Simple Linear Inverse for Complex Sources in Early Warning

In this project we aim to determine a foreshock/mainshock pair in real time. That is, we are trying to determine the timing and amplitudes of events in a complex sequence (e.g. foreshock/mainshock pair) in an ongoing earthquake. We are assuming random phase in seismograms so that energy envelopes add linearly which provides the opportunity to pose the problem as a linear least squares inverse problem. We are working on a methodology to solve the least squares problem in real time. The advantage is that the inverse problem can be regularized (unlike traditional deconvolution). We parameterize earthquakes as moments in time windows. Then the problem becomes linear as the determination of the amplitude of the moment in each time window. We first construct the Green's functions for the energy envelopes derived from the predicted envelopes of channels of ground motion of the Virtual Seismologist (VS) (Cua, G. and Heaton, T. 2007). Then, these Green's functions are deconvolved (in a generalized way) from the energy envelopes of seismic records using a damped least-squares inversion in order to determine the amplitude of the moment in each time window. There is a positivity constraint that the amplitude of the moments be positive everywhere. Because the VS predicted envelopes are defined for earthquakes of magnitude 6.5 and below, this technique is not used for very large events.