GEOS 597e PW4

GEOS 597e
Spatiotemporal Data Analysis Workshop

Prework 4: Empirical orthogonal function calculation

This week we'll learn the rudimentary algebra of EOF calculation, and we'll calculate the empirical orthogonal functions and principal components of the covariance matrix of GOSTA sea surface temperatures for 1961-1990.

Last updated 9/14/06. To be completed prior to class session Weds., Sept. 20th.

Introduction:

The mathematics and analysis of empirical orthogonal functions are very profound. This week we'll just touch on the most important features of the technique, and see how it works in practice.

Reading:

Peixoto and Oort, 1992: Physics of Climate, Appendix B: Analysis in Terms of Empirical Orthogonal Functions. New York: American Institute of Physics.
Weare et al., 1976: Empirical Orthogonal Function Analysis of Pacific Sea Surface Temperatures, J. Phys. Oceanogr., 6, 671-678.

Reading questions:

Why is this statistical decomposition called "empirical" and "orthogonal"? How is an EOF analysis similar and different to the analogous features of a Fourier analysis?
Why did Weare and colleagues get different results with and without the annual cycle left in the data series?
What exactly is plotted in Weare et al., Figure 2, and how is it related to the results of the EOF analysis results plotted in Figures 3 and 4?
Do you think Weare and colleagues were justified in giving a physical interpretation to the statistical results shown in Figures 3 and 4? How about for Figures 5-10? Why/why not?
What are the greatest sources of uncertainty in the analysis of Weare et al.? Why?

Products to hand in (keep a copy for yourself to use in class):

Write down Peixoto and Oort's equations B1, B4 (let the left hand side of equation B4 be equal to S, the cost function, or sum of squares), B5, B6, B7, B9, B10, B11 in matrix form for two cases:

General case (as in P&O): data matrix F is composed of M time series and N observational stations.
Specific case: data matrix F is composed of 2 time series f1 and f2, each observed at 3 times t1, t2, and t3, organized as rows in a 2x3 matrix:

F = [ f₁₁ f₁₂ f₁₃

f₂₁ f₂₂ f₂₃ ]

(Note: Once you have written the elements of the 2 x 2 covariance matrix R in terms of the elements of the data matrix F, let the elements of R be called [r1 r2; r3 r4]. Note also that the determinant of a 2 x 2 matrix [a b; c d] is ad-bc.)

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