Relating stratigraphic position to numerical time using age–depth models plays an important role in determining the rate and timing of geologic and environmental change throughout Earth history. Astrochronology uses the geologic record of astronomically derived oscillations in the rock record to measure the passage of time and has proven to be a valuable technique for developing age–depth models with high stratigraphic and temporal resolution. However, in the absence of anchoring dates, many astrochronologies float in numerical time. Anchoring these chronologies relies on radioisotope geochronology (e.g., U–Pb,

In this study, we present a new R package,

We test the

Linking the rock record to numerical time is a crucial step when investigating the timing, rate, and duration of geologic, climatic, and biotic processes, but constructing chronologies (age–depth modeling) from the rock record is complicated by a variety of factors. The premier radioisotopic geochronometers enable direct determination of a numerical date from single mineral crystals (e.g., sanidine, zircon) to better than 0.1 % throughout Earth history (Schmitz and Kuiper, 2013). However, rocks amenable to radioisotopic dating, mostly volcanic tuffs, may only occur as a few dispersed horizons within a stratigraphic section. This leads to the problem of a small number of high-precision dates scattered throughout stratigraphy with limited chronologic information between these horizons. Consequently, chronologies developed using only radioisotopic dates have widely varying uncertainties throughout a given stratigraphic record, with precise ages near the position of the dates and increasing uncertainties with distance from the dated horizons (Blaauw and Christen, 2011; Parnell et al., 2011; Trachsel and Telford, 2017; Trayler et al., 2020).

Adding more chronological information is the best way to improve age–depth model construction (Blaauw et al., 2018). In particular, including stratigraphically continuous data can significantly reduce model uncertainties. Astrochronology uses the geologic record of oscillations in Earth's climate system (“Milankovitch cycles”) to measure the passage of time in strata (Hinnov, 2013; Laskar, 2020). Some of these oscillations can be linked to astronomical physics with well-understood periods, including changes in the ellipticity of Earth's orbit (eccentricity;

Combining floating astrochronologies and radioisotopic dates into an integrated model of age is an attractive prospect, as it leverages the strengths and overcomes the limitations of both data sources. Here we present a freely available R package (

The

Quasiperiodic variations in Earth's orbital and rotational parameters impact the spatial and temporal distribution of sunlight on the planet's surface and thus have the potential to alter regional and global climate. Such quasiperiodic climate changes can influence sedimentation and be preserved in the geologic archive, providing a dating tool for developing astronomical timescales, or astrochronologies. The astronomical variations include orbital eccentricity with modern periods of 0.405 and

Radioisotope geochronology utilizes the radioactive decay of a long-lived parent isotope to its daughter product within a closed geologic system to the determine its age. Temporal information is quantified in the evolving ratio of daughter to parent as a function of the decay constant(s) of the constitutive nuclear reactions. In the case of sedimentary strata in deep time, these geologic systems are either radioisotopes captured in rapidly erupted and deposited igneous mineral grains in discrete interbedded volcanic tuff horizons (U–Pb in zircon or K–Ar – implemented as the

The Bayesian statistical approach aims to determine the most probable value of unknown parameters given data and prior information about those parameters. This is formalized in Bayes' equation.

Existing Bayesian methods for age–depth model construction rely on sedimentation models that link stratigraphic position to age through mathematical functions that approximate a sedimentation process conditioned through dated horizons throughout a stratigraphic section, which are then used to estimate the age and uncertainty at undated points (Blaauw and Heegaard, 2012). A variety of Bayesian approaches have been proposed to construct age–depth models including

Previous Bayesian approaches for linking astrochronology and radioisotopic dates have taken numerous approaches, including (1) solely focusing on improving the ages of radioisotopically dated horizons using astrochronology (Meyers et al., 2012), (2) relying on post hoc comparisons of computed astrochronologic and radioisotopic durations to accept or reject accumulation models in the Markov chain Monte Carlo process (De Vleeschouwer and Parnell, 2014), or (3) “transforming” astrochronologic durations into age likelihoods via anchoring to other radioisotopically dated horizons (Harrigan et al., 2021). Meyers et al. (2012) modified the Bayesian “stacked bed” algorithm of Buck et al. (1991) to incorporate known astrochronologic durations between dated horizons, allowing for the improvement of Cretaceous radioisotopic age estimates using astrochronology, and the age of the Cenomanian–Turonian boundary. Their approach, however, did not explicitly model posterior age estimates for intervening strata in the Bayesian inversion. De Vleeschouwer and Parnell (2014) recalibrated the Devonian timescale and calculated new stage boundaries using a two-step process. First the authors generated a continuous Bayesian age–depth model using the

The inputs for

Existing Bayesian age–depth modeling approaches approximate sedimentation as a relatively large number of piecewise linear segments. Sedimentation rate can vary substantially between segments, leading to the sausage-shaped uncertainty envelopes that characterize these models (Trachsel and Telford, 2017; De Vleeschouwer and Parnell, 2014; Parnell et al., 2011). However, this model of sedimentation is not ideal for the construction of astrochronologies because fluctuations in sedimentation rate can be constrained by preserved astronomical frequencies as spatial stretching or compression of the preserved rhythm. As our nominal approach, we adopt a sedimentation model with a small number (

Malinverno et al. (2010) presented a simple sedimentation model appropriate for astronomical tuning of sedimentary records, and we use their framework as the starting basis for the joint inversion. The sedimentation model consists of two sets of parameters. The first is a vector of sedimentation rates (

Schematic of model parameters.

Summary of model parameters.

The selection of layer boundary positions is an important user-defined step that is informed by detailed investigation of the cyclostratigraphic data. Evolutive harmonic analysis (EHA) is a time-frequency method that can identify changes in accumulation rate by tracking the apparent spatial drift of astronomical frequencies. Expressed as cycles per depth, high-amplitude cycles may “drift” towards higher or lower spatial frequencies throughout the stratigraphic record. Assuming these spatial frequencies reflect relatively stable astronomical periodicities, the most likely explanation for those spatial shifts is therefore stratigraphic changes in sedimentation rate (Meyers et al., 2001). That is, stability in spatial frequencies reflects stability in sedimentation rate, allowing sedimentation to be approximated by a small number of piecewise linear segments.

We visually inspected EHA plots to develop simple sedimentation models (e.g., Fig. 1b) for our testing datasets. We choose layer boundary positions (

Synthetic testing datasets used for model validation.

Together

Together the vectors of sedimentation rates (

We follow the approach of Malinverno et al. (2010) to calculate the probability of our data given a sedimentation rate and set of target astronomical frequencies (

Astronomical frequencies used for model testing and validation for the two synthetic testing datasets (discussed below). The precession and obliquity terms are based on the LA04 solution (Laskar et al., 2004), and the eccentricity terms are based on the LA10d solution (Laskar et al., 2011).

The anchored age–depth model now consists of two paired vectors that relate stratigraphic position (

The overall likelihood function of an anchored age–depth model is now the joint probability of Eqs. (2) and (3). We use a vague uniform prior distribution where sedimentation rate may take any value between a specified minimum and maximum value.

We tested

The second dataset (CIP2) was originally published by Sinnesael et al. (2019) as a testing exercise for the Cyclostratigraphy Intercomparison Project, which assessed the robustness and reproducibility of different cyclostratigraphic methods. The CIP2 dataset was designed to mimic a Pleistocene proxy record with multiple complications including nonlinear cyclical patterns and a substantial hiatus. For full details on the construction of the CIP2 dataset see Sinnesael et al. (2019) and

We assessed model performance using two metrics. First, we assessed model accuracy and precision by calculating the proportion of the true age–depth model that fell within the 95 % credible interval (95 % CI) of our model posterior. We assume that a well-performing model should contain the true age model in most cases. This method has been used previously to assess performance of existing Bayesian age–depth models (Parnell et al., 2011; Haslett and Parnell, 2008). Second we monitored the variability of the model median (50 %) and lower and upper bounds (2.5 % and 97.5 %) of the credible interval.

To assess the reproducibility and stability of

Dates used as inputs for reproducibility and stability testing of the synthetic test cases (TD1 and CIP2).

We tested the sensitivity of our age–depth model results to both the number and stratigraphic position of radioisotopic dates. We randomly generated a set of dates from the underlying sedimentation model using Monte Carlo methods. The uncertainty (1

Since the CIP2 dataset includes a significant hiatus (Sinnesael et al., 2019) we also investigated the influence of the number and stratigraphic position of radioisotopic dates on the quantification of the hiatus duration. Estimating hiatus duration requires at least one date above and below the stratigraphic position of a hiatus. Consequently, we added an additional constraint when generating synthetic dates from the CIP2 dataset to ensure that the hiatus was always bracketed by at least two dates. For each of the sensitivity validation models (two, four, six, and eight dates) we benchmarked the stratigraphic distance between the hiatus and the nearest date.

We also tested the sensitivity of

Reproducibility tests indicate that the

Model accuracy does not appear to be particularly sensitive to the number or stratigraphic position of dates as the true age–depth model fell within the 95 % credible interval of the

Example age–depth models of the synthetic TD1 and CIP2 test datasets with randomly placed dates shown as colored Gaussian distributions. Interior tick marks on the vertical axis of each panel indicate the layer boundary positions (see also the horizontal dashed lines in Fig. 2c and f). The dates were randomly generated from the true age–depth model (dashed red line). The black line and shaded gray region are the

Hiatus duration versus the stratigraphic distance between the hiatus and the nearest radioisotope date for the CIP2 dataset. The points are the model median, and the error bars are the 95 % credible interval. The red line is the true hiatus duration of 0.203 Myr.

The number of radioisotopic dates appears to have the strongest effect on overall model uncertainty (see also Blaauw et al., 2018). As the number of dates increases, the width of the 95 % credible interval shrinks and approaches the input uncertainty of the radioisotopic dates (Fig. 3). Crucially however, the uncertainties never “balloon” (e.g., compare

Clearly, our choice of a simple sedimentation model for Bayesian inversion influences age–depth model construction. Since Eq. (2) is calculated layer by layer, a limitation of our model is that each layer must contain enough time and astrochronologic data to resolve the astronomical frequencies (

A potential criticism of our approach is that our choice of a simple Bayesian sedimentation model artificially reduces overall model uncertainties. Since we do not allow sedimentation rate to vary randomly at all points throughout the stratigraphy, our model avoids the inflated (“ballooning”) credible intervals that characterize dates-only age–depth models (i.e.,

The ability to estimate hiatus durations is a significant strength of the

However, it should be noted that there are two potential weaknesses of this approach to estimating hiatus duration. First, since hiatus positions are user-defined, the stratigraphic position of a hiatus must be known a priori and must be informed by geologic (i.e., a visible unconformity) or cyclostratigraphic data (Meyers and Sageman, 2004). In both the CIP2 testing dataset and the Bridge Creek Limestone Member case study (discussed below), the stratigraphic positions of the hiatuses were known in advance. The second weakness is that

Because

For a simple example of an inappropriate use of

Results of

The Bridge Creek Limestone Member is the uppermost member of the Greenhorn Formation of central Colorado. It is primarily composed of hemipelagic marlstone and limestone couplets that extend laterally for over 1000 km in the Western Interior Basin (Elder et al., 1994). These couplets are characterized by alternations from darker organic-carbon-rich laminated clay and mudstones to lighter carbonate-rich, organic-carbon-poor limestone facies. Previous work has reported Milankovitch-scale cyclicity in the Bridge Creek Limestone Member through the application of statistical astrochronologic testing methods (Sageman et al., 1997, 1998; Meyers et al., 2001, 2012, 2008). Using U–Pb and

We used

Astronomical target periods used for the Bridge Creek Limestone Member

Radioisotopic dates used as model inputs for the two Bridge Creek Limestone Member age–depth models shown in Fig. 6.

Results of

Results for both the

Finally, we calculated the age of the Cenomanian–Turonian boundary using both age depth models. The

The

Modeled

Radioisotopic geochronology and astrochronology underlie the development of age–depth models that translate stratigraphic position to numerical time. In turn, these models are crucial to the evaluation of climate proxy records and the development of the geologic timescale. Existing Bayesian methods for age–depth modeling generally rely only on radioisotopic dates and as a consequence do not explicitly incorporate astronomical constraints on the passage of time. However, astrochronology is a rich source of chronologic information and its explicit inclusion in the calculation of age–depth models can substantially improve model accuracy and precision. Here we have presented a new joint Bayesian inversion approach for radioisotopic and astronomical data,

The

RBT and MDS conceived the project and developed the modeling framework with input from SRM. RBT wrote the code for the

The contact author has declared that none of the authors has any competing interests.

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.

We thank Matthias Sinnesael for providing and Christian Zeeden for developing the Cyclostratigraphy Intercomparison Project CIP2 data used for model testing. We also thank Jacob Anderson and Alberto Malinverno for insightful discussions during the development of this project. Finally, we thank Maarten Blaauw, David De Vleeschouwer, Niklas Hohmann, and Matthias Sinnesael for their comments during the open review and discussion of this paper. This work was supported by National Science Foundation grants EAR-1813088 (MDS) and EAR-1813278 (SRM).

This research has been supported by the Directorate for Geosciences (grant nos. EAR-1813088 and EAR-1813278).

This paper was edited by Michael Dietze and reviewed by David De Vleeschouwer and Maarten Blaauw.