Fig. 4

PDF distribution of observed parameters and areas where the solutions exist for the combined equation. a PDF distribution of observed parameters (\(f_\text {0}\) and \(\Delta t\)). This PDF is normalized so that the sum is 1. The PDF distribution of \(f_\text {0}\) is centered at \(\sim\)0.5 Hz. The spectral structure of the amplitude spectrum of \(f_\text {0}\) for each event is reflected in this distribution. The PDF distribution of the \(\Delta t\) axis shows the variation in \(\Delta t\) of all events. b, c The range of the observed parameters (\(f_\text {0}\) and \(\Delta t\)), which can provide realistic depths and sound velocities inside the vent (colored arcuate areas). The colors show the values of estimated b depth and c sound velocity. These colored arcuate areas are narrow compared with the observed PDF distribution (gray contours). When the observed parameters included in the uncolored areas are substituted into the combined equation, the equation does not produce appropriate solutions. These figures show the cases in which the closed radius of the frustum pipe (\(a_\text {c}\)) is 58 m