<p>Data assimilation (DA) has tried to incorporate GNSS data into physics-based fault slip models to estimate frictional properties and predict future slip evolution on faults. For unstable slip events, such as ordinary fast-slip earthquakes and slow slip events (SSEs), accurately estimating both the time-varying frictional strength and the time-invariant frictional parameters is crucial for reliable slip prediction. However, because frictional strength has not been directly observed, previous DA studies estimating the frictional parameters often assumed a steady-state strength at the initial time of assimilation, which limited the accuracy of long-term slip prediction. In the present study, we propose a new adjoint-based DA method that estimates an appropriate initial frictional strength along with the frictional parameters to assimilate long-term SSEs. The key idea is to impose an additional constraint on DA, assuming that the current SSE recurs periodically, though the exact interval is unknown. This approach reflects the observed recurring nature of SSEs. This new method is validated through numerical experiments focusing on long-term Bungo Channel SSEs in southwest Japan. The results demonstrate that our proposed method provides reasonable estimates for both the frictional strength and the frictional parameters, enabling accurate predictions of slip evolution and the timing of subsequent SSEs, along with estimating the unknown recurrence interval. The method proves effective even with data windows shorter than the recurrence interval, overcoming the limitations of previous DA methods.</p> Graphical Abstract <p></p>