Recently I received the following question in an email:

I used the CRONUS earth calculator v2.3 to calculate the Be-10 exposure ages of the samples from [some location], but I did not find the numerical value of Be-10 reference production rate used in CRONUS v2.3. On the webpage it only said it is “derived from the primary calibration data set of Borchers and others (2016)”. Can you kindly tell me the value ? Thank you very much.

This also comes up in a lot of paper reviews. Here is an example. The text of the paper is as follows:

Reference Be-10 production rates were determined using the CRONUS-Earth “primary” calibration dataset (Borchers et al., 2016).

And the reviewer said:

…regardless of citing the source, for completeness the SLHL 10Be prod rate should be listed ( with its error)…

The purpose of this posting is to explain why one should state the production rate calibration data set used, but one should not state a value of the reference production rate. As usual, it’s a long posting, and it’s tedious because I’m going to try to do this just in words and not in math, but getting this right is important in understanding how exposure dating works and accurately communicating it in papers.

To begin with, I’ll review the overall structure of an exposure-age calculation one more time. To compute a cosmogenic-nuclide exposure age, you need three things. One, a scaling method that describes how production rates vary with location and time. Two, a set of calibration data that consists of nuclide concentrations measured at sites whose exposure age is already known from a different, independent, dating method such as carbon-14 or argon-argon, such that we know what the true average production rate at those sites has been during the period they were exposed. Three, the nuclide concentration measured at the unknown-age site that we wish to date. Having these three things, one takes the following steps.

1. Fit the scaling method to the calibration data set. In most cases, the scaling method generates a nondimensional “scaling factor” describing how the average production rate for a particular place and age is related to the production rate at some reference location and time. So for each sample in the calibration data set, the scaling method predicts a scaling factor for that place and exposure age. We then perform a one-parameter fitting exercise in which we choose a value of the “reference production rate” — the production rate at the reference location that is used in that scaling method — that, when multiplied by the scaling factors for all the calibration sites, best duplicates the actual production rates that we measured at those sites. A simple way to think about this is to consider a particular site: at this site we divide the true production rate that we measured by the scaling factor predicted by the scaling method; the result of this calculation is an estimate of the production rate at the reference location. If we have a number of different calibration sites, we have a number of different estimates of the reference production rate, and (given certain assumptions) the value of the reference production rate that best fits the entire calibration data set is just the average of the estimates from each site.
2. Now that we have an estimate of the reference production rate, we take the scaling factor predicted for the unknown-age site and multiply it by the best-fitting reference production rate in step (1). Then we use this production rate to compute the exposure age from the measured nuclide concentration. Note that for a scaling method in which the production rate varies in time, this calculation must be iterated rather than done only once, but the principle is the same.

There are several important aspects of this procedure.

One, nowhere in this procedure have we ever measured the reference production rate. The actual measurements that we made are the average production rates at the calibration sites. None of the calibration sites are at the reference location (except by accident). For some scaling methods, the reference location is defined such that it does not exist at any real place and time. Thus, the “reference production rate” is not a measurement. It is a parameter estimate that is the result of a model-fitting exercise, and what the parameter actually means is chosen by convenience and not for any physically meaningful reason.

Two, different scaling methods have different reference locations. Historically, for scaling methods (primarily that of Lal, 1991) in which production rates are assumed to be constant over time, the reference location was defined to be at “sea level and high latitude.” What this means is kind of vague, but usually it is assumed to mean something like (i) an atmospheric pressure of 1013.25 hPa (the defined international standard “sea level pressure”, although this is not the actual atmospheric pressure at sea level anywhere (except by accident), and (ii) geomagnetic latitude above about 60°, where the variation in magnetic field strength with latitude becomes negligible. There is no reason the reference location has to be at “SLHL.” It could be on the lawn of the Mar-a-Lago Club in Palm Beach, Florida, or the summit of Mont Blanc. It could be anywhere. For more recent scaling methods, in particular that of Lifton and others (“LSD” or “LSDn”) that is currently in common use, the production rate is predicted to change over time with changes in the magnetic field strength and solar output. This makes it necessary to choose both a reference location and reference values of magnetic and solar properties for the reference location. For the LSDn method, this is typically taken to be 1013.25 hPa, 0 GV, and a zero value of a parameter describing solar modulation. This is a different reference condition than the “SLHL” reference condition historically used for non-time-dependent scaling. Thus, even if we used the same calibration data set and both scaling methods were perfectly accurate, the best-fitting “reference production rates” to each scaling method would be different, because they pertain to different reference conditions. They could not be compared to each other, because they are not estimates of the same thing. They are estimates of different things.

Three, different implementations of the same scaling method will yield different best-fitting estimates of the “reference production rate” for that scaling method. All the commonly used scaling methods are numerically complicated, and different people have written different computer codes that use different numerical approximations, integration methods, interpolation methods, etc. to perform the calculations. In general, if you take one calibration data set and two implementations of the same scaling method and you perform both steps (i) and (ii) above, you will get the same exposure age for the unknown-age site. However, the intermediate value you have computed — the best-fitting value of your fitting parameter — will probably not be the same (except by accident). Thus, if you perform step (1) with one computer code, but then take the resulting fitted value of the reference production rate and perform step (2) with a different computer code, you will probably get the wrong exposure age for the unknown-age site.

So what does this mean for how you should describe how exposure-age calculations were carried out?

Historically, when most exposure-age calculations were carried out using the non-time-dependent scaling method of Lal (1991) or the slightly different implementation of the same method by Stone (2000), it was common in papers for authors to report the value of the reference production rate that they used to compute the exposure ages. Because the calculations involved in these scaling methods are relatively simple, it was possible for readers to easily produce their own implementation of the scaling method, take the value of the reference production rate stated in the paper, and closely reproduce the authors’ calculations.

With more complex, time-dependent, scaling methods in use at present, this is much more difficult. Software implementations of the LSDn scaling method involve thousands of lines of computer code and extensive background data sets describing paleomagnetic and solar variability as well as particle interaction cross-sections, all of which are updated periodically. All such implementations are slightly different. As noted above, if you take a fitted reference production rate derived, as in step (1) above, using one implementation, and then perform step (2) with a different implementation, you will generally get the wrong answer. In addition, different implementations (e.g., “CRONUSCalc” and “the online exposure age calculator formerly known as the CRONUS-Earth online exposure age calculator”) define different reference states for some scaling methods (or don’t define a reference state at all and use a different fitting parameter entirely), so the best-fitting values of the “reference production rates” are not even estimates of the same thing.

What this means for documenting exposure-age calculations in papers or elsewhere is that reporting the value of the reference production rate that you used doesn’t help readers to reproduce your calculations. Although certainly it is possible that your readers can obtain the exact computer code that you used, it is, in general, difficult due to the continued evolution and improvement of both the scaling methods themselves and the background data. Reporting a value for the reference production rate, in contrast, has two negative effects. First, it gives readers the impression that the value of the reference production rate is a measured physical constant. As discussed above, this is not true. Second, it creates a situation where readers who want to replicate your calculations can only perform step (2) above; they cannot perform step (1). Thus, they cannot start from the same actual measurements that went into your calculations and therefore cannot correctly and consisntently reproduce your calculations.

What does help readers to correctly and consistently reproduce your calculations is to report what calibration data set you used. Remember, the calibration data set consists of actual measurements and observations. Unlike the reference production rate, these measurements and observations are the same no matter what scaling method you are using. They are the same no matter what numerical implementation of a particular scaling method you are using. They are the same no matter what the “reference location” is. It is easy to obtain these measurements; all published calibration data sets are publicly accessible in a standardized form at this website. If you tell readers what calibration data set they used, they can reproduce your exposure age calculations with any implementation of any scaling method in an internally consistent way that begins with the same source measurements. They can perform both steps (1) and (2) above, BOTH of which are necessary for an internally consistent exposure-age calculation, instead of only step (2). Although it may seem difficult or intimidating for readers to perform both a production rate calibration and an unknown-age calculation, this task is simple with existing online exposure age calculation systems: CREp allows a choice of a wide variety of published calibration data sets; the online calculator formerly known as the CRONUS-Earth online calculator allows production rate calibration with any calibration data set.

To summarize,

1. Computing an exposure age is a two-step process that starts with a calibration data set. The calibration data set consists of actual measurements. The “reference production rate” is an inconsistently defined, and never actually measured, parameter that is estimated halfway through this process. Its value is completely dependent on the scaling method used and the specific implementation of that scaling method; it is not a measured physical constant.
2. Emphasizing the “reference production rate” in data reporting or in a description of calculations does not, in fact, help readers to correctly reproduce the calculations in a paper. In fact, it is more likely to lead to internally inconsistent, and likely misleading, reproduction of the calculations.
3. In contrast, reporting the calibration data set that was used does allow the reader to reproduce calculations in an internally consistent way.

To say this a slightly different way, there are two independent ingredients that go into computing an exposure age: a calibration data set and a particular implementation of a particular scaling method. The reader needs to know what these are. The reference production rate isn’t one of these things, and the reader doesn’t need to know it.