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.

S U M M A RYAn iterative and constrained deconvolution technique, called projected Landweber deconvolution (PLD), is applied to a local earthquake data set recorded in the southwestern Alps (NW Italy) in order to estimate the relative source time function (RSTF) of the events. The magnitude range analysed is 1X4`M L`4 X3. The smallest events (M L`2 X0) are used as empirical Green's functions (EGF) to deconvolve the larger earthquakes. The crucial choice of appropriate EGF is tackled using highprecision relative locations. Results demonstrate that PLD successfully overcomes the instability e¡ects of the deconvolution process and provides stable and reliable RSTFs. Moreover, this method allows us to obtain an objective determination of the duration of the RSTF, which is essential to estimate correctly the source size. Combining the information inferred from both RSTFs and displacement spectra, source parameters are computed for all the events. We obtain seismic moments ranging from 2X9|10 11 to 3X1|10 14 N m and source radii between 70 and 700 m. Taking the instrumental limits into account, no breakdown in constant stress drop scaling is seen in our estimates.

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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.

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