We analyzed stress drops of 686 earthquakes with 4.0 ≤ *M* ≤ 5.0 off the south-east of Hokkaido, Japan, that took place from June 2002 to December 2015 (Fig. 1). Note that, in this paper, M expresses the magnitude of an earthquake as determined by the Japan Meteorological Agency (JMA). The hypocenters of the 686 earthquakes were located from 40.5° N to 43.5° N in latitude and from 141.0° E to 146.5° E in longitude with a depth of ±15 km from the interface of the Pacific Plate deduced by Kita et al. (2010). The distance of 15 km in depth direction is within the range of uncertainty in the hypocenter determination in the study area because of the poor azimuthal coverage of seismic stations. We used waveforms observed at stations maintained by National Research Institute for Earth Science and Disaster Resilience, Japan (NIED), Hokkaido University, and JMA.

An observed waveform as a function of time *W*(*t*) includes the effects of the source *S*(*t*), the path from a hypocenter to a receiver *P*(*t*), site amplification effects *St*(*t*), and the instrumental features of a seismometer *I*(*t*), that is:

$$ W(t)=S{(t)}^{\ast }\ P{(t)}^{\ast }\ St{(t)}^{\ast }\ I(t), $$

(1)

where ^{∗} denotes convolution. In the frequency domain, the convolution can be expressed as a scalar product, and the following equation holds:

$$ W(f)=S(f)\cdotp P(f)\cdotp St(f)\cdotp I(f), $$

(2)

where *W*(*f*), *S*(*f*), *P*(*f*), *St*(*f*), and *I*(*f*) are expressions in the frequency domain of *W*(*t*), *S*(*t*), *P*(*t*), *St*(*t*), and *I*(*t*), respectively. If we know the expressions of *P*(*f*), *St*(*f*), and *I*(*f*), we can calculate the Green’s function and estimate the source effect from the observed waveform. As it is actually difficult to estimate the Green’s function precisely, we used the method of empirical Green’s function (EGF) (e.g., Hartzell 1978).

The observed waveforms of two earthquakes at a station can be expressed as follows:

$$ {W}_1(f)={S}_1(f)\cdotp {P}_1(f)\cdotp {St}_1(f)\cdotp I(f), $$

(3)

$$ {W}_2(f)={S}_2(f)\cdotp {P}_2(f)\cdotp {St}_2(f)\cdotp I(f). $$

(4)

If the hypocenters of the two earthquakes are identical, no velocity change takes place during the two earthquakes, and the soil beneath the station acts linearly independent of the amplitude of the incoming waveforms, then the path and site effects are exactly the same, *P*
_{1}(*f*) = *P*
_{2}(*f*) and *St*
_{1}(*f*) = *St*
_{2}(*f*). In this case, we can extract the ratio of the source effect by calculating the ratio of the observed waveforms in the frequency domain,

$$ \frac{W_1(f)}{W_2(f)}=\frac{S_1(f)\cdotp {P}_1(f)\cdotp {St}_1(f)\cdotp I(f)}{S_2(f)\cdotp {P}_2(f)\cdotp {St}_2(f)\cdotp I(f)}=\frac{S_1(f)}{S_2(f)}. $$

(5)

Equation (5) gives the spectral ratio of each pair of an analyzed earthquake and an EGF earthquake. We assume that source spectrum of an earthquake *S*
^{C}(*f*) can be expressed by the omega-squared model of Boatwright (1978), which is shown as the following equation:

$$ {S}^C(f)={R}^C(f)\cdotp {M}_0^C(f)\cdotp {\left\{\frac{1}{1+{\left(f/{f}_0^C\right)}^4}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, $$

(6)

where *R*, *M*
_{0}, and *f*
_{0} are the coefficient of the radiation pattern, the seismic moment, and the corner frequency of the earthquake, respectively. Here, *C* indicates the wave type, which is either P or S. This assumption means that we approximated the fault plane as a circular plane. The deconvolved velocity amplitude spectra \( \left|{\dot{u}}_r^C(f)\right| \) can then be expressed as follows:

$$ \left|{\dot{u}}_r^C(f)\right|=\left|\frac{S_A^C(f)}{S_E^C(f)}\right|=\frac{R_A^C\cdotp {M}_{0A}}{R_E^C\cdotp {M}_{0E}}\cdotp {\left\{\frac{1+{\left(f/{f}_{0E}^C\right)}^4}{1+{\left(f/{f}_{0A}^C\right)}^4}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$

$$ ={R}_r^C{M}_{0r}\cdotp {\left\{\frac{1+{\left(f/{f}_{0E}^C\right)}^4}{1+{\left(f/{f}_{0A}^C\right)}^4}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, $$

(7)

where subscripts *A* and *E* indicate analyzed and EGF earthquakes, respectively. Moreover, \( {R}_r^C \) and *M*
_{0r
} express the relative values of \( {R}_A^C/{R}_E^C \) and *M*
_{0A
}/*M*
_{0E
}, respectively. The value of \( {R}_r^C \) is equal to 1 if the focal mechanisms and hypocenters of the analyzed and EGF earthquakes are exactly the same. The sampling rate of waveforms analyzed in the present study was 100 Hz. We used waveforms of earthquakes in 2004 through 2015 with M3.5 that were closest to the hypocenters of the analyzed earthquakes as EGFs. A list of analyzed and EGF earthquakes is available as an Additional file 1 (see eqlist.txt).

We selected earthquakes with M3.5 as EGFs and analyzed earthquakes in a relatively narrow magnitude range of 4.0 ≤ *M* ≤ 5.0 based on the following considerations. In order to ensure a good signal-to-noise ratio of EGFs, especially for spectra of lower frequencies, we selected earthquakes with M3.5 as EGFs. The lower limit (M4.0) of the analyzed earthquakes was fixed in order to maintain a difference in magnitude of 0.5 compared to the EGFs and to ensure quality in estimating the corner frequencies. As stress drops are estimated for individual earthquakes, the values for large earthquakes would represent the average characteristics of individual large fault planes and would not reflect local frictional characteristics. In addition, as waveforms of some large earthquakes were clipped, we set the upper limit of the earthquake size to M5.0.

We calculated the spectral ratios of P and S waves for individual pairs of an earthquake and an EGF. The spectral ratios were calculated for three time windows with a length of 10.23 s, or 1024 data points. The start times of the first window were 0.50 s prior to the arrival times for either the P or S waves. The elapsed times of the two successive time windows were 1.28 and 2.56 s, respectively. Taking the logarithm of Eq. (7), the spectral ratio can be approximated as follows:

$$ \mathit{\ln}\left|{\dot{u}}_r^C(f)\right|\approx g\left(f;{f}_{0A}^C;{f}_{0E}^C\right), $$

(8)

$$ g\left(f;{f}_{0A}^C;{f}_{0E}^C\right)=\mathit{\ln}\left({R}_r^C{M}_{0r}\right)-\frac{1}{2}\mathit{\ln}\left\{1+{\left(f/{f}_{0A}^C\right)}^4\right\}+\frac{1}{2}\mathit{\ln}\left\{1+{\left(f/{f}_{0E}^C\right)}^4\right\}. $$

(9)

Before fitting each analyzed spectral ratio with the theoretical ratio expressed by Eqs. (8) and (9), we resampled the frequencies so that the interval was equal to 0.05 on a log_{10} scale. As a result, we obtained 20 frequency bands (data points) for each order of frequency. This procedure made it possible to treat high- and low-frequency data equally. We also calculated the standard deviation of the spectral ratio for each frequency band and used the value in fitting data, as explained below.

We estimated the values of \( {R}_r^C{M}_{0r} \), \( {f}_{0A}^C \), and \( {f}_{0E}^C \) in Eq. (9) for each station by a grid search that gave the minimum residual for the spectral ratios of three time windows, similar to Imanishi and Ellsworth (2006):

$$ res={\sum}_i\frac{{\left\{\mathit{\ln}{\dot{u}}_r^C(f)-g\left(f;{f}_{0A}^C;{f}_{0E}^C\right)\right\}}^2}{\sigma_i^2}, $$

(10)

where *σ*
_{
i
} is the standard deviation for each frequency band calculated in resampling the spectral ratio. All of the earthquakes have four or more stations available for the corner frequency estimation. We used data from 0.7 to 20 Hz to calculate the residual in Eq. (10) and estimated corner frequencies by grid search between 0.3 and 20 Hz. This frequency range and the length of the time window (10.23 s) were selected so that we were able to analyze corner frequencies of earthquakes with 4.0 ≤ *M* ≤ 5.0, which would be around 1–3 Hz, as expected by the self-similarity of earthquakes. Figures 2 and 3 show examples of velocity waveforms, their spectra, deconvolved spectra with a resampling, and the best-fit curves of spectral ratio that were used for estimating a corner frequency. We also provide examples for another earthquake as supplemental figures (Additional files 2 and 3: Figures S1 and S2). We can see that waveforms have a signal-to-noise ratio larger than a factor of five even for low frequencies between 0.7 and 2 Hz for EGF earthquakes.

Finally, we calculated the values of stress drop following Madariaga (1976):

$$ \Delta {\sigma}^C=\frac{7}{16}{M}_{0A}{\left(\frac{f_{0A}^C}{k{V}_S}\right)}^3, $$

(11)

where *V*
_{
S
} is the shear wave velocity (=4.5 km/s, from Matsubara and Obara 2011), and *C* indicates the wave type. We used *k* = 0.32 and *k* = 0.21 for P and S waves, respectively, assuming that the rupture speed corresponds to 0.9*V*
_{
S
} (Madariaga 1976). We will discuss the effect of the value *k* in the section “Discussion.” We assumed that *M* is equivalent to the moment magnitude *M*
_{
W
} in calculating stress drops. We also discuss the validity of this assumption in the section “Discussion.” The seismic moment *M*
_{0} in newton meters (Nm) can be calculated from *M*
_{
W
} using the following equation (Hanks and Kanamori 1979): log_{10}
*M*
_{0} = 1.5*M*
_{
W
} + 9.1.