Blind deconvolution of seismograms regularized via minimum support

Abstract

The separation of earthquake source signature and propagation effects (the Earth’s ‘Green’s function’) that encode a seismogram is a challenging problem in seismology. The task of separating these two effects is called blind deconvolution. By considering seismograms of multiple earthquakes from similar locations recorded at a given station and that therefore share the same Green’s function, we may write a linear relation in the time domain ui(t)*sj(t) − uj(t)*si(t) = 0, where ui(t) is the seismogram for the ith source and sj(t) is the jth unknown source. The symbol * represents the convolution operator. From two or more seismograms, we obtain a homogeneous linear system where the unknowns are the sources. This system is subject to a scaling constraint to deliver a non-trivial solution. Since source durations are not known a priori and must be determined, we augment our system by introducing the source durations as unknowns and we solve the combined system (sources and source durations) using separation of variables. Our solution is derived using direct linear inversion to recover the sources and Newton’s method to recover source durations. This method is tested using two sets of synthetic seismograms created by convolution of (i) random Gaussian source-time functions and (ii) band-limited sources with a simplified Green’s function and signal to noise levels up to 10% with encouraging results.