Empiricist Geophysics

Data, Models, and Bayesian Inference. Geophysical inverse problems, by picture and by computation. The notebooks I keep to refresh my own thinking and to build my own map of how data, models and inference fit together in geophysics, shared as an open course. Every idea on a small problem whose answer is known; every method actually run; every figure from that run. The notebooks were developed with Claude Opus 5.5 (Anthropic). The course will continue to evolve as my understanding of the subject grows and changes.

Topic 1 · Geophysical forward problems and numerical methods planned

• Elastic dislocation models: point sources and finite faults in an elastic half-space (Okada 1985, 1992), triangular dislocations and layered Earth models; how the Green's-function matrix G of a slip inversion is built, and how its errors become the prediction covariance C_p (2.13).
• Numerical methods: finite-difference, finite-element and boundary-element methods; meshing a curved fault from a slab model; checking accuracy by mesh refinement.
• Earthquake cycle and postseismic processes: interseismic locking (backslip), rate-and-state and rate-strengthening friction, stress-driven afterslip (the forward model used in Topic 3), viscoelastic relaxation in Maxwell and Burgers materials, power-law flow laws, and poroelastic rebound.
• Tsunami modelling: from seafloor deformation to the initial sea surface (including the contribution of horizontal motion on sloping seafloor), the shallow-water equations, dispersion (Boussinesq equations), adaptive mesh refinement, and tsunami Green's functions.
• Seismic waves: the elastic wave equation, body and surface waves, synthetic seismograms, kinematic rupture models, and back projection of array recordings.

Topic 2 · Inverse problems and Bayesian inversion

Knowledge map: how the ideas of this topic connect, with what each one builds on and a self-check question for each.

A · Probability for inverse problems

• 2.01 Basics: deterministic physics, uncertain answers
• 2.02 Bayes in pictures: what a posterior looks like
• 2.03 Probability distributions: describing numbers you cannot predict exactly
• 2.04 Joint, marginal and conditional probability; Bayes' rule
• 2.05 Law of Large Numbers and Central Limit Theorem
• 2.06 Matrix and notation primer: the linear algebra an inversion actually uses

B · Linear inversion, ill-posedness, priors

• 2.07 Least squares: the maximum-likelihood answer, and why it is not enough
• 2.08 Bayesian linear inversion and the MAP
• 2.09 Regularisation and smoothing are priors
• 2.10 Loss functions and optimisation: what a fitting program minimises, and how it walks downhill
• 2.11 Choosing a prior: families, shapes and prior-predictive checks
• 2.12 Gaussian processes: priors on whole functions

C · Errors and data weighting

• 2.13 Model errors: C_d and C_p
• 2.14 Building C_d: noise in GNSS, InSAR and tsunami data

D · Computing a posterior: sampling

• 2.15 The map: five methods, six posterior shapes
• 2.16 Metropolis: the random walk
• 2.17 JAX, differentiability and gradient-based methods
• 2.18 Gradients, HMC and NUTS: sliding instead of stumbling
• 2.19 Convergence, mixing, R-hat and Monte Carlo error
• 2.20 Tempering and SMC: turning the data up slowly
• 2.21 Variational inference and normalizing flows: inference as optimisation
• 2.22 Gradients through physics: finite differences, autodiff and adjoints

E · Time-dependent and ensemble inversion

• 2.23 The Kalman filter: inversion through time
• 2.24 Ensemble Kalman methods: EnKF, EKI, EnKI
• 2.25 Nonlinearity: how to measure it, and how EKF, EnKI and the samplers cope
• 2.26 Localized ensemble Kalman inversion: small ensembles, many unknowns

F · Hyperparameters and weights

• 2.27 Entropy and information: how much did the data teach us?
• 2.28 KL divergence: how different are two beliefs?
• 2.29 Choosing hyperparameters and comparing models
• 2.30 Hierarchical Bayes: who sets the weights?
• 2.31 To smooth or not: smoothing, objectivity and model error

G · Checking and synthesis

• 2.32 Using and checking a posterior
• 2.33 Capstone: one fault-slip inversion, every choice

Topic 3 · Forward surrogate models under development

• 3.01 Surrogates 101: a cheap stand-in for the forward model
• 3.02 Neural networks and ResNets from scratch
• 3.03 GP surrogates and Calibrate-Emulate-Sample
• 3.04 Physics-informed neural networks
• 3.05 Neural operators: learning the map from a friction profile to afterslip
• 3.06 Reduced-order models and real-time inference
• 3.07 Inference with surrogates: putting it together

Topic 4 · Data and data processing planned

• GNSS: daily positions and velocities, high-rate GNSS displacement waveforms, reference frames, common-mode error, offsets, coloured noise (white, flicker and random walk; 2.14), and time-series models of postseismic motion.
• InSAR: interferograms, phase unwrapping, atmospheric and orbital errors, decimation, and time-series methods; Sentinel-1, ALOS-2 and NISAR.
• Seafloor and ocean observations: DART buoys; cabled seafloor networks such as S-net along the Japan Trench; ocean-bottom pressure and differential pressure gauges (DPG); GNSS-acoustic seafloor positioning; coastal tide gauges; and removing tides, instrument drift and non-tectonic signals.
• Satellite altimetry: SWOT wide-swath sea-surface height and conventional nadir altimeters, as tsunami observations.
• Seismic data: teleseismic body and surface waves, strong-motion accelerograms, instrument response, filtering and alignment.
• From data to C_d: noise models, correlated errors and data weighting for each data type (2.14, 2.30).

Also: the sampler playground (runs in the browser). Code MIT, text and figures CC BY 4.0.