This is the first of a two-part post: the first part explores the necessity for a new way to probe signals within the context of a generalized Fourier Transform. The second part, which is more rooted in examples, explores the difficulty of doing so in a full, spherical Earth and highlights some of the measures that those who map the deep Earth take to remedy it.

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As we grow, we inevitably learn that the world is more complex than we think it is. From valence shells to orbitals, from Newtonianism to Quantum Mechanics and from Ray Theory to Finite-Frequency Kernels, the truth gradually unravels as we prove ourselves able to handle it, and we are forced to accept that the natural world is infinitely more involved, chaotic, and exciting than we believed it to be. Consequently, describing it requires us to turn to tools that are equally complex and highlight flaws in their predecessors-exhilaratingly so! Here, I’ll explore the context behind why we need one of these tools: wavelets.

I hope to leave you with a basal excitement for some of the limitations and nuances of the signal processing tools we can employ to describe the Earth.

Inherent in Geophysical research in the fundamental problem of deconstructing and reconstructing complex signals generated by the Earth. When I began delving into the magical world of signal processing, I rapidly imbibed the ideology that the Fourier transform (a mathematical process that enables us to view signals in both time and frequency domains) was the panacea for any and all problems one might face while trying to analyze a range of waveforms.

It took me nearly 3 years to convince myself I had even a basic grasp on the Fourier Domain. I believed that this last, magical conceptual leap was finally a solution I could be happy with- a solution that finally elucidated convolution, deconvolution, and a range of operations in both time and frequency domains.

Oh, how wrong I was.

While a tremendous breakthrough, the Fourier transform finds itself sorely limited in several regions- the source of this limitation lies within its core ideology: the idea that a basis set of orthogonal sines and cosines of varying frequencies can represent any signal. These sines and cosines are infinitely repeating waveforms- as a result, the Fourier transform is designed to deal with signals that are periodic- signals that repeat themselves after a period of time.

This means that when signals are impulsive in nature- defined by a sharp burst of energy, or a non-repeating anomaly, the Fourier transform has trouble reconstructing this signal. A primitive manifestation of this issue can be seen in something known as the Gibbs Phenomenon- when signals are forced to rapidly oscillate near sharp discontinuities in the signal.

If we want to deconstruct signals that are localized or vary rapidly in frequency, we’re forced to turn to the wavelet transform- a procedure by which we represent signals using a basis composed of wavelets- pulses which are localized in both time and frequency. These pulses differ in their “scale”, a property loosely related to the duration of the pulse that we might loosely equate to wavelength for a repeating signal, and thus form a basis. Unlike with the rigid sinusoids used in the fourier transform, these basis functions need not necessarily be orthogonal, and can take a range of general shapes, offering greater flexibility.

Although wavelets were used early in the Twentieth Century to analyze quantum systems, their real utility was only explored when engineers realized that wavelets behaved uniquely in the frequency domain, spanning it in a nonlinear fashion. This means that when visualized in Matrix representation, wavelets tend to favor depictions that approach, but don’t quite mimic, sparse matrices. However, if these matrices are truncated, we experience unprecedented gains in computational speed and power. In addition, the wavelet transform can flit more freely between time and frequency domains, favoring neither and allowing joint representations of both, such as in this spectrogram. This process allows us to deal powerfully with signals, such as those from seismic tomography.

However, the complications have just begun for us. In the field of global tomography, we’re forced to deal with signals that are defined on the surface of the Earth- in other words, defined on a sphere, or more technically, on the ball. This means that the traditional bias functions used to deal with these signals aren’t simply sines and cosines, but Spherical Harmonics, which involve a series of Polynomials known as Legendre Polynomials as basis functions. Here, too, wavelets triumph, as you’ll see next time.

By jonah_bartrand on July 5th, 2017

Anant, I'm very excited for your next blog. As a math and physics major, I have a great deal of experience with the Fourier transform and have come to appreciate both its amazing utility and its limitations in time-dependent frequency analysis. I have, for a while now, been surprised by the limited application of wavelet analysis to seismology, for which it seems ideally suited.

By Brady Flinchum on July 10th, 2017

"Inherent in Geophysical research in the fundamental problem of deconstructing and reconstructing complex signals generated by the Earth." This is such a cool sentence! I minored in electrical engineering, which is where most of my signal processing background came from. I always liked to explain geophysics to the engineers like we know (sometimes..) the input signal and can measure the output signal and we want to figure out the specifics of the circuit (i.e. the earth).
I also really nerded out and thoroughly enjoyed this post. I am glad you pointed out that Fourier Analysis applies to only periodic signal, it is something that we often overlook. It also took me about 3 years to fully grasp this powerful concept.

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