Analysis and Indications on Long-term Forecasting of the Oceanic Niño Index with Wavelet-Induced Components

Abstract

The present paper provides an analysis and a long-term forecasting scheme of the Oceanic Niño Index (ONI) using the continuous wavelet transform. First, it appears that oscillatory components with main periods of about 17, 31, 43, 61 and 140 months govern most of the variability of the signal, which is consistent with previous works. Then, this information enables us to derive a simple algorithm to model and forecast ONI. The model is based on the observation that the modes extracted from the signal are generally phased with positive or negative anomalies of ONI (El Niño and La Niña events). Such a feature is exploited to generate locally stationary curves that mimic this behavior and which can be easily extrapolated to form a basic forecast. The wavelet transform is then used again to smooth out the process and finalize the predictions. The skills of the technique described in this paper are assessed through retroactive forecasts of past El Niño and La Niña events and via classic indicators computed as functions of the lead time. The main asset of the proposed model resides in its long-lead prediction skills. Consequently, this approach should prove helpful as a complement to other models for estimating the long-term trends of ONI.

Appendices

Appendix A: Iterations of the CWT

When it comes to extracting components from a signal s as accurately as possible, the CWT can be used several times to sharpen the desired modes. More precisely, if the components of interest are located at scales $(a_i)_{i\in I}$ (for some set of indices I), then at the first iteration one can extract $(c_i^1)_{i\in I}$ as

$$\begin{aligned} c_i^1(t)=2\mathfrak {R}(W_s(t,a_i)). \end{aligned}$$

Then repeat the process, i.e. the CWT and extraction at the same scales $(a_i)_{i\in I}$ but with

$$\begin{aligned} s_1=s-\sum _{i\in I}c_i^1, \end{aligned}$$

get the modes $(c_i^2)_{i\in I}$, repeat with $s_2=s_1-\sum _{i\in I}c_i^2$, and so on. Stop the process when the components extracted are not significant anymore, i.e. at iteration J if

$$\begin{aligned} \max \left\{ \left\| c_i^{J}\right\| ,i\in I\right\} <\alpha \ \max \left\{ \left\| c_i^{1}\right\| ,i\in I\right\} \end{aligned}$$

where $\left\| .\right\|$ denotes the energy (square of $L^2$ norm) of a signal and $\alpha$ is a threshold typically chosen as 0.01. The final components $(c_i)_{i\in I}$ are then obtained as

$$\begin{aligned} c_i=\sum _{j=1}^J c_i^j. \end{aligned}$$

Appendix B: Details for $y_{43}^1$

For simplification, let us write s instead of $s_3$ and y instead of $y_{43}^1$. If y is already known up to time $t-1$, here are the steps describing how to obtain y(t). We note $p(t-1)$ the position of the peak used to generate $y(t-1)=A_{43}\cos (2\pi (t-1-p(t-1))/43)$. We use a variable called lock to prevent abrupt changes from $p(t-1)$ to p(t). To obtain p(t), proceed as follows.

Then $y(t)=A_{43}\cos (2\pi (t-p(t))/43)$. Special mention for $y_{61}$ in order to better synchronize the forecasts with EN events: if it comes that $y_{61}$ reaches a peak before $s_1$, impose that $y_{61}$ stays at $A_{61}$.