Earth-Moon dynamics from cyclostratigraphy reveals possible ocean tide resonance in the Mesoproterozoic era

Abstract

Cyclostratigraphy is an important observational window into the history of the Earth-Moon system. However, there is limited information from the Mesoproterozoic era (1.0 to 1.6 billion years ago); accordingly, only weak constraints on Earth-Moon separation and tidal dissipation are available for this time. To close this knowledge gap, we analyze cyclostratigraphy from the Yemahe Formation (~1.2 billion years ago), Wumishan Formation (~1.5 billion years ago), and Chuanlinggou Formation (~1.6 billion years ago) in China. We use a Bayesian inversion method to analyze the three cyclostratigraphic sections. We combine previous results with these three estimates to construct an updated Earth-Moon system evolution and tidal dissipation history after 2.5 billion years ago. The results show a tidal dissipation peak that is consistent with the model predictions within the error range but also that there may be an additional resonance fluctuation in the Mesoproterozoic era.

Inversion calculation of rhythmic strata in the Mesoproterozoic era reveals a possible tidal dissipation resonance.

INTRODUCTION

Because of tidal interactions in the Earth-Moon system, Earth’s rotational motion experiences deceleration, resulting in the transfer of angular momentum to the Moon’s orbital angular momentum (1–3). Consequently, the Moon gradually moves away from Earth (4). Previously, this process has been constrained by records of fossils and tidalites (5–8). Recently, cyclostratigraphy has supplemented these records by providing information on Earth’s precession frequency, denoted as k, which allows for independent evaluation of the evolution of Earth-Moon distance and Earth length of day (9–22). Compared to the fossil records and tidalite records, Milankovitch cycles recorded in cyclostratigraphy offer a much higher and potentially continuous distribution throughout geological history (23). Thus, cyclostratigraphy is well suited for establishing the history of Earth-Moon evolution (24–26).

Recently, the theoretical model F2022 has provided insights into the tidal resonance evolution and lunar separation of the Earth-Moon system since ~4.4 billion years ago (Ga) (24). The F2022 model, characterized solely by parameterized conditions and not accounting for geological constraints, exhibits a remarkable level of alignment with cyclostratigraphy (9, 18). This alignment validates the efficacy of cyclostratigraphic constraints for understanding the Earth-Moon system and further suggests that cyclostratigraphy holds promise as a standardized observational framework for investigating the dynamics of the greater solar system. The F2022 model predicts two ocean tidal resonances at ~310 and ~540 million years ago (Ma), which is supported by geological evidence (18). However, for the Mesoproterozoic era (~1.0 to 1.6 Ga), the existence of only one reference point from the Xiamaling Formation (~1.4 Ga) limits our understanding of the evolution of the Earth-Moon system during this time (9, 27).

In this study, we obtained proxy records from the Yemahe (1215 Ma) and Chuanlinggou (1634 Ma) formations, and a pre-existing proxy record from the Wumishan Formation (1480 Ma) (28) in China (figs. S1 and S2). Cyclostratigraphic analysis demonstrates that these proxies were controlled by astronomical cycles. Through Bayesian inversion using the Time Scale Optimization Markov Chain Monte Carlo (TimeOptMCMC) method (9), we obtained posterior estimates of the precession frequency and the secular frequencies of the orbital perihelia of five planets (g₁ to g₅, Mercury to Jupiter). These results, when combined with calculations of Earth-Moon distance and length of day, provide crucial constraints for understanding the Mesoproterozoic evolution of Earth-Moon separation and tidal dissipation. We further integrated all Bayesian inversion evidence to date from cyclostratigraphy, synthesizing an updated evolution curve for Earth-Moon distance, length of day, and Earth’s obliquity since ~2.5 Ga using stochastic monotone interpolation. This leads to the recognition of a tidal dissipation peak from ~1.3 to 1.5 Ga, indicating a possible unfound resonance in the Mesoproterozoic era.

RESULTS

Cyclostratigraphy

The stratigraphic depth domain 2π MTM (multitaper method) power spectra of the three sections (details in Materials and Methods) reveal a number of statistically significant cycles. The thorium-uranium ratio (Th/U) sequence of the Yemahe Formation (Fig. 1A) reveals potential astronomical cycles, including a 6.5-m cycle, 2.6-m cycle, 1.6-m cycle, 0.34-m cycle, and 0.2- to 0.24-m cycles (fig. S3A). The magnetic susceptibility (MS) sequence of the Wumishan Formation (Fig. 1B) reveals a 30-m cycle, 7.8-m cycle, 2.1-m cycle, and a 1.3-m cycle (fig. S4A). The thorium (Th) sequence of the Chuanlinggou Formation (Fig. 1C) reveals a 21.3-m cycle, 8.7-m cycle, 4.9-m cycle, a 2.9- to 1.6-m cycle and a 0.53- to 0.47- m cycle (fig. S5A).

Fig. 1. Cyclostratigraphic datasets of the Yemahe, Wumishan, and Chuanlinggou formations.

(A) Yemahe Formation: 52-m section with a U-Pb isotope age of 1215.8 ± 2.4 Ma at 46.7 m (58). The depth increases sequentially from stratigraphic Lower to Upper, the same below. The gray line represents 52-m Th/U sequence data, with the red line showing a LOWESS trend line, and the blue line indicates data after long (405 kyr) orbital eccentricity filtering. The horizontal black line shows the segment used in TimeOpt and TimeOptMCMC (34 to 50 m). (B) Wumishan Formation: 144.25-m section. The gray line represents MS sequence data, the red line shows a LOWESS trend line, and the blue line indicates data after short (100 kyr) orbital eccentricity filtering. The horizontal black line shows the segment used in TimeOpt and TimeOptMCMC (13 to 63 m). (C) Chuanlinggou Formation: 68-m section with an isotope age of 1634.8 ± 6.9 Ma at 58.7 m (59). The gray line represents Th sequence data, the red line shows a LOWESS trend line, and the blue line indicates data after short eccentricity filtering. The horizontal black line shows the segment used in TimeOpt and TimeOptMCMC (16 to 45 m).

Considering the most likely sedimentation rates corresponding to lithology, for the Yemahe Formation we assign the prominent 6.5-m cycle with a period of 405 thousand years (kyr; the long orbital eccentricity cycle). For the Wumishan Formation, the 100-kyr (short orbital eccentricity) cycle is assigned to the 7.8-m cycle and in the Chuanlinggou Formation to the 4.9-m cycle. On the basis of these assignments, we filtered and tuned the original stratigraphic depth sequences to the time domain (details in Materials and Methods). The time-domain power spectrum of the Yemahe Formation shows 405-kyr long orbital eccentricity cycles (the tuning target), 109-kyr short orbital eccentricity cycles, obliquity cycles with periods ranging from 25 to 27 kyr, and climatic precession cycles with periods ranging from 12 to 16 kyr (fig. S3B). For the Wumishan Formation: 416-kyr long orbital eccentricity cycles, 100-kyr short orbital eccentricity cycles (the tuning target), obliquity cycles with periods of about 18 kyr, and climatic precession cycles with periods ranging from 12 to 15 kyr (fig. S4B). In addition for the Chuanlinggou Formation: a 412-kyr long orbital eccentricity cycle, a 100-kyr short orbital eccentricity cycle (the tuning target), a 17-kyr obliquity cycle, and 11- to 12-kyr climatic precession cycles (fig. S5B). In addition, there is a 187-kyr cycle in the Chuanlinggou Formation (fig. S5B), possibly corresponding to the amplitude modulation cycle of obliquity (s₃-s₆) (29–31).

The ratios among the periodicities in these sections roughly align with prior estimates for the astronomical cycle periods predicted for the Mesoproterozoic era (table S1) (24, 32). We applied evolutive harmonic analysis (EHA) to identify the portions of the sections that have the most stable frequency structure in the stratigraphic depth domain, with strong climatic precession and orbital eccentricity cycles, to enable effective Time Scale Optimization [TimeOpt (33) and TimeOptMCMC analyses] (fig. S6). Accordingly, TimeOpt analysis was applied to the Yemahe Formation between 34 and 50 m, the Wumishan Formation between 13 and 63 m, and the Chuanlinggou Formation between 16 and 45 m. The results indicate an optimal sedimentation rate of 1.79 cm/kyr with a null hypothesis confidence (P value) of 0.02 in the Yemahe Formation (fig. S7), an optimal sedimentation rate of 7.87 cm/kyr, with a P value of 0.005 for the Wumishan Formation (fig. S8) and 4.91 cm/kyr with a P value of 0.005 in the Chuanlinggou Formation (fig. S9). The climatic precession envelope and modeled orbital eccentricity curves have a strong fit in all three sections, which together with the low P values indicates the likely preservation of astronomical signals and provides support for the TimeOptMCMC inversion in the next step (9).

Bayesian inversion

After running 50 chains of 600,000 Markov chain Monte Carlo (MCMC) iterations on each section, and thoroughly exploring the parameter space using test programs, we removed some chains that were trapped in low-likelihood regions (see Materials and Methods). Subsequently, we merged the burn-in results of selected chains to generate the final posterior distribution.

TimeOptMCMC inversion estimates sedimentation rates and precession rate k for the three formations as follows: Yemahe Formation with a sedimentation rate at 1.96 cm/kyr and mean precession rate k of 84.15 ± 1.74 (1σ) arc sec per year (″/year); the Wumishan Formation with a sedimentation rate of 8.54 cm/kyr and mean precession rate k of 92.62 ± 2.39 (1σ) ″/year; and the Chuanlinggou Formation with a sedimentation rate of 5.24 cm/kyr and mean precession rate k of 96.40 ± 1.94 (1σ) ″/year (Fig. 2). The posterior burn-in probabilities have converged to narrower ranges compared to their prior probabilities, indicating that the cyclostratigraphic observations successfully constrained the estimates. The sedimentation rates are in reasonable agreement with the original estimates (table S1). Other astronomical parameters and results are shown in figs. S10 to S15 and table S2.

Fig. 2. Bayesian inversion: Basic information and histograms of the prior and posterior distributions for sedimentation rate and precession rate k.

Prior, red; posterior, blue. Yemahe Formation: (A) sedimentation rate and (B) precession rate k. Wumishan Formation: (C) sedimentation rate and (D) precession rate k. Chuanlinggou Formation: (E) sedimentation rate and (F) precession rate k.

Earth-Moon separation and tidal dissipation

Using online tools provided by the AstroGeo Project (24), we obtained estimates for Earth-Moon distance and Earth’s length of day based on the k values obtained from the above inversions (Fig. 3). The results indicate that at 1215 Ma (the Yemahe Formation), the Earth-Moon distance was estimated as 341.37 ± 1.70 (1σ) × 10⁶ m, with a length of day of 18.86 ± 0.17 (1σ) hours. At ~1480 Ma (Wumishan Formation), the Earth-Moon distance was estimated as 333.56 ± 2.10 (1σ) × 10⁶ m, with a length of day of 18.12 ± 0.19 (1σ) hours. Last, at ~1634 Ma (Chuanlinggou Formation), the Earth-Moon distance was estimated as 330.29 ± 1.64 (1σ) × 10⁶ m, with a length of day of 17.82 ± 0.15 (1σ) hours. The detailed results above with Earth’s obliquity results are all shown in table S3.

Fig. 3. Reconstructed precession rate k inversion points, interpolation sequence, and models.

Red circles with error bars (± 2σ) represent constraint points from this study (A: Yemahe Formation, C: Wumishan Formation, and D: Chuanlinggou Formation). Red diamond-shaped points with error bars (±2σ) depict results obtained through robust inversion for reconstruction (including a crucial control point in the Mesoproterozoic, B: Xiamaling Formation). Black diamond-shaped points with error bars (±2σ) indicate inversion results from other cyclostratigraphy, shown for reference and discussion. Detailed information on all data can be found in table S3. The red shaded area represents the range (±2σ) of reconstructed precession rate k using a stochastic monotonic interpolation method. The blue line with the shaded area shows the mean precession rate k and error range from the F2022 model. The light-yellow shaded area displays the error range of precession rate k from the W2015 model. The black dashed line represents the precession rate k estimate from the L2021 model. [In equation 40 in (29), p represents the precession rate k.]

Using the “Bchron” package in R, we performed a stochastic monotonic interpolation of the TimeOptMCMC inversion results based on reliable cyclostratigraphic evidence (italic items in table S3) since ~2.5 Ga (18). First, we obtained the evolutionary curve of the precession rate k, which is presented in Fig. 3 with several models. This allowed us to update the empirical Earth-Moon separation history over the past 2.5 billion years (Fig. 4A). The same method is used to calculate length of day (Fig. 4B) and obliquity (Fig. 4C).

Fig. 4. Reconstructed Earth-Moon distance, length of day and obliquity inversion points, interpolation sequence, and F2022 model.

(A) Reconstructed Earth-Moon distance. Red diamond-shaped symbols with error bars (± 2σ) depict results from the cyclostratigraphic inversion points, and the red line with a shaded area represents the reconstructed Earth-Moon distance using the stochastic monotonic interpolation method, with the line indicating the mean value and the shaded area showing the ±2σ error range. Detailed information on all data points can be found in table S3. The blue line with shaded area shows the mean Earth-Moon Distance and error range for the F2022 model. (B) Reconstructed length of day. (C) Reconstructed Earth obliquity. The F2022 model here has low errors, so the light blue shading here may not be obvious.

Furthermore, using the classical formula for tidal dissipation (Eq. 1 in Materials and Methods), we estimated the variation in the average tidal drag factor based on the evolution of Earth-Moon distance (Fig. 5). We provide the average tidal dissipation for every 100 million years of the Earth-Moon distance sequence obtained by the stochastic monotonic interpolation (with the first point representing the average for 0 to 100 Ma and a total of 24 points, shown in table S4). Considering the relatively unchanged results between several cyclostratigraphic tie points, we display only those results that overlap with the existing 10 cyclostratigraphic tie points (marked with red arrows) in Fig. 5. This estimate shows possible tidal dissipation maxima around 300 to 600 Ma and 1300 to 1500 Ma. The dissipation strength for the first maximum reaches approximately 1.8 to 2.1 × 10³⁸ m^6.5 s⁻¹, while the second maximum is around 1 × 10³⁸ m^6.5 s⁻¹.

Fig. 5. Reconstructed tidal drag factor and comparison with several representative models.

The red points with error ranges represent the reconstructed history of the tidal drag factor based on this study’s Earth-Moon distance reconstruction; each point represents the average value between 100 Ma and the margin of error (±1σ) (only the results that overlap the cyclostratigraphic tie points marked with the red arrows are shown here, and the flatter results between these results are not shown, detailed in table S4). The blue line shows the tidal drag factor evolution calculated according to the F2022 model, labeled with two tidal resonances: 1 and 2; the pink dashed line represents the tidal drag factor history based on the T2021 model, marked with one tidal resonance.

DISCUSSION

We compared the TimeOptMCMC inversion results obtained in this study for the Mesoproterozoic era with several other cyclostratigraphic estimates (Fig. 3 and table S3). At 1100 Ma, Nanfen Formation results indicate an estimated 18.9 ± 0.4-hour length of day (15) and a precession rate k that aligns well with the F2022 model and this study, which may represent conditions in the early Mesoproterozoic. By contrast, results from the Hongshuizhuang Formation at 1450 Ma (34) and the Gaoyuzhuang Formation at ~1560 Ma (35) differ from the F2022 model (24) and this study but align with the intermediate values of the W2015 model (32). However, the W2015 model assumes only a smooth change in tidal dissipation factors between two end points (4.6 Ga and present day). Considering that these two studies primarily relied on the W2015 model when validating the cyclostratigraphy, we have reservations about the reliability of their results. Another recent study also focused on the Chuanlinggou Formation at 1640 Ma using the TimeOptMCMC method (20). That result appears to deviate from the F2022 model and has a lower k than our study. This previous work did not provide information on the number and scale of MCMC chains, making it difficult to assess whether they adequately explored the parameter space. The unusually low error range (±0.25 in k) further supports this point. The Rocknest Formation at 1890 Ma also provides an estimate of a 17.9-hour length of day (36, 37). However, this estimate was roughly derived from a cycle thickness ratio of 8.75:1 compared with estimated ratios of Earth’s orbital eccentricity and axial obliquity (37). The result may be coincidental (26), but it aligns well with the F2022 model and this study. Future updates in cyclostratigraphy or tidalite studies for this period may help validate the results presented here. We noticed that Meyers and Malinverino (MM2018) (9) used the posterior distribution results synthesized from multiple MCMC chains for Earth-Moon evolution derived from cyclostratigraphy, to obtain robust results. Consequently, some of the aforementioned studies are considered for reference only in this study (black points in Fig. 3); we favor the more robust inversion results that follow MM2018’s method for synthesizing our final curve (red points in Fig. 3). Nota bene: MM2018 has a new 2024 guide (see Materials and Methods).

Some studies have proposed that Lamb resonance caused “day-length locking” in the Proterozoic (38–40), citing evidence including cyclostratigraphy (37). It is important to note that the length-of-day estimates in this and similar studies are derived from k and assume conservation of angular momentum in the Earth-Moon system. Therefore, if we assume the existence of a Lamb resonance, which leads to an imbalance in the angular momentum of the Earth-Moon system, an additional independent constraint is needed for length-of-day estimation from cyclostratigraphy (41). Fortunately, recent theoretical investigations have re-evaluated the theoretical foundation of this model (25), suggesting that an equivalent to lunar tidal dissipation, the Lamb resonance, is unlikely in the Mesoproterozoic, thus challenging the idea of “day-length locking”. On the basis of the results presented above, we do not observe evidence of “day-length locking” in the Proterozoic (Fig. 4B). Furthermore, whether inferred from the precession rate k or related parameters such as Earth-Moon distance, Earth’s length of day and axial obliquity, our results are fully consistent with the F2022 model within the error range (24).

Next, we compared the tidal dissipation derived from the reconstructed Earth-Moon separation history with various models (Fig. 5). The T2021 model, based on two parameters, effective ocean depth and nondimensional dissipation timescale, predicted an ocean resonance at ~330 Ma (42). On the other hand, the F2022 model, also based on two free model parameters, takes into account the change in the position of the tectonic plates’ center since 1 Ga and predicts two ocean resonances (24). The high values around 300 to 600 Ma in our study may correspond to peaks 1 and 2 in the F2022 model, at approximately 550 and 310 Ma, respectively (Fig. 5). The results for 400 to 500 Ma are higher than in F2022 while those for 500 to 600 Ma are lower, indicating that the interpolation has smoothed the peak here. For this paper, we provide three crucial constraint points in the relatively data-scarce Mesoproterozoic that, together with the existing result in the Xiamaling Formation (9), reveal potential fluctuations. These fluctuations are concentrated within the possible ranges of 1300 to 1500 Ma. This is noteworthy, but at the same time, the existing uncertainties still limit us from fully confirming the existence of resonance. Nevertheless, this result still offers more detailed insights into the evolution of the Earth-Moon system during the Mesoproterozoic, representing potential resonances that may have existed.

The exact causes of ocean tidal resonances remain a mystery, but through our observational results and model comparisons, we attempt to speculate on possible factors. Continental shelves and specific supercontinent distributions are considered factors enhancing dissipation effects (24). Numerical tidal simulations also propose that the tides are controlled by supercontinent configuration and provide simulation indicating higher magnitude M₂ tides during the Mesoproterozoic (43). Here, we find that the dissipative peaks inferred for the Mesoproterozoic may correspond to the formation of the Nuna supercontinent (Fig. 6A), and in the Phanerozoic, the Pangaea supercontinent corresponds to the peak around 200- to 400-Ma range (44–46). The breakup of the Rodinia supercontinent and formation of the Gondwana megacontinent are complex processes (47), which may have generated the largest resonance between 400 and 700 Ma but in this study appear as a sustained high value.

Fig. 6. Comparison of reconstructed tidal drag factors with the number of continental plates and atmospheric oxygen levels through time.

(A) Changes in the number of continental plates, according to (44, 45), have gone through three major supercontinent cycles: Nuna (~1.6 to 1.3 Ga), Rodinia (900 to 720 Ma), and Pangaea (320 to 170 Ma). (B) Atmospheric oxygen level change over time with increases from the GOE (Great Oxygenation Event), MOE (Mesoproterozoic Oxygenation Event), NOE (Neoproterozoic Oxygenation Event), and POE (Paleozoic Oxygenation Event). The MOE was originally proposed in (53) and displayed here according to the version presented in (54). (C) The tidal drag factor (red points with ±1σ error ranges and related cyclostratigraphic tie points marked by red arrows) reconstructed in this article and F2022 model (mean value, blue line).

Aside from supercontinent evolution, rapid changes in Earth’s length of day and axial obliquity may be correlated with Earth’s climate system and climatic events (Fig. 6B). (24). Resonances since 700 Ma appear to be correlated with the Paleozoic Oxygenation Event (POE, 350 Ma) and the Neoproterozoic Oxygenation Event (NOE, 600 Ma) (24, 48). The Proterozoic includes the so-called “Boring Billion” (0.8 to 1.8 Ga) (49–51), characterized by slow evolutionary rates and low oxygen levels (52). However, some studies have also proposed fluctuations in oxygen levels during this time, referred to as the Mesoproterozoic Oxygenation Event (MOE, 1.6 to 1.3 Ga) (53, 54). Our study indicates times of relatively high tidal dissipation during the Mesoproterozoic, and the timing and intensity of these events coincide with the characteristics of the MOE. We speculate that the evolution of the Earth-Moon system, including rapid changes in Earth’s length of day and axial obliquity during the Mesoproterozoic era, may have led to changes in Earth’s climate system, thereby triggering the aforementioned oxygenation events. The results of this study reveal that the evolution of the Earth-Moon system may have had an impact on the paleoenvironment and habitability of Earth, although more evidence is needed to confirm this.

MATERIALS AND METHODS

Cyclostratigraphic datasets

The Yemahe Formation is in the Shennongjia Forest District, Hubei Province, South China (fig. S1). The strata in this region span the Mesoproterozoic to Neoproterozoic and belong to the Shennongjia Group. The Tiechanghe section, which is the study section in this paper, is situated on the northwestern margin of the Shennongjia Group outcrop area, primarily exposing the upper portions of the Wenshuihe Formation, Yemahe Formation, and Taizi Formation. It is located along National Highway 209 (55). In the middle of the Yemahe Formation, a precise laser ablation–multicollector–inductively coupled plasma mass spectrometry zircon U-Pb age was obtained at 1215.8 ± 2.4 Ma from volcanic ash, with geographic coordinates 31°36′2.35″N 110°23′30.26″E (56). We delineated a 56-m section around this age point for this study, placing this ash layer at 47 m. Apart from some purple tuff intercalations at 53 to 55 m, the section predominantly consists of gray, blocky, medium-crystalline to fine-crystalline dolomite. We selected the 0- to 52-m section to isolate the influence of the tuff. We used a handheld gamma spectrometer with a spacing of 0.05 m to measure Th/U in this segment. This proxy represents changes in seawater redox conditions during the deposition of the dolomite (57).

The Wumishan Formation is in Jixian, Tianjing, North China (fig. S2). The basic research and datasets of this section are provided by previous studies (28, 58). The formation age is constrained by two sensitive high resolution ion microprobe (SHRIMP)-dated isotope dates, 1483 ± 13 Ma and 1487 ± 16 Ma, and is tentatively assigned to 1480 Ma (59). MS data were measured from 0 to 144.25 m, which represents sea-level variations, supported by the succession of shallow marine carbonate facies (28).

The Chuanlinggou Formation is in Kuancheng County, Hebei Province, North China (fig. S2). This region is characterized by the presence of Mesoproterozoic Changcheng Group stratigraphy (20). The Shenxianling section, the focus of this study, is situated on the eastern outskirts of Kuancheng County, primarily exposing the Chuanlinggou Formation and Tuanshanzi Formation along National Highway 508. In the upper part of the Chuanlinggou Formation, a zircon U-Pb age was determined using the SHRIMP method, yielding an age of 1634.8 ± 6.9 Ma from a volcanic tuff with geographic coordinates 40°35′2.4″N and 118°32′15.4″E (60). We measured a 68-m section around this age point, locating the age point at 58.7 m. Below the age point, the section is primarily composed of gray-black shales, and above the age point, fine-grained silty sandstones start to appear. We used a handheld gamma spectrometer with a spacing of 0.05 m to measure Th in this section. This proxy represents variations in the clay mineral content within the shale (57). All the field measurements data for the three sections mentioned above can be found in the dataset attached to this paper (see Data and materials availability).

Cyclostratigraphic analysis

We conducted cyclostratigraphic analysis by removing trends, performing spectral analysis, filtering and tuning with the Acycle software (61). The Yemahe Formation U/Th sequence was detrended using an 18.2-m locally weighted scatterplot smoothing (LOWESS) window (35%). A Gaussian band-pass filter was applied to extract signals in the 0.11 to 0.21 cycle/m frequency band (405-kyr cycle band). The maxima of the filtered signal corresponding to 405-kyr intervals were used to generate an age model. The MS series from 0 to 144.25 m of the Wumishan Formation was first interpolated to equally spaced intervals of 0.2 m and detrended using a 72.1-m LOWESS window (50%). A Gaussian band-pass filter was applied to extract signals in the 0.11 to 0.17 cycle/m frequency band (100-kyr cycle band). The maxima of the filtered signal corresponding to 100-kyr intervals were used to generate an age model. The Th sequence from 0 to 68 m of the Chuanlinggou Formation was detrended using a 34-m LOWESS window (50%). A Gaussian band-pass filter was applied to extract signals in the 0.16 to 0.25 cycles/m frequency band (100-kyr cycle band). The maxima of the filtered signal corresponding to 100-kyr intervals were used to generate an age model. All the time-domain datasets were further interpolated to uniform 3-kyr intervals.

We carried out EHA and TimeOpt analysis using the astrochron package in R (62). The EHA identified sections with orbital eccentricity and climatic precession power, leading us to select 34 to 50 m for the Yemahe Formation, 13 to 63 m for the Wumishan Formation, and 16 to 45 m for the Chuanlinggou Formation as appropriate data segments for TimeOpt and TimeOptMCMC analyses (figs. S6 to S9).

Bayesian inversion by TimeOptMCMC

We conducted Bayesian inversion using the TimeOptMCMC algorithm in the astrochron package in R (9, 62) (Supplementary Text Part 1). In this process, we used “target.m” for prior estimates of obliquity and eccentricity (23). The precession rate (k) used in the analysis was derived from the F2022 model (24), and the prior uncertainty was based on the W2015 estimates (32). The data used in the Bayesian inversion are directly extracted from the corresponding segments without undergoing preprocessing such as LOWESS detrending.

To ensure effective exploration of the parameter space, we conducted tests on the parameters before running the analyses. The test results are presented in figs. S16 to S18. These results indicate that the set parameters can effectively explore the prior parameter space. We ran 50 chains of 600,000 iterations each for all three sections. The evolution of representative chains for the three sections are shown in fig. S19. After extracting the burn-in results, we present the overall situation of these chains in figs. S20 to S22. It can be observed that seven chains, six chains, and seven chains were trapped in low-likelihood spaces for the Yemahe Formation, Wumishan Formation, and Chuanlinggou Formation, respectively. Therefore, we excluded them from the results. The acceptance rates for all chains range from 7 to 15%. The above methods reference a new guide, “Bayesian Inversion with TimeOptMCMC”, by S. Meyers, available at https://eartharxiv.org/repository/view/7134/.

The posterior output results are presented in table S2. Brief summaries of the scenarios, sedimentation rates, and k are shown in Fig. 2, while the complete posterior parameter distributions can be found in figs. S10 to S15. Last, we used the AstroGeo web calculator (24) to convert the estimated k values into Earth-Moon distance, Earth’s length of day, and axial obliquity.

Stochastic monotonic interpolation by Bchron

We applied the simple monotone procedure proposed by Haslett and Parnell (63) for the stochastic monotonic interpolation of Earth-Moon distance, length of day, and obliquity. This method is widely used in isotope age interpolation (64) and has been used in previous orbital parameter studies for Earth-Moon distance interpolation (18). A key consideration in adopting this method is ensuring that the inversion of Earth-Moon evolution parameters using cyclostratigraphic evidence, such as precession rate k, relies on the dominance of Earth-Moon tidal dissipation, thereby ensuring the conservation of angular momentum in the Earth-Moon system. This requires that the entire process of Earth-Moon distance, length of day, and increase in obliquity is monotonic, which is a crucial factor distinguishing it from other interpolation processes. Our interpolation was carried out using the “Bchron” package in R (Supplementary Text Part 2, with an example of the configuration in table S5).

Characterization of tidal dissipation by tidal drag factor

The gravitational torque exerted on nonequilibrium tides results in angular momentum transferred from Earth’s rotation to the Moon’s orbit, and this process is described as (65)

$$\frac{\mathit{da}}{\mathit{dt}}=f{a}^{-5.5}$$ (1)

where a is the Earth-Moon distance, t is time, and the f is tidal drag factor. An analytical solution is obtained by integrating Eq. 1 in time (32)

$$a^{6.5}=a_0^{6.5}-6.5F_{t}t$$ (2)

where a is a past time Earth-Moon distance, a₀ is a later time (e.g., present-day) Earth-Moon distance, t is the time difference between the two separations, and F_t is the mean drag factor for this period of time t. Because of the stochastic monotonic interpolation providing 10,000 possible Earth-Moon distance paths (random interpolation), we calculated the average tidal drag factor for each 100-million-year interval, i.e., 0 to 100 Ma, 100 to 200 Ma, and so on. These results are presented in Fig. 5 and table S4. The results for 700 to 1200 Ma and 1700 to 2300 Ma do not overlap with the existing tie points and are relatively unchanged; therefore, they are not displayed in Figs. 5 and 6C. The Supplementary Materials provide the code for this calculation (Supplementary Text Part 3).

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S22

Tables S1 to S5