Canonical model of volcano acoustics

A full wave-theoretical model is developed for the acoustic field generated by an explosive point source embedded in a magma column that is open to the atmosphere. The Green's functions for the field in the magma and the atmosphere are derived on the basis of several simplifying assumptions concerning the geometry of the conduit, the boundary conditions, and the geoacoustic properties of the magma. A kinematic model for the acoustic signature of the explosive source is proposed, which, when combined with the Green's functions, provides full analytical expressions for the acoustic field in the magma and in the atmosphere as (complex) functions of frequency. By Fourier inversion the airborne pressure spectrum is transformed into a pressure time series. The predicted sound pulse in the atmosphere and its energy spectrum ate highly dispersive, showing complicated structure that arises from the coherent addition of many normal modes of oscillation (i.e. depth and radial resonances) in the magma column. At high frequencies, for which the aperture of the vent is many wavelengths across, each mode is launched into the atmosphere as a parallel-sided beam of sound with a characteristic angle of elevation, which, through Snell's law, is determined by the speed of sound in the magma relative to that in air. At somewhat lower frequencies,the modal beams of sound undergo angular spreading due to diffraction at the edge of the vent. In the lowest-frequency regime, where the wavelength is comparable with the aperture, the airborne field shows little angular structure. A comparison between airborne acoustic data that we recorded in July 1994 at the western vent of Stromboli Volcano and the predictions of the theory, using parameters that are characteristic of Stromboli, show compelling agreement. The theoretical and observed power spectra both display the following features: (1) a concentration of energy below 20 Hz, associated with the first four longitudinal resonances; (2) radial resonances between 35 and 65 Hz; and (3) a broad minimum around 30 Hz, arising because the source lies near nulls in longitudinal modes that would otherwise be excited. The conclusion is that the airborne sound signature from an explosive volcanic event may be inverted to provide estimates of the depth and radius of the magma conduit, the depth, spectral shape and peak shock-wave pressure of the source, and the viscosity of the magma.