Emerging Directions in Geophysical Inversion

Valentine, Anne; Sambridge, Malcolm

doi:10.1017/9781009180412.003

Cited by 2 publications

(2 citation statements)

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“…We fit g 1 ( m ) using a traditional variational objective function (Blei et al., 2017; C. Zhang et al., 2018). Note that in a linear problem optimizing a single component (e.g., a single Gaussian distribution) by maximizing ELBO[ q 1 ( m )] gives precisely the Bayesian least squares solution (Tarantola & Valette, 1982; Valentine & Sambridge, 2023). In each subsequent step t = 2, 3, …, n , BVI adds one new component g t to the mixture model, with weight w t ∈ [0, 1].…”

Section: Methodsmentioning

confidence: 99%

“…If only a single Gaussian component is involved in Equation 14, BVI becomes automatic differentiation variational inference (ADVI—Kucukelbir et al., 2017)—another well‐established variational method that tries to fit (approximate) the true posterior pdf with a Gaussian distribution (Tarantola & Valette, 1982; Valentine & Sambridge, 2023). ADVI also provides an analytic approximation to the posterior distribution, and usually seems to estimate the mean model accurately.…”

Section: Methodsmentioning

confidence: 99%

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Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Zhao,

Curtis

2024

JGR Solid Earth

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Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non‐linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so‐called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimized. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non‐linear, high‐dimensional Bayesian full waveform inversion problems.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Zhao,

Curtis

2024

JGR Solid Earth

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Variational prior replacement in Bayesian inference and inversion

Zhao,

Curtis

2024

Geophysical Journal International

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Summary Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically by calculating the posterior probability distribution function. Prior information is often described by a prior probability distribution. Situations arise in which we wish to change prior information during the course of a scientific project. However, estimating the solution to any single Bayesian inference problem is often computationally costly, as it typically requires many model samples to be drawn, and the data set that would have been recorded if each sample was true must be simulated. Recalculating the Bayesian inference solution every time prior information changes can therefore be extremely expensive. We develop a mathematical formulation that allows the prior information that is embedded within a solution, to be changed using variational methods, without recalculating the original Bayesian inference. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2D seismic full waveform inversion example, in which VPR provides similar posterior solutions to those obtained by solving independent inference problems using different prior distributions. The former can be completed within minutes on a laptop computer, whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information: uniform, smoothing and geological prior distributions.

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Emerging Directions in Geophysical Inversion

Cited by 2 publications

References 144 publications

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Variational prior replacement in Bayesian inference and inversion

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