## 3 Study Case and Method
### Point Force History Inversion
ZHF: vertical component for a horizontal force in the radial direction,
4. RHF: radial component for a horizontal force in the radial direction, and
5. THF: tangential component for a horizontal force in the transverse direction. From the waveform perspective, the first four components correspond to the velocity system of P wave and S wave, while THF corresponds to the SH system. [PERSON] [22] described the displacement seismograms for any point force time series by  
\[u_{z}=\left(f_{N}\cos\phi+f_{E}\sin\phi\right)\mathrm{ZHF}+f_{Z}\mathrm{ZVF} \tag{4}\]  
\[u_{r}=\left(f_{N}\cos\phi+f_{E}\sin\phi\right)\mathrm{RHF}+f_{Z}\mathrm{RVF} \tag{5}\]  
\[u_{t}=\left(f_{N}\sin\phi f_{E}\cos\phi\right)\mathrm{THF} \tag{6}\]  
where \(u_{z}\), \(u_{r}\), and \(u_{t}\) are the synthetic ground displacement in the vertical, radial, and transverse directions, and \(\phi\) is the source to station azimuth measured clockwise from the north. The single force \(\overline{F}=\left(f_{N},f_{E},f_{Z}\right)\) is in the north, east, and vertical direction, respectively. The displacement seismograms are defined as positive in the following directions: up, away from the source, and right angle clockwise in the vertical, radial, and transverse components, respectively. The displacement seismogram \(d_{i}\left(t\right)\) at each component out of the \(i\) station is the result of the force-time function \(f_{j}\left(t\right)\) in the \(j\) direction convoluted (*) with the Green's functions \(G_{ij}\left(t\right)\), which can be expressed as  
\[d_{i}\left(t\right)=G_{ij}\left(t\right)\;*f_{j}\left(t\right) \tag{7}\]If the displacement seismogram and Green's functions are available, the landslide force-time function can be extracted by the waveform inversion. To execute the inversion, Equation (7) should be written as a multiplication matrix rather than a convolution. A convolution is defined as the integral of the product of two functions after one is reversed and shifted. To convolve a Green's function with a force-time series through matrix multiplication, the Green's functions were shifted by one sample in each successive row to generate a convolution matrix. An example of the convolution through matrix multiplication between a four-sample Green's function \(g\) and a four-sample force-time function \(f\) to obtain a seven-sample long seismogram \(d\) can be written as  
\[\left[ \begin{array}{c}d1\\ d2\\ d3\\ d4\\ d5\\ d6\\ d7\end{array} \right]=\left[ \begin{array}{cccc}g1&0&0&0\\ g2&g1&0&0\\ g3&g2&g1&0\\ g4&g3&g2&g1\\ 0&g4&g3&g2\\ 0&0&g4&g3\\ 0&0&0&g4\end{array} \right]\times\left[ \begin{array}{c}f1\\ f2\\ f3\\ f4\\ \end{array} \right] \tag{8}\]  
Equation (8) illustrates a component of the seismogram of a station, showing the convolution of the force-time function and the Green's function. To better characterize the relationship among the seismograms, Green's functions, and the force-time function, the convolution matrix of the Green's functions may incorporate the equations considering the azimuth relationship between the source and the station. The detailed convolution of the forward equation of the station can be written as  
\[\left[ \begin{array}{c}dz\\ dr\\ dt\end{array} \right]=\left[ \begin{array}{c}\star\text{ZVF}\\ \star\text{RVF}\\ 0\end{array} \right]\times\left[ \begin{array}{c}fz\\ fn\\ fe\end{array} \right] \tag{9}\]  
where \(dz\), \(dr\), and \(dt\) are the column vectors that contain the seismograms for each station at the vertical (\(z\)), the radial (\(r\)), and the transverse (\(t\)) components, respectively. The 3 \(\times\) 3 matrix (G matrix) consists of the five Green's functions and the azimuth calculation. Each element of the G matrix is a Green's function convolution matrix for each station. Since large-scale landslides often occur during typhoons, the seismic records can be affected by high-level ambient noise. Thus, different weighting coefficients were given to the seismograms according to their SNR values (Table 2).