Temperatures in Tc1D start from a thermal steady state and evolve forward in time as the result of heat advection, conduction, and production. The general form of the transient heat transfer equation in 1D, including the effects of advection, conduction, and heat production, is 1ρCp∂T∂t+vz∂T∂z=-∂qz∂z+H, where ρ is rock density, Cp is rock heat capacity, T is temperature, t is time, vz is the vertical advection velocity, z is the spatial coordinate (depth in Tc1D), and H is heat production. The vertical heat flux term is qz, which can be calculated as 2qz=-k∂T∂z, where k is thermal conductivity. The time evolution of temperature in Tc1D is calculated using either an explicit or implicit finite-difference formulation of Eq. () using a user-defined time step dt. Heat produced by radioactive decay is currently the only heat source for the production term H, and the material properties ρ, Cp, k, and H can vary in the model crust, mantle lithosphere, and asthenospheric mantle materials. Heat production H in the crust can be constant or decay exponentially to simulate the concentration of heat-producing elements in upper crustal rocks. To ensure the heat flux is conserved across node points in the finite-difference grid, a staggered grid is used for thermal conductivity e.g.,.

Node points are fixed in space during Tc1D calculations and the evolution of temperature in the models is constrained by constant temperature boundary conditions at the surface and base of the model. The initial temperatures in Tc1D are calculated by setting the time-dependent term in Eq. () to zero, yielding 3ρCpvz∂T∂z=-∂qz∂z+H. Equation () is solved using an implicit finite-difference formulation, and it is important to note that the initial conditions can thus include the effects of an initial advection velocity or variable material properties (including heat production). Values used to define the boundary condition temperatures are assumed to be constant over the duration of the simulation with the present version of Tc1D, however they could vary in time in principle.

Because the node points do not move in Tc1D, the vertical advection velocity vz defined in different user-selectable erosion models affects both the model thermal field and the thermal history of tracking particles exhumed or buried during the simulation (see Tc1D documentation at https://tc1d.readthedocs.io/en/latest/erosion-models.html (last access: 30 April 2026) for an overview of available erosion models). Starting from a position at the model surface at 0 Ma, a particle's starting depth is back-calculated by running the erosion model backwards in time for the duration of the numerical experiment. The particle's position is then updated forward in time during each model time step by vz⋅dt, and the current depth, pressure, and temperature are recorded. Depth, pressure, and thermal histories can be calculated for particles ending at the surface at the end of the numerical experiment or at various user-defined times during the experiment.

The modeled thermal history (or histories) can be used to calculate thermochronometer ages for the (U–Th)/He and fission-track systems in apatite and zircon (e.g., Fig. ). (U–Th)/He age prediction includes the effects of radiation damage and annealing through the RDAAM model of for apatite (U–Th)/He (AHe) ages and the ZRDAAM model for zircon (U–Th)/He (ZHe) ages. Both use an external program from . Apatite fission-track (AFT) ages can be predicted either using the fission track annealing model of or that of . The model has the Cl content set to 0 ppm and uses the approach of as implemented in . The AFT model from can use the kinetic parameters from or the Durango or fluorapatite kinetics from . Zircon fission-track (ZFT) age prediction uses the approach of with the kinetics of either or following the implementation in .

In addition to the age prediction model, there are several variables in Tc1D that affect age prediction. The equivalent spherical radius can be specified for AHe and ZHe age prediction, or will default to an equivalent spherical radius of 60 µm. Uranium and thorium concentrations can also be defined for AHe and ZHe ages, or will default to concentrations of 10 and 40 ppm for uranium and thorium in apatite or 100 and 40 ppm for uranium and thorium in zircon. In addition, it is possible to define a time duration that a tracked particle resides at the initial model temperature prior to the start of a Tc1D thermal simulation. This can be useful for including the effects of radiation damage accumulation without requiring transient temperature calculations during that interval.

The predicted thermochronometer ages can be compared to measured ages in two different ways in Tc1D. First, it is possible to read age data from a data file. In this case, the user must input at least the data type (e.g., AHe), the measured age, and its standard deviation. The user can optionally include a sample ID and depositional age for dated sedimentary samples. If non-zero depositional ages are provided, Tc1D will calculate predicted ages for a tracking particle reaching the surface the time of deposition in the model, assuming zero transport time from the source area. For (U–Th)/He ages it is also possible to input an effective uranium (eU) concentration and grain radius. When provided, the sample grain radius and eU values will be used for age prediction, otherwise the default U, Th, and radius values given above will be used. The second option for providing measured ages is to provide one or more measured ages and standard deviations to Tc1D using function parameters or command-line flags. In this case, however, the sample-specific grain radius and/or eU values are not utilized for (U–Th)/He age prediction and it is not possible to specify depositional ages. In either case, it is possible to output a summary csv-formatted text file containing the measured and predicted ages and relevant other values such as the eU concentration and grain radius.

A quantitative measure of goodness-of-fit between measured and predicted thermochronometer ages in Tc1D can be calculated using a misfit function (e.g., Fig. ). By default, the misfit Φ uses the L2 norm between the measured and predicted ages divided by the square root of the summed measured age uncertainties and normalized to the number of samples e.g., Eq. (8) of, 4Φ=1N∑i=1NOi-Ci2σi2, where N is the number of ages, Oi is the ith measured (or observed) age, Ci is the ith predicted (or calculated) age, and σi is the standard deviation (uncertainty) for the ith measured age. This default goodness-of-fit approach will produce misfit values that are less than or equal to 1 when the predicted ages fall within the measurement uncertainties on average and greater than 1 otherwise. Another misfit function option is described below for use with inverse models in Tc1D.

In addition to individual forward models, groups of models can be run in Tc1D to either explore parameter sensitivity or invert data to determine ranges of model parameters that best reproduce the measured ages. Both options are briefly described in the sections below.

This first example considers a pluton that is emplaced in the crust at the start of the numerical experiment, then undergoes conductive cooling and later crustal exhumation similar to, for example, or. The pluton is emplaced with a temperature of 1000 °C and a thickness of 2 km between 8–10 km depth at 30 Ma and held at that temperature for 1 Myr. The effects of the latent heat of crystallization are not included. No erosion is applied (advection velocity equals zero) during 30–10 Ma and then 9 km of exhumation is imposed at a constant rate of 0.9 kmMyr-1 from 10–0 Ma, displacing the Moho from its initial depth of 44 km to a final depth of 35 km. Default values are used for all other model parameters.

The thermal effects of magmatism on upper crustal temperatures are clear in this example, but only the ZFT thermochronometer records cooling directly related to magmatism. The thermal distrubance caused by the pluton increases temperatures by more than 800 °C at the emplacement depths of 8–10 km when active and by a smaller amount in the crust above and below the pluton for several million years afterward (Fig. ). By 25 Ma, however, this thermal effect has diminished considerably, leaving temperatures at most ∼150°C higher than their conductive steady state across the entire crust (e.g., geotherms at 25–10 Ma in Fig. ). The final phase of exhumation from 10–0 Ma increases temperatures across the entire lithosphere as heat is advected to shallower depths in the model (e.g., geotherms at 5–0 Ma in Fig. ). The combination of pluton-related heating and later exhumation produces a thermal history in which the tracked particle within the pluton is initially heated to 1000 °C, cools rapidly to ∼200°C from ∼29–24 Ma, and then undergoes a final phase of gradual cooling during exhumation from 10–0 Ma (Fig. ). This cooling history results in a ZFT age of ∼24Ma, roughly 5 Myr younger than the cessation of plutonic activity, and AHe, AFT, and ZHe ages of ∼7–3 Ma, corresponding to the later exhumation phase (Fig. ). This example illustrates the sensitivity of low-temperature thermochronometers to processes such as crustal magmatism, but also the challenge of interpreting age data due to the combination of long timescales for conductive cooling and interactions of magmatism and crustal exhumation that can produce ages that vary considerably among the different chronometers.

As a second example, we consider the evolution of thermochronometer ages during exhumation of a thrust sheet over 50 Myr. In this case, erosion type 4 is used in Tc1D, which involves instantaneous emplacement of a thrust sheet of a specified thickness atop the existing crust e.g., and its subsequent erosion. Thermally, this model duplicates temperatures from the upper section of the model crust equal to the thrust sheet thickness and “stacks” those temperatures atop the existing crust. After stacking, the temperatures re-equilibrate during exhumation of part or all of the thrust sheet. For this example, a 10km-thick thrust sheet is emplaced at 30 Ma and 15 km of rock (the thrust sheet plus five kilometers of the footwall) are eroded during 30–0 Ma. In this example, however, ages are predicted for tracked particles reaching the model surface every 1 Myr, showing how the various chronometers are reset during exhumation and how ages vary between the hanging wall and footwall of the thrust system. Default values are used for all other model parameters and additional plots showing the evolution of geotherms and the thermal history for this model can be found at the link under the heading Code and data availability.

The evolution of predicted ages follows the expected pattern of resetting of the lower temperature chronometers (AHe, AFT) prior to the higher temperature systems (ZHe), but also a jump in ages for some systems as the thrust sheet is fully eroded and footwall rocks are exposed via erosion. Prior to 30 Ma, no exhumation occurs and the predicted ages are all unreset (Fig. ). As early as ∼27Ma, partially reset AHe ages begin to appear, as indicated by a deviation from the unreset reference line, and by ∼23Ma the AHe ages are fully reset and less than 10 Ma when exposed at the model surface (e.g., Fig. a). As exhumation continues, partially reset AFT ages appear by ∼20Ma, followed shortly afterwards by partially reset ZHe ages at ∼18Ma. Fully reset AFT and ZHe ages are exposed consistently by 17 and 14 Ma, respectively. At 10 Ma, however, ages suddenly increase for the AFT and ZHe systems as the footwall is exhumed and rocks that have not been hot enough to be fully reset are exposed. For ZHe, these ages correspond to unreset ages at 10 Ma and gradually young during exhumation, as seen earlier during exhumation of the thrust sheet (Fig. ). A similar trend is observed for the AFT ages, but only partially reset ages appear at 10 Ma, and subsequent ages are fully reset. No jump in age is seen for the AHe system, as the 10km-thick thrust sheet fully resets ages in the footwall after its emplacement. Finally, no reset ZFT ages are observed, reflecting insufficient exhumation depths to expose reset ages. A notable result from this example is that the jump in age during exhumation only occurs for some systems and depends on the thickness of the emplaced thrust sheet. Preservation of such age signals in foreland basin sediments, for example, could provide a means to constrain the timing and thickness of thrust sheet emplacement and exhumation in convergent tectonic settings.

Here we consider an example of a geological history that involves several stages of exhumation and burial in a cratonic setting over a long duration (>1Gyr). The model includes 10 stages of erosion or burial and uses an exhumation history defined in the input file for Tc1D rather than using command-line flags. This flexible approach to describing the exhumation history can be used to explore more complex geological and exhumation histories than the general erosion models in Tc1D. This example is based on the exhumation history of the Slave Craton in Canada as described in and . A summary of the exhumation history for this example and related geological constraints can be found in Table .

The thermal history for this example (Fig. ) supports the proposal by that Ordovician to Devonian (450–375 Ma) burial is essential for resetting the AHe ages in their dataset, yet using Tc1D we also find a minimum magnitude of burial of at least 5 km is required to achieve the age reset. The early exhumation phase following the last significant tectonic activity (1900–1800 Ma) provides the largest magnitude of cooling and produces ZFT and ZHe ages that are >1800Ma (Fig. ). Subsequent gradual exhumation and cooling brings the tracked particle to the surface at 550 Ma, based on geological constraints Table ,. Phases of burial from 550–375 Ma increase the temperature of the tracked particle to ∼100°C before exhumation and cooling from 375–100 Ma bring the tracked particle back to the surface. Exhumation during these stages produces respective AFT and AHe of ∼330Ma and ∼240–290 Ma, consistent with the age range for the measured AHe ages of 337–188 Ma. In this example, the magnitude of burial from 450–375 Ma is 5 km, which is sufficient to reset both the AFT and AHe ages. However, reducing the burial magnitude to 4.5 km produces only partially reset AFT and AHe ages that are ∼725 and ∼1350–275 Ma, respectively (plot can be found in the supplementary plots linked under the heading Code and data availability). Thus, although states that the magnitude of Ordovician to Devonian burial is >3km, this can be refined using Tc1D to be ≥5km. Further exploration of the timing and magnitudes of burial and exhumation using Tc1D in inverse mode (see Sect.  for an example) could further refine the proposed geological history of this cratonic region.

To illustrate the Markov chain Monte Carlo (MCMC) inverse modeling functionality in Tc1D, we selected sample 18PE28 from the Marcapata transect in southern Peru from (Fig. ). This sample provides a useful natural example since it includes four thermochronometric systems that can be directly explored with Tc1D: AHe, AFT, ZHe, and ZFT. Sample 18PE28 is a Permian–Triassic granite collected at 2416 m elevation and yields young to intermediate cooling ages of 1.3±0.4Ma (AHe), 3.3±0.3Ma (AFT), 5.5±0.5Ma (ZHe), and 16.7±1.4Ma (ZFT), offering relatively strong constraints on late-stage cooling and exhumation history.

The aim of this example is not to reproduce exactly the inverse time-temperature histories presented by , but rather to demonstrate how Tc1D can be used to invert a natural multi-system thermochronological dataset using an explicit thermo-kinematic parameterization. In contrast to inverse approaches that primarily return time-temperature paths, Tc1D links the observed ages to a prescribed thermal structure and parameterized erosion history. This allows the inversion results to be interpreted directly in terms of geologically meaningful model parameters, such as the timing and magnitude of changes in erosion rate.

For this illustrative example, we inverted a two-stage erosion history using three free parameters: (1) the total amount of erosion during the first stage (E1∈[0.01,25.0]km), (2) the timing of the transition between the two stages (tswitch∈[0.01,30.0]Ma), and (3) the total amount of erosion during the second stage (E2∈[0.01,25.0]km) (Fig. ). The total model time spans 30 Ma, such that tswitch effectively controls the timing of a potential break in slope in the erosion history. These parameters are internally converted into time-dependent erosion rates (kmMyr-1) during forward modeling. The MCMC inversion was performed using 16 walkers and 5000 steps per walker, with the first 500 steps discarded as burn-in and a thinning factor of 3 applied to reduce sample autocorrelation, resulting in a total of 24 000 retained models used for posterior analysis.

The thermal structure and crustal properties were fixed based on representative values for the region following , including crustal density, heat capacity, thermal conductivity, heat production, surface temperature, and initial Moho depth, while other model parameters were set to commonly used default values.

The inversion recovers a family of solutions that reproduce the observed AHe, AFT, ZHe, and ZFT ages (Fig. a), with the best-fitting model providing a close agreement with the measured ages within uncertainty. The corresponding erosion histories (Fig. b) consistently indicate an acceleration of erosion around 2 Ma. This timing is in good agreement with the time-temperature paths reported by , which show an increase in cooling rates starting at approximately 2 Ma (Fig. c). The MCMC diagnostics (Fig. d and e) indicate good exploration of parameter space, with well-mixed chains and well-defined posterior distributions for all inverted parameters.

The examples presented above demonstrate a range of geological scenarios in which Tc1D can be used to understand how geological processes affect thermochronometer age data. We have also suggested that Tc1D is complementary to available thermal history modeling and more complex thermo-kinematic modeling software, such as Pecube. This raises a natural question: What are the best cases to consider using Tc1D?

Although Tc1D could be used in many cases where it is desirable to explore the effects of varying magnitudes of exhumation, burial, magmatic heating, or the impact of other geological processes, we feel there are four scenarios in which use of Tc1D is most appropriate. First, Tc1D is recommended in cases where the aim is to quantify the magnitudes of burial and/or exhumation, and the primary motion of material is vertical. Such cases could include tectonically inactive regions and regions where the horizontal component of tectonic motion is limited. Because Tc1D does not simulate horizontal motion of rock mass, there may be scenarios where its use could result in misinterpretations of age data due to a large horizontal component of heat transfer, such as in active fold-and-thrust belts. Tc1D could still be used as an exploratory tool in such settings, but it may be more appropriate to consider a more complex thermo-kinematic numerical modeling software package such as Pecube in such situations.

A second scenario where the use of Tc1D is recommended is when working with age data from regions with limited topographic relief or with samples collected across areas with limited relief. The reason for this recommendation is similar to that for the first case: Tc1D uses a 1D heat transfer model and thus may not capture the thermal effects of topography on subsurface isotherms, which could lead to data misinterpretation. Mountain topography typically perturbs the geometry of shallow crustal isotherms e.g.,, though not always e.g.,, and such an effect is not possible to directly incorporate in Tc1D due to its geometry. As a result, it is possible that modeling thermochronometer data in regions with significant topographic relief could be problematic. Thus, Tc1D may be more safely applied in areas with limited topographic relief, such as areas that are no longer tectonically active or where the topographic effects on crustal isotherms are thought to be minimal. Future versions of Tc1D may include the possibility to correct for the effects of topography on crustal temperatures following the apporach of , for example.

A third case where we recommend using Tc1D is for the interpretation of multi-chronometer datasets or groups of samples thought to have experienced similar thermal histories. In this case, the benefit for use of Tc1D mainly relates to the potential to quantify rates of exhumation and/or burial using Tc1D's inverse mode. Having multiple temperature sensitivities from multiple chronometers provides better constraint on the possible thermal histories that could produce the measured age data. The same concept applies to thermal history models like HeFTy, QTQt, or Thermochron.jl, but the strength of Tc1D in this case is the potential to use a data-driven inverse approach to quantify rates of geological processes more directly. This type of inverse modeling could also be done together with thermal history modeling or as a preparatory step in more complex modeling using Pecube.

The final case where using Tc1D is suggested is to simply explore the effects of various geological processes on thermochronometer data. Tc1D provides a tool with which researchers can explore how varying rates and magnitudes of burial and/or exhumation affect predicted ages, as well as the impact of different thermal processes (e.g., magmatism) and rock thermo-physical properties (e.g., radiogenic heat production). Varying different parameter values can illustrate when and by how much different processes impact predicted ages. This can aid not only in data interpretation, but also in planning where to sample, which thermochronometer systems to consider, and other factors that can help ensure the signal sought in collected samples is likely to be present. Similarly, the effects of different geological processes can be introduced to students interactively using Tc1D to help them gain intuition about what matters when studying geological processes using thermochronology.

The thermal histories produced in Tc1D can be exported and read directly into the Tcplotter software to further explore the combined effects of the geological processes simulated in Tc1D and factors affecting individual mineral grains such as variability in grain size or eU concentration. Tcplotter uses the same FT annealing and He diffusion equations as Tc1D but can calculate variations in cooling ages and effective closure temperatures simultaneously for desired ranges of grain size and eU (and hence variability in radiation damage accumulation) to simulate their effect on He diffusion and resulting calculated AHe and ZHe ages. While grain sizes and eU concentrations can be varied for individual mineral grains in Tc1D, the Tcplotter software produces contour plots for ranges of these values that can be illustrative when considering the possible impact of radiation damage or variation in grain size on predicted thermochronometer ages or effective closure temperatures. The thermal histories produced by Tc1D are stored in plain text files, which can also be readily adapted to be read into other modeling software such as HeFTy.

In the preceding test we have noted some of the benefits and limitations of Tc1D, and we summarize the main limitations of the Tc1D here. First, Tc1D uses a one-dimensional thermal and kinematic framework in which heat transport and rock movement occur only in the vertical direction. Consequently, the model cannot take into account lateral variations in temperature, topography, or material properties. In particular, processes such as lateral heat transfer due to dipping faults or topography-driven perturbations of isotherms are unable to be simulated, as noted above.

Second, the current version of Tc1D imposes constant temperature boundary conditions at the surface and base of the model that do not vary in time. This is not a limitation of the numerical method, but there is not yet support for time-varying boundary condition temperatures. Similarly, rock thermo-physical properties for the crust and lithosphere are prescribed using simplified layer-based values that also do not generally evolve through time, except when using specific processes implemented in the model, such as magmatic intrusion or lithospheric removal. These restrictions may limit its applicability in settings with long and complex thermal evolutions, such as cratonic regions or orogenic systems with multiple deformation phases.

Third, Tc1D simulates the thermal history of a single vertical column of rock and is therefore best suited to the interpretation of thermochronological data from individual samples or from locations where samples are thought to experience similar burial or exhumation histories. While multiple samples can be analyzed individually, the model does not explicitly account for spatial relationships between them. As a result, it cannot directly represent lateral variations in exhumation history or thermal structure, nor test spatially varying processes such as fault activity and/or lateral exhumation gradients. Similarly, although vertical sampling profiles (e.g., age-elevation relationships) can be explored on a sample-by-sample basis, the model cannot explicitly reproduce the continuous structure of such profiles.

Recent major developments in Tc1D have focused on increasing flexibility in defining burial and exhumation histories and extending the capabilities of Tc1D for inverse modeling. These features are currently under active development and testing, and are briefly outlined here to highlight the evolving capabilities of the model.

The first development concerns the introduction of the flexible, multi-stage erosion framework (ero_type=0), in which exhumation or burial histories are defined as a sequence of successive stages. Each stage can be assigned a different erosion style (e.g., constant, linear, or exponential) and corresponding parameterization, allowing complex and more realistic time-dependent histories to be represented within a single model run. The same framework can also be used in inverse mode, where selected stage parameters are treated as free variables within prescribed bounds. This new erosion model type in Tc1D is usable from the Tc1D input file.

A second development is the implementation of a trans-dimensional (i.e., variable-dimension) Markov chain Monte Carlo (Reversible Jump MCMC) approach for inversion of multi-stage erosion histories. In contrast to fixed-dimension inversion methods, this approach allows both the number of erosion stages and the parameters defining the erosion laws of each stage to be inferred simultaneously from the data. The current implementation uses a set of stochastic moves that include parameter perturbations as well as changes in model dimension (i.e., the number of erosion stages), enabling exploration of models with varying levels of complexity within a single inversion workflow. This provides a flexible framework for balancing model fit and model complexity (parsimony) when modeling thermochronological data. A working version of a first implementation of this framework is in the development version of Tc1D, will be available in a future release of the code after further testing and refinement.

In addition to the ongoing code development, there are several planned features in Tc1D that are worth noting. First, we intend to improve support for modeling thermochronometer ages of sedimentary deposity in Tc1D. It is possible at present to include the time of deposition for predicted ages in the Tc1D data file, but the only effect this has in the current implementation is to calculate an age at the surface at the time corresponding to deposition and then add time to that age between when it is deposited and the end of the simulation. We hope to improve this in future versions of Tc1D by allowing subsequent burial and heating to enable users to explore complex thermal histories in foreland basins, for example. Second, we plan to explore implementing a means for incorporating the effects of topography on predicted ages. Although Tc1D uses a 1D thermal model, it is possible to account for the effects of surface topography on crustal temperatures e.g.,. Thus, we intend to test various options for handling surface topography to enable the use of Tc1D in regions with significant topographic relief or with elevation transects. Finally, a longer term goal is to develop a flexible age prediction library for use with Tc1D that would suport more thermochronometer systems. This is not likely to be implemented in the near future, but is planned as a future development goal to broaden the potential range of chronometers that can be modeled using Tc1D.