1 The Venzke et al. * Optimal Detection Analysis Jeff Knight * Venzke, S., M. R. Allen, R. T. Sutton and D. P. Rowell, The Atmospheric Response over the North Atlantic to Decadal Changes in Sea Surface Temperature, J. Clim, 12, , 1999.

2 Ensembles Initial-value ensembles of climate model calculations allow estimation of the climate response to forcings, as multiple realisations of the chaotic noise inherent in the climate system are available. The independence of these noise components in each simulation allows it to be ‘averaged out’ by the formation of an ensemble mean.

3 Ensemble means For data X from calculation k: X k = X M + X Nk, where X M is the ensemble mean and X Nk is the noise in k. For large enough ensemble size n, the forced signal in X is X F = E(X M )  X M.

4 EOF analysis A common way to analyse the forced model response is principal component analysis (PCA, or EOF analysis) of the ensemble mean. PCA aims to find the covarying spatial patterns that have maximum variance in the data. In essence we can treat the mean X M as a set of time series for each point, X M (i, j) ; i = 1, l ; j = 1, m i.e. the arrangement of points irrelevant.

5 SVD analysis In general, an l×m matrix can be decomposed X l,m = E l,m S m,m P T l,m, where the columns of E are the eof patterns of X, S is a diagonal matrix of the variance coefficients (square eigenvalues) and P contains the associated principal component time series. A Singular Value Decomposition.

6 The trouble with this… EOF analysis of means of realistic size climate ensembles will only give accurate estimates of the EOFs of X F if the noise is not spatially coherent. But this is not the case. Even worse, it is likely that the patterns of noise are in fact quite similar to the signal (e.g. NAO). So the ensemble mean EOFs are heavily biased away from the forced response towards the noise.

7 The way forward Ensemble averaging throws out a lot of the information about the spatial characteristics of the noise. The aim of the Venzke technique is to estimate the properties of the noise so better account of its influence can be made in estimating the forced EOFs.

8 Technique We can identify the noise in each experiment: X Nk = X k - X M, Rewrite the noise X N as the concatenation of each run X Nk : X N = {X N1 (i, j), X N2 (i, j), …, X Nn (i, j)}, i = 1, l ; j = 1, m, making a matrix of size l  nm.

9 Noise EOFs Next, compute the EOFs of the noise by SVD: X N = E N S N P N T, so the columns of E N are the EOF patterns of the noise. These patterns can be used to define a new BASIS for the ensemble mean, i.e. we seek to express the ensemble mean in terms of these EOF patterns. We choose the  best defined patterns to do this.

10 Prewhitening Function If we define a matrix function, F = n ½ E N (  ) (S N (  ) ) -1, and its (pseudo-) inverse, F (-1) = n -½ E N (  ) S N (  ), we notice that multiplying by F projects onto the noise EOFs and scales the result by the inverse singular values, or the variance.

11 Prewhitening So the transform X M = F T X M converts the ensemble mean into a set of time series of projections of the mean on the noise EOFs, such that each EOF has the same weight in the result. In other words, we have transformed the data into a basis made of these EOFs.

12 EOFs of transformed mean As we now have a mean in which the signals lie amongst unit-variance noise EOFs, we can isolate them by further EOF analysis: X M = ESP T, so that E’ contains the EOFs under this new basis. But beware! The weighting at each point in the EOF relates to the strength of each noise pattern, not the spatial weight.

13 Back to physical space To get back to a physical basis from the noise EOF basis, we need to invert the transform based on noise EOFs, Ê = F (-1) E, which also rescales, giving us back an l×  matrix i.e. the truncated number of optimal EOF patterns of l points each. Projection on the mean gives the principal components.

14 To summarise Concatenate the noise estimates for each realisation, X Nk Derive joint noise EOF patterns, E N Project these patterns onto the ensemble mean, scaling the EOFs by their variance Find the signal by deriving EOFs in this basis Transform these EOFs back to physical space

15 An Example Courtesy of Rowan Sutton We create a synthetic signal on a 20×20 grid (l = 400) by taking a simple harmonic pattern and modulating through m = 50 times with a sine of period 20.

16 Signal Pattern

17 Noise Patterns Generate random noise by associating a random time series with each 2D harmonic pattern H i,j, for i,j = 1,5, and weighting the variance by e -(i+j). Repeat this for n = 6 ensemble members.

18 Noise Patterns

19 Synthetic Data

20 Ensemble Mean EOFs

21 Noise EOFs

22 EOFs in the noise basis

23 Optimised 1st EOF

24 Conclusions Venzke et al. find that Principal Component Analysis of the mean of realistic ensembles is heavily contaminated by the noise and so provides a poor estimate of the EOFs of the forced response. By using information about the spatial characteristics of the noise in the ensemble, these estimates can be improved.