Geophysics is the science of inference about Earth's interior from surface observations. Seismic waves, gravity anomalies, magnetic fields, and electromagnetic signals all carry information about subsurface structure — but the relationship between observations and structure is indirect, nonlinear, and non-unique. Bayesian inversion treats the subsurface model as a random variable with a prior distribution, computes the likelihood of observed data given any proposed model, and delivers a posterior distribution that captures the full range of structures compatible with the data.
Seismic Inversion
Seismic tomography — imaging Earth's interior from earthquake arrival times or waveforms — is the geophysical analogue of medical CT scanning. Bayesian seismic inversion replaces the single "best-fit" velocity model with an ensemble of models sampled from the posterior distribution. This is critical because seismic data typically constrain some features well (large-scale structure) while leaving others poorly resolved (small-scale heterogeneity), and the posterior naturally reveals these resolution differences.
where m is the Earth model, g(m) is the forward model (wave propagation),
d_obs is observed data, and C_d is the data covariance matrix.
Transdimensional Bayesian inversion, pioneered by Malcolm Sambridge and colleagues, treats the number of model parameters as itself unknown. For Earth structure, this means the number and location of layer boundaries are inferred from the data alongside the layer properties — the model complexity adapts to the information content of the data. Reversible-jump MCMC is the computational engine for this approach.
Earthquake Location and Hazard
Bayesian methods improve earthquake location by properly accounting for uncertainty in the velocity model through which seismic waves travel. The NonLinLoc software uses Monte Carlo sampling to produce posterior probability density functions for earthquake hypocenters, revealing the elongated, irregular uncertainty regions that arise from poor station geometry — information lost in traditional linearized inversions that report only error ellipses.
PSHA estimates the probability of exceeding a given ground motion level at a site over a specified time period. Bayesian PSHA incorporates uncertainty in earthquake occurrence rates, magnitude-frequency relationships, and ground motion prediction equations through a logic tree framework that is increasingly formalized as Bayesian model averaging. This uncertainty quantification directly informs building codes and insurance pricing.
Resource Exploration
In petroleum geophysics, Bayesian inversion of seismic reflection data estimates subsurface properties — porosity, fluid content, lithology — relevant to hydrocarbon exploration. The prior encodes geological knowledge: sedimentary layers are typically laterally continuous, porosity decreases with depth, and certain rock types tend to co-occur. Bayesian rock physics models link seismic velocities to reservoir properties, enabling probabilistic reserve estimation.
Full Waveform Inversion
Full waveform inversion (FWI) uses the complete seismic waveform rather than just arrival times, extracting far more information but at enormous computational cost. Bayesian FWI using Hamiltonian Monte Carlo or variational inference is an active frontier, promising uncertainty-aware subsurface images that could transform both academic geophysics and the energy industry.
"The Earth does not yield its secrets easily. Bayesian inversion is our most honest way of stating what we know, what we don't know, and what the data actually tell us about the subsurface." — Albert Tarantola, author of Inverse Problem Theory and Methods for Model Parameter Estimation
Current Frontiers
Machine learning surrogates for expensive geophysical forward models are enabling Bayesian inversion at previously infeasible scales. Bayesian optimal experimental design guides the placement of seismic stations and survey geometries. And the integration of multiple geophysical data types — seismic, gravity, magnetic, electromagnetic — through joint Bayesian inversion produces more tightly constrained subsurface models than any single data type alone.