Figure 2 shows example of the OBP time series after removing the tidal and drift components from the observed time series. These time series indicate annual variations: a higher-pressure value appeared in the summer season (July to August) and a lower pressure value appeared in the winter season (January to February). This annual variation was evident even before 2016 in DONET1 network (KMB–KMD time series), prior to our analyses period (Fig. 2). The annual variation has an amplitude of up to 5 hPa, and these characteristics are within the same node and display similar features over a wide area. Similar characteristics can be observed throughout the network, even for shorter time-constant variations. In the following sections, we present the PCA results that separate these spatially coherent oceanographic fluctuations.

## Spatiotemporal characteristics of each principal component

In PCA, PC1 has the largest eigenvalue. PC1 is more highly correlated with the original features than PC2, while PC2 is better correlated than PC3, and so on. PCA reduces the dimensionality of the data by selecting the first \(k\) PCs to represent the original data. Figure 3 (a–d) shows the extracted PCs at the DONET OBP sites. The eigenvector and eigenscore of each PC correspond to its spatial distribution and temporal variation, respectively. The contributions of each PC were 75.7% for PC1, 10.3% for PC2, 3.5% for PC3, and 2.5% for PC4 (Fig. 3). The cumulative contribution reached 92% through PC1–PC4, and the spatial patterns and other characteristics of the higher-order components after PC5 were not clear, therefore, the sections hereafter will utilize PC1–PC4 to interpret their characteristics. PC1 and PC2 explain the majority (86.0%) of the time series characteristics; PC1 is the common component across the entire observational network and includes the annual variability (Figs. 3 (a) and 4(a)). PC2 has a spatially inclined characteristic, reversing the positive and negative eigenvector values between the shallow and deep stations (Figs. 3(b) and 4(c)). This sea depth dependence characteristic is clear in the time series of PC2, whose amplitude is notably large in shallow or deep stations (Fig. 5(b)). PC3 does not have a clear sea-depth dependency like PC2, but it does have a longitudinal dependence characteristic (Fig. 3(c) and Fig. 4(f)). PC4 has parabolic spatial characteristics with positive and negative eigenvector values at shallow and deep-water depths, respectively (Figs. 3(d) and 4(g)). Based on the characteristics of each of these PCs, PCs 1–4 were determined to be the components that needed to be removed from the oceanographic fluctuation components in this study.

Previous studies have noted the sea depth dependency of the seafloor water-pressure characteristics based on actual observed data and oceanographic models (Fredrickson et al. 2019; Inoue et al. 2021). The PCA results obtained in this study were consistent with those from previous studies. The results also suggest that not only can the depth dependent component be extracted as the main component, but it can also be separated into two different components: PC2 and PC4.

## Contribution of the PCs in the OBP time series

Results of subtraction of the PCs from the original observed data in different combinations were evaluated to confirm how the noise in the OBP time series can be reduced using the extracted PCs (Fig. 6). After removing PC1 (Fig. 5(a)), which included the annual variation component, the ASD of the time series was considerably reduced from 1.98 hPa to 0.91 hPa (the ASD value was reduced by 46%), while the spatially correlated variations were remained (Fig. 6 (b)). The characteristics of the remaining components are not common to the entire observational network but are identified at specific sites, which suggests that they reflect depth dependent oceanographic fluctuations. Figure 6(c) shows the results obtained by subtracting PC1 and PC2 from observed data. Figure 6(b) depicts elimination of the remaining variations, where the ASD shows a decrease to 0.69 hPa (the ASD value was further reduced by 24%, for an overall reduction of 65%). By removing PCs 1–3, the ASD decreased to 0.61 hPa (Figure S1(b)), and removing PCs 1, 2, and 4 resulted in an ASD of 0.62 hPa (Figure S1(c)). Finally, removing PCs 1–4 resulted in an ASD of 0.53 hPa (Figure S1(d)), reducing the noise level by 73% compared to the time series prior to the PCs’ removal. PCA is a method of identifying as much information as possible using as few synthetic variables. Naturally, subtracting the PCs with high contributions from the original data reduces their noise levels; however, as previously mentioned, PCs 1–4 showed distinct spatially correlated variations that could be attributed to oceanic origins. As such, subtracting these PCs appeared to be a reasonable procedure. In particular, PC1 and PC2 reflected most of the common or depth dependent gradients of oceanographic fluctuations in the OBP time series, given that their sum contributions of both reached 86.0% and that of PC3 and PC4, which also reflect the oceanic signal, were only 6.0% (Fig. 4). These results also suggest that PCA can efficiently remove depth dependent oceanographic fluctuations (Fredrickson et al., 2019; Muramoto et al. 2019; Inoue et al. 2021). In addition, reducing the noise level via pressure values at individual sites rather than calculating the pressure differences between sites is considered a significant advantage. Ensuing section further confirms the details of the extent to which the PCA removes these depth dependent oceanographic fluctuations.

## Reduction of sea-depth dependency based on PCA

In this section, we quantitatively discuss the extent to which these depth dependencies are reduced by PCA. First, we calculated the relative pressure time series for all the stations. This was followed by calculating the RSDs and plotting them as a function of the difference in depth or distance between sites. In addition, correlation coefficients were calculated to quantitatively evaluate the dependence.

Figure 7(a–b) shows the sea depth and distance dependency, respectively, before PCA was applied to the time series. The results for Hikurangi, New Zealand, by Inoue et al. (2021) have also been included for comparison (Fig. 7(a–b)). This figure shows that the dependence on depth is greater than that on distance. This trend is also consistent with that of off-Hikurangi margin. Figure 7(c–d) shows the correlation plot when PC1 was subtracted from the original observed time series. It is interesting to note that the depth dependence visible in the original time series, was not reduced. This result suggests that PC1 contains a common component throughout the observational network, which contributes little to the spatial variation of the observed data even after its influence is removed. The results of subtracting PC1 and PC2 (Fig. 7(e–f)) indicate that subtracting PC2 significantly reduced the dependence on sea depth. The correlation coefficient when only PC1 was subtracted was 0.78, which was almost halved to 0.40 when both PC1 and PC2 were subtracted. Interestingly, for distance dependence, the ASD itself decreased significantly (from 0.91 to 0.69), although the value of the correlation coefficient itself did not change significantly (from 0.35 to 0.30). This suggests that PC2 can adequately remove only the sea depth dependent component of the OBP time series. The fact that the DONET observation network has a slightly elongated shape in the direction of the isobaths (Fig. 1) also helps ensure that the removal of the sea depth dependence does not reduce the distance dependence. Figure S2(e–f) shows the results obtained after subtracting PCs 1, 2, and 4. Consequently, the depth dependence was further reduced with a correlation coefficient of 0.33, approximately 60% smaller than that of the original time series. However, Figure S2(c–d) shows the result after subtracting PCs 1–3. PC3 does not reduce the sea depth dependence (from 0.40 to 0.40), although it slightly reduces the distance dependence (from 0.30 to 0.26). After removing PCs 1, 2, and 4 or PCs 1–4, the correlation coefficient of the sea-depth dependence is almost equivalent to the distance dependence, indicating that removing PCs 1, 2, and 4 can significantly reduce the depth dependence of oceanographic fluctuations.

Previous studies have noted that applying PCA may reduce the detection accuracy of crustal deformation owing to the seepage of crustal deformation components into the PCs (Watts et al. 2021). Therefore, in the next section, we discuss the accuracy of crustal deformation detection by PCA using synthetic rectangular faults in the observation region.

## Physical oceanographic interpretation of each principal component

In the previous section, PCA was used to successfully extract components with multiple spatial characteristics. The observed pressure time series include both oceanographic fluctuations and tectonic signals but the oceanographic circulation models include only the oceanographic fluctuations. To examine the characteristics of oceanographic fluctuations, we applied PCA to ocean models and compared the spatial distribution of each PC with the observation. The ocean models have advantages and disadvantages in the region and time scale used herein, depending on the input data and model structure (Dobashi and Inazu 2021). We chose a single-layer ocean model (SOM) (Inazu et al. 2012) and ECCO2 (Menemenlis et al. 2008) for comparison with observations. SOM assumes a single-layer barotropic ocean model and is driven by atmospheric pressure as well as wind stress, whereas ECCO2 assumes a multiple-layer (50 layers) baroclinic ocean model (that is, better vertical resolution than SOM) and is driven by wind stress, heat, and freshwater flux, although the atmospheric pressure is not considered. The details of each model have been rlisted in Table 2 by Dobashi and Inazu (2021). The expected pressure time series was calculated based on each model. As each numerical model provides calculations for each grid, the grid nearest to the DONET observation region (77 grids in longitude of 134 °E–137 °E and latitude of 32 °N–34 °N) was selected, and PCA was applied to the time series data to confirm the characteristics of the model.

After applying PCA to SOM, the PC1 contribution reached 96.7% (Figure S3(a)). This is clearly larger than the 75.7% observed (Fig. 3), indicating that ocean variations show similar fluctuations (Figure S3). Overall, the oceanographic fluctuations expressed pressure variations, but their contribution was more exaggerated than the observation. In contrast, higher-order terms, such as the sea-depth-dependent component identified in the observational data, were poorly extracted from the SOM. Importantly, when applying PCA to ECCO2, PC1, PC2, PC3, and PC4 contributed 80.8%, 9.9%, 2.9%, and 1.8%, respectively. The contribution ratio and the spatial distribution (Figure S6) were similar to the observational data (Fig. 3). The reproduced oceanographic fluctuations calculated by ECCO2 are more realistic than SOM.

The extracted PC characteristics of ECCO2 were as follows. PC1 was the overall pressure variation with a large spatial scale (several hundred km extent) driven by wind stress and atmospheric pressure, which was also reproduced by SOM. PC2 was the depth dependent component (that is, a pattern orthogonal to isobath). PC3 and PC4 were the depth dependent patterns but their frequency was higher than PC2 (Figures S6 and S8). The longitudinal dependence component seen in PC3 of the observed data is part of the parabolic component seen in PC3 or PC4 of ECCO2 and is considered to be included in these components.

Following confirmation that the PCs extracted from the observed time series were due to oceanographic fluctuations, we attempted to interpret them qualitatively. The Nankai Trough, where DONET stations are installed, is a channel of the Kuroshio Current (e.g., Nitani 1972), which meanders widely on time scales of several years or more (e.g., Kawabe 1985). For example, one of the largest Kuroshio meanders occurred in 2017 and continued through 2022, contributing to pressure fluctuations on the seafloor (Nagano et al. 2019; 2021). Therefore, oceanographic phenomena such as the strength and meandering of the Kuroshio Current should be recorded in the OBP data for the DONET region. The meandering of the geostrophic current including Kuroshio is interrelated with the mesoscale eddy (Qiu and Miao 2000; Hasegawa et al. 2019) at approximately 100 km radius and their contribution to the sea bottom pressure perturbation cannot be well separated. Note that the sea level variation due to the meso-scale eddies is considerably attenuated to the ocean bottom pressure variation because the sub-sea surface water column imperfectly compensates for the pressure change by sea surface variation (Dobashi and Inazu 2021; Hasegawa et al. 2021).

Based on the results of the application of PCA to the observed data and ocean models, and the previous studies on oceanographic fluctuations and the Kuroshio Current (Hasegawa et al. 2021), each PC in the observed data was interpreted as follows. PC1 is the large spatial scale variation associated with wind stress and atmospheric pressure variation. PC2 corresponds to the strength and weakness, and flow axis position of the geostrophic current (that is, Kuroshio Current) (Fig. 8(a)). Both PC3 and PC4 can be compatible with the spatial pattern depending on the location of the mesoscale eddy, which has a radius less than 50km (Fig. 8(b, c)). If the centroid of the cold vortex is located east (Fig. 8(b)) or west (Fig. 8(c)) of the Kii Peninsula, the longitudinal dependence variation can be reflected in the observed DONET data. A parabola-like spatial distribution can occur when the cold vortex migrates eastward or westward, orthogonal to the direction of isobath (that is, changes from Fig. 8(b) to (c) or Fig. 8(c) to (b)). As mentioned above, the sea-surface level fluctuations are not directly reflected as seafloor pressure variations, but fluctuations caused by mesoscale eddies are thought to affect ocean bottom pressure.

In these analyses, PCA was applied to a long-term continuous time series of more than 3 years. However, if the components reflected in the PCs have characteristic time constants, changing the time window may change the extracted components. Nevertheless, applying PCA to the OBP time series shows that pressure fluctuations of oceanic origin can be separated based on their spatial characteristics. This suggests that the method can improve the quality of time series data when extracting pressure variations caused by crustal deformation. Further section discusses the potential usefulness of PCA when transient crustal deformations are included in a time series.