Combining High Rate GPS and Strong Motion Data: A Kalman Filter Formulation for Real-Time Displacement Waveforms

Displacement waveforms in seismology are traditionally obtained by integration, polynomial fitting and filtering of strong motion records. The displacements so obtained are band-limited since the low frequency components (including the static deformation) are not accurately determined. Additionally there is no objective way of determining the proper parameters for this numerical process (filter corner frequency, polynomial order, etc.) and they have to be tailored by each researcher to what “works best” for every station-event pair. This hinders the automatization of the process, the application to large networks, and real-time processing. We demonstrate a new technique that utilizes elements from the theory of stochastic estimation and control to derive a multi-rate Kalman filter that fuses data from strong motion and GPS instruments in order to obtain real-time total (dynamic and static) displacement waveforms. The filter allows one to combine data streams with different sampling rates. Our formulation assumes Gaussian white noise in both accelerometer and GPS observations, whose variances are determined objectively from pre-event windows for each data type. We demonstrate the multi-rate Kalman filter with two examples: (1) Outdoor experiments at the NEES Large High-Performance Outdoor Shake Table at UCSD in which real recorded seismograms from the 1971 San Fernando and 1994 Northridge earthquakes were utilized as input, with observations taken by 250 Hz accelerometers and 50 Hz GPS receivers; (2) Data from the April 4, 2010 El Mayor-Cucapah earthquake (Mw 7.2) collected by 100 Hz strong motion recordings from stations of the California Integrated Seismic Network (CISN) and 1 Hz GPS data from the California Real Time Network (CRTN), at co-located stations as far as 300 km from the epicenter. Spectral analysis of the resulting displacement waveforms shows some limitations in the Kalman filter’s real-time epoch-by-epoch formulation that depend on the slower sampling rate of the GPS receivers. However, as we show these limitations have little practical consequence, and near-real-time Kalman filter smoothing can recover the full spectral content. We conclude that this formulation is superior to traditional methodologies by providing total displacement waveforms at the sampling rate of the higher-rate accelerometers. As such, we propose that the future of strong motion sensing should include a co-location of accelerometers and high rate GPS and the corresponding integrated processing of both data streams. This configuration is ideal as part of earthquake early warning systems.