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Solving Geophysical Inversion Problems with Intractable Likelihoods: Linearized Gaussian Approximations Versus the

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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.

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HydrogeophysicsIntractable likelihoodInverse theoryLatent variable model

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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.