Seismic energy envelopes in volcanic media: in need of boundary conditions

Seismogram envelopes recorded at Campi Flegrei caldera show diffusive characteristics as well as steep amplitude increases in the intermediate and late coda, which can be related to the presence of a non-uniformly scattering medium. In this paper, we first show the results of a simulation with a statistical model considering anisotropic scattering interactions, in order to match coda-envelope duration and shape. We consider as realistic parameters for a volcanic caldera the presence of large square root velocity fluctuations (10 per cent) and two typical correlation lengths for such an heterogeneous crust, a = 0.1 and 1 km. Then, we propose the inclusion of a diffusive boundary condition in the stochastic description of multiple scattering, in order to model intermediate and late coda intensities, and particularly the sharp intensity peaks at some stations in the caldera. Finally, we show that a reliable 2-D synthetic model of the envelopes produced by earthquakes vertically sampling a small region can be obtained including a single drastic change of the scattering properties of the volcano, that is, a caldera rim of radius 3 km, and sections varying between 2 and 3 km. These boundary conditions are diffusive, which signifies that the rim must have more scattering potential than the rest of the medium, with its diffusivity 2-3 orders of magnitude lower than the one of the background medium, so that the secondary sources on its interface(s) could enhance coda intensities. We achieve a good first-order model of high-frequency (18 Hz) envelope broadening adding to the Monte Carlo solution for the incident flux the secondary source effects produced by a closed annular boundary, designed on the caldera rim signature at 1.5 km depth. At lower frequencies (3 Hz) the annular boundary controls the intermediate and late coda envelope behaviour, in a way similar to an extended diffusive source. In our interpretation, the anomalous intensities observed at several stations and predicted by the final Monte Carlo solutions are mainly due to the diffusive transmission reflection from a scattering object of increased scattering power, and are controlled by its varying thickness.