A novel blind-deconvolution method with an application to seismology

The empirical Green function (EGF) model assumes that the recorded far-field waveform of an earthquake is the output of a linear system whose impulse response function is approximated by the waveform of a suitable small earthquake (the EGF) with the same focal mechanism and location as the larger one. The input of the system is the so-called source time function (STF) which describes the energy release and the rupture evolution. In a previous paper the projected Landweber method was applied to this deconvolution problem, i.e. to the estimation of the STF being given the EGF and the recorded waveform of the seismic event. The results obtained are more realistic and qualitatively much better than those provided by linear regularization methods, as a consequence of the beneficial effect of the constraints on the STF (positivity, causality, etc) introduced by means of the projected Landweber method. However, the STFs obtained in this way do not reproduce the observed seismograms within the experimental errors. This effect is presumably due to the modelling error introduced when approximating the exact (but unknown) Green function by means of the EGF so that the problem arises of improving such an approximation. To this purpose we propose a nontrivial modification of an iterative blind-deconvolution method used for image identification. The main feature of our method, which is based on the projected Landweber method, is that the use of different constraints for the EGF and STF is allowed. The convergence of the method is very fast and the results obtained in the case of synthetic and real data are quite satisfactory. Even if described and validated in the specific problem of seismology we are considering, it can be applied to any deconvolution problem where a rough approximation of the point spread function is available and different constraints must be used for the impulse response function and the input of the system.