# Wavelets

#### Wavelets

Wavelet analysis is the cornerstone for processing satellite data. The information obtained can be made ​​very synthetic, which facilitates the interpretation of measurements. Although little used by the community of oceanographers and climatologists, the wavelet technique allows analysis of phenomena in frequency and time. During the 19th century and until recently, Fourier analysis was the only technique for the decomposition of a signal and its reconstruction without loss of information; unfortunately, it provides a frequency analysis but does not allow the temporal location of abrupt changes, such as the advent of a second musical note after a first note was played.

Thus, this technique reflects the evolution of a phenomenon as a function of periodicity (or frequency) observed as sound engineers are accustomed to do. It is relatively recent since, as used today, it was introduced by Jean Morlet and Alex Grossmann in 1984. It is still the subject of academic research published in the « Journal of Wavelet Theory and Applications », journal which remains open to applications.

Wavelets, which were originally developed for the analysis of seismic signals, are well suited to pseudo periodic signals as the sea surface height, the speed of surface currents and sea surface temperature. These signals typically have a large bandwidth ΔT, compared with the mean period T, whence a significant temporal variability. Also in this case temporal analysis, in addition to frequency analysis, can reveal changes in behavior over time.

#### Cross-wavelets

However, in the present case we will have a restrictive use of wavelets, in limiting it to the expression of the amplitude and phase anomalies. Indeed, whether altimetry, speed of surface currents or sea surface temperature, only the deviations from the mean values ​​of signals matter. They are generally called anomalies, whatever the magnitude of the deviation. If T is the mean period and ΔT the bandwidth, the wavelet is averaged over the TT/2, TT/2 band to representatively account for the phenomenon during the cycle (in the relevant frequency band).

When it is crossed, the wavelet is expressed relative to a reference signal used as a time base, which has to have the same periodicities as the signal analyzed. If we look at the sea surface height, the altimeter signal, which depends on the longitude x and latitude y, can be compared to the same signal measured at a particular location at longitude x0 and latitude y0 or any indicator such as SOI or NAO, or even a sine wave when it comes to represent the solutions of the equations of motion.

In addition to the representation of the amplitude of anomalies, the cross-wavelet determines their phase. The latter can be expressed either in relation to the reference signal, or as a calendar time. In the first case it is the time elapsed between when the sea surface height anomaly (longitude x and latitude y) and the reference signal reach their respective maximum in the TT/2, TT/2 band. In the second case it is the date on which the maximum of the sea surface height anomaly is reached during the cycle.

By construction, if a positive anomaly is reached at time t, a symmetrical negative anomaly occurs at time t+T/2 (offset by a half period).

But the main asset of cross-wavelets is the accuracy with which they estimate altimetry and current speed anomalies, the error obtained is about 0.001 m in the first case and 0.003 m/s in the second. The error in the phase is nearly one month for the most significant anomalies, which is the sampling rate (monthly data are used). Then spatial and temporal information is rich, even collected from noisy data (undergoing erratic variations).

#### Wavelet analysis and complex EOF (Empirical Orthogonal Functions)

A wavelet analysis is preferred to a complex EOF analysis to investigate the resonance of long waves in the oceans. If the two methods are similar for typical frequency domain analyses, i.e. the power spectral and coherency analyses, the time domain EOF analysis, which is basically the computation of eigenvector and eigenvalue of a covariance or a correlation matrix computed from a group of original time series data, is far different from the wavelet analysis, which is well suited to highlight the time lag between time series when it varies continuously. Applied to sea surface temperature, sea surface height and surface current velocity data, the wavelet analysis brings out the propagation of the waves as well as their variability from a cycle to another.

##### Glossary

Bandwidth of a quasi-periodic phenomenon of period T

If T is the average period of quasi-periodic phenomenon, the width of the band ΔT indicates the limits between which the period varies, i.e. T – ΔT/2 and T+ ΔT/2.

SOI (Southern Oscillation Index)

The SOI is the amplitude of the Southern Oscillation; it is a measure of the monthly change in the normalized atmospheric pressure difference at sea level between Tahiti and Darwin (Australia).

NAO (North Atlantic Oscillation)

The North Atlantic Oscillation refers to a phenomenon affecting the climate system of the North Atlantic Ocean. It describes changes in the ocean-atmosphere system in the region and is measured as the difference in atmospheric pressure between the Azores High and the Icelandic Low.