The goal of this study is to reconstruct a locally varying sedimentation rate profile with errorbars from multiple geochemical depth series, under the assumption that certain stratigraphic signals (e.g., elemental variability) reflect astronomical forcing. We achieve this by inverting a forward model of signal distortion under varying sedimentation, using a formulation rooted in probabilistic inversion.

This method builds upon the principles of astrochronology, particularly the spectral identification of Milankovitch-band periodicities in sedimentary records, but extends them to allow for instantaneous, non-stationary sedimentation rates. AstroComb performs a sliding-window probabilistic fit of observed geochemical spectra to a predefined target astronomical spectrum, yielding a sedimentation rate for the mid-point in the window that optimally “stretches” or “compresses” the time axis at each position to recover astronomical frequencies. The fit is performed in frequency domain, which allows multi-channel (e.g. multiple elements) that will not be obscured if displaced by a phase-lag.

We begin by formalizing the relationship between sedimentation rate and spectral distortion, then describe the inversion procedure and implementation of AstroComb. Finally, as an example, we discuss its application to the Alum Shale data and compare to previous cyclostratigraphic results .

Let d(z) be a geochemical depth series (e.g., Al concentrations), where z is stratigraphic depth. If sediment accumulated at a locally constant rate, r(z)=dzdt, then in the time domain, the corresponding series is dt(t). However, in practice we only observe the signal in the depth domain.

Let k be a frequency in the depth domain (wave number in cycles per mm) and f its time-domain counterpart (cycles per kyr). The transformation between the two is: 1k=fr Thus, a fixed astronomical frequency f will appear in the stratigraphic record as a peak at k, depending on the local sedimentation rate. Variations in r(z) distort the signal, shifting, splitting, or smearing spectral peaks . Our objective is to recover r(z) such that the observed power spectrum matches that predicted from astronomical forcing (see Fig. ).

For a given depth interval (window) centered at z0, we obtain power spectra Dobs of multiple, observed signals (Dobs is a matrix of spectra with a spectrum of one channel in each column). For each trial sedimentation rate r, we convert the target time-domain frequencies f to expected depth-domain frequencies (wave numbers) k=f/r.

In our computation of the likelihood, data uncertainties are indirectly provided by the user as the probability that a peak in the observed spectrum at a given trial wavenumber is a Milankovitch peak. Existing data-fitting-based methods (; ; TimeOpt: ; TimeOptMCMC: ; AstroGeoFit: ) are purely based on misfits between data (elemental records) and synthetic signals (generated from assumed Milankovitch periods), but AstroComb is instead aimed at simulating the spectral peak identification and data uncertainty evaluation carried out by practitioners analyzing observational data. In traditional misfit calculations you not only need experimental uncertainties, but also the variance of components of the data that are not modeled by the Milankovitch cycles. These components are significant, but not directly available (for example, The TimeOptMCMC algorithm estimates them through a Hierarchical Bayes approach ). However, in our approach we are automatically taking all data uncertainty sources into account, using the practitioner's confidence in identifying spectral peaks.

The most (almost universally adopted) definition of the likelihood would, in our case, read 2L(m)=pn(Dobs-g(m)) where pn is the noise distribution, Dobs is the observed multi-channel data, and g(m) is the forward function mapping the unknown model parameters m into the data. We can see from Eq. () that the likelihood depends on m through the function g, so we can define a non-negative function pd such that 3pd(g(m))≡L(m). Since g(m) is the “noise-free data”, we can view pd as the (possibly unnormalized) distribution of the noise-free data, given that we only know the observations Dobs and the data uncertainties pn (). However, Eq. () also shows that a likelihood can be calculated by any procedure that measures the goodness of fit, based on a forward function g and observations Dobs. The underlying uncertainty need not be given in advance through a noise distribution pn. If the goodness of fit is a priori acceptable for the analyst (e.g., based on experience), pn can be determined afterwards, if needed. In the following we follow a procedure where we define the likelihood without an initial definition of pn.

Our likelihood calculation is as follows:

As prior information about the data uncertainties, we mimic the 'manual' process of spectral line identification by defining a probability that there is a potential Milankovitch peak at a given wavenumber k in the spectrum. Between user-defined lower and upper bounds a (the lower limit for being identified as a “peak”) and b (the value above which we are sure that we have found a potential peak), the probability increases from a user defined, small positive number pa at the lower limit to pb=1-pa at the upper limit. Typical values are a=〈F〉+σF and b=〈F〉+3σF, where 〈F〉 is a local mean of the spectrum around the considered wavenumber k, and σF is the local standard deviation of the spectrum. In our implementation, 〈F〉 was found as the value of a background spectrum at wavenumber k, where the background was calculated as a smooth function fitted to the power spectrum. We fitted the spectrum with a sum of three decreasing exponentials, but other ways are possible. Our threshold probability pa is 0.01. The probability function p(x) is chosen from user experience, and in our implementation it was chosen as a smooth, monotonically increasing sigmoid function satisfying4p(a)=pa,pb(b)=1-pa,with the constraints5b>a,0<pa<0.5,with the form6p(x)=11+exp⁡(-k(x-x0))with7x0=a+b2.

If prior information about (r,a) is encoded by a probability density ρ(r,a) we can – in a Bayesian formulation – express the complete, probabilistic solution as a posterior probability density 9σ(r,a)=C⋅L(r,a)ρ(r,a). where C is a proportionality constant. In the example below, where data from Fågeltofta-2 is analyzed, the prior is uniform and assigns constant values to all sedimentation rates within user-defined boundaries of 1.0 mm kyr⁻¹ <r<10.0 mm kyr⁻¹.

From the marginal sedimentation rate distribution 10σ(r)=∫σ(r,a)da at each stratigraphic position, we can now compute a floating geological time, t, and its uncertainties as a function of depth, z. If the absolute time, t0, is known at a given depth z0, and the sedimentation rate in layer n is rn, the time at z0+NΔz is 11t(z0+NΔz)=t0+Δz∑n=1N1rn. From the posterior probability densities of the sedimentation rates rn, and hence Δz/rn, we can now calculate the uncertainties of the times, t. From the variances of the distributions of Δz/rn for all layers n, we can obtain the variance of t(z0+NΔz) by a simple summation: 12Var(t(z0+NΔz))=∑n=1NVarΔzrn. Working with variances, instead of the full probability distributions for the inverse sedimentation rates, saves computer storage and time when processing large data sets. Furthermore, due to the central limit theorem, the distribution of the sum of random time increments will converge to a Gaussian distribution when the number of layers increases, making the variance an adequate description of the uncertainty.

Finally, we calculate the Kullback-Leibler information measure at each depth z showing how much Milankovitch information is present in the data as a function of depth . This is obtained from: 13I=∫Rσz(r)ln⁡σz(r)u(r)dr where σz(r) is the calculated probability distribution of r at depth z, R is the range of sedimentation rates, and u(r) is a uniform (constant, and hence non-informative) distribution. Theoretically, the Kullback-Leibler information measures the expected number of yes/no questions needed to update our state of information from what is given by u(r) (here total ignorance) to what is given by σz(r) (here, the posterior distribution).

AstroComb implements the above approach through the following steps:

Preprocessing: Detrend and normalise the depth series, optionally re-sample or smooth.

Because no closed form astronomical solution exists beyond about 50 Ma , we ignore the relative phases and adopt theoretical periods constructed from studies of deep-time orbital mechanics . For the upper Cambrian ∼ 497 Ma, the target includes:

Long eccentricity: 405 kyr

A test on synthetic data was carried out to study the accuracy and stability of the algorithm. More importantly, this allows us to validate whether we can reproduce a known sedimentation rate profile using a known target astronomical spectrum.

A severe test with artificial data forced by a sinusoidal Cambrian Milankovitch forcing overlain by 200 % red noise and showing significant sedimentation rate discontinuities, was carried out (Fig. ). The synthetic data were calculated by adding sinusoidal signals, using only 4 approximate Cambrian Milankovic frequencies fM=1/TM where TM=405,99,32.4,19.5 (all in kyr), all with amplitudes aM set to 1. The synthetic sedimentation rate model (Fig. ) is characterized by layers with sedimentation rates separated by significant discontinuities which could reflect abrupt changes in the climate such as going from coarse grained quartz sand to fine-grained mudrock and/or a pause in sedimentation (hiatus) that may also leave a geochemical imprint on the XRF data. From this model, the synthetic data were calculated (Fig. ). The 200 % noise was red noise with a 1/f2 power spectral density, simulating a background spectrum of geological signal that cannot be explained as astronomical forcing. Figure  shows the posterior probability densities of sedimentation rates for all depths, computed by AstroComb using the synthetic data. The true result (the synthetic reference model) is overlain to show the fidelity of the reconstruction.

Considering the relatively high noise level, the reconstruction of the true sedimentation rates is excellent. Abrupt changes in the sedimentation rate of about a factor three occur at 70, 74, 81, and 83 m, all of which are accurately captured by the AstroComb algorithm (Fig. ). Sedimentation rates are not “smoothened”, but the timing of the abrupt change appears uncertain.

However, the test also illustrates how the probability distribution outside the correct sedimentation rate is not invariable or zero. In fact, artifacts (mathematically possible, but incorrect sedimentation rates) are introduced by the noise, the variability of the true model, window size, and the non-linearity of the problem. Some of these artifacts show up as spurious parallel features that could potentially be misinterpreted as representing true sedimentation rates. This highlights a key challenge in cyclostratigraphic analyses: a given cycle may be interpreted as corresponding to different target periods, resulting in a multimodal sedimentation-rate probability distribution. Nevertheless, the highest probability remains associated with the true reference model (Fig. ). The Kullback-Leibler information content varies along stratigraphic depth with high information content at intervals where the sedimentation rate is constant across the window size (3.3 m) and low where the sedimentation rate changes within the window. In any case, we note that the algorithm identifies the true (reference) sedimentation rates despite the high noise level, and that it reconstructs the large discontinuities between layers of constant rates quite well, with lower probability near abrupt sedimentation rate changes. For comparison, we also processed a pure noise signal (see Fig. ) to demonstrate that the algorithm is stable under such conditions, and will not produce spurious solutions with any significant probability.

This study demonstrates that the AstroComb method extends the spectral misfit minimization strategy in a probabilistic framework, handling uncertainty propagation and parameter tuning with statistical rigor.

To illustrate the application of the multichannel, probabilistic Milankovitch period detector, AstroComb, we analysed high-resolution XRF data from a ∼ 21 m long interval (67.2520–88.0744 m) of the Fågeltofta-2 drill core through the Alum Shale Formation in southern Sweden obtained by . Briefly, the XRF data was collected using an ITRAX core scanner from Cox Laboratories with 30 kV voltage and 50 mA current on a Rh tube. The XRF measurements were collected with a high scanning resolution of 0.2 mm along the core and relatively long exposure time (7 s per scan), producing signal reported as concentrations for 22 elements, incl. Al₂O₃, SiO₂, S, K₂O, CaO, TiO₂, V, MnO, Fe₂O₃, Ni, Cu, Zn, Ge, As, Rb, Sr, Y, Zr, Mo, Ba, Hf, and U. This extensive elemental dataset formed the basis for the spectral analysis. Figure  shows a subset (the top 40 mm) of the data from ∼ 19 m of core, yielding ∼ 90 000 measurements.

The assumed orbital target periods are model-based. While the tuning strategy often relies on the 405 kyr eccentricity cycle as a stable chronometer, potential shifts in Earth’s rotational parameters due to e.g. tidal dissipation effects could introduce biases.

Our algorithm is based on the assumption that prior information about the spectral peak identification process can be provided, and that it adequately describes the way Milankovitch periods are identified by human analysts. Little prior information about the unknown sedimentation rates is used, meaning that the effective prior probability density is uniform (constant in a wide interval).

The fundamental output of the algorithm is the posterior probability densities shown in Figs.  and . It is apparent from these distribution that multiple, valid solutions to the problem exists. For example, in the single-channel data, in parts of the depth interval 74–78 m, there is a choice between rather low (∼3.2 mm kyr⁻¹) sedimentation rates and rather high values (∼9.8 mm kyr⁻¹). When calculating the geological timescales (time-depth curves) a choice between these solutions must be made, and the current version of the algorithm calculates the time independently for every layer based on the maximum a posteriori (MAP) estimates, that is, the sedimentation rates with the highest values of the posterior probability density (see Figs.  and ). However, expert knowledge about the sedimentary system could be added as a non-uniform prior on the sedimentation rates, and this could well change the MAP solution. For example, one may add that the sedimentation rate cannot jump arbitrarily between adjacent layers in the case where two sedimentation rates are roughly equally probable within a stratigraphic interval. Using such information can be seen in Figs.  and , where we have plotted sedimentation rates from previous studies , where higher values for the sedimentation rates in the depth interval 74–78 m has been chosen based on prior knowledge not included in AstroComb. As a result, the age-depth models deviate in this interval (Fig. ).

There are also some fundamental assumptions behind Milankovitch analysis that apply to any method. Milankovitch cycles are (quasi-)periodic, but not sinusoidal. This means that a hypothetical, noise-free Milankovitch spectrum will, in general, contain spectral components outside the main Milankovitch frequencies . These components are generally disregarded in Milankovitch analysis programs, including AstroComb. More seriously, the Earth's sedimentary system is highly nonlinear and therefore could produce additional periods that are sums and differences of the main Milankovitch cycles. The amplitudes of such periods could be significant, but they are ignored by our algorithm. For linear, as well as nonlinear sedimentary systems, even if we know the amplitudes of the input Milankovitch signal, the output amplitudes observed in boreholes are entirely unknown. This was the background for the way that we designed our spectral fit algorithm, e.g. making an analytical tool which is somewhat agnostic to the amplitude of the recorded periodicities.

Last, but not least, it is important to remember that Milankovitch cycle identification is a difficult problem. The signal-to-noise ratio is generally very low, and the problem is often similar to finding a needle in a hay stack. Such problems are prone to detrimental human biases, but AstroComb is designed with the aim of quantifying the analysis, and casting the problem as a probabilistic inference problem, so as to avoid such biases. The probability distribution presented here generally provides several plausible solutions for the sedimentation rates at given depths, allowing the user to choose based on geological/stratigraphic knowledge and clearly identify such choices.

The AstroComb algorithm is a new unsupervised approach for identifying Milankovitch cycles in multi-channel data. The algorithm has discovered abrupt changes in sedimentation rate in the Fågeltofta-2 core, as opposed to smooth transitions, potentially guiding sedimentologist in reconstructing pulses of sedimentation from geological archives. This opens tantalizing new prospects for understanding relative sea level changes because the shifts in sedimentation rate in the Alum shale Formation (incl. Billegrav-2 and Fågeltofta) are correlated with the changes in relative sea level obtained from the sequence stratigraphy . Where previous sedimentation rate estimates obtained using eCOCO and a smoothing spline fit on visually profound periods in the wavelet spectrum unavoidably induced some degree of smoothing of the sedimentation rate that may give a false impression of the actual sedimentation rate changes, AstroComb delivers sedimentation rate changes that are not necessarily changing smoothly across stratigraphy.

As sedimentation rate is a fundamental parameter characterizing a sedimentary environment and a factor that serves as a first order control on organic matter preservation, AstroComb may help engaging sedimentologists with a new relative chronometer. Also, we envision to refine geochronology further by using it in combination with geochemical information about sedimentation rate changes.