Kalman Filter Algorithm for the joint Processing of GNSS PPP and Accelerometer Data, EEW parameters from the unbiased displacement time-series

The filter owes its name to R. E. Kalman at the Institute for Advanced Studies in Baltimore (Kalman, 1960), developed well before the start of the digital age. It has since found numerous applications in control, signal estimation and general filtering problems. Although similar proposals have been made before (c.f. Mayhew, 1999) in a different context, the proposal to apply a Kalman filter to the problem of combining observations from the Global Navigation Satellite Systems (GNSS) with data from a three axis accelerometer was initially proposed by Smyth and Wu, 2007. The general idea of the Kalman filter is to predict a system’s behaviour from incomplete observations. The problem here is to predict high resolution displacement and velocity from acceleration and Precise Point Position (PPP) data which are available at different times and different rates from the accelerometer and corrected GNSS data respectively. The idea to apply this to real-time seismology and earthquake early warning (EEW) goes back to Bock, Melgar, and Crowell, 2011 and has been validated in several studies (Melgar et al., 2013; Li, 2015; Niu and Xu, 2014) where various data-sets from large earthquakes were processed off-line, after the event.