# Internal vs. external uncertainties, again

The issue of “internal” vs. “external” uncertainties is something that seems fairly straightforward, but a lot of people have questions about it. For example, a recent email:

Dear Dr. Balco,

I have a short question — maybe it is ridiculous, but I am little bit confused about uncertainties. I have read the Balco et al., 2008 paper and played with test data in the CRONUS website. The question is, how should we use those internal and external uncertainties in the exposure age results? For example; I saw some papers says “all ages include 1 sigma analytical errors.” Is this the external uncertainty? Thank you.

Best Regards,

Is this question ridiculous? Probably not, because many people seem confused about it. Also, I thought this was reasonably well explained in the 2008 paper, but maybe not. So here is another try, starting from the basic idea of how error propagation works, and I’m going to try to explain this in fairly simple English without resorting to math.

The concept of “error propagation” just means that commonly one makes a few measurements, each of which has some measurement uncertainty, and then uses these measurements to calculate some quantity that is actually of interest. In exposure-dating, the simplest possible such situation involves sampling a surface that has experienced a single period of surface exposure with no erosion, and measuring the concentration of a cosmic-ray-produced nuclide that is either stable or has a half-life much longer than the exposure-age of the surface. The thing we want to know is the exposure age of the surface, and to calculate it we need two measurements: the nuclide concentration in the surface and the production rate. Call the exposure age t (years), the production rate P (atoms per gram target mineral per year), and the nuclide concentration N (atoms per gram target mineral). Then we can calculate the exposure age simply from the relationship t = N/P.

Measuring the nuclide concentration N involves some measurement uncertainty, which mostly reflects how many atoms were actually counted in a mass-spectrometric measurement. We are not really “measuring” the production rate P — in fact we are estimating it from a large number of other measurements in a fairly complicated way — but there is also an uncertainty associated with it. So we are using two measured quantities that have uncertainties to compute a third quantity — the exposure age — and we need to decide what the uncertainty in this computed quantity is. There is a mathematical formula for doing this, but the point here is to discuss how we choose which of the uncertainties on the input parameters should be included in the final uncertainty on the exposure age. How to answer this question depends on what you want to do with the final uncertainty.

First, let’s say we have exposure-dated two boulders on the same landform and we want to know if they have the same ages. For an example with actual numbers, let’s say that the nuclide concentration in sample 1 (N1) is 230000 +/- 10000 atoms/g, that in sample 2 (N2) is 260000 +/- 8000 atoms/g, and the production rate P = 10 +/- 1 atoms/g/yr. Thus, the exposure age of sample 1 (t1) is 23 ka, and the exposure age of sample 2 (t2) is 26 ka. To compute an uncertainty for one of these ages, we need to include two uncertainties: that on N and that on P. Carrying through this calculation yields t1 = 23.0 +/- 2.5 ka and t2 = 26.0 +/- 2.7 ka.

These values for the age uncertainties, at face value, appear to indicate that the two ages differ by approximately the same amount as their respective uncertainties. If true, this would make it very likely that they are both measurements of the same thing and they differ only because of measurement uncertainties. This would be an important conclusion, because it would mean that we could average the two measurements to yield a more accurate and more precise age for the landform than we can get from either one of the measurements alone. Unfortunately, this conclusion would be wrong. The reason has to do with the idea of whether uncertainties are independent between samples or not. In this case, the uncertainty on the production rate measurement is not independent between samples. We may not know what the production rate is very accurately, but because these samples are located in the same place, we know that the production rate is exactly the same for both samples. If we make an error in estimating the production rate, this could cause us to get the wrong age for either one of the samples, but it could not cause us to get ages for the two samples that are more similar, or less similar, to each other. If we concluded that the two samples were likely to be the same age, we would basically be arguing as follows: i) measurement errors in both N and P could result in two measurements of the same thing showing an apparent difference; ii) the difference we observe is similar to the difference we expect if the two samples are really the same age and point i) is true; so we conclude that iii) the samples are likely to be the same age. In reality, point i) is wrong — uncertainty in P could not cause two samples, that in reality have the same age, to appear to have different ages. Only measurement uncertainty in N can have that effect. Thus, point iii) is also wrong.

The important thing here is that if we are asking the question of whether two samples, that are located in the same place and therefore have the same production rate, are the same age, we can’t include the production rate uncertainty in calculating the age uncertainty. If we recalculate these ages using only the uncertainty in N, we get t1 = 23.0 +/- 1.0 ka and t2 = 26.0 +/- 0.8 ka. These ages differ by significantly more than their respective uncertainties, which leads us to the correct conclusion that they are not likely to both be measurements of the same landform age, so we cannot average them to get a more accurate landform age.

Now ask a different question: is the exposure age of sample 1 the same as that of a nearby landform that has been radiocarbon dated at 20 +/- 0.5 ka? Our estimate of the production rate P is not involved in computing the radiocarbon age — the exposure age and the radiocarbon age are completely independent — so we need to include production rate uncertainty in computing the uncertainty on t1. Comparing 20 +/- 0.5 ka with t1 = 23 +/- 2.5 ka shows that the difference between the ages is more or less the same as the uncertainties in the ages, so there is a good chance that the two things being dated are coeval. If we incorrectly used an uncertainty on t1 that included the uncertainty on N but not on P, we would be comparing 20 +/- 0.5 ka with 23 +/- 1.0 ka and we would conclude, incorrectly, that these ages are unlikely to be the same.

Summary: which input uncertainties must be included in the final age uncertainty depends on what question you are trying to ask. Now to answer the questions in the email above in more detail.

What are the different uncertainties? The “internal” uncertainty includes only measurement uncertainty, that is, uncertainty on N, in computing the uncertainty on the exposure age t. The internal uncertainty could be sometimes called the “measurement uncertainty,” although that is less clear and could mean several different things. The “external” uncertainty includes all input uncertainties — uncertainties on both N and P in the example above — and is also often called the “total uncertainty.”

When to use them? Depends on what question you are trying to answer. See above.

What does “all ages include 1 sigma analytical errors” in a paper mean? OK, well, it is hard to tell. There is really not enough information here to determine whether the author is talking about internal or external uncertainties. “Analytical errors,” I would imagine, is intended to mean measurement errors in determining the number of atoms of Be-10 present. However, both the internal and external uncertainties include this uncertainty. What we really want to know is whether the uncertainties being described also include production rate uncertainties, or not. So, not enough information. If you are reviewing this paper, send it back for more information.

Your explanation is clarifying! Thank you, I will suggest it to my colleagues.