Fig. 3

Example of Houston plot. a Theoretical tidal response (gray circles, sampling interval 15 min). Black circles indicate earthquake occurrences. b Example of a Houston plot in which the tidal stress is divided into two bins: positive or negative. The gray bars show the distribution of the expected background relative frequency \({N}_{{\text{exp}}}\) of tidal stress (relative frequency distribution of tidal stress during 1 day before and 1 day after the earthquake, sampling interval 15 min), and the bars outlined by thick black lines show the distribution of the relative frequency \({N}_{{\text{obs}}}\) of tidal stress at the time of the earthquake (bottom axis). The error bars show the \(1\sigma\) error (assuming a total of 100 events), where \(\sigma =\sqrt{n{p}_{i}\left(1-{p}_{i}\right)}\) for the binomial distribution \(B\left(n, {p}_{i}\right)\), \(n\) is the total number of earthquakes, and \({p}_{i}\) is the probability of random events in the \(i\)th bin obtained from the frequency distribution of theoretical tidal values. The blue diamonds, which show the ratio of \({N}_{{\text{obs}}}\) to \({N}_{{\text{exp}}}\) (top axis), are connected by a straight blue line. c Example of a Houston plot in which the tidal stress is divided into six bins. The orange line indicates the value obtained by maximum likelihood method (Yabe et al. 2015) of \({N}_{{\text{obs}}}\left(\Delta S\right)/{N}_{{\text{exp}}}\left(\Delta S\right)={e}^{\alpha \Delta S}\) (Eq. (4)) to the \({N}_{{\text{obs}}}\)/\({N}_{{\text{exp}}}\) values in the six bins. In this example, tidal sensitivity \(\alpha =0.5\)