A heuristic approach for modeling the surface histories of cratons using apatite fission-track ``super samples''

Understanding the long-term erosion and burial history of cratons is often challenging due to the incompleteness of the geologic record. Low-temperature thermochronology has been used to provide constraints on these histories and apatite fission-track dating has long been one of the preferred methods. In terms of analytical protocol the convention has been to measure ∼100 confined track lengths and to produce ∼20 single-grain ages. These data are then inverted for thermal history along with sparse constraints and other assumptions pertaining to the regional geologic evolution. However, imposing constraints will influence the form of the inferred thermal histories. In some cases this step may limit impartial assessment of the unknown history in terms of what features are required by the data and those that the data are consistent with (or at least do not contradict). Here we present a study present involving apatite fission-track data collected from central Canadian Shield basement rocks with more dated age grains and ∼3–7× the number of track-length measurements when compared to a conventional analysis. We refer to these data as “super samples” (AFTSS) and show such data can improve resolution of complex histories involving episodic reheating and partial annealing. Importantly for the data we present, AFTSS can also be used to independently infer past geologic conditions without the enforcement of many a priori constraints during modeling—such as the approximate times of past regional basement exposure. Modeling in this way is guided by a heuristic philosophy regarding the use of thermal history constraints. This allows us to examine the ability of the model to independently infer geologically plausible time-temperature paths from the fission-track data alone. Inversions of these data ∗Corresponding author: kalin.t.mcdannell@dartmouth.edu Preprint submitted to Earth & Planetary Science Letters January 18, 2022 establish that the currently exposed basement near the Hudson Bay Basin was buried in the middle Paleozoic and late Mesozoic, in agreement with the preserved regional rock record and adding further evidence to suggest that the basin is an erosional remnant. The AFTSS data alone imply two reheating events and indirectly require periods at cooler (near-surface) conditions in the latest Neoproterozoic to early Paleozoic and in the Jurassic to early Cretaceous—the timing of which are consistent with known Hudson Platform unconformities. We recommend that cratonic basement rocks that may have experienced episodic burial reheating (∼60 to < 100◦C) and partial annealing over hundreds of millions of years should have a minimum of 250–300 track lengths collected to provide adequate time-temperature information for thermal history modeling.

understanding thermal history. of geologic constraints (or geologic assumptions/interpretations) for the given study area. However, when 52 trying to reconstruct the time-temperature (t-T ) histories from these data, the lack of physical geologic 53 constraints to inform modeling is problematic (McDannell and Flowers, 2020;. 54 This is commonly addressed by utilizing whatever geologic information we do have-however, unless samples 55 are taken directly from well-constrained locations (e.g., near basement unconformities), there is typically 56 some degree of regional extrapolation of, or uncertainty in, assumptions about past geologic conditions. In 57 some situations these regional inferences may be warranted, whereas in others they may not, and it is difficult 58 to know which is the case before carrying out modeling. The issue then relates to our ability to resolve more 59 complex thermal histories in the absence of firm geological constraints. 60 One option to constrain low-temperature histories (< 150 • C) is to better exploit the information contained 61 in the horizontal confined track-length distributions provided by the AFT method, as track lengths are 62 sensitive indicators of thermal history style (Crowley, 1985;Gleadow et al., 1986;Duddy et al., 1988;Green 63 et al., 1989). The convention has been for analysts to count the number of spontaneous tracks (N s ) from up to 64 20 grains for age determination and measure a minimum of ∼50-100 track lengths to obtain a representative may then ask how many tracks are necessary for adequate t-T resolution? While these questions are specific 126 to the examples discussed here, the requirement of a representative number of track lengths is axiomatic to 127 any AFT thermal history reconstruction and may generally apply to other regions that experienced similar protracted histories involving episodic reheating and partial track annealing from sedimentary burial. We first use synthetic AFT data to explore a simple case and so avoid the problems inherent to natural samples with 130 unknown histories. We use a thermal history in which the maximum reheating temperature is relatively low 131 but still high enough to cause some annealing of the tracks present at that time. The aim is to demonstrate 132 the sensitivity of the length distribution (and the number of tracks defining it) to subtle reheating events and 133 the our ability to recover the thermal history with different amounts of track-length data. The reheating history shown in Figure 1A (dashed gray path) was forward modelled to produce synthetic 135 AFT data (Fig. 2). The AFT age data were held constant, whereas the number of track lengths sampled 136 from the predicted length distribution was progressively increased from 100 to 400. Increasing the number of 137 tracks represented in the histograms improves characterization of subtle features of the predicted distribution, 138 namely the tails and the 'shoulder' at 14-15 µm (Fig. 2). We are demonstrating that with a greater number 139 of tracks, we more accurately represent the 'true' length distribution (Fig. 2). These synthetic samples were 140 then inverted in an attempt to recover the true history used to generate the track data (Fig. 3). While these 141 results are conditional on rejecting more complex models-100 measured track lengths are not enough data to 142 fully resolve the t-T path. These models illustrate that > 200 track lengths are required to properly capture 143 the details of the thermal history in a case involving minor thermal annealing. Inverse models with a greater 144 number of tracks improved the resolution of the early cooling episode, which is to be expected since more 145 tracks experienced this cooling event-however, as more tracks are utilized, the reheating event is better 146 resolved by a larger proportion of the accepted model paths (Fig. 3). A general point learned from this 147 demonstration is that if the thermal history is complex, then we will likely require more track lengths to 148 properly define the (similarly complex) distribution for modeling. We examined these concepts further using 149 real AFT data from the Canadian Shield.  Fig. 1D).
Results are shown as heat maps of t-T path density, where brighter colors are higher relative posterior probability. The difference between each model going from A to D is the increased number of tracks that define the true predicted distribution. (A) results for 100 tracks; (B) results for 200 tracks; (C) results for 300 tracks; (D) results for 400 tracks. The gray path is the true history and the pink lines are the QTQt Expected model with 95% credible interval. The latter model is the average of the marginal distribution and is shown as a summary of all accepted post burn-in solutions and does not represent an individual history sampled during the inversion. Figure S1 in the SI shows forward models of the same t-T path for a typical endmember fluorapatite where the thermal peak is progressively increased from 65 • C (shown here) to 115 • C in 10 • C increments. This demonstrates how the AFT central age and track-length distribution evolve with increased heating into the PAZ until the sample is thermally reset near ∼110 • C. Ga Tran-Hudson Orogen (THO) basement to the south (Hoffman, 1989;Fig. 4) and the Paleozoic-Mesozoic

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• The Williston Basin lies to the southwest of our samples (Fig. 4) and contains thick basin fill of > 4 km 172 due to deposition during most of the Phanerozoic (Burrus et al., 1996), beginning with the platform 173 onlap of the Sauk sequence (Sloss, 1963;Burgess, 2019).  23-5 Ma) unconformably overlying the Paleozoic section (Galloway et al., 2012).

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This information suggests that Precambrian basement was exposed by 450 Ma. An interval of regional 178 subaerial exposure during the Early-Middle Jurassic was possible, followed by deposition during the Cretaceous 179 and exhumation by approximately Miocene. There is also the question of whether this part of the currently 180 exposed shield basement was buried during deposition of the Hudson Bay sequence. We present our new 181 analytical results, which are then modelled to assess whether our AFTSS data can yield thermal histories 182 that are independently consistent with the accepted regional geological evolution.

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Apatite grains were double-dated (AFT and U-Pb) by the LA-ICPMS method (Chew and Donelick,185 2012; Cogné et al., 2020 previously published values (see Chew and Donelick, 2012). All analytical methods are the same as those

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Single guided laser-ablation spots were chosen within minimized grain counting areas to avoid potential U 192 zonation and all analytical results are shown in Table 1 and Table 2. The high N s track densities make U zoning on the etched grain surface easily detectable, and neither example showed evidence for strong zoning.

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One procedural difference for the data discussed here is that the AFT samples were split into two aliquots, 195 and thus different grain mounts. The first aliquot was analysed using the typical, faster LA-ICPMS operation 196 where track lengths are measured on all grains from which an age was measured and also other (undated) 197 grains. Whereas, the second aliquots had track lengths measured only from grains that had ages measured.

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The latter approach is more time consuming, since all lengths within a count area are measured to avoid the LA pit, and the other located in a different area of the grain to assess potential compositional heterogeneity.

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Complete EPMA data are provided in the Supplementary Information (SI ), and are summarized in Table   208 1 and Table 2. Elemental analyses with wt% oxide totals are 98.8 ± 1.8% for 225 analyses (including REA is a real phenomenon, it is evidently not a concern for apatite (Li et al., 2021). Outstanding questions 229 relate to how and if time plays a role in this process with respect to radiation damage and fission-track 230 accumulation for ancient apatites, or if over long timescales, accumulated alpha-radiation damage lowers 231 thermal annealing resistance (Ketcham, 2019;McDannell et al., 2019a). Here the data were interpreted 232 as overdispersed single populations. High dispersion is likely attributable to a continuous distribution of 233 ages rather than the typically assumed discrete age components (Vermeesch, 2019)-which may be at least 234 partially due to the protracted slow cooling (and differential annealing) these samples experienced, the greater 235 number of analyses relative to conventional AFT of ≤ 20 age grains, and higher relative LA-ICPMS age 236 precision (Ketcham et al., 2018;Vermeesch, 2019;McDannell, 2020).   probability. The 'unconstrained' model does not include t-T constraints. The geologic information being evaluated includes two distinct times in the past that we can reasonably assume basement was at near-surface 242 conditions (15 ± 15 • C) based on the regional geologic information discussed in Section 4.2. This information 243 was subsequently added to the model as constraint boxes, namely at: (i ) 450 ± 10 Ma and (ii ) 175 ± 25 Ma 2 .

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We refrain here from showing the individual maximum likelihood, maximum posterior, maximum mode, 245 or expected model paths (Gallagher, 2012), so as not to draw undue attention to a single t-T path since 246 they are single models or a representative summary for a broader range of solutions (i.e., mode or expected 247 model). Those models can be found in the SI or refer to the data repository for QTQt output (McDannell,248 2022). We focus on the entire stationary distribution of paths, particularly the 'unconstrained' model without constraints-yet both models independently corroborate the known cratonic geology by requiring two reheating 264 events. Given the imposition of simple models, the time-temperature paths also indirectly need periods 265 at low temperatures in the mid Mesozoic to produce a heating event. Thus, the thermal histories suggest 266 similar, albeit poorly resolved surface conditions in the late Precambrian to early Paleozoic (Fig. 5). The low 267 temperatures are required to form a population of tracks that are then shortened by reheating to produce the 268 observed lengths-without this, a certain component of lengths cannot be generated that are needed to fit 269 2 Placement of a Miocene surface constraint at 14 ± 9 Ma did not significantly change the results when compared to the 'unconstrained' or 'Ordovician/Jurassic box' models, and was therefore excluded for simplicity. The AFT data independently allow cooling to near-surface temperatures by Miocene time. A notable result is that the general features of the two-peak thermal history are visible in the unconstrained models. The high-quality track length data resolved the heating events and the t-T solutions independently support the regional geologic information.
The 'unconstrained' model in panel A clearly illustrates the penalization of more complex histories due to more simple, 'linear' paths being accepted or retained preferentially between the two thermal peaks. All QTQt models are available in McDannell (2022).
the observations. It may be that time or duration at higher temperatures become increasingly important over 270 long timescales, which we can similarly observe in Figure 3 where maximum temperatures are slightly lower 271 than the true peak and a subset of paths remain nearly isothermal, therefore producing similar amounts of 272 annealing as the true history with high maximum temperatures for a geologically instantaneous duration.

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The AFTSS models best resolve a broad thermal peak between approximately latest Devonian to Triassic Pennsylvanian strata on the Hudson platform, which was controversially posed by Tillement et al. (1976).

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The Michigan and Williston basins also contain a few hundred meters of Pennsylvanian and Jurassic strata 284 (e.g., Burrus et al., 1996;Burgess, 2019), perhaps suggesting a regionally common history for interior North

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America. The final cooling event in the model takes place in the Oligocene-Miocene. The White River Group are significant because they further establish that burial extended across the currently exposed basement of 289 the Canadian Shield, that the Hudson Bay sedimentary succession is an erosional remnant (Pinet et al., 2013;290 McDannell et al., 2021), and the Hudson Bay and Williston basins were probably intermittently connected. The inferred complexity of a thermal history is related to the number of track lengths (Fig. 3). Our 299 simulations clearly show that our AFTSS data have enough lengths to independently require two thermal 300 events (i.e., without requiring t-T boxes) during the Phanerozoic for the exposed Precambrian basement of 301 the central Canadian Shield-but adding the constraints improves the resolution on the timing of maximum 302 temperatures. However, it seems clear that 100 measured tracks for a single kinetic AFT population are not 303 enough to resolve complicated deep-time thermal histories without applying interpretation-based constraints 304 (e.g., . To further explore this with the real data, we 305 took the entire length dataset for each AFT example and randomly downsampled it using a simple Monte 306 Carlo method, retaining ∼10%, ∼20%, and ∼50% of the original length distributions, while maintaining 307 a stable mean length within uncertainty (Fig. 6). This was done to determine how well we resolve the 308 two thermal peaks in the full model t-T history from Figure (Galloway et al., 2012;Fig. 4). In this particular 321 instance, we have geologic information to empirically validate our model, whereas in more frontier regions 322 where less Phanerozoic strata are known or preserved, a t-T model such as this may be more difficult to 323 justify or be considered an artifact. To that end, AFTSS data may be extremely valuable for inferring and 324 resolving the timing of unrecognized or poorly recorded geologic events on cratons.

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The results of our modeling emphasize that amount of track length data is possibly too low in many 326 cratonic t-T modeling applications and that inadequate characterization of length distributions may affect 327 our ability to recover thermal history information. While this is not conceptually novel-what constitutes a 328 robust track length dataset and if those data can independently support geologic observations has gone mostly 329 unrecognized. While the mean track length is often a useful summary statistic, it is the width and shape of 330 the track length distribution that are critical for modeling (Crowley, 1985;Gleadow et al., 1986). The main 331 body of the distribution needs to be well defined with many tracks, but the tails of the true distribution also 332 need to be well represented. Namely, any shorter lengths that provide key temperature information must 333  be included, which will typically require more measurements because they have a lower probability both 334 of being observed and measured accurately . C-axis angle projection of track lengths 335 also plays a role in improving resolution by reducing length dispersion due to track orientation (Donelick 336 et al., 1999), yielding a better defined length distribution (Ketcham et al., 2018;Ketcham, 2019)-thereby 337 taking advantage of the extra information contained in the annealing dependence on track orientation. If the 338 distribution shape is well characterized then the thermal model can deconvolve the mixed length components 339 generated by the different heating-cooling cycles.

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In detail, many different thermal histories can satisfy a given track length distribution. However, even if 341 the distribution looks similar between an example with many tracks and fewer tracks, the possibility to resolve 342 multiple heating-cooling events in a history is reduced in the latter case. A good example of this is apparent 343 in the downsampling results shown in Figure 6. Here the increased number of tracks tends to broaden the 344 overall distribution, implying (or requiring) greater history complexity-which is then verified in the Figure   345 7 inversion results. The same limitations can apply to different forms of thermal histories as reflected for 346 the example shown in Figure 1. The real slow-cooling history may be misinterpreted as rapid and/or recent 347 cooling if the skewed distribution (Fig. 1C) were undersampled such that shorter lengths were not measured.

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The same generally applies to the broad distribution (Fig. 1D) if there are not enough intermediate (∼12-13 349 µm) and/or long (∼15-16 µm) c-axis projected lengths collated to distinguish between a narrow or wide 350 unimodal track population. In addition, the synthetic AFT inversions (Fig. 3) suggest to us that exploratory forward modeling potentially offers a means to practically estimate the number of track lengths required for a robust AFT analysis if a 'schematic' burial history can be surmised from the regional geology or other data.
apatite composition) at any time during the Phanerozoic-then the additional t-T information normally 355 provided by an AFTSS analysis will diminish in relation to the timing of the resetting event (i.e., a thermal 356 pulse late in the history will tend to erase or at least reduce the information provided by additional lengths).

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The examples and model results presented here demonstrate that a minimum of ∼250-300 confined length 358 measurements are required for robust thermal history recovery for single-age population samples in cratonic 359 regions where rocks experienced modest thermal annealing over the past 500-600 million years. dating has traditionally been a preferred method for constraining aspects of these complex burial and erosion 364 events. However, due to the absence of physical geologic constraints, detailed thermal history reconstruction 365 is often difficult. This issue leads to a thermal history modeling approach that incorporates interpretive 366 assumptions about the geologic history that may be invalid or at least difficult to validate independently.

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New apatite fission-track data were presented from the central Canadian Shield that included many more 368 confined track-length measurements than a typical fission-track analysis. Inversions of these data yield results 369 that are consistent with the regional shield geology without requiring the imposition of t-T 'constraint boxes'.

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Subsequently, consideration of known geologic constraints with either forward or inverse modeling approaches 371 allows an assessment of the impact of constraints relative to the unconstrained thermal histories. While the 372 appropriate number of tracks lengths to collect is a function of the thermal history, our results demonstrate 373 that the conventional approach of measuring around 100 track lengths may be inadequate for long duration 374 (500-1000 My) thermal history scenarios involving a higher level of history complexity and/or episodic minor 375 annealing. Ultimately, each problem is unique and analyses should be tailored to optimize the amount of 376 information available for modeling since a standardized approach may not yield sufficient data to clearly 377 resolve significant thermal events. We suggest that 250-300 confined track lengths (with c-axis angles) may 378 be considered an effective minimum-suitable for thermal history inversion in cratonic settings for rocks that 379 contain a single kinetic population and have experienced low-to-moderate thermal annealing. This simple 380 change in analytical protocol may improve thermal history recovery and lend more credence to geologic 381 interpretations in slowly cooled continental interiors.