Highlight-6

Highlight 6: TimeOpt methodologies for cyclostratigraphy

PIs Meyers and Malinverno developed a family of methodologies for cyclostratigraphy to jointly estimate sedimentation rate u, Earth’s axial precession frequency k, and Solar System fundamental frequencies gi and si, together with their uncertainties. The methods are collectively known as “TimeOpt” in reference to their mathematical optimization approaches to determine a timescale for the cyclostratigraphic data. The methods are described below, followed by a review of their application to a well-known cyclostratigraphic dataset from the Early Eocene. The development of TimeOptB and TimeOptBMCMC (Items 3 and 4 below) was supported by Heising-Simons Foundation.

1. TimeOpt (Meyers, 2015) uses a regression model framework to evaluate a stratigraphic series of proxy data, exploring a range of plausible sedimentation rates. The approach seeks to maximize (1) the expected orbital eccentricity modulation of climatic precession, and (2) the concentration of spectral power at the orbital eccentricity and climatic precession frequencies. The optimal u is identified for a set of prescribed orbital eccentricity and climatic precession frequencies. Statistical significance is evaluated with Monte Carlo simulations. Meyers (2019) describes the extension of TimeOpt for obliquity, and the use of templates to evaluate variable sedimentation rates (TimeOptTemplate), as well as a moving window implementation of the method (eTimeOpt).

2.  TimeOptMCMC (Meyers and Malinverno, 2018) is a Bayesian implementation of the TimeOpt approach, which seeks to fully account for uncertainties in gi, k  and u. It applies a Markov Chain Monte Carlo (MCMC) procedure that samples the posterior probability density function (PDF) of u, k, and five gi. The posterior PDF combines a prior PDF of the parameters and a likelihood function to quantify how closely stratigraphic data predicted by the parameters fit observed data. TimeOptMCMC is computationally expensive, requiring days to weeks of simulation for a single dataset.

3.  TimeOptB (Malinverno and Meyers, 2024) is a Bayesian implementation of TimeOpt to calculate posterior PDFs of u and k, while fixing five gi and five si to values determined in numerical solutions of Solar System history. The statistical significance of the fit to the observed data for the estimated u and k is also evaluated with Monte Carlo simulations. Whereas TimeOpt assumed a fixed value for the axial precession frequency k, TimeOptB estimates k and its uncertainty from the cyclostratigraphic data. These independent estimates have been used to derive the history of k from many data sets as far back as 650 Ma (Wu et al. 2024; see Highlight 1).

4. TimeOptBMCMC (Malinverno and Meyers, 2024) is a Bayesian implementation of TimeOpt that samples the posterior PDF of u, k, five gi and five si. An adaptive sampling MCMC strategy no longer requires a preliminary setup (as in TimeOptMCMC), and the procedure is orders of magnitude faster than TimeOptMCMC (typically minutes rather than days).

TimeOptB and TimeOptBMCMC account for the possible presence of obliquity cycles, implement Bayesian priors for gi and si (Hoang et al., 2021) and k (based on Farhat et al., 2022 and Waltham, 2015), and include improvements in calculation of the likelihood of the data. 

Early Eocene color reflectance a* and XRF Fe-count cyclostratigraphic data from ODP Site 1262, Walvis Ridge, South Atlantic Ocean are shown in Fig. 1. All four TimeOpt methods have been used to analyze a* across the PETM-ELMO interval; results are summarized in Table 1. For comparison, Farhat et al. (2022) calculate model k=51.806902±0.025322 arcsec/yr at 54 Ma (LINK).  In Zeebe and Lourens (2022), the best fit astronomical solution for the study interval is ZB18a(1,0.9), for which k=51.60 to 51.63 arcsec/yr. In both cases, modeled k is higher than the estimated mean k values presented in Table 1, Rows B-D. Lastly, Boulila and Hinnov (2022) estimated k=51.546 ± 0.168 arcsec/yr from a 405-kyr-tuned XRF Fe-count time series of the study interval (red curve in Fig. 1), assuming fixed gi and si from the La2004 model (Tables 4 and 5 in Laskar et al., 2004).

Fig. 1. ODP Leg 208, Site 1262 a* and XRF Fe-count cyclostratigraphic series, shown with magnetic reversal stratigraphy (from Figure 2 in Westerhold et al., 2007). The ELMO-PETM study interval for the four TimeOpt methods, Zeebe and Lourens (2022) and Boulila and Hinnov (2022) reviewed here is indicated by the blue bar, 117.4-139.7 mcd (meters composite depth). The stratigraphic positions of base ELMO at 117 mcd and top PETM at 140 mcd, just outside the study interval, have ages of 54.1 Ma and 56.01 ± 0.05 Ma, respectively (Zeebe and Lourens, 2019).  

Table 1. TimeOpt estimates of sedimentation rate u and Earth’s axial precession frequency k for Site 1262 a* data in the study interval (Fig. 1). Row A is from Meyers (2015), Row B is from Meyers and Malinverno (2018), and Rows C and D are from Malinverno and Meyers (2024). † Assumed value of k at 54 Ma in the La2004 model (Equation 40, Laskar et al., 2004).

References:

Boulila, S., Hinnov, L.A., 2022, Constraints on Earth-Moon dynamical parameters from Eocene cyclostratigraphy, Global Planetary Change, 216, 103925, https://doi.org/10.1016/j.gloplacha.2022.103925 

Farhat, M., Auclair‐Desrotour, P., Boué, G., Laskar, J., 2022, The resonant tidal evolution of the Earth‐Moon distance. Astronomy and Astrophysics, 665, L1, https://doi.org/10.1051/0004‐6361/202243445

Hoang, N. H., Mogavero, F., Laskar, J., 2021, Chaotic diffusion of the fundamental frequencies in the Solar System. Astronomy & Astrophysics, 654, A156, https://doi.org/10.1051/0004‐6361/202140989

Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A.C.M., Levrard, B., 2004, A long-term numerical solution for the insolation quantities of the Earth: Astronomy & Astrophysics, 428, https://doi.org/10.1051/0004-6361:20041335

Malinverno, A., Meyers, S.R., 2024, Bayesian Estimation of Past Astronomical Frequencies, Lunar Distance, and Length of Day From Sediment Cycles, Geochemistry, Geophysics, Geosystems, 25, e2023GC011176, https://doi.org/10.1029/2023GC011176

Meyers, S.R., Malinverno, A., 2018. Proterozoic Milankovitch cycles and the history of the solar system. Proceedings of the National Academy of Sciences, 115(25), 6363–6368, https://doi.org/10.1073/pnas.1717689115

Meyers, S.R., 2015. The evaluation of eccentricity-related amplitude modulation and bundling in paleoclimate data: An inverse approach for astrochronologic testing and time scale optimization, Paleoceanography, 30, https://doi.org/10.1002/2015PA002850

Meyers, S.R., 2019. Cyclostratigraphy and the problem of astrochronologic testing, Earth-Science Reviews 190, 190-223, https://doi.org/10.1016/j.earscirev.2018.11.015

Waltham, D., 2015. Milankovitch period uncertainties and their impact on cyclostratigraphy, Journal of Sedimentary Research, 85, 990-998, https://doi.org/10.2110/jsr.2015.66  

Westerhold, T., Röhl, U., Laskar, J., Bowles, J., Raffi, I., Lourens, L.J., Zachos, J.C., 2007. On the duration of magnetochrons C24R and C25n and the timing of early Eocene global warming events: implications from the Ocean Drilling Program Leg 208 Walvis Ridge depth transect. Paleoceanography 22, PA2201, https://doi.org/10.1029/2006PA001322

Zeebe, R.E., Lourens, L.J., 2022, A deep-time dating tool for paleoapplications utilizing obliquity and precession cycles: The role of dynamical ellipticity and tidal dissipation. Paleoceanography and Paleoclimatology, 37, e2021PA004349, https://doi.org/10.1029/2021PA004349

Zeebe, R.E., Lourens, L.J. (2019). Solar system chaos and the Paleocene-Eocene boundary age constrained by geology and astronomy. Science, 365, 926–929, https://doi.org/10.1126/science.aax0612