## 1 Introduction  
Earthquakes(EQs) are one of the most serious natural disasters. There have been hundreds of Ms \(\geqslant 7\) EQs worldwide from 2000 to now [1]. EQs cause direct damage to the buildings in the residential areas in the center of the EQ and cause different levels of damage to buildings of different materials and structures [2, 3]. Secondary disasters such as tsunamis and landslides triggered by EQs cause enormous human casualties, construction damages, and property losses [4, 5, 6, 7]. Increasingly vital research on pre-EQ disaster phenomena was conducted by researchers around the world to avoid significant losses caused by EQ disasters. For instance, [PERSON], [PERSON], and [PERSON] have found that low-frequency electromagnetic radiation is emitted in the preparation period for EQs in the ionosphere [8]. The abnormal phenomenon has been observed by ground-based stations and satellite detectors [8, 9]. Moreover, the movement of the crust during the preparation period of many EQs also caused anomalous increases in radon concentrations, including water radon and gas radon [10, 11, 12, 13, 14]. In addition, there were also other observable anomalies that the researchers studied before the EQ, such as thermal anomalies and ground light phenomena [15, 16]. However, the pre-EQ anomalies mentioned above generally occur on a time scale of a few days to a few months during the preparation period of an EQ, and it is impossible to predict the occurrence time and location of an EQ accurately. Furthermore,the measurement of radon concentration and low-frequency electromagnetic waves are associated with high uncertainties. In recent decades, EQ forecasters have repeatedly detected negative abnormal signals in the atmospheric vertical electric field in front of the EQ. During fair weather and excluding weather conditions such as haze, thunder, and dust, this signal has occurred at the hourly level ahead of an EQ, and numerous observations have demonstrated the accuracy of this anomalous signal [17; 18; 19]. Although the atmospheric electric field anomaly was proven to be an important precursor before EQs, the cause of its formation was not clearly understood. In previous work [20], researchers claimed that atmospheric vertical electric field anomalies were caused by radon ionization of the air; however, the authors failed to propose a specific mechanism of interaction from radon concentration to atmospheric electric field. A mechanism for the formation of anomalous electrode effects by near-surface air ionization was proposed by [PERSON], [PERSON], and [PERSON] in 1997 [21]. They obtained an anomalous reduction effect of the atmospheric electric field within 1 m of the surface. Existing studies have confirmed that radon ionization of air contributes to the atmospheric vertical electric field anomaly, and some explorations of radon ionization atmospheric processes have also been conducted. However, up to now, there is no quantitative study on the signal of atmospheric vertical electric field anomaly caused by radon concentration. In this paper, the physical mechanism of the formation of negative atmospheric vertical electric field anomalies during the EQ preparation period is proposed by studying the pre-EQ radon gas concentration data. This paper aims to obtain detailed information on the variation of the atmospheric electric field with height obtained by solving the proposed physical model using radon gas and atmospheric electric field data before the Wenchuan EQ. The first part of this paper briefly analyzes the formation mechanism of the atmospheric vertical electric field before the earthquake and describes in detail the formation process of anomalous atmospheric electric fields in a progressive way, including the variation of radon concentration with height, simulating the ionization rate of radon stable decay daughter at different heights and solving the convective diffusion equation of positive and negative particles to obtain the atmospheric electric field.