the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Oct 2024
The quantification of downhole fractionation for laser ablation mass spectrometry
Abstract. Downhole fractionation (DHF), a known phenomenon in static spot laser ablation, remains one of the most significant sources of uncertainty for laser-based geochronology. A given DHF pattern is unique to a set of conditions, including material, inter-element analyte pair, laser conditions, and spot volume/diameter. Current modelling methods (simple or complex linear models, spline-based modelling) for DHF do not readily lend themselves to uncertainty propagation, nor do they allow for quantitative inter-session comparison, let alone inter-laboratory or inter-material comparison.
In this study, we investigate the application of orthogonal polynomial decomposition for quantitative modelling of LA-ICP-MS DHF patterns with application to an exemplar U–Pb dataset across a range of materials and analytical sessions. We outline the algorithm used to compute the models and provide a brief interpretation of the resulting data. We demonstrate that it is possible to quantitatively compare the DHF patterns of multiple materials across multiple sessions accurately, and use uniform manifold approximation and projection (UMAP) to help visualise this large dataset.
We demonstrate that the algorithm presented advances our capability to accurately model LA-ICP-MS DHF and may enable reliable decoupling of the DHF correction for non-matrix matched materials, improved uncertainty propagation, and inter-laboratory comparison. The generalised nature of the algorithm means it is applicable not only to geochronology but also more broadly within the geosciences where predictable linear relationships exist.
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RC1: 'Comment on egusphere-2024-2908', Anonymous Referee #1, 15 Nov 2024
While I am not in a position to critically evaluate the mathematical models used to quantify Differential Heating Factor (DHF) in this extensive dataset, the results presented here are compelling. Having previously analyzed a similar dataset and examined DHF patterns in U-Pb dating across various minerals, I am often struck by the surprising variability between them. The physical mechanisms underlying DHF remain somewhat ambiguous, which makes this study’s methodology and findings—particularly in the context of “Big Data”—valuable to the community engaged in LA-ICP-MS U-Pb dating methods.
Drawing from my experience, I believe certain aspects warrant additional discussion. Ideally, expanding this impressive dataset with data from alternative instrumentation (both laser ablation and ICP-MS systems) would mitigate some current limitations of the study, though it would likely raise additional questions as well.
Several potential limitations and influential factors in DHF quantification, as presented in this work, could benefit from a more detailed discussion. These include aspects such as signal duration, differences in instrumentation and operator handling, sample heterogeneity, inclusions, focal position accuracy, and variation in detector cross-calibration and dead time.
Lines 82-85: “The independence and physical meaning of the lambda coefficients allows them to be used to quantitatively compare independent fits (e.g., single analyses, materials, analytical sessions, differing laboratories) so long as other parameters (e.g., fluence, spot diameter/volume, laser wavelength) are considered.”
How are these other parameters accounted for within the analysis framework? What impact would neglecting them have on the method’s accuracy and reliability?Is the centering of time-dependent ratios influenced by signal duration, or is it only lambda 0 that is sensitive to such changes, potentially due to ICP tuning dependencies? Furthermore, how does total analysis time (with data presented at 30 and 40 seconds) impact the DHF pattern once data are centered for further calculation of lambda components (lambda 1, 2, 3, and 4) and subsequent UMAP visualization? If lambda components exhibit different characteristics during different parts of the signal—such as a linear trend dominating the initial 10 seconds and a higher-order trend thereafter—then the lambda coefficients would differ for signal durations of 20, 30, or 40 seconds. Consequently, the same analysis might plot differently in UMAP depending on signal duration. This warrants further discussion.
The statement, “and for Wilberforce, the steeper linear DHF component and larger uncertainty are due to inclusion of several points from some analyses that are highly leveraging the fit, even with automated outlier removal being applied,” suggests that inclusions and heterogeneity (particularly variable initial Pb content, as in Apatite) within reference materials can influence the DHF pattern beyond what outlier removal can address. How is this handled in your analysis? A more in-depth discussion would be beneficial.
This study is based on data from a single laboratory using a single laser ablation system and two similar ICP-MS instruments likely operated or trained by a single Lab Manager. The robustness of DHF quantification could be enhanced by incorporating data from different laser ablation systems, which vary in wavelength, pulse width, energy density, and ablation cell design, as well as different ICP-MS instruments. Please discuss how this limitation might affect the generalizability of the quantification.
Additionally, ICP-MS instruments employ various detection modes that require cross-calibration. How is it ensured that ablation signals—where intensity generally decreases during single-hole ablation—are unaffected by potential cross-calibration errors that could impact ratio measurements?
Minor comments:
Line 33: “As DHF is a volume- dependant spot-ablation phenomena…” I would rather state “ As DHF is a crater geometry dependant ….”
Line 216 there is an E missing in “(𝜆3) range from -1.08E-5 to +2.15-5,”
Citation: https://doi.org/10.5194/egusphere-2024-2908-RC1 -
RC2: 'Comment on egusphere-2024-2908', Noah M McLean, 18 Feb 2025
This manuscript, which details a method to quantify the shape of laser ablation downhole fractionation profiles and compares them across minerals and instrument parameters from a single lab, is a solid contribution. Its quantitative, technical scope, and its methodological slant are well suited to Geochronology. In my opinion, the article needs somewhere between minor and major revisions, mostly to do with the clarity of the writing and quantitative explanations and figures. The orthogonal polynomial regression approach is sound, and the dataset is interesting, although applications for the technique have not been thoroughly explored.
The orthogonal polynomial regression approach used here is appropriate, and its use to quantify the shape of the DHF is an interesting contribution. I’ve never heard of UMAP, but it seems to be used appropriately. A quick survey of the GitHub repository shows that the code is relatively easy to follow, and major functions have informative docstrings. However, there are some issues with the regression framing (see comments around equations 7-9).
What’s missing from this publication is some explanation and clarity – on occasions throughout the text, I’m not sure what is being explained or plotted. I’ve enumerated those below. Another missing point, brought up by the previous review, is that the specific results discussed in this paper (most of the figures and discussion) are applicable to the authors’ analytical setup and choices. For instance, the shape parameters for DHF will depend not just on the mineral and the spot size but also on the choice of fluence. How were the fluences in Appendix A chosen? From lab to lab, these will vary based on sample cell setup, gas flows. It’s clear from reading the manuscript that the authors know this, but it’s not clear upfront to the reader.
It is also unclear how the authors propose to use the results of their orthogonal polynomial regression to correct for down-hole fractionation in the Applications section. On line 293, the authors suggest that “We envisage that this algorithm could be implemented in data reduction software to self-correct the DHF pattern of well-behaved materials…” I don’t know what this means, but it sounds like a very different approach than the two common DHF correction schemes, the ‘intercept’ method (used, e.g., by AZ LaserChron), or the ‘downhole’ method from Paton et al. implemented in Iolite. Can the authors describe such a self-correction and how it might be different from what’s used currently? This point is obtuse, and if the authors wish to write a forthcoming paper on a new algorithm, I’d suggest leaving it out here rather than the current vague mention.
Finally, I don’t see any documentation of how this algorithm was tested. Did you create synthetic data with a known data covariance matrix and known regression parameters, fit the data, and recover the input parameters? How does the scatter between regression parameters fit to many synthetic datasets compare with the uncertainties in the fit parameters estimated by the model?
Line-by-line comments follow.
Figure 2b: The trace of lambda_4 here is extraneous or in error. Based on the description in this portion of the manuscript, I would expect to see a fourth-order polynomial. If the coefficient is so small that the trace won’t plot on the y-axis of this figure, then it is not a good illustration. If it has been set to zero by an AICc cutoff explained much later in the text, then it’s quite confusing here. Is there some mathematical significance to the dot between the lambda_j and the variable x in the legend?
Figure 2a and caption: The ‘gmean-centered ratio’ y-axis label needs a detailed explanation for those who don’t regularly center their data with a geometric mean and are likely confused by this axis label. The first sentence of the caption doesn’t help much – please sacrifice some brevity for the sake of clarity. The x-axis label and color scale for 2a are confusing – do you need all those dates on the color scale? It’s the same length as the x-axis, which is confusing – I thought for a long time that the color scale corresponded to the ‘time since laser start’ and couldn’t figure out what was being plotted. Maybe make the color scale vertical and place it to the right of the axes with fewer dates?
Line 81: there is some confusion here about whether lambda_0 represents an arithmetic or geometric mean of the data. I like that this manuscript uses geometric means extensively. Why do the regression on ratios instead of log-ratios, though? The authors seem aware (e.g. in the appendix) about the challenges of dealing with compositional data. Those challenges extend not just to taking means, but to regression problems (same idea, more parameters). See countless Aitchison publications for details.
Line 115: I don't understand this paragraph as written. Specifically, what numbers are discrepant? Are lambda_1 and higher different for centered and non-centered fits of the same (e.g., 30-second) laser ablation analyses? This seems like a rounding or numerical precision problem. If the labda_1 and higher coefficients are different among separate laser ablation analyses (e.g., the first reference material analysis and the second), then this implies that the shape of the DHF is changing during the session, or that the ref mat is heterogeneous in the parameters that impart the DHF behavior.
Figure 4: Maybe thin this data out by randomly selecting 10% or 25% of the data to plot? That would prevent data overlap and occlusion problems and avoid aliasing effects.
Line 132: The example here is counts per second, but the figure shows a ratio on the y-axis. Is there a time when you'd fit an intensity rather than a ratio, and when would that fit be useful?
Line 136: What is N?
Equations 7-9: There is some lack of clarity here, along with some errors or typos.
In equation 7, if you want to leave that Sigma term in, you’ll want to take the hat off of the y. Usually, hat means the predicted values, which are given by multiplying the design matrix by the best-fit parameters (here, Lambda), usually also given a hat, which means there’s no error term in the equation. The error term here (Sigma in equation 7) would need to be a vector, not a matrix, and it is not given by equation 9, which describes the uncertainties in the best-fit parameters in Lambda, not the measurements in y.
In the generalized least squares framing, that Omega matrix in equation 8 is the covariance matrix for the measurements in y (or the residuals in a differently formulated equation 7). This matrix goes unmentioned in the main text, until line 425 in the Appendix. Where do the analytical uncertainties in equation 8 come from? I’ve looked at the code and the appendix and I’m still unsure. What terms go into the uncertainties? For ratios, are the numerator and denominator intensity measurements accounted for? Are detector effects like dead time included? What sort of assumptions are made here and how might they affect the results?
As far as I can tell from the appendix, these uncertainties are estimated from the variability of the measured ratios about the… geometric mean? But this wouldn’t make much sense for a time series where you expect a trend (and you want to measure its shape) – some of the variability would come from the analytical uncertainty and some from the trend itself. I can’t piece this together.
Looking at the code and the appendix, Omega is a diagonal matrix. My understanding is that this technically makes your algorithm a weighted least squares, rather than a generalized least squares, regression problem. Also, I think most folks would put a hat on Lambda here.
In equation 9, the Sigma contains the variances and covariances for the fit parameters (here, the lambdas). This step only works for generating uncertainties, confidence bands, etc, if the Omega is an inverse covariance matrix for the data in y. Note that in your GitHub code, lines 286-300 look like they're weighting by integer ones and maybe a few other approaches? I can't follow the code exactly and it's not commented. If you use ones as the diagonal of Omega, then equation 9 only gives you a Sigma that contains variances and covariances if the uncertainties are all independent (e.g., no covariance terms from detector effects like dead time) and identical to one in whatever the units of y are (unclear here and elsewhere).
Are the lambda uncertainties plotted throughout this paper calculated using input uncertainties to fit_orthogonal() or are some or all calculated with the default unit weights?
Figure 5, line 212: It is unclear to me what is going on here. I think I get (c), which describes the results of 5478 different orthogonal polynomial regression analyses (of what exactly?) versus time? Maybe 206/238? Or maybe an unspecified intensity in cps, per line 132?
What then are the 188(?) points in b? Maybe the (cps? ratio?) data from a single reference material (e.g. GJ1) have been lined up to a common time datum (the laser turning on?), and all the (centered?) data have been fit by the same orthogonal polynomials? If multiple datasets are combined, do they agree? One of those chi-squared values you mention in the model selection portion of the paper should tell you, but those are not reported here. The way that you’ve set up the regression in equations 7-9, the uncertainties in Sigma will get smaller with more data, no matter how scattered the data are about any one trend. Try it and see! This is the same effect as the weighted mean (a special case of weighted least squares) getting more precise when adding more analyses, even when those analyses don’t agree and the reduced chi squared grows large. Just like a dataset can have a weighted mean with a large reduced chi squared and a tiny uncertainty, your regression could have a large scatter about the trend you’ve described with orthogonal polynomials but tiny uncertainties.
Continuing on the comments from Figure 5b, what does this analysis tell us? Spell out where a user would one use these lambdas, rather than the lambdas from (c)? What about (a)??
Line 215: Spell out scientific notation here and in the figure axis labels, but I'd say just leave them out of the text altogether unless you mean to explain the physical significance (starting with the units) for each number.
Line 227: Describe very clearly here what you mean by “without needing reference material calibration,” or rephrase this. Maybe something along the lines of "the lambdas quantify the shape of the DHF for each of the unknown analytes and reference materials in a way that is independent of a shape derived from an average of reference material analysis (ref Paton).
Citation: https://doi.org/10.5194/egusphere-2024-2908-RC2
Data sets
Raw and derived data: The quantification of downhole fractionation for laser ablation mass spectrometry Jarred Lloyd and Sarah Gilbert https://doi.org/10.25909/26778298
Supplementary analyte signal figures - The quantification of downhole fractionation for laser ablation mass spectrometry Jarred Lloyd https://doi.org/10.25909/26778592
Supplementary Figures - The quantification of downhole fractionation for laser ablation mass spectrometry Jarred Lloyd https://doi.org/10.25909/27041821
Model code and software
Julia scripts - The quantification of downhole fractionation for laser ablation mass spectrometry Jarred Lloyd https://doi.org/10.25909/26779255
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