First we will finish up with a couple of more exercises using the central england temperature record.

1. Download the following the seasonal CET data set. Note that the time period for this set of data starts in 1659 and the data is the mean daily temperature per season.
   
   
   • Seasonal Temperature Data
   
   
   
2. Using the basline optimization program that you just wrote, produce the seasonal tempearatue anomalies (that maximize the warming residual) for the two seasons DJF and JJA). In this case use a baseline of 50 years. Comment on which season you believe shows the strongest effect.
   
   
3. For the period prior to 1960, determine the number of winter seasons which exhibit an average temperature of less than 2.3 degrees. Divide that number by the number of decades you sample. Use that expectation value and poisson statistics to determine the probability that over the 15 year period 2003-2017 there would be 0 events.
   
   Simple tutorial on Poisson statistics
   
   

Power Spectral Density Exercises:

This assignment will exposure you to Fast Fourier Transforms (FFT) and how that can be used to make power spectra to search for specific frequencies that have the most power and to find periodic signals in data. You will be applying this technique to several real data sets, some of which, may or may not contain signals, but the first data set does for sure.

For this exercise you will only be dealing with 2-3 data sets with more to follow for the following week.

Steps:

1. Download the file
   
   
   • Icecore
     
     This data is organized as follows:
     
     
     • Column 1: Depth (meters)
     • Column 2: Ice Age -in units of years before present
     • Column 3: Deuterium content of ice (measured with respect to standard mean Ocean sea water) - we will call this δD
     • Column 4: "Global" Temperature difference compared to current time
   
   
   
2. Starting with basics:
   
   
   • a) Report the number of lines in the file and how you did this
     
     
   • b) Make a plot of δD vs time. Submit that plot and make a comment on its overall shape and structure.
     
     
   • c) Demonstrate that the data are not evenly spread out in time by submitting some kind of plot that shows non-even time spacing
     While your eye does well at spotting periodic behavior, we want to be quantitative about this using a Fourier transormation, specifically the Fast Fourier Transform (FFT). FFT works most efficiently for data sets that have evenly speced data points and the total number of data points is a power of two. However, nature does not provide such controlled data sets.
     
     
   • d) In order to feed the data to the FFT package you will need to transfrom and rebin the data using some interpolation function (or you could write your own). So rebin δD into 2ⁿ data points that are evenly spaced in time. The higher that N is, the more data points you have but the more noise you are likely to include; the lower N is, the lower time resolution you have - so its a balance. For this exercise let's use N=7, N=11, N=15 - start at zero and go back as far as you can with your choice of N (for instance N=11 will truncate the data at 414000 years).
     
     Report the length of the data series for each value of N to check that your data really is 2^N long. For each value of N plot your now synthetic data over the original time series of δD and include all plots. Comment on how well your interpolation seems to reproduce the data as well as on any places there it doesn't seem to do particularly well
     
     
   • e) Now feed your rebinned arrays into the Fourier Transform. That would produce an array of 2^N complex numbers. The first element of that array is the 0th frequency term which is just the sum of all the δD values. compare and report on this sum for the 3 values of N compared to the raw sum of the original data
     
     The first half of the remaining returned array (elements 2 through 1024 for the case of N=11) contains complex numbers which are just the coefficients of the cosine and sine terms in the Fourier series, where the n-1 element corresponds to the nth period.
     
     Lets call the returned values in the Fourier array fft. fft(2) corresponds to the lowest frequency, which should be 414 kiloyears (the whole time period); fft(3) would be 1/2 of that, fft(4) would be 1/3 of that, and so on. fft(1023) is called the Nyquist frequency is is the highest frequency that can be resolved with your time sampling (404.3 years in the case of N=11); the second half of the array contains unphysical negative frequency values so don't worry about that.
     
     Now fft(2) has a real and imaginary part. One-half of the square of the absolute value of say (r_n^2 +i_n^2)/2 is the contribution of that frequency to the variance (this is called a power spectrum when plotted).
     
     
   • f) Produce a plot of magnitude vs frequency but make sure you use the ABS value of your frequencies. (depending on which fourier package you used, you might have worry about 2π). Submit a line plot of magnitude vs frequency (this plot may be messy).
     
     
   • g) You should see that most of the power is in the first few components so submit another plot with just the first 50 components plotted. Report the frequencies of the 4 biggest relative peaks and comment on how clean this signal is and which value of N (7,11 or 15) shows this the best
     
     
3. Now repeat the following steps to determined if there is periodicity and/or characteristic frequencies in the following data sets.
   
   
   • a) download the respective data sit from the list below
     
     
   • b) submit a smoothed plot of the time series (smoothing helps to possibly identify a periodicity) and plot on top of that what you think a period might be
     
     
   • c) Perform the FFT analysis and produce the magnitude vs frequency plot
     
     
   • d) Comment on what aspects of the data that you think are causing the FFT to fail (or to suceed if it actually can)
     
     
   
   
   Other Data Sets to Download:
   
   
   • Cascadia Megathrust earthquakes Note there are 18 events here - the age entries after each event are in -years from 1990 (we know the last one occurred in 1700). Multiple age entries per event represent different ages for different turbidity cores - average them. Do the analysis but also repeat it with event T2 removed as this event is somewhat controversial
     
     
   • Extinction events column 1 is million years before present; column 2 is number of genera on the planet. An extinction event is a relative reduction in the number of genera between time steps. Good luck