## 1 Introduction  
A large number of experimental studies show that rocks under high stress will experience the development process of \"micro fracture initiation (micro scale and micro energy)--fracture propagation, penetration (small-scale and small energy fracture)--local rock instability (mesoscale and medium energy impact)--rock mass structure failure and large-scale instability (large-scale and large-energy release) [1]\". One significance of extracting the low-frequency seismic signals is to explore the abnormal information in the above process. After the huge energy generated by the source is released, most of the high-frequency energy is absorbed by the medium or continuously weakened in the form of aftershocks, and a small part of the low-frequency energy has the characteristics of long wave length and slow attenuation, which is not easily absorbed and can be transmitted to thousands of kilometers away. Materials with strong high-frequency absorption will have positive prospects in seismic fortification [2, 3]. The low-frequency signal from seismic waves also belongs to a kind of underground pulsation information, which may be the external manifestation of slow earthquakes (also known as intermittent deep tremors, slow slip events, low-frequency earthquakes, etc.) [4, 5]. Compared with the conventional seismic radiation energy, the energy radiated from slow earthquakes is less than that from conventional earthquakes [6]. This micro fracture until rock mass instability and macro failure play an important role in the process of seismic nucleation [7, 8]. The micro fracture behavior is different from that of rigid structural media [9]. The mechanism explanation of slow earthquakes may be a slow slip. This mechanism explanation has been proved by the rock fracturing in the laboratory [10, 11], that is, the stable slip and stick slip phenomena before a rock fracture in the laboratory may correspond to the stable slip and stick slip phenomena near the subduction zone of a seismic fault. The occurrence of these phenomena mainly depend on the critical stiffness coefficient of the fault [6]. When the fault is unstable or conditionally stable, it leads to the phenomenon of a slow slip. However, the mechanical model of moderately strong earthquakes in nature is very complex and may be triggered by the joint action of many factors [12]. The occurrence of a slow slip event may not necessarily produce a destructive earthquake, but if the slow slip event occurs in the locking area deep within a plate or fault, resulting in a continuous increase in stress accumulation in the locking area, a strong earthquake may be triggered. Extracting the low-frequency abnormal signals from the continuous seismic waveforms is an attempt to find the phenomenon of a slow slip caused by the continuous stress accumulation in this locked area. It is certain that the strongest earthquakes in history showed a locked state underground before the main earthquake: for example, the Wenchuan earthquake of magnitude 8 that occurred in the Chinese mainland on 12 May 2008, and the epicentral area of the Inner Mongolia Left Banner 5.8 earthquake on 15 April 2015. By studying the low-frequency abnormal signals from seismic waves before and after a main earthquake, we can identify the characteristics of a slow slip and of the energy release from the stress accumulation in the locking area [7; 13]. Therefore, it is very necessary to make a retrospective study of historical moderate and strong earthquakes. This study can first determine the location of the locking area. If a slow earthquake occurs in the locked section of the fault, when the stress accumulates to the critical value, the locked section of the fault will be unlocked. At the moment of unlocking, the fault friction will suddenly decrease and produce rapid sliding or fracturing, which explains the hypothesis that a slow earthquake in the locked area will trigger the occurrence of a normal earthquake.