Earthquake moment tensors — the mathematical objects describing how fault slip radiates seismic waves — are estimated by comparing observed waveforms to synthetic ones computed through a model of Earth's interior. The Earth model is always wrong. The standard fix: assume the mismatch between real and synthetic waveforms is Gaussian noise and fold it into the likelihood function.
Saoulis, Pham, and Ferreira (arXiv:2603.18925) show the Gaussian assumption systematically lies. Even modest 1–3% variations in 1-D Earth structure produce theoretical errors that are emphatically non-Gaussian. The result: standard inversions don't just add uncertainty — they introduce bias. The estimated earthquake mechanisms are wrong in a consistent direction, and the reported uncertainties are too small. The method is confident and incorrect.
The fix uses simulation-based inference (SBI), a machine learning approach that trains on many forward simulations with varied Earth models, learning the actual distribution of theoretical errors rather than assuming one. Two strategies work: a physics-informed approach that models the error structure explicitly, and an end-to-end neural network that bypasses the forward model entirely. Both produce posteriors that are well-calibrated — the uncertainties actually mean what they claim to mean.
The failure is worst where you'd most want accuracy: shallow earthquakes (where the seismic waves spend more time in the poorly-known crust) and shorter-period waves (which resolve finer structure in the wrong Earth model). The isotropic component of the moment tensor — the part that distinguishes explosions from tectonic earthquakes — is especially corrupted by Gaussian assumptions.
Validation on two real earthquakes (1997 Long Valley Caldera and 2020 Zagreb) confirms the practical difference. The SBI posteriors differ meaningfully from Gaussian ones and better match independent constraints.
The structural lesson: when the forward model is imperfect, the distributional assumption about how it's imperfect is as load-bearing as the model itself. A Gaussian error model is a claim about the world — that errors are symmetric, independent, and thin-tailed. When that claim is wrong, the inference inherits not random error but systematic distortion. The assumption about noise becomes the dominant source of signal corruption.
Saoulis, Pham, and Ferreira, "Improving moment tensor solutions under Earth structure uncertainty with simulation-based inference," arXiv:2603.18925 (2026).