## 1 Introduction  
Atmospheric 3D electric field detection technology plays a crucial role in various fields, including atmospheric science, space physics, aviation safety, and meteorological disaster warning [1, 2, 3, 4]. For instance, in atmospheric science, it enables the analysis of charge distribution within thunderclouds and their evolution mechanisms, facilitating lightning prediction [5]. In space physics, electric field measurements contribute to studying geomagnetic storms and plasma fluctuations, which are essential for space environment monitoring and satellite communication security [6]. In aviation safety, real-time electric field information is used to assess the impact of atmospheric electric field disturbances on aircraft, ensuring the safety of aerospace missions [7]. Due to the complexity of the atmospheric electric field, which is influenced by the charge distribution within clouds and exhibits a 3D dynamic non-uniform distribution [8], it is necessary to employ electric field sensors for accurate 3D field measurements. Currently, 3D electric field sensors for aerial atmosphere primarily include airborne, field-mill, and micro-electro-mechanical system (MEMS) types [9; 10; 11]. Airborne electric field sensors are typically mounted on aircraft, allowing for 3D field measurements by adjusting the aircraft's attitude. However, their measurement accuracy is often affected by the stringent calibration requirements and interference from aircraft-induced charging effects [12]. Field-mill electric field sensors detect electric field signals by periodically exposing and shielding electrodes, thereby enhancing their anti-interference capability. Nevertheless, their complex mechanical structure and susceptibility to wear over prolonged operations can compromise measurement stability [13; 14]. In contrast, MEMS 3D electric field sensors have attracted significant attention in recent years for their compact size, low power consumption, and high integration, making them a focal point in airborne electric field measurement research [11]. However, in practical measurements, the output signal of the sensing electrodes is influenced not only by the electric field component perpendicular to the electrode surface but also by interference from the remaining two directional components caused by inter-axis coupling. This coupling effect significantly degrades the accuracy of electric field measurements. To minimize such interference, various optimization strategies have been proposed. For example, [PERSON] et al. [15] designed a MEMS 3D electric field sensor with low inter-axis coupling, employing a symmetric structure and differential circuits to suppress electric field coupling interference along non-measurement axes, achieving an inter-axis coupling sensitivity better than 3.48%. Similarly, [PERSON] et al. [16] introduced a 3D electric field sensor chip featuring a single-shielded electrode, utilizing differential detection to enable 3D electric field measurements with a maximum measurement error of less than 5.3%. While these methods have effectively mitigated inter-axis coupling interference through structural optimization at the chip level, they still face limitations in achieving high-precision atmospheric 3D electric field detection, as inter-axis coupling is not completely eliminated, making it challenging to fully meet the requirements for accurate 3D field measurements. To further enhance electric field measurement accuracy and mitigate coupling interference, researchers have proposed a series of decoupling calibration methods. [PERSON] et al. [12] developed a decoupling approach for distributed airborne field models by establishing a relationship matrix between sensor output signals and actual electric field components based on the inverse of the sensitivity coefficient matrix. Although this method enables 3D field decoupling, its calibration process is complex and requires high-precision instrumentation. To improve decoupling accuracy while simplifying the calibration procedure, [PERSON] et al.