Fig. 7

Ranges of the observed parameters that can provide realistic solutions for two source positions. a, b The colored arcuate areas show the ranges of the observed parameters (\(f_\text {0}\) and \(\Delta t\)) with realistic solutions of the combined equation, assuming that the source positions of the explosions are at the center (\(r = 0\)) and the margin (\(r = 58\)) of the lava pond. The colors show the values of the estimated depth, as in Fig. 4b. The gray contours show the PDF distribution of the observed parameters (Fig. 4). Here, two cases are shown, where the seismic propagation velocities \(v_\text {p0}\) are a 23 m/s and b 100 m/s. For simplicity, focusing on a case of \(f_\text {0} = 0.5\) Hz (dotted line), \(\Delta t\) when the source is at the center is smaller than that when the source is at the margin. If the source position changes within \(0\le r \le 58\), the range of \(\Delta t\) between the upper and lower bounds (black circles) can be reproduced in our model. c Ranges of \(\Delta t\) between the bounds. The range narrows as \(v_\text {p0}\) increases. However, the upper bound remains the same with increasing \(v_\text {p0}\). The range also depends on the assumed radius of the lava pond \(a_\text {c}\)