## 3 Study Case and Method
### Point Force History Inversion  
The point force history (PFH) inversion method was proposed by [PERSON] [17]. Even if the landslide material gradually breaks up during the movement, it can still be regarded as a single point under the large terrain scale. Complicated and local dynamic characteristics changes in collapsed materials are not easily reflected in distant seismic records at large spatial scales. Based on a hypothesis that a landslide can be simplified as a block of constant mass m sliding on a slope, the slope parallel force \(F_{\parallel}\) from the driving force of gravity opposed by the friction force can be expressed as  
\[F_{\parallel}=mg\ sin\ \theta-mg\ \mu^{\prime}\ cos\ \theta \tag{1}\]  
where \(g\) is the gravity acceleration, \(\theta\) is the slope angle, and \(\mu^{\prime}\) is the dynamic frictional force. If the sliding force is larger than the frictional force, the landslide block will be unstable and start to slide. Based on [PERSON]'s second law, Equation (1) can be modified as  
\[F_{\parallel}=ma=mg(sin\ \theta-\mu^{\prime}\ cos\ \theta) \tag{2}\]  
According to [PERSON]'s third law, when body A exerts a force on body B, body B simultaneously exerts a force equal in magnitude and opposite in direction on body A. Therefore, as the landslide  
Figure 4: Vertical seismic signal of the Taimali and landslide recorded at the MASB station. (**a**) Original waveform, (**b**) spectrogram, (**c**) spectrum, (**d**) 0.01-0.033 Hz filtered waveform, and (**e**) 1-5 Hz filtered waveform. The signal recording begins at local time 2009–08–09 16:31:00 and the sampling rate is 20 Hz. block receives a force downslope, the sliding surface receives an equal point force \(Fe\) in the opposite direction. This force \(Fe\) is a time-dependent three-component vector that can be expressed as  
\[Fe\left(t\right)=-\;m\;a\left(t\right) \tag{3}\]  
The long-period seismic signals generated by landslides are attributed to the single force \(Fe\)[10; 17; 23]. Equation (3) dictates that the direction of the force exerted on the sliding surface will be in the opposite direction to the landslide acceleration. That is, as the landslide accelerates downslope, the direction of the force heads upslope; as the landslide decelerates, the sliding surface receives a force in the downslope direction. After the instrument responses are removed, the seismograms recorded by seismometers represent the results of the seismic sources that pass through radiation paths. The responses of an impulse force between each source and station are called Green's functions, and they consider all seismic waves and attenuation on the radiation paths [17; 24]. If the velocity structure of the material through which the seismic waves are passing is known, Green's functions can be calculated. Due to the long-period band-pass filter (30-100 s) used in this study, the seismic waveforms radiated by the landslide had wavelengths of tens to hundreds of kilometers and had low sensitivity to small-scale heterogeneities in the velocity structure and regional topography. Hence, a generalized one-dimension velocity model is sufficient to calculate Green's functions. In this study, the Taiwan one-dimension velocity model proposed by [PERSON] and [PERSON] [25] was used. At the top 2 km of the velocity structure, the velocity of the P wave reaches 4.64 km/s with a velocity gradient of 0.37 km/s per km, and the velocity of the S wave is 2.64 km/s. The velocity structure was derived from P and S wave travel time and was confirmed that it can indicate the thickness of recent sediment on the Taiwan island. The Computer Program in Seismology (CPS) software was used to calculate the Green's functions between each station and the landslide location. For the single-force mechanism of the landslide processes suggested by [PERSON] and [PERSON] [7; 8], five Green's functions are required [24]. The five Green's functions are listed as follows:  
1. ZVF: vertical component for a downward vertical force,
2. RVF: radial component for a downward vertical force,
3.