Spectral analysis is a key tool for identifying periodic patterns in sedimentary sequences, including astronomically related orbital signals. While most spectral analysis methods require equally spaced samples, this condition is rarely achieved either in the field or when sampling sediment core. Here, we propose a method to assess the impact of the uncertainty or error made in the measurement of the sample stratigraphic position on the resulting power spectra. We apply a Monte Carlo procedure to randomise the sample steps of depth series using a gamma distribution. Such a distribution preserves the stratigraphic order of samples and allows controlling the average and the variance of the distribution of sample distances after randomisation. We apply the Monte Carlo procedure on two geological datasets and find that gamma distribution of sample distances completely smooths the spectrum at high frequencies and decreases the power and significance levels of the spectral peaks in an important proportion of the spectrum. At 5 % of stratigraphic uncertainty, a small portion of the spectrum is completely smoothed. Taking at least three samples per thinnest cycle of interest should allow this cycle to be still observed in the spectrum, while taking at least four samples per thinnest cycle of interest should allow its significance levels to be preserved in the spectrum. At 10 and 15 % uncertainty, these thresholds increase, and taking at least four samples per thinnest cycle of interest should allow the targeted cycles to be still observed in the spectrum. In addition, taking at least 10 samples per thinnest cycle of interest should allow their significance levels to be preserved. For robust applications of the power spectrum in further studies, we suggest providing a strong control of the measurement of the sample position. A density of 10 samples per putative precession cycle is a safe sampling density for preserving spectral power and significance level in the Milankovitch band. For lower sampling density, the use of gamma-law simulations should help in assessing the impact of stratigraphic uncertainty in the power spectrum in the Milankovitch band. Gamma-law simulations can also model the distortions of the Milankovitch record in sedimentary series due to variations in the sedimentation rate.

Spectral analysis methods have become a key tool for identifying Milankovitch
cycles in sedimentary series and are a crucial tool in the construction of
robust astronomical timescales

Illustration of the problem.

Although uncertainties exist regarding the actual position of samples, few case
studies document their effect on the identification of periodic patterns.

In this paper, the term “stratigraphic uncertainty” refers to the uncertainty of the sample positions. Testing the impact of the stratigraphic uncertainty on the spectral analyses requires a randomisation procedure that reflects typical errors made during measurements of the stratigraphic position of samples. Figure 1c to e illustrate the consequences of the stratigraphic uncertainties on a geological series (here the La Charce series; see Sect. 3.1). Figure 1c compares the real sampling made on this series (in red) to an ideal sampling in which samples are taken at a strictly even sample distance (in black). Errors in the sample positions distort the sedimentary series: some intervals are compressed while some others are increased. Ideally, all sample distances should be strictly the same, so that the distribution of sample distances should be concentrated on only one value (Fig. 1d). In reality, as uncertainties exist regarding the sample positions, the sample distances show a distribution over a certain range of values, which depends on the accuracy with which the distance measurements have been taken (Fig. 1e). In the case of the La Charce series, the standard deviation of the sample distances is assessed at 12.5 % of the average sample distance (the method to estimate this standard deviation is provided in Sect. 4). If the error in the distance measurement was systematic, one should expect the same level of error in the total length of the series. However, in total, the difference in the length of the series between the ideal case (all sample distances strictly the same) and the real case is only 1.4 % of the total length of the series (Fig. 1c). Each sample distance is measured independently of the other sample distances, so that each measurement can overestimate or underestimate the real distance between two successive samples. The errors thus compensate for each other, implying that the process at the origin of the error measurements is not systematic but random.

Three conditions must be respected to design the error model: (i) the stratigraphic order of samples is hard-set and must not be changed by the randomisation process (e.g. Fig. 1c); (ii) the average and standard deviation of sample steps must be maintained during the randomisation process; (iii) the error model must be random. These conditions can be achieved if the error model randomises the sample distances rather than the sample positions. In that case, the probability density function should have a positive and continuous distribution (i.e. values obtained after randomisation are continuous and positive). In addition, the average sample step and the standard deviation of the distance between two successive samples are known and should be parameterised.

The gamma distribution fulfils all these conditions. The gamma distribution
is continuous and has a positive support. Parameters

Gamma probability density functions (PDFs). All Gamma PDFs have a positive support, which is a crucial characteristic to realistically simulate sample steps. The gamma density probability functions were generated with the Matlab gampdf function.

In the opposite case, a low variance-to-mean ratio corresponds to a low

Two published geological datasets were used here to assess the effect of stratigraphic uncertainty on power spectra.

A total of 555 gamma-ray spectrometry measurements were performed in
situ on the La Charce section

Gamma-ray spectrometry measurements were performed directly in the field with a sample step of 0.20 m that is as regular as possible. Before each measurement,
rock surfaces were first cleaned to remove reworked material and flattened to
prevent any border effects that could affect the measurement value. Each
measurement was performed using a SatisGeo GS-512 spectrometer, with a
constant acquisition time of 60 s

The second case study consists of the 184 m thick continuous early Givetian
to early Frasnian sequence of the La Thure section

Based on the assessments summarised in the previous paragraph, we tested
three different levels for the error on the measurement of sample steps (5,
10 and 15 %), which we consider realistic scenarios for geological sampling
during fieldwork. We applied our Monte Carlo based procedure for randomising
sample steps to a sinusoidal series, as well as to the two previously
published geologic datasets described in Sect. 3

Spectral analyses were performed using the multi-taper method

Lomb–Scargle spectra were calculated with the REDFIT algorithm

During each test, both MTM and REDFIT Lomb–Scargle power spectra were calculated for each of the 1000 Monte Carlo distorted series. Subsequently, the average power spectra and the range of powers covered by 95 % of the simulations were calculated for the MTM and Lomb–Scargle analyses. The confidence levels of the datasets deduced from the red-noise fit of the spectral background were calculated after each simulation. The average power of the confidence levels and the range of powers of the confidence levels covered by 95 % of the simulations were calculated and directly plotted on top of the simulated spectra. The sum of sinusoids series does not need correction for red noise, and the raw Lomb–Scargle spectra are shown. The two geological datasets show a red-noise background, and the REDFIT-corrected Lomb–Scargle spectra were shown.

We finally provide a quantification of the relative change in spectral power,
using the following criterion:

Effect of the gamma-law randomised sample distances on the

The effect of randomising the sample positions within the section is first
tested on a sum of pure sinusoids. A dataset of 600 points is generated with
a sample step of 1 arbitrary unit. The series is a sum of 24 sinusoids,
having equal amplitudes and different frequencies: frequencies range from
0.02 to 0.48 cycles (arbitrary unit)

Figure 4 notably shows the relative change in power of the average spectrum
after applying the 1000 simulations. At 5 % uncertainty, a decrease of 50 %
in the power spectrum is observed in the

Relative change in power in the

Stratigraphic uncertainty does not only trigger loss of power of the spectral
peaks, it also increases the power spectral background (Fig. 3). At 5 and
10 % uncertainties, the average and background spectrum still preserve the
structure of individual peaks in both

It should be noticed that in the case of pure sinusoids, the signal is only
composed of pure harmonics concentrating the spectral power at specific
frequencies. This implies that a small shift in the sample position triggers
a strong decrease in the average power spectrum at these specific
frequencies. In addition, in this theoretical example, the sample distance
before the randomisation procedure was strictly constant (1 arbitrary unit).
More realistically, spectra of geological datasets are rather composed of a
mixture of harmonics, narrowband and background components, and the sample
distances are not strictly constant. For instance, because of variations in
the sedimentation rates, the sedimentary expression of the orbital cycles is
not focused on specific frequencies but rather expressed in ranges of
frequencies

Detrending procedure of the gamma-ray series from the La Charce
section.

Prior to performing

The

Results of red-noise background estimates from the La Charce and
the La Thure series with the

Spectra of the La Charce and La Thure series before Monte Carlo
simulations of the sample distances.

The autoregressive coefficient, a measure of the redness of the spectrum, is
assessed at 0.440 in the

Prior to performing

The

Detrending procedure of the magnetic susceptibility (MS) series from
the La Thure section.

The autoregressive coefficient of the red-noise background level is assessed
at 0.657 in the

Synthesis of the results of highest frequencies before smoothing of the spectra when applying the Monte Carlo simulations and of highest frequency at which the spectra before and after simulation are practically identical.

At 5 % uncertainty, the average

Effect of the gamma-law randomisation of the sample distances on the

In the REDFIT spectrum with 5 % of stratigraphic uncertainty, the periods at
20.5 m and around 1 m still exceed the 99 % CL (Fig. 9a). Like in the

The average autoregressive coefficients of the 1000 simulations (with

At 5 % uncertainty, the

Effect of the gamma-law randomisation of the sample distances on the
REDFIT spectrum of the La Charce series. Panels

Effect of the gamma-law randomisation of the sample distances on the

At 5 % uncertainty, the REDFIT analysis still displays significant periods at 30–40 m exceeding the 99 % CL and a period at 2.3 m exceeding the 95 % CL (Fig. 11a). The peak at 1.5 m does not exceed the 90 % CL anymore, while the peaks at 1.1 and 0.9 m do not exceed the 95 % CL anymore. At 10 % uncertainty and 15 % uncertainties, spectral peaks in the precession and the obliquity bands do not reach the 95 % CL anymore. The tendency of the Lomb–Scargle analysis to produce high-power peaks in the high frequencies prevents strong smoothing of the power spectrum at 5 % uncertainty. However, at 10 and 15 % uncertainties, all fluctuations of the power spectrum at frequencies higher than 53 % Nyquist frequency are flattened and not distinguishable (Table 2). The significance level in the eccentricity band is still preserved in the average spectrum. At 10 and 15 % uncertainty, the power spectrum displays spectral peaks with reduced powers compared to the spectrum of the original series, which impacts the significance levels in the obliquity and precession bands (Fig. 11d–f). At 5 % uncertainty the REDFIT spectrum of the La Thure series remains practically unchanged compared to the spectrum of the original series from 0 to 58 % Nyquist frequency (Fig. 11d), while at 10 and 15 % uncertainty this range is respectively restricted to 0–22 and 0–19 % Nyquist frequency (Fig. 11e–f).

The average autoregressive coefficients of the 1000 simulations are assessed
for 5, 10 and 15 % of stratigraphic uncertainties at 0.658

In the

Effect of the gamma-law randomisation of the sample distances on the
REDFIT spectrum of the La Charce series. Panels

Comparisons between original and average simulated spectra show that at 5 %
uncertainty, both are practically identical from 0 to 27 % of Nyquist
frequency in the La Charce series and from 0 to 52 % of Nyquist frequency in
the La Thure series. At 10 and 15 % uncertainties, these ranges dramatically
shift from 0 to 20–22 % Nyquist frequency. Although differences exist in the
variance of the average spectrum and in the frequency resolution between the

As an example, if the targeted range of frequencies are the Milankovitch cycles, the shortest period of interest is the precession cycles. A density of one sample per 4 kyr should allow the detection of the spectral peaks in the precession band. A density of sampling of one sample per 2 kyr should then ensure the detection of significant peaks in the precession band, even in the case of strong red noise and medium-to-high levels of stratigraphic uncertainty. The minimum density of sampling being dependant on the level of red noise and stratigraphic uncertainty, we strongly recommend applying the simulations developed here to assess the impact of stratigraphic uncertainty on the identification of significant spectral peaks in the sedimentary record.

Uncertainties in the measurement of sample position can practically not be
avoided in outcrop conditions. The similarity between the topographic slope
and the sedimentary dip, the absence or scarcity of marker beds, or the need
to move laterally in a section can trigger disturbances in the sampling
regularity. In core sedimentary sequences, non-destructive automated
measurements such as X-ray fluorescence, gamma-ray spectrometry or magnetic
susceptibility should limit errors in the sample position. However, physical
samplings (e.g. for geochemistry or mineralogy) are subject to small
uncertainties, especially when the sampling resolution is very high. Core
sedimentary series can in addition be affected by the expansion of sediment
caused by the release of gas or the release of overburden pressure

Linking sedimentary cycles to orbital cycles or assessing the quality of an
orbital tuning procedure often requires a good matching between the
sedimentary period ratios and the orbital period ratios

Also in the evaluation of the relative contribution of precession and
obliquity-related climatic forcing, an accurate assessment of the respective
spectral power is essential

Errors made during the measurement of the stratigraphic position of a sample significantly affect the power spectrum of depth series. We present a method to assess the impact of such errors that is compatible with different techniques for spectral analysis. Our method is based on a Monte Carlo procedure that randomises the sample steps of the time series, using a gamma distribution. Such a distribution preserves the stratigraphic order of samples and allows controlling the average and the variance of the distribution of sample steps after randomisation. The simulations presented in this paper show that the gamma distribution of sample steps realistically simulates errors that are generally made during the measurement of sample positions. The three case studies presented in this paper all show a strong decrease in the power spectrum at high frequencies. Simulations indicate that the power spectrum can be completely smoothed for periods less than 3–4 times the average sample distance. Thus, taking at least three to four samples per thinnest cycle of interest (e.g. precession cycles for the Milankovitch band) should preserve spectral peaks of this cycle. However, the decrease in power observed in a large portion of the spectrum implies a decrease in the significance level of the spectral peaks. Taking at least 4–10 samples per thinnest cycle of interest should allow their significance level to be preserved, depending on the level of stratigraphic uncertainty and depending on the redness of the power spectrum. Robust reconstruction of the power spectrum in the entire Milankovitch band requires a robust control of the sample step in the field and requires a high density of sampling. To avoid any dispersion of the power spectrum in the precession band, taking 10 samples per precession cycles appears to be a safe density of sampling. For lower-resolution sampling, we recommend applying gamma-law simulations to ensure that stratigraphic uncertainty only has limited impact on the spectral power and significance level of the targeted cycles. Gamma-law simulations can also be used to simulate the effect of variations in the sedimentation rate on insolation series, which should help in modelling the transfer from insolation series to sedimentary series.

Data used in this study are available via the following links:

When interpolating an unevenly sampled time series to an even sample
distance, part of the amplitude is lost in the high frequencies because the
sample positions in the interpolated series do not necessarily correspond to
the position of the maxima and minima of the original time series (Fig. A1a
and b). Oversampling has been suggested to limit the loss of amplitude during
the interpolation process

To test which resampled time series fits best with the original time series,
various depths are tested as starting points to resample the time at the
average sample distance (Fig. A1d). The various scenarios of starting points
tested increase by d

The best-fit curve is the one for which M is minimised.

An example of application is shown for the La Thure section in Fig. A2.
Differences in the resulting spectrum between the best-fit and the worst-fit
resampled time series are displayed in this figure. Main differences in the
spectra of the two cases are observed in the middle and high frequencies.
Compared to the worst-fit resampling, the spectra of the best-fit resampling
show decreased power and confidence levels in the middle frequencies (from
0.2 to 0.7 cycles m

Scheme of the procedure of the optimised linear interpolation of time series.

Comparison of spectra of the resampled time series for the worst-fit
case (

ERC Consolidator Grant “EarthSequencing” (Grant Agreement no. 617462) funded this project. We acknowledge Christian Zeeden and Linda Hinnov for their thorough and insightful reviews. We also thank Anna-Joy Drury for English proof reading. The article processing charges for this open-access publication were covered by the University of Bremen. Edited by: H. Fischer Reviewed by: C. Zeeden and L. A. Hinnov