Abstract: This paper tries to formulate the C-response of geomagnetic depth sounding(GDS)on an Earth model with finite electrical conductivity. The computation is performed in a spherical coordinate system. The Earth is divided into a series of thin spherical shells. The source is approximated by a single spherical harmonic P₁⁰ due to the spatial structure of electrical currents in the magnetosphere. The whole solution space is separated into inner and external parts by the Earth surface. Omitting displacement current, the magnetic field in the external space obeys Laplacian equation, while in the inner part, due to the finite conductivity, the electromagnetic fields obey Helmholtz equation. To connect the magnetic fields in the inner and external space, the continuity condition of magnetic fields is used on the Earth surface. The external magnetic fields are expressed by the inner and external source coefficients, from which a new parameter called C-response is computed from the inner coefficient divided by the external coefficient, thus normalizing the actual source strength. The inner magnetic fields in each layer can be recursively derived by the continuity boundary condition of both normal and tangential components of the magnetic field from the initial boundary condition at core-mantle-boundary. The consistency of our C-responses with that from a typical 1-D global model validates the accuracy of the proposed algorithm. Numerical results also show that the C-response estimated from the geomagnetic transfer function method will deviate exceeding 5%from the actual response at longer periods than about 10⁶s, which means that ignoring the curvature of the Earth at extreme long periods will make inversion result unreliable. Therefore, an accurate C-response should be computed in order to lay a solid foundation for reliable inversion.

Key words: geomagnetic depth sounding, C-response, electrical conductivity, spherical coordinate system, 1-D model

摘要： 对地磁数据的反演是获取地球深部电性结构的1种重要方法，其反演结果的可靠性必须以准确的正演模拟为基础。文中详细介绍了球坐标系中导电薄球层状地球模型的C-响应计算理论，并对典型的地球模型进行了数值模拟。地磁测深的激发源为磁层中的电流体系，其形态由球谐函数P₁⁰近似表示。地球内部，导电层中电矢量位满足亥姆霍兹方程；通过各层界面上磁场法向分量和切向分量满足的连续边界条件，由超导地核确定的核幔边界系数向上逐层递推，进而获得地表的边界系数，最终将地下电性结构和地表磁场各分量联系起来。通过磁场分量比值得到与源强度无关而与地球内部导电性相关的地表C-响应，实现地磁测深一维正演计算。一维典型模型的C-响应与前人结果的一致性验证了本文算法的有效性；通过与直接计算的C-响应曲线进行对比，发现由于忽略了地球曲率的影响，利用地磁函数转换方法获得的C-响应在大周期时（>10⁶s）与理论响应存在一定的偏差，会造成反演结果的不准确。用文中的数值模拟方法能获得精确的C-响应，进而支撑地磁测深一维反演结果的可靠性。

关键词: 地磁测深, C-响应, 电导率, 球坐标系, 一维模型