Figure 4d shows the P-wave velocity model of the J-SHIS (NIED 2019b) along the transects. Two distinct boundaries lie between the low- (\(\le\) 2400 m/s; red to orange) and intermediate- (3200 m/s; yellow) velocity layers and between the intermediate- and high- (5500 m/s; blue) velocity layers. There are also minor boundaries with smaller velocity contrasts within the low-velocity layer. Likely candidates for the low-, intermediate- and high-velocity layers are the Kazusa and Miura layers and the basement (e.g., Koketsu et al. 2009). Figure 4e shows the boundaries superposed on the reflection response obtained using our method. The reflectors are clearly imaged at the sites where the intermediate (Miura) layer is thin enough to be negligible (from 139° 55′ E to 140° 20′ E along both transects); the subsurface approximates a two-layer medium with a large velocity contrast at these sites. The reflectors are less prominent at sites that approximate a three-layer medium. This could be because the intermediate (Miura) layer reduces the impedance contrast at the interfaces. Additionally, the western part of the ESE–WNW transect is close to the study area of Yoshimoto and Takemura (2014), who reported gradual rather than abrupt increases in the velocity in the sedimentary layer. These three-layer or gradually increasing velocity structures are possible explanations for the absence of prominent reflectors in our reflection responses. The eastern part was not imaged well mainly because the number of earthquakes was small (Fig. 4a).

We assumed that the noise obeyed a normal distribution. This was a good approximation, as shown in Fig. 5a and b; the cumulative distribution of absolute amplitudes of the observed (whitened) waveform in the noise window (black) was fitted well by that from a normal distribution (green). Our statistical tests showed that the assumption of the normal distribution is not dismissed at 5% significance level (Additional file 1: Text S4). Maeda et al. (2020) indicated that the normal distribution was a good approximation for the noise waveforms in a different region. The covariance in the noise waveform is another factor that needed to be evaluated. As we used a narrow frequency band (0.0305-Hz width) for whitening (“Method for estimating the ACFs and errors” section), the covariance was expected to be small. Additional file 1: Text S3 and Fig. S4 indicate that the covariance was indeed small, except for \(\tau \le 0.075\) s. As we used \(\tau >\) 0.075 s for the interpretation of the subsurface structure (Figs. 2 and 3), the covariance would not affect the result significantly, although it is ideal to design a noise waveform that has the same covariance as that of the data.

We generated \({u}_{i,j}^{noise}(t)\) using a normal distribution, calculated \({a}_{i,j}^{eq}\left(\tau \right)\) from Eqs. (2) and (3) and used three times the standard deviation of \({a}_{i,j}^{eq}\left(\tau \right)\) as a measure of the errors in the ACFs. This threshold is consistent with the 99% confidence level if the ACFs obey a normal distribution. Theoretically, \({a}_{i,j}^{eq}\left(\tau \right)\) derived from Eqs. (2) and (3) obeys the normal distribution because \({a}_{i,j}^{eq}\left(\tau \right)\) is expressed by:

$${a}_{i,j}^{eq}\left(\tau \right)=\frac{\left[{a}_{i}^{oo}\left(\tau \right)-{a}_{i,j}^{on}\left(\tau \right)-{a}_{i,j}^{no}\left(\tau \right)+{a}_{i,j}^{nn}\left(\tau \right)\right]}{\int {u}_{i,j}^{eq}{\left(t\right)}^{2}dt},$$

(6)

$${a}_{i}^{oo}\left(\tau \right)=\int {u}_{i}^{obs}\left(t\right){u}_{i}^{obs}\left(t+\tau \right)dt,$$

(7)

$${a}_{i,j}^{on}\left(\tau \right)=\int {u}_{i}^{obs}\left(t\right){u}_{i,j}^{noise}\left(t+\tau \right)dt,$$

(8)

$${a}_{i,j}^{no}\left(\tau \right)=\int {u}_{i,j}^{noise}\left(t\right){u}_{i}^{obs}\left(t+\tau \right)dt,$$

(9)

and

$${a}_{i,j}^{nn}\left(\tau \right)=\int {u}_{i,j}^{noise}\left(t\right){u}_{i,j}^{noise}\left(t+\tau \right)dt$$

(10)

\({a}_{i}^{oo}\left(\tau \right)\) is not a random value, and \({a}_{i,j}^{on}\left(\tau \right)\) and \({a}_{i,j}^{no}\left(\tau \right)\) obey the normal distribution because they are linear combinations of \({u}_{i,j}^{noise}\left(t\right)\) that were generated using the normal distribution. To evaluate \({a}_{i,j}^{nn}\left(\tau \right)\) (Eq. 10), we arrange the equation as:

$$\frac{{a}_{i,j}^{nn}\left(\tau \right)}{T}=E\left[{u}_{i,j}^{\mathrm{noise}}\left(t\right){u}_{i,j}^{\mathrm{noise}}\left(t+\tau \right)\right],$$

(11)

where \(T\) is the length of the time window of the integration in Eq. (10) and \(E[]\) is an expected value. Although \({u}_{i,j}^{noise}\left(t\right){u}_{i,j}^{noise}\left(t+\tau \right)\) does not obey the normal distribution, the distribution of \({u}_{i,j}^{noise}\left(t\right){u}_{i,j}^{noise}\left(t+\tau \right)\) is independent of \(t\), meaning that \(E\left[{u}_{i,j}^{noise}\left(t\right){u}_{i,j}^{noise}\left(t+\tau \right)\right]\) asymptotes to the normal distribution by increasing the number of samples in \(T\) according to the central limit theorem. In summary, all terms in Eq. (6) obey the normal distributions, so that \({a}_{i,j}^{nn}\left(\tau \right)\) obeys the normal distribution. Figure 5c and d shows the cumulative distribution of \(|{a}_{i,j}^{eq}\left(\tau \right)-{a}_{i}^{\mathrm{ave}}\left(\tau \right)|\) and that from the normal distribution. The excellent fit between the two distributions and our statistical tests (Additional file 1: Text S4) indicate that \({a}_{i,j}^{eq}\left(\tau \right)\) indeed obeys the normal distribution.

A trade-off exists between avoiding the detection of false reflectors and discovering possible minor reflectors. This trade-off can be controlled by the threshold. For example, using two times the standard deviation instead of three increases the reflectors imaged, including both true and false ones. The threshold should be chosen based on purposes. In poorly studied areas, a high threshold is preferable because even the identification of only major reflectors is valuable, and other information to validate the existence of the reflectors is limited. In extensively studied areas where the discovery of minor reflectors is more essential and abundant knowledge and data other than ACFs are available for verification, the threshold can be lowered. Note that the threshold only affects the visual images; numerical values for the ratio of the amplitude to standard deviation are available in all depths regardless of the choice of the threshold.

In the conventional method, the meaning of the amplitudes of reflection responses from ACFs was unclear. For example, the amplitude of 0.1 only meant that the correlation coefficient was −0.1; whether this value was large or small has been evaluated in relative manner, often qualitatively. Our method realizes an evaluation of the amplitude; for example, if the amplitude is 2.0 times the standard deviation, it means that the reflector is significant at 95% confidence level. Because this evaluation is possible from a single station, the method is available in various areas regardless of the density of stations.

In this study, we focused on ACFs for seismic daylight imaging. Following this work, future studies can apply our method to other earthquake-based imaging techniques, including cross-correlation and receiver functions, to calculate errors, although a validation study is needed. Our method is not applicable to ambient noise-based techniques, and alternative approaches will need to be developed in future studies.