Fig. 4

Effect of background seismic signals, \(v_s\), in the seismic data, \(d_v\), on the estimation of \(H_{ps}\). The colors indicate the frequency bands: 1–3.5 Hz (red), 3.5–7 Hz (green), 7–12 Hz (blue), and 12–18 Hz (purple). One-hundred random functions are bandpass filtered and used as \(v_s\). The horizontal axes in a, c, and d are \(\Gamma \), the square-root of power ratio of \(v_s\) to the original \(d_v\) that is approximated as \(v_a\). It is noted that the actual power ratio \(E(v_s)/E(v_a)\) is larger than \(\Gamma ^2=E(v_s)/E(d_v)\) as \(d_v\) itself contains a significant power of background signals. The lines and error bars show the means and the standard deviations of \(H_{ps}\) (a), the maximum correlation coefficient, \(R_{max}\) (d), and the time delay, \(\tau _{max}\), that gives \(R_{max}\) (c) for the hundred random functions. The closed circles on the vertical axies are the values without \(v_s\). The relation between the individual \(R_{max}\) and \(H_{ps}\) for all the cases are shown with circles in b