the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 27 May 2026
Technical Note: The Enhanced Controlled Random Search (CRS) Algorithm for Thermal History Analysis in HeFTy
Abstract. The thermal history modeling software HeFTy is one of several widely used tools for the numerical analysis of low-temperature thermochronologic data. HeFTy version 2 includes a new optional Controlled Random Search (CRS) strategy for posing the candidate time-temperature (tT) paths that are tested against data. Unlike the Monte Carlo (MC) strategy that is the default algorithm, the CRS procedure attempts to converge toward a solution. In order to overcome known limitations of CRS convergence (e.g., returning an artificially narrow range of tT solutions that exaggerates the capacity of the data to document a specific thermal history), this ‘enhanced’ CRS randomizes, expands, and reconverges on candidate tT-paths. Here, we use both synthetic and real apatite and zircon (U-Th)/He data, previously analysed using the MC algorithm in HeFTy, to explore the utility of this new tool. Overall, we demonstrate that the enhanced CRS can substantially improve computation time and is useful for finding families of tT paths that fit thermochronologic data. However, users should still expect the CRS to produce an uneven distribution of paths (i.e., ‘clustering’) in the permissible tT space. Therefore, we suggest that systematically performing multiple CRS and/or MC inversions is essential for building a robust assessment of how CRS inversion results are produced by the data and inversion design choices, or else it may not be clear what features of a CRS result are produced by the convergence algorithm rather than the data and a sample’s geologic context. Diagnosing such behaviors is essential for using thermal history inversion results for geological interpretations.
Competing interests: KM and ALSG are co-coordinators (but not associate editors) for the special issue to which this paper belongs. Otherwise, the authors declare that they have no conflict of interest.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-2525', Olivia Thurston, 16 Jun 2026
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AC1: 'Reply on RC1', Kendra Murray, 21 Jul 2026
We thank Olivia for her support and feedback. Below we respond briefly in bold to the suggestions she provides in the PDF of the manuscript.
Comments in the PDF:
1. Page 4, line 110 — highlighted: "2.3 Updates" (section heading)
"I would explicitly state that this update has been released in the software and is NOT an update that was generated by the authors for this technical note."
Agreed; in the revised manuscript, we intend to clarify this point.2. Page 4, line 118 — highlighted: "'reboot'"
"I imagine many readers will use this a supplement to the user manual so it may be helpful to state weather or not this 'reboot' is a selected parameter or automatically utilized in the updated software."
Agreed; in the revised manuscript, we intend to clarify this point. This reboot is a feature that is built-in to the CRS procedure as of version 2.3 and is not a behavior selected by the user.3. Page 18, line 429 — highlighted: "is used." (end of the sentence "...regardless of which inversion method is used.")
"In this final recommendation paragraph, you might state which version of HeFTy 2 you would recommend."
Agreed; in the revised manuscript, we intend to clarify this point. It is generally good practice to use the most up-to-date version of HeFTy available when embarking on a thermal history analysis—currently this is v 2.3. However, we also see the opportunity to re-iterate here that users should pay attention to version control (e.g., not arbitrarily bounce back and forth between versions while developing a relationship with data) to help avoid unnecessary complexities during the analysis of inversion results.Citation: https://doi.org/10.5194/egusphere-2026-2525-AC1
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AC1: 'Reply on RC1', Kendra Murray, 21 Jul 2026
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RC2: 'Comment on egusphere-2026-2525', Chelsea Mackaman-Lofland, 05 Jul 2026
General comments:
This contribution by Murray et al. presents a rigorous exploration of the Controlled Random Search (CRS) strategy for posing candidate time-temperature (t-T) histories, newly implemented as an inverse modeling option in HeFTy version 2. The CRS approach differs from HeFTy’s default Monte Carlo (MC) path generation mechanism through implementation of an optimization algorithm that iteratively converges on t-T paths that represent the best-fitting solutions to thermochronology data. In addition, and in contrast to other thermal history modeling programs that use convergence algorithms to generate t-T paths, HeFTy 2’s ‘enhanced’ CRS procedure incorporates a series of randomization, expansion, and reconvergence steps designed to overcome the tendency of the algorithm to produce artificially narrow ranges of t-T solutions that may exaggerate the resolving capability of the data.
Murray et al. summarize recent updates to the enhanced CRS procedure used in HeFTy v2.3, then, via a series of sensitivity tests based on synthetic, then real (deep-time) (U-Th)/He data, examine and characterize the advantages and pitfalls of the enhanced CRS algorithm in comparison to the default MC approach. Their most important findings include 1) significant improvements to computation time using CRS, which is to be expected with the use of a convergence algorithm relative to random (MC) path generation but still extremely useful to quantify; and 2) the persistent tendency of the ‘enhanced’ CRS algorithm to produce uneven distributions or “clusters” of paths, even with the randomization and other steps implemented to help expand t-T solutions to more realistically represent the resolving power of the data. They also highlight several informative cases in which CRS inversions – especially multiple inversions of the same dataset as facilitated by the rapid CRS computation time – reveal t-T solution clusters/families that trade off specific time and temperature conditions to yield equally good fits to the data.
I found this to be a very strong manuscript: the above findings, and the authors’ recommendations, will be of high value to the thermochronology community, and will certainly inform how I use inverse thermal history modeling tools and interpret thermochronology data in my own research. I also want to commend their thorough reporting and presentation of the sensitivity testing process, from posing questions based on thermal history inversion results, to modifying their model design to address those questions, to evaluating subsequent changes in the context of geologic- or algorithm-based processes. I can absolutely see this narrative serving as a valuable blueprint for future modeling studies.
The following comments/suggestions are overall minor, and intended to help expand the discussion and recommendations presented in this contribution. I encourage the authors to contact me if they have any questions, and look forward to seeing this work published. -Chelsea Mackaman-Lofland
Specific comments:
The authors do a great job of comparing results using the CRS search option to identical thermal history inversions using the MC approach, and of cautioning that use of CRS requires a careful revisitation of the questions that HeFTy was designed for. E.g., because the enhanced CRS algorithm tends to generate clustered t-T path families/solutions, the results may not effectively answer “What is the range of thermal histories that are consistent with my data and assumptions?” (Ketcham, 2024).
My biggest suggestion to improve the manuscript is to encourage the authors to also, at least briefly, discuss their CRS inversion results alongside those of other thermal history models that implement optimization algorithms. For example, a short discussion of how the enhanced CRS inversions for the “single 40 Ma crystal” and “synthetic age-eU” models compare with QTQt inversion results for the same scenarios (I’m thinking of those already performed and published in Abbey et al., 2023) could help define how the enhanced CRS algorithm performs in relation to existing (learning and non-learning) path search approaches, and provide a framework for highlighting the types of questions CRS inversions may be uniquely well-suited to address. I can envision such a discussion motivating future studies specifically designed to leverage the capabilities, and principles, backing the enhanced CRS path search approach. This suggestion to include one or two synthetic model comparisons hopefully doesn’t extend too far beyond the scope envisioned for this Technical Note.
Lines 65–66 (“We demonstrate that this solution clustering arises from the fundamental 65 trade-offs between time and temperature that are inherent in thermochronology”): Consider providing a little more information/a very brief summary here, for readers who may have less intuition for such long/cold hot/fast thermal history tradeoffs in reproducing thermochronology datasets?
Lines 67–68 (“leverage clustered solutions to find families of tT paths…”): This seems like a unique capability of the enhanced CRS path search approach over other learning and non-learning path generation mechanisms. It would be really interesting, and could help inform the design of studies that specifically leverage the capabilities of enhanced CRS, to see how results compare not just with fully random (MC) results but also those of other optimization algorithms that prefer the simplest solution.
Line 130: Typo in “Single”
Fig. 1: Consider adding a panel that diagrams out how “solution space” is evaluated based on the paths, constraint points (end-points), and half-points as illustrated by visualizations of inversion results as t-T paths and constraint points?
Lines 151–152 (and related): I think it would be worth explicitly defining how the authors are evaluating “coverage of the solution space” based on the lines (paths) and constraint points visualized in the HeFTy results figures – especially for complex cases like the synthetic age-eU model results, where the path envelopes for CRS-3 encompass a smaller window of t-T space relative to CRS-1 and MC, but the time range defined by the CRS-3 constraint points in box 2 is greater than for these other models.
Lines 156–165, and Fig. 2: These results demonstrating/cautioning the extent to which different CRS settings can affect model solutions are extremely valuable. Given some of the results presented for the other model use cases, I’m wondering if repeated CRS inversions using the most up-to-date default parameters (v2.3) are able to consistently reproduce the broad solution space compatible with the MC inversion?
Lines 201 and 204: Maybe specify “the CRS-1 constraint points”, and “CRS-2 and CRS-3 returned fewer good-fit paths … but constraint points spanned a larger solution space” ? Suggesting because the paths and associated envelopes encompassing t-T solution conditions for CRS-2 actually take up less of the box 2 t-T space...
Lines 247–248: Since the peak temperature of the ‘true’ path doesn’t have associated error, may be better to revise language to something like “but also encompassed the peak T …” ?
Lines 373–375: This discovery – of CRS model results revealing a broader range of permissible thermal histories than previously characterized by MC models of a complex, deep-time thermochronology dataset – is really interesting, and certainly makes a compelling case for running several CRS inversions alongside a MC inversion of the same dataset. It might also be worth highlighting that this finding emerged based on the random chance that the CRS convergence, randomization, expansion, and reconvergence procedure highlighted that family of t-T paths? If I’m understanding the CRS procedure correctly, the discovery of this path family is not necessarily any more likely than a MC model generating a path in that family, but once a CRS paths is generated the optimization algorithm assists in populating a greater range of thermal histories that fit characteristics of this family (right?)?
Line 399 (“sometimes less complete, though sometimes more”): Expand on this just a little to briefly summarize most important case studies/characteristics of the pros and pitfalls illustrated by these CRS models?
Citation: https://doi.org/10.5194/egusphere-2026-2525-RC2 -
AC2: 'Reply on RC2', Kendra Murray, 21 Jul 2026
We thank Chelsea for her support and feedback. Below we respond briefly in bold text to her specific comments.
Specific comments:
The authors do a great job of comparing results using the CRS search option to identical thermal history inversions using the MC approach, and of cautioning that use of CRS requires a careful revisitation of the questions that HeFTy was designed for. E.g., because the enhanced CRS algorithm tends to generate clustered t-T path families/solutions, the results may not effectively answer “What is the range of thermal histories that are consistent with my data and assumptions?” (Ketcham, 2024).
My biggest suggestion to improve the manuscript is to encourage the authors to also, at least briefly, discuss their CRS inversion results alongside those of other thermal history models that implement optimization algorithms. For example, a short discussion of how the enhanced CRS inversions for the “single 40 Ma crystal” and “synthetic age-eU” models compare with QTQt inversion results for the same scenarios (I’m thinking of those already performed and published in Abbey et al., 2023) could help define how the enhanced CRS algorithm performs in relation to existing (learning and non-learning) path search approaches, and provide a framework for highlighting the types of questions CRS inversions may be uniquely well-suited to address. I can envision such a discussion motivating future studies specifically designed to leverage the capabilities, and principles, backing the enhanced CRS path search approach. This suggestion to include one or two synthetic model comparisons hopefully doesn’t extend too far beyond the scope envisioned for this Technical Note.
We see the value of adding one simple comparison to QTQt or another commonly used thermal history modeling software that uses a converging algorithm, to further place the CRS in context. However, as the reviewer points out, there are some useful results already published in Abbey et al., 2023 (and in Murray et al., 2022, their Fig. 3). More importantly, the way that other software with learning algorithms—such as QTQt—handle convergence and thermal history path complexity is challenging to comprehensively represent in one or two simple examples. Indeed, there is another Technical Note submitted to this Special Volume on this topic for QTQt (https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2328/). So, we feel that adding such an example is outside the scope of this Technical Note.
We propose that instead of adding more inversion results from QTQt, we revise the manuscript to refer readers more explicitly to existing resources in the literature.
However, if the Associate Editor agrees with the reviewer, we can pursue running a simple QTQt comparison for the single 40 Ma age and consult with the authors of the in-review QTQt technical note to get their advice about how to best represent the capacities of the current version of QTQt.
Lines 65–66 (“We demonstrate that this solution clustering arises from the fundamental 65 trade-offs between time and temperature that are inherent in thermochronology”): Consider providing a little more information/a very brief summary here, for readers who may have less intuition for such long/cold hot/fast thermal history tradeoffs in reproducing thermochronology datasets?Additional extensive descriptions and demonstrations of these trade-offs are later in the paper (starting a line 225), so we think this mention in the introduction overview is sufficient.
Lines 67–68 (“leverage clustered solutions to find families of tT paths…”): This seems like a unique capability of the enhanced CRS path search approach over other learning and non-learning path generation mechanisms. It would be really interesting, and could help inform the design of studies that specifically leverage the capabilities of enhanced CRS, to see how results compare not just with fully random (MC) results but also those of other optimization algorithms that prefer the simplest solution.Please see response to “biggest suggestion” above.
Line 130: Typo in “Single”Agreed.
Fig. 1: Consider adding a panel that diagrams out how “solution space” is evaluated based on the paths, constraint points (end-points), and half-points as illustrated by visualizations of inversion results as t-T paths and constraint points?This is part of what readers can glean from Figure 2 and later figures that visualize inversion results in the tT space described in Figure 1. We propose that we add some additional descriptive text in the Figure 2 caption to identify the “solution space” (totally fills box 1, only partially fills box 2). It is possible to include the half-points as dots in inversion result visualizations, but we find it makes the plots extremely busy looking. However, we would consider adding a 4th panel to Figure 2 that demonstrates what this looks like, to more directly relate back to the elements of the inversion design in Figure 1.
Lines 151–152 (and related): I think it would be worth explicitly defining how the authors are evaluating “coverage of the solution space” based on the lines (paths) and constraint points visualized in the HeFTy results figures – especially for complex cases like the synthetic age-eU model results, where the path envelopes for CRS-3 encompass a smaller window of t-T space relative to CRS-1 and MC, but the time range defined by the CRS-3 constraint points in box 2 is greater than for these other models.Agreed; we can add several sentences that describe this in that paragraph.
Lines 156–165, and Fig. 2: These results demonstrating/cautioning the extent to which different CRS settings can affect model solutions are extremely valuable. Given some of the results presented for the other model use cases, I’m wondering if repeated CRS inversions using the most up-to-date default parameters (v2.3) are able to consistently reproduce the broad solution space compatible with the MC inversion?For this simple “one 40 Ma age” example paired with the inversion design in Fig 1, repeated CRS results are indeed consistent. This is why we are only showing one CRS rep here. We will add a sentence here that describes this explicitly, given that this is not the case for our later examples and this is a useful thing to point out.
It would be possible to also add more repeated CRS results to figure 2, or as a supplementary figure. However, we think it would be more effective to encourage interested readers to download the related hefty file that we provide and run repeated CRS inversions themselves.
Lines 201 and 204: Maybe specify “the CRS-1 constraint points”, and “CRS-2 and CRS-3 returned fewer good-fit paths … but constraint points spanned a larger solution space” ? Suggesting because the paths and associated envelopes encompassing t-T solution conditions for CRS-2 actually take up less of the box 2 t-T space...Agreed; we will adjust the language in the paragraph to better parse the different ways of describing the solution space (paths vs points).
Lines 247–248: Since the peak temperature of the ‘true’ path doesn’t have associated error, may be better to revise language to something like “but also encompassed the peak T …” ?The ‘true’ path does not have error, but the inversion results do, as listed in the following discussion.
Lines 373–375: This discovery – of CRS model results revealing a broader range of permissible thermal histories than previously characterized by MC models of a complex, deep-time thermochronology dataset – is really interesting, and certainly makes a compelling case for running several CRS inversions alongside a MC inversion of the same dataset. It might also be worth highlighting that this finding emerged based on the random chance that the CRS convergence, randomization, expansion, and reconvergence procedure highlighted that family of t-T paths? If I’m understanding the CRS procedure correctly, the discovery of this path family is not necessarily any more likely than a MC model generating a path in that family, but once a CRS paths is generated the optimization algorithm assists in populating a greater range of thermal histories that fit characteristics of this family (right?)?
Yes; the other factor here is simply the difference in the number of paths returned by the MC (<200) vs. the CRS (~30,000). More paths, more opportunities to map out the solution space. We agree that this point can be emphasized more here.
This comment has also prompted us to revisit the comparison we are making here (between the published MC and the new CRS results). The published MC result was generated using HeFTy v1.9.3. The transition to HeFTy version 2 was accompanied by an adjustment to how the composite GOFs (and thus the classification of attempted paths as good, acceptable, or not accepted) are calculated, to make it slightly more forgiving. Therefore, this new statistical schema, and not the CRS, may be contributing to finding the new family of acceptable-fit solutions (e.g., permissible hotter peak T in late K time). In order to make a more clear comparison, we intend to re-run the MC inversion in HeFTy 2.3 and use that result here instead of the published one. We do not anticipate this will change the discussion here substantially.
Line 399 (“sometimes less complete, though sometimes more”): Expand on this just a little to briefly summarize most important case studies/characteristics of the pros and pitfalls illustrated by these CRS models?
Agreed.
Citation: https://doi.org/10.5194/egusphere-2026-2525-AC2
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AC2: 'Reply on RC2', Kendra Murray, 21 Jul 2026
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This technical note explores the limitations, biases, and advantages of the thermal history modeling software, HeFTy 2, with a special focus on the difference between the two underpinning algorithmic approaches, namely Monte Carlo (MC) and Controlled Random Search (CRS). The authors break down the logic and computational steps of the CRS approach in a way that is both succinct and easily digestible, which is critical to any researcher who is trying to decide which algorithm to use on their own data. The real utility of this note is that it provides guardrails for researchers to use the new CRS algorithm in HeFTY 2, which has substantially faster computing times than the MC algorithm, while remaining aware of how CRS might bias the results to show a limited range of possible time-Temperature paths. By providing clear evidence of the difference in computational labor (total runs), number of Good-Fit paths found, and clustering of results using MC and CRS approaches the authors create the framework needed to develop the most efficient modeling approach to get the largest number of possible solutions while reducing overall modeling time. I do appreciate that the authors make a point to state that though they have provided a framework for identifying biases within results from MC and CRS derived models, there is no substitute for hands on modeling when it comes to mastering the identification of biases within your own results and understanding the workings of HeFTy 2.
In addition to providing HeFTy specific modeling information, this technical note has created a platform around which discussion can be generated on “best practice” for thermal history modeling methods. Though one could argue that the results presented in this technical note would invariably have been worked out in piece-meal by any competent researcher over time, there is a tendency within the thermochronology community to interrogate the biases within a given thermal history modeling software only within the confines of their own lab group and become rigid in their modeling approach. In documenting these results and presenting them in a clear manner, the authors have saved many researchers from having to develop these HeFTy bias tests themselves and have hopefully opened up a public forum in which to debate methodology across lab groups and philosophies.
Though I do not see a need for any major revisions, I have provided a few comments on the attached PDF that would allow this note to be more easily utilized by novice modelers in developing their own HeFTy thermal histories.