### Numerical model of landslide mass and tsunami

A generation of tsunami by submarine landslide is a complex process, and various numerical models have been developed to analyze the propagation of the resultant tsunamis (see review by Heidarzadeh et al. 2014; Yavari-Ramshe and Ataie-Ashtiani 2016). Two practical models are known for modeling the landslide part of the landslide tsunami model: a viscous fluid model and a granular material model. The former viscous fluid models include Imamura and Imteaz (1995), Yalciner et al. (2014), and Baba et al. (2019). It has reproduced well the run-up heights of tsunamis generated by submarine landslides in application to the case of Papua New Guinea earthquake in 1998 (Imamura and Hashi 2003). Pakoksung et al. (2019) analyzed the 2018 Sulawesi earthquake using this type of model. The latter granular material models include Iverson and George (2014), Ma et al. (2015), Grilli et al. (2019) and Paris et al. (2020), whose models’ landslide part governed by Coulomb friction is originated from Savage and Hutter (1989). Grilli et al. (2019) and Paris et al. (2020) applied this type of model to the case of Krakatau in 2018, demonstrating mass and tsunami behavior well. Grilli et al. (2019) also showed that tsunami waveforms at points far away from the source did not show much difference between the two models, dense Newtonian fluid model and granular material model.

In this study, we adopted the latter granular material model and the method similar to Paris et al. (2020). Titan2D (Pitman et al. 2003; Patra et al. 2005; Titan2D 2016) was used for the calculation of the granular mass material, and JAGURS (Baba et al. 2017) was used for the calculation of the tsunami propagation. Titan2D is a depth-averaged model that stably calculates the rheological motion of a continuum granular mass using real topographic data. JAGURS is a tsunami simulation model that can solve a depth-averaged nonlinear long-wave equation.

The equations of the mass model used in this study are shown in (1)–(3), adding a buoyancy effect to the original Titan2D model:

$$\frac{\partial h}{\partial t} + \frac{{\partial hv_{x} }}{\partial x} + \frac{{\partial hv_{y} }}{\partial y} = 0,$$

(1)

$$\begin{aligned} & \frac{{\partial hv_{x} }}{\partial t} + \frac{\partial }{\partial x}\left( {hv_{x}^{2} + \frac{1}{2}k_{\text{ap}} k_{\text{b}} g_{z} h^{2} } \right) + \frac{{\partial hv_{x} v_{y} }}{\partial y} \\ & = k_{\text{b}} g_{x} h - hk_{\text{ap}} {\text{sgn}}\left( {\frac{{\partial v_{x} }}{\partial y}} \right)\frac{\partial }{\partial y}\left( {k_{\text{b}} g_{z} h} \right)\sin \phi_{\text{int}} - \frac{{v_{x} }}{{\sqrt {v_{x}^{2} + v_{y}^{2} } }}\hbox{max} \left( {k_{\text{b}} g_{z} + \frac{{v_{x}^{2} }}{{r_{x} }},0} \right)h \tan \phi_{\text{bed}} , \\ \end{aligned}$$

(2)

$$\begin{aligned} & \frac{{\partial hv_{y} }}{\partial t} + \frac{{\partial hv_{x} v_{y} }}{\partial x} + \frac{\partial }{\partial y}\left( {hv_{y}^{2} + \frac{1}{2}k_{\text{ap}} k_{\text{b}} g_{z} h^{2} } \right) \\ & = k_{\text{b}} g_{y} h - hk_{\text{ap}} {\text{sgn}}\left( {\frac{{\partial v_{y} }}{\partial x}} \right)\frac{\partial }{\partial x}\left( {k_{\text{b}} g_{z} h} \right)\sin \phi_{\text{int}} - \frac{{v_{y} }}{{\sqrt {v_{x}^{2} + v_{y}^{2} } }}\hbox{max} \left( {k_{\text{b}} g_{z} + \frac{{v_{y}^{2} }}{{r_{y} }},0} \right)h \tan \phi_{\text{bed}} , \\ \end{aligned}$$

(3)

where *t* represents time, the *x*-axis and the *y*-axis are along the slope, and the *z*-axis is a direction perpendicular to the *x*–*y* plane. *h* is the mass thickness, *v*_{x} and *v*_{y} are the mass velocity, and \(g_{x}\), \(g_{y}\), and \(g_{z}\) are the components of gravitational acceleration to each axis, respectively. \(\phi_{\text{int}}\) and \(\phi_{\text{bed}}\) are the internal and bed friction angles of the mass, and \(r_{x}\) and \(r_{x}\) are the curvature of the local basal surface. *k*_{ap} is the coefficient that changes depending on the state of the active and the passive mass pressure, and is a function of \(\phi_{\text{int}}\) and \(\phi_{\text{bed}}\). Note that *k*_{ap} is negative if the mass is spreading. The description is omitted here [see the references of Savage and Hutter (1989), Pitman et al. (2003)]. The original Titan2D model is a dry avalanche model. We used the calculation code of Titan2D after multiplying the gravitational acceleration by the coefficient \(k_{\text{b}}\) in the same way as Paris et al. (2020) to take into account the buoyancy of mass moving in water. Here, \(k_{\text{b}} = \left( {\rho_{\text{s}} - \rho_{\text{w}} } \right)/\rho_{\text{s}}\). \(\rho_{\text{s}}\) and \(\rho_{\text{w}}\) are the density of soil in water and water. \(k_{\text{b}} = 0.5\) and \(\frac{{\rho_{\text{s}} }}{{\rho_{\text{w}} }} = 2\) in this study with reference to debris flow density in Denlinger and Iverson (2001).

The equations of the tsunami propagation used in this study are shown in (4)–(6), which are depth-averaged nonlinear long-wave equations. Although the tsunami calculations were performed in a latitude and longitude spherical coordinate system, equations in a Cartesian coordinates system are shown for simplicity:

$$\frac{\partial \eta }{\partial t} + \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} = \frac{\partial d}{\partial t},$$

(4)

$$\frac{\partial M}{\partial t} + \frac{{\partial \left( {\frac{{M^{2} }}{D}} \right)}}{\partial x} + \frac{{\partial \left( {\frac{MN}{D}} \right)}}{\partial y} = - gD\frac{\partial \eta }{\partial x} - \frac{{{\text{gn}}^{2} }}{{D^{{{\raise0.7ex\hbox{$7$} \!\mathord{\left/ {\vphantom {7 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}M\sqrt {M^{2} + N^{2} } ,$$

(5)

$$\frac{\partial N}{\partial t} + \frac{{\partial \left( {\frac{MN}{D}} \right)}}{\partial x} + \frac{{\partial \left( {\frac{{N^{2} }}{D}} \right)}}{\partial y} = - gD\frac{\partial \eta }{\partial y} - \frac{{{\text{gn}}^{2} }}{{D^{{{\raise0.7ex\hbox{$7$} \!\mathord{\left/ {\vphantom {7 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}N\sqrt {M^{2} + N^{2} } ,$$

(6)

where *t* represents time, the *x*-axis and the *y*-axis are on a horizontal plane. *η* is the sea level, and *M* and *N* are flow rates in *x* and *y* directions. *D* is the total depth and *D *=* h *+* η*, *h* is the depth, *g* is the gravitational acceleration, and *n* is the Manning roughness coefficient. *d* is the input sea level displacement, which is set to equal to the sea bottom displacement in this study. The results of our mass motion calculations were given as seabed deformations of tsunami calculations. For stable calculation, the landslide mass motion is calculated in advance, as in Paris et al. (2020) and Baba et al. (2019), and it is used as input to the tsunami propagation calculation as the temporal deformation of the seabed. Figure 2 shows a conceptual diagram of the tsunami generation associated with mass motion. Here, seabed deformation was converted from universal transverse Mercator coordinates to latitude and longitude coordinates. The resistance between mass and water was ignored. This is because the details of landslide such as degree of interaction that is greatly affected by the shape of the mass and the degree of mixing with water have been uncertain in this event.

### Conditions of landslide mass calculation

The numerical model of mass is discretized using the finite volume method and solved by the Godunov solver (Pitman et al. 2003). The initial shape of the flowable landslide mass is given as a paraboloid, and the mass motion is calculated according to its rheology (here, we selected a Coulomb-type rheology) under the driving force of gravity. The parameters to be given are limited to the radii of the major and minor axes (here, for convenience, *r*_{1} and *r*_{2}, respectively, in km), the orientation angle *θ* (counter-clockwise rotation from east), the maximum thickness *h* (m), and the horizontal location of the paraboloid, and the internal and bed friction angles of the mass, \(\phi_{\text{int}}\) and \(\phi_{\text{bed}}\), respectively. \(\phi_{\text{int}}\) represents the frictional resistance of the mass to collapse; we used a fixed value of 30° because this parameter does not strongly affect the results (Ogburn and Calder 2017). \(\phi_{\text{bed}}\) represents the mobility of the mass, with smaller values representing higher mobility, i.e., faster downslope flow of the mass. The flowing mass tends to stop when the slope approaches this angle. Figure 3 shows the diagram of the calculation procedure of mass motion.

### Conditions of tsunami calculation

The numerical model of tsunami propagation is discretized using the finite difference method and solved explicitly by the staggered-leaf-frog difference scheme (Baba et al. 2017). The spatial grid length for the calculation was set to 0.27 arc-seconds (about 10 m), the time interval to 0.04 s, and the prediction time to 30 min. Calculations were performed in the area between 0.52° S and 0.91° S and between 119.685° E and 119.91° E. The matrix size of the spatial domain for calculation was 3000 × 5200. Sea level fluctuations were assumed to reflect the seafloor deformations without delay at intervals of 10 s for 3 min after the mass started to move. The Manning roughness coefficient of friction was set to 0.025 in all regions uniformly. Land-side boundary conditions were set to run-up. On land, the friction coefficient generally increases due to structures and coastal forests. Therefore, the result obtained may slightly be overestimated. Six output points were used to compare the calculated waveforms with the video-inferred waveforms and the waveform observed at the tide gauge station (Additional file 1: Figure S1). For comparison, we also used the JAGURS model to simulate the propagation of a tsunami generated by the earthquake alone. Vertical fluctuations due to horizontal motions on steep slopes were not quantitatively large (Heidarzadeh et al. 2019) and are omitted here. Because the local sea level was about 1 m higher than mean sea level at the time of the earthquake (Fig. 1b), we lowered the topography and bathymetry by 1 m when calculating the tsunami propagation, and restored the topography, bathymetry, and sea level after the simulation.

### Comparison index

Aida’s (1978) correlation values for tsunami amplitudes: geometric mean, *K*, and its variability, \(\kappa\), were used to quantitatively compare the observed and calculated run-up heights:

$$\log K = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \log k_{i} ,$$

(7)

$$\log \kappa = \left[ {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {\log k_{i} } \right)^{2} - \left( {\log K} \right)^{2} } \right]^{1/2} ,$$

(8)

where \(k_{i} = R_{i} /H_{i}\), \(R_{i}\) is the field survey height at the *i*th point, \(H_{i}\) is the calculated height at the *i*th point, and *n* is the number of data. A strong match between observations and calculations is indicated by a small \(\kappa\) value and *K* close to 1. Calculated run-up heights were evaluated as the heights of the farthest west (on the western coast of the bay) or east (on the eastern coast) inundated points within every 0.002° of latitude, averaging the calculated mesh values.