Abstract
Difference methods have been routinely used to compute velocity and acceleration from precise positioning with global navigation satellite systems (GNSS). A low sampling rate (say a rate not greater than 1 Hz, for example) has been always implicitly assumed for applicability of the methods, because random measurement errors are significantly amplified, either proportional to the sampling rate in the case of velocity or square-proportional to the sampling rate in the case of acceleration. Direct consequences of a low sampling rate are the distortion of the computed velocity and acceleration waveforms and the failure to obtain almost instantaneous values of velocity and acceleration. We reformulate the reconstruction of velocity and acceleration from very high-rate (50 Hz) precise GNSS as an inverse ill-posed problem and propose the criterion of minimum mean squared errors (MSE) to regularize solutions of velocity and acceleration. We successfully apply the MSE-based regularized method to reconstruct the very high-rate velocity and acceleration waveforms, the peak ground velocity (PGV) and the peak ground acceleration (PGA) from 50 Hz precise point positioning (PPP) position waveforms for the 2013 Lushan Mw6.6 earthquake. The reconstructed results of velocity and acceleration are shown to be in good agreement with the motion patterns in the PPP position waveforms and correctly recover the earthquake signal. The reconstructed GNSS-based PGA values are a few hundred times smaller than those from the strong motion seismometers.