed half-life does fully and correctly describe an AMS standard — but it’s a lot less confusing to keep separate the question of “how many atoms of Be-10 are in my sample” from “what is the half-life of Be-10.” IT IS LESS CONFUSING to describe standards in terms of an assumed 10/9 ratio. Keep things simple — first, determine how many atoms of Be-10 there are in your sample. Then, after you’ve figured that out, decide what value of the Be-10 half-life should be used to interpret your results (see this post).

The second is to use AMS standards whose absolute isotope ratios are linked to the Nishiizumi implantation experiment, not to a decay-counting measurement. This is why the online exposure age calculator renormalizes all data to the “07KNSTD” standardization, which is derived from the implantation experiment. Again, this keeps the question of “how many atoms” separate from “what is the half-life.” This issue is alleviated somewhat now that accurate measurements of the Be-10 half-life exist — and in principle, using the new half-life measurement with existing activity measurements for AMS standards most likely yields more precise absolute isotope ratios for many AMS standards than does referencing them to the Nishiizumi implantation standards — but converting this half-life measurement into new absolute isotope ratios for decay-counting-based standards has not really made it into published form yet. So I may take back this recommendation in future.

So I’m going to make one more attempt to clear this up, from the beginning. First, there will be a long section on basic concepts, then I will try to answer the two questions above.

Long section on basic concepts. 

Definition of half-life and decay constant.  The rate at which Be-10 decays can be described by either a decay constant or a half-life, which are related as follows:

                  (1)

where is the Be-10 half-life (1.39 x 10^6 yr) and is the Be-10 decay constant (5.00 x 10^-7 /yr).

How does one measure the decay constant? One actually measures the decay constant by obtaining a large quantity of Be-10 and using a beta counter to measure the number of decays per time interval. The decay rate, also called activity, is related to the amount of Be-10 and the decay constant by:

                       (2)

where, to keep the units consistent, is the number of atoms of Be-10 and A is the activity in decays per year.

It will therefore be obvious that to determine the decay constant, one must know how many atoms of Be-10 one has. Usually, this is done by obtaining a sample of beryllium that has been neutron-irradiated so that unnaturally large amounts of Be-10 are present. One uses any standard chemical analytical method to determine the total amount of Be that is present, and a mass-spectrometric measurement of some sort to determine the Be-10/Be-9 ratio. So the sequence of events to determine the Be-10 half-life is as follows: determine how much Be you have, determine its 10/9 ratio, use this information to compute the amount of Be-10 present ( above), measure its activity by decay counting (A above), and then apply equation (2) above to compute the decay constant/half-life. All the steps of this process are fairly easy except for determining the 10/9 ratio — making absolute isotope ratio measurements with a mass spectrometer at high accuracy is very difficult, because different isotopes of the same element are usually transported through the machine and detected with different efficiencies. So this is the hard part of the measurement.

How does one actually make AMS measurements of the amount of Be-10 in a sample? In analyzing a geological sample that we want to exposure-date using Be-10, we add a measured quantity of Be-9 (the “carrier”), dissolve the sample, extract the now-mixed carrier Be-9 and sample Be-10, then measure the Be-10/Be-9 ratio of the Be that we extracted. Because we know how much Be-9 we added (and because the sample contributes a negligible amount of Be-9), we can multiply this amount (in number of atoms) by the measured ratio to get the number of atoms of Be-10 that were present in the sample.

What is an “AMS standard” and why do you need it? As mentioned above, it is very difficult to measure the absolute isotope ratio of anything with any mass spectrometer, and AMS technology is no different — it is probably worse, in fact, because beams of Be-9 and Be-10 are not even detected in the same type of detector on an AMS. Thus, what we actually do in an AMS measurement is compare the apparent Be-10/Be-9 ratio of a sample with the apparent Be-10/Be-9 ratio of a standard material for which we already know the actual isotope ratio. I will call a material like this an “AMS standard.” Let’s say we have a Be AMS standard that is known to have an absolute isotope ratio of Be-10/Be-9 = 1 x 10^-13. We alternate measurements of this and a sample, and we find that the apparent Be-10/Be-9 ratio of the sample is twice that of the standard. We can use these two pieces of information to determine that the absolute isotope ratio of the sample is 2 x 10^-13. The point of all this is that all Be-10 measurements depend on an AMS standard whose true Be-10/Be-9 ratio is already known. You can’t analyze samples unless you have an AMS standard.

How do  we know the absolute 10/9 ratio of the AMS standard? The obvious question, then, is how we determine the absolute 10/9 ratio of the standard. Well, making an absolute isotope ratio measurement by mass spectrometry is really hard, but beta counting is easy, and we already know the half-life for Be-10, right? So if we obtain a sample of Be-10-enriched Be, we can determine how much Be-10 is present by measuring the activity by beta counting, and applying Equation (2) to determine the number of atoms of Be-10 that are present. We can then dilute our sample with Be-9 to make an AMS standard with any 10/9 ratio that we want.

This describes how most AMS standards are prepared. The actual measurements involved in determining the 10/9 ratio of the AMS standard are i) how much Be is present, and ii) what Be-10 decay rate is observed. One then must assume a value for the Be-10 half-life to come up with an estimate of the absolute 10/9 ratio.  If one assumes a different half-life, then one comes up with a different ratio. Thus, defining an absolute 10/9 ratio for an AMS standard that will then be used as a basis for measuring Be-10 concentrations in samples is the same as choosing a value for the Be-10 half-life. The actual measurements that were made on the standard material — the Be concentration and the Be-10 activity — don’t change. However, if one updates, or revises, the Be-10 half-life, this implies a corresponding update or revision of the absolute isotope ratio of an AMS standard prepared in this way.

Summary so far: the absolute 10/9 ratio of the AMS standard is what we really need to know to make the measurements of Be-10 concentration in samples that we want to make. However, in most cases, the absolute 10/9 ratio of the AMS standard involves an activity measurement, which can only be interpreted as a 10/9 ratio if you know the Be-10 half-life. So for AMS standards that are based on activity measurements, the half-life of Be-10 and the absolute isotope ratio of the standard are linked — if you know one, you know the other, and if you change one, you have to change the other.

OK, now to an example. One commonly used AMS standard was prepared by Kuni Nishiizumi by diluting a sample of Be-10-enriched Be called the “ICN standard.” This dilution is called the “01-5-4″ dilution. When used as an AMS standard this material is usually called “KNSTD3110.”
It has a measured Be-10 activity of 97.3 decays/yr per mg Be.  This activity measurement is related to the Be-10 half-life and to the absolute 10/9 ratio of the Be in the solution by:

where R is the 10/9 ratio, A is the number of Be-10 decays per year per mg Be, and is the number of atoms of Be-9 per mg Be (6.68 x 10^19). If we assume that the Be-10 half-life is 1.5 Myr (i.e. is 4.62e-7), then the 10/9 ratio of this material is 3.153 x 10^-12, and when comparing samples to this standard during AMS measurements we would compute the 10/9 ratio of samples based on this value. For example, if we observed that the apparent 10/9 ratio during AMS measurement of a sample was 0.1 times that of this standard, we would conclude that that 10/9 ratio in our sample was 3.153 x 10^12 x 0.1 = 3.15 x 10^-13.

What if you’re not sure what the Be-10 half-life is? The fact that the half-life of Be-10 is linked to the absolute isotope ratio inferred from activity measurements on AMS standards has been important because, until relatively recently, there was some doubt about what the half-life of Be-10 actually was. Another measurement of the half-life had concluded that it was actually not 1.5 Myr but 1.34 Myr (i.e.,   is 5.17e-7). If we assume this half-life when computing the 10/9 ratio of the KNSTD3110 standard from the measured activity, we would conclude that the absolute 10/9 ratio of this standard was 2.81 x 10^-12. Then given the same AMS measurements on our sample, we would conclude that the 10/9 ratio in our sample was 2.81 x 10^-13, or 11% lower than we concluded before.

Confusion about the Be-10 half-life = confusion about how much Be-10 is in your sample. As shown in the example above, because the absolute isotope ratio of an AMS standard is computed from an activity measurement and an assumed half-life for Be-10, what half-life you choose affects the absolute isotope ratio of the standard and thus the absolute isotope ratio of your sample. Doubt about the Be-10 half-life implies equal doubt about how many atoms of Be-10 are actually in a sample.

Attempt to summarize and clear up the main areas of confusion. 

What is the relationship between the half-life of Be-10 and the Be-10 concentration in my sample, i.e. how can my measurements be described as being normalized to a particular value of the half-life?

This should be clearer now. Really, your measurements are normalized to an AMS standard whose absolute 10/9 ratio is known. But that “absolute ratio” is (in most cases…see more below) not directly measured.  The Be-10 activity is what is actually measured, and that measurement plus the value of the Be-10 half-life is used to compute the absolute 10/9 ratio. So if you change the half-life, you change the absolute ratio for the standard (which doesn’t sound so “absolute” now…this is part of the confusion).

This is why stating that a measurement is ”normalized to the NIST SRM 4325 standard material with a Be-10/Be-9 ratio of 2.68e-11.” is the same as saying that it is “normalized to the NIST SRM 4325 standard material with a Be-10 half-life of 1.34 Myr.” A critically important point here, however, is that there IS NO WAY for a user to know that these two statements are equivalent, unless he or she carries out additional research into what the actual measured Be-10 activity in the standard is. In my view, the first statement is much more clear and much more meaningful for the user: if you know the sample-standard relationship measured by AMS, you can use the information in the first statement to compute how much Be-10 is in your sample. You can’t use the second statement to do this unless you carry out additional research.

What is the relationship between the Be-10 half-life used to compute the absolute 10/9 ratio in a standard and the Be-10 half-life that I will use to compute exposure ages and erosion rates? 

Well, they should be the same. Whatever you think the Be-10 half-life is, it can’t be two different things at the same time, so if you use one value of the half-life to compute the number of atoms of Be-1o present in your sample, you must use the same value of the half-life to compute its exposure age or erosion rate. Extensive discussion of this is beyond the scope of the present post, but if you use inconsistent values in these two parts of an exposure age calculation, you create systematic errors that are often quite important. Thus, if you are working from an AMS standard whose absolute isotope ratio is defined by reference to a particular value of the Be-10 half-life, you must use the same value of the half-life in computing exposure ages or erosion rates.

This statement is pretty simple, but there are two important aspects to it that are sometimes confusing. First, to apply this statement you need to determine whether or not your AMS standard is defined in relation to an activity measurement and a half-life, and if so, determine what this half-life is. Second, even if you correctly apply this statement and use the same half-life to define your AMS standard and to interpret your measured Be-10 concentrations, if this value of the half-life is incorrect, you will obtain the wrong answer.

There is one other very important point here: if you are using an AMS standard whose absolute 10/9 ratio does NOT involve any assumptions about the half-life, then you can choose any value of the half-life you think is most accurate to carry out further calculations. I will cover this issue in the next post.

Summary remarks

This post attempts to explain why the half-life of Be-10 is related to the number of atoms of Be-10 that you think are present in your sample based on an AMS measurement. This issue is confusing. In a subsequent post I will try to explain how AMS specialists and users of Be-10 measurements can best clear up confusion and make this issue easier to deal with. The main advance that makes this possible is a recent paper by Kuni Nishiizumi and others in which they created a series of AMS standards whose absolute isotope ratios are NOT determined by reference to any particular value of the Be-10 half-life. This advance separates the question of “how many atoms of Be-10 are in my sample” from that of “what is the half-life of Be-10,” which in turn i) makes things a lot simpler, and ii) makes most geological applications of Be-10 measurements more accurate.

The general concept of cosmogenic-nuclide burial-dating is that one has a pair of cosmogenic nuclides that are produced at a fixed ratio in some rock or mineral target, but have different decay constants. If a sample is exposed at the surface for a time, no matter what the production rate or how long the exposure, the concentrations of the two nuclides conform to the production ratio. Then if you bury the sample deeply enough to stop new nuclide production, inventories of both nuclides (or at least one of the nuclides, if the other is stable) decrease due to radioactive decay. Because they decay at different rates, the actual ratio of the two nuclides gradually diverges from the production ratio. Measuring this ratio tells you the length of time the sample has been buried.

So far, nearly all applications of burial dating have used the Al-26/Be-10 pair in quartz. The 26/10 production ratio is 6.75. The half-lives of Al-26 and Be-10 are 0.7 and 1.4 Ma, respectively. This turns out to be a very useful nuclide pair because quartz is so common — nearly all sedimentary deposits contain quartz that has been exposed for a time and then buried as the deposit accumulated.

However, there are a lot of other nuclide pairs that could potentially be used for this purpose. There are really two reasons one might want to use a different nuclide pair — first, one can’t find quartz; second, the new nuclide pair would give a more precise age in the time range of interest. Because it’s pretty rare not to be able to find quartz, the second reason — potentially reduced uncertainty or a wider age range — is really the key reason to think about burial-dating with other nuclide pairs. The uncertainty of a cosmogenic-nuclide burial age is set by a number of factors: measurement precision for the nuclides in question; the actual values of the production ratios and decay constants; how precisely the decay constants of the nuclides in question are known; how precisely the production ratios are known; and geological factors, mainly to do with the burial history of the sample. Geological factors don’t depend on the nuclides that are being used — they would have a similar effect no matter what the nuclide pair — so because the point of this discussion is to discuss why you would want to use one nuclide pair rather than another, we’ll ignore geological factors and assume that whatever we’re dating gets immediately buried at infinite depth and stays there until the time of measurement.

So to compare the precision of burial dates with various nuclide pairs over different age ranges, a few ingredients are needed.

One is the precision of the half-life determinations. I’ll consider four commonly used cosmogenic nuclides: Ne-21, Be-10, Al-26, and Cl-36 (both He-3 and Ne-21 are stable, so they’re pretty much equivalent for purposes of this discussion and I’ll only talk about Ne-21). Of these, Ne-21 is stable, so there is no uncertainty in its half-life. The half-life of Be-10 has recently been very precisely measured to about 0.8% precision. That of Cl-36 is also fairly accurately known (0.7%). That of Al-26 is somewhat less well known (ca. 2.5%).

Another is measurement precision. The following plot shows the concentration-measurement uncertainty relationship for all the Al-26 and Be-10 concentrations I could assemble from readily available data.

Red and blue dots are actual Be-10 and Al-26 measurements from the past few years. Nearly all these measurements include chemical processing at the University of Washington, and AMS analysis at LLNL-CAMS. The solid lines show model uncertainty relationships fit to these data; they have approximately a square-root dependence at high concentrations (the log-linear part of the curve) and then diverge upward from that relationship at low concentrations as one approaches the detection limit. I don’t have similar data readily to hand for Cl-36, so I’ll assume for now that Cl-36 measurements have the same statistics as Al-26. Ne-21 measurement precision depends on the geomorphic situation and will be discussed later. The final ingredient we need is an estimate of the uncertainty in the production ratios of these nuclides. This is hard to estimate in a general way — it depends on the rock or mineral in question and its composition — so for purposes of the following discussion I’ll assume that we know these ratios accurately (however, this is a major issue for some nuclide pairs).

Given these ingredients, we can make an uncertainty estimate for all six nuclide pairs implied by these four nuclides. My point in doing this calculation in the first place was to carry out a feasibility study for burial-dating of sediments derived from ignimbrites in the western US, so the basic assumptions are tailored to that scenario. The target mineral for Cl-36 production is a K-rich feldspar. That for all other nuclides is quartz. The sediment source is at 1300 m elevation and is eroding at 3 m/Myr. This results in a cosmogenic Ne-21 concentration near 10 Matoms/g, which we can measure with approximately 5% precision. I use the measurement uncertainties for the other nuclides as discussed above, assume no geologic uncertainty, and assume we know the production ratios accurately. This yields the following burial age-uncertainty relationship for the six nuclide pairs we are considering:

Here is the same plot with a different axis, focusing on the Pleistocene:

OK, what do we learn from this? First of all, the general structure of this plot is as follows. For a particular nuclide pair, relative age uncertainties are large at young ages (this is just a consequence of the radioactive decay equation and the fact that if the age uncertainty is more or less constant in absolute terms, it blows up in relative terms as the age approaches zero), and then become large at old ages again (because at least one of the nuclides decays to concentrations too low to measure accurately). There is a relatively flat “sweet spot” in the middle. The location, width, and uncertainty of the sweet spot depend in a fairly complicated way on the production ratios and decay constants themselves as well as on the measurement uncertainty characteristics of each nuclide. One thing that is interesting is that it generally pays to pick a stable nuclide (Ne-21) as one of the pair, for three reasons: first, it doesn’t decay, so there’s no loss of measurement precision with burial age; second, it doesn’t decay, so one less half-life uncertainty gets propagated into the burial age; third, its ‘decay constant’ is zero, which maximizes the difference between decay constants (an important part of the uncertainty) relative to anything else you could choose. So this is a good reason to focus on Ne-21 measurements (or on He-3 measurements, which would work similarly except that He-3 is not retained in quartz).

Regardless, two things are clear, at least in theory for this particular scenario:

First, with this scenario, for any time period it is (again, theoretically) possible to improve on the precision of Al-26/Be-10 burial dating by choosing a different nuclide pair. Mainly this is for two reasons: i) the uncertainty is inversely proportional to the difference between decay constants (this falls out of the math) and the difference between Al-26 and Be-10 decay constants is not as large as for other nuclide pairs; ii) the half-life of Al-26 is the least precisely measured of all the nuclides.

Second,  different nuclide pairs are the optimal choice for different time ranges. Pairs where the half-life difference is larger are useable at younger ages; pairs that include one nuclide with a short half-life become unusable faster. So this sort of a plot can serve as a guide for which nuclide pair one ought to apply to a particular problem. In this example, we’re looking at Pleistocene alluvial terraces so in theory we should like either the Cl-36/Ne-21 or the Cl-36/Be-10 pair.

Of course, there are a couple more points here. Besides the fact that I am ignoring geologic complications, all of these pairs might not be feasible because the targets don’t occur together, or because of complications in figuring the production ratios. Nuclide pairs involving Cl-36 are only feasible for targets where Cl-36 production by thermal neutron capture is negligible; this most likely means K- or Ca-rich feldspars with very low Cl-35 concentrations. Ne-21 can only be measured precisely at relatively high concentrations, such as in this example; at low concentrations precision degrades rapidly because of interference from non-cosmogenic Ne-21 trapped in the target mineral. Al-26, Be-10, and Ne-21 all can be measured and have well-characterized production rates in quartz, so those three nuclides can commonly be used together. Cl-36 and Ne-21 both occur in K-feldspars, although production rates are not as well characterized as in quartz. This pair is potentially quite precise, and could be used to good effect in sanidine from ignimbrites. However, if a nuclide pair has different target minerals, for example if combining Cl-36 in feldspar with something else in quartz,  then the sample must consist of a rock that contains both minerals together, to ensure that they have the same exposure history.

Summary: in theory one can tune the burial-dating method to have better precision, and a much wider range of applicability, by using a range of nuclide pairs beyond Al-26/Be-10. This is really interesting.

Other important point: I haven’t discussed at all the possibility of using three nuclides in the same sample. This gets complicated fast, but it is a really interesting idea because it can potentially allow one to be less dependent on the geological assumptions that go into two-nuclide burial dating — so the method would be useful in more geologic situations.

More information:

Balco G., Shuster D.L., 2009. Al-26 – Be-10 – Ne-21 burial dating. Earth and Planetary Science Letters, v. 286, pp. 570-575. doi:10.1016/j.epsl.2009.07.025

Granger, D., 2006. A review of burial dating methods using Al-26 and Be-10. In: Siame, L., Bourlés, D., Brown, E., eds., In-situ-produced cosmogenic nuclides and quantification of geological processes: Geological Society of America Special Paper 415. Geological Society of America, Boulder, CO. pp. 1-16.

This plot shows relative uncertainty (one standard error, expressed in percent) on measurements of Be-10 concentrations in quartz. It’s hard to say for sure what is included in these uncertainties because I didn’t compute all of them myself, but to the best of my knowledge they include i) AMS measurement uncertainty, ii) uncertainty in the amount of Be-9 added as carrier, and iii) uncertainty in the blank correction.

The red circles show measurements made by Paul Bierman and his co-workers in the late 1990′s (various papers by Bierman and Marsella selected to include a good range of Be-1o concentrations; citations below). These measurements represent the approximate state of the art at the time, and share two important features. First, they used a commercially-available Be-9 carrier that had a Be-10/Be-9 ratio somewhere around 1 x 10^-14. Second, the isotope ratio measurements were made at LLNL-CAMS on accelerator targets prepared by mixing BeO with silver powder as the conductive binder.

The black circles are a compilation of measurements on samples analysed in the University of Washington lab by me, Greg Balco, between approximately 2002 and 2009. The black triangles are measurements from Schaefer et al. (2009) that extend to lower Be-10 concentrations. These measurements represent the approximate state of the art at the present time and share two important features. First, they employed a low-blank Be-9 carrier prepared from deep-mined beryl according to a recipe by John Stone. This carrier has a Be-10/Be-9 ratio somewhere around 2-6 x 10^-16. Second, the isotope ratio measurements were made at LLNL-CAMS on accelerator targets prepared with Nb rather than Ag as the binder. Sometime around 2002, the LLNL-CAMS staff discovered that this modification significantly increased Be beam currents and thus Be-10 count rates. There have also been a handful of other incremental improvements over the years in increasing beam currents at CAMS.

This plot makes no attempt to control for sample size, which of course affects the Be-10/Be-9 ratio that is actually being measured, in either of these data sets. As the available range of adjustment of sample size is about one order of magnitude, that presumably accounts for most of the half-an-order-of-magnitude scatter in relative uncertainty.

These two major improvements — the low-blank Be-9 carrier and the Nb-BeO targets — make a difference. In the 1990′s data it is evident that precision is blank-limited at lower concentrations…uncertainties diverge from the overall log-linear relationship and become large as sample ratios approach blank levels somewhere around a few tens of thousands of atoms per gram. The low-blank carrier removes this limitation and permits maintenance of counting statistics at much lower concentrations. In addition, higher beam currents push this relationship down across the entire range of Be-10 concentrations.

Homework: plot your own measurements on this figure.

Some references:

Bierman, P., and Turner, J., 1995, 10Be and  26Al evidence for exceptionally low rates of Australian bedrock erosion and the likely existence of pre-Pleistocene landscapes: Quaternary Research, v. 44, p. 378-382.

Bierman, P.R., Gillespie, A., and Caffee, M., 1995, Cosmogenic age-estimates for earthquake recurrence intervals and debris-flow fan deposition, Owens Valley, California: Science, v. 270, p. 447-450.

Marsella, K.A., 1998. Timing and extent of glaciation in the Pangnirtung Fjord region, Baffin Island: determined using in situ produced cosmogenic 10Be and 26Al. Ms thesis, University of Vermont.

Marsella, K.A., Bierman, P.R., Davis, P.T., Caffee, M.W., 2000. Cosmogenic 10Be and 26Al ages for the last glacial maximum, eastern Baffin Island, arctic Canada. Geol. Soc. Am. Bull. 112, 1296-1312.

J. Schaefer, G. Denton, M. Kaplan, A. Putnam, R. Finkel, D. Barrell, B. Andersen, R. Schwartz, A. Mack- intosh, T. Chinn, and C. Schlu ̈chter. High frequency glacier fluctuations in New Zealand differ from the northern signature. Science, 324:622–625, 2009.

Email received by Balco:

Hi Greg,

I wonder if you could explain to me how you have calculated the production rate by high-energy nucleogenic spallation in the Stone scheme (P_St = 4.49).

There is a difference between “nuclear disintegrations” in the Stone/Lal and Dunai schemes and “nucleogenic spallations” in the other production schemes. I checked your code, made some small calculations with your production rates, and saw that the fast muon induced contribution is potentially added twice in the Stone/Lal scheme. Could you please check my considerations below and give me your opinion?

First, I always assumed that you have applied the same approach to all scaling schemes:
Get P_total_local
Subtract P_smu_local
Subtract P_fmu_local
Scale the rest to SLHL

Then the values in Table 6 of your “calculator” article (or Table 1 in update 2.2) would be nucleogenic spallations only. However, my calculations give me the impression that P_St consists of the nucleogenic spallations plus fast muon interactions (so, +0.1 atom/g/yr).

The calculations referred to here are in this PDF.

Balco’s response:

Yes, you are right, there is an inconsistency there. One problem I ran into in putting this together is that in addition to the surface production rates, the calculator needs to calculate the depth dependence, P(z), for all the different production pathways. If the Lal polynomials describing “nuclear disintegrations” include both spallation and fast muon interactions in an unknown proportion, then there is no way to know how this combined production rate decreases with depth. It is not clear: i) if all fast muon interactions are included as part of “nuclear disintegrations,” or ii) what proportion of “nuclear disintegrations” represent fast muons as opposed to spallation. I suppose this is similar to asking if fast muon interactions produce disintegration stars in film, which is another question that I don’t know the answer to.

Therefore, it seemed that the most sensible thing to do was to pretend that the Lal polynomials describe spallation only, have this “spallation” production rate decrease with depth according to a spallation length scale, and then to have the fast muon production be separate so that it can have the correct depth dependence. So, yes, in a way fast muons are included twice in this part of the code. However, they are not “added twice” in the sense that the production rate is systematically too high, because the fast muon production rate computed according to Heisinger is removed before determining the best-fitting “spallation” production rate, if that makes any sense. Instead, the effect of this simplification is that the altitude dependence of the production rate calculated with the St and Lm scaling schemes is slightly weaker than Lal may have originally intended. Whether this is more or less correct, or good or bad for calculator users, is not clear. First of all, remember that the original Lal scaling scheme gave much more importance to negative muon capture than we now think is correct, so the (incorrect) Balco implementation of Lal actually has a stronger altitude dependence of the total production rate than Lal originally intended. Also, for example, the St scaling scheme, even including the inconsistency we are talking about here, fits the 2008 calibration data set better than the other scaling schemes that are more “correct.” Of course, this could be because the scaling scheme and the calibration data are both wrong in offsetting ways, but there is no way to determine this.

So the summary is that I am not sure what the best thing to do here is — it seemed like the only other option was to guess how much of “nuclear disintegrations” is accounted for by fast muons, and I didn’t want to guess.

One other thing is that in the calculations you sent, you noted that the total high latitude, 1013.25 hPa production rate for the Stone scaling scheme comes out higher than the value of 5.06 used in Stone (2000). Those values are not comparable — the calibration data set used in the 2008 paper is similar but not exactly the same as the data set used in the 2000 paper, and the methods of averaging are also different. So you should not expect those two values to agree.

Response to response:

Hi Greg,

Thank you for this explanation.

I see that my initial understanding of your procedure was correct:
Get P_total_local
Subtract P_smu_local (Heisinger)
Subtract P_fmu_local (Heisinger)
Scale the rest to SLHL
The average of these “rests” gives you the SLHL production by the nucleogenic spallation.

There is a potential danger here that a simple user of the production rates may mix your values and the procedures from the Stone/Lal papers, because the names of scalings are still the same. I can imagine that someone takes Stone PR from your update 2.2 (“Aha, good, the production rate corrected to the new standards!”) and puts it into his old Excel file for the Stone2000 scaling (“Like in old good days! I still understand how I calculate!”). Do you think it would help if you advise people to use Heisinger with the Stone production rate from your update? Or rename the scaling into Balco-Stone.

Balco’s response to response to response:

I see that my initial understanding of your procedure was correct:
Get P_total_local
Subtract P_smu_local (Heisinger)
Subtract P_fmu_local (Heisinger)
Scale the rest to SLHL
The average of these “rests” gives you the SLHL production by the nucleogenic spallation

Yes, (if you make sure to include the other stuff like thickness and geometric scaling) that is what I did to get the production rates in the 2008 paper. Now I have changed things a little bit — I do the calibration with a minimization method where I consider the calculator as a function with one input parameter (the SLHL production rate for “spallation”) and one output parameter (some sort of misfit statistic, like MSWD or chi-squared, that compares the true ages of calibration sites with the ages predicted by the calculator). Then for each scaling scheme, I choose the value of the SLHL production rate that minimizes the misfit between predicted and actual ages. This method allows me to only have one piece of code — the “forward” code that calculates ages from Be-10 concentrations — rather than also requiring a piece of “backward” code that computes production rates from calibration sites of known age. This reduces the potential for errors.

As you may have noticed, I am working on a web page that actually does this — you input a calibration data set, it finds the best-fitting production rates, and then lets you compute ages that are consistent with the calibration data set. It is on the ‘developmental’ page.

There is a potential danger here that a simple user of the production rates may mix your values and the procedures from the Stone/Lal papers, because the names of scalings are still the same. I can imagine that someone takes Stone PR from your update 2.2 (“Aha, good, the production rate corrected to the new standards!”) and puts it into his old Excel file for the Stone2000 scaling (“Like in old good days! I still understand how I calculate!”). Do you think it would help if you advise people to use Heisinger with the Stone production rate from your update? Or rename the scaling into Balco-Stone.

Well, there are many dangers. The world is a dangerous place. Seriously, you are right, people do get confused between the as-published scaling scheme, that includes negative muon production as well, and the online calculator that only uses the “spallation” polynomials. However, the important thing — and one of the main purposes of the calculator — is that if you think an author has made this mistake, you can enter his measurements into the online calculator and check. In addition, you may notice that I try very hard not to talk about the “production rate” used to compute exposure ages. Instead I talk about the “calibration data set and scaling scheme.” Obviously it is important to focus people’s attention on the thing that is actually measured (Be-10 concentrations at calibration sites) and not the scaling-scheme-dependent parameter that no one can ever measure directly (the “SLHL production rate.”).

enjoy,

–greg

The red squares show the location of exposure-age samples and the plot at left shows the exposure-age/elevation relationship. The important thing here is that these nunataks are bare of snow or soil cover. Once glacial debris is deposited by ice retreat from a particular surface elevation, it stays put and accurately records the deglaciation history.

So that’s why exposure-dating works so well in most of Antarctica. I spent much of last winter on a ship near the Antarctic Peninsula trying to carry out a similar study on the east side of the Peninsula. This is an area where we have very little information about the actual LGM-to-present deglaciation chronology: only the very broad outlines are know from radiocarbon-dated marine sediments. It should be possible to do a much better job with exposure-dating, because the geomorphology of the east side of the Peninsula is similar to the rest of Antarctica in that  ice-free nunataks separate major outlet glaciers: as the surface of these glaciers lowered, glacial debris should have been preserved more or less undisturbed. This view looks east from the center of the Antarctic Peninsula over the east-side outlet glaciers:

Once you get off the summit ice cap, there are many ice-free areas where we can reasonably expect exposure-dateable glacial debris to be preserved.  So this project should go very much like other, mostly pretty successful, exposure-dating efforts in Antarctica.

The west side of the Peninsula is totally different. This is also an area where we would really like to be able to learn about the deglaciation chronology through exposure dating. Again, the overall deglaciation chronology is known in broad terms from marine radiocarbon dates, but there is very little information about the history of ice thickness change, which of course is the important thing from the point of view of sea-level impacts. However, the west side of the Peninsula is notable because of all the world’s glacial landscapes, it is the one where we have the least chance of using exposure dating to learn anything about past glacier change.

Here is what the west side of the Peninsula looks like:

This is cruise-ship Antarctica, where massive glaciers calve into crystal blue water filled with frolicking penguins. From the perspective of LGM-to-present ice sheet history, this landscape is deglaciated. The marine geology shows very clearly that all the open water visible in this image was filled with thick flowing ice at approximately the LGM. However, from the perspective of exposure-dating, almost nothing in this landscape is exposed. Nearly all rock surfaces are covered with tens to hundreds of meters of ice:

This is a consequence of the nearly unique climate in this region: the west side of the Peninsula is exposed to the southern hemisphere westerlies and receives large amouts of precipitation, but the temperature is cold enough that much of the snow and ice accumulation can remain frozen to rock surfaces. There is almost nowhere else in the world where a landscape this rugged is so comprehensively ice-covered. The only rock surfaces that are not ice-covered are nearly vertical, and these vertical faces are calving rock almost as fast as the adjacent glaciers are calving ice. Sure, maybe that is a little bit of an overestimate, but we observed daily rockfall on many of these faces. Here is a large fresh rockfall that postdates the snow that fell 24 hours before this photo was taken:

It is pretty clear that even if some glacial deposits were present on these near-vertical faces at the time the ice sheet surface lowered through this landscape, they’re not there now.

The only relatively flat ice-free areas in this landscape are small islands and peninsulas very close to sea level:

And, as evidenced by the granite erratic in the center of the photo, these sites do preserve glacial deposits that record occupation of this fjord by through-flowing ice. The difficulty is that these tiny ice-free sites are  fundamentally no different from nearby sites that are covered by ice caps:

so there is no way to know whether the exposure age of an erratic on this tiny bit of rock records the time that major fjord glaciers retreated from their LGM positions (which we would like to know) or the time that the last 20-meter-thick bit of ice slid off the outcrop in an unusually warm summer (which would be sort of interesting, but not very relevant to the main problem). So to summarize, even though the western Antarctic Peninsula is deglaciated in a sense, there are few if any rock surfaces that were permanently exposed by this deglaciation. Those rock surfaces that are likely to have been exposed since deglaciation are nearly vertical and demonstrably disintegrating at extreme rates. This is one of the few glacial landscapes in the world where cosmogenic-nuclide exposure-dating is unlikely to help make any progress on understanding past glacier and ice sheet change.

Dear Greg,

Could you, please, update your web-update to the article and to the calculator to the new 10Be half-life of 1.39 Ma? Your latest re-calculations of Be-10 PR are made with Kuni’s 1.36 Ma half-life, and this contradicts to 1.39 Ma. Since several people known to me refer to your web-update as to the latest Be-10 PR, to keep the old Be-10 PR is to add more to the already existing confusion in the community. The update should include, naturally, Heisinger’s constants (*1.39/1.51). I guess this will be a quick effort for you, and the community will be very grateful to you for keeping the calculator and its reference paper updated.

The part about the half-life made sense to me, but not the part about the production rates. Fortunately , it was quickly followed by this email:

Hi Greg,

sorry for messing things up myself…

I realized that the Be-10 production rates (PR) must stay as you cite them in the version 2.2 update, because they are normalized to an absolutely calibrated Be-10 standard, and it does not matter if Kuni has calculated 1.36 Ma or whatever.

Just to be sure — I assume that you have used 1/1.096 scaling (ETH->Kuni) of the ETH data that were used in calculation of the Be10 PR in your update 2.2?

If this is correct, then the only new changes for the calculator and update would be a new decay constant (ln(2)/1.39) and 1/1.096 scaling (ETH->Kuni) of old Be10 production constants from muogenic reactions.

What do you think?

This unnamed person is correct, at least in his second email, on all counts. First, the great value of the Nishiizumi et al., 2007 paper is that they calibrated AMS standards without reference to an activity measurement, thus decoupling measurements of AMS standard isotope ratios from measurements of the decay constant. Second, all this stuff has recently been published — which is the main criterion for inclusion in the calculators — so should be included in the calculators.

Thus, I have updated the constants file used by version 2.2 of the calculators to use the Chmeleff/Korschinek value of the Be-10 decay constant (discussed at greater length here) as well as to update the values for the muon interaction cross-sections to reflect a new ETH-KNSTD intercomparison. It is important to note that these two changes will have a negligible effect on exposure ages and erosion rates in nearly all situations. There is more information about the update on the documentation page for version 2.2 of the calculators.

Anyone who talked to me at any length in the last few years, or had their paper reviewed by me, has probably heard this lecture before. The same lecture appears in the paper describing the CRONUS-Earth online exposure age/erosion rate calculators. One of the goals of these calculators, in fact, was to improve the data-reporting situation by providing a clear list of information that readers would need to have in order to recalculate exposure ages using the online calculators. In the last few months, however, a number of other people have started to harass cosmogenic-nuclide users on this subject in a variety of print outlets. Here I review these various publications and try to make sense of the resulting forest of instructions.

First, in October 2007, myself, Nat Lifton, Joerg Schaefer, and Tibor Dunai sent in a manuscript to EOS that was intended to highlight the whole issue and tell people how to do it right. The last draft of this manuscript can be downloaded here. This manuscript was a bit more confrontational than others to be discussed subsequently. First, it featured a census of the current literature that showed that zero out of ten randomly selected papers that involved Be-10 exposure ages or erosion rates actually contained enough information to recalculate them. Second, it included a discussion of why the term ‘Be-10 years’ should not be used, which appears in a similar form here.  Alas, this manuscript became trapped in some sort of Kafka-esque editorial limbo at AGU, and was never published. Here I reproduce my 2009 letter to EOS editor John Geissman withdrawing it:

Dr. Geissman:

I regret that I and my co-authors would like to withdraw this manuscript, for the following reasons.

We originally submitted this manuscript to EOS in October, 2007, as  a proposal for a ‘Feature’ article. It was sent to three reviewers, who expressed general approbation of the content but indicated that it should rather be published as a ‘Forum’ article. In May, 2008, we submitted a manuscript in the ‘Forum’ format. In September, 2008, we received an additional three reviews that all found it acceptable for publication, but suggested it would be better as a ‘Feature’ article. Thus, EOS staff instructed us to revise it into ‘Feature’ format and resubmit it. We sent in a revised manuscript in January, 2009 which contained the same content that had been approved for publication in six previous reviews, expanded into the ‘Feature’ format. At that point, we anticipated that it would be published. However, your letter appended below indicates that you have subjected this manuscript to a seventh review, which has recommended rather extensive changes prior to publication.

The extraordinary length of this editorial process means that the purpose and content of this paper have largely been overtaken by events. The examples in the paper are obsolete, and a variety of other publications and online exposure-age calculation schemes have mitigated the condition which originally motivated us.

We regret that Paul Bierman, the most recent reviewer, has been put to the inconvenience of reviewing this manuscript to no purpose. We appreciate Paul’s efforts and we apologize that they will go to waste.

regards,

Greg Balco, Berkeley Geochronology Center

Simultaneously with the tail end of this notable failure, Tibor Dunai and Fin Stuart published a similar manuscript in Quaternary Geochronology. Here is a link to this article (which unfortunately requires a library subscription to QG. It is hard to imagine, however, that any reader on this blog is not part of a major university with such a subscription). This is basically the same as the Balco and others manuscript discussed above, with the sarcastic and/or confrontational bits removed and an increased emphasis on describing not only the direct measurements and observations but also the method that was used to calculate the exposure ages or erosion rates from the measurements.

Even more recently, Kurt Frankel, Bob Finkel, and Lewis Owen published this article in EOS (again, AGU login needed). It covers more or less the same material, follows the Balco  line rather than the Dunai line in emphasizing observational data over information about how ages and erosion rates were calculated, and adds its own emphasis on reporting more of the internal laboratory information, including sample weights, isotope ratios, and blank concentrations, that go into computing a cosmogenic-nuclide concentration.

Finally, I was contacted a couple of months ago by the editors of GSA Bulletin, who wanted to add some material to the `Instructions for authors’ outlining what information ought to appear in papers involving cosmogenic-nuclide measurements. These instructions were to be as compact as possible. I crushed my former three-page diatribe into four sentences as follows:

“For cosmogenic-nuclide exposure ages, authors must report: i) the location, elevation, thickness, density, and geologic context of each sample; ii) any corrections made for surface erosion and topographic or other shielding (or a statement that no such corrections were applied); iii) nuclide concentrations and measurement uncertainties; and iv) for nuclides whose production rates are composition-dependent, the major element composition of bulk rock and target mineral separates as well as the concentrations of important trace elements. For nuclides measured by accelerator mass spectrometry, authors must state the name and assumed isotope ratio of the standard to which the measurements were normalized.  For example, the input data required to compute Be-10 or Al-26 exposure ages with the CRONUS-Earth online exposure age calculator (http://hess.ess.washington.edu) meets these requirements. Authors should also describe the production rate scaling scheme and calibration data set used in computing the ages, as well as the date and version number of any software used for this purpose. “

To summarize, if we include Dunai and Stuart, Frankel et al., the remarks in the Balco et al., 2008 exposure age calculator paper, and the abortive EOS article, but not the proposed GSA instructions for authors, we have three published and one unpublished instruction manuals for cosmogenic-nuclide data reporting. For the most part these say basically the same thing with varying levels of smugness, but they are inconsistent in some of the important details.

Why all this attention now? One possibility is that this sort of instructional article, although mind-numbingly boring, is likely to be highly cited. If you live and die by your citation index, it is to your advantage to publish articles of this type. From a less cynical perspective, however, this effort is actually a useful public service. It is important to get this right if the vast amount of money being spent on exposure-dating is not to be wasted from the perspective of future utility of the measurements. With adequate data reporting, the set of exposure ages that can in future be recalculated according to a common method and applied to answer large-scale paleoclimate questions continues to grow. My guess is that as more people try to do this — synthesize existing exposure-dating studies to answer larger questions — and discover how pathetic and scattershot is the data reporting in the existing literature, they become inspired to improve the situation.

A more important issue for the person who is actually going to publish exposure ages is: which of these only partially consistent prescriptions should you follow?

There are two main differences among them. The first is that Balco and Frankel emphasize reporting the actual measurements and observables, and don’t emphasize reporting what method, what production rates, etc. you used in computing ages from these measurements. Dunai on the other hand puts a lot more emphasis on reporting how you got from measurements to ages. From my perspective this issue is a no-brainer. I don’t care how you got from measurements to ages. When I read an exposure-dating paper, the first thing I am going to do is extract the measurements and feed them to the online exposure-age calculator so that I can compare them to other ages on a common basis. So in my opinion authors need only give the barest minimum of information as to how the ages were actually calculated. My emphasis is firmly on reporting all the measurements and observations that only the author knows and I can’t reproduce myself. This is the critical part.

The second difference is that Frankel, Finkel, and Owen would like you to report a lot more lab data, specifically the sample weight, measured isotope ratios, carrier amounts, and blank concentrations; whereas Balco and Dunai are happy with the final nuclide concentration and some summary information about how important the blanks are. The former position is presumably based on two considerations — determining how important the blank correction is from raw data, and determining what sort of ratio was actually being measured. Both of these are important in evaluating the reliability of the concentration measurement. The latter position trusts the author to get this right, and asks only for a summary value without the nitty-gritty of the author’s internal lab operations. In the past I’ve followed the latter line — just give me the summary result — but I am sympathetic to the desire to verify the author’s blank correction and error-propagation scheme. It is possible that sometimes people don’t get error propagation quite right.

Summary: You must include every observation or measurement needed to compute the exposure ages. It is helpful to include additional laboratory information that would help readers to evaluate the importance of the blank corrections and what sort of isotope ratios were actually being measured. Once these goals are accomplished, it is not necessary to go into very much detail about the method used to calculate the ages, or the production rates used…because, as discussed at length above, these things will most likely be obsolete soon.

Incomplete list of references:

Balco G., Stone J., Lifton N., Dunai T., 2008. A simple, internally consistent, and easily accessible means of calculating surface exposure ages and erosion rates from Be-10 and Al-26 measurements.Quaternary Geochronology 3, pp. 174-195.

Dunai T.J., Stuart F.M., 2009. Reporting of cosmogenic nuclide data for exposure age and erosion rate determinations. Quaternary Geochronology, v. 4, p. 437.

Frankel K.L., Finkel R.C., Owen L.A., 2010. Terrestrial Cosmogenic Nuclide Geochronology data reporting standards needed. (why are they not needed for extraterrestrial cosmogenic-nuclide geochronology?). EOS Transactions AGU, v. 91, p. 31.