Blind deconvolution of seismograms regularized via minimum support

The separation of earthquake source signature and propagation effects (i.e. the Earth’s Green’s function) that encode a seismogram is a challenging problem in seismology, and is termed blind deconvolution. By considering records of multiple earthquakes at a given station that share the same Green’s function, we can write the linear relation uk(t) ● sj(t) - uj(t) ● sk(t) = 0 where uk is the seismogram for the kth source and sj is the jth unknown source. The symbol ● represents the convolution operator. Using two or more seismograms, we obtain a homogeneous linear system where the unknowns are the sources (note from symmetry, we might equally well consider seismograms representing the same source but different Green’s functions and consider the Green’s functions as unknowns). This system is augmented through specification of scale to deliver a non-trivial solution. Two issues require careful consideration for this problem to be solved. First, source durations are not known a priori and must be determined. If the specified source duration is too short, the system becomes over-determined, and if it is too long, the system is under-determined. Accordingly, we introduce source durations as unknowns and solve the combined system (sources and source durations) using separation of variables (Golub & Pereyra, 1973) regularized such that source durations are the shortest necessary to solve the system, i.e. minimum support. The solution is derived iteratively using least squares to recover the sources and the Gauss-Newton algorithm to recover source durations. Second, to resolve realistic signals requires seismograms and solutions of dimension ≥104, hence iterative matrix inversion (LSQR) has been employed in conjunction with fast matrix multiplication operators based on the Fast Fourier Transform (FFT). This method has been tested using synthetic seismograms created by convolution of random sources with simplified Green’s functions and added noise. In numerical simulations to date, the method accurately recovers sources at noise levels up to 10% peak signal strength. Results from more realistic synthetic simulations and applications to real data will be presented.