Full waveform inversion reconstructs subsurface velocity models by matching simulated and recorded seismic waves. The inversion is nonlinear, and multiple velocity structures can produce similar waveforms -- a non-uniqueness problem that demands uncertainty quantification. Xuebin Zhao and Andrew Curtis compare two approaches: a linearised method that approximates the posterior as a Gaussian around the best-fit model, and a nonlinear method using variational inference with transformed Gaussians. Both recover similar posterior means. But their uncertainty structures diverge dramatically, especially near layer interfaces where the physics is most nonlinear.
The linearised approach produces tighter, more confident uncertainty bounds -- and these bounds are wrong. They generate poor data fits and biased property estimates because linearising the wave equation eliminates precisely the nonlinear interactions that create the complex posterior geometry. The nonlinear method produces wider, fuzzier uncertainty estimates that are harder to interpret but correspond to the actual range of solutions consistent with the data. This is a case where apparent precision degrades the answer. A geologist using the linearised uncertainty might drill with high confidence in the wrong location; the same geologist using the nonlinear uncertainty would know to hedge. The sharper image is the less truthful one. The lesson extends beyond seismology: in any inverse problem where the forward model is nonlinear, the uncertainty structure inherits that nonlinearity. Approximating the uncertainty as linear while keeping the inversion nonlinear produces a chimera -- a point estimate from one mathematical world and error bars from another. The honest answer is blurry, and the blur is information.
(arXiv:2603.11711)