This exercise is intended to help students to understand how to detect cycles in a time series and design filters to eliminate individual cycles.
The file odp659data.txt contains marine oxygen isotope data from foraminifera sampled from a deep-see sediment core. Such isotope data reflect changes in global ice volume, in the case of planktonic foraminifera, also local sea surface temperature and salinity of the sea water on time scales of thousands of years. All three parameters are strongly influenced by orbital cyclicities according to Milankovitch theory. The data series contains three main periodicities of 100 kyrs (kilo years, thousands of years), 40 kyrs and 20 kyrs. The first column of the data represent age (in kyrs), the second column represent the oxygen isotope value, i.e., the ratio of the oxygen isotopes O18 and O16 converted to a percentage difference from the ratio found in a carbonate standard.
(1) Interpolation – The transformation of evenly-spaced length-parameter data to time-parameter data in an environment with changing length-time ratios results in unevenly-spaced time series. Fast-Fourier Transform-based spectral analysis tools such as the Blackman-Tukey method or the Periodogram method require evenly-spaced data. Please interpolate the unevenly data upon an evenly-spaced time axis. Plot the original data and the interpolated data together within a single plot and compare the results.
(2) Spectral analysis – Use the Periodogram method to detect the three main Milankovitch cycles in the interpolated data series. Use detrending prior to the application of the periodogram method to remove a temporal trend. Optional: Compute an evolutionary spectral method to detect changes of the cyclities through time.
(3) Filtering – Design a filter to eliminate the 40 kyr cycle in the data. Display the amplitude response of the filter. Filter the data and display the filtered time series in both time and frequency domain.
Please use the publishing feature of MATLAB to publish the results of your experiment as a PDF file with all graphs included.
Solution
Here is a possible solution of the exercise. You need to rename to file exercise_3_solution.m before using it with MATLAB. Please send me your comments, improvements, corrections!