5.1 Validation of the tsunami sediment transport simulation
To better simulate tsunami-induced sediment erosion, transport, and resulting morphological changes, an accurate reproduction of tsunami hydrodynamics (e.g., tsunami height, waveform, and flow velocity) is necessary. The tsunami source model used in this study (Satake et al. 2013) could reproduce the observed tsunami waveforms at offshore stations and coastal tide gauges. However, these tsunami records were sparsely distributed spatially; additionally, coastal inundation was highly dependent on the resolution of the DEM used for the simulation. Thus, the validity of the tsunami source model does not necessarily guarantee a successful prediction of coastal inundation by a high-resolution simulation. Therefore, the simulation was calibrated with a flux factor of 115% based on the comparison results of the simulated and measured tsunami heights (Mori et al. 2012). Consequently, the simulated tsunami heights were close to the observations (see Sect. 3.2.2 and Additional file 1: Fig. S1). Additionally, the calibrated simulation well reproduced the observed inundation limit (white line in Fig. 4a; Haraguchi and Iwamatsu 2013).
Nevertheless, a better hydrodynamic simulation does not necessarily indicate a successful reproduction of sediment transport and sediment transport simulations must be validated through comparisons with field measurements. However, a point-by-point comparison is not feasible considering the local variability of deposit thickness, which cannot fully be addressed during the field observations. Thus, we compared the simulated and observed depositional volumes that could serve as a robust metric for validating the simulations.
The simulated deposition volume \(\left( {V_{{{\text{sim}}}} = 86,645\;{\text{m}}^{3} } \right)\) was almost consistent with the observed deposition volume \(\left( {V_{{{\text{obs}}}} = 85,336\;{\text{m}}^{3} } \right)\), indicating that the magnitude of sediment transport was correctly simulated in the reference simulation. The overall deposition trend in both simulations and observations was similar, except in the Suijin-numa pond (Fig. 2). Qualitatively, the observed (Abe et al. 2020; Fig. 2d) and simulated (Fig. 2c) shore-normal thickness distributions were similar. Further, the simulation demonstrated that the erosion due to the tsunami devastated the beach (Fig. 4b), and the cross-shore erosion changed the coastline significantly. Previous studies deduced that the concentrated backwash eroded the beach deposits, and the scour, similar to a small bay (so-called tsunami bay), remained even after the tsunami (Yoshikawa et al. 2015, 2017, 2018; Abe et al. 2020). Specifically, the simulation results indicated that the concentrated erosion on the beach was qualitatively consistent with such a deduction. Based on these quantitative and qualitative comparisons of the deposition volumes, the calibrated tsunami sediment transport simulation successfully reproduced the general characteristics of the tsunami inundation and sediment deposition, thus suggesting the possibility of investigating the tsunami source based on the simulated deposit volume. Despite the successful reproduction of the onshore deposition, the simulated volume of the beach erosion was \(6.85 \times 10^{4} \;{\text{m}}^{3}\), whereas the observed erosion volume was \(1.34 \times 10^{5} \;{\text{m}}^{3}\) (Abe et al. 2020). Considering the consistency in onshore deposition, this difference in erosion may influence the offshore rather than the onshore deposition process. Thus, the results of the calibrated tsunami sediment transport simulation can be regarded as valid as long as it is focused on the onshore deposition.
5.2 Tsunami source model inferred from the tsunami deposit and its source
Among the models se3510 and se4005, it is difficult to judge based solely on the deposit volume which is the better one because both models well reproduce the deposition volume (Fig. 5). Even if measured tsunami heights are available, the judgment is still uneasy because the difference in the K (1.15 for se3510 and 1.13 for se4050) and κ (1.21 for se3510 and 1.22 for se4050) metrics is quite small. Similarly, the comparison of the inundation limit is not promising to find their difference (Fig. 7). Therefore, another metric other than those is necessary to further constrain the tsunami source.
Masaya et al. (2021) highlighted that simulated sediment source provides additional information to estimate paleotsunami sources. They proposed the ratio of marine-origin deposits to whole deposits, \(P_{{{\text{sea}}}}\), as defined:
$$P_{{{\text{sea}}}} = \frac{{V_{{{\text{sea}}}} }}{{V_{{{\text{all}}}} }}$$
where \(V_{{{\text{sea}}}}\) and \(V_{{{\text{all}}}}\) denote the volumes of marine-origin deposits and whole tsunami deposits in the area of comparison, respectively. A numerical experiment shows that the \(P_{{{\text{sea}}}}\) ratio varies depending on the tsunami waveform and can be a constraint of deposit-based tsunami source modeling (Masaya et al. 2021). In the present approach of Eulerian simulations, exact value of \(V_{{{\text{sea}}}}\) cannot be calculated directly. Thus, \(V_{{{\text{sea}}}}\) was approximated based on the deposit volume calculated assuming that the entire land is non-erodible.
Although an exact \(P_{{{\text{sea}}}}\) may not be obtained from real tsunami deposits, contributions of different sediment sources can be elucidated by detailed analyses. The sandy tsunami deposits in the Sendai Plain formed by the 2011 Tohoku-oki tsunami have been estimated to originate onshore (Szczuciński et al. 2012; Takashimizu et al. 2012; Putra et al. 2013). For example, marine diatom species occupy only ~ 2% of the entire diatom assemblage in the sandy fraction of the tsunami deposits in the northern Sendai Plain (Takashimizu et al. 2012). Existing tsunami sediment transport simulations support such micropaleontological findings (Sugawara et al. 2014b). In the present study area, Abe et al. (2020) elucidated the onshore source of the tsunami deposits by comparing pre- and post-tsunami DEMs and grain-size distributions of the deposits. \(P_{{{\text{sea}}}}\) in the reference simulation is only 3% (Fig. 8), which fits the estimation by Abe et al (2020).
Deposit distributions are in general similar among the models se4005 and se3510 (Fig. 9). However, seafloor erosion near the coastline differs significantly. In the case of se4005 (Fig. 9c), a shore-parallel band of seafloor erosion was found ~ 200 m offshore of the coastline, whereas such a band was located much closer to the land and was relatively narrow in se3510 (Fig. 9a). The difference in offshore erosion can affect the contribution of the marine sediment (i.e., \(P_{{{\text{sea}}}}\)). Figure 8 shows that se4005 shows a greater contribution \(\left( {P_{{{\text{sea}}}} = 37\% } \right)\) comparing with se3510 \(\left( {P_{{{\text{sea}}}} = 23\% } \right)\).
Figure 10 compares the tsunami waveforms by three different source models (Satake et al. 2013, se3510, and se4005) at a virtual tide station SJN1 (Fig. 1). The waveforms by the models se3510 and se4005 are in general similar, except for a period between 45 and 63 min. Among a whole range of waveforms, se3510 and se45005 have different features compared to Satake et al. (2013). Major differences are the duration of the main wave and its precedent sea-level drop. With hypothetical source models, an abrupt sea level drop (receding wave 45–63 min) occurred prior to the arrival of the main wave. The simulation by model se4005 yielded a larger receding wave compared with that by se3510. In the reference model (Satake et al. 2013), a gradual and moderate decrease in the sea level appeared prior to the main wave. Masaya et al. (2021) pointed out that the period of the main wave and the amplitude of the precedent receding wave control the erosion of the shallow seafloor. With a larger precedent receding wave, nearshore seafloor is exposed extensively and the main wave with a longer period exerts a persisting bottom shear stress on the seabed, resulting in greater sediment erosion and entrainment into the flow.
Considering the minor contribution of seafloor sediments (Abe et al. 2020), simulations with a lower \(P_{{{\text{sea}}}}\) value are preferable. Thus, the model se3510 better explains the sediment source of the 2011 Tohoku-oki tsunami deposits in the study area. Note that \(P_{{{\text{sea}}}}\) of the hypothetical models is much higher than that of the reference simulation \(\left( {P_{{{\text{sea}}}} = 3\% } \right)\). The hypothetical models may have a limitation on the reproducibility of the waveforms and resulting \(P_{{{\text{sea}}}}\) due to their insufficient spatial and temporal resolution of the slip distribution.
The above approach to constrain the tsunami source model based on \(P_{{{\text{sea}}}}\) relies on the assumption that differences between beach and shoreface deposits can be identified and fractions of respective deposits can be quantified. Distinguishing the deposits based on geological proxies is difficult because the differences may not be evident in general cases as the littoral drift system mixes and homogenizes these deposits. Therefore, to apply \(P_{{{\text{sea}}}}\) for general cases, a new methodology to quantify fractions of beach and shoreface deposits is required.
5.3 Comparison with existing source models
Figure 11 compares the slip models inferred from this study and previous studies. Iinuma et al. (2012), Satake et al. (2013), and Yamazaki et al. (2018) estimated the slip distribution based on geodetic inversion, tsunami inversion, and a combination of iterative forward modeling of tsunamis and inversion of teleseismic P-waves, respectively. The fault model of Yamazaki et al. (2018) represents a self-consistent slip that can explain seismic and tsunami waveforms and tsunami inundation. In this study, the slip distribution estimated based on the tsunami deposits (se3510; Fig. 11a) showed a larger slip on S4 near the trench axis than in other models. Conversely, the slip amount on the landward S9 was consistent with that in the other models. Although the slip amount is smaller than that near the trench axis, a “deep large slip” (Satake et al. 2013), shown as subfaults surrounded by a thick frame in Fig. 11, is necessary to explain the deposit volume in the study area. This was consistent with the results of Satake et al. (2013), who indicated the significant contribution of the “deep large slip” to the extended inundation in the Sendai Plain.
Satake et al. (2013) also reported that the “shallow huge slip” near the trench axis does not influence the inundation in the Sendai Plain. In fact, the tsunami sediment transport simulation based on the source model proposed by Satake et al. (2013) without the “shallow huge slip” can explain not only the observed inundation but also deposit volume (Additional file 1: Fig. S3). Here, the question that arises is whether the deposit-based fault model se3510 was inconsistent with the findings of Satake et al. (2013). In this study, the assumption of different ranges of slips on deep (0–10 m) and shallow (0–40 m) subfaults was used for slip estimation, based on the slip distributions of the 2011 Tohoku-oki earthquake determined by Hayashi and Koshimura (2012), which attributed larger slips to the shallow subfaults. If the ranges of the slip to be tested are not constrained based on the existing fault model, the balance of the estimated slip on the deep and shallow subfaults may differ. Thus, in this study, the empirical assumption of the slip range may have resulted in the selection of the slip distribution with a larger shallow slip. In addition to the assumption about the maximum slip, the subfault size might explain this conflict. Here, only a seaward half of subfault S4 belongs to the “shallow slip” domain defined by Satake et al. (2013). A combination of narrow and wide subfault in shallow and deep parts, respectively (an example can be found in Ioki and Tanioka 2016), may be more suitable for this kind of discussion.
Compared with se3510, se4005 had a slip distribution that was larger on S4 and smaller on S9. This implies that the larger slip on the landward S9 (se3510) contributed to the smaller receding wave (Fig. 10). A huge slip near the trench axis caused significant subsidence of the sea floor and the sea surface landward of the rupture area (Satake and Tanioka 1999). However, with a larger slip on the deep portions of the plate interface, such negative sea surface displacement is negated due to larger crustal uplifting directly below. Thus, the smaller amount of marine-origin materials in the onshore tsunami deposit by the se3510 source model can be associated with the larger slip on Subfault S9 and the resulting smaller receding wave near the coastline.
Ignoring the contribution of other subfaults might affect the slip amounts on S4 and S9. In fact, S3, which is equivalent to the segment with the largest slip during the Tohoku-oki earthquake, was located directly north of S4. Although the sensitivity of the fault slip to the deposit volume in the study area was low, a huge slip (over 30 m) on S3 similar to previous models may affect the sediment erosion and deposition in the study area and reduce the slip amount on S4. However, without any prior knowledge of the slip distribution, it may be difficult to assume such a distant large slip in a paleoseismic event. The paleotsunami source model as well as se3510 (Fig. 11a) can be incomplete if reconstruction relies on spatially limited tsunami deposit data.
It is noteworthy that se3510 is a solution based on several assumptions from the result of the sensitivity analysis and known general along-dip slip distribution. In this study, the optimal solution seemed not to be determined without any a priori assumptions because the geological data for tsunami source modeling are quite limited. Two problems in this study, the incomplete spatial extent of the reconstructed source and strong dependency on the assumptions, may be conquered by using extensive geological data along the tsunami-prone coast. The estimation of the slip on each subfault can also be more accurate by properly solving trade-off relationships among all subfault slips in the source region. Thus, for better deposit-based tsunami source modeling, the availability of tsunami deposit data from a wide area along the tsunami-prone coast is essential.
5.4 Uncertainty in deposit-based source estimation
Various factors, such as inputs and model setups, associated with tsunami sediment transport modeling can introduce uncertainties in the numerical simulations (Jaffe et al. 2016; Sugawara 2019). The results of the simple sensitivity analysis (Table 1) showed the fluctuated simulated deposit volume with changes in grain size. In general, the grain size of the sediment source is difficult to constrain despite its importance in the simulations.
Figure 12 demonstrates that the uncertainty in grain-size setting propagated via the numerical simulation to the estimated fault slip. With a fixed fault slip, the deposit volume was ~ 15,000 m3 smaller for every 0.05 mm increase in the grain size. Such a change was nearly equivalent to the variations in the deposit volume, due to a decrease in the fault slip on S4 by ~ 4.3 m with the fixed grain size. Although it is just an example, this comparison indicated that the uncertainty due to sediment grain size should be carefully considered in paleotsunami source estimation.
5.5 Implications for paleotsunami studies
Deposit-based source modeling requires the accurate simulation of tsunami deposit formation at surveyed sites. The bottom deposits of coastal ponds and lakes are important candidate sites for field surveys in paleotsunami research due to the high preservation potential of tsunami deposits (Sawai et al. 2008; Furumura et al. 2011; Kempf et al. 2017). Previously, the bottom sediments of the Suijin-numa pond have been surveyed to investigate the paleotsunami history (Sawai et al. 2008). Thus, the successful simulation of lake-bottom tsunami deposit formation will greatly contribute to the assessment of tsunami magnitude. Despite this importance, the simulated thickness distribution in the Suijin-numa pond was significantly different from the observed distribution even in the reference simulation (Fig. 2). This could be caused by the limitation of 2DH hydrodynamic simulation (TUNAMI-N2), which cannot simulate three-dimensional flow and the resulting sedimentary process. Thus, this limitation will hinder the application of such numerical models in paleotsunami source estimation based on lake-bottom tsunami deposits.
Regarding the paleotsunami source modeling, we have more difficulties than in this study. Available data for inputs, such as topography, land condition, and initial tide level, can be poorly constrained. Moreover, post-depositional processes can modify the thickness of tsunami deposits, leaving them unidentifiable near the distribution limit (Szczuciński 2012; Spiske et al. 2020). Even if deposits can be identified, the consolidation of tsunami deposits can decrease the deposit thickness and volume and cause an underestimation of the slip. These uncertain inputs, boundary conditions, and constraints hinder the accurate estimation of a paleotsunami source. Thus, the uncertainty in the estimated source should be demonstrated even though the estimation appears to be successful.
The combination of forward and inverse models (i.e., hybrid modeling; Sugawara et al. 2014a; Jaffe et al. 2016) is a promising approach for deposit-based tsunami source modeling. Inverse models of tsunami-induced sediment transport (Jaffe and Gelfenbaum 2007; Tang and Weiss 2015; Naruse and Abe 2017) are useful to extract tsunami flow conditions, such as flow velocity, inundation depth, and concentration of suspended sediments, from sedimentary features of tsunami deposits. Inverse modeling is mainly a contrasting approach to forward modeling of tsunami-induced sediment transport (e.g., TUNAMI-STM), whereas hybrid modeling utilizes both models to complement each other; for example, forward simulation can be used to validate an assumption of the inverse model and inverse calculations can be used to estimate the hydraulic constraint of the forward model. Although the methodology is not yet established, this approach might be effective to mitigate the uncertainty of the tsunami source and the accurate estimation thereof.
Furthermore, considering the uncertainties of tsunami deposit characteristics, using the deposit thickness or volume as the only constraint of source estimation can create problems. Although difficult to implement, including constraints from various kinds of sedimentary data is a possible solution. In addition to thickness, volume, and sediment source, grain-size distribution of tsunami deposits may provide clues on the characteristics of the tsunami source (Gusman et al. 2018). Tsunami source modeling with sediment transport simulations has the potential to fully utilize such sedimentary data. Other paleoseismological information, such as coseismic crustal deformation and historical records, can be considered as additional constraints (Sugawara et al. 2019; Dourado et al. 2021). Moreover, seismological scaling laws (Murotani et al. 2013; Skarlatoudis et al. 2016), which are empirical relationships between fault parameters, are useful to construct hypothetical fault models (Minamidate et al. 2022). Such additional constraints should be examined from the perspectives of both geology and numerical modeling.