An estimation of ENSO predictability from its seasonal teleconnections

Abstract

We study here the potential predictability of the El Niño-Southern Oscillation (ENSO) state, represented by the Niño3.4, the Niño1+2 and the Niño4 Indexes. We choose the predictors among a variety of Teleconnection Indexes. We use a linear statistical relationship and focus on leads from one season to one year. Highest potential predictability levels are scored by the tropical predictors, in particular, the equatorial Pacific Warm Water Volume Index or the Pacific Meridional Mode Index. Moreover, our analysis finds and explores interesting potential predictors for the Niño1+2 and Niño4 conditions that were never pointed at before and test their prediction skill in a series seasonal hindcast experiments. Finally, we compare the results obtained for the potential predictability of the recent 1980–2012 years with those obtained for the 1950–1979 period using the same methodology. We find statistically significant differences in the linear relationship between the Niño3.4 Index and the South Tropical Zonal Gradient Index in the recent 1980–2012 period as compared with the earlier 1950–1979 one.

Appendix

The linear feedback model

There are physical systems where the variables can be labeled as ‘slow’ or ‘fast’ according to their very distinct characteristic time scales. In these systems, a change in the value of one of the slow variables can produce an influence on the others that these would return. In the case of systems where the coupling between the ‘slow’ and the ‘fast’ variables is weak, the effect of the fast variables on the evolution of the slow ones can be modeled by a stochastic forcing term. Let us assume that the future system state is determined only by its present state (the system is markovian) and that it can be represented by a reduced number of n variables that we collected into a vector state ${\overrightarrow{y}}^s$ (s denote the season).

We collect m samples of this vector, corresponding to m observations in time of the system state, into a m x n matrix Y. Its evolution in time can be represented by the linear Langevin equation:

$$ {d}_t{\overrightarrow{y}}^s+\mathbf{A}{\overrightarrow{y}}^s(t)={\displaystyle \sum_{j=1}^p}{\overrightarrow{N}}_j(t) $$

(A1)

where the system dynamics is represented by some feedback coefficients collected into the A matrix. The residual term ${\displaystyle \sum_j}{\overrightarrow{N}}_j(t)$, representing the forcings, is commonly referred to as ‘noise’.

The Principal Oscillation Pattern (POP) analysis (Hasselmann 1988) obtains a set of eigenvectors and eigenvalues that characterize the system dynamics and its time evolution from the analysis of the A matrix. From the equation (A1), if we assume that the noise is white and with a gaussian and delta-correlated statistics, we can determine the A matrix

$$ \mathbf{A}=\frac{\left\langle {\overrightarrow{y}}_{i+1}^s{\overrightarrow{y}}_i^s\right\rangle -\left\langle {\overrightarrow{y}}_i^s{\overrightarrow{y}}_i^s\right\rangle }{\left\langle {\overrightarrow{y}}_i^s{\overrightarrow{y}}_i^s\right\rangle } $$

(A2)

where the angular brackets denote averages to the m samples. This can be written in terms of C ₁ and C ₀, the covariance matrices at 1 and 0 lags, respectively

$$ \mathbf{A}=\frac{C_1-{C}_0}{C_0} $$

(A3)

In this way, we take out of the feedbacks coefficients the part of the cross-covariance matrix that is determined by its own autocovariances, and the part introduced by their common cross-covariance with a third variable. We can express the general solution of the equation (A1) as:

$$ {\overrightarrow{y}}_i^s={\displaystyle \sum_{k=1}^n}{C}_k{e}^{\mu_k\varDelta {t}_i}{\overrightarrow{u}}_k; \kern1em \varDelta {t}_i={t}_i-{t}_0,\kern0.75em i=1,..k $$

(A4)

where the characteristics exponents μ _k can be obtained from the eigenvalues λ _k , and the ${\overrightarrow{u}}_k$ are the corresponding eigenvectors of the A matrix. If the number of complex eigenvectors is l _i , (A4) can be written as:

$$ {\overrightarrow{y}}_i^s={\displaystyle \sum_{k=1}^{l_i}}{C}_k{e}^{\mu_k\varDelta {t}_i}\left[Re\left({\overrightarrow{u}}_k\right)+Im\left({\overrightarrow{u}}_k\right)\right]\kern0.5em +{\displaystyle \sum_{k={l}_i+1}^n}{C}_k{e}^{\mu_k\varDelta {t}_i}\ {\overrightarrow{u}}_k $$

(A5)

If we denote as γ _k ≡Re(μ _k ) and ${\tilde{\omega}}_k\equiv Im\left({\mu}_k\right)$, it is easy to work out that for each pair of complex conjugated eigenvectors, μ _k and μ _k + 1, there is a pair of real patterns ${\overrightarrow{p}}_k=Re\left({\overrightarrow{u}}_k\right)$ and ${\overrightarrow{q}}_k=Im\left({\overrightarrow{u}}_k\right)$ associated to a frequency ${\tilde{\omega}}_k$, a period T _k and a damping factor γ _k . These are the Principal Oscillation Patterns (POP) of the system at that frequency. For each pair of POP derived from a pair of complex eigenvectors, eq. A5 prescribes the following evolution in time

$$ {\overrightarrow{p}}_k\kern0.75em \overset{\kern0.75em \frac{T_k}{4}\kern0.75em }{\Rightarrow}\kern0.75em -{\overrightarrow{q}}_k\kern1.25em \overset{\kern0.75em \frac{T_k}{4}\kern0.75em }{\Rightarrow}\kern1.25em -{\overrightarrow{p}}_k\kern1em \overset{\kern0.75em \frac{T_k}{4}\kern0.75em }{\Rightarrow}\kern1em {\overrightarrow{q}}_k $$

(A6)

Alternatively, the evolution in time of a POP can be obtained empirically from the vectors ${\overrightarrow{y}}^s$. If we collect the eigenvectors or patterns into a matrix W, with elements $\left({\overrightarrow{w}}_j={\overrightarrow{p}}_j,{\overrightarrow{w}}_{j+1}={\overrightarrow{q}}_j,\kern0.5em j=2k-1,\kern0.5em k=1,\ {l}_i\right)$ and ${\overrightarrow{w}}_j=Re\left({\overrightarrow{u}}_k\right),\ k={l}_i+1,n\Big)$, then ${\overrightarrow{y}}^s=\mathbf{W}{\overrightarrow{s}}_i$and the temporal coefficients ${\overrightarrow{s}}_i$ can be obtained as:

$$ {\overrightarrow{s}}_i={\mathbf{W}}^{-1}{\overrightarrow{y}}^s $$

(A7)

where the eigenvectors of the adjoint equation ${\mathbf{W}}^{-1}\overrightarrow{v}=\varLambda \overrightarrow{v}$, are known as the ‘associated patterns’ to the POP.

From the preceding analysis, a predictive scheme can be derived. From the inspection of the POP exponent and time coefficients, one or two pair of useful POP can be identified (von Storch et al. 1995). Thereafter, the root mean square error given by the expression below is minimized

$$ \varepsilon ={\overrightarrow{y}}^s(t)-{\displaystyle \sum_i}{s}_i(t){{\overrightarrow{p}}_i}^2 $$

(A8)

where the sum extends to the POP pair (pairs) identified as useful.

The other approach (Penland and Magorian 1993; Penland and Matrosova 1998) considers the complete solution to the inhomogeneous equation given by

$$ \begin{array}{c}\hfill {\overrightarrow{\ y}}^s\left(t+\varDelta t\right)={\overrightarrow{y}}^s(t)+G\left(t+\varDelta t\right){\overrightarrow{y}}^s(t)\hfill \\ {}\hfill +G\left(t+\varDelta t\right){\displaystyle \underset{t}{\overset{t+\varDelta t}{\int }}}{G}^{-1}\left(\tau \right){\displaystyle \sum_j}{N}_j\left(\tau \right)d\tau \kern1em \hfill \end{array} $$

(A9)

where

$$ \mathbf{G}\left(\tau \right)={\displaystyle \sum_{k=1}^n}{e}^{\mu_k\tau }{\mathbf{W}}^{-1} $$

(A10)

is the Green function of the equation (A1), obtained from the solution of the homogeneous equation in the POP space. Given the noise covariance matrix Q:

$$ \mathbf{Q}=\left\langle {N}_j(t){N}_i{(t)}^T\right\rangle $$

(A11)

We can express the noise as:

$$ {N}_j(t)={\displaystyle \sum_{d=1}^n}{\overrightarrow{\psi}}_{jd}{\pi}_d{R}_d(t) $$

(A12)

where the ${\overrightarrow{\psi}}_{jd}$ and π _d are respectively the eigenvectors and eigenvalues of Q, while the R _d (t) coefficients must satisfy the white noise requirements. Then, the time evolution of the state vector can be approximated by

$$ \begin{array}{c}\kern1em {\overrightarrow{y}}^s\left(t+\varDelta t\right)={\overrightarrow{y}}^s(t)+G\left(t+\varDelta t\right){\overrightarrow{y}}^s(t)\kern1em \\ {}\kern1em +G\left(t+\varDelta t\right){\displaystyle \underset{t}{\overset{t+\varDelta t}{\int }}}{G}^{-1}\left(\tau \right)\left({\displaystyle \sum_{j=1}^p}{\displaystyle \sum_{d=1}^n}{\overrightarrow{\psi}}_{jd}{\pi}_d{R}_d(t)\kern.2em d\tau \right)\kern1em \end{array} $$

(A13)

In this way, some characteristics of the noise are introduced into the hindcast.