Wavelet footprints

In the mid-1980s, the mathematician Yves Mayer from the University of Marseille and the petroleum engineer Jaean Morlet worked with the analysis of data from petroleum surveys at Elf-Aquitaine. In their efforts to find better methods for frequency analysis, they rediscovered a set of a new type of transformations which they called Wavelets. The wavelet transform solved some of the weaknesses of the Fourier transform. It required less computing power; it was possible to identify period and phase relations in time-series, and non-stationary periodic variations in nature. When the method was presented, Morlet received the comment, “A method, not described in any textbook, cannot be of great importance”. In 1988, Ingrid Daubechies published the article “Orthogonal Bases of Compactly Supported Wavelets”. This is perhaps the most important contribution to frequency analysis of time series since Fourier published his book in 1822.

I started using wavelet spectrum analysis around the year 2000. The problem was to find coincidences between the cod recruitment and the temperature variations in the Barents Sea. There are, however, a variety of wavelet functions, which calculate results variations. The best choice of wavelet function had to be adapted to the statistical properties from the time series. This led to my own investigations of all wavelet function in the MATLAB Toolbox.

When I submitted the manuscript for review, wavelets spectrum analysis was unknown to editor. To find a reviewer, I visited the EGU conference in Vienna. Here I found two young PhD students who studied the wavelet transform. Gradually, more people began to use the wavelet transform to study climate time series. Still most scientists of are using the wavelet transform as a substitute for Fourier analysis. To me this is a wrong use of this powerful method.

Wavelet transform

Figure 1 Wavelet pulse function h(t).

A Fourier transform is based on a correlation between a time series x(t) and a set of sinus functions sin(2pt/T). A wavelet transform is based on a moving correlation between a time series x(t) and an impulse function h(t) (Figure 1.5.1). The correlates wavelet function s(t) will respond when there is a match between x(t) variation and the pulse function h(t). Longer pulse periods will have correlation to longer periods in x(t). A set wavelet pulses computes a wavelet spectrum of all pulse variations in the time series x(t). Each wavelet reveals the position of a pulse variations in the time series x(t). Pulse positions maxima and minima reveals the period phase. The phase information is needed to have a complete time series signature S(T, F(t-t₀)), where t₀ represents the time when a cycle period T has a maximum.

Example 1. Identify two sine functions

Figure 2 Sum of two sinus periods

A time series is produces by the simple model:

x(t) = 1.9 +0.4sin(2pt/T₁ + F₁) + 0.6sin(2pt/T₂ + F₂)

where the cycle period T₁ = 18.6/3 = 6.2 years, the phase angle F₁= -0.29(rad), T₂ = 18.6 years, and F₂ = 1.52(rad). The question is how a wavelet spectrum analysis can help us to identify time series signature S(T, F(t-t₀)).

Figure 13. Computed wavelet spectrum

The first thing we should do is subtract the mean by setting x(t) = x(t)-E[x(t)] to divide the time series. This avoids false periods at the endpoints of the time series. You can then normalize the time series by dividing by the variance.

Figure 3 shows calculated wavelet spectrum of the time series x(t), where the x-axis represents the development over time and the y-axis represents moving correlation between x(t) and a scalable pulls h(t). The y-axis has a maximum when the pulse width h(t) coincides with the period times T₁ and T₂. The time of maximum correlation reveals the phase of periods T₁ and T₂.

Figure 4. Time series x(t) and identified periods in the wavelet spectrum

Figure 4 shows the time series, x(t), and identified periods of approximately T1 = 6 years and T2. = 18 in the wavelet spectrum. The cycle periods are identified by computing the autocorrelation of the wavelet spectrum. The cycle period phase has maxima and minima in the wavelet spectrum. It is clear from the figure that the wavelet spectrum has revealed the x(t) signature S(T, F(t-t₀)), where T = [6, 18] and F(t-t₀) = [39, 39].

Example 2: Total solar irradiation

Figure 5: Total solar irradiance from A.D. 1700 to 2013 (Scafetta & Willson, 2014).

The Total solar irradiation (TSI) represents the measured irradiation Wm^-2 at the average distance from the Sun to the Earth. Figure 5 shows an annual mean Total solar irradiance time series that covers the period from A.D. 1700 to 2013 (Scafetta & Willson, 2014). A simple visual inspection of this data series shows that TSI from the sun has variations, and TSI has a mean growth from 1700 to 2014. The research question is to reveal the source of TSI variations.

Figure 6: Computed wavelet spectrum W(s, t) of the TSI data series (Figure 1.5.5) (Yndestad and Solheim 2017).

The transformed wavelet spectrum Whs(s,t) (Figure 1.5.6) represents a set of separated wavelet periods from the TSI data series from 1700 to 2013. In this presentation, the wavelet scaling range is s = 1 . . . 0.6N , and the data series contains N = (2013-1700) = 313 data points. A visual inspection of the TSI wavelet spectrum shows the dominant periods in the TSI data series in the time window between 1700 and 2013. The long wavelet period has a maximum in 1760, 1840, 1930, and 2000, with a mean time difference of approximately 80 years.

Figure 17: Autocorrelations R(m) of the TSI wavelet spectrum (Figure 1.5.7) (Yndestad and Solheim 2017).

The autocorrelation of the wavelet spectrum W(s, t), or R(m) = E[W(s, t)W(s, (t+m)], is shown on Figure 1.5.7 computes the correlations R = [0.55, 0.55, 0.65, 0.70] to the stationary periods T = [11, 30, 86, 164]. The Jovian planets Jupiter, Saturn, Uranus, and Neptune have the stationary periods: [11.86, 29.45, 84.02, 164.79] (yr.). (Yndestad and Solheim 2017). The computed wavelet specter revealed, for the first time, that the Jovian planets are the source of TSI variations. The result was confirmed by a wavelet spectrum analysis of the suns movement around the solar system Barycenter. A study at Jovian planet movements has revealed a direct relations between Jovian planet movement, solar positions movement and TSI variations in periods up to 4450 years (Yndestad 2022).

Wavelets in climate science

The wavelet spectrum paradigm computes periods and period phase-relations in a time-series x(t). When periods and phase are known:

we can compute phase-lag between time-series,
we can compute period and phase-relation in a chain of events,
we can compute interference between a period maximum and minima,
we can compute extreme climate maxima and deep minima,
we can compute future maxima and minima in upcoming events

It has been possible to compute a Climate Clock of events, from the oscillating solar system to the oscillating earth axis movement, tide oscillation, lunar forced climate period, solar forced climate periods, marine eco system oscillations, and much more. The difference between deterministic astronomic periods and estimated wavelet periods in climate data series is only 0-5 years.