Proterozoic Milankovitch cycles and the history of the solar system Stephen R. Meyers a,1 and Alberto Malinverno b a Department of Geoscience, University of Wisconsin–Madison, Madison, WI 53706; and b Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY 10964-1000 Edited by Paul E. Olsen, Columbia University, Palisades, NY, and approved March 30, 2018 (received for review October 9, 2017) The geologic record of Milankovitch climate cycles provides a rich conceptual and temporal framework for evaluating Earth system evolution, bestowing a sharp lens through which to view our planet’s history. However, the utility of these cycles for constrain- ing the early Earth system is hindered by seemingly insurmount- able uncertainties in our knowledge of solar system behavior (including Earth–Moon history), and poor temporal control for val- idation of cycle periods (e.g., from radioisotopic dates). Here we address these problems using a Bayesian inversion approach to quantitatively link astronomical theory with geologic observation, allowing a reconstruction of Proterozoic astronomical cycles, fun- damental frequencies of the solar system, the precession constant, and the underlying geologic timescale, directly from stratigraphic data. Application of the approach to 1.4-billion-year-old rhythmi- tes indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ), an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length of day of 18.68 ± 0.25 hours (2σ), with dominant climatic precession cycles of ∼14 ky and eccentricity cycles of ∼131 ky. The results confirm reduced tidal dissipation in the Proterozoic. A complemen- tary analysis of Eocene rhythmites (∼55 Ma) illustrates how the approach offers a means to map out ancient solar system behavior and Earth–Moon history using the geologic archive. The method also provides robust quantitative uncertainties on the eccentricity and climatic precession periods, and derived astronomical time- scales. As a consequence, the temporal resolution of ancient Earth system processes is enhanced, and our knowledge of early solar system dynamics is greatly improved. Milankovitch cycles | astrochronology | Bayesian inversion | Earth–Moon history | fundamental frequencies Q uasiperiodic variations in insolation, known as Milankovitch cycles, serve as a primary control on climate change over timescales of 10 4 –10 6 y (1). Their expression in the stratigraphic record provides a powerful tool for reconstructing geologic timescales, or astrochronologies, and evaluating Earth history. Extending this astronomical metronome into the Precambrian, however, has proven challenging due to shortcomings in both theory and geologic data. From the perspective of the geologic archive, a major limitation is the lack of sufficient independent time control (e.g., radioisotopic dates) to unambiguously cali- brate the observed spatial rhythms to astronomical (temporal) periods. In terms of theory, the periods of Earth’s astronomical cycles also become more poorly constrained during the Pre- cambrian due to uncertainties in the evolution of the solar sys- tem (2). Although it is established that the dominant eccentricity and climatic precession cycles derive from fundamental fre- quencies associated with the orbits of the five innermost planets (g 1 to g 5 ; ref. 2) and the precession constant k, these values are not precisely determined because of the chaotic nature of the solar system (2, 3) and because the history of tidal dissipation of the Earth–Moon system is not well known (2, 4). In fact, the validity of theoretical astronomical solutions that underpin astrochronology are limited to the past 50 My (2, 5), although “floating” astrochronologies have been proposed for older intervals, and the 405-ky-long orbital eccentricity cycle is expected to be relatively stable with an uncertainty of 0.2% by 250 Ma (2). Recent advances in astrochronologic assessment yield a partial solution to the challenges noted above (6–8), in providing sta- tistical approaches that explicitly consider and evaluate timescale uncertainty in terms of the accumulation rate of a given sedi- mentary record. However, these methods require assumptions about the astronomical frequencies associated with the Earth’s orbital eccentricity, axial tilt, and climatic precession (the Milankovitch cycles). In the present study, we build upon prior work to formulate a Bayesian inversion approach that quanti- tatively links astronomical theory with geologic observation, thus overcoming limitations associated with each. At the core of this approach are three components: (i ) the TimeOpt method (8), which explicitly considers timescale uncertainty, and utilizes multiple attributes of the astronomical signal to increase statis- tical reliability; (ii ) the underlying astronomical theory, which links observed climatic precession and orbital eccentricity rhythms to fundamental frequencies of the solar system and Earth–Moon evolution (2, 4) (Table 1); and (iii ) a Bayesian Markov Chain Monte Carlo approach that allows explicit ex- ploration of the data and model space and uncertainties. The result is a robust methodology for astrochronology that is suit- able for the Proterozoic, and greatly enhances the astronomical knowledge that we can obtain from younger strata (e.g., the early Significance Periodic variations in Earth’s orbit and rotation axis occur over tens of thousands of years, producing rhythmic climate changes known as Milankovitch cycles. The geologic record of these climate cycles is a powerful tool for reconstructing geo- logic time, for understanding ancient climate change, and for evaluating the history of our solar system, but their reliability dramatically decreases beyond 50 Ma. Here, we extend the analysis of Milankovitch cycles into the deepest stretches of Earth history, billions of years ago, while simultaneously reconstructing the history of solar system characteristics, in- cluding the distance between the Earth and Moon. Our results improve the temporal resolution of ancient Earth processes and enhance our knowledge of the solar system in deep time. Author contributions: S.R.M. initiated the project; S.R.M. and A.M. designed research; S.R.M. and A.M. performed research; S.R.M. and A.M. contributed new analytic tools; S.R.M. and A.M. analyzed data; and S.R.M. and A.M. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. Published under the PNAS license. Data deposition: The function “timeOptMCMC” has been deposited in the Comprehen- sive R Archive Network (CRAN) repository (https://cran.r-project.org), as a component of the package “astrochron.” 1 To whom correspondence should be addressed. Email: [email protected]. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1717689115/-/DCSupplemental. Published online June 4, 2018. www.pnas.org/cgi/doi/10.1073/pnas.1717689115 PNAS | June 19, 2018 | vol. 115 | no. 25 | 6363–6368 EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Downloaded by guest on August 1, 2020
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Proterozoic Milankovitch cycles and the history of thesolar systemStephen R. Meyersa,1 and Alberto Malinvernob
aDepartment of Geoscience, University of Wisconsin–Madison, Madison, WI 53706; and bLamont-Doherty Earth Observatory, Columbia University,Palisades, NY 10964-1000
Edited by Paul E. Olsen, Columbia University, Palisades, NY, and approved March 30, 2018 (received for review October 9, 2017)
The geologic record of Milankovitch climate cycles provides a richconceptual and temporal framework for evaluating Earth systemevolution, bestowing a sharp lens through which to view ourplanet’s history. However, the utility of these cycles for constrain-ing the early Earth system is hindered by seemingly insurmount-able uncertainties in our knowledge of solar system behavior(including Earth–Moon history), and poor temporal control for val-idation of cycle periods (e.g., from radioisotopic dates). Here weaddress these problems using a Bayesian inversion approach toquantitatively link astronomical theory with geologic observation,allowing a reconstruction of Proterozoic astronomical cycles, fun-damental frequencies of the solar system, the precession constant,and the underlying geologic timescale, directly from stratigraphicdata. Application of the approach to 1.4-billion-year-old rhythmi-tes indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ),an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length ofday of 18.68 ± 0.25 hours (2σ), with dominant climatic precessioncycles of ∼14 ky and eccentricity cycles of ∼131 ky. The resultsconfirm reduced tidal dissipation in the Proterozoic. A complemen-tary analysis of Eocene rhythmites (∼55 Ma) illustrates how theapproach offers a means to map out ancient solar system behaviorand Earth–Moon history using the geologic archive. The methodalso provides robust quantitative uncertainties on the eccentricityand climatic precession periods, and derived astronomical time-scales. As a consequence, the temporal resolution of ancient Earthsystem processes is enhanced, and our knowledge of early solarsystem dynamics is greatly improved.
Quasiperiodic variations in insolation, known as Milankovitchcycles, serve as a primary control on climate change over
timescales of 104–106 y (1). Their expression in the stratigraphicrecord provides a powerful tool for reconstructing geologictimescales, or astrochronologies, and evaluating Earth history.Extending this astronomical metronome into the Precambrian,however, has proven challenging due to shortcomings in boththeory and geologic data. From the perspective of the geologicarchive, a major limitation is the lack of sufficient independenttime control (e.g., radioisotopic dates) to unambiguously cali-brate the observed spatial rhythms to astronomical (temporal)periods. In terms of theory, the periods of Earth’s astronomicalcycles also become more poorly constrained during the Pre-cambrian due to uncertainties in the evolution of the solar sys-tem (2). Although it is established that the dominant eccentricityand climatic precession cycles derive from fundamental fre-quencies associated with the orbits of the five innermost planets(g1 to g5; ref. 2) and the precession constant k, these values arenot precisely determined because of the chaotic nature of thesolar system (2, 3) and because the history of tidal dissipation ofthe Earth–Moon system is not well known (2, 4). In fact, thevalidity of theoretical astronomical solutions that underpinastrochronology are limited to the past 50 My (2, 5), although“floating” astrochronologies have been proposed for older
intervals, and the 405-ky-long orbital eccentricity cycle isexpected to be relatively stable with an uncertainty of 0.2% by250 Ma (2).Recent advances in astrochronologic assessment yield a partial
solution to the challenges noted above (6–8), in providing sta-tistical approaches that explicitly consider and evaluate timescaleuncertainty in terms of the accumulation rate of a given sedi-mentary record. However, these methods require assumptionsabout the astronomical frequencies associated with the Earth’sorbital eccentricity, axial tilt, and climatic precession (theMilankovitch cycles). In the present study, we build upon priorwork to formulate a Bayesian inversion approach that quanti-tatively links astronomical theory with geologic observation, thusovercoming limitations associated with each. At the core of thisapproach are three components: (i) the TimeOpt method (8),which explicitly considers timescale uncertainty, and utilizesmultiple attributes of the astronomical signal to increase statis-tical reliability; (ii) the underlying astronomical theory, whichlinks observed climatic precession and orbital eccentricityrhythms to fundamental frequencies of the solar system andEarth–Moon evolution (2, 4) (Table 1); and (iii) a BayesianMarkov Chain Monte Carlo approach that allows explicit ex-ploration of the data and model space and uncertainties. Theresult is a robust methodology for astrochronology that is suit-able for the Proterozoic, and greatly enhances the astronomicalknowledge that we can obtain from younger strata (e.g., the early
Significance
Periodic variations in Earth’s orbit and rotation axis occur overtens of thousands of years, producing rhythmic climatechanges known as Milankovitch cycles. The geologic record ofthese climate cycles is a powerful tool for reconstructing geo-logic time, for understanding ancient climate change, and forevaluating the history of our solar system, but their reliabilitydramatically decreases beyond 50 Ma. Here, we extend theanalysis of Milankovitch cycles into the deepest stretches ofEarth history, billions of years ago, while simultaneouslyreconstructing the history of solar system characteristics, in-cluding the distance between the Earth and Moon. Our resultsimprove the temporal resolution of ancient Earth processesand enhance our knowledge of the solar system in deep time.
Author contributions: S.R.M. initiated the project; S.R.M. and A.M. designed research;S.R.M. and A.M. performed research; S.R.M. and A.M. contributed new analytic tools;S.R.M. and A.M. analyzed data; and S.R.M. and A.M. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Published under the PNAS license.
Data deposition: The function “timeOptMCMC” has been deposited in the Comprehen-sive R Archive Network (CRAN) repository (https://cran.r-project.org), as a component ofthe package “astrochron.”1To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1717689115/-/DCSupplemental.
Cenozoic). We refer to this approach as TimeOptMCMC.We emphasize that although TimeOptMCMC provides arigorous quantification of the uncertainties in astrochronologicresults, the method does not by itself reduce these uncer-tainties. Ultimately, uncertainties in astrochronology can only bedecreased by additional information provided by measureddata.We apply the TimeOptMCMC method to evaluate two
cyclostratigraphic records that are of special importance. Thefirst is the 1.4-billion-year-old Xiamaling Formation from theNorth China Craton (9), one of the oldest proposed records ofastronomical forcing (Fig. 1A). The second is the well-studied∼55-million-year-old record from Walvis Ridge (ref. 10 and Fig.1E), which is notable because it includes the Paleocene–EoceneThermal Maximum, and it just exceeds the temporal limits of theavailable theoretical astronomical solutions [<50 Ma (2, 5)]. Themethodology allows us to address two primary research objec-tives: (i) to provide well-constrained geologic estimates of theclimatic precession and eccentricity periods for both the earlyEocene and Proterozoic (including uncertainties); and (ii) toquantify length of day and Earth–Moon distance during theProterozoic (via the precession constant k), at a time when ex-trapolation of the present-day rate of tidal dissipation wouldimply a condition near Earth–Moon collision (11).
ResultsWe focus our analysis of the Xiamaling Formation on a 2-m-thick section of rhythmically bedded black shale and chert (“unit3” of ref. 9) that has been interpreted to reflect changes in up-welling and biological productivity. Paleogeographic recon-structions place this site in a subtropical/tropical marineenvironment that was under Hadley cell influence, suggesting anastronomical forcing scenario involving migration of the in-tertropical convergence zone (9). We investigate the publishedCu/Al record (9), a proxy for productivity/redox state (12), whichdemonstrates high fidelity (SI Appendix, Fig. S6 and SI Appen-dix). Initial screening of the high-resolution dataset using theTimeOpt method with tentative (nominal) Proterozoic values forthe climatic precession and eccentricity periods (2, 4) (SI Ap-pendix, Table S2) reveals a highly significant astronomical signal(r2opt = 0.300; P < 0.005, 2,000 simulations; SI Appendix, Fig. S6)at a sedimentation rate of 0.33 cm/ky. This sedimentation rate isconsistent with radioisotopic data in an overlying 52-m-thickinterval (SI Appendix). The statistically significant TimeOpt re-sult is an important finding, as it overcomes the problem of falsesignal detection that complicates spectrum evaluation (13, 14)and provides an independent confirmation of the astronomicalinterpretation of Zhang et al. (9).
Bayesian inversion of the Xiamaling Cu/Al record is con-strained by prior distributions for the fundamental frequencies g1to g5, the precession constant k, and sedimentation rate (SIAppendix, Tables S3 and S4). Prior distributions for the funda-mental frequencies g1 to g5 are based on the full range of vari-ability in the model simulations of Laskar et al. (2) computedover 500 My. The prior distribution for the precession constant isderived from the recent study by Waltham (ref. 4; 78 ± 28 arcsec/y,2σ), and sedimentation rate is permitted to vary across allplausible values for which it is possible to robustly identify a fullastronomical signal, given the available data resolution. Theposterior distribution from the TimeOptMCMC analysis indi-cates a precession constant of 85.79 ± 2.72 arcsec/y (2σ; Fig. 2B),consistent with an Earth–Moon distance of 340,900 ± 2,600 km(2σ) and length of day of 18.68 ± 0.25 h (2σ; Fig. 2C and Table2). Climatic precession periods range between 12.5 and 14.4 ky(Fig. 2F and Table 2), with a dominant cycle of ∼14 ky in thestudy interval (Fig. 1D). The Proterozoic analog of the long ec-centricity cycle, which has a duration of 405 ky in theoreticalmodels for the Cenozoic (2), and is expected to be the mostregular of the eccentricity cycles because it involves interactionbetween the very stable Jupiter and relatively stable Venus, has aduration of 405.1 ky (401.3–408.9 ky, 2σ; Fig. 2D). Finally, thereconstructed Proterozoic short eccentricity periods (Fig. 2E) areconsistent with those observed in the theoretical models for theCenozoic (2) (95–131 ky), with a dominant period of ∼131.4 kyin the study interval (Fig. 1D). It is notable that the posteriordistributions for sedimentation rate (Fig. 2A) and the precessionconstant (Fig. 2B) are much narrower than their prior distribu-tions, and the prior and posterior distributions of the funda-mental frequencies g1 to g5 are nearly identical (SI Appendix, Fig.S7). Notwithstanding little improvement in the posterior uncer-tainties of the fundamental frequencies, the coupled nature ofthe eccentricity and climatic precession cycles, which sharecommon g terms (the climatic precession terms also share acommon k term) allows resolution of the Proterozoic Milanko-vitch periods with low uncertainty (Fig. 2 D–F and Table 2).To provide a baseline assessment from the early Cenozoic, we
investigate Eocene reflectivity data (a*, red/green) from theWalvis Ridge (10) (Ocean Drilling Program Site 1262; Fig.1E). This dataset has been previously evaluated with theTimeOpt approach (8), and a statistically significant astro-nomical signal (P < 0.005) is identified at a sedimentation rateof 1.33 cm/ky (SI Appendix, Fig. S9). Application of theTimeOptMCMC algorithm allows a rigorous assessment of theuncertainty in the Eocene Milankovitch periods, in the un-derlying g and k terms, and in the sedimentation rate (Fig. 2,Table 2, and SI Appendix, Figs. S10 and S11). In this case, theposterior distributions for sedimentation rate (Fig. 2G) andfor the fundamental frequencies g3 (Earth) and g4 (Mars) showthe greatest change relative to their prior distributions (SIAppendix, Fig. S10 E and G). Most notably, the posterior meanvalue for g4 (Mars) is greater than the maximum value ob-served in the modeling study of Laskar et al. (2). This dis-crepancy in g4 is also expressed in the e2 (91.98 ky) and e3(118.95 ky) eccentricity terms at Walvis Ridge (Fig. 2K and SIAppendix, Fig. S11 C and E), both of which share the g4 termand are notably shorter than those observed in the astro-nomical model simulations of Laskar et al. (2) (Table 2). Apossible explanation for these differences in g3 and g4 is thatthe frequencies reported in figure 9 of Laskar et al. (2) areaveraged over 20-My intervals, whereas the Walvis Ridge re-cord spans a shorter interval of about 1.7 My.
DiscussionComparison of the Proterozoic and Eocene results highlightshow the TimeOptMCMC approach combines cyclostratigraphicdata and astronomical theory to improve model parameters. In
Table 1. Source of the climatic precession and eccentricityperiods, as derived from the fundamental frequencies andprecession constant
Parameter Source Period (ky)—Today*
p1 k + g5 23.678p2 k + g2 22.371p3 k + g4 18.951p4 k + g3 19.103p5 k + g1 23.120e1 g2 − g5 405.091e2 g4 − g5 94.932e3 g4 − g2 123.945e4 g3 − g5 98.857e5 g3 − g2 130.781
*Precession and eccentricity estimates from ref. 2.
6364 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno
the case of the Proterozoic, where the deviation of k from itspresent value is expected to be substantial due to the large un-certainties in the Earth–Moon history, the Xiamaling Cu/Al dataprovides strong constraints to improve our knowledge of theprecession constant. In the case of the Eocene, the expectedchanges in k are much smaller, and the cyclostratigraphic datamore strongly improve our knowledge of the fundamental fre-quencies g3 and g4. For both the Proterozoic and Eocene ex-amples, the sedimentation rate (and hence the duration of thestratigraphic interval) is highly constrained by the Bayesianinversion.Although stratigraphic-based estimates of the fundamental
frequencies of the solar system are rare [g1 to g5; however, seeOlsen and Kent (15)], numerous studies have attempted to
reconstruct the precession constant and/or the Earth–Moondistance and length of day using geologic data. These approachesinclude inferences from the evaluation of tidal deposits and ofgrowth patterns in marine invertebrate fossils and stromatolitesthroughout the past 2.5 billion years (16), and for the lateCenozoic (<25 Ma), the application of astronomical-basedmethods (1, 17, 18). In addition, a number of theoretical mod-eling exercises have been conducted to constrain the Earth–Moon history (2, 4, 16, 19, 20). In Fig. 3, we compare ourastronomical-based results for the Proterozoic and Eocene to anumber of Earth–Moon separation models, and also to two tida-lite datasets that are considered to be of high quality (16):rhythmites from the Big Cottonwood Formation (∼900 Ma; refs.21 and 22), and the Elatina Formation and Reynell Siltstone
Frequency (cycles/ky)0.00 0.02 0.04 0.06 0.08 0.10
-1
Fig. 1. TimeOptMCMC results for the ∼1.4 Ga Proterozoic Xiamaling Formation Cu/Al data (9) and the ∼55 Ma Eocene Walvis Ridge a* (red/green) data (10).(A) Xiamaling Cu/Al data versus stratigraphic height. (B) Astronomically tuned Xiamaling Cu/Al data, using the TimeOptMCMC derived posterior meansedimentation rate (Table 2). The data series has been standardized to unit variance, and a linear trend has been removed. (C) Xiamaling Cu/Al data pre-cession envelope (red line) for the posterior mean sedimentation rate, and precession filter output (blue line). The black line illustrates the TimeOptreconstructed eccentricity model. (D) Xiamaling Cu/Al data power spectrum (squared Fourier transform) using the posterior mean sedimentation rate. Thevertical dashed red lines indicate the reconstructed target periods (mean posterior values in Table 2) for climatic precession and eccentricity, and theblue line illustrates the frequency response of the bandpass filter for precession modulation evaluation. (E ) Eocene Walvis Ridge a* data versus meterscomposite depth (mcd). (F ) Astronomically tunedWalvis Ridge a* data, using the TimeOptMCMC derived posterior mean sedimentation rate (Table 2). The dataseries has been standardized to unit variance, and a linear trend has been removed. (G) Walvis Ridge a* data precession envelope (red line) for the posterior meansedimentation rate, and precession filter output (blue line). The black line illustrates the TimeOpt reconstructed eccentricity model. (H) Walvis Ridge a* datapower spectrum (squared Fourier Transform) using the posterior mean sedimentation rate. The vertical dashed red lines indicate the reconstructedtarget periods for climatic precession and eccentricity (Table 2), and the blue line illustrates the frequency response of the bandpass filter for precessionmodulation evaluation.
Meyers and Malinverno PNAS | June 19, 2018 | vol. 115 | no. 25 | 6365
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(∼620 Ma; refs. 16 and 23). It should be noted that the in-terpretation of tidalite (and also bioarchive) datasets in terms ofEarth–Moon history remains a contentious issue due to problemswith cycle recognition and the potential for missing laminations (4,16), thus the TimeOptMCMC approach provides an independentmeans for their validation. To supplement our comparison, SIAppendix, Fig. S12 includes some additional more contro-versial estimates from Phanerozoic bioarchives, and a datumfrom the Weeli Wooli Formation rhythmite (∼2,450 Ma; refs. 11,16, and 24).The TimeOptMCMC reconstructed Earth–Moon distance
from the Xiamaling Formation (340,900 ± 2,600 km, 2σ; Table2) is consistent with that derived from ocean models (20) thatimply smaller torques, reduced tidal dissipation, and slowerlunar retreat rates in the distant past, ultimately related to aless efficient excitation of the ocean’s normal modes by tidalforcing on an Earth with a faster rotation rate (Fig. 3). TheXiamaling result is also compatible with a model that employsan average tidal dissipation rate that is 60% of the presentvalue (SI Appendix, Fig. S12). If the Elatina and Cottonwood
tidalite data are taken at face value, either of these Earth–Moon separation models are possible, depending on thetidalite record considered. However, the small uncertainty ofthe Xiamaling estimate excludes a range of other potentialmodels that are permitted by the tidalite data, such as one thatemploys a tidal dissipation rate that is 40% of the presentvalue, and furthermore, the 60% model is inconsistent withestimates from the Weeli Wooli Formation (SI Appendix, Fig.S12). It should also be noted that the 40% and 60% ratemodels violate the constraint provided by modern observedMoon retreat rate, in contrast to the ocean model (20) andpresent rate model. Finally, the astronomical-based Bayesianreconstruction is consistent with a length of day (18.68 ±0.25 h, 2σ), which is shorter than that of published Proterozoicestimates from geologic data (16), and is at the low end ofexisting model length of day estimates (4), but has a greatlyreduced uncertainty (Fig. 2C).The methodology presented here is not affected by problems
inherent in previous estimates of the precession constant, asso-ciated with ambiguity in the interpretation of bioarchives and
Sedimentation Rate (cm/ky) Precession Constant (arcsec/y) Length of Day (hours)
Long Eccentricity Period (ky)
g2-g5
Short Eccentricity Period (ky) Precession Period (ky)
g4-g5 g3-g5
g4-g2 g3-g2
k+g4 k+g3 k+g2k+g1
k+g5
Proterozoic Xiamaling Formation
Eocene Walvis RidgeI
L
D E
A B
G H
J K
Fig. 2. Summary of TimeOptMCMC prior and posterior distributions for the ∼1.4 Ga Proterozoic Xiamaling Formation Cu/Al data (9), and the ∼55 Ma EoceneWalvis Ridge Site 1262 a* (red/green) data (10). (A–F) Prior (red line) and posterior (histogram) probability distributions for Xiamaling Formation sedimentationrate, precession constant, length of day, long eccentricity period, short eccentricity periods, and climatic precession periods (C). (G–L) Prior (red line) and posterior(histogram) probability distributions for Eocene Walvis Ridge Site 1262 sedimentation rate, precession constant, length of day, long eccentricity period, shorteccentricity periods, and climatic precession periods (I). See Table 1 for the relationship between g, k, and observed astronomical periods. See Table 2 for the meanposterior values associated with each distribution shown in this figure.
6366 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno
tidal deposits (4, 16). Furthermore, the technique should bewidely applicable, given the abundance of relatively continuousrecords of astronomically forced sedimentation. An importantfeature of this quantitative approach is a comprehensive treat-ment of uncertainties, facilitated by the explicit coupling of as-tronomical theory with geologic observation. The quantificationof prior and posterior distributions allows for a rigorous treat-ment of astrochronologic uncertainties, addressing a majorweakness in prior work and providing an objective way to inte-grate astrochronologies with radioisotopic data, from the Pro-terozoic to the Cenozoic. Application of this methodology tosedimentary records that span Earth history will facilitate im-provement in the calibration of the geologic time scale, willconstrain the history of the Earth–Moon system in deep time,and shows promise of reconstructing the evolution of the fun-damental orbital frequencies of the solar system over billionsof years.
Materials and MethodsWe utilize the recently developed TimeOpt regression framework (8) toevaluate two features that are diagnostic of an astronomical fingerprint instrata: the concentration of spectral power at the proposed astronomicalfrequencies (25) (e.g., climatic precession and eccentricity), and the am-plitude modulation of climatic precession (26), which is caused by varia-tions in eccentricity. For the Bayesian inversion, TimeOpt is reformulatedin terms of likelihood functions (27) (SI Appendix), and Markov ChainMonte Carlo (MCMC) simulation is utilized to sample values of the solarsystem secular frequencies g1 to g5, precession constant k, and sedimen-tation rate that are physically plausible and agree with the stratigraphicdata. This allows evaluation of the five dominant eccentricity and fivedominant climatic precession cycles that are observable in sedimentarystrata, which depend on sums or differences of the g terms and/or k (Table1). For example, the 405-ky-long eccentricity term of the Cenozoic origi-nates from the difference between g2 (Venus) and g5 (Jupiter), and one ofthe strongest climatic precession cycles (23.7 ky in the Cenozoic) corre-sponds to g5 + k.
Table 2. TimeOptMCMC reconstructed sedimentation rate, precession and eccentricity frequencies, fundamentalfrequencies (g terms), precession constant (k), and Earth–Moon distance, for the Proterozoic Xiamaling Formationand the Eocene Walvis Ridge
*Precession, eccentricity, g, and k estimates from ref. 2.†Results from the Xiamaling Formation are based on 50 MCMC simulation chains of length 1 × 106 each.‡Results from the Walvis Ridge are based on 150 MCMC simulation chains of length 2 × 105 each.§Semimajor axis.{Sidereal day.
-1500 -1000 -500 0
280
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ElatinaCottonwood
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Fig. 3. TimeOptMCMC reconstructed Earth–Moon distance, compared withtwo tidalite-based estimates and two models. Uncertainties in Earth–Moondistance are ±2σ and age uncertainties span minimum and maximum values.The Bayesian posterior TimeOptMCMC estimates for the ProterozoicXiamaling Formation and Eocene Walvis Ridge are indicated with bluesymbols. Note the significant improvement in precision between posterior(blue) and prior estimates for the Xiamaling data (prior = 319,743 to380,309 km; 2σ). Age uncertainties for the Xiamaling and Walvis Ridge re-sults fall within the size of the blue symbols. Tidalite estimates from the BigCottonwood Formation (∼900 Ma; ref. 22) and Elatina Formation and Rey-nell Siltstone (∼620 Ma; ref. 23) are shown with dark green symbols (basedon the updated analyses of ref. 16). The light green Big Cottonwood For-mation estimate (364,192 km) is an alternative value reported by ref. 21 withuncertainties from the 348,884-km estimate (dark green symbol). The oceanmodel (red line), and a model using the present rate of tidal dissipation(black line), derive from ref. 20.
Meyers and Malinverno PNAS | June 19, 2018 | vol. 115 | no. 25 | 6367
A complete description of the approach, including evaluation and calibrationwith a synthetic astronomical test series, is in the SI Appendix. All analyseswere conducted using the free software R (28), and an implementation ofthe TimeOptMCMC algorithm is available in the Astrochron package (29).
ACKNOWLEDGMENTS. We thank the reviewers and the editor for theirconstructive remarks. This study was supported by NSF Grant EAR-1151438(to S.R.M.), and by a sabbatical leave from the University of Wisconsin—Madison (S.R.M.) to conduct research at Lamont-Doherty Earth Observatory.
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