Resolution analysis by random probing

Abstract

We develop and apply methods for resolution analysis in tomography, based on stochastic probing of the Hessian or resolution operators. Key properties of our methods are (i) low algorithmic complexity and easy implementation, (ii) applicability to any tomographic technique, including full‐waveform inversion and linearized ray tomography, (iii) applicability in any spatial dimension and to inversions with a large number of model parameters, (iv) low computational costs that are mostly a fraction of those required for synthetic recovery tests, and (v) the ability to quantify both spatial resolution and inter‐parameter trade‐offs. Using synthetic full‐waveform inversions as benchmarks, we demonstrate that auto‐correlations of random‐model applications to the Hessian yield various resolution measures, including direction‐ and position‐dependent resolution lengths, and the strength of inter‐parameter mappings. We observe that the required number of random test models is around 5 in one, two and three dimensions. This means that the proposed resolution analyses are not only more meaningful than recovery tests but also computationally less expensive. We demonstrate the applicability of our method in a 3D real‐data full‐waveform inversion for the western Mediterranean. In addition to tomographic problems, resolution analysis by random probing may be used in other inverse methods that constrain continuously distributed properties, including electromagnetic and potential‐field inversions, as well as recently emerging geodynamic data assimilation.