ATMO 632: Week 3

       Time Series Analysis: Spectral Analysis

 

                    Course Home Page: http://www.cgd.ucar.edu/~svn/atmo632

 

Reading Material

 

  1. Wilks: Chapter 8.4-8.5 (Harmonic and Spectral Analysis)
  2. North: Chapter 6 (Spectral Analysis of Time Series)
  3. Hartmann: Chapter 6.2 (Harmonic Analysis)

 

 

Concepts

 

            Continuous Fourier Transform

                        Box function

                        Gaussian

                        Delta function

                        Convolution

                        Parseval’s theorem

 

            Discrete Fourier Transform

                        Fourier series

                        Discrete Parseval’s theorem

                        Data windowing

 

            Power spectrum

                        White noise

                        Red noise

                        Aliasing

                        Plotting conventions

 

            Significance of spectral peaks

                        Chi-square distribution

                        a priori vs. a posteriori confidence intervals (see Hartmann’s notes: 6.2.6)

 

           


ATMO 632: Homework for week 3 (due Sep. 21)

           

NOTE: You can obtain the datasets used below by visiting the course home page

                        http://www.cgd.ucar.edu/~svn/atmo632 and clicking the appropriate links.

You do not need to type in the URLs given below, which are provided for reference.

 

 

  1. Analysis of the spatial structure of El Nino using Empirical Orthogonal Functions obtained from Singular Value Decomposition: Obtain the Northern winter (Dec-Feb) average sea surface temperature (SST) for the spatial domain (160E-90W, 12.5S-12.5N) in the Tropical Pacific region for the 50 year period 1951 to 2000, where 1951 refers to year of the January in the winter 1950-51 and so on. The spatial grid size is 40x10, with 40 equally spaced longitude values from 160E to 90W and 10 equally spaced latitude values from 12.5S to 12.5N. The data is available as a plain text file containing 10 SST values per line (in degrees C), with each set of four lines corresponding to the 40 longitudinal values (=> forty lines correspond to data for each year, latitude values ordered from south to north). There are 2000 lines of data in the file, corresponding to 50 years of SST data, starting from 1951. The file is available from the URL http://www.cgd.ucar.edu/~svn/atmo632/sst-pacific-djf-1951-2000.txt

 

    1. Write a program to read the SST data and store it as a MxN matrix X, where M=400(=40*10) denotes all the spatial grid points and N=50 denotes the time values.
    2. Remove the time-average from each row, by averaging the N values in each row and subtracting it from each element of the row. We will refer to this as the anomaly data matrix X’.
    3. Compute the “total variance” of X’, defined as the sum of all its squared elements.
    4. Compute the singular value decomposition of X’: X’ = U S VT , where U is a MxM orthonormal eigenvector matrix and V is a NxN orthonormal eigenvector matrix, and the N diagonal values of S are the singular values. Order the singular values from the largest value to the smallest value.
    5. Compute the sum of all the squared singular values; compare it to the “total variance” of X. Note that the squared singular values, divided by, are the same as the eigenvalues of EOF analysis. (For extra credit, you may compute the variance matrix (1/N)X’ X’T, compute its eigenvalues explicitly and verify this statement.)
    6. For the first five largest singular values, compute the sum of the squared values and compare it to the “total variance”. How much of the total variance does the first singular value “explain”? How much do the first five singular values together explain?
    7. Plot the first two columns of U, i.e., the left eigenvectors corresponding to the two largest singular values, as a longitude-latitude plot, by interpreting the M values in each column as corresponding to a 40x10 grid. These correspond to EOF1 and EOF2.
    8. Plot the first column of V, i.e., the right eigenvector corresponding to the largest singular value, as a time series, after multiplying by . This corresponds to the dimensionless expansion coefficient for EOF1. Compute the mean and standard deviation of the time series. Compare it to the NINO3 time series plotted in the Week 2 homework.
    9. Compute the regression of the original SST data against the above time series, for each grid point, and plot the slope of the regression at each grid point as a longitude-latitude plot. (Treat the SST values as the dependent variable for regression.)
    10. Re-plot the first column of U, after multiplying each element by s1/, where s1 denotes the first (largest) singular value. Compare it to the above regression plot.
    11. Discuss the spatial structure of the SST variability associated with El Nino based upon the above results.