### Comparison with Mo–SI relationship

Combining the moment magnitude scale law \({\mathrm{log}}_{10}\left({M}_{o}\right)\approx 1.5{M}_{w}+9.09\) (Hanks and Kanamori 1979) with the Gutenberg–Richter law \({\mathrm{log}}_{10}N=\mathrm{a}-\mathrm{b}M\) (Gutenberg and Richter 1956), we have

$${M}_{o}\approx \mathrm{c}{T}^{k}={10}^{\frac{3a}{2b}+9.09}{T}^{\frac{3}{2b},}$$

(1)

where \({M}_{0}\) is the seismic moment, c is the earthquake moment rate, \(\mathrm{c}={10}^{\frac{3a}{2b}+9.09}\), T is the average seismic interval (unit: s), a and b are the coefficients in the Gutenberg–Richter law, and \(\mathrm{k}=\frac{3}{2b}\) (see Additional file 1: “Explanation for Eq. 1”). Equation 1 interprets the relationship we observed between the logarithm of the seismic moment and MSI in the observed seismically dense spheres (Figs. 2, 3, 4, 5, 6, 7, 8). In addition, assuming that \(\mathrm{k}\) is 1 and \(\mathrm{c}=\frac{{M}_{o}}{T}{\approx 10}^{4}{-10}^{6}\) (Nms^{−1}), which is equivalent to fewer than 100 earthquakes with Mw ≥ 1.0 recurring every day, we have

$${\mathrm{log}}_{10}\left({M}_{o}\right)={\mathrm{log}}_{10}\left(T\right)+{\mathrm{log}}_{10}\left(\mathrm{c}\right)\approx {\mathrm{log}}_{10}\left(T\right)+4\sim 6,$$

(2)

which shows that if a giant earthquake occurs within 500 km, the value of log c may reach 7–8, resulting in a larger gap between the curves of SI and magnitude (e.g., seismically dense sphere 24).

If we consider the definition of the seismic moment \({M}_{o}=\mu D{L}^{2}\), where \(\mu\) is the fault rigidity or fragility (Pa), *D* is the average slip amount (m), and *L* is the rupture length of a fault (m), and assume a seismic load rate \(v\) (m/s) during the period of a seismic interval SI (T), then \(D=vT\), and the seismic moment can be described as:

$${M}_{o}=\mu {L}^{2}vT.$$

(3)

If the seismic load rate *v* and the fault rupture length *L* are constant, then the seismic moment \({M}_{0}\) will scale to \(T\)(SI) by c (c = \(\mu {L}^{2}v)\). The c-value is proportional to the seismic load rate *v*, the fault rigidity or fragility \(\mu\) and the fault rupture length *L*, which are assumed to be correlated with the fault strength. This correlation can be used to interpret the observed Mo–SI relationship coincidentally from the slope of the seismic moment. Assuming that \(\mu\) is 30 GPa, \(L\) is 30 m for Mw = 1 (i.e., logMo = 10.6, which is the rough average magnitude in seismically dense sphere 49, as shown in Fig. 2), and v is 0.05 m/year, we can infer \(\mathrm{c}=\mu {L}^{2}\mathrm{v}=4.28 \times {10}^{4}\) (Nms^{−1}), whose logarithm is 4.63, which complies with the empirical estimate in Eq. 2. The other seismically dense spheres also follow this estimate. For example, logMo \(\approx\) 11 in SDS5 (Fig. 3), then *L* \(\approx\) 41.1 m, \(\mathrm{c}=\mu {L}^{2}\mathrm{v}=8.03 \times {10}^{4}\) (Nms^{−1}), whose logarithm is 4.90; logMo \(\approx\) 10 in SDS7 (Fig. 3), then *L* \(\approx\) 19 m, \(\mathrm{c}=\mu {L}^{2}\mathrm{v}=1.73 \times {10}^{4}\) (Nms^{−1}), whose logarithm is 4.24.

### Comparison of the different dimensions and b-values of the fault systems

Different fault systems loaded along various fault planes account for various fault slip behaviors. The direction of the load rate v is dependent more on the strike and dip of the fault system than on the relative motion of the plate direction. More importantly, the slip behavior probably changes its direction along a curved fault plane during the rupture occurrence. For a 3-D curved fault plane, the rupture length and the area of the ruptured zone likely change during fault slip. In this case, \(L\) (fault rupture length) is a variable similar to D (slip), and we must consider whether it should be divided by the seismic interval (T). Therefore, the seismic moment rate could also be two- or three-dimensional, involving full ranges of dip, rake and strike.

Compared with \({M}_{0}=\mathrm{c}{T}^{k}\), if we consider the dimension to be *k* \(=\) 3 (corresponding to *b* = 0.5) and the fault rupture length \(L\) to be approximately proportional to the seismic slip amount D, then we would have a scaling law related to two- and three-dimensional slip behaviors:

$${M}_{o}=\alpha \mu {\tau }^{-2}{v}^{3}{T}^{3},$$

(4)

where \(\tau =D/L=E/{M}_{0}\) is the ratio of the seismic energy and seismic moment, which is proposed to be a constant of \(5.0 \times {10}^{-5}\) (Kanamori 1983), and \(\alpha\) is a proportional coefficient between the rupture length and characteristic length (e.g., Leonard 2010) and is approximately a constant for M < 6 earthquakes (e.g., Weng and Yang 2017). \(\alpha \mu {\tau }^{-2}{v}^{3}\) is the seismic moment rate (c-value), and v is the seismic load rate. Equation 4 has an advantage in that it does not involve the characteristic fault length *L,* which further suggests that the key factor influencing the relationship between the seismic moment and SI is not simply the moment rate but is also the seismic energy–moment ratio over a long period. For the MMSI dataset (red circles), both equations are applicable to the DMSI catalog (orange circles) with a lower resolution. Earthquakes with \({M}_{0}>{10}^{15}\) Nm (M > 4) have a smaller MMSI than the values predicted because the M > 4 events are partially contaminated by the clustered small events.

Equations 3 and 4 correspond to the empirical maximum and minimum b-values of 1.5 and 0.5 (e.g., Wiemer and Benoit 1996), respectively, which indicates that the dimensions of slip and rupture are complex and that the dimensions probably play an important role in determining the MO–SI coefficient. The recorded events are likely accompanied by two- or three-dimensional slips dominated by Eq. 3 or 4. In addition, M < 4 (small size) earthquakes are likely more two- and three-dimensional than M > 4 events due to their smaller b-values. The range of b-values from 0.5–1.5 for earthquakes has also been estimated in other regions, such as Alaska (Freymueller et al. 2008), South Asia (Kayal 2008) and Yellowstone (Farrell et al. 2009).

### Comparison with the seismic sequences

The deviation of the seismic slip rate from the average seismic slip rate may account for the long-term variation in the value of *c*, as seen from the gap size between the pink and cyan curves of \({M}_{0}\) and MSI, respectively, in Figs. 2, 3, 4, 5, 6, 7, 8. This deviation possibly suggests coseismic slip or a slip deficit along a seismogenic fault. A diverging trend of the two curves appears after the 2011 Tohoku-oki great earthquakes at the seismically dense spheres close to Tohoku, such as seismically dense spheres 20, 24 and 40. This trend might suggest a decreasing seismic slip rate, from the coseismic peak to a postseismic normal rate, possibly representing fault recoupling. This kind of aftershock sequence has also been found to follow the modified Omori law \(\mathrm{n}\left(\mathrm{t}\right)={\mathrm{K}\left(\mathrm{t}+\mathrm{c}\right)}^{-\mathrm{p}}\) (Utsu et al. 1995) or epidemic-type aftershock sequence model (Ogata et al. 2003). Most seismic sequences experience an immediate increase in Mo and an immediate decrease in SI and a subsequent decay as Mo and SI approach prequake durations, which show that the degree of seismic variation in Mo and SI is a function of the radius (*r*) and nucleation zone size (*h**) of the velocity-weakening patch (e.g., Chen and Lapusta 2009; Chen et al. 2010; Uchida et al. 2015). If we use T = 1/n(t) to replace T in Eq. 3, then we easily have \(\mathrm{n}\left(\mathrm{t}\right){\mathrm{M}}_{0}(\mathrm{t})=\mu {L}^{2}v\), and log \(\mathrm{n}\left(\mathrm{t}\right){+\mathrm{log M}}_{0}(\mathrm{t})=\mathrm{log }\mu {L}^{2}v\), which reasonably interprets the simultaneous decay of Mo and SI (T) for most of the seismic sequences. The sparsely distributed stage of seismicity usually accompanies a narrow gap between the pink and cyan curves and is interpreted as a slip deficit (e.g., Fig. 2, before September 2001), indicating a-value of \(\mathrm{log }\mu {L}^{2}v\).

### Comparison with the seismic rate and global seismic moment

Based on the calculated background seismicity rate (Mw > 4.5) for 117 subduction zones globally, a proportionality relationship, in which the relative plate motion velocity correlates with the seismicity rate, has been demonstrated (Ide 2013). If the total seismic energy release rate is hypothesized to be less variable, then Eq. 4 is equivalent to

$$R\propto {\left({{\mu }^{-1}{\tau }^{2}M}_{ot}\right)}^{-\frac{1}{3}} V,$$

(5)

where \(R=1/T\) is the seismic rate (s^{−1}), *V* is the plate seismic load rate (m/s), and \({M}_{ot}\) is the total seismic moment (Nm) during a long period in the subduction zone or the global system. The advantage of Eq. 5 is a potential proportionality of the seismic rate with the relative plate motion velocity.

### Calculation of the scaling coefficient in Japan

We further enlarged the modeled region to include all of Japan using the same catalog (Fig. 9). If a radius of 20 km and a threshold of ten thousand events for each sphere are applied, the resulting distribution of the c-values of the earthquake moment rates is shown in Fig. 9. Along offshore Tohoku and the Japan Trench, high c-values are identified despite other parts of inland Japan showing a predominantly low c-value. This difference might be caused by the different plate seismic load rates and the different rigidities or fragilities of the continental crust, according to Eq. 3. In contrast, along with the Izu-Bonin island arc chain, the high dominant c-value is possibly caused by the thinner Philippine Sea (PHS) oceanic crust over the subducted Pacific (PAC) plate. Figure 10 depicts the calculated extrapolated distribution minimum c-values for every sphere from 2009 to 2010, implying a larger c-value (shorter seismic interval) for the regions abundant in M5 + earthquakes. This provides a potentially important way to study the details of the featured regional seismicity. Although the seismic load rates and c-values vary greatly in seismogenic zones, the regional seismicity is probably controlled by fault interaction properties such as fault strength and patch size (e.g., Taira et al. 2009; Chen et al. 2010) that are influenced by a tectonic background, which has the potential to provide insight into earthquake prediction provided that a sufficient number of recorded earthquakes enable such a study.