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
(3232 KB) - Metadata XML
-
Supplement
(8472 KB) - BibTeX
- EndNote
Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-2525', Olivia Thurston, 16 Jun 2026
-
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
Viewed
| HTML | XML | Total | Supplement | BibTeX | EndNote | |
|---|---|---|---|---|---|---|
| 55 | 10 | 6 | 71 | 4 | 6 | 8 |
- HTML: 55
- PDF: 10
- XML: 6
- Total: 71
- Supplement: 4
- BibTeX: 6
- EndNote: 8
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
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.