### Forward modeling

We used a Bing debris-flow model (Imran et al. 2001) to conduct a numerical analysis of the landslide. This model assumes two layers of debris (i.e., an upper plug-layer and a lower shear-layer) that flow laminarly with rheological soil properties described by a viscoplastic Herschel–Bulkley model:

$$\left| {\frac{\gamma }{{\gamma_{{\text{r}}} }}} \right|^{n} = \left\{ {\begin{array}{*{20}ll} {0 \quad {\text{if}}\; \left| \tau \right| < \tau_{{\text{y}}} } \\ {\frac{\tau }{{\tau_{{\text{y}}} {\text{sgn}} \left( \gamma \right)}} - 1 \quad {\text{if}} \; \left| \tau \right| \ge \tau_{{\text{y}}} } \\ \end{array} } \right.,$$

(1)

where \(\tau\) is the shear stress, \(\tau_{{\text{y}}}\) is the yield stress, and \(\gamma\) is the strain rate. A reference strain rate, denoted by \(\gamma_{{\text{r}}}\), is defined as follows:

$$\gamma_{{\text{r}}} = \left( {\frac{{\tau_{{\text{y}}} }}{\mu }} \right)^{\frac{1}{n}} ,$$

(2)

where \(\mu\) is the dynamic viscosity. The volcanic soil of the studied landslide is assumed to behave as a Bingham fluid, which is a limiting case for the Herschel–Bulkley model with *n* = 1. The Bingham model is the most commonly used viscoplastic model that describes the behavior of debris or mud flows (Jiang and LeBlond 1993; Wan and Wang 1994; Julien 1995; Naruse 2016). The integral equations of the debris flow on a Lagrangian framework are described by the following three conservation equations for mass and momentum (Jiang and LeBlond 1993; Huang and Garcia 1998; Pratson et al. 2001; Imran et al. 2001; Naruse 2016):

$$\frac{\partial D}{{\partial t}} + \frac{\partial }{\partial x}\left[ {U_{{\text{P}}} \left( {D_{{\text{P}}} + \frac{2}{3}D_{{\text{S}}} } \right)} \right] = 0,$$

(3)

$$\frac{\partial }{\partial t}\left( {U_{{\text{P}}} D_{{\text{P}}} } \right) + U_{{\text{P}}} \frac{{\partial D_{{\text{S}}} }}{\partial t} + \frac{\partial }{\partial x}\left( {U_{{\text{P}}}^{2} D_{{\text{P}}} } \right) + \frac{2}{3}U_{{\text{P}}} \frac{\partial }{\partial x}U_{{\text{P}}} D_{{\text{S}}} = - gD_{{\text{P}}} \left[ {1 - \frac{\rho }{{\rho_{{\text{m}}} }}} \right]\frac{\partial D}{{\partial x}} + gD_{{\text{P}}} \left[ {1 - \frac{\rho }{{\rho_{{\text{m}}} }}} \right]S - \frac{\mu }{{\rho_{{\text{m}}} }},$$

(4)

$$\frac{2}{3}\frac{\partial }{\partial t}\left( {U_{{\text{P}}} D_{{\text{S}}} } \right) - U_{{\text{P}}} \frac{{\partial D_{{\text{S}}} }}{\partial t} + \frac{8}{15}\frac{\partial }{\partial x}\left( {U_{{\text{P}}}^{2} D_{{\text{S}}} } \right) - \frac{2}{3}U_{{\text{P}}} \frac{\partial }{\partial x}U_{{\text{P}}} D_{{\text{S}}} = - gD_{{\text{S}}} \left[ {1 - \frac{\rho }{{\rho_{{\text{m}}} }}} \right]\frac{\partial D}{{\partial x}} + gD_{{\text{S}}} \left[ {1 - \frac{\rho }{{\rho_{{\text{m}}} }}} \right]S - 2\frac{\mu }{{\rho_{{\text{m}}} }}\frac{{U_{{\text{P}}} }}{{D_{{\text{S}}} }},$$

(5)

where \(U_{{\text{P}}}\) is the plug-layer velocity, \(D_{{\text{P}}}\), \(D_{{\text{S}}}\), and *D* are the plug-layer, shear-layer, and total thickness of the debris, respectively, *S* is the slope angle, *t* is time, *x* is the position along the slope (downward is positive), \(g\) is gravitational acceleration, \(\rho\) is the density of air, and \(\rho_{{\text{m}}}\) is the density of the debris material. According to Li and Wang (2020), the clay-rich volcanic soil near the base of the landslide (known as Ta-d pumice MS) has a specific gravity of 2.631 (\(\rho_{{\text{m}}}\) = 1177 kg/m^{3}) and a water content of 206% at a degree of saturation (*S*_{r}) of 92.8%. The upper volcanic soil (Ta-d pumice with volcanic ash) has a specific gravity of 2.553 (\(\rho_{{\text{m}}}\) = 1122 kg/m^{3}) and a water content of 121% at a degree of saturation (*S*_{r}) of 76.2%. If these soils were fully saturated at failure (i.e., *S*_{r} = 100%), \(\rho_{{\text{m}}}\) is estimated to be 1240 kg/m^{3} for the upper layer and 1310 kg/m^{3} for the lower layer. Therefore, we assume \(\rho_{{\text{m}}}\) = 1300 kg/m^{3} in our model.

We developed a MATLAB code to numerically solve Eqs. (3)–(5). The equations were assembled in a staggered lattice using a finite difference method with a deformable moving grid system. Following Imran et al. (2001), a numerical viscosity of 0.001 was used to prevent the solution from becoming unstable. More detailed information on the solution procedure can be found in Imran et al. (2001). The calculation was stopped when the frontal velocity of the debris flow decreased to 1 cm/s.

### Inversion analysis

Based on the geometry of the landslide deposit, inversion analysis was conducted to optimize the parameters \(\tau_{{\text{y}}}\) and \(\gamma_{{\text{r}}}\) in Eq. (2). Since our analysis is 1-D, lateral flow is ignored. However, the volume of debris before and after failure is not equal along the constructed cross-section (Fig. 1c), with the latter being larger than the former by 17%. This implies that soils from outside the survey line contributed to the final geometry of the deposit. For this reason, we use a volume of the post-failure deposit that is artificially reduced (shortened vertically), so that it is identical to the pre-failure volume. Due to these limitations, we aim at reproducing key features of the debris flow such as run-out distance and position of the hump of the deposit.

In the inversion analysis, we defined an objective function as a residual sum of squares between the observed (corrected) and modeled deposit thickness

$$F = \mathop \sum \limits_{i = 1}^{n} \left( {D_{{{\text{o}}i}} - D_{i} } \right)^{2} ,$$

(6)

where *n* is the number of grid points and \(D_{{{\text{o}}i}}\) is the observed thickness of the deposit at *i* grid point. Based on the result of the forward model, the thickness of the modeled deposit at each grid point (\(D_{i}\)) was calculated using the one-dimensional data interpolation method. The analysis was done over the length of the deposit, from *x* = 76.5 to 184.68 m, which consisted of 58 grid points. The source area was excluded from the inversion analysis (*x* = 0 to 76.5 m), because we assumed the thickness of the deposit on this area as described above.

Since the *F* function is non-linearly dependent on the fitting parameters, a Markov Chain Monte Carlo (MCMC) method was applied to optimize the fitting parameters over a range of values. Details of this method are outlined by Metropolis et al. (1953). For an MCMC simulation, a candidate value for one of the unknown parameters is tested by adding a random number (*n*_{r} = [− 1,1]) to the old candidate, namely log_{10}(*P*_{i,can}) = log_{10}(*P*_{i}) + *n*_{r}, where *P*_{i} and *P*_{i,can} are the old and current candidates, respectively. Therefore, the candidate parameter vector for one local step can be described as \(\varphi_{{{\text{can}}}} = \left\{ {P_{{1,{\text{can}}}} , P_{2} , \ldots , P_{N} } \right\}\). \(F\left( {\varphi_{{{\text{can}}}} } \right)\) is then calculated from these parameters and \(\Delta F\) is evaluated from \(\Delta F = F\left( \varphi \right) - F\left( {\varphi_{{{\text{can}}}} } \right).\)

The candidate vector \(\varphi_{{{\text{can}}}}\) is accepted if the probability is \(\min \left( {1, \exp \left( { - \Delta F\beta_{N} } \right)} \right)\), where \(\beta_{N}\) is the “inverse temperature”, which controls the acceptance or rejection of the candidate values. At low \(\beta_{N}\), the candidate value is updated when \(- \Delta F\) > 0. At high \(\beta_{N}\), this is not the case. The appropriate \(\beta_{N}\) is problem-dependent, and in this work, \(\beta_{N}\) was set to 0.5–1.0 by trial-and-error (*β*_{N} was fixed during one series of calculations). After one local update, the next unknown parameter is tested in a similar way. One Monte Carlo Step (MCS) is defined as *N* trials (i.e., the number of unknown parameters examined) of the local update, and *E*_{sum} is obtained by the arithmetic sum of *F* of each MCS. Schematic flowchart of the MCMC method for our inversion analysis is shown in Fig. 2.