Geophysical imaging using trans-dimensional trees

Abstract

In geophysical inversion, inferences of Earth's properties from sparse data involve a trade-off between model complexity and the spatial resolving power. A recent Markov chain Monte Carlo (McMC) technique formalized by Green, the so-called trans-dimensional samplers, allows us to sample between these trade-offs and to parsimoniously arbitrate between the varying complexity of candidate models. Here we present a novel framework using trans-dimensional sampling over tree structures. This new class of McMC sampler can be applied to 1-D, 2-D and 3-D Cartesian and spherical geometries. In addition, the basis functions used by the algorithm are flexible and can include more advanced parametrizations such as wavelets, both in Cartesian and Spherical geometries, to permit Bayesian multiscale analysis. This new framework offers greater flexibility, performance and efficiency for geophysical imaging problems than previous sampling algorithms. Thereby increasing the range of applications and in particular allowing extension to trans-dimensional imaging in 3-D. Examples are presented of its application to 2-D seismic and 3-D teleseismic tomography including estimation of uncertainty.

Trans‐Dimensional Surface Reconstruction With Different Classes of Parameterization

“…We jointly invert for two surfaces: absolute land motion and absolute see level rise. This test further illustrates the fact that the choice of parameterization affects both the recovered structure and its estimated uncertainties as reported by Hawkins and Sambridge ().…”

“…These criteria provide an approximation of Bayes factors or evidence ratios that can be used to select which model best fits our observations. The deviance information criteria (DIC) has the advantage that it can be applied in trans‐dimensional inversion (Hawkins & Sambridge, ; Steininger et al, ). The DIC variant we use for trans‐dimensional inversions is given by where D ( m ) is called the deviance and given by where the constant is a function of the data and cancels for model comparison purposes.…”

On the use of sensitivity tests in seismic tomography

“…For example, Bayesian trans-dimensional tomography, in which the number of unknowns is an unknown itself, and the parametrization is adaptive, is starting to become increasingly popular. It has been shown to be computationally tractable for most 2-D and some 3-D problems (Bodin & Sambridge 2009;Bodin et al 2012;Young et al 2013;Galetti et al 2015;Piana Agostinetti et al 2015;Hawkins & Sambridge 2015), and generates a large ensemble of solutions which can be used to quantitatively assess solution reliability. In general, these methods have an intrinsic parsimony which results in a (variable) spatial resolution that contains only structure that is 'required' by the data; the uncertainty associated with structure at this spatial resolution can be estimated by taking the standard deviation of the solution ensemble.…”

Taming uncertainty in geophysical inversion

“…These include seismic travel time and waveform tomography (Bodin and Sambridge, (2009), Bodin et al, 2012a, Hawkins & Sambridge (2015; geophysical imaging of the crust with seismic, ground based and airborne EM, Piana-Agostinetti and Malinverno (2010), Brodie, and Sambridge, (2012), Bodin et al, (2012b); Local earthquake tomography, Piana-Agostinetti et al (2015); Geoacoustic acoustic imaging of sea-bed and near surface properties, Dettmer and Dosso (2010), Dettmer et al (2012a&b), Steininger et al (2014) and more recently seismic and Tsunami source studies, Dettmer et al (2014). Trans-D approaches have also begun to make a mark in exploration geophysics.…”