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Solving Geophysical Inversion Problems with Intractable Likelihoods: Linearized Gaussian Approximations Versus the
1Institute of Earth Sciences, University of Lausanne, Lausanne, Switzerland.
Mathematical Geosciences
|January 29, 2024
Summary
This study introduces a new Bayesian inversion method for hydrogeology that treats unobservable geophysical properties as latent variables. The approximate Gaussian method is fast but less accurate with high uncertainty, unlike the correlated pseudo-marginal method.
Area of Science:
- Geophysics
- Hydrogeology
- Bayesian inference
Background:
- Geophysical Bayesian inversion aims to estimate subsurface parameters from geophysical data.
- Petrophysical relationships linking geological parameters to geophysical properties often exhibit scatter.
- Treating intermediate geophysical properties as latent variables addresses this uncertainty.
Purpose of the Study:
- To develop and evaluate methods for geophysical Bayesian inversion within a latent variable model.
- To estimate the intractable likelihood function of geological parameters given geophysical data.
- To compare a new approximate Gaussian method with the correlated pseudo-marginal method.
Main Methods:
- Approximation of the likelihood function using a Gaussian probability density function based on local linearization.
- Incorporation of petrophysical relationship noise into the data covariance matrix.
- Comparison with the general correlated pseudo-marginal method using Monte Carlo averaging over latent variable samples.
Main Results:
- Both methods yielded similar estimates for a synthetic crosshole ground-penetrating radar travel time inversion with low petrophysical uncertainty.
- Ignoring petrophysical uncertainty led to biased estimates.
- The linearized Gaussian approach's accuracy decreased with increasing petrophysical scatter, while the correlated pseudo-marginal method remained accurate.
Conclusions:
- The correlated pseudo-marginal method is robust for geophysical inversion with significant petrophysical uncertainty.
- The linearized Gaussian approximation offers computational efficiency but is sensitive to petrophysical scatter.
- Accurate geophysical inversion requires accounting for petrophysical uncertainty.
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