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已有的均匀磁化长方体磁场计算方法大都假设观测点位于上半无源空间,对于起伏地形条件而言,这些方法可能存在解析“奇点”。为此,本文基于地磁场基本理论,采用变量替换的积分方法,导出了改进的磁场表达式,详细讨论并有效解决了整个无源空间的所有奇点问题。相比前人的方法,其积分过程更自然、简单,最后的积分结果形式更加统一,并且不需要坐标变换即可求出无源区任意点处的磁场值,从而简化了正演过程。对比模型试验表明,新导出的磁场无解析“奇点”理论表达式是正确的,并能适应地形起伏的情况。
Most existing methods of calculating the magnetic field of a uniformly magnetized cuboid assume that the observation points are located in the upper semi-passive space. For the undulating topographic conditions, these methods may have analytic “singularity ”. Therefore, based on the basic theory of geomagnetic field, this paper adopts the integral method of variable replacement to derive the improved magnetic field expression, and discusses and resolves all the singularities of the entire passive space in detail. Compared with the previous methods, the integration process is more natural and simple, and the result of the integral is more uniform. The magnetic field at any point in the passive region can be obtained without coordinate transformation, which simplifies the process of forward modeling. Comparative model tests show that the newly derived magnetic field has no analytical “singularity ” theoretical expression is correct and can adapt to the situation of terrain relief.