Based on the unscented Kalman filter, we develop a time‐dependent inversion filter combining Global Positioning System (GPS) and Interferometric Synthetic Aperture Radar (InSAR) time series observations for modeling volcano deformation. We use the Variance Component Estimation method as means to assign the relative weights for GPS and InSAR data. Then we use the inversion filter to model the posteruptive deformation at Okmok volcano, Alaska. We find that a Mogi source at 3–4 km depth fits the InSAR data well, while the best fit to the GPS data is an oblate spheroid source at about 2.5 km depth. Our final model consists of a shallow sill at ~0.9 km and a Mogi source at ~3.2 km depth, which well fit both the GPS and InSAR data simultaneously. We think the Mogi source obtained here is the same source account for the preeruptive deformation. The shallow sill is a new structure that was not seen before the 2008 eruption. From 2008 to 2019, we have observed five inflation episodes, each of which decays exponential in time. We find that the characteristic timescale of those inflation episodes decreases with respect to time. The total volume change from the two sources is 0.068 km3, which recovers 50–60% of the volume decrease during the 2008 eruption.

Using ~25 years of GPS and InSAR observations and the time‐dependent inversion filter, we invert for the volume change history of three active volcanoes in the east central Alaska Aleutian arc and the secular velocity of the region. The inferred time‐dependent volume change shows (1) two exponentially decaying pulses followed by a nearly linear inflation period at Westdahl volcano, (2) multiple episodic inflation events at Akutan volcano, and (3) transient volcanic inflation and long‐term deflation signals at Makushin volcano. Comparing our regional velocity estimates with recently published block models, we find a small signal that can be explained by slip deficit on the megathrust, either from a shallow (<20 km depth) almost fully locked zone or a weakly locked zone confined to 25–50 km depth.

There has been a tremendous expansion of automated processing and open data from the Global Positioning System (GPS). For example, the Nevada Geodetic Laboratory currently acquires geodetic GPS data from more than 17,000 stations and processes daily solution of more than 10,000 stations every week (Blewitt et al., 2018). Such large data sets provide new research opportunities for geophysical problems, but pose challenges in how to efficiently extract signals of interest, such as transient deformations due to various geophysical processes.During the past two decades, many methods have been proposed to detect transient deformation in GPS time series. Examples include, but are not limited to, methods built upon some kind of Kalman filter (Bekaert