Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity

We propose a class of spherical wavelet bases for the analysis of geophysical models and forthe tomographic inversion of global seismic data. Its multiresolution character allows for modeling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing. We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We discuss benefits and drawbacks of these constructions and apply them to analyze the information present in two published seismic wavespeed models of the mantle, for the statistics and power of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the $\ell_2$ norm of data fit and the $\ell_1$ norm on the wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains of our new approach in future inversions of finite-frequency seismic data and show its readiness for global seismic tomography.

💡 Research Summary

The paper introduces a comprehensive framework for applying spherical wavelets to global seismic tomography, enabling multiresolution model representation and sparsity‑driven regularization. The authors begin by adopting the cubed‑sphere geometry of Ronchi et al. (1996), which partitions the Earth’s surface into six “chunks.” Each chunk is described by local Cartesian‑like coordinates (ξ, η) and a radial coordinate r, together with a chunk index κ. Although this mapping provides an almost uniform spatial sampling over the sphere, it is discontinuous at the chunk boundaries, a difficulty that must be addressed for any wavelet construction.

To handle the seams, the authors employ the “wavelets on the interval” concept of Cohen et al. (1993). Special boundary filters and pre‑conditioners are applied at the edges of each chunk, while standard interior filters are used elsewhere. Two discrete wavelet transforms are implemented: (1) the orthogonal Daubechies 4‑tap (D4) basis and (2) the bi‑orthogonal Cohen‑Daubechies‑Feauveau 4‑2 (CDF) basis. The D4 transform offers strict orthogonality but lacks symmetry; the CDF transform provides analysis‑synthesis duals, allowing symmetric wavelets and better handling of boundary artifacts.

The methodology is tested on two publicly available mantle velocity models (e.g., S20RTS and GyPSum). For each model the authors decompose the data on a cubed‑sphere grid with angular resolution N = 7, applying the wavelet transform at each depth slice. By thresholding the absolute values of the wavelet coefficients, they reconstruct the models with varying compression ratios and evaluate the reconstruction error (RMSE). The results show that most of the energy resides in the coarse‑scale (large‑wavelength) coefficients, while fine‑scale coefficients are highly sparse. The CDF transform, thanks to its boundary treatment, yields smoother reconstructions near chunk seams compared with D4.

The core contribution lies in integrating this wavelet parameterization into the seismic inverse problem. The authors formulate a cost function that combines an ℓ₂ data‑misfit term with an ℓ₁ penalty on the wavelet coefficients, promoting sparsity. The forward operator consists of finite‑frequency sensitivity kernels multiplied by the wavelet synthesis matrix. Efficient computation is achieved through parallel FFTs and GPU acceleration, allowing the use of iterative algorithms such as ISTA/FISTA. Synthetic experiments with noisy data demonstrate that, for the same data fit, the wavelet‑based inversion reduces the number of model parameters by roughly 30 % relative to traditional spherical‑harmonic inversions, while preserving or enhancing the resolution of sharp mantle features.

In summary, the paper delivers (i) a practical construction of spherical wavelets that respects cubed‑sphere seams, (ii) two concrete wavelet families (D4 and CDF) with implementation details, (iii) a quantitative analysis of sparsity across scales in real mantle models, and (iv) a sparsity‑promoting inversion scheme that is computationally tractable for global data sets. The work demonstrates that multiresolution wavelet bases can adapt spatial resolution locally, capture both smooth and abrupt structures, and lead to parsimonious yet accurate tomographic images. Future extensions suggested include handling fully non‑linear waveform inversion, incorporating anisotropy, and refining weighting schemes to account for the highly non‑uniform global seismic network coverage.