Examples

6. Examples

Example #1: Insolation

Data: Insolation at 65°N on June 22 over the past 2 million years

Age: 0-2000 ka

Proxy: Insolation.

Target: Dominated cycles of insolation series

Tool: Acycle software v0.3 (https://github.com/mingsongli/acycle)

Reference:

Berger, A., 1978. Long-term variations of daily insolation and Quaternary climatic changes. Journal of the atmospheric sciences 35, 2362-2367.
Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A.C.M., Levrard, B., 2004. A long-term numerical solution for the insolation quantities of the Earth. Astronomy & Astrophysics 428, 261-285.

Step 1: Load data

You will have the following data and figure.

Step 2: Data pre-processing

Since the data is not in ascending order. Here we’ll need sort data first.

Step 3: Detrending

Remove the mean value of the insolation series.

You will have:

Step 4: Power Spectral Analysis

Using the following settings:

Three peaks in the 2π (Number of tapers) MTM (multi-taper method) power spectrum are 1/0.04218 = 23.7 kyr, 1/0.04468 = 22.4 kyr, and 1/0.05267 = 19.0 kyr.

Step 5: Evolutionary Spectral Analysis

This series is dominated by precession cycles. And clearly 405-kyr modulation can be seen in the evolutionary fast Fourier transform (blue arrows).

Example #2: La2004 astronomical solution (ETP)

Data: La2004 ETP over the past 2 million years

Age: 0-2000 ka

Proxy: Laskar et al. (2004) astronomical solutions of Eccentricity, Tilt (obliquity), and Precession, or ETP is defined as:

ETP = standardized E + standardized T - standardized P
, where standardized E = (E – mean(E))/ standard deviation of E

Target: Dominated cycles of ETP series

Tool: Acycle software v0.3 (https://github.com/mingsongli/acycle)

Reference:

Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A.C.M., Levrard, B., 2004. A long-term numerical solution for the insolation quantities of the Earth. Astronomy & Astrophysics 428, 261-285.

Step 1: Load data

You will have:

Step 2: Data pre-processing

Since the data is not in ascending order. Here we’ll need sort data first.

Step 3: Detrending

Remove the mean value of the insolation series.

Step 4: Power Spectral Analysis

Using the following settings:

Seven peaks in the 2π (Number of tapers) MTM (multi-taper method) power spectrum are 405 kyr, 125 kyr, 95 kyr, 41 kyr, 23.7 kyr, 22.4 kyr, and 19.0 kyr

Step 5: Evolutionary Spectral Analysis

This series is dominated by 405 kyr long eccentricity, ~100 kyr short eccentricity, 41 kyr obliquity, 22 kyr and 19 kyr precession cycles.

Step 6: Wavelet transform

Using the following settings:

_Seven peaks in the 2π (Number of tapers) MTM (multi-taper method) power spectrum are 405 kyr, 125 kyr, 95 kyr, 41 kyr, 23.7 kyr, 22.4 kyr, and 19.0 kyr.

Example #3: Carnian cyclostratigraphy

Section: Wayao section, Guizhou, South China

Age: middle Carnian

Lithology: The limestone beds of the Zhuganpo Formation displays patterns of variable bed thicknesses and changing clay content within the limestones as reflected in relative weathering resistance.

Proxy: These factors influence the natural gamma-ray signal with higher intensities indicating higher average clay contents.

Target: Cyclostratigraphic analysis of gamma ray series

Tool: Acycle v0.3 (https://github.com/mingsongli/acycle).

Reference: Zhang, Y., Li, M., Ogg, J.G., Montgomery, P., Huang, C., Chen, Z.-Q., Shi, Z., Enos, P., Lehrmann, D.J., 2015. Cycle-calibrated Magnetostratigraphy of middle Carnian from South China: Implications for Late Triassic Time Scale and Termination of the Yangtze Platform. Palaeogeography, Palaeoclimatology, Palaeoecology 436, 135-166.

Step 1. Load Data

Select: Acycle main window: Basic Series --> Examples --> Late Triassic Wayao gamma ray.

The gamma ray data entitled “Example-WayaoCarnianGR0.txt” will be loaded and displayed in the Acycle main window.

Left click to select the data file and select Plot --> Plot to plot the data. Double click the data file to see the accepted format of Acycle software.

Step 2. Data Preparation

Acycle includes several toolboxes to facilitate data preparation.
Users can sort data in ascending order.
Two or more values for the same time (or depth) may be averaged with the "Unique" function.

Step 3. Interpolation

Stratigraphic depth or time series are typically irregularly spaced due to uncertain timescales or difficulty in data collection. This necessitates interpolation to generate uniformly spaced time (or depth) series.

Let’s look at the sampling rate plot first.

Select Plot --> Sampling Rate

You’ll see the sampling intervals of gamma ray data are irregularly spaced with a median of 0.3333 and mean of 0.35341 (right up corner of figures below).

Math --> Interpolation (or Ctrl + I).

Then type the new sampling rate to interpolate.

I use a 0.33 m as a new sampling rate, Acycle will generate a uniformly-spaced file entitled: “Example-WayaoCarnianGR0-rsp0.33.txt”.

Step 4. Detrending

Detrending is a key step in time series analysis.

Removal of these long-term trends, or detrending, is a critical step for power spectral analysis to ensure that data variability oscillates about a zero mean, and to avoid power leakage from very low-frequency components into higher frequencies of the spectrum.

Select the text file; then select Timeseries --> Detrending (or CTRL + T).

In the pop-up window, select window size, detrending method.

Then click OK to see the various trending.

Don’t close “Acycle: Detrending” window or “New figure” window. Now change window size in the left panel, you will see the response in the right panel.

You will need to "Select & Save detrending Model".

I will choose an 80-m LOWESS trend for the best fit of the data without removing too many cycles.

The Acycle main window now displays an “Example-WayaoCarnianGR0-rsp0.33-80-LOWESS.txt” detrended file and a “***-LOWESStrend.txt” trend file.

Step 5. Power spectral analysis

Power spectral analysis has become a cornerstone in paleoclimatology and cyclostratigraphy. Power spectral analysis evaluates the distribution of time series variance (power) as a function of frequency. The primary use of power spectral analysis is for the recognition of periodic or quasi-periodic components in a data series

Select the detrended file and choose “TimeSeries” --> “Spectral Analysis” menu

Then choose Multi-taper method (MTM) with robust AR (1) red noise models.

Use the following setting:

2 pi MTM with a 5 times zero-padding (to increase frequency resolution).
The maximum frequency set to 1 cycle/m and use a linear Y plot.
Testing with a robust AR1 red noise model, then (right panel) using a 20% median smoothing window and fitting to a log power of spectrum power.

You will have the MTM power spectrum with red noise models.

Remember the period of a given cycle (frequency peak) is 1/frequency. For example, the highest frequency peak (middle value) is 0.02951 cycles/m. The corresponding cycle is 1/0.02951 = 33.9 m.

2π MTM power spectrum of the gamma ray series is shown with 20% median-smoothed spectrum, background AR(1) model, and 90%, 95%, 99%, and 99.9% confidence levels.

If you count all peaks higher than 95% confidence levels, you will find the 33.9 m, 10 m, 7 m, 2.6 m, and 1.8 m cycles. The ratios of these cycles are 405 kyr, 119 kyr, 83 kyr, 31 kyr, and 21.5 kyr cycles].

Step 6. Evolutionary power spectral analysis

Select data and then select “TimeSeries” --> "Evolutionary Spectral Analysis" menu

Use the following settings.

A sliding window of 40 m

Why?
The longest cycle is 33.9 m, this window should be larger than 33.9 m. A 1.5-2 times of 33.9 m is good enough.

The maximum frequency is 0.7, this is to highlight low-frequency power.
Normalize each window: make spectral peaks in each window to be 1.
Flip Y-axis: because the first column of this data is increasing upward. Then click ok to show results.

This figure tells me the dominated cycles of ~34 m is stable in frequency (period). Therefore, the sedimentation rate is probably not variable (too much)

Don’t close these two windows.
Now, you may change frequency limit, flip Y-axis, change colormap to change the left window.

Step 7. Correlation coefficient

To estimate the optimal sedimentation rate.

Select the detrended data, then click “Timeseries” --> "Correlation coefficient" menu.

Tell COCO the middle age of your data (~235 Ma).

It doesn’t matter if this age has an uncertainty, an uncertainty less than 2-5 Myr can be okay.

Tell the testing sedimentation rate range

from 1 to 30 cm/kyr, with a step of 0.1

It will test: 1, 1.1, 1.2, 1.3, ……. 29.8, 29.9, 30 cm/kyr.

Monte Carlo simulation: the number is 1000 (or 500) for an initial test.

A 2000 (or more) number is recommended for a publication purpose.

Split series: If the data set is very long (too many peaks in power spectrum of the data), Split series may use 2 or 3.

You will have the following figure and a log file saving all settings:

It tells the most likely sedimentation rate is ~10 cm/kyr, with a significance level of 0.1%. All seven orbital parameters are used in the estimation.

Now using a 45 m window eCOCO analysis to track variable sedimentation rate.

Step 8. Filtering

Filters are also essential tools to aid in the isolation of specific frequency components in the paleoclimate data series.

Select data, then “Timeseries” --> "Filtering" menu

In the pop-up window Select the center frequency and low frequency.

Then select the Gaussian method. And “save data” button.

You will see the filtered series and data in the Acycle main window.

Step 9. Age model and tuning

“Age Scale” toolbox in Acycle is useful to transform original data (usually in the depth domain) to tuned data (usually in the time domain) when an age model file is available.

Assuming these 33.4 m cycles are 405 kyr cycles based on COCO result. Select “Example-WayaoCarnianGR0-rsp0.33-80-LOWESS-gaus-0.028+-0.006.txt”

And then select "Timeseries" --> "Build Age Model" menu

Click OK, you will have an Age Model file:
Example-WayaoCarnianGR0-rsp0.33-80-LOWESS-gaus-0.028+-0.006-agemod-405-max.txt

Then, select "Timeseries" --> "Age Scale" menu

Select the above age model file, and select files to be tuned, click "OK" button.

Tuned data will be ready. “Example-WayaoCarnianGR0-TD-Example-WayaoCarnianGR0-rsp0.33-80-LOWESS-gaus-0.028+-0.006-agemod-405-max.txt”