A novel blind-deconvolution method with an application to seismology

Abstract

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.

A Study of the Adaptive Method for Decoupling Overlapping Seismic Records

“…Originating from existing adaptive algorithms frequently encountered in 2-D to 3-D image reconstruction, our approach has significant potentials for the analysis of earthquakes and explosions. This algorithm incorporates techniques known for seismic imaging and verification algorithms (e.g., DARGAHI-NOUBARY, 1999): for example, the use of a characteristic seismic event for signal separation is intrinsically similar to the Empirical Greens Function analysis (HATZEL, 1978;HUTCHINGS and WU, 1990;BERTERO et al, 1998) in strong motion studies and event relocation procedures. For this reason, our adaptive method is only a viable alternative, rather than a replacement, to many of the other signal processing approaches.…”

“…For instance, XIE (2002) uses Pn and Lg spectra to discriminate between nuclear and chemical explosions and effectively extract seismic moments (M 0 ), corner frequencies and Q. Other researchers have applied a simple deconvolution approach based on a characteristic earthquake/explosion taken place at the study region at an earlier time (e.g., HUTCHINGS and WU, 1990;BERTERO et al, 1998). While the success of our adaptive method still impinges on the choices of relatively clean records, the built-in iterative process can sufficiently suppress noise levels and achieve superior signal separation result under appropriate conditions.…”

Variational Methods in Imaging

“…Blind deconvolution is concerned with the simultaneous identification of the kernel k and the function u (see, for example, [48,66,78,101,102,229], which are concerned with variational methods for deconvolution). ♦…”

Untitled

“…For a causal system the support of the kernel function k is restricted to R , which can be interpreted so that the convolution takes into account data of the input signal only from the past. Blind deconvolution problems of this type occur, e.g., in seismology (see [3]). Appropriate function spaces for input signals are, e.g., the function space H s 0Y T , s b 0, or BV 0Y T .…”