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 employ an Ensemble Kalman Filter (EnKF) to estimate shear stresses, slip rates, and the state theta acting on a fault point governed by rate-and-state friction embedded in a 1D 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 the shear stress at the fault of 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 theta. For earthquakes, the shear stress observations improve the estimation of shear stress, slip rates and the state theta, while the velocity observation improves in particular the slip-rate estimation. Data assimilation significantly improves the estimates of temporal occurrence of slow slip events and to a large extent also for earthquakes. The latter is reduced due to large, abrupt changes in velocity and shear stress during earthquakes, which lead to non-Gaussian priors for subsequent assimilation steps. These challenge, but do not undermine the effectiveness of the EnKF. 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% 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.

<p>Our ability to forecast earthquake events is hampered by limited information of the state of stress and strength of faults and their governing parameters. Ensemble data assimilation methods provide a means to estimate these variables by combining physics-based models and observations taking into account their uncertainties. In this study, we estimate earthquake occurrences in synthetic experiments representing a meter-scale laboratory setup of a straight-fault governed by rate-and-state friction. We test an Ensemble Kalman Filter implemented in the Parallel Data Assimilation Framework, which is connected with a 1D forward model using the numerical library GARNET. A perfect-model test shows that the filter can estimate shear stresses, slip rates and state θ acting on the fault even when simulating slip rates up to m/s and can thus be used for estimating earthquake occurrences. We assimilate shear stress and slip-rate observations, representing measurements obtained from shear strain gauges and piezoelectric transducers sensors, and their uncertainties acquired at a small distance to the fault in the homogeneous elastic medium. In this study we evaluate how the Ensemble Kalman filter estimates the state and strength of the faults using these observations, and assess the relative influence of assimilating various observations. The results suggest that the data assimilation improves the estimated timing of the earthquake occurrences. The assimilation of the shear stress observed in the medium improves in particular the estimates of the state θ and the shear stress on the fault, while assimilating observations of velocity in the medium improves the slip-rate estimation.</p>

<p>The highly nonlinear dynamics of earthquake sequences and the limited availability of stress observations near subsurface faults make it very difficult, if not impossible, to forecast earthquakes. Ensemble data-assimilation methods provide a means to estimate state variables and parameters of earthquake sequences that may lead to a better understanding of the associated fault-slip process and contribute to the forecastability of earthquakes. We illustrate the challenges of data assimilation in earthquake simulation with an overview of three studies, each with different objectives and experiments.</p> <p>In the first study, by reconstructing a laboratory experiment with an advanced numerical simulator we perform synthetic twin experiments to test the performance of an ensemble Kalman Filter (EnKF) and its ability to reconstruct fault slip behaviour in 1D and 3D simulations. The data assimilation estimates and forecasts earthquakes, even when having highly uncertain observations of the stress field. In these experiments, we assume the friction parameters to be perfectly known, which is typically not the case in reality.</p> <p>A bias in a friction parameter can cause a significant change in earthquake dynamics, which will complicate the application of data assimilation in realistic cases. The second study addresses how well state estimation and state-parameter estimation can account for friction-parameter bias. For this, we use a 0D model for earthquake recurrence with a particle filter with sequential importance resampling. This shows that in case of intermediate to large uncertainty in friction parameters, combined state-and-parameter estimation is critical to correctly estimate earthquake sequences. The study also highlights the advantage of a particle filter over an EnKF for this nonlinear system.</p> <p>The post- and inter-seismic deformations following an earthquake are rather gradual and do not pose the same challenges for data assimilation as the deformation during an earthquake event. To estimate the model parameters of surface displacements during these phases, a third study illustrates the application of the Ensemble Smoother-Multiple Data Assimilation and the particle filter with actual GPS data of the Tohoku 2011 earthquake.</p> <p>Based on the comparison of the various experiments, we discuss the choice of data-assimilation method and -approach in earthquake simulation and suggest directions for future research.</p>

<p>Our ability to forecast future earthquakes is hampered by the very limited information on the fault slip that produce them. In particular the current state of stress, strength, and parameters governing the slip of the faults are highly uncertain. Ensemble data-assimilation methods provide a means to estimate these variables by combining physics-based models and observations while considering their uncertainties. Perfect model experiments with an Ensemble Kalman Filter (EnKF), connected with one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) earthquake cycle models, demonstrate the ability to estimate the state variables of shear stress, slip velocity, and state (θ) of a straight fault governed by rate-and-state friction surrounded by a homogeneous elastic medium. The models represent a direct-shear laboratory setup in one, two and three dimensions, with an array of shear-strain gauges and piezoelectric transducers located at a small distance to the fault. In this research, we compare the recurrence interval and earthquake occurrence of the EnKF across the different models to better understand the challenges associated with a space-time systems with increasing dimensions and increasingly complex earthquake sequences. The assimilation of synthetic shear-stress and slip-rate observations improves in particular the estimates of shear stress and slip rate on the fault, despite the very low accuracy of the observations. We get reasonable estimates when modelling long-duration earthquakes or slow slip events . Interestingly, we also obtain very good estimates when simulating earthquakes with fast slip rates (up to m/s). The large, nonlinear, changes in stress and velocitiy  during the fast transition from the interseismic to the coseismic phase cause the distributions of the state variables to become bi-modal. The EnKF still provides a reasonable estimate of the time of occurrence of the earthquakes in the synthetic experiments, despite the inherent assumption on the Gaussianity of these distributions.</p>

Summary 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 employ 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 1D 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, while 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% 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.