Compressive inversion in an overcomplete basis

Turunçtur, Buse1, Valentine, Andrew1, and Sambridge, Malcolm1

1Research School of Earth Sciences, Australian National University, Canberra, Australia

The concept of compressive sensing has promised a revolution in data collection. Rather than a traditional sampling of a temporal or spatial signal with uniformly distributed samples, compressive sensing enables an exact recovery of the signal with fewer but randomly chosen samples. Provided that the target signal is `sparse’, i.e. has only a few non-zero Fourier components, it can be recovered with high fidelity using inversion algorithms designed to minimize the L1 norm of the recovered solution. Compressive sensing allows signal recovery beyond the Nyquist limit, allowing high-frequency information to be recorded using relatively few samples.

We aim to adapt this concept to suit inverse problems of the form commonly encountered in geophysics. To guarantee that models have a sparse representation, we employ an ‘overcomplete’ basis, comprising a complete set of functions with global support, and another complete set with local support. We compare results for L1– and L2-regularised inversion in synthetic examples, and show that the former enables excellent recovery of input models. We suggest that this concept has a variety of promising applications in geophysics, including low-artefact imaging of systems containing features at multiple scale lengths, and image denoising.


Buse Turunctur is a PhD student in Seismology and Mathematical Geophysics group at the Australian National University. She is working with Malcolm Sambridge and Andrew Valentine since 2019. Their study is on finding new sparsity constrained approaches to geophysical inversion.

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