Inversion (geobrain.inverse, geobrain.optim)

geobrain.inverse states the problem; geobrain.optim searches it.

Stating the problem

# sketch: runnable version in the quick start
InverseProblem(forward=..., observed=..., likelihood=..., prior=...)

Piece	Options
Likelihood	GaussianLikelihood, StudentTLikelihood via ChannelLikelihood for multi-channel data
Prior	GaussianPrior, or any object with log_prob and sample
Waveform misfits	L2Waveform, L1Waveform, HuberWaveform, EnvelopeWaveform, GlobalCorrelationWaveform, NormalizedIntegrationWaveform, TravelTimeWaveform, WassersteinWaveform
Filtering	FrequencyFilteredMisfit, butterworth_lowpass
Joint	JointProblem, JointForward, JointModelSpace

The waveform misfits are not interchangeable. Least squares is the one that falls into a local minimum when the starting model is a cycle out; envelope, travel-time and Wasserstein misfits are the ones that do not, and a multi-scale schedule usually starts with one of those.

Searching it

Inverter is the loop; AdamConfig and LBFGSConfig choose the optimizer.

Piece	Options
Regularizers	smallness, smoothness, smoothness_second_order, total_variation, total_variation_second_order, tikhonov, l1, l2
Structural coupling	cross_gradient, gmm_prior
Weighting	depth_weighting, DepthWeight, Weight, Mask, Freeze
Gradient processing	GaussianSmooth, NormClip, NaNGuard, BoundsClamp, StepProjection
Bookkeeping	InversionResult, IterationRecord, StopReason, CancellationToken, log_every

A regularizer is a prior seen from the other side

A smoothness term in the objective is a Gaussian prior on the model’s roughness. Writing it as a prior= on the problem rather than as a term in the loss means the Bayesian door sees it too, automatically, and without a second copy that can drift out of step.

Where your geology goes

Regularization is not a numerical detail; it is where you say what you believe about the earth before the data arrive. Smoothness says gradational contacts. Total variation says blocky units with sharp boundaries. cross_gradient says two properties change where together but not how much together, which is what you want when the physics are independent but the geology is not. gmm_prior says the cells belong to a known set of rock types, which is the strongest statement of the four and the one that turns two property images into a rock-unit map.

The knobs this page describes, and what each one buys: regularizers, bounds and an IRLS sparsity pass. From examples/01_architecture/05_inversion_toolbox.py.

Stopping

Data carry noise. Driving the misfit to zero fits the noise and grows structure to do it, so the honest stopping rule is chi-squared, the misfit measured in units of the noise, reaching 1. The physics examples stop there and show what carrying on looks like.

See also

• examples/01_architecture/05_inversion_toolbox.py: regularizers, bounds and IRLS, compared on one problem.

• examples/03_physics/02_dc_resistivity.py: the stopping rule, and what happens past it.

• examples/03_physics/04_petrophysical_joint_inversion.py: gmm_prior as the coupling that gives a rock-unit map.