Issues concerning S/N optimization procedure Preliminary analysis of ECHAM4.5

October 5, 2001

1 S/N optimization procedure

Truncations.

Three truncations made in the calculation of the statistical atmosphere:

• number of noise EOFs,

• number of S/N EOFs,

• number of singular vectors.

Number of noise EOFs for S/N EOF calculation

Find the patterns that maximize the ratio:

λ = \frac{x^{T} C_{M}x}{x^{T} C_{N}x} = \frac{y^{T} C_{N}^{−1/2}C_{M}C_{N}^{−1/2}y}{y^{T} y} , y = C_{N}^{1/2}x .  (1)

C_{N} is noise covariance. C_{M} is ensemble mean covariance. Inverse of noise covariance C_{N} is poorly known.

How to truncate C_{N}?

Look at time-series. Individual patterns. Local measure.

• Look at correlation of time-series as a function of the number of noise EOFs.

• Look at error of time-series as a function of the number of noise EOFs.

• Look at convergence. |T s_{k+1} − T s_{k}|.

Global measure? Look at sum of S/N eigenvalues ∑_{i} λ_{i} where

λ_{i} = \frac{x_{i}^{T} C_{M}x_{i}}{x_{i}^{T} C_{N}^{T}x_{i}} .

2 Preliminary analysis of ECHAM4.5

Ratio of ensemble mean variance to noise variance.

EOFs. Ratio signal variance to ensemble mean variance. Note: ECHAM4.5 still seems noisier. Large inter-ensemble variability in the

the ECHAM4.5 precipiation relative to the CCM and COLA models was noted in Camargo et al, Seasonal Comparison of the Response of CCM3.6, ECHAM4.5 and COLA2.0 Atmospheric Models to Observed SSTs , (e.g.,see Fig. 14)

Time-series