### 3.1 Reduction in temperature dependence

As apparent in the pressure and temperature records in Fig. 4, high correlations were observed between the pressure and temperature variations. Histograms of the cross-correlations and the lag times that produce the maximum correlations are shown in Fig. 5. Lag (h) is positive when the pressure record appears to result from a temperature lag. The cross-correlation coefficients were nearly 1 at most stations. It should be noted that negative lag times dominated the results. The clear peak at ~ –8 h indicates that changes in temperature lead to pressure changes after ~ 8 h. Such significant temperature correlations were not observed in the OBPR data and are considered to be inherent in S-net records.

It is unlikely that inaccurate temperature compensation for the pressure sensor output accounts for the high correlation. If it were, the time lag would be shorter. The temperature-dependent pressure variation would be in the form of mechanical noise resulting from the structure of the observation device. Since the correlation seems to be more prominent at noisier stations (Fig. 4), we can assume that the dependence of pressure fluctuation on temperature changes is the major source of noise in the S-net pressure data. If it can be eliminated, the signal-to-noise ratio of the pressure data may be improved.

To reduce the effect of temperature variation, prediction filtering was applied to the pressure and temperature records. A filter that predicts the apparent temperature-dependent pressure variation from the temperature record and subtracts the output of the filter from the pressure record was expected to remove any noise resulting from temperature disturbances. We assume that the observed pressure fluctuation consists of the sum of the apparent pressure fluctuation and the true pressure fluctuation:

$$p\left( t \right) = \hat{p}\left( t \right) + p^{\prime}\left( {t,T} \right)$$

(2)

where \(p\left( t \right)\) is the observed value,\(p^{\prime}\left( {t,T} \right)\) is the apparent pressure fluctuation due to the temperature disturbance, and \(\hat{p}\left( t \right)\) is the true pressure fluctuation.

We further assume that the apparent pressure variation can be obtained from the temperature time series \(T\left( t \right)\) by applying filter *F*:

$$p^{\prime}\left( {t,T} \right) = F\left( t \right) * T\left( t \right) .$$

(3)

Here, the asterisk denotes convolution. OBPR observations indicate that the actual seafloor pressure does not correlate with temperature. Therefore, the equation below can be used to estimate the filter coefficients by the least-square method.

$$F\left( t \right) * T\left( {t - \tau } \right) = p\left( t \right) .$$

(4)

Note that time shift *τ* (< 0) needs to be applied to ensure causality in the S-net data because \(p\left( t \right)\) leads to \(T\left( t \right)\) lag. Once a filter that can predict the apparent pressure variation from temperature is obtained, the true pressure variation can be determined using the predicted error and is expressed using the following equation:

$$\hat{p}\left( t \right) = p\left( t \right) - F\left( t \right) * T\left( {t - \tau } \right) .$$

(5)

Since the filter estimation is equivalent to the division in the frequency domain,

$$F\left( f \right) \propto \frac{P\left( f \right)}{{T\left( f \right)}},$$

(6)

the filter coefficients can diverge if the temperature spectrum \(T\left( f \right)\) is much smaller than pressure spectrum \(P\left( f \right)\) in a certain frequency range. In such cases, a component proportional to the time-shifted temperature \(T\left( {t - \tau } \right)\) is subtracted from \(p\left( t \right)\).

To evaluate the effect of the temperature correction using least-square filtering, the corrected S-net pressure records were compared with the pressure record obtained by an OBPR located ~ 4 km from station 'c' (Fig. 6). The LPR which was 0.50 before the temperature change decreased to 0.28 as a result of the temperature correction. Although there is a high similarity between the corrected waveform and the OBPR waveform, the long-period component is not in good agreement. However, we found that the similarity increased after a band-pass filter that covered the 48–720-h period component was applied to both this station and the S-net data following temperature correction.

Figure 2b shows the probability density of the power spectra obtained by applying the temperature correction to the results from all stations. The power was significantly reduced for stations in Group 2, to which most S-net stations belong. The Group 1 stations also showed improvement, as demonstrated in the above example (station 'c'), and the noise level was significantly reduced by removing the components correlated with temperature for many of the stations. However, the gap between Groups 1 and 2 remains, although it has been reduced. In addition, at stations where long-period irregular variations that do not correlate with temperature predominate [e.g., station 'b' (Fig. 4b)], the noise level has increased due to the temperature correction, leading to a relative increase in the frequency of higher power compared to the levels prior to correction.

The spatial distribution of LPRs in the temperature-corrected data is shown in Fig. 3c. Compared to Fig. 3b, the number of stations with LPR < 1, which is the noise level expected for OBPRs, increased from 40 before the correction to 51 after the correction. However, the LPRs are still > 1 at many stations even after correction, and stations that can detect small tectonic transients are limited and sparsely distributed along the Japan and Kuril Trenches.

### 3.2 Reduction in common-mode noise

Examples of the temperature-corrected records of stations with a LPR < 1 are shown in Fig. 7a. As data gaps can introduce noise, time periods were selected in which continuous records from all stations with a LPR < 1 were recorded without any data gaps. Although a significant variation was observed in the noise level among all stations, the temperature-corrected time series shows a coherent component attributed to pressure disturbances of a non-tidal oceanographic origin. Several methods have been proposed to reduce such disturbances in order to increase the detectability of tectonic signals in the seafloor pressure data. Examples include a deterministic approach in which the seafloor pressure fluctuations calculated by ocean circulation models were subtracted from the results of observation (Inazu et al., 2012; Muramoto et al., 2019; Dobashi and Inazu, 2021), and estimation using obtained auxiliary oceanographic observations (Watts et al., 2021). Another is to find a proxy of the oceanographical fluctuation using a combination of pressure observations obtained in different locations. Here, we try to apply the latter approach.

One method that is often used for common-mode noise reduction is to use the pressure difference from a reference station (Ito et al., 2013) located far from the source location of the tectonic event representing the oceanographic signals. Wallace et al. (2016) set a reference station on the incoming plate side of the trench, where no vertical variations were expected, to detect an SSE signal in the OBPR data distributed on the landward slope. Fredrickson et al. (2019) and Inoue et al. (2021) showed that the non-tidal oceanographic fluctuations tend to be coherent among the sites located at stations with similar water depth and pointed out the advantage of using common-depth station pairs to obtain pressure differences with reduced noise.

Records that were obtained from stations on the incoming plate and landward sides with similar water depths were extracted from the high-quality S-net records obtained at 51 quiet stations (LPR < 1 after the temperature correction) in Fig. 7a and are displayed in Fig. 7b. The waveforms at stations 3–20 and 4–22 on the landward side of the trench were notably similar; however, few correlations were observed between the other stations. S-net does not provide a high degree of freedom in the combination of stations used for common-mode noise reduction via pressure difference. The difference between the records from stations 3–20 and 4–22, with relatively high similarity, was therefore investigated to test the theory, and we found that the RMS of the pressure data at station 3–20 was 1.14 and that at station 4–22 was 1.25, showing a reduction of 1.02 hPa.

Hino et al. (2014) showed that the common-mode noise can be reduced by subtracting the first principal component obtained by principal component analysis (PCA) without the use of pressure differences from a reference station. Similarly, we attempted to remove the common-mode noise in the S-net data in the same way. In applying PCA to the S-net data, it should be noted that the noise level varies greatly across stations. Here, we performed PCA weighted by the inverse of the variances of input temperature-corrected pressure data. Additional file 3: Figure S3 shows the results of PCA using all the records shown in Fig. 7a. The common-mode variations were reduced by subtracting the estimated first principal component. To evaluate the effect of PCA-based noise reduction, we determined the reduction rate of RMSs at all the stations and calculated their median, which was 81%. At stations 3–20 and 4–22, where we attempted to reduce the common-mode noise by taking pressure differences, the RMS decreased from 1.14 to 0.82 (72% reduction) and from 1.25 to 1.07 (86%), respectively.

From Fig. 7a, the non-tidal oceanographic component was not necessarily coherent over a wide area, and the correlation of waveforms decreased with increasing distance between stations. We, therefore, expected the noise reduction using PCA to be more effective if the range of stations used was limited to a relatively narrow area. Below, we applied the PCA-based common-mode noise reduction to the pressure records obtained at 13 quiet stations belonging to the S-4 system in the northern part of the Japan Trench. We chose these stations because their number and density of quiet stations were the largest in the region (Fig. 7b).

Figure 8a shows the temperature-corrected pressure records and the first principal components obtained from the temperature-corrected pressure records at the S-4 quiet stations selected from Fig. 7a. The color of the traces changes according to the depth of the stations, and it is apparent that the correlation was high between stations at similar depths, whereas the similarity of the waveforms decreased when stations at different depths were compared. Figure 8b shows the pressure records after the contribution of the first principal component was subtracted. The median of the RMS reduction rate was 66%, indicating a better performance when using stations within a narrow subarea than that when using broad network data. However, the noise reduction rate at station 4–22 was 89% (from 1.25 to 1.11), slightly worse than that after the noise reduction using broad network data.

One possible reason for the poor performance at the station can be the water depth dependence of waveform similarity. Station 4–22 was the deepest one among the quiet S-4 stations, the only station deployed at ~ 4,000 m depth, whereas six more stations located at depths similar to 4–22 were included in the PCA using all the quiet stations. The water depth-dependent components would be extracted as principal components of second or higher orders. However, subtracting the second principal component did not substantially reduce the RMS values. The reduction rates between the records after removing the first component and those after removing the first and second components were 96% and 86%, when applied to all the quiet stations and to the S-4 stations, respectively. The high level of instrumental noise likely remained in the record even after the temperature correction, and the noise hinders the extraction of depth-dependent components.