the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Combined linear regression and Monte Carlo approach to modelling exposure age depth profiles
- 1Department of Earth and Planetary Sciences, UC Davis, Davis, 95616, USA
- 2Earth Observatory of Singapore, Nanyang Technological University, 639798, Singapore
- 1Department of Earth and Planetary Sciences, UC Davis, Davis, 95616, USA
- 2Earth Observatory of Singapore, Nanyang Technological University, 639798, Singapore
Abstract. We introduce a set of methods for analyzing cosmogenic-nuclide depth profiles that formally integrates surface erosion and muogenic production, while retaining the advantages of the linear inversion. For surfaces with erosion, we present solutions for both erosion rate and total eroded thickness, each with their own advantages. For practical applications, erosion must be constrained from external information, such as soil-profile analysis. By combining linear inversion with Monte Carlo simulation of error propagation, our method jointly assesses uncertainty arising from measurement error and erosion constraints. Using example depth profile data sets from the Beida River, northwest China and Lees Ferry, Arizona, we show that our methods robustly produce comparable ages for surfaces with different erosion rates and inheritance. Through hypothetical examples, we further show that both the erosion rate and eroded-thickness approaches produce reasonable age estimates so long as the total erosion less than twice the nucleon attenuation length. Overall, lack of precise constraints for erosion rate tends to be the largest contributor of age uncertainty, compared to the error from omitting muogenic production or radioactive decay.
Yiran Wang and Michael E. Oskin
Status: final response (author comments only)
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RC1: 'Comment on gchron-2021-34', Anonymous Referee #1, 03 Dec 2021
This paper presents two approaches modelling concentrations of cosmogenic nuclide concentrations based on linear regression combined with a Monte Carlo approach.
Despite being in a field I really appreciate, I have some difficulties to judge what is the value added by this paper. This is probably by the fact that too many assumptions or to be more precise too many shortcuts are used to simplify the main equation governing the cosmogenic production equation as a function of denudation rate and time. Some of these shortcuts are dangerous and some other can be avoided with the used of numerical calculations. I will thus ask for a revision of this paper
In the entire paper I suggest changing erosion by denudation that is more appropriate for cosmo.
At the end of abstract you mention “compared to the error from omitting muogenic production…” I totally agree so, why do you present a linearization that ignores muons contributions?
Line 35-40: despite muons contributions are small at surface compared to the neutron one, ignoring their contributions and considering only neutrons will yield to multiple time/denudation pairs that can model a depth distribution.
Line 44: If you want to be totally objective you should live all parameters free and in a second step consider the solutions that can match the field observations. If you constrain at the first step your unknowns, time or denudation you may miss the real solutions.
Line 55 Legend of Figure 1: you should update the muon contributions; since Braucher 2003, these contribution have been updated (Braucher 2011,2013, Balco 2008, 2017 ). More it has been also shown that Heisinger muons contributions were too high. You should correct them in your matlab code and in the Hidy one.
Line 90-91: again do not omit muon contributions! In a high denudation environment, their contributions are far from being negligible.
Line 67: I think Nishiizumi, 2007 is not appropriate as in this paper he proposed a half-life of 1.36±0.07 Ma.
Line 100 and following paragraph: I think this is not the right approach. First I will have a look to the distribution as a function of depth (in g/cm2) to see within the first two meters what is the value of the “slope” of the exponential decrease. Lower than 250 g/cm2 will traduce a contribution mainly due to neutrons with moderate denudation rate. If higher muon contributions are more important due to higher denudation rate or can be due to a recent rejuvenation of the profile making deep samples to be now closer to surface. In this latter case, running an inversion model with density as free parameter will probably propose high values for density >3 g/cm3 making clear that the profile has been perturbed. This can be the case when loess covers are rapidly eroded by wind deflation, so fast that the cosmo production cannot be at equilibrium.
Therefore I will let run the model with totally free parameters and then cut the Time/ denudation space by probable eroded thickness to reduce this space. By imposing since the beginning of the modelling a constrain as important as the eroded thickness may be dangerous to my point of view.
Line 108: which muon contribution do you used as Tem? Fast or slow? Is this choice important?
Please change the * by × in the tables. Please use uniform values for concentraions ( at/g or 105at/g )
Line 174: why this denudation rate of 0.3±0.05 cm/kyr ?
With this loess covered surface, probably the use of two nuclides will be better than one.
Line 177: I am not convinced by the fact that you authorize inheritance to be negative. This is as you mentioned “non-physical”. Therefore what will happen if you restrict the modelling to inheritance ≥ to zero? Is the overall space of solution affected?
Paragraph 4.2.5 : I agree but using variable production rates implies adding more uncertainties and this is not the fact in the actual calculators !
If you think to revise this contribution you should try to add a second nuclides (26Al for example) and try to remodel the depth profile with two nuclides. Inheritance can thus be variable and this can probably be a great value to the modelling of depth profile.
- AC2: 'Reply on RC1', Yiran Wang, 13 Mar 2022
- AC4: 'Reply on RC1', Yiran Wang, 13 Mar 2022
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EC1: 'Comment on gchron-2021-34', Pieter Vermeesch, 26 Jan 2022
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AC1: 'Reply on EC1', Yiran Wang, 13 Mar 2022
1. Linear fitting with muons.
We agree with the reviewers that it was inappropriate to claim that our method does not require any prior knowledge: erosion depth or rate must be estimated. What we meant to convey is that linear regression inverts for exposure age and inheritance directly, without needing to define a pre-set boundary or initial prior distribution for these. The erosion rate (or eroded thickness) is a required input for our inverse modelling approach.
Our motivation for introducing this approach is, first, to expand the application of linear regression to model exposure-age depth profiles. Prior published linear regression methods ignore muons and do not adequately address erosion. Our approach incorporates muogenic production, making it possible to estimate ages for sites with steady rates or a defined amount of erosion. As demonstrated in the discussion section, this method can provide exposure age estimations with high precision for profiles with less than 2 times attenuation length erosion, suggesting it may be applied for most surface exposure dating scenarios.
Our second motivation is that as an inverse approach, the least squares linear regression approach directly solves for age and inheritance, while treating the erosion rate/eroded thickness as an input instead of an output of the model. This characteristic makes it a convenient tool for exposure age estimation. It can be used with Monte Carlo sampling to explore the full distribution of possible ages and inheritance from the variation of input parameters (including erosion). Linear regression is also useful as a starting point for forward (e.g. Bayesian) models. Inverse-modeled age and inheritance may thus help researchers to tune the boundary values for the forward models to get better simulation results. Therefore, instead of replacing forward methods, we will argue that our approach complements forward methods. In addition, as an added benefit of our inverse model approach, the effective age Te provides a straightforward way to explore the trade-off between erosion rate and exposure age.
By comparing results from our least squares linear inversion with a Bayesian inversion using Metropolis-Hastings sampling, we find the age estimation results have similar precision for both methods, which supports the robustness of our approach. We are considering including this comparison into our revision.
2. Monte Carlo error estimation
The editor’s description of the procedure of our inversion and Monte Carlo simulation approach is correct. We would like to modify the procedure as follows:
1. Generate all the input parameters (10Be concentration, sample depth, eroded thickness, production rate, attenuation length, etc.) from pre-defined distributions.
2. Fit the data with eq. 10 to get Ten and inheritance.
3. Calculate the exposure age use Ten from step 2 and parameters generated from step 1 using eq. 11.
4. Repeat step 1-3 for many times, sampling the underlying distributions of each parameter, to produce a distribution of results.
In our inversion, we treat the erosion as an input of the model, therefore the uncertainty related to the erosion can be treated the same way as other parameters. We treat the erosion as a known value instead of an inversion output for several reasons. First, it simplifies the inversion process, so that one can inverse linearly in 2D space, and the Te value resulted from the first step could be used to explore the trade-off between erosion rate and exposure age. Second, though theoretically it is possible to find a unique set of solutions for erosion rate and exposure age for a certain depth profile, in practice, the sample uncertainties always exceed the resolution that required to define that unique solution. This means including erosion as an unknown is unlikely to increase the precision of the estimation results. As mentioned earlier, we are considering including comparisons of our least squares linear inversion and Bayesian inversion with pseudo depth profiles to demonstrate that a simple Monte Carlo sampling with two-step estimation can provide sufficiently accurate results.
3. Negative Inheritance
Negative inheritance estimations can be prevented from inversion process, and we have incorporated that into our earlier published work (e.g. Wang et al., 2020). However, we realized in the writing of this manuscript that those negative results, though physically impossible, are necessary for mathematical reasons.
For example, suppose we have a sample group with a true age of 100 kyr and a true inheritance of 0 atoms/g. Because of the uncertainties within the sample measurements, ideally, we want to have the estimated distribution of the exposure age to be centered around 100 kyr, with approximately half of the estimation older than 100 kyr, the other half younger than 100 kyr. The estimations younger than 100 kyr correlate to some positive inheritance, while those older than 100 kyr correlate to negative inheritance. The overall distributions of the inheritance should be centered around 0, meaning that approximately half of the estimated inheritance should be negative. If we require non-negative inheritance during the inversion, it will end up with deviated estimation results: the inheritance would center around a positive value, while the central age would be younger than 100 kyr.
We do not agree with Dr. Hidy’s comment that it may be an overstatement and the effect of negative inheritance may not be an issue for other methods. We tested the two methods (least square linear inversion described in our manuscript and a Metropolis-Hastings Monte Carlo Bayesian inversion) using pseudo depth profiles with zero inheritance. We find that requiring non-negative inheritance during inversion leads to underestimation of the exposure age for both approaches. We are considering including this into the revision.
4. Scaling methods
Thank you for the suggestion. We will use a more complicated muon production model instead.
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AC1: 'Reply on EC1', Yiran Wang, 13 Mar 2022
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RC2: 'Comment on gchron-2021-34', Alan Hidy, 14 Feb 2022
The comment was uploaded in the form of a supplement: https://gchron.copernicus.org/preprints/gchron-2021-34/gchron-2021-34-RC2-supplement.pdf
- AC3: 'Reply on RC2', Yiran Wang, 13 Mar 2022
Yiran Wang and Michael E. Oskin
Yiran Wang and Michael E. Oskin
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