Seismic interferometry is a powerful tool to monitor the seismic velocity change associated with volcanic eruptions. For the monitoring, changes in seismic velocity with environmental origins (such as precipitation) are problematic. In order to model the environmental effects, we propose a new technique based on a state-space model. An extended Kalman filter estimates seismic velocity changes as state variables, with a first-order approximation of the stretching method. We apply this technique to three-component seismic records in order to detect the seismic velocity change associated with the Shinmoe-dake eruptions in 2011 and 2018. First, ambient noise cross-correlations were calculated from May 2010 to April 2018. We also modeled seismic velocity changes resulting from precipitation and the 2016 Kumamoto earthquake, with exponential type responses. Most of the results show no significant changes associated with the eruptions, although gradual inflation of the magma reservoir preceded the 2011 eruption by one year. The observed low sensitivity to static stress changes suggests that the fraction of geofluid and crack density at about 1 km depth is small, and the crack shapes could be circular. Only one station pair west of the crater shows the significant drop associated with the eruption in 2011. The gradual drop of seismic velocity up to 0.05% preceded the eruption by one month. When the gradual drop began, volcanic tremors were activated at about 2 km depth. These observations suggest that the drop could be caused by damage accumulation due to vertical magma migration beneath the summit.