the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modeling apparent Pb loss in zircon UPb geochronology
Matthew A. Malkowski
Abstract. Although the loss of radiogenic Pb from zircon is known to be a major factor that can cause inaccuracy in the UPb geochronological system, the distribution of Pb loss in natural samples has not been well characterized. Treatment of zircon by chemical abrasion (CA) has become standard practice in isotope dilutionthermal ionization mass spectrometry (IDTIMS), but CA is much less commonly employed prior to insitu analysis via laser ablationinductively coupled plasmamass spectrometry (LAICPMS) or secondary ionization mass spectrometry (SIMS). Differentiating the effects of low levels of Pb loss in Phanerozoic zircon with relatively low precision insitu UPb dates, where the degree of Pb loss is insufficient to cause discernible discordance, is challenging. We show that UPb dates that have been perturbed by Pb loss may be modeled by convolving a Gaussian distribution, that represents the unperturbed UPb date distribution, with a distribution that characterizes Pb loss. We apply this mathematical framework to model the distribution of apparent Pb loss in 10 igneous samples that have both nonCA LAICPMS or SIMS UPb dates and an estimate of the crystallization age, either through CA UPb or ^{40}Ar/^{39}Ar geochronology. All but one sample showed negative age offsets that were unlikely to have been drawn from an unperturbed UPb date distribution. Of the eight continuous distribution types we considered, modeling apparent Pb loss using the Weibull distribution produced, on average, the closest match with the nonCA UPb date distributions. We show two contrasting patterns in apparent Pb loss: samples where most zircon UPb dates undergo a bulk shift and samples where most zircon UPb dates exhibited low age offset but fewer grains had more significant offset. Our modeling framework allows comparison of relative degrees of apparent Pb loss between samples of different age, with the first and second Wasserstein distances providing useful estimates of the total magnitude of apparent Pb loss. Given that the large majority of insitu UPb dates are acquired without the CA treatment, this study highlights a pressing need for improved characterization of apparent Pb loss distributions in natural samples to aid in interpreting nonCA insitu UPb data and to guide future data collection strategies.
Glenn R. Sharman and Matthew A. Malkowski
Status: final response (author comments only)

RC1: 'Review of gchron20236', Anonymous Referee #1, 08 May 2023
The fundamental concept underlying this contribution by Sharman and Malkowski  that observed UPb ages can be considered as a convolution an a true age distribution (i.e., a distribution representing analytical uncertainty around the true mean age of the analyzed material) with a distribution representing Pbloss  is certainly reasonable, though the form of these distributions may vary widely. The analytical age distribution of a single analysis in the absence of Pbloss is frequently assumed to be Gaussian, so this is a reasonable assumption; the distribution of Pbloss is at present much less well understood. To better understand this latter distribution, the authors start with independent (arguably Pblossfree, CA or non UPb) ages for ten Phanerozoic samples, and convolve each with different potential Pbloss distributions to see which best reproduces the observed nonCA UPb distribution. While I have a number of questions and suggestions, overall this is a worthwhile contribution.
The authors represent Pbloss as a negative percentage offset from the true crystallization age. This is fine mathematically for the purposes of modelling Pb loss in a single decay system, but perhaps it is worth emphasizing that
1) this is not equivalent to the percent of Pb lost, and
2) this percentage age difference will not generally be the same for the 206Pb/238U and 207Pb/235U ages, and for each system will depend on both the time of Pbloss as well as actual amount of Pb lost
In this context, how do the authors propose to deal with the fact that different "Pbloss" proportional age distributions must be convolved for the 206Pb/238U and 207Pb/235U systems? Would it be possible to consider instead a convolution between a Gaussian distribution representing the isotopic ratios at the time of Pbloss and a distribution representing the actual amount of Pb lost? This would allow the same convolution or deconvolution to apply to both systems simultaneously (and even in principle 208Pb/232Th).
One other issue arising from the fact that Pbloss happens in terms of atoms rather than ages is that of common Pb corrections. In CAIDTIMS, common Pb from inclusions is generally thought to be removed by CA, so only a lab blank subtraction is performed. However, in insitu analyses some form of common Pb correction is commonplace; this may have secondary consequences in the case that a sample is also discordant (e.g., discussion in Andersen et al. 2019, which you currently cite in the context of the general problem of Pbloss in insitu datasets). Fully dealing with this may be outside the scope of the current paper, but perhaps bears some consideration.
One other conceptual concern involves the form of the distributions chosen to represent Pbloss; a number of parametric distributions are tested, and all are better than no correction (with Weibull performing best), it seems possible that the true distribution of Pbloss may diverge from any of these (i.e., be a combination of multiple distributions, or nonparametric). Ideally, it might be possible to invert for the true form of the Pbloss distribution.. have the authors considered if a deconvolution / inverse approach is feasible? Absent that, is there perhaps any underlying quantitative or intuitive rationale to explain the relative success of the Weibull distribution?
A few other more minor notes:
While the authors do provide several nice illustrations of convolution, one point which may be worth noting to help make the concept more intuitive to nonspecialists may be that convolving distributions is equivalent to adding random variables  so for example convolving an exponential Pbloss distribution with a Gaussian analytical distribution yields a third distribution which is the same one you would draw from by drawing a random variable (i.e., a random age) from the Gaussian and another from the Exponential and adding them together.
Another point which bears some note: while both CAIDTIMS UPb ages and Ar/Ar ages are likely to avoid the influence of Pbloss, daughter loss is not unheard of in the Ar/Ar system. How analogous is the HF leaching sometimes conducted by Ar labs to CA? Is this equally effective in eliminating daughter loss?
I was glad to see that the authors provided their full code via a persistent DOI (in this case, Zenodo), in line with best practices. The supplementary video illustrating convolution was a fun addition.
Citation: https://doi.org/10.5194/gchron20236RC1 
AC1: 'Reply on RC1', Glenn Sharman, 07 Jun 2023
The comment was uploaded in the form of a supplement: https://gchron.copernicus.org/preprints/gchron20236/gchron20236AC1supplement.pdf

AC1: 'Reply on RC1', Glenn Sharman, 07 Jun 2023

RC2: 'Review of gchron20236', Anonymous Referee #2, 11 May 2023
The work by Sharman and Malkowski presents a modelbased consideration of the effects of radiogenicPb loss in zircon. Such effects are well known in the UPb community and a discussion on the diagnosis of open system behaviour of widespread importance for UPb geochronology. Nonetheless, there are some significant concerns with aspects of the study that preclude me recommending publication in its current form.
Specifically, the work apparently seeks to better characterise radiogenicPb loss in situations that it may be cryptic. However, there are already wellestablished, more appropriate, and more powerful mechanisms to do this. For example, simple comparison of isotopic ratios to geochemistry (uranium, iron, calcium, REE, raman, OHO, etc) and / or internal mineral texture will already provide a much simpler but much more powerful way to demonstrate the presence of Pb loss. In short, it is unclear how the proposed models provide a tool that will be used to advance geochronology interpretations.
I am sorry to do this, but I think this work needs to be considered in the historical context of UPb geochronology because it is relevant to perceptions around modelbased UPb approaches and (as I get to) has implications for key tests for this work. In the 1960s UPb isotopic analyses of zircon clearly demonstrated that in many cases zircon behaves in an open system fashion (e.g. is discordant). Now many researchers at that time also attempted to extract primary ages (and secondary overprinting) by interpreting linear and indeed nonlinear arrays on concordia diagrams using models that rapidly increased in complexity (for example; Tilton 1960 JGR, Silver and Deutsch 1963 Journal of Geology, Steiger and Wasserburg 1966 JGR). Other developments also happened at around the time modelbased interpretations were in vogue. Namely, isotope dilution analysis of single zircon grains with air abrasion and magnetic separation (e.g. Krogh) and of course insitu dating via ion microprobe dating (e.g. Compston). These analytically based developments set zircon UPb geochronology on the pathway of identification, extraction, and dating of grain domains with closed UPb systems (or specific targeting of open system domains where geochemical evidence could also be brought to bear on the subject).
Now my point (and I am aware of this from my own experience in reviews) the general community has a strong preconception that modelbased approaches are generally unreliable to the point of being unproductive (given the numerous processes that can lead to the same distribution). Hence, works that try to revive a modelbased approach to UPb geochronology, in an effort, to enhance understanding and make such models helpful to better understand geology, must allay this perception. In order to achieve this outcome of an advance then what can be done: Well, it would seem logical to this reviewer, that any new modelbased approach needs to satisfy two conditions:
1/ It must be quantitatively calibrated against high quality closedsystem geochronological data AND known times of disturbance. The choice of the samples where both primary and secondary ages are determined by precise, accurate and modelindependent methods for such tests is crucial. Unfortunately, the sample choice in this work failed this criterion as the same grains were not analysed after LAICPMS by TIMS and in fact, in some cases the choosen studies have used even a different isotopic system to constrain the “true” age. Moreover, the timing of overprinting processes has not been clearly independently determined on the same material to the level needed. Hence, to demonstrate the use of this work and continue this study, such condition really needs to be passed. Such tests would significantly benefit from including detailed geological and petrologic information so the geological context and implications of the proposed models can be understood. This would necessitate detailed characterization of the grains, for example CL and BSE images before and after analyses, the latter showing ablation spots (and potentially also Raman spectroscopy) so any relationship between these grain level observations and isotopic ratios could be made, as they would serve as prima facie evidence of open system conditions.
2) It must be demonstrated that the new approach yields new information that is not available and unobtainable with modern closed system methods or simple relationships already at hand. This is a big challenge because by combination of mineral chemistry with isotopic ratios already can yield much more rigorous insight into geological processes than by this strongly model based example of age distribution fitting. Furthermore, any ages calculated, or more specifically in this case, distributions proposed with such new methods really needs to be accompanied by uncertainty intervals that include the modelrelated uncertainty around the distribution. This is a very difficult goal to achieve.
In this current study, there appears to be a signficant way to go to satisfactorily address both these conditions.
Significant issues
Precision in the language. There are numerous cases where the level of precision in the text could lead to missinterpretation by a reader. Moreover, there are specific inaccuracies. Please refer to the specific points below which document some of these.
The discussion of the causes of radiogenic Pb loss appears incomplete. While a damaged crystal structure is clearly a factor it isn’t the sole prerequisite for open system processes. Please see the work of Silver / Pigeon which clearly demonstrates that fluids are also needed to strip Pb. In short, a more accurate description of radiogenicPb loss is needed.
Assumption of a gaussian distribution for the undisturbed zircon state of UPb ratios. There are several primary processes that could lead to a nongaussian distribution that should at least be mentioned. While the simplifying assumption of a gaussian distribution is a reasonable starting position for certain growth processes, the work would be improved with a consideration of the natural complications to this situation. For example: Common Pb – it’s presence and form of correction. Specifically, a nonuniform common Pb composition (while unlikely to be of significant concern in zircon and of more relevance for minerals with typically higher common Pb loads e.g. apatite and titanite) will invalidate the assumption of a gaussian distribution. Furthermore, there would be expected to be a complex interrelationship between radiogenicPb loss, discordance, and common Pb amount and composition that would have an implication for the model. Moreover, as precision increases so a natural outcome of this will be a nongaussian distribution, the point where this nongaussian distribution appearance breaks down would be a function of the growth duration of a population of zircon which is highly magma (size, temperature, cooling rate, chemistry, etc) dependent. A more sophisticated realisation of what zircon growth is, would benefit this work (there are several new mineral equilibrium model papers that deal with zircon growth rates that clearly are relevant in this regard). It is highly simplistic, without any caveats, to assume zircon growth is instantaneous – there are many environments where prolonged zircon growth has been demonstrated and these sorts of environments are entirely unsuited to a model assumption of a normal distribution.
Overlooked published similar populationbased approaches in geochronology:
The work makes quite a few claims of novelty. While aspects of the proposed model are indeed new, there is quite a body of existing work that uses ostensibly, very similar, to similar, to quite similar approaches to understand: 1/ the most likely timing of radiogenicPb loss, 2/ mixing between different compositional domains and 3/ common Pb correction.
Specifically, the comparison between a model distribution and a measured UPb distribution has in fact been frequently previously utilized and a recognition of this foundation to the present study clearly required to provide context to this work and demonstrate the advance it makes.
The following works are only those I am aware of, but they may provide some useful context from which the current model appears developed. It is odd they are not considered and implies some limitation in the survey of existing literature relevant to this work.
Pb loss modelling
1/ Morris et al., 2015, Lithosphere, 138143; Kirkland et al., 2017, GR, v. 52, 3947; Kirkland et al., 2020, GR, v. 77, 223237. There are probably other publications from this research group that use distribution comparison techniques to understand Pb loss as well.
Of note here is that the similarity test for the model distribution to the measured distribution is essentially the same as this work proposes. Surely, this should be acknowledged. The major difference in these works and the current approach is that they used the observed concordant distribution in the model whereas the approach proposed in this work is to compare the age distribution to theoretical distributions.
Unmixing
2/ Olierook et al., 2021, GR, v. 92, 102112.
A similar approach in some regards to address the potential of mixing between different zircon domains. It also uses a comparison between a reconstructed (e.g. model) distribution and a known distribution.
Common Pb correction
3/ Andersen 2002, CG, v. 192, 5979.
The common Pb correction approach of Andersen uses some of the same concepts.
The proposed procedure would be able to provide more geological insight if the various distributions (gamma, Weibull, lognormal, uniform, half normal, pareto etc) compared to the data were firmly rooted in some dominant geological process. Specifically, the discussion of the distribution shapes relative to geological processes needs to be significantly enhanced. For example, even simple end member distributions can be linked to likely geological processes; radiogenicPb loss / uranium gain / Pb gain / U loss, discrete or episodic, common Pb gain, heterogeneous common Pb, recent Pb loss, ancient Pb loss. In short, more geological context is required for the patterns that are compared to the measured data.
Specific points
Abstract: the authors claim that Pb loss in natural samples has not been well characterized. I would dispute this, the simplest measure of this process (discordance) is the primary filter essentially every UPb geochronology work uses, there are numerous works considering the process of radiogenicPb loss from the pioneering work of Silver, Pigeon, Krough, Black etc, the field of UPb geochronology has been focused around addressing open system processes (just consider the formulation of the concordia and TeraWasserburg diagrams even). So is it really “not well characterized”? However, is radiogenicPb loss difficult to characterise, absolutely it can be, depending on the measurement precision (which itself can be a function of age). This latter aspect is worth focusing on, to indicate where the proposed modelling approach may have benefits.
Line 26>. Very limited referencing to UPb geochronology concepts that appear to favour a specific author. Suggest providing a more balance and historically accurate list of references that recognises the contributions to the field.
Line 34. Inaccurate statement, depending on when radiogenic Pb loss has occurred (and the measurement precision) and the degree of radiogenic Pb loss (e.g. if complete) data may not be off the concordia curve.
Section 5.3 has specifically been addressed in other works (using a similar more tailored approach) it seems highly unusual that this context isn’t provided here.
Also, the proposed approach for DZ seems incomplete as it is unclear what the purpose of this modelling is for; is it to better understand the primary crystallization ages, the timing of Pb loss, or the degree of mixing between different age components in any distribution? Furthermore, the proposition is somewhat cryptic and certainly difficult to apply to a detrital situation. I really don’t see the contribution this paragraph of text makes to the overall presentation.
A major assumption of this work is that radiogenicPb loss is an impediment to understanding. Yet the reality is that tracking open system processes is possible with radiogenicPb loss and depending on the geological question posed, a very useful way of gaining otherwise difficult to access geological information. Moreover, the whole point of insitu dating is to characterize the full range of (texturally / geochemically defined) age components thus providing an understanding of the full range of geological processes a sample may have undergone. CA work clearly has its place but it is inevitable that such approach is removing some element of geological information in favour of another. The text is strongly one sided in its appraisal of CA and its merits or otherwise.
The discussion of strategies for future data collection needs to be very specific about what the aim of any data collection is; is it to date igneous crystallization, metamorphism, fluid mediated recrystallization, overprinting thermal events? What? Such fundamental information is necessary first before the strategy can be evaluated for the proposed purpose because such underlying geological question would affect everything from required temporal resolution to the most likely manifestation of radiogenicPb loss. Simply arguing for greater number of analyses to better characterise apparent age distributions seems a rather weak suggestion. The more dominant age components (be they detrital or caused by radiogenicPb loss) will be more likely to be sampled (assuming random sampling) for any n selected. This aspect appears to be overlooked but the statistics in some of the DZ work of Anderson and others demonstrate this point.
It is incorrect to appeal to increasing precision alone to identify radiogenicPb loss. The natural extension of this argument ends, rather, with being able to identify the timeframes of which zircon itself grows; there are plenty of zircon growth models about based on modified equilibrium pseudosections that demonstrate zircon has variably prolonged growth intervals in certain environments. Again, the geological environment that the strategy is proposed for needs to be much better presented (e.g. rapid volcanic crystallization).
Furthermore, it would seem useful to consider the model in the context of thermochronology considerations where timing through closure temperature is of relevance (e.g. growth within a magma chamber versus explosive removal from that chamber).
The reality is that strategies should be developed that integrate geochemical parameters of the zircon to better understand the growth or modification process the UPb systematics have been potentially affected by. Considering the age distribution alone seems a simplistic and potentially highly misleading approach given the numerous cofounding variables that could give rise to the same distribution.
 Does the paper address relevant scientific questions within the scope of GChron? Yes
 Does the paper present novel concepts, ideas, tools, or data? In part Yes, but it is strongly overstated in the text to the point where it implies a lack of understanding of the current state of the field.
 Are substantial conclusions reached? The advance is incremental and can not in its current form be described as substantial. Given the current form of the model I can not see it being applied to understand the potential of Pb loss by many in the community as there are already better approaches (faster, more accurate, with more apparent geological meaning)
 Are the scientific methods and assumptions valid and clearly outlined? There are numerous caveats to the application of the model. More discussion of the geological relationship of the distributions to physical processes is needed.
 Are the results sufficient to support the interpretations and conclusions? Yes, but the application of the “tool” should be better defined. Where can this be applied? And why would one what to use this? Both these questions need to be addressed in a simple manner.
 Is the description of experiments and calculations sufficiently complete and precise to allow their reproduction by fellow scientists (traceability of results)? Generally, yes, there are some elements that are cryptic especially the DZ section that is undocumented and its relevance to this work unclear.
 Do the authors give proper credit to related work and clearly indicate their own new/original contribution? No (see the detailed information above)
 Does the title clearly reflect the contents of the paper? Yes
 Does the abstract provide a concise and complete summary? Yes
 Is the overall presentation well structured and clear? Yes
 Is the language fluent and precise? No (refer to the request for precision in the way sentences are formulated above).
 Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated? The relevance of discussion around DZ is unclear. The use of the model for this sort of sample provides little advance and should probably simply be removed as it does not advance this text.
 Are the number and quality of references appropriate? No
 Is the amount and quality of supplementary material appropriate? Yes
Citation: https://doi.org/10.5194/gchron20236RC2 
AC2: 'Reply on RC2', Glenn Sharman, 07 Jun 2023
The comment was uploaded in the form of a supplement: https://gchron.copernicus.org/preprints/gchron20236/gchron20236AC2supplement.pdf
Glenn R. Sharman and Matthew A. Malkowski
Data sets
Supplemental materials for "Modeling apparent Pb loss in zircon UPb geochronology" Glenn R. Sharman and Matthew A. Malkowski https://doi.org/10.5281/zenodo.7783226
Model code and software
Pb_loss_modeling (GitHub) Glenn R. Sharman https://doi.org/10.5281/zenodo.7783243
Video supplement
Supplemental Video 1 Glenn R. Sharman and Matthew A. Malkowski https://doi.org/10.5281/zenodo.7783226
Glenn R. Sharman and Matthew A. Malkowski
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