Probabilistic forecasts are regarded as the highest achievable goal when predicting earthquakes, but limited information on stress, strength, and governing parameters of the seismogenic sources affects their accuracy. Ensemble data-assimilation methods, such as the Ensemble Kalman Filter (EnKF), estimate these variables by combining physics-based models and observations. While the EnKF has demonstrated potential in perfect model experiments using earthquake simulators governed by rate-and-state friction (RSF) laws, challenges arise from the non-Gaussian distribution of state variables during seismic cycle transitions. This study investigates the Adaptive Gaussian Mixture Filter (AGMF) and the Particle Flow Filter (PFF) as alternatives for improved stress and velocity estimation in earthquake sequences compared to Gaussian-based methods like the EnKF. We test the AGMF and the PFF's performance using Lorenz 96 and Burridge–Knopoff 1D models which are, respectively, standard simplified atmospheric and earthquake models. This approach, using widely recognized and commonly used testbed models in their fields, makes the methods and findings accessible to both the data assimilation and seismology communities, while supporting comparisons and collaboration. We test these models in periodic, and aperiodic conditions, and analyze the impact of assuming Gaussian priors on the estimates of the ensemble methods. The PFF demonstrated comparable performance in chaotic scenarios, yielding lower RMSE for the estimates of the Lorenz 96 models and stronger resilience to underdispersion for the Burridge–Knopoff 1D models. This is vital given the limited and sparse historical earthquake data, underscoring the PFF's potential in enhancing earthquake forecasting. These results emphasize the need for careful data assimilation method selection in seismological modeling.

Our ability to forecast earthquakes and slow slip events is hampered by limited information on the current state of stress on faults. Ensemble data assimilation methods permit estimating the state by combining physics-based models and observations, while considering their uncertainties. We use an ensemble Kalman filter (EnKF) to estimate shear stresses, slip rates and the state θ acting on a fault point governed by rate-and-state friction embedded in a 1-D elastic medium. We test the effectiveness of data assimilation by conducting perfect model experiments. We assimilate noised shear-stress and velocity synthetic values acquired at a small distance to the fault. The assimilation of uncertain shear stress observations improves in particular the estimates of shear stress on fault segments hosting slow slip events, while assimilating observations of velocity improves their slip-rate estimation. Both types of observations help equally well to better estimate the state θ. For earthquakes, the shear stress observations improve the estimation of shear stress, slip rates and the state θ, whereas the velocity observations improve in particular the slip-rate estimation. Data assimilation significantly improves the estimates of the temporal occurrence of slow slip events and to a large extent also of earthquakes. Rapid and abrupt changes in velocity and shear stress during earthquakes lead to non-Gaussian priors for subsequent assimilation steps, which breaks the assumption of Gaussian priors of the EnKF. In spite of this, the EnKF still provides estimates that are unexpectedly close to the true evolution. In fact, the forecastability for earthquakes for the same alarm duration is very similar to slow slip events, having a very low miss rate with an alarm duration of just 10 per cent of the recurrence interval of the events. These results confirm that data assimilation is a promising approach for the combination of uncertain physics and indirect, noisy observations for the forecasting of both slow slip events and earthquakes.