Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion

Many flexible parameterizations exist to represent data on the sphere. In addition to the venerable spherical harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper we focus on the latter two: spherical wavelets developed for geophysical applications on the cubed sphere, and the Slepian “tree”, a new construction that combines a quadratic concentration measure with wavelet-like multiresolution. We discuss the basic features of these mathematical tools, and illustrate their applicability in parameterizing large-scale global geophysical (inverse) problems.

💡 Research Summary

This paper presents two complementary mathematical tools for representing and inverting geophysical data on the sphere: spherical wavelet transforms built on the cubed‑sphere grid, and a multiscale “Slepian tree” dictionary that blends the concentration properties of Slepian functions with wavelet‑like multiresolution.
The authors first describe how the six‑patch cubed‑sphere mapping (Rončhi et al., 1996) can be used to apply standard Cartesian wavelet constructions (orthogonal and bi‑orthogonal D2 (Haar), D4, D6) to spherical data. By treating each patch (“chunk”) as a rectangular domain, they employ boundary filters and pre‑conditioning to mitigate seam artifacts. Experiments on the North‑American chunk show that hard thresholding at the 85th percentile eliminates roughly 85 % of the wavelet and scaling coefficients while preserving the map with only 5–9 % root‑mean‑square error, depending on the basis. The D4 basis is highlighted as the best compromise between sparsity and visual smoothness, avoiding blocky artifacts seen with D2.
The same pipeline is applied to global seismic velocity models (P‑wave and S‑wave anomalies at 670 km depth). After transforming the models with the D4 wavelet, coefficients are hard‑thresholded at 85 % and 95 % levels. The authors report the number of retained coefficients, the ℓ₂ reconstruction error, the ℓ₁ norm reduction, and a “total variation” metric. Errors remain below single‑digit percentages until about 80 % of coefficients are removed; beyond that the degradation accelerates. Correlation analysis between the wavelet coefficients of the two models reveals scale‑dependent relationships that can be interpreted as regional variations of the logarithmic ratio δln VS/δln VP, a key diagnostic for distinguishing thermal from compositional anomalies.
The second contribution is the Slepian tree transform. Classical Slepian functions are optimally concentrated within a spatial region R while being band‑limited to degree L, but they form a global orthogonal set that lacks multiscale locality. To overcome this, the authors construct a binary subdivision tree of the region R. Each node of the tree corresponds to the first n Slepian functions on the node’s sub‑region, preserving the bandlimit L. The tree height H is chosen so that the node capacity n satisfies n ≥ N₂⁻ᴴ|R|,L, where N is the Shannon number; analytically, H≈log₂(|R|·(L+1)²/(4π n)). The resulting dictionary D_R,L,n contains n·(2^{H+1}−1) functions that are simultaneously band‑limited, spatially localized, and multiscale. Because child nodes occupy disjoint sub‑regions, their associated functions are nearly incoherent, making the dictionary well‑suited for sparse regularization (ℓ₁) in inverse problems. The authors illustrate the construction on the African continent, showing how the first Slepian function of each sub‑region provides a natural binary split (the zero‑crossing of the second Slepian function).
Numerical experiments demonstrate that using the Slepian tree as a prior yields lower residual errors than classical Slepian‑only inversions, especially when combined with modern sparsity‑promoting techniques (e.g., FISTA). The tree’s multiscale nature allows the inversion to adapt resolution locally, preserving fine‑scale features where data support is strong while maintaining stability in poorly sampled regions.
In summary, the paper delivers a practical workflow for global geophysical inverse problems: (1) map the sphere to a cubed‑sphere grid, apply standard wavelet transforms, and exploit sparsity via hard or soft thresholding; (2) construct a Slepian‑tree dictionary that respects both band‑limit and spatial concentration, providing a flexible multiresolution basis for regularization. The combined approach achieves substantial data compression, reduces computational cost, and improves the fidelity of recovered Earth models, with potential applications ranging from seismic tomography to gravity and magnetic field inversions, and even to cosmological data analysis on the celestial sphere.