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

Frederik J Simons¹, Ignace Loris², Guust Nolet³, Ingrid C. Daubechies⁴,
Sergey Voronin⁴, J. Stephen Judd⁴, Philip A. Vetter⁴, Jean Charléty³ & Cédric Vonesch⁴

1 Geosciences Department
Princeton University
Princeton NJ 08544, USA

2 Mathematics Department
Université Libre de Bruxelles
1050 Brussels, Belgium

3 Géosciences Azur
Université de Nice
06560 Sophia Antipolis, France

4 Program in Applied and Computational Mathematics
Princeton University
Princeton NJ 08544, USA

Abstract

We propose a class of spherical wavelet bases for the analysis of geophysical models and for the 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. Localization and sparsity properties allow finding a sparse solution to the inverse problem by iterative minimization of a combination of the l2 norm of data fit and the l norm on the wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains of using our new approach in future inversions of finite-frequency seismic data.

Figures

1. Figure 01 Geometry of the cubed sphere in a three-dimensional view
   
2. Figure 02 Geometry of the cubed sphere in a two-dimensional view, and nomenclature
   
3. Figure 03 The looks of the D4 wavelets as they are being used on the cubed sphere
4. Figure 04 The looks of the CDF4.2 wavelets as they are being used on the cubed sphere
5. Figure 05 Preconditioned interval wavelet transforms on the faces of the cubed sphere: thresholding, compression, and coefficient statistics
6. Figure 06A Analysis of wavelet sparsity in the seismic model of Montelli using preconditioned D4 wavelets on the cubed sphere
7. Figure 06B Analysis of wavelet sparsity in the seismic model of Ritsema using preconditioned D4 wavelets on the cubed sphere
8. Figure 07A Analysis of the scale and sparsity in the seismic model of Montelli using preconditioned D4 wavelets on the cubed sphere
9. Figure 07B Analysis of the scale and sparsity in the seismic model of Ritsema using preconditioned D4 wavelets on the cubed sphere
10. Figure 08 Scale lengths of seismic heterogeneity in the seismic models of Montelli and Ritsema, broken down per chunk
11. Figure 09 Correlation between velocity anomalies in the seismic models of Montelli and Ritsema
12. Figure 12 Performance of the algorithm on a semi-realistic synthetic test

Frederik Simons