Empirical Orthogonal Functions

The climate state is a time-dependent multivariate system. The eigenfunctions of the covariance matrix of the various state parameters (temperature, pressure, sea ice, precipitation, etcetera) are termed "empirical orthogonal functions" or "EOFs" and the set of corresponding eigenvalues are the "EOF variance spectra". EOFs and their variance spectra are of particular interest, because they represent basic information on the parent probability distribution of the climate.

Empirical estimates of these are based on a finite number, say n, realizations of the the instantaneous state of a geophysical field during a limited time period, and the field is sampled at a finite number of points, say p, covering a limited spatial domain. Errors in these estimates depend on the true spectrum of the field, which determines whether p and n have been appropriately chosen. There are two interesting contrasting cases:

All true eigenvalues are equal
No true eigenvalues are equal

The first case is equivalent to the statement that the true covariance between climatic variables at any two different points vanishes. There is both good news and bad news for this case: The good news is that addition of any such uncorrelated variations (which might arise, for example, from measurement error or other noise sources) to the true correlated climatic variations does not alter the shape of the true eigenvalue spectrum, but only increases each eigenvalue by a constant. The bad news is that this "nondistortion" theorem holds only when n > > p, otherwise most of the noise goes into the first few eigenvalues. Fortunately, the exact noise spectrum is known analytically as a function of p and n.

Here are some comparisons between the exact noise spectrum for 91 years of monthly mean temperature and precipitation measured at 62 stations distributed fairly uniformly over the continental United States:

January temperature. The first EOF has about 40% of the variance. The first few are well above the noise spectrum for the total variance (the dashed curve). When the variance of the first 11 is removed, an addition few are still marginally above the noise spectrum for the remaining modes (the dotted curve), but modes 9 through 14 are all effectively "degenerate" with each other. i.e. Their eigenvalues are not distinguishable, and thus any linear combination of their eigenfunctions are equivalent.
January precipitation. The first EOF has about 19% of the variance. The first four or five are significantly above the noise spectrum, but modes 2 and 3 are effectively degenerate, as are modes 6 and beyond.
July temperature. The first EOF has about 30% of the variance. Modes 2, 3, and 4 are effectively degenerate, as are modes beyond 6.
July precipitation. The first EOF has about 9% of the variance. Only the first two eigenvalues are distinguishable from the noise, and they are effectively degenerate.