Quantifying Magma Overpressure Beneath a Submarine Caldera: A Mechanical Modeling Approach to Tsunamigenic Trapdoor Faulting Near Kita‐Ioto Island, Japan

Submarine volcano monitoring is vital for assessing volcanic hazards but challenging in remote and inaccessible environments. In the vicinity of Kita‐Ioto Island, south of Japan, unusual M ∼ 5 non‐double‐couple volcanic earthquakes exhibited quasi‐regular recurrence near a submarine caldera. Following the earthquakes in 2008 and 2015, a distant ocean bottom pressure sensor recorded distinct tsunami signals. In this study, we aim to find a source model of the tsunami‐generating earthquake and quantify the pre‐seismic magma overpressure within the caldera's magma reservoir. Based on the earthquake's characteristic focal mechanism and efficient tsunami generation, we hypothesize that submarine trapdoor faulting occurred due to highly pressurized magma. To investigate this hypothesis, we establish mechanical earthquake models that link pre‐seismic magma overpressure to the size of the resulting trapdoor faulting, by considering stress interaction between a ring‐fault system and a reservoir of the caldera. The trapdoor faulting with large fault slip due to magma‐induced shear stress in the submarine caldera reproduces well the observed tsunami waveform. Due to limited data, uncertainties in the fault geometry persist, leading to variations of magma overpressure estimation: the pre‐seismic magma overpressure ranging approximately from 5 to 20 MPa, and the co‐seismic pressure drop ratio from 10% to 40%. Although better constraints on the fault geometry are required for robust magma pressure quantification, this study shows that magmatic systems beneath calderas are influenced significantly by intra‐caldera fault systems and that tsunamigenic trapdoor faulting provides rare opportunities to obtain quantitative insights into remote submarine volcanism hidden under the ocean.

• Non-double-couple earthquakes with seismic magnitudes of 5.2-5.3recurred in the vicinity of a submarine caldera near Kita-Ioto Island • A mechanical model of trapdoor faulting based on tsunami data of the 2008 earthquake infers pre-seismic overpressure in a magma reservoir • Uncertainty in fault geometry varies our estimate of pre-seismic overpressure (5-20 MPa) and co-seismic pressure drop ratio (10%-40%) Supporting Information: Supporting Information may be found in the online version of this article.

Tsunami Signal From a Volcanic Earthquake at Kita-Ioto Submarine Caldera
Kita-Ioto Island is an inhabited island in the Izu-Bonin Arc, to the northwest of which a submarine caldera with a size of 12 km × 8 km is located, hereafter called Kita-Ioto caldera (Figures 1a-1c).While no historical eruption on the island has been reported, past submarine eruptions were found at a submarine vent called Funka Asane on a major cone within the caldera structure (Figure 1c).According to Japan Meteorological Agency (2013), the latest eruptions of Funka Asane were reported between 1930 and 1945, and its volcanic activity has been recently inferred from sea-color changes and underwater gas emission near the vent (Ossaka et al., 1994).In March 2022, Japan Meteorological Agency (2022) reported ash-like clouds near Kita-Ioto Island and suggested the possibility of an eruption, but it is not clear whether the clouds were caused by an eruption or by meteorological factors.Thus, the volcanic activity of the submarine caldera has not been understood well.
Active volcanism of Kita-Ioto caldera shows unique seismic activity characterized by shallow earthquakes near the caldera repeating every 2-5 years, in 2008, 2010, 2015, 2017, and 2019, in addition to that in 1992 (Figure 1c; Table S1 in Supporting Information S1).As the focal mechanism of the earthquake in 2008 represents in Figure 1c, these six earthquakes reported in the Global Centroid Moment Tensor (GCMT) catalog (Ekström et al., 2012) similarly had seismic magnitudes of M w 5.2-5.3 and non-double-couple moment tensors with large compensated-linear-vector-dipole (CLVD) components (Figure S1 in Supporting Information S1).Such types of earthquakes at a shallow depth in volcanic or geothermal environments are often called vertical-CLVD earthquakes (e.g., Sandanbata, Kanamori, et al., 2021;Shuler, Nettles, & Ekström, 2013), which can be categorized into two types: vertical-T CLVD earthquakes with a nearly vertical tension and vertical-P CLVD earthquakes with a nearly vertical pressure axis.In recent caldera studies, vertical-T earthquakes were observed in caldera inflation phases (Bell et al., 2021;Glastonbury-Southern et al., 2022;Jónsson, 2009;Sandanbata, Kanamori, et al., 2021;Sandanbata, Watada, et al., 2021), whereas vertical-P earthquakes coincided with caldera collapse and formation (Gudmundsson et al., 2016;Lai et al., 2021;Michon et al., 2007;Riel et al., 2015;Rodríguez-Cardozo et al., 2021).The earthquakes near Kita-Ioto caldera fall into the vertical-T type, implying their association with the caldera inflation. 10.1029/2023JB027917 3 of 23 Yet, the mechanisms of shallow vertical-CLVD earthquakes are often indistinguishable only from the seismic characters, due to weak constraint on parts of moment tensor components (M rθ and M rϕ ) (Kanamori & Given, 1981;Sandanbata, Kanamori, et al., 2021) and a tradeoff between the vertical-CLVD and isotropic components (Kawakatsu, 1996).These ambiguities leave room for different interpretations for the earthquake mechanism, such as fault slips in calderas, deformation of a magma reservoir, or volume change due to heated fluid injection, as previously proposed for similar vertical-CLVD earthquakes (Shuler, Ekström, & Nettles, 2013, and references therein).
Following the earthquake that occurred at 13:10 on 12 June 2008 (UTC), a tsunami-like signal was recorded by an ocean-bottom-pressure (OBP) gauge with a sampling interval of 15 s of the station 52404, ∼1,000 km away from the caldera, of Deep-ocean Assessment and Reporting of Tsunamis (DART) system (Bernard & Meinig, 2011) (Figure 1a). Figure 1d shows the OBP data, which we obtain by removing the tidal component from and by applying the bandpass (2-10 mHz) Butterworth filter to the raw record.The OBP data demonstrates that clear oscillations with the maximum pressure of ∼2 mm H 2 O started ∼5,000 s after the earthquake origin time.Our calculation using the Geoware Tsunami Travel Time software (Geoware, 2011) estimates that the tsunami would  (Geoware, 2011).Solid gray line represents the data length for calculating the root-mean-square amplitudes (Equation 15).This waveform data is obtained by removing the tidal component from and by applying the bandpass (2-10 mHz) Butterworth filter to the raw OBP data for 12,000 s after the earthquake origin time.Note that oscillations of OBP changes with a few mm H 2 O are recorded after the estimated arrival time, indicating tsunami signals.

Methodology
In this section, we describe the methodology to construct a 3-D mechanical model of trapdoor faulting and to apply it to the tsunami data of the 2008 Kita-Ioto caldera earthquake.Through the application, we attempt to reproduce the tsunami data and estimate the sub-caldera magma overpressure that drove the tsunamigenic earthquake.

Mechanical Model of Trapdoor Faulting
We consider the 3-D half-space elastic medium of the host rock with an intra-caldera ring fault and a horizontal crack filled with magma (Figure 2).The ring fault and the horizontal crack are discretized into small triangular meshes, or sub-faults and sub-crack (with N F and N C meshes), respectively.The crack is assumed to have a finite inner volume and to be filled with compressible magma.Note that we do not consider viscoelasticity or heterogeneous rheology of the host rock, as the limitations are discussed later in Section 6.5.3.
We assume that trapdoor faulting is driven by magma overpressure in the crack, as follows; before trapdoor faulting, continuous magma input into the crack gradually increases the inner pressure and volume, and causes elastic stress in the host rock, accumulating shear stress on the ring fault; when the shear stress on the fault overcomes its strength, trapdoor faulting takes place.In the following, we model trapdoor faulting as a dislocation model that combines sudden and interactive processes of dip-slip on the fault with stress drop, deformation (vertical opening/closure) of the crack with volume change, and pressure change of the magma in the crack.Note that, some previous studies used the terminology of trapdoor faulting to refer to only the fault part (e.g., Amelung et al., 2000), while we consider it as the composite process involving both the fault and the magma-filled crack.

Pre-Seismic Elastic Stress in the Host Rock
As a reference state, we consider that the magma pressure p 0 in the crack is in equilibrium with the background stress   0  in the host rock due to the lithostatic and seawater loading, and that the background differential stress as zero.If we take the stress in the host rock as positive when it is compression, the background stress at an arbitrary position in the reference state is expressed as: where ρ h and z are the host rock density and the arbitrary depth in the host rock, respectively, ρ s and H are the seawater density and the thickness of the overlying seawater layer, respectively, g is the gravitational acceleration, and δ ij is the Kronecker's delta.The magma pressure in the reference state is expressed as follows: where z 0 is the depth of the horizontal crack, respectively.
We assume that long-term magma input into the crack increases the magma overpressure and opens the crack vertically, and that the resultant crack deformation changes the stress in the host rock.Thus, the shear stress is accumulated on the fault, which eventually causes trapdoor faulting.Magma pressure in the pre-seismic state, just before trapdoor faulting, is assumed to be spatially uniform within the crack and expressed as p = p 0 + p e , where p e is the pre-seismic magma overpressure.If we denote the spatial distribution of the crack opening in the pre-seismic state as    , the equilibrium relationship between the normal stress on the surfaces of sub-cracks and the inner magma pressure reduces to: where    is the N C × 1 column vector of the pre-seismic normal stress on sub-cracks, P is the interaction matrix, with a size of N C × N C , that map the tensile opening of sub-cracks into the normal stress on sub-cracks, and    is the N C × 1 column vector of ones.The distribution of the crack opening in the pre-seismic state    can be obtained from the second equality of Equation 3.Then, the pre-seismic shear stress along the dip direction on the surfaces of sub-faults (denoted as    ) created by the magma overpressure p e can be expressed as: where Q is the interaction matrix, with a size of N F × N C , that maps the tensile opening of sub-cracks into the shear stress on sub-faults.With Equation 3, Equation 4 can be rewritten as: The part in the bracket,   −1   , represents the shear stress on the surfaces of sub-faults caused by the crack opening due to unit magma overpressure.If we denote it as    , Equation 5 can be rewritten as: (6)

Occurrence of Trapdoor Faulting
Trapdoor faulting is caused by sudden stress drop of the shear stress accumulated on the fault.The motion involves dip-slip of the fault, and deformation (opening/closure) of the crack.To determine the motion of trapdoor faulting, we here derive two boundary conditions on the surfaces of the ring fault and the horizontal crack. 10.1029/2023JB027917 6 of 23 Assuming that the shear stress along the dip direction on the fault decreases by a stress drop ratio α due to trapdoor faulting, the boundary condition on the surface of the fault can be expressed as: where  Δ is the N F × 1 column vector of the shear stress change on sub-faults during trapdoor faulting.Q and R, with sizes of N F × N C and N F × N F , map dip-slip of sub-faults into the normal stress on sub-crack and the shear stress on sub-faults, respectively (Q is the same as that in Equation 4).
Sudden stress change in the host rock due to dip-slip of the fault interactively accompanies deformation (opening/ closure) of the crack, and the resultant normal stress change on the crack induces horizontal movement of the inner magma.For simplicity, we assume that the magma movement finishes and the magma pressure becomes spatially uniform in the crack quickly.Under this simplification, the boundary condition on the surface of the horizontal crack is derived from the equilibrium relationship between the normal stress on sub-cracks and the inner magma pressure, as follows: where  Δ and Δp are the N C × 1 column vector of the normal stress change on sub-cracks and the scalar of the magma pressure change during trapdoor faulting, respectively.P and U are the interaction matrices, with sizes of N C × N C and N C × N F , that map the tensile opening of sub-cracks into the normal stress on sub-cracks and into the shear stress on sub-faults, respectively (P is the same as that in Equation 3).
The magma pressure change Δp during trapdoor faulting can be related to the crack volume change ∆V through the mass conservation law, as follows: where ∆m is the magma influx and β m is the compressibility of magma.Since previously observed trapdoor faulting occurred within less than ∼10 s (Geist et al., 2008;Sandanbata et al., 2022Sandanbata et al., , 2023)), we can disregard magma mass influx during trapdoor faulting to reduce Equation 9 to: where   is the N C × 1 column vector of the areas of sub-cracks.
By substituting Equations 6 and 10 into Equations 7 and 8, respectively, we obtain the following equations: Equation 11 can be rewritten by: where Equations 12 and 13 represent N C + N F equations with N C + N F unknown values (   ,   ), if we priorly assume the pre-seismic magma overpressure p e , the stress drop ratio α, the source geometry determining the interaction matrices, and the parameters β m and V 0 .In this study, the source geometry and the parameters are assumed as described in Section 4.2.Also, the stress drop ratio is simply assumed as α = 1; in other words, the pre-seismic shear stress on the fault completely vanishes to zero due to trapdoor faulting.In this case, Equation 12 is reduced to: By solving Equation 14with Equation 13for (   ,   ), we can determine the motion of trapdoor faulting generated by pre-seismic magma overpressure p e .Also, we can estimate the co-seismic changes of magma pressure and crack volume due to trapdoor faulting by substituting   into Equation 10, and the stress drop by substituting   into Equation 7.

Model Setting
The source geometry employed for main results is shown in Figure 2. A partial ring fault is along an ellipse with a size of 3.6 km × 2.6 km on seafloor; the center is at (141.228°E, 25.4575°N), and its major axis is oriented N60°E.The fault is on the NW side of Kita-Ioto caldera with an arc length of 90° and dips inwardly with a dip angle of 83°; this fault setting on the NW side is based on our moment tensor analysis that suggests a ring fault orientated in the NE-SW direction (see Text S1 in Supporting Information S1, for details).The fault's down-dip end connects to a horizontal crack at a depth of 2 km.The crack is elliptical in shape, 15% larger than the size of an ellipse traced along the fault's down-dip end.After discretizing the source geometry into sub-faults and sub-cracks, the four interaction matrices (P, Q, R, and U) between sub-faults and sub-cracks are computed by the triangular dislocation (TD) method (Nikkhoo & Walter, 2015), when we assume the Poisson's ratio of 0.25 and the Lame's constants λ and μ of 5 GPa.
The product V 0 β m controls how the magma-filled crack responds to stress perturbation by faulting, as explained by Zheng et al. (2022).For main results, we assume the crack volume V 0 and the magma compressibility β m as 1.5 × 10 10 m 3 (corresponding to a crack thickness of ∼500 m) and 1.0 × 10 −10 Pa −1 (from a typical value for degassed basaltic magma (e.g., Kilbride et al., 2016)), respectively, thereby, V 0 β m = 1.5 m 3 /Pa.This product value is similar to Zheng et al. (2022)'s estimates for a magma reservoir of Sierra Negra caldera.
We emphasize that the model setting above, which is used to obtain the main results shown in Section 5, is just an assumption.The location of the ring fault cannot be constrained from the earthquake information of the GCMT catalog, since the solutions can contain horizontal location errors up to ∼40 km (Hjörleifsdóttir & Ekström, 2010;Pritchard et al., 2006).The bathymetry data containing several cones found on the NW side of the caldera floor (Figure 1c) may suggest the existence of a fault system, given such structures often formed over a sub-caldera ring fault (e.g., Cole et al., 2005), but this is not decisive information.Also, we have no constraint on the magma compressibility and the reservoir depth.In Section 6.1, we will test the sensitivity to those possible uncertainties in the model setting.

Constraint From the Tsunami Data of the 2008 Kita-Ioto Caldera Earthquake
We apply the mechanical model of trapdoor faulting to the tsunami data of the 2008 Kita-Ioto caldera earthquake.Utilizing the linear relationship between (   ,   ) and p e through Equation 14, we estimate the pre-seismic magma overpressure p e causing the earthquake by constraining the magnitude of trapdoor faulting from the tsunami data.
For estimation of p e , we prepare a model of trapdoor faulting due to unit pre-seismic magma overpressure p e = 1 Pa, which we call unit-overpressure model, and then simulate a tsunami OBP waveform at the station 52404 from the model (see the methodology in Section 4.4).We denote the synthetic waveform as    and consider it as the tsunami OBP amplitude due to unit overpressure, whose unit is (mm H 2 O/Pa).Because of the linearity of the tsunami propagation problem we employ, the amplitude of tsunami waveform is linearly related to the magnitude of trapdoor faulting, and thereby to the pre-seismic magma overpressure p e through Equation 14.Therefore, the synthetic tsunami waveform from trapdoor faulting due to an arbitrary p e can be expressed as O/ Pa]), respectively.The time window for calculating the RMS amplitudes is set as it includes major oscillations in earlier parts of the observed waveform (see the gray line in Figure 1d).

Tsunami Waveform Simulation
A tsunami waveform from the unit-overpressure model    is synthesized as follows.Assuming (   ,   ) of the unit-overpressure model, we compute the vertical seafloor displacement by the TD method, and convert it to vertical sea-surface displacement by applying the Kajiura filter (Kajiura, 1963).We then simulate the tsunami propagation over the time of 12,000 s from the sea-surface displacement over Kita-Ioto caldera, generated instantly at the earthquake origin time, by solving the linear Boussinesq equations (Peregrine, 1972) in the finite-difference scheme of the JAGURS code (Baba et al., 2015).The simulation is done with a two-layer nested bathymetric grid system, composed of a broad-region layer with a grid size of 18 arcsec (∼555 m) derived from JTOPO30 data, and a caldera-vicinity-region layer with a grid size of 6 arcsec (∼185 m), obtained by combining data from M7000 series and JTOPO30.The computation time step is 0.5 s, as the Courant-Friedrichs-Lewy (CFL) condition is satisfied.The outputted 2-D maps of sea-surface wave heights, every 5 s, are converted into maps of OBP perturbation by incorporating the reduction of tsunami pressure perturbation with increasing water depth (e.g., Chikasada, 2019).The synthetic waveform of OBP perturbation at the station 52404 is obtained from the OBP maps.
The linear Boussinesq equations employed above do not include the effects of the elastic Earth, the seawater compressibility, and the gravitational potential change, and are less accurate for computation of higher-frequency waves due to the error of dispersion approximation (Sandanbata, Watada, et al., 2021).Hence, we apply a phase correction method for short-period tsunamis (Sandanbata, Watada, et al., 2021) to improve the synthetic waveform accuracy by incorporating the effects (i.e., elastic Earth, compressible seawater, and gravitation potential change) and by correcting the approximation error.

Source Model of the 2008 Kita-Ioto Caldera Earthquake
Under the model setting explained in Section 4.2 (Figure 2), we obtain a trapdoor faulting model for the 2008 Kita-Ioto caldera earthquake that explains the OBP tsunami data (Figure 3).The pre-seismic magma overpressure p e constrained from the OBP tsunami data is 11.8 MPa.Figures 3b and 3c show the spatial distributions of the ring-fault slip   and the crack opening/closing   during trapdoor faulting.Large reverse slip at maximum of 8.9 m is on the ring fault, near which the inner crack opens by 5.5 m at maximum and the outer closes by 2.7 m.In the SE area, the crack closes broadly with a maximum value of 0.86 m.In total, the crack volume increases by ΔV = 0.0030 km 3 .The co-seismic magma pressure change ∆p is −1.97 MPa during trapdoor faulting, meaning that the magma overpressure drops by 16.7% relative to the pre-seismic state and makes additional storage for magma.The response of the magmatic system to faulting may have postponed eruption timing; on the other hand, post-seismic magma overpressure is estimated to remain at a high level (∼9.8 MPa) even after trapdoor faulting.
The obtained trapdoor faulting model is predicted to cause large asymmetric caldera-floor uplift, thereby generating a tsunami efficiently.The large seafloor displacement is concentrated near the fault, with the maximum uplift of as large as 5.6 m and outer subsidence of 2.8 m (Figure 3d).The sea surface displacement is smoothed by the low-pass effect of seawater, resulting in seawater uplift of 3.6 m within the caldera rim with the exterior subsidence of 1.1 m (Figure 3e). Figure 3f compares the synthetic tsunami waveform from the model with the OBP tsunami signal recorded at the station 52404, which demonstrates good waveform agreement, including later phases that are not used for the amplitude fitting.In addition, the spectrogram analysis confirms quite similar tsunami travel times and dispersive properties of the synthetic and observed waveforms (Figures 3g and 3h).These results support the reasonability of our mechanical model for the 2008 Kita-Ioto caldera earthquake.

Pre-Seismic State Just Before Trapdoor Faulting
From the mechanical model, we consider how trapdoor faulting is caused by the inflated crack.In the pre-seismic state just before trapdoor faulting, the crack has inflated with vertical opening    of 12.1 m at maximum due to the pre-seismic magma overpressure p e (Figure 4a).The inner volume has been increased by 0.21 km 3 relative to that in the reference state.This pre-seismic crack opening generates the shear stress on the fault    , which takes  15).(g, h) Spectrograms of the (g) synthetic and (h) observed waveforms.In (f-h), black dashed line represents the tsunami arrival time.its maximum value of 11.6 MPa (Figure 4b); this value corresponds to the stress drop during trapdoor faulting, because we assume that the stress totally vanishes co-seismically.
In a simple earthquake paradigm of the stick-slip motion, which assumes that slip occurs when the shear stress overcomes the static frictional stress (e.g., p. 14 of Udías et al. (2014)), the fault requires friction to remain stationarity until faulting occurrence.The total normal stress on the fault    0 is the sum of the effects of the crack opening     (Figure 4c) and the lithostatic and seawater loading    lit +   sea , as shown in Figure 4d (see the caption).By taking a ratio of the area-averaged values of    and     , the static friction coefficient on the ring fault can be estimated as 0.31.The frictional fault system may enable the caldera system to accommodate the high magma overpressure without fault slip until trapdoor faulting.Note that, however, sophisticated modeling approaches including realistic fault friction law will be needed for investigation of the dynamic initiation process.

Deformation and Elastic Stress Change in the Host Rock
Our model demonstrates how trapdoor faulting deforms the host rock and changes its elastic stress.With the model outputs, we compute the displacement, stress and strain fields in the host rock along an SE-NW profile across the caldera (see the dashed line in Figure 3c) by the TD method; the pre-seismic state is from    , the co-seismic change is from (   ,   ), and the post-seismic state is the sum of the pre-seismic state and the co-seismic change.We also calculate the shear-strain energy from the stress and strain fields (e.g., Saito et al., 2018).When we denote the stress tensors in the host rock as: where   ′  is the deviatoric components, the shear-strain energy density W in the elastic medium can be expressed as: Note that the shear-strain energy density is zero in the reference state (p = p 0 ), where the deviatoric stress is assumed to be zero.Using Equation 17, the shear-strain energy density in the pre-and post-seismic states, W pre and W post , can be calculated with the deviatoric shear stress.The co-seismic change in the shear-strain energy density is obtained by: Figures 5a-5c show displacement in the host rock along the SE-NW In the pre-seismic state (Figure 5a), since the fault accommodates no slip, the host rock deforms purely elastically from the reference state due to the opening crack and causes large uplift of the caldera surface by 8.8 m at maximum at the caldera center.During trapdoor faulting, the co-seismic displacement is concentrated along the fault (Figure 5b).The inner caldera block uplifts by 5.7 m at maximum, while the outer host rock moves downward by 3.2 m.The fault motion accompanies crack opening beneath the NW side of the caldera block, whereas slight downward motion is seen in the SE part of the caldera block, which can be attributed to elastic response to magma depressurization.Figure 5c shows the displacement in the post-seismic state, where the upward displacement is confined within the caldera block, with cumulative uplift of 9.9 m at maximum, from the center to near the fault, while notable deformation is not found outside the fault.As shown in Figure 5d, the pre-seismic seafloor displacement takes its uplift peak in the center, while after trapdoor faulting the seafloor becomes almost flat on the NW side near the fault.This indicates that if we take a long term including the pre-seismic inflation and trapdoor faulting, the caldera causes a block-like motion with a clear boundary cut by the fault.
In terms of the stress and the shear-strain energy, trapdoor faulting can be considered as a process that releases the shear-strain energy accumulated in the host rock.Figures 5e-5g show the shear-strain energy density with the principal axes of the stress field in the host rock along the same SE-NW profile.In the pre-seismic state, the shear-strain energy density is concentrated around the crack edge, or near the fault (Figure 5e).The plunge of the maximum compressional stress near the fault ranges from ∼50° in the middle of the fault, which preferably induces a reverse slip on a steeply dipping fault.During trapdoor faulting (Figure 5f), the shear-strain energy density near the fault on the NW side dramatically decreases.Eventually, in the post-seismic state (Figure 5g), the shear-strain energy density almost vanishes near the fault.Note that, on the other hand, the shear-strain density is only slightly reduced on the SE side in response to co-seismic magma depressurization and remains high even after trapdoor faulting.We speculate that the remaining shear-strain energy may be released by other causes, such as aseismic fault slip, a subsequent trapdoor faulting, or viscoelastic deformation of the host rock, which are not incorporated in our modeling; we will discuss the limitations of our models in Section 6.5.

Model Uncertainties
Our source model has been constructed in the model setting as described in Section 4.2.However, since a single tsunami waveform data at a distant location has low sensitivity to the source details, we do not have enough data to constrain the sub-surface structure and magma property.Hence, our model outputs vary depending on how the model setting is assumed priorly.

Depth of a Horizontal Crack
The depth of a horizontal crack, or a magma reservoir, significantly influences our pre-seismic magma overpressure estimation.When a deeper crack is assumed at a depth of 4 km below seafloor (Figure 6), the estimated magma overpressure p e is 22.26 MPa, almost a factor of two larger compared to our main result assuming a depth of 2 km (Figure 3).The obtained model with a 4-km deep crack explains the tsunami data well, even better than that with a 2-km deep crack (compare waveforms and spectrograms in Figures 3f-3h and 6f-6h), implying preference of the deeper crack model.When a crack is located deeper in the crust, the magnitude of the crack opening per unit magma overpressure becomes smaller because it is farther from the free-surface seafloor (Fukao et al., 2018).This lowers the shear stress on the fault generated per unit magma overpressure, and thereby larger pre-seismic magma overpressure is required to cause a similar-sized earthquake and tsunami.Despite the large difference in pre-seismic magma overpressure, the estimated co-seismic parameters for the 2008 earthquake, such as magnitudes of fault slips, crack deformation, and changes in magma pressure and crack volume, do not change largely.

Arc Length of a Ring Fault
The arc length of a ring fault is also an important factor affecting our modeling.As shown in Figure 7, when we assume a ring fault with an arc length of 180°, or a half-ring fault, on the NW side, pre-seismic magma overpressure p e is estimated as 4.84 MPa, less than half of the value from our main results assuming an arc length of 90° (Figure 3).This large difference can be attributed to two main causes.First, the average fault slip amount is known to be proportional to the fault length when the stress drop is identical (Eshelby, 1957); therefore, a longer ring fault causes large slip efficiently, compared to that on a shorter arc length.Additionally, trapdoor faulting 10.1029/2023JB027917 13 of 23 with a longer fault uplifts larger volume of seawater over a broader area (compare Figures and 3e), making its tsunami generation efficiency higher.
Although smaller magma overpressure (p e = 4.84 MPa) is estimated in the case with a ring-fault arc angle of 180°, we emphasize that the co-seismic magma pressure change ∆p is as large as −1.99 MPa.The magma overpressure efficiently drops by 41.1% from the pre-seismic state, in contrast to the ratio of only 16.7% in the case of an arc length of 90° (see Section 5.1).The difference arises from the fact that the fault slip along a longer segment induces the crack opening in a broader area and increases the inner volume more, resulting in more efficient pressure relief.The two models with different ring-fault arc lengths produce very similar tsunami waveforms at 10.1029/2023JB027917 14 of 23 the station 52404 (compare Figures 7f and 3f), indicating the difficulty in distinguishing the arc length from our data set.However, these results provide an important insight that the magma pressure drop ratio strongly depends on a fault length ruptured during trapdoor faulting, suggesting importance to investigate the intra-caldera fault geometry for robust quantification of magma pressure change due to faulting.

Other Uncertainties
We discuss on effects of the product V 0 β m , which controls how the magma-filled crack responds to stress perturbation by faulting.The effects in extreme cases are discussed by Zheng et al. (2022); when V 0 β m → 0, the crack involves no total volume change (ΔV → 0), while a magnitude of magma pressure drop becomes the largest; on the other hand, when V 0 β m → ∞, the net volume change of the crack is at maximum, while no pressure change occurs (Δp → 0).In previous studies of the 2018 Kilauea caldera collapse and eruption sequence, the estimated product ranges 1.3-5.5 m 3 /Pa (Anderson et al., 2019;Segall & Anderson, 2021).We assumed V 0 β m = 1.5 m 3 / Pa for our main results, which is close to the lower end of the range.To examine the model variations, we try the source modeling alternatively by assuming V 0 β m = 6.0 m 3 /Pa, near the upper limit of the range estimated in the case of Kilauea.For the larger V 0 β m , the area of the crack opening becomes broader, while a magnitude of the closure on the other side becomes smaller (Figures S4a-S4c in Supporting Information S1; compare them with Figures 3a-3c).The sea-surface displacement is thereby broader (Figure S4e in Supporting Information S1), exciting long-period tsunamis more efficiently that arrives as earlier waveform phases used for the amplitude fitting (Figure S4f in Supporting Information S1).Thus, in this case, our estimation of the pre-seismic magma overpressure, p e = 9.11 MPa, becomes slightly smaller than the main result (p e = 11.8MPa); on the other hand, we estimate smaller magma pressure drop (Δp = −1.27MPa) and a larger crack volume increase (ΔV = 0.0076 km 3 ).These suggest that if we take a plausible range of V 0 β m , variations of our estimations are insignificant.
It is uncertain on which side of the caldera the ruptured fault is located.Based on our moment tensor analysis (Text S1 in Supporting Information S1), the fault ruptured during the 2008 earthquake can be estimated to be oriented manly in the NE-SW direction, allowing us to assume two different fault locations, either of the NW or SE sides of the caldera; for our main results, we chose the model with a fault on the NW side.Here, we alternatively assume a fault on the SE side to obtain another source model, and consequently estimate the pre-seismic magma overpressure p e as 15.36 MPa (Figure S5 in Supporting Information S1).Despite the fault location difference, the tsunami data is explained well by the model with a SE-sided fault (Figure S5f in Supporting Information S1).The change of the estimated magma overpressure can be attributed to effects of tsunami directivity and complex bathymetry in the source region on the wave amplitude of a tsunami arriving at the station.Thus, our limited data set is not sufficient to determine well the fault location, but the uncertainty in fault location influences our estimations insignificantly.

Comparison With Previous Studies
Our quantification of pre-seismic magma overpressure before trapdoor faulting in Kita-Ioto caldera (p e = 4-22 MPa) is of the same order of magnitude as those estimated geodetically for the subaerial caldera of Sierra Negra.Gregg et al. (2018) applied a thermomechanical finite element method (FEM) model to long-term geodetic data and estimated that magma overpressure of ∼10 MPa in the sill-like reservoir induced a trapdoor faulting event that occurred ∼3 hr before the eruption starting on 22 October 2005.Another trapdoor faulting event on 25 June 2018 (M w 5.4) also preceded the 2018 eruption of Sierra Negra by 10 hours; Gregg et al. (2022) employed the thermomechanical FEM approach to the long-term deformation and suggested that a similar magma overpressure <∼15 MPa had been accumulated to cause the failure of the trapdoor fault system.Zheng et al. (2022), on the other hand, quantified co-seismic magma pressure change by trapdoor faulting with an m b 4.6 earthquake on 16 April 2005.By modeling the interaction between the intra-caldera fault system and the sill-like reservoir, Zheng et al. geodetically estimated the trapdoor faulting event with a maximum fault slip of 2.1 m reduced magma overpressure by 0.8 MPa; the slightly smaller pressure change, relative to our estimation (|Δp| = 1-3 MPa) for the 2008 Kita-Ioto earthquake, may be explained by the discrepancies in the earthquake size or the length of a ruptured fault.Sandanbata et al. (2023) compiled the seismic magnitude and the maximum fault slip of trapdoor faulting events and demonstrated their atypical earthquake scaling relationship; in other words, trapdoor faulting accompanies larger fault slip by an order of magnitude than those for similar-sized tectonic earthquakes.Source models presented in this study for the 2008 Kita-Ioto caldera earthquake also accommodate large fault slip ranging 5-10 m at maximum, which are clearly larger than those empirically predicted for M w 5.3 tectonic earthquakes; for example, the empirical maximum slip for M w 5.3 earthquake is only ∼0.1 m, following Wells and Coppersmith (1994).This indicates the efficiency of intra-caldera fault systems in causing large slip, possibly due to their interaction with magma reservoirs and shallow source depth (Sandanbata et al., 2022).

Long-Period Seismic Waveforms
For validation from a different perspective, we consider long-period seismic excitation by the mechanical source model that we have obtained based on the tsunami data.For this analysis, we follow the methodology used in Sandanbata et al. (2022Sandanbata et al. ( , 2023)), as the detailed procedures are described in Text S2.We here briefly summarize the method.We first approximate the trapdoor faulting model (Figure 3a) as a point-source moment tensor M T by summing up partial moment tensors of the ring fault M F and the horizontal crack M C (Figures 8a-8c).We then compute long-period (80-200 s) seismic waveforms from the moment tensor M T by using the W-phase package (Duputel et al., 2012;Hayes et al., 2009;Kanamori & Rivera, 2008) and compare the synthetic waveforms with broad-band seismic data from F-net and global seismic networks.In Figure 8d and Figure S6 in Supporting Information S1, we show synthetic seismic waveforms from the moment tensor (Figure 8a), which reproduce well the observed seismograms.This supports that our trapdoor faulting model is plausible in terms of seismic excitation, as well as tsunami generation.
We note that the theoretical moment tensor obtained from our model (Figure 8a) is different from the GCMT solution; our theoretical solution has a seismic magnitude (M w 5.6) and is characterized by large double-couple and isotropic components, while the GCMT solution is with a smaller magnitude M w 5.3 and a dominant vertical-T CLVD component (Figure 1c).The difference can be explained by very inefficient excitation of long-period seismic waves by specific types of shallow earthquake sources (Fukao et al., 2018;Sandanbata, Kanamori, et al., 2021).As demonstrated in Figure S7 in Supporting Information S1, major parts of the long-period seismic waves of the trapdoor faulting model arise from limited moment tensor components that constitute a vertical-T CLVD moment tensor, equivalent to M w 5.2 (Figure S7b in Supporting Information S1), whereas the contribution by the horizontal crack M T , and M rθ and M rϕ components in M F are negligibly small.Hence, the GCMT solution determined with the long-period seismic waveforms becomes a vertical-T CLVD moment tensor with a smaller magnitude than that of the theoretical moment tensor of our model.The gap between theoretical and observed moment tensors of trapdoor faulting is discussed in more detail by Sandanbata et al. (2022).

Tsunami Generation by Other Kita-Ioto Caldera Earthquakes
We have conducted a survey of OBP data from the station 52404 to determine if there were any tsunami signals following the other Kita-Ioto caldera earthquakes (Figure S1 in Supporting Information S1), apart from that in 2008.Available data was found only for the event on 15 December 2015 (Figure 9a), for which a clear tsunami signal was recorded in the OBP data with a 15-s sampling interval (Figure 9b).On the other hand, we were unable to obtain OBP data to confirm tsunami signals from the earthquakes in 1992, 2010, 2017, and 2019.The station 52404 had not been deployed yet as of the 1992 event.For the other events, the bottom pressure recorders have been lost, preventing our access to its 15-s sampling-interval data.Although low-sampling data (15-min interval) sent via a satellite transfer are available, they are not useful for confirming tsunami signals with dominant periods of 100-500 s.
We further investigate the tsunami signal from the 2015 earthquake in comparison with that from the 2008 event.Note that the station location (20.7722°N, 132.3375°E) as of 2008 had shifted about 20 km northward to a new location (20.9478°N, 132.3122°E) as of 2015.To examine the similarity between the two earthquake events, we simulate a tsunami waveform at the station location as of the 2015 event from a model similar to that of the 2008 event.We assume the model setting with a deeper crack at a depth of 4 km, based on that presented in Section 6.1.1 (Figure 7).Since the GCMT catalog reports a smaller seismic moment for the 2015 event Although the observed tsunami waveforms from the two earthquakes look different (compare the waveforms in Figures 7f and 9b), the trapdoor faulting model, based on the tsunami data form the 2008 earthquake, also explains that from the 2015 earthquake overall (Figure 9), simply by changing the station location.The nonnegligible waveform difference at the two nearby locations can be attributed to the focusing/defocusing effect by complex bathymetry along the path (Figure S8 in Supporting Information S1; see the figure caption for details).This suggests that the 2015 earthquake was caused by trapdoor faulting, in a similar way to the 2008 earthquake.The similarity is further supported by our moment tensor analysis (see Text S1 in Supporting Information S1).Thus, we confirmed tsunami signals from both of the two events.Therefore, we propose that the quasi-regularly repeating earthquakes with similar magnitudes and vertical-CLVD characters reflect the recurrence of trapdoor faulting in Kita-Ioto caldera, as observed in the three calderas of Sierra Negra, Sumisu, and Curtis, where trapdoor fault ing events have recurred (Bell et al., 2021;Jónsson, 2009;Sandanbata et al., 2022Sandanbata et al., , 2023)). 10.1029/2023JB027917 17 of 23

Limitations of Our Mechanical Trapdoor Faulting Model
Our mechanical model of trapdoor faulting has been developed under some simplifications to focus on the essential mechanics.In this subsection we discuss some factors simplified or ignored in our model, which may influence our results.

Stress Drop Ratio
The stress drop ratio during earthquakes has been controversial in general.Some studies reported complete or near-complete stress drop during tectonic earthquakes (Hasegawa et al., 2011;Ross et al., 2017), while the stress drop ratio can be partial and vary from earthquake to earthquake (Hardebeck & Okada, 2018).For intra-caldera earthquakes, several recent studies estimated stress drop during caldera collapses (Moyer et al., 2020;T. A. Wang et al., 2022), but our knowledge on the stress drop ratio in calderas is poor and the ratio may vary from caldera to caldera.
We have avoided the problem by simply assuming the complete stress drop as an extreme case (Equation 14, obtained by assuming α = 1 in Equation 12); this assumption can influence our estimation of the pre-seismic magma overpressure p e .Because   and   are determined by the stress drop on the fault, not directly by pre-seismic magma overpressure (Equation 3), if a partial stress drop ratio α (0 < α < 1) is instead assumed in Equation 12, the trapdoor faulting size due to the same pre-seismic magma overpressure becomes smaller proportionally to α, and the tsunami amplitude does.In this case, larger magma overpressure by a factor of 1/α is required to explain the observed tsunami amplitude.Hence, the complete stress drop assumption provides lower-limit estimation of pre-seismic magma overpressure in the model setting.On the other hand, estimations of co-seismic parameters, such as fault slip   and crack opening   , and changes of magma pressure Δp and crack volume ΔV, do not change regardless of our assumption of the stress drop ratio α, since they are constrained form the tsunami amplitude.

Pre-Slips and Earthquake Cycles
We have attributed the shear stress that generates trapdoor faulting to an inflating crack alone and neglected other factors that may also cause the stress on the fault.First, different segments of the intra-caldera ring fault may have caused microseismic or aseismic slips prior to the occurrence of M w ∼ 5 trapdoor faulting.In Sierra Negra caldera, high microseismicity was observed along the western segment of the intra-caldera fault, leading to trapdoor faulting on the southern segment before eruption (Bell et al., 2021;Shreve & Delgado, 2023).Similarly, during the 2018 eruption and summit caldera collapse sequence of Kilauea, large collapse events accompanying M w ∼ 5 earthquakes were located on the southeastern and northwestern sides of the summit caldera, while high microseismicity was found on other segments (Lai et al., 2021;Shelly & Thelen, 2019).T. A. Wang et al. (2023) further suggested non-negligible effects on large collapses of Kilauea by intra-caldera fault creep in the inter-collapse period.Such high microseismicity or creeping on other fault segments, adjacent to the ruptured segment of trapdoor faulting, may impose additional shear stress.
Additionally, the recurrency of trapdoor faulting can play an important role in the stress accumulation on the fault.Similar earthquakes have been repeated near Kita-Ioto caldera (Figure S1 in Supporting Information S1), strongly suggesting recurrence of trapdoor faulting, as supported by the tsunami signal from the 2015 earthquake (see Section 6.4).If a similar earthquake repeated on the same segment of the fault and the stress drop is only partial, the remaining stress may influence subsequent trapdoor faulting events.Also, assuming that the earthquakes occur on different segments of the ring fault, an event on a segment increases the shear stress on its adjacent segment.Thus, in the presence of additional shear stress by pre-slips or creeping on different segments or previous trapdoor faulting events, the ring fault may be ruptured by smaller pre-seismic magma overpressure.For better understanding of the physics of trapdoor faulting, further studies of the earthquake cycle in calderas are crucial.

Other Factors
Other factors simplified in our model, such as magma reservoir geometry, and viscoelastic and heterogeneous rheological properties of the host rock, may influence the mechanics of trapdoor faulting.While we have modeled a magma reservoir simply as an infinitely thin crack that lies horizontally, the reservoir should have a finite thickness and the geometry may not be flat, as estimated for that beneath Sierra Negra caldera (Gregg et al., 2022).The host rock has been also simplified as a homogeneous elastic medium, but the viscoelastic effects and thermal dependency of the rheological property may impact the deformation and stress and strain states in hot volcanic environments.For example, Newman et al. (2006) showed that the viscoelastic effect significantly reduces the estimated magma overpressure using surface deformation data at Long Valley caldera, compared to that based on a purely elastic model.The viscoelastic effect can be more important in the stress accumulation process, particularly during a long-term caldera inflation phase.Additionally, the temperature-dependency of the host-rock rheology is shown to have an impact on the stress accumulation process in the host rock, impacting on estimation of the timing of host-rock failures and eruption (Cabaniss et al., 2020;Zhan & Gregg, 2019).For further studies, it would be critical to incorporate these effects on the deformation and the stress-strain accumulation in the host rock, as done by previous studies employing the FEM modeling approach (e.g., Gregg et al., 2012;Le Mével et al., 2016;Zhan & Gregg, 2019).

Conclusions
We have presented a new mechanical model of trapdoor faulting that quantitatively links pre-seismic magma overpressure in a sill-like reservoir and the size of trapdoor faulting.We applied this model to a tsunami-generating submarine earthquake in 2008 around Kita-Ioto caldera, for quantifying the caldera's mechanical states.Our trapdoor faulting model explains well the tsunami signal recorded by a single distant ocean bottom pressure gauge, as well as regional long-period seismic waveforms.Although we acknowledge that other possible mechanisms (e.g., fluid-flow or volumetric-change source in magma reservoir) are not tested in this study, and that there is no direct observation of an active fault system in the caldera, our results suggest the plausibility of our hypothesis of the submarine trapdoor faulting in Kita-Ioto caldera.This is also supported by the similarity to trapdoor faulting events found recently in better-investigated submarine calderas (Sumisu and Curtis calderas).Repeating vertical-T CLVD earthquakes and another tsunami signal following the 2015 earthquake suggest the recurrence of trapdoor faulting in Kita-Ioto caldera.
Our mechanical models enable us to infer the pre-seismic magma overpressure beneath the submarine caldera, through quantification of the trapdoor faulting size.In an example case with a ring fault with an arc length of 90° and a horizontal crack at a depth of 2 km in the crust as the main model setting, we estimate that pre-seismic magma overpressure over ∼10 MPa causes the trapdoor faulting event, and that the co-seismic magma pressure drops by ∼15%.Yet, since uncertainty on the source geometry remains due to our limited data set, or a single tsunami record, these estimated values related to magma overpressure vary by a factor of half to twice, depending on model setting; the pre-seismic magma overpressure ranges approximately from 5 to 20 MPa, and the co-seismic overpressure drop ratio from 10% to 40%.For example, a longer ring fault with an arc angle of 180° requires less magma overpressure to generate the similar-sized tsunami but more effectively reduces the overpressure; on the other hand, larger magma overpressure is estimated when the source has a crack at a deeper depth of 4 km.The significant variations suggest that magmatic systems beneath calderas can be strongly influenced by source properties of trapdoor faulting.Therefore, it is critical to study trapdoor faulting in active calderas and its source properties, which would help us obtain more robust estimation of magma overpressure or stress states, providing rare opportunity to achieve comprehensive understanding of how inflating calderas behave in the ocean.

Figure 1 .
Figure 1.Vertical-T CLVD earthquakes near Kita-Ioto caldera.(a) Map of the southern ocean of Japan.Orange triangle represents the ocean-bottom-pressure (OBP) gauge of Deep-ocean Assessment and Reporting of Tsunamis (DART) 52404.(b) Map of the region near Kita-Ioto Island.(c)Bathymetry of the region near Kita-Ioto caldera, a submarine caldera with a size of 12 km × 8 km, near Kita-Ioto Island.Funka Asane is the summit of a cone structure within the caldera rim.Red circle represents the location of the 2008 Kita-Ioto earthquake with its moment tensor, whereas black circles represent locations of similar events; the earthquake information is from the Global Centroid Moment Tensor catalog(Ekström et al., 2012).The focal mechanism is shown as projections of the lower focal hemisphere, and the orientation of the best double-couple solution is shown by thin lines.(d) Tsunami waveform recorded at the OBP gauge of DART 52404.Dashed gray line represents the tsunami arrival time estimated using the Geoware Tsunami Travel Time software(Geoware, 2011).Solid gray line represents the data length for calculating the root-mean-square amplitudes (Equation15).This waveform data is obtained by removing the tidal component from and by applying the bandpass (2-10 mHz) Butterworth filter to the raw OBP data for 12,000 s after the earthquake origin time.Note that oscillations of OBP changes with a few mm H 2 O are recorded after the estimated arrival time, indicating tsunami signals.

Figure 2 .
Figure 2. A source structure for the mechanical model of trapdoor faulting viewed from top (left) and southeast (right).Gray lines are plotted every water depth of 200 m.

Figure 3 .
Figure 3. Mechanical trapdoor faulting model of the 2008 Kita-Ioto earthquake.(a) Mechanical model viewed from the southeast, represented by dip-slip of the ring fault   and vertical deformation of the crack   .Red color on the ring fault represents reverse slip, while red and blue colors on the horizontal crack represent vertical opening and closure, respectively.(b, c) Spatial distributions of (b) the ring fault and (c) the horizontal crack.In (b), the fault is viewed from the caldera center, and the azimuth from the caldera center to arbitrary point on the fault is measured clockwise from the midpoint of the fault.In (c), dashed line represents a profile shown in Figure 5. (d, e) Vertical displacement of seafloor (d) and sea surface (e) due to the model.Red and blue colors represent uplift and subsidence, respectively, with white lines plotted every 1.0 m.Black lines represent water depth every 100 m.(f) Comparison between a synthetic tsunami waveform from the model (red line) and the observed ocean-bottom-pressure waveform (blue line) at the station 52404.Solid gray line represents the data length for calculating the root-mean-square amplitudes (Equation15).(g, h) Spectrograms of the (g) synthetic and (h) observed waveforms.In (f-h), black dashed line represents the tsunami arrival time.

Figure 4 .
Figure 4. Pre-seismic state of the fault-crack system just before trapdoor faulting.(a) Distribution of the crack opening,    .(b) Critical shear stress along dip-slip direction on the ring fault,    .(c) Normal stress on the ring fault induced by the critically opening crack,     .In (b, c), blue and red colors represent compressive and extensional normal stress, respectively.(d) Total normal stress on the ring fault,    0 =    +   lit +   sea ; here,    lit = ℎ, where ρ h , z, and g are the host rock density (2,600 kg/m 3 ), the depth of each mesh, and the gravitational acceleration (9.81 m/s 2 ), respectively, and    sea =  , where ρ s and H are the seawater density and the approximated thickness of the overlying seawater layer (1,020 kg/m 3 and 400 m), respectively.

Figure 5 .
Figure 5. Displacement and shear-strain energy density in the host rock, along a SE-NW profile shown in Figure 3c.(a-c) Displacement, relative to the reference state (p = p 0 ): (a) the pre-seismic state just before trapdoor faulting, (b) the co-seismic change due to trapdoor faulting, and (c) the post-seismic state after trapdoor faulting.(d) Vertical seafloor displacement in each state shown in panels (a-c).(e-g) Shear-strain energy density W: (e) the pre-seismic state, (f) the co-seismic change, and (g) the post-seismic state.Color represents shear-strain energy density, and bars represent principal axes of compression projected on the profile, whose thickness reflects half the differential stress change (σ 1 − σ 3 )/2, where σ 1 and σ 3 are the maximum and minimum stress, respectively.

Figure 6 .
Figure 6.Same as Figure 3, but for a model with a horizontal crack at a depth of 4 km.See details in Section 6.1.1.

Figure 7 .
Figure 7. Same as Figure 3, but for a model with a longer ring fault of an arc angle of 180°.See details in Section 6.1.2.
10 18 Nm), we adjust the source model assuming a smaller pre-seismic overpressure of p e = 16.41MPa (=22.

Figure 8 .
Figure 8. Long-period (80-200 s) seismic waveform modeling.(a) Moment tensor of the model, composed of partial moment tensors of (b) the ring fault and (c) the horizontal crack.(d) Comparison between synthetic waveforms (red line) and the observation (black line) at representative stations.In inset figures, a large red circle and a blue star represent the station and the earthquake centroid, respectively.On the top of each panel, the network name, station name, record component, station azimuth, and epicentral distance are shown.Note that waveform comparisons in all the tested seismic records are shown in Figure S6 in Supporting Information S1.

Figure 9 .
Figure 9. Tsunami waveform data from the 2015 earthquake.(a) The Global Centroid Moment Tensor solution of the Kita-Ioto caldera earthquake on 15 December 2015.(b) Comparison between a synthetic tsunami waveform from a source model adjusted from the 2008 earthquake model (red line; see Section 6.4) and the observed ocean-bottom-pressure (OBP) waveform (blue line) at the station 52404.(c, d) Spectrograms of the synthetic waveform (c) and the OBP waveform (d).In (b-d), black dashed line represents the tsunami arrival time.Note that the location of the 52404 station as of the 2015 earthquake has been shifted by ∼20 km southward from the location as of the 2008 earthquake (see text and Figure S8 in Supporting Information S1).
and    are the root-mean-square (RMS) amplitudes of   and    (in units of [mm H 2 O] and [mm H 2 =  Â .Supposing that the tsunami signal from the 2008 earthquake recorded in the OBP data (denoted by   , we can estimate the pre-seismic magma overpressure p e from: