The evaluated sensitivity *α* is 0.78 under the reference velocity structure. This value indicates that almost 80% of the ACF energy senses changes in the medium below the Hi-net sensor installation depth of 150 m. Using this *α* value and Eq. (3), we plot *ΔV*/*V*
_{deep} in Fig. 7 (blue circles), as well as the relative velocity change detected from the DCW of the KiK-net records (*ΔV*/*V*
_{shallow}, red circles) and from the ACF of the Hi-net records (*ΔV*/*V*
_{Hi}, black circles). In period 1, the estimated relative velocity changes are −3.1 and −1.4% in the shallow and deep zones, respectively. In period 2, the relative velocity changes recover to −1.9 and −1.1% in the shallow and deep zones, respectively.

The value of *α* = 0.78 is evaluated using the reference velocity structure, which may include biases from the true velocity structure. Therefore, it is important to evaluate how much the *α* value could be perturbed due to uncertainty in the reference velocity structure. We verify the range of sensitivities *α* using different reference velocity structures reproduced by decreasing and increasing the reference velocities above 2.5 km depth by up to 20%, as shown by the red and blue 1D velocity profiles in Fig. 6, respectively. The evaluated *α* values are 0.75 and 0.82 for decreased and increased velocities, respectively. As the velocity of the shallow zone decreases, more of the energy of the ACF is trapped by the shallow layers; consequently, the ACF becomes more sensitive to the velocity change in the shallow zone and the *α* value decreases.

Using Eq. (3), we draw the *α* dependence of *ΔV*/*V*
_{deep} in Fig. 8, where *ΔV*/*V*
_{shallow} and *ΔV*/*V*
_{Hi} are fixed at the observed values. As derived from Eq. (3), the value of *ΔV*/*V*
_{deep} is equal to *ΔV*/*V*
_{Hi} when *α* = 1. In this case, the ACF is sensitive only to the velocity change in the deep zone. As *α* becomes smaller, *ΔV*/*V*
_{deep} and its standard deviation increase, and both diverge to infinity when *α* = 0. In this case, the ACF has no sensitivity to change in the deep zone, which means we cannot obtain any information about the velocity change below 150 m. From Fig. 8, we confirm that the value of *ΔV*/*V*
_{deep} does not change drastically within the evaluated range of *α* (from 0.75 to 0.82, as indicated by the white background).

### Contribution of dynamic and static strain changes to velocity change

To investigate the cause of the velocity changes observed at different depths, we now examine the contributions of dynamic and static strain changes due to the N. Nagano earthquake as the candidate of the cause.

For the N. Nagano earthquake, the maximum dynamic strains (MDS) recorded at station NIGH17 are \( {\varepsilon}_{\mathrm{surface}}^D=5.3\times {10}^{-4} \) and \( {\varepsilon}_{\mathrm{borehole}}^D=1.4\times {10}^{-4} \) at the surface and the borehole bottom, respectively. These values can constrain the range of the MDS in the shallow zone. In the deep zone, on the other hand, we indirectly evaluate the MDS through the results of the numerical wave propagation simulation. Using the reference velocity structure shown in Fig. 6 and setting a 0.5 Hz Küpper wavelet as a source waveform (since 0.5 Hz is the dominant frequency of the observed velocity seismogram) to the position that corresponds to the hypocenter of the N. Nagano earthquake (*x* = –25.9 km, *z* = 4.6 km), we synthesize the numerical waveforms of the horizontal component at positions *x* = 0 km, *z* = 150 m and *x* = 0 km, *z* = 5 km. The 5 km depth is considered because below which the ACF has almost no sensitivity, according to our numerical tests. We obtain a ratio of computed MDS values at 5 km depth/MDS at 150 m depth of \( {\varepsilon}_{5\mathrm{km}}^D/{\varepsilon}_{150\mathrm{m}}^D=0.091 \). We then calculate an MDS value at 5 km depth of \( {\varepsilon}_{5\mathrm{km}}^D=0.091\times {\varepsilon}_{\mathrm{borehole}}^D=1.3\times {10}^{-5} \) for the N. Nagano earthquake. We note that this evaluation may include a large bias due to the difference between geometrical spreading factors for the 2D (numerically simulated) and 3D (observed) media. Using these values, we evaluate the possible ranges of MDS values in the shallow and deep zones as \( 1.4\times {10}^{-4}\le {\varepsilon}_{\mathrm{shallow}}^D\le 5.3\times {10}^{-4} \) and \( 1.3\times {10}^{-5}\le {\varepsilon}_{\mathrm{deep}}^D\le 1.4\times {10}^{-4} \), respectively.

Next, we compute the coseismic static strain change beneath station N.MKGH using the code of Okada (1992) and a rectangular plane fault model (20 km long and 14 km width) for the N. Nagano earthquake (quadrangle in Fig. 1). Here, the focal mechanism is adopted from the CMT solution (strike = 22°, dip = 51°, rake = 62°) of F-net (the Full Range Seismograph Network of Japan, operated by NIED). A slip of 0.4 m is uniformly distributed on the rectangular fault. The corresponding seismic moment is 3.4 × 10^{18} (Nm) = *M*
_{
W
} 6.3, which is close to the reported value. Our computed ranges of the static volumetric strain change (SVSC) are \( -3.73\times {10}^{-7}\le {\varepsilon}_{\mathrm{shallow}}^S\le -3.68\times {10}^{-7} \) and \( -3.7\times {10}^{-7}\le {\varepsilon}_{\mathrm{deep}}^S\le -1.5\times {10}^{-7} \) in the shallow (0–150 m depth) and deep (150 m–5 km) zones, respectively, where the minus sign indicates contraction. The distributions of SVSC are mapped in Fig. 1a by colored grids.

Figure 9 compares the observed relative velocity change in period 1 and the evaluated ranges of MDS (solid circles) and SVSC (open symbols) in the shallow and deep zones. Although the evaluated SVSC has negative values, we show their absolute values in Fig. 9 for the sake of discussion. The values of MDS are larger than SVSC in both the shallow and deep zones by ~10^{2}–10^{3}. In the same figure, we also show the values of susceptibility (relative velocity change with respect to the applied strain change, given by Δ*V*/*V*/*ε*) with gray curves. If the observed velocity reduction is caused mainly by the dynamic strain change, then the value of susceptibility should be approximately −1000 to −100 for both zones. On the other hand, if the observed velocity reduction is caused mostly by the static strain change, then the susceptibility should be around −10^{5}.

We now compare these calculated values with susceptibilities measured in rock experiments. According to Prioul et al. (2004), the values of the third-order elastic constants (TOE), which give the variability of material with respect to the applied static stress change (e.g., Murnaghan 1967), are *c*
_{111} = −11300, *c*
_{123} = 5800, and *c*
_{111} = −3100 GPa, *c*
_{123} = 40 GPa under hydrostatic stresses of 5–30 and 30–100 MPa, respectively, for a rock sample of North Sea Shale (Table 2 of Prioul et al. 2004). Because hydrostatic stresses of 5–100 MPa correspond to depths of ~200–4000 m, these values are applicable to the medium in the deep zone. Using these TOE values, the susceptibility is given by

$$ \frac{{\varDelta V/V}^S}{\varepsilon }=\frac{1}{12{c}_{44}}\left({c}_{111}-{c}_{123}\right)+\frac{1}{2}, $$

(4)

where *c*
_{44} is the shear modulus. Equation (4) is derived by modifying the second formula of Eq. (13) of Sawazaki et al. (2015), where we took the average of the relative S-wave velocity change for all three propagation directions; i.e., *ΔV*/*V*
^{S} = (*ΔV*
_{
S12}/*V* + *ΔV*
_{
S23}/*V* + *ΔV*
_{
S31}/*V*)/3. Using Eq. (4), the TOE constants from Prioul et al. (2004) and the *c*
_{44} (shear modulus) value at depths of 200 to 4000 m, we calculate a possible range of susceptibilities of −688 to −8 in the deep zone. This range is drawn in yellow in Fig. 9; it mostly covers the estimations from the MDS, but does not cover the estimates from the SVSC. Because the TOE constants measured by Prioul et al. (2004) are obtained for loading of the static strain change, we admit that these values are not necessarily applicable to loading of the dynamic strain change. Also, we note that the well log core sample at station NIGH17 is composed of a mixture of topsoil, tuff breccia, sandy gravel, and andesite (see Fig. 1b), which is not necessarily similar to the North Sea Shale used by Prioul et al. (2004). Nevertheless, we think that the dynamic strain change is more likely to be the primary cause of the velocity reduction than static strain change, because the susceptibilities evaluated from the static strain change are completely outside of the possible range of the rock experiments.

### Cause of velocity recovery

The velocity recovery also appears differently in the shallow and deep zones, as the shallow zone recovers more quickly. We consider that “slow dynamics” (i.e., the behavior of material that slowly approaches its equilibrium state; originally presented in the rock experiments such as TenCate et al. 2000) is a convincing mechanism that can explain the observed velocity recovery.

Firstly, we note that the horizontal static displacement due to 3 months of postseismic deformation is less than 4% of the estimated coseismic deformation (Yarai et al. 2015). Because the estimated coseismic static strain change is on the order of 10^{–7}–10^{–6}, the static strain change due to postseismic deformation is on the order of 10^{–8}. Such a small strain change alone cannot explain an observed velocity recovery of up to 30% in the shallow zone. Therefore, we exclude postseismic deformation as the primary cause of the observed velocity recovery.

According to the rock experiments of TenCate et al. (2000), the relative resonance frequency shift of a rock sample, Δ*f*/*f*, recovers in proportion to the logarithm of the lapse time *t* after unloading a dynamic strain, as follows:

$$ \frac{\varDelta f(t)-\varDelta f\left({t}_0\right)}{f}=m{ \log}_{10}\left(\frac{t}{t_0}\right), $$

(5)

where *t*
_{0} and *m* are the reference time and the parameter that controls the speed of the recovery, respectively. This log(*t*)-type recovery has been widely confirmed by field data (e.g., Sawazaki et al. 2009; Gassenmeier et al. 2016). Applying Eq. (5) to the observed relative velocity changes shown in Fig. 7, where the relative velocity change is used instead of the relative resonance frequency shift because they are equivalent, we obtain *m* values of (9.8 ± 3.8) × 10^{− 3} and (2.2 ± 4.4) × 10^{− 3} in the shallow and deep zones, respectively. These values are plotted in Fig. 10 along with the *m*–*ε*
^{D} relationship *m* ≅ 11.1 × (|*ε*
^{D}| − 4.7 × 10^{− 7}) obtained by TenCate et al. (2000). We note that their experiment was carried out for dynamic strains ranging from 4.0 × 10^{−7} to 2.64 × 10^{−6} (gray solid curve in Fig. 10), which are much smaller than the estimated MDS values in our study. Although the error range is large, especially for the deep zone, the *m* values obtained in our study seem to lie along the extrapolation curve of TenCate et al.’s (2000) empirical relationship (gray broken curve in Fig. 10). This result indicates that a similar recovery mechanism underlies both the laboratory scale (centimeters) and field scale (hundreds of meters), and the recovery rate is controlled primarily by the strength of the applied MDS.

The MDS dependence of the recovery rate can be explained as follows. After unloading the dynamic strain, the actual contact area on the surface of microcracks begins to increase through a creep-like deformation process. As the actual contact area becomes larger, the normal stress applied to the contact area becomes smaller and the rate of the deformation decreases. This decelerates the expansion of the actual contact area. As a consequence of this feedback process, the initial contact area controls the average deformation speed: if the initial contact area is smaller, then the deformation speed is faster (e.g., Brechet and Estrin 1994). Because the initial contact area is related to damage in the medium, which would be excited more by stronger dynamic strains, the recovery speed will be faster for the material affected by a larger MDS.

### Other factors that control velocity change and recovery

As stated above, we consider that the observed velocity reduction is controlled mainly by damage due to the dynamic strain change, and recovery is controlled by healing of microcracks, analogous to slow dynamics. However, we note that this explanation is only applicable to areas distant from the faulting zone (>20 km). Because dynamic and static strains attenuate geometrically following *r*
^{–2} and *r*
^{–3}, respectively, where *r* is the distance from the hypocenter (Aki and Richards 2002), the contribution of the static strain change due to coseismic deformation would be non-negligible near the fault zone. For the same reason, the static strain change due to postseismic deformation may also contribute to velocity recovery near the fault zone (e.g., Brenguier et al. 2008). Another important factor in the recovery process is aftershocks. Because the magnitude of the largest aftershock during period 2 was only 4.4 for the N. Nagano case, we do not consider additional damage from aftershocks. However, if aftershocks are large and active, secondary damage from large aftershocks could cause additional velocity reduction (Rubinstein and Beroza 2004) and recovery speed could be slowed. Thus, the contributions of dynamic and static strain changes may depend on both distance from the fault zone and aftershock activity. If these factors are contaminated, the features of the velocity change and the recovery processes will become more complex.

We also need to consider that if the damage in the medium is severe and the shapes of the DCWs and ACFs before and after the mainshock become dissimilar, a stretching technique might be inapplicable. For such cases, an irreversible medium change would be excited and the recovery process would be different from the case of weaker damage. In fact, in this study, the peak amplitude of the DCW mostly recovers to pre-mainshock level in period 2 for the E−W component (Fig. 2a), but not for the N−S component (Fig. 2b). This indicates that the N−S component was more severely damaged by the N. Nagano earthquake than the E−W component.

Because we examined only one station for one earthquake, it is important to collect case studies of velocity changes at different depths for different situations. In particular, a comparison between the case of significant strong motion with small crustal deformation (e.g., a large, deep earthquake) and that of large crustal deformation without seismic motion (e.g., a slow earthquake) is interesting. Because the susceptibility of the subsurface medium and recovery speed may each differ due to conditions in the medium, such as temperature, confined stress, saturation ratio, and permeability (e.g., Brenguier et al. 2014), it is also important to perform time-lapse velocity monitoring at a variety of sites.