Frederik J. Simons

Frederik J. Simons, Ignace Loris, Guust Nolet et al.

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.

Frederik J. Simons, Dong V. Wang

We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.

Arthur P. Guillaumin, Adam M. Sykulski, Sofia C. Olhede et al.

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(n log n) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and real-world settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.