Homework 3 – taken from Gidon Eshels Homeworks 6-7

The dataset for this work, Z700.mat, is here. (You might need to push the Shift key before, and during, clicking on this link in order to download the file.) It is the 700 mb geopotential height. When you load it to your workspace, you will find 4 variables, of which 3 are grids that the height dataset depends on. T is the 168-long time grid in units of calendar years and fractions thereof. Its points are season centers, 3 months apart. X is the 25-long longitude grid in degrees east of Greenwich (that is, X=2, the last X value, is 2^oE, while X=-94, the first X value, is 94^oW). Y is the 20-long latitude grid in degrees north, spanning 10^o-67^oN in 3^o increments. Finally, Z7(Y,X,T) is the three-dimensional array containing the height data (in so-called `dynamic meters'). The MATLAB command A = squeeze(Z7(:,:,i)); will put in a matrix A the geopotential height field of season i. Thus the matrix A will depend only on X and Y, i.e., it is a simple map (in the colloquial sense).

1. Obtain the climatological height field for the four seasons. For each season, this means averaging all the available seasons the dataset comprises, thus reducing the entire dataset to 4 maps, one for each season. Plot the 4 maps on one sheet [using axes('Position',v), or subplot(2,2,plot number)]. Try to figure out how to plot the coastline contour, so the map will be easier to examine. See mapwcoast.m for an example which uses the matlab m_map package. Documentation of this toolbox is found here: M_Map: A Mapping package for Matlab.

2. Obtain anomalies. That is, from each of the 168 available seasons, subtract the appropriate climatological map you obtained before. Plot the anomaly field for the last 4 available seasons (again on one sheet).

3. Use reshape to turn the entire (anomalies) dataset into one big matrix with dimensions 500x168. Use svd to decompose the matrix. Plot the first 100 singular values as a function of mode number, and comment on the plot. Plot the spatial pattern of the first mode.

4. Obtain the EOFs of the seasonal cycle of anomalies. This is done by creating annual 3-D fields consisting of the latitude-longitude-seasonally varying fields, and then reshaping each annual field to a long vector. You now have a matrix of 42 (years) by 2000 (four seasons of 20x25 maps), on which you do your EOF analysis. The EOF patterns are a series of 4 seasonal maps.

5. Download Atlantic sea-surface temperature anomalies (SSTAs). Trim the timegrids of both the SSTAs and the Z700 anomalies so that they both cover the same time period, that of the overlap. Now construct the covariance matrix, and decompose it. Plot the singular spectra, and the 2 leading covariability modes of the 2 fields. Plot loading timeseries of the 2 leading covariability modes for both fields (4 all together).