the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Sep 2024
Nonparametric estimation of age-depth models from sedimentological and stratigraphic information
Abstract. Age-depth models are fundamental tools used in all geohistorical disciplines. They assign stratigraphic positions to ages (e.g., in drill cores or outcrops), which is necessary to estimate rates of past environmental change and establish timing of events in sedimentary sequences. Methods to estimate age-depth models commonly use simplified parametric assumptions on the uncertainties of ages of tie points. The distribution of time between tie points is estimated using simplistic assumptions on the formation of the stratigraphic record, for example that sediment accumulates in discrete events that follow a Poisson process. In general, age-depth models are a crude simplification that fail to provide a comprehensive implementation of all empirical data or expert knowledge (e.g., from sedimentary structures such as erosional surfaces or from basin models). In other words, many information sources that can potentially provide geochronologic information remain un- or underused.
Here, we present two non-parametric methods to estimate age-depth models from complex sedimentological and stratigraphic data. The methods are complementary as they use different sources of information (sedimentation rates and observed tracer values), are implemented in the admtools package for R Software and allow the user to specify any error model and distribution of uncertainties.
As use cases of the methods, we
- construct age-depth models for the Late Devonian Steinbruch Schmidt section in Germany and use it to estimate the timing of the Frasnian-Famennian boundary and the duration of the Upper Kellwasser event.
- use measurements of extra-terrestrial 3He from ODP site 960 (Maud Rise, Weddell Sea) to construct age-depth models for the Paleocene–Eocene thermal maximum (PETM).
The first case study suggests that the Upper Kellwasser event lasted 89 kyr (IQR: 84 to 97 kyr) and places the Frasnian-Famennian boundary at 371.834 ± 0.101 Ma (2 σ), whereas the second case study provides a duration of 41 to 48 kyr for the PETM recovery interval.
These examples show how information from a variety of sedimentological and stratigraphic sources can be combined to estimate age-depth relationships that accurately reflect uncertainties of both available data and expert knowledge.
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RC1: 'Comment on egusphere-2024-2857', Maarten Blaauw, 25 Oct 2024
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This manuscript discusses two modules that could be useful for age-depth models - one that enforces sedimentation to be in chronological order, and another one that adjusts age-depth curves so that they even out fluctuations in proxies that could be assumed to have been subject to constant influxes over time. Neither of the modules are particularly novel, with many developments in this area having taken place decades ago. It also seems that the authors have an axe to grind with regard to existing approaches to age-modelling, and I disagree with most of the arguments made against them.
The description of the availability, development, versioning and use of the R software is described very well and reflects best practice. Much is made of the process not being automatic and requiring much user input. I wonder though how many researchers will have the expertise to select the parameters and settings appropriately/correctly. Is there a danger that users will (inadvertedly) choose settings such that they get age-models based on wishful-thinking? Lines 537-541: Many of the other age-depth models out there, be they Bayesian or classical, are also explicit in their modelling assumptions. As long as the users report what settings and versions were used, the models are replicable, documented, and can run under a wide range of computational environments.
The usage in the abstract of terms like 'simplistic' and 'crude simplification' should be toned down. Many of the currently available techniques presented a significant improvement on what was done before. In lines 512-517, the authors disparage currently popular mathematical distributions for age-depth modelling such as Poisson or gamma distributions as simplistic mathematical shortcuts with little if any real-world relevance. I don't think that is fair criticism - instead, as one of the authors of these methods, I can vouch that these distributions were indeed selected with the aim to provide more realistic deposition models. Everybody would agree that models are simplifications of real-world processes. It is frustrating that the manuscript rejects these mathematical distributions but then proposes a uniform distribution for sediment accumulation rates, which to me seems much less geologically robust/defendable and almost entirely made for ease of coding. Lines 403-6, limits on sedimentation rates: how are these decided? For your chosen limits of 0.1 to 0.6 cm/kyr, is it really geologically likely that any value inbetween that range is equally likely, but that any value outside of that range has a probability of 0%? I don't think so. Why not use say a gamma distribution which by itself enforces sedimentation to never be <0, and which can downweigh sedimentation rates that are considered to be unlikely, without the need to enforce such strict borders? With more information available, the gamma distribution can be made to peak more (or less). Line 70, all models are partly driven by data, and partly by assumptions.
Lines 531-536, yes, ages often scatter beyond their lab uncertainties, but Bayesian models deal very well with such scatter - much better than classical models can. Same for lines 551-551: both are examples of classical models, yet real-world data and simulations have shown that Bayesian models provide much more pessimistic/realistic uncertainty estimates (e.g., Blaauw et al. 2018 <doi:10.1016/j.quascirev.2018.03.032>). How do the uncertainties of the proposed age-depth model compare with existing Bayesian/classical models?
Line 54, note that orbital matching (as opposed to age-depth modelling based on e.g. radiometric dates) makes assumptions that could be seen as simplistic, hard to prove, and causing a degree of circularity with and dependence on other records. See Blaauw 2012 <doi:10.1016/j.quascirev.2010.11.012> for a critical review of climatic tuning - although I agree that sometimes one would need to be pragmatic and choose tuning if no absolute age estimates are available.The CON algorithm is presented using many equations, but in the end defines nothing fundamentally new - this has all been stated before in different notation or terms in previous age-depth model papers. As for FAM, another classical use of correcting for supposed constant influx assumed a constant pollen rain and then applied this assumption to stretch and compress a core's accumulation rate. See Middeldorp 1982 <doi:10.1016/0034-6667(82)90003-3> and Young et al. 1999 <doi:10.1016/S0034-6667(98)00060-8>. Of course, on long time-scales it is unlikely that any proxy will have a constant flux (Figure 4 and the accompanying discussion are interesting in this respect). The interpretation that the authors' FAM model essentially resembles the CRS model doesn't make the new model novel either - the CRS stems from the 1970s.
Line 301, "Timing and positions of tie points can follow arbitrary probability distributions as long as they are strictly ordered." Do you mean that closely spaced dates/tie-points cannot have overlapping age distributions? That would be a drawback of the method. This type of problems is exactly why Bayesian models which use Poisson/Gamma distributions are so successful in modelling sedimentation - they guarantee that reversals are absent, and moreover, they can take advantage of high dating densities; they "learn" with precision becoming ever higher as more dates are added. This is not necessarily the case for classical age-models (see Blaauw et al. 2018 <doi:10.1016/j.quascirev.2018.03.032>).
Fig. 2, isn't the floating age-model in panel B and the anchored one in panel A? If not, then the terminology used is confusing (at least to me). If a core's age is measured relative to an otherwise dated layer (i.e., it is tied), how can it be floating? Surely the age uncertainty of the Bentonite layer is not 0, so there is no point in presenting Figure 2A. (Later on in the Discussion, a point is made that one can set the uncertainty at a layer to 0 in order to estimate the durations of an interval. However, that can also be done by simply subtracting the bottom ages from the top ages for each iteration of say a Bayesian age-depth model, be it floating or anchored.) For Figure 2A, I also don't see a 'sausage-shape' but rather a nematode-shape. Are the reconstructed 95% ranges of Fig. 2B considered realistic? Do they compare well with established methods such as BChron? This should be discussed, e.g. after line 511.
Details:
Line 29 presents a very interesting example of how an updated age-depth model affected evolutionary interpretations.
Relying on R's integrate function could be problematic, as explained in its help function.
36, and further on in the manuscript, the author of Oxcal is Bronk Ramsey (not Ramsey)
60, CONdensation (not CONndensation)
68, superposition
91, the assumption that accumulation always happens (no hiatuses) would fail easily on long time-scales.
158, congruent
345, missing full stop
378, two assumptions
380, length
501, shown
516, the the
557, full use of these
Table 1, Bacon additionally uses sed.rates and its variability as (prior) information sources.Citation: https://doi.org/10.5194/egusphere-2024-2857-RC1 -
CC1: 'Problematic', Richard Zeebe, 03 Nov 2024
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The manuscript discusses age-depth models in general, as well as age models for specific events, including the Paleocene-Eocene Thermal maximum (PETM). Unfortunately, the omission of references to immediately relevant work on the subject and the lack of discussion of the authors' results in relation to previous work is highly problematic.
The authors' PETM age model uses 3He data from Farley and Eltgroth (2003). Omission of the immediately related study by Murphy et al. (2010) using 3He for the PETM is hence rather incomprehensible:
Murphy, B. H., K. A. Farley, and J. C. Zachos. An extraterrestrial 3He-based timescale for the Paleocene-Eocene Thermal Maximum (PETM) from Walvis Ridge, IODP Site 1266. Geochimica et Cosmochimica Acta 74, no. 17 (2010): 5098-5108.
Murphy et al. (2010) designated the end of the PETM CIE recovery to occur at 217 (+44/-31) kyr, a much longer time interval than suggested in the present manuscript (Fig. 6).
Zeebe and Lourens (2019) derived a PETM duration of 170 (+-30) kyr from onset to recovery inflection, again a much longer time interval - and at odds with the present manuscript.
Zeebe, R.E. and Lourens, L.J., 2019. Solar System chaos and the Paleocene-Eocene boundary age constrained by geology and astronomy. Science, 365(6456), pp.926-929.
Regrettably, none of the studies above is mentioned or referenced in the ms (there are more). Neither are the large discrepancies discussed relative to previous work. Perplexingly, there is also no reference to any of the PETM landmark papers led by Zachos in the reference list.
Given the above, the manuscript should not be considered for publication without an in-depth analysis and discussion of relevant previous work on the PETM and PETM duration, most notably those studies at odds with the present manuscript, including the references mentioned above.
Richard E. Zeebe
Citation: https://doi.org/10.5194/egusphere-2024-2857-CC1 -
AC2: 'Reply on CC1', Niklas Hohmann, 22 Dec 2024
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Dear Richard
Thank you for your comment and feedback on this manuscript.
The aim of the PETM example was to examine how deviations from the assumptions of constant 3He flux change the inferred age estimates. The assumption of constant flux is an obvious weakness of previous age-depth models as acknowledged in the respective publications, which is why we chose this example to demonstrate the flexibility of FAM. We provide all code and documentation for this example, ensuring it is transparent and reproducible, and show that short-term variability has only a minor effect on the estimated ages.
We apologize if this example was perceived as problematic – as stated in the manuscript, our intention was not to determine new durations for the PETM, but rather explore the robustness of the model assumptions used in previous age-depth models (lines 450f). We are happy to adjust the phrasing of the results and problem in the abstract and the example, and include more references for context on the discussion on the PETM duration. As preliminary analyses, we applied FAM to the 3He record from ODP site 1266 presented by Murphy et al (2010) (https://doi.org/10.1016/j.gca.2010.03.039) and arrive at very similar durations as they do (e.g., duration clay layer: Murphy et al. (2010): 167 kyr, our results: 170 kyr). If both you and the editor agree that the example from site 1266 provides a more suitable example to demonstrate the methodology and adds to the manuscripts clarity and reception, we would be happy to substituting site 690 with site 1266.
Best regards,
Niklas HohmannCitation: https://doi.org/10.5194/egusphere-2024-2857-AC2
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AC2: 'Reply on CC1', Niklas Hohmann, 22 Dec 2024
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CC2: 'Comment on egusphere-2024-2857', Robin Trayler, 20 Nov 2024
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I have one minor comment regarding my own work. On like 500 of the preprint the authors state:
“For example, the computations accompanying Trayler et al. (2023) publication of astroBayes will take “several days or weeks” to compute on a laptop. In contrast, the computation time for the examples show here is typically within minutes.”.
The “days or weeks” quote does not come from the main manuscript, and is instead found in the github repository (https://github.com/robintrayler/astroBayes_manuscript) associated with model testing and validation. This code generates ~10,000 individual models (each made up of thousands of MCMC iterations) and was intended to test and validate overall astroBayes model performance. Individual astroBayes models take a few minutes to maybe an hour, depending on complexity and the number of parameters. It is somewhat disingenuous to compare the time it takes to compute thousands of models to the time it takes to generate single models using the authors code.
These computational times are also true for the other Bayesian models mentioned by the authors. Bchron, Bacon, OxCal, etc can all be run on a personal laptop in a few minutes to hours depending on complexity and the number of user specified MCMC iterations.
Best,
Robin Trayler
Citation: https://doi.org/10.5194/egusphere-2024-2857-CC2 -
AC1: 'Reply on CC2', Niklas Hohmann, 27 Nov 2024
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Thank you for taking the time to read our manuscript, and for your comment.
We apologize for this mistake, and will remove the corresponding part when revising the manuscript.Best regards,
Niklas HohmannCitation: https://doi.org/10.5194/egusphere-2024-2857-AC1
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AC1: 'Reply on CC2', Niklas Hohmann, 27 Nov 2024
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CC3: 'Comment on egusphere-2024-2857', Jarno Huygh, 21 Nov 2024
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The manuscript aims at providing stratigraphers with two non-parametric methods for constructing age-depth models, taking into account uncertainty and integration of a wide(r) range of information available to the researcher into the age-depth model. Both the manuscript and the supplementary material (R-code etc.) are very detailed and well organized. The quality of the included figures is good. This manuscript is targeted at stratigraphers and the following comments are from the perspective of a (cyclo)stratigrapher. No comments will be given on the selection of the case studies and the discussion thereof.
Even though the manuscript is aimed at stratigraphers, it is very mathematics-oriented, very abstract, and, requires a considerable statistical background and large time-investment to fully grasp. Figuring out how to (correctly) apply the developed methods (using the admtools package) would equally demand a very large time-investment. These factors create a very high barrier and may likely result in few researchers using the methods.
In addition, the manuscript fails to clearly highlight the added value of using these methods and thus to convince researchers to invest their time in learning how to apply them. Especially when working on Paleozoic cyclostratigraphic case studies (that inherently have very large time-domain uncertainties), the added value of the proposed methods ‘appears to be’ negligible. Lastly, age-depth models are generally not just crude simplifications (as is mentioned in the abstract) – in fact, a vast array of information, data and insight is combined by the researcher to construct them.
I would strongly recommend to drastically reduce the size of subchapter 2 (lines 90 to 341) and to remove all formulas to the supplementary material. In addition, I would recommend emphasizing why stratigraphers need these methods, when they are most useful and, what pitfalls to watch out for when usign them (use of plain language may be useful here). Lastly, this manuscript would, after publication, greatly benefit from an online course on how to correctly apply given methods. Clearly, great effort and talent was put into this work and it would be great to see it applied in research.
Citation: https://doi.org/10.5194/egusphere-2024-2857-CC3 -
AC3: 'Reply on CC3', Niklas Hohmann, 15 Jan 2025
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Dear Jarno
Thank you for taking the time to read the manuscript and for your comments.
Thank you for your praise of the structure and detail of the manuscript and supplementary material such as the R code. We value code documentation and clarity to make methods and results as well documented as possible to facilitate methodological clarity.
One of the main points you raise is that the manuscript is very mathematics oriented (“Even though the manuscript is aimed at stratigraphers, it is very mathematics-oriented, very abstract[…]”) and suggest moving most of the methods description into the supplementary materials (“I would strongly recommend to drastically reduce the size of subchapter 2 (lines 90 to 341) and to remove all formulas to the supplementary material.”)
We believe that scientific methods should be documented in detail, so that the tool is not a “black box” to the users. In doing this, we follow the example of previous articles in this field, which served as guidance for us, including Blaauw & Christen (2011) or Haslett & Parnell (2008). It would be patronizing towards the readers to assume they cannot follow the derivation of the method. We support the expectation that one should aim at understanding the method one uses, and the premise of what we propose is in fact delegating more agency to the users in choosing the parameters and assumptions. Removing the equations would go against this intent. The brief summary of the underlying logic at the beginning of each section (lines 120 to 125 and 165 to 180) gives readers the opportunity to skip the technical parts of the manuscript. We leave the choice whether to move the equations to the supplementary materials or to keep them in the main text body to the editor.You raise the point that the methods are hard to apply and will take time-investment to learn (“Figuring out how to (correctly) apply the developed methods (using the admtools package) would equally demand a very large time-investment”), and suggest that we should organize workshops on the methods (“this manuscript would, after publication, greatly benefit from an online course on how to correctly apply given methods.”).
We intend to develop such a workshop, and are committed to improve the documentation and vignettes associated with the R package. We welcome all specific comments and suggestions for these components, and invite the scientific community to improve them or suggest improvements to them via GitHub according to the package contribution guidelines.You state that the added value of the presented methods is not clear, “especially when working on Paleozoic cyclostratigraphic case studies (that inherently have very large time-domain uncertainties)”. We are sorry if this is not clear from the current version of the manuscript, and are happy to clarify this when revising it. However, considering that the lack of systematic reporting of uncertainty by cyclostratigraphic studies was one of the challenges listed in the seminal reproducibility review study by Sinnesael et al. (2019), we are surprised to hear that the added value of incorporating the uncertainty of sedimentation rate estimate is not clear. We believe that the presented methods provide the opportunity to include previously unused data (e.g., semi-quantitative estimates of sedimentation rates, knowledge on tracer fluxes) into the construction of age-depth models. And we stand by our claim that admtools allows incorporating uncertainty from evolutive cyclostratigraphic methods, which is otherwise nowhere exploited in current literature.
Best regards,
On behalf of the authors
Niklas HohmannCitation: https://doi.org/10.5194/egusphere-2024-2857-AC3
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AC3: 'Reply on CC3', Niklas Hohmann, 15 Jan 2025
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RC2: 'Comment on egusphere-2024-2857', Robin Trayler, 31 Dec 2024
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Note
As a quick note, I was asked to provide a Refereed Comment on this manuscript after I left a short community comment and had read the comments by Dr. Blaauw and Dr. Zeebe. In general I agree with their assessments as well, and have tried not to duplicate their comments here.
Overview
This manuscript presents some interesting methods for constructing age depth models, and the underlying code appears to be very flexible and allows users to specify arbitrary distributions and uncertainty structures. I view this as both a positive and negative. As is, it is difficulty to tell exactly how one might actually generate an age depth model and what information is needed. For example, the ICON model relies on sedimentation rate estimates, but it is unclear where these come from and what they are based on. Given that one of the most common uses of age-depth modeling is to estimate sedimentation rate, it is unclear what a user is gaining from using these methods over more established models.
I also agree with Dr. Blaauw’s comment that the manuscript “[has] an axe to grind with regard to existing approaches to age-modeling” which I think harms the message of the manuscript. I feel like the manuscript would be well served to focus on comparing and contrasting their methods with existing methods, rather than dismissing the existing methods out of hand as “crude simplifications”.
Introduction
Both methods are nonparametric in the sense that they do not make assumptions on the processes or probability distributions that govern sediment accumulation and the structure of the stratigraphic record. They are assumption explicit in the sense that they assume the law of superpositon holds, but all other assumptions must be provided by the user. This ensures that model assumptions are specified verbatim and clearly communicated, and the estimated age-depth models are data-driven rather than assumption-driven.
- This manuscript makes some strong statements about “assumptions” throughout that I find problematic. especially since the methods presented within essentially contain equivalent assumptions about sediment accumulation. I appreciate that the manuscript lays out their basic assumptions about sedimentation in section 2.1, but this does not mean they are not being made nor does it inherently make these methods more data driven. It only means that they are being stated clearly, as they have been the papers detailing past age-depth models [@haslett2008; @blaauw2011]. For example, I view assumption 1 as a valid compromise for constructing age depth models, but it is not strictly true in a geologic sense. Hiatuses and negative accumulation (erosion) are both clearly observable processes but are difficulty to deal with from a modeling standpoint. Nevertheless, multiple existing models (Bacon, my own astroBayes) have at least rudimentary ways of dealing with these problems.
Model Formulation
- The manuscript does a very thorough job laying out their mathematics for both the ICON and FAM models. However found much of this information difficult to follow. As written, it’s difficult to tell what information is needed to actually construct a model. Is it a series of dates with stratigraphic positions? Continuous or discrete estimations of sedimentation rate or tracer flux? I would strongly suggest the inclusion of a few conceptual figures that show how the models are actually constructed in practice.
Implementation
- The manuscript follows best practices for code and data availability. The linked Zenodo repository contains a fixed version of the R package and the CRAN/ Github version are also well documented.
- I would like to see more details on how the models were tested however.
- First how has it been verified that equivalent results are generated each time the models are run? As far as I can tell, there are stochastic, random number generators “under-the_hood”. Do these always produce the same answer (allowing for small amounts of change due to the nature of random number generation)?
- Second, how do these new methods compare to established age-depth models? For example the Bacon and Bchron R packages, and OxCal have 10,000+ of citations between them and have become the de facto standards for age-depth modeling in many contexts. How well does admtools compare to these packages? Does it produce more precise models, allow fitting of information not allowed by them, etc? I’d like to see this explored more systematically to help guide users towards the most appropriate models for their data. It might be worth doing comparisons of admtools with Bacon/Bchron/ OxCal for the Devonian or PETM examples. This could be added to the “comparison with other methods” section
Examples
- Line 375: There is a probability based version of timeOpt included in astrochron. I am unsure if it has been officially implemented in eTimeOpt however. See [Meyers and Malinverno, 2018] for details.
- Figure 2: I think the floating and anchored subplots are labeled backwards or It is not clear how the anchoring date is being used. I would suggest double checking the labeling and adding a point + error bars to the anchored panel to visually show the anchoring date and its uncertainty. As is, it’s very unclear why the “floating” model would converge to near zero uncertainty in the bentonite layer and the anchored one would not. Is this convergence because the “floating model” is actually assuming the bentonite has a fixed age with no uncertainty?
“In the floating age-depth model, time is measured relative to the bentonite layer dated by…”
- The bentonite uncertainty (±0.053 Ma; 1 or 2 σ?) should have error bars that cover slightly less than the distance between the a pair of grid-lines on the x-axis.
- Line 395: how can the model be “floating” if it has a tie point?
Limitations
- Line 519: A minor comment but it cannot incorporate gaps in the stratigraphic record. The presence of fossils in the rocks seems to play no role in model construction.
- I agree that hiatuses are very difficult to deal with without a priori knowledge of their position and or duration.
Citation: https://doi.org/10.5194/egusphere-2024-2857-RC2
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