Mathematical tools are needed to analyze complex, oscillating signals generated by physical and biological systems, from brain waves to seismic activity. Wavelets are mathematical functions used to break down a signal into its constituent parts, offering a view of the signal in both time and frequency. The Morlet wavelet is the most widely used choice in scientific analysis. It provides an optimal balance between time and frequency localization, allowing researchers to pinpoint precisely when and at what frequency an event occurs. The Morlet wavelet is essential for studying signals that change their frequency content over time.
The Limitations of Traditional Frequency Analysis
Older methods of signal analysis, such as the Fourier Transform, assume that a signal’s frequency content remains constant over the entire duration of the measurement. This technique decomposes a signal into the various frequencies that compose it, allowing researchers to determine which frequencies are present. However, the Fourier Transform provides this information only for the signal as a whole, losing the ability to determine when those frequencies occur in time.
This limitation means the traditional approach is poorly suited for analyzing non-stationary signals, such as a sudden burst of brain activity or a fleeting seismic event. The trade-off in this fixed-basis analysis is that achieving high-frequency resolution requires analyzing a long segment of the signal, which drastically reduces the temporal resolution. Conversely, attempting to pinpoint an event in time sacrifices the precision needed to identify the exact frequency. This inability to simultaneously resolve both time and frequency created a significant analytical gap.
The Design and Advantage of the Morlet Wavelet
The Morlet wavelet was engineered to overcome the limitations of fixed-basis analysis by introducing a movable, adaptable function that can tune its resolution. Conceptually, the Morlet wavelet is created by multiplying a pure sine wave—the carrier wave—by a Gaussian window, which acts as an envelope. The sine wave component determines the frequency being analyzed, while the Gaussian envelope gives the wavelet its distinctive localized shape, ensuring that the analysis is restricted to a small area of the signal.
This Gaussian envelope is the source of the Morlet wavelet’s power, as it localizes the function in both the time and frequency domains simultaneously. When the wavelet is mathematically passed over the original signal, it acts like a filter tailored to detect a specific frequency at a specific moment in time. The process of analyzing a signal involves creating a family of Morlet wavelets, each scaled, or stretched and compressed, to analyze a different frequency band.
This scaling process grants the wavelet its multi-resolution capability. A Morlet wavelet stretched wide in time analyzes low-frequency, long-duration trends, providing excellent frequency resolution but poor temporal resolution. Conversely, a Morlet wavelet compressed narrowly in time analyzes high-frequency, short-lived events, providing excellent temporal resolution at the expense of frequency specificity. By systematically adjusting the width of the Gaussian envelope, the Morlet transform allows the researcher to control the inherent trade-off between time and frequency precision.
How to Interpret a Wavelet Map
The practical output of a Morlet wavelet analysis is a visual representation called a scalogram, or time-frequency map, which provides a three-dimensional view of the signal’s activity. This visualization maps the signal’s energy onto a two-dimensional plot, with a third dimension represented by color or intensity. The horizontal axis of the scalogram represents time, and the vertical axis represents the frequency components derived from the different scales of the Morlet wavelet family. The color or intensity at any point on the map represents the power of the signal at that specific time and frequency.
A scientist interprets the scalogram by looking for bright, concentrated areas of color, which indicate a strong match between the wavelet and a feature in the signal. These bright spots pinpoint a transient event, revealing exactly when a specific frequency component was most active. For instance, a bright horizontal line indicates a consistent frequency that lasted for a period of time, whereas a vertical smear of color indicates a very short-lived event that contained energy across many different frequencies.
Scientific Fields Relying on Morlet Analysis
The ability of the Morlet wavelet to precisely localize events in time and frequency has made it a tool across several scientific disciplines dealing with complex, non-stationary data.
Neuroscience
In neuroscience, researchers use the Morlet transform to analyze electroencephalography (EEG) and magnetoencephalography (MEG) data, where the brain’s electrical activity is constantly changing. The transform tracks rapid, short-lived neural events, such as transient bursts of gamma-band oscillations believed to underlie cognitive processing.
Seismology
The Morlet wavelet was originally introduced by a geophysicist. It remains relevant in seismology because seismic signals, such as those generated by earthquakes, are inherently non-stationary. Morlet analysis allows researchers to distinguish between true reflection events and coherent noise, like ground roll, by isolating their unique time-frequency signatures.
Climate Science
In the atmospheric and climate sciences, the Morlet wavelet analyzes long-term, non-linear climate records, such as surface temperature or precipitation data. Scientists identify significant periodic fluctuations, revealing underlying patterns and their transformation characteristics over time. This analysis helps link observed climate cycles to large-scale phenomena, such as the El Niño-Southern Oscillation (ENSO).