Direct Comparison of the Tsunami‐Generated Magnetic Field With Sea Level Change for the 2009 Samoa and 2010 Chile Tsunamis

The motion of conductive seawater by tsunamis can generate magnetic fields in the presence of the background geomagnetic main field. Previous studies found that, using the tsunami‐generated seafloor magnetic field, it is possible to predict the propagation direction and wave height prior to the actual arrivals of tsunamis. This study correlates the tsunami magnetic field and the tsunami sea level change using observed data and three‐dimensional simulations of the 2009 Samoa and 2010 Chile tsunamis. Our direct comparison of the tsunami observed magnetic field and tsunami sea level change illustrate that the vertical tsunami magnetic component, bz , arrived earlier than the sea level change. The “initial rise” signal in the observed horizontal tsunami magnetic component, bh , which was arrived even earlier than bz also is found by combing the observation with the three‐dimensional simulations. We further examine the precision of conversion of the tsunami magnetic field to the sea level change and find that the magnetic field derived tsunami sea levels are as precise as those obtained from differential pressure gauge data. However, our simulation shows that existing tsunami source models are incompatible with our tsunami magnetic data. Therefore, it is necessary to include magnetic field derived tsunami sea level changes to improve those source models.

Chile earthquakes (red stars), the DART stations (yellow circles) and the TIARES area (the yellow rectangle). Both maps were drawn by Generic Mapping Tools (GMT) v.6.1.1  https://www.generic-mapping-tools.org) based on the global relief (SRTM15 + V2.1, Tozer et al., 2019). the analysis of the tsunami z E b arrival time at all nine SOC stations as shown in Supporting Information S1.
(b) The tsunami z E b was converted into the upward direction which is the same as the sea level change. In    (Toh et al., 2011) and the 2011 Tohoku tsunami . The amplitudes are as small as about 0.2 and 0.3 nT for the 2009 Samoa and 2010 Chile tsunamis, respectively. On the other hand, when the tsunami passed by (black solid vertical lines in Figure 2), the tsunami magnetic signals at Pamatai (PPT), the closest land observatory to the TIARES area, were not visible even after the same bandpass filtering as the SOCs data. It may be because the land station is exposed to large magnetic noises of external origin without a thick cover of conductive seawater compared with the seafloor stations. It is evident that the observation on the seafloor is more sensitive to the tsunami magnetic field than on land.
The data at the SOC8 station are very unique in the sense that it is the world's first observatory of simultaneous ocean bottom pressure and magnetic field data (Suetsugu et al., 2012). This enabled us to study the following two topics: 1. The phase lead of tsunami z E b and the initial rise (actually "retreat" in our case) in h E b w.r.t. the tsunami sea level change. 2. The conversion of tsunami magnetic field to tsunami sea level change.
The former can help us to investigate the tsunami early warning ability by the tsunami magnetic field. Moreover, the conversion of the tsunami magnetic field to the tsunami sea level allows us to examine the validity of our procedure to extract the tsunami magnetic field.
The phase lead of tsunami z E b and the initial rise in h E b have been simulated by several previous studies (Minami & Toh, 2013;Minami et al., 2015Minami et al., , 2017. However, the results have never been verified by comparison of observed tsunami sea level changes with observed tsunami magnetic fields, which has done for the first time by this study as shown in Figure 3.
As can be seen from the direct comparison of the observed sea level change and magnetic field, the observed tsunami z E b arrived earlier than the sea level change, while h E b were delayed compared with tsunami z E b . We calculated the phase differences between the magnetic components and the sea level change for the main periods of both events and illustrated them in Figure 4. The longer the magnetic variations' periods are, dependent on a 2-D analytical solution. In order to investigate the discrepancy further, 3-D tsunami kinetic/ electromagnetic simulations will be introduced in the next section.
It is also shown in the SOC8 panel that the results of two different conversion equations have limited difference, which means that the effect of linear dispersion is small at least locally. Furthermore, the longwave approximation offers a very simple way to estimate the tsunami sea level change by tsunami z E b and h E b (and vice versa). For example, the amplitude ratio and phase lead angle of tsunami z E b with sea level change at SOC8 is 4.112 nT/m and 20.9° according to the longwave equation. Hence, the tsunami sea level change can be recovered by simply multiplying the amplitude factor and making the phase shift accordingly to the tsunami z E b time series.  (Minami et al., 2021) using the filtered magnetic fields. The observed sea level changes at SOC8 and the converted sea level changes calculated by the linear longwave solution (Tyler, 2005) are also shown in the top left panel for each tsunami. The comparison at SOC8 shows the excellent agreement between the converted sea level changes from two different equations and the observed tsunami sea level changes, and consistency between two converted sea level changes using the different tsunami magnetic components except for SOC6.

Simulation of Tsunami Magnetic Fields for the 2009 Samoa and 2010 Chile Events
The 3-D time domain tsunami magnetic simulations were conducted to explain the two issues in the observation: (a) The missing initial rise in the Samoa event. (b) The 3-D effect found by the comparison of the observed sea level change and the tsunami magnetic field derived sea level change. To those ends, the 3-D simulation results will be compared with the observation. On the other hand, the observed sea level change along with the magnetic field at SOC8 allows us to estimate the error of tsunami velocity and magnetic field calculation separately. In other words, we will discuss the main error source in the simulations of both tsunami velocity and magnetic field.
The simulation of the tsunami magnetic field of the 2009 Samoa and 2010 Chile events over the TIARES area consisted of the following two steps: First, the tsunami velocity field in the TIARES area were simulated kinetically using an open-source software, JAGURS (Baba et al., 2017), and then the tsunami magnetic field were calculated using the velocity field thus simulated. The outputs from JAGURS were appraised by the sea level change observed at SOC8. Finally, the simulated tsunami magnetic fields were compared with the observed magnetic fields at all SOC stations in the TIARES area.

Simulation of the Tsunami Velocity Field
The TIARES area is considerably distant from the two epicenters as listed in Table 2, and the two tsunamis contained multiple frequencies (Schnepf et al., 2016), which means that the dispersion of the tsunami propagation should be considered. In this study, an open-source tsunami simulation code, JAGURS, by Baba et al. (2017) was used for calculation of particle motions associated with the tsunamis in concern, because it can account for the effects of Boussinesq dispersion as well as elastic loading.
JAGURS simulates the velocity field based on earthquake fault models that are normally inverted from the seismic and tsunami data. Proper fault models are indispensable for decent tsunami velocity simulations. Several source models reported by previous research have been considered in this study. Our guiding principle of the fault model selection was that it can fit the sea level changes around the TIARES area most.
For the 2009 Samoa tsunami, we used the "Mega 1" fault model of Fritz et al. (2011), which is a single normal fault model and can explain the observed sea level of the DART stations near the TIARES area (see Figure 11 of Fritz et al., 2011). As for the 2010 Chile tsunami, we used the "all DART only" fault model of Yoshimoto et al. (2016), because it is a multiple-fault model obtained by the inversion of the tsunami data at 26 DART stations over the Pacific. These fault models show the best fit to the tsunami sea level around the TIARES area among the existing source models, although they did not include the sea level change data at SOC8.
Properties of the velocity field simulations by JAGURS for the 2009 Samoa and 2010 Chile tsunamis are shown in Table S1 in Supporting Information S1. The velocity fields of tsunamis were simulated from the epicenters through the TIARES area for time spans of 7 hr for the Samoa event and 14 hr for the Chile event. In order to examine our simulation results, the DART stations of 51425 and 51426 were included in the simulation in the case of the 2009 Samoa tsunami, while the DART stations of 32412 and 51406 were incorporated in the 2010 Chile tsunami simulation. Comparisons of the simulated and observed sea levels at the DART stations and Site SOC8 in the TIARES area for both events are shown in Figure 6. For the Samoa tsunami, the simulated sea level at DART 51425 and 51426 were successfully reproduced as those in Fritz et al. (2011). However, this fault model can explain the observation at DART 51426 but not at 51425. As for the Chile tsunami, the simulated sea levels at DART 32412 and 51406 make good reproduction of those by Yoshimoto et al. (2016) and fit the observation well except for somewhat attenuated amplitudes.
Although we successfully reproduced the simulation results of the previous works using the respective fault models at the DART stations, the simulated sea levels at SOC8 in the TIARES area still show obvious discrepancies both in amplitude and phase from the observation as indicated in Figure 6. Better velocity field simulations are definitely necessary here. For example, a better fault model to match the amplitude of the observed sea level change, or a proper dispersion to yield better agreement of the arrival times. However, it is beyond the scope of the present study.
On the other hand, to explain the first peak of the observation, combinations of amplitude gain and time shift were calculated: In the Samoa tsunami event, the simulated sea level was divided by 2.65 and delayed by 99s. As for the Chile tsunami, the simulated result was divided by 1.04 and delayed by 171 s. After the modification, the discrepancies between observation and simulation were significantly reduced as illustrated in Figure 6. It is noteworthy that the same modification can be applied to the simulated magnetic field, because of the linearity of the magnetic variation to the tsunami seawater velocity (e.g., see Equation 4 in   Minami et al., 2021). Using the linear relationship, it is no need to calculate the velocity field of the modified sea level change. The most likely sea level change can be obtained by applying the same modification to the original sea level change. For example, if we want to derive the proxy of the tsunami magnetic field of the Samoa tsunami sea level change, we need only to divide the simulated magnetic field by 2.65 and further delay it by 99s. This method actually used in this study to get the better magnetic field simulation result as shown in Figure 6.
which is in harmony with the report by Minami et al. (2015). The initial rise in tsunami h E b is about −0.03 nT, which corresponds to the following a 3.4-cm-high tsunami. However, the observed tsunami wave height at the first arrival is 1.3 cm as shown in Figure 3 (the upper panel), so that the actual amplitude of initial rise must be smaller than the simulated amplitude.
The initial rise in Chile tsunami h E b was simulated as small as −0.04 nT. In contrast to the tsunami first arrival consisting of a single peak in the 2009 Samoa tsunami, the first arrival of 2010 Chile tsunami is more complex to have a small dual peak before the larger peak (see the lower left panel of Figure 6). That simulated value of −0.04 nT should be similar to the actual amplitude of the initial rise, because we succeeded in reproducing the observed tsunami wave height by simulation (see Figure 6). Although the amplitude in the Chile event was larger than that in the Samoa event, the initial rise is still hard to recognize, because the Chile event started with a small dual bump that makes it difficult to isolate the initial rise.
The phase lead of tsunami z E b , however, is still clearly seen in the simulated z E b for both events. The tsunami z E b have a small but evident phase lead with respect to the tsunami sea level change.
In the simulation results, the initial rises may exist in both h E b of the 2009 Samoa and 2010 Chile tsunamis, although that of the latter is less prominent due to the complexity of its first arrival. Our simulation results suggest that the initial rise is a little bit too small for the Samoa tsunami and too complicated for the Chile tsunami to recognize clearly in the observation data. We will further argue this issue in the next subsection by direct comparison of the simulation with the observation.

Comparison of the Simulated Tsunami Magnetic Field With the Observation
In Figure 8, the simulated magnetic fields without and with modification at SOC8 are compared with the observation, as well as the magnetic field converted from the observed sea level change using the 2-D analytical solution (Minami et al., 2021).
Comparison of the simulated magnetic field (black solid lines) and observation (red and blue solid lines) shows poor agreement of the simulation with the observation. The phase and amplitude discrepancies between simulation and observation are notable, similar to that between the simulated and observed sea level change shown in Figure 6. As mentioned before, the simulated tsunami sea level change can be modified by a combination of amplitude gain and time shift to match the observation better. Because the accuracy of the tsunami magnetic calculation was examined intensively, it suggests that the discrepancy between the simulated magnetic field with the observation is mainly due to the inaccuracy in the tsunami velocity field simulation (see Figure 3 and Table 1 of Minami et al., 2017). The much better fit of the modified magnetic simulation to the observation by the same sets of amplitude gain and time shift as the velocity simulation also supports this claim. In other word, we need a better velocity field to improve the tsunami magnetic simulation significantly.
By inspection of the converted h E b , the initial rise in h E b may be present for both events. However, the actual initial rise in the Samoa event was as small as −0.012 nT, much smaller than the background noise. On the other hand, in the Chile tsunami, we may claim that the initial rise can be seen in observed h E b (indicated by arrow in Figure 8). However, the initial rise in the observation, in turn, is much larger than the simulation, which may be because the simulation could not fit the amplitude of the small dual peak.
Comparison of the modified simulation with the converted z E b and h E b tells us that the fit is fairly well. The first arrival of simulation agrees moderately with the converted magnetic components for both amplitude and phase in Samoa event. Also, the modified z E b of the Chile tsunami fits the converted z E b well. It means that the simulation results were compatible with the 2-D analytical solution especially for the Samoa tsunami.
Finally, we compared the modified magnetic simulations with the observations at the remaining SOC sites (SOC1-6, 9) for both events in Figure 9. It can be seen that the modified simulations seem to fit the observations well. Although the initial rise is hard to recognize in 2009 Samoa tsunami, small initial rises may present at SOC1-4 and 9 in the 2010 Chile tsunami.

Discussion and Conclusion
Presence of the phase lead in tsunami z E b and the initial rise in h E b has been shown by simulations in previous studies. However, no research has confirmed their presence by observation so far. In this study, we examined these two tsunami magnetic signals, which are arrived earlier than tsunami sea level change, using the combination of observation and simulation of both tsunami sea level change and magnetic field at the times of the 2009 Samoa and 2010 Chile earthquakes within the TIARES area.
First of all, the tsunami signals were successfully extracted from the observed ocean bottom pressure change and magnetic field of both 2009 Samoa and 2010 Chile tsunamis in spite of the very large epicentral distances. The phase of tsunami z E b always leads by ∼90° with respect to h E b , which is in line with the observation of other tsunami events Toh et al., 2011). Also, the converted sea level changes using the observed tsunami magnetic field agree with the observed sea level change very well. The tsunami magnetic signals on the seafloor were identified clearly at each SOC station, even though the amplitudes of tsunami magnetic field were small, 0.2 and 0.3 nT for the 2009 Samoa and 2010 Chile tsunamis, respectively. It was Figure 9. Comparison of the observation and modified simulation for the tsunami magnetic field at SOC1-6, 9 at the time of the 2009 Samoa tsunami (upper eight) and the 2010 Chile tsunami (lower eight). SOC7 is not included due to failure in proper mech generation around SOC7. The observed tsunami magnetic field has passed through the 3-30 and 3-45-min bandpass filter for the Samoa and Chile tsunamis, respectively and the external field has been subtracted. Also, the simulated tsunami magnetic field was divided by 2.65 and delayed by 99s for the Samoa tsunami and divided by 1.04 and delayed by 171 s for the Chile tsunami.
also illustrated that the seafloor observation was more sensitive to the tsunami magnetic field than the observation on land.
The phase lead in tsunami z E b and the initial rise in tsunami h E b were studied using the unique simultaneous observation of both ocean bottom pressure and magnetic field at SOC8. Comparison of the observed sea level change with the observed magnetic field confirmed that tsunami z E b has its phase lead with respect to the sea level change, which is compatible with the prediction by Minami et al. (2015). This means that we can exploit the significant phase lead in tsunami z E b for the purpose of tsunami early warning. We also showed that the phase lead of tsunami z E b w.r.t. the sea level change in each main frequency is in harmony with the leading time at each frequency using the analytical solution described in Minami et al. (2021). Our results indicated that the value of the phase lead is almost identical over the tsunami period band, while the leading time gets larger linearly against the tsunami periods. It is noteworthy that this study investigated the tsunami magnetic field at the averaged 4,800 m water depth, which is classified as "Self-induction dominant case" or "Intermediate case" according to Minami et al. (2015). On the other hand, the behavior of tsunami magnetic field in shallow water, the "diffusion dominant case" in Minami et al. (2015), is still not confirmed and needs further investigation.
The initial rise in tsunami h E b was not detected clearly in the observation. By comparison with the simulation results, we found that, in the 2009 Samoa tsunami, the first wave of tsunami was too small to produce an observable initial rise in h E b , although our 3-D time domain magnetic simulation predicted its presence. As for the 2010 Chile tsunami, we probably captured the initial rise because a small "retreat" in h E b is recognizable at several SOC stations. However, the first wave of the 2010 Chile tsunami was rather complicated to identify the initial rise clearly using the first arrival. These results suggest that further study on the initial rise in tsunami h E b is necessary in order to apply it to tsunami early warning.
Precision of the conversion equations of tsunami magnetic field to sea level change (and vice versa) by Tyler (2005) and Minami et al. (2021) were also verified using the simultaneous data of different physical quantities. Comparison of the observed and converted sea level changes indicates that both equations can precisely predict the tsunami sea level change based on the known tsunami z E b or h E b (and vice versa), provided that there are good approximations for the ocean depth as well as the subseafloor electrical structure. It also implies that the tsunami magnetic field is equivalent to the tsunami sea level change and useful in tsunami research such as the earthquake fault model inversion, since the tsunami z E b and h E b are proved very good proxies of the sea level change by this study. We created a data set of sea level changes converted from the observed tsunami magnetic field in the TIARES area (see Data Availability Statement). However, there still exists discrepancy between the converted and observed magnetic field in the 2010 Chile tsunami due mainly to the 3-D effect of this tsunami.
Using the 3-D tsunami velocity and magnetic field simulation in time domain, the presence of the initial rise and 3-D effect were investigated in the magnetic field of both tsunami events. The tsunami velocity fields were calculated from the fault model of Fritz et al. (2011) and Yoshimoto et al. (2016) for Samoa and Chile events, respectively. Comparison of the simulated and observed tsunami sea level change indicated that simulated velocities based on the given source models have too large amplitudes around the TIARES area to explain the observed tsunami magnetic fields. Very good fits of the observed magnetic signal to the converted magnetic signal described in Section 3 encouraged us to calculate the best fit magnetic time series to the observation from the simulated time series by optimizing two parameters, that is, the amplitude gain and the time shift. The results show that the major source of simulation error lies in the inaccurate velocity simulation, which may also explain the reason for the discrepancy between the simulated magnetic field and the observation in previous studies Zhang et al., 2014). Better source fault models using the converted tsunami sea level change in the TIARES area are desired here. Further studies in search for the best tsunami simulation model by better earthquake source models are essential to explain the TI-ARES observation, which is, unfortunately, beyond the scope of the present study.
The 3-D effects of both tsunamis were studied by our 3-D magnetic field simulation as well. The converted sea level change fit the observation well for the Samoa event, while the converted amplitude is significantly smaller than the observation for the Chile event. It means that the 3-D effect is not very large in the case of the 2009 Samoa tsunami, while it is significant in the 2010 Chile tsunami. It should be noted that our velocity simulation is 2-D in the sense that JAGURS outputs horizontal velocity components alone. According to Minami et al. (2021), the vertical velocity component may produce small ( z E b and h E b ) even in the equatorial region, which is seen only when the tsunami propagates mostly in a direction parallel to the horizontal component of the geomagnetic main field. However, both tsunami events treated here are mainly traveled in the east-west direction, which is perpendicular to the direction of the background geomagnetic main field around the TIARES area. It, therefore, is unlikely that contribution of the vertical velocity component amends results of this study drastically.