Finite-source waveform inversion of volcanic LP/VLP seismic signals

Long-period (LP) events with typical oscillation periods of 0.2-2 s, and very-long-period (VLP) events with typical durations in the range 2-100 s, are frequently observed in active volcanoes. These signals are related to acoustic vibrations and/or volumetric changes of a fluid-filled resonator, in which the fluid has a hydrothermal or magmatic origin. The observed characteristic periods of LP and VLP events suggest source dimensions ranging from tens of meters up to a few kilometers, with actual length estimates depending on both the physical properties of the fluid and oscillation mode. So far, most waveform analyses for these events have been carried out under the assumption of a point source, allowing quantification of the source mechanism but providing limited information about source size. However, quantitative investigations of the size of the resonator, along with a knowledge of the physical properties of the fluid and oscillation characteristics of such sources, are critical to our understanding of volcanic fluid dynamics. % Here, we propose a method based on waveform inversion carried out for a finite source to investigate the size and oscillation characteristics of the source of LP/VLP signals. The hypocenter, geometry, and orientation of the source are first estimated based on waveform inversion assuming a point source. A finite source is then realized by a set of point sources distributed on a grid surrounding the hypocenter in accordance with the known source geometry and orientation. The mechanism of each point source is fixed by the mechanism obtained from waveform inversion for the initial single point source, and the source-time functions for all point sources are estimated simultaneously by waveform inversion carried out in the frequency domain. As the number of free parameters and attendant noise increase with the number of sources, we apply a smoothing constraint to suppress short-scale noisy fluctuations of moment release between adjacent sources. The smoothing parameter we select is that which minimizes the Akaike Bayesian Information Criterion (ABIC). % The capability of our method is tested by using synthetic seismograms which mimic oscillating vertical or horizontal cracks. A set of 3x3 point sources spaced 50 m apart is arranged on a vertical or horizontal plane centered at a depth of 200 m. A decaying monochromatic oscillation with frequency of 2 Hz and Q=25 is used as source-time function for each point source. A crack oscillation of the mode with wavelength 2W/3, in which W is the width of the crack, is then synthesized by multiplying by a factor -1/2 the amplitudes of the source-time functions of all the point sources distributed symmetrically off the main axis of the modeled crack. The ground response to the crack is obtained by summing the synthetics obtained for each point source, assuming a source embedded in a homogeneous half space. Waveform inversion for a finite source, using 14 receivers distributed within 1 km from the source, can resolve the 2W/3 mode well. The oscillation of a horizontal crack is better resolved than that of a vertical crack, and the horizontal oscillation of a vertical crack is better resolved than the vertical oscillation. These results suggest that this approach may be potentially applicable to observed LP and VLP events.