This is the second 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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By now, your interest in the wavelet transform is hopefully piqued enough for us to discuss the process of actually applying it to the real Earth. However, this requires us to focus on a new problem- that of the original parametrization in the spatial domain.

In order to apply the wavelet transform (or any transform, for that matter) we must have the signals parameterized over a coordinate system. The signals that describe the Earth inherently exist in a three or four-dimensional system, and are thus only accurately defined when considered over a framework that is accurate over the surface of the sphere (or more accurately, the ellipsoid).

A range of attempts have been made to do this. For completeness, I must mention the three most popular bases over which signals that geophysicists deal with on the surface of the Earth are defined: cubic splines, spherical harmonics, and spatially-variant and -invariant grids. Cubic splines simply use represent any single signal as the summation of multiple cubic polynomials constrained at certain boundary points. Spherical harmonics are a similar set of polynomials, Legendre Polynomials, that solve Laplace’s Equation for a range of orders for a given degree; these are just indices used to categorize the different solution. The Spherical Harmonic basis is much more like the fourier domain in that it is orthogonal, and also inherits some of the fourier domain’s convolution/deconvolution properties.

Intuitively, however, we best visualize functions when considering them as defined over a grid of some sort- this is why latitude and longitude are so popular; they combine a spherical co-ordinate system with the intuitiveness of a grid. Over regional and local spatial scales, this is a perfect solution. However, on a global scale, their inhomogeneity is brought to light, and their coverage is rendered inaccurate, particularly at the poles. As a result, we have to formulate the wavelet transform within what’s known as the cubed sphere- a sphere gridded via a spherical coordinates on 6 separate chunks. This has the advantage of incredibly homogenous coverage, though functions are forced to not be smooth across the edges of the chunks. The image below (Simons et al., 2011) best describes the parameterization we use.

That’s the hardest part taken care of! Now, we’ve established the parameterization we use to view signals and conduct the wavelet transform in. But how does this help us to understand the Earth? In two ways: Firstly, it helps us understand if natural systems are sparse when viewed in the wavelet basis. And more importantly for our purposes, the localizable property of wavelets in both space and wavelength lets us see how heterogeneous certain functions (such as velocity models) describing the Earth are on certain length scales. This is extremely important! If we can better constrain the dominant length scales of Geodynamic processes acting within the interior of the Earth, we can gain incredible insight onto the physics of the solid Earth and better constrain the source of heterogeneities within the Earth’s interior. For example, the length scale of purely thermal heterogeneities is vastly different from that of thermo-chemical anomalies.

How do we view heterogeneities within the Earth’s interior? With tomography, of course!

Here, I present an analysis of MIT’s 2008 (Li et al.) P-wave tomographic model within the D4 (Daubechies-4) wavelet basis. You can see that the scale-4 processes are more dominant than those at lower scales, indicating that processes on the scale of 20 degrees are more dominant in the lower mantle than those at lower degrees.

This is, of course, only the beginning. Global processes have their appeal, but what about regional scales? What can wavelets tell us when applied to the smaller scales of, say, subduction zones? The journey begins now!

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