Fig. 2

Estimation of ACFs and errors using the largest earthquake at E.STHM as an example. a Raw waveform. The lateral axis is taken from the earthquake origin time. The arrow indicates the manually identified P-wave arrival time \({T}_{i}^{p}\). b Whitened waveform. The arrow indicates the noise window, which is defined as 10.5–0.5 s before \({T}_{i}^{p}\). c Bandpass-filtered (1–10 Hz) waveform. The orange square indicates the P-wave window, which is defined as between 0.5 s before and 9.5 s after \({T}_{i}^{p}\). d Fourier amplitude spectra of the raw (brown), whitened (gray), and bandpass-filtered (blue) waveforms. e Filtered waveform in the P-wave window multiplied with a 0.5 s cosine taper to both sides (\({u}_{i}^{obs}(t)\)). f Filtered noise traces (\({u}_{i,j}^{noise}(t)\)) generated by applying the same bandpass filter and taper that is used to generate \({u}_{i}^{obs}(t)\) to 1000 candidates of random traces. g 1000 candidates of \({u}_{i,j}^{eq}(t)\) (Eq. 2). h 1000 candidates of \({a}_{i,j}^{eq}(\tau )\) (Eq. 3; pink), their ensemble average (\({a}_{i}^{ave}(\tau )\); black), and three times their standard deviation (\(3{\sigma }_{i}^{a}(\tau )\); purple). i \(3{\sigma }_{i}^{a}(\tau )\) for 100, 1000, and 10,000 random traces. j Effects of the noise level on the estimates of \(3{\sigma }_{i}^{a}(\tau )\). The results from two and three times the actual noise level are shown. Common amplitude scales are used for e–g and h–j