Oceanography 569 Oceanographic Data Analysis Laboratory Kathie Kelly Applied Physics Laboratory 515 Ben Hall IR Bldg class web site: faculty.washington.edu/kellyapl/classes/ocean569_ 2014/ Kathie Kelly Applied Physics Laboratory 515 Ben Hall IR Bldg class web site: faculty.washington.edu/kellyapl/classes/ocean569_ 2014/

Exercise 5: Taylor Diagram NCEP2 fluxes have too large amplitudes and slightly poorer correlation with observed OAFlux slightly underestimates the anomalies with similar correlation but smaller normalized error Skills are surprisingly small... it sets a high bar. OAFlux skill is much higher.

Empirical Orthogonal Functions (EOFs) or Principal Component Analysis F – typically a spatial pattern A - typically a time series, “principal component” (PC)

Linear Algebra: special matrices

EOFs by Singular Value Decomposition associate magnitudes S with V (time series or principal components) The squares of the diagonals of matrix S are the variance in each mode

EOFs: Practical Issues EOFs are a statistical tool in which the object is to understand random variables (mean?? periodic signals??) Data matrix can only be 2D (usually, space and time). But space has two dimensions and missing regions (land). What to do?

EOFs: Practical Issues Data matrix can only be 2D (usually, space and time). But space has two dimensions and missing regions (land). 1) Make 2D space into a vector by ordering locations: dat=reshape(data,nx*ny,nt); 2) Save only valid regions tmp=sum(dat,2);indv=find(~isnan(tmp)); D=dat(indv,:);

EOFs: Practical Issues What if the data are vectors? The easy way: enter both components (u,v) into single array D = (with M columns for locations x and 2 N rows for N times) When unpacking the modes U, remember that the top M rows correspond to the u-component and the bottom M rows correspond to the v-component.

EOFs of SSH using SVD EOF 1 mostly one sign EOF 2 “dipole”

EOFs of SST modes can be reversed: change sign of both F i and A i

EOFs of SST EOF 1 (only) reversed: similar to SSH PC 1 (only) reversed

Significance of EOF Modes Fraction of variance in random numbers * 9 significant modes Monte Carlo Simulation 1.find degrees of freedom in data matrix D (N*, M*) 2.generate matrix R(N*,M*) of random numbers 3.compute EOFs on R 4.compare variance in data EOFs with simulated

EOFs Comparing Fields How similar are these amplitudes? Could the longer record of SST be used to infer historical values of SSH?

EOFs by Singular Value Decomposition associate magnitudes S with V (time series or principal components) The squares of the diagonals of matrix S are the variance in each mode

Covariance Matrix Methods for EOFs for data with gaps Diagonals of covariance matrix are the variance But because there are gaps the amplitudes A are found by doing a least-squares fit of the data to the significant modes

Equivalence of EOF Methods For data without gaps the covariance method gives the same result as SVD on data matrix D

Covariance Method Bookkeeping device for covariance computation NN = N*N’covariance: D*D’/NN N number of valid entries at each location (x,y) D data matrix with 0 for invalid data (NaN)

Covariance Matrix Eigenvectors EOFs on wind vectors with gaps EOF computation: Remove seasonal cycle Covariance matrix Find PCs by least- squares fit of data to EOFs (U) Plot only times with at least 3 buoys operational

Different Results from EOFs