Application of the projected Landweber method to the estimation of the source time function in seismology

The empirical Green function (EGF) model, which is used in this paper for the analysis of the waveforms of low-energy earthquakes, consists in assuming that the propagating medium and the recording instrument can be treated as a linear system and that the impulse response function of the system can be approximated by the waveform of a very small earthquake. The deconvolution of the Green function event from the waveform of a larger one, located at approximately the same position, provides information about the source time function (STF) of the latter. Linear inversion methods do not yield satisfactory estimations of the STF which must be positive and causal. Moreover, an estimate of the duration (support) of the STF should be desirable. In this paper we apply to this problem the so-called projected Landweber method, which is an iterative nonlinear method allowing for the introduction of constraints on the solution. The implementation of the method is easy and efficient. We first validate the method by means of synthetic data, generated by the use of waveforms of a seismic swarm that occurred in the Ligurian Alps (north-western Italy) during July 1993. Then, taking into account the indications provided by the simulations, the method has been applied to the inversion of real data, yielding satisfactory results also in the case of quite complex events.