总强度磁异常各阶垂向导数换算新方法

磁异常垂直导数换算在磁性目标解释推断过程中具有重要意义。分析了总强度磁异常各阶垂向导数频率转换因子的滤波特性，指出常规傅立叶变换法在求解总强度磁异常各阶垂向导数时，观测数据中高频噪声会显著放大，甚至会淹没掉真实信息。从理论上证明了总强度磁异常沿垂直方向的积分和各阶导数均为调和函数，在此基础上，提出联合采用空间域和频率域运算求解其沿垂直方向的各阶导数。该方法的基本思路是：利用频率域转换关系，计算平面上总强度磁异常沿垂直方向的积分值；选择三点二阶中心差分或双三次样条曲线函数法计算总强度磁异常垂向积分值沿水平方向的二阶导数；最后将二阶水平导数代入拉普拉斯方程求解出总强度磁异常的垂向一阶导数；以总强度磁异常及其垂向一阶导数沿水平方向的二阶导数为基础，结合拉普拉斯方程，可进一步求解出总强度磁异常任意阶垂向导数。同时，为了研究提出方法的有效性，采用球体磁场模型进行验证，并首次推导了地磁场方向和磁化强度方向不一致时，球体总强度磁异常沿垂直方向的一阶和二阶导数表达式。经研究表明：提出的方法换算得到的垂向导数结果精度明显优于常规傅立叶变换法换算结果，且具有较强的抗噪能力，尤其是在计算高阶导数时效果更加明显。

A New Method to Calculate the Vertical Derivatives of Total Field Magnetic Anomaly

The vertical derivative of the total field magnetic anomaly(TMA) plays an important role in the interpretation process of magnetic objects.Owing to the filter characteristics of vertical derivative operator in frequency domain,the conversion of vertical derivative is inherently unstable and any high-frequency noise presented in the surveying data gets strongly magnified in the transformed map in such a way to mask any useful signal.Based on the harmonic properties of the vertical integral and derivatives of TMA,an algorithm is presented to perform calculation of vertical derivatives using both frequency and space domain transformations.The effectiveness of the suggested techniques has been illustrated by synthetic sphere magnetic model whose TMA’s first and second vertical derivative expressions are deduced when the geomagnetic direction is different from magnetization direction.The conclusion indicates that the presented vertical derivatives calculation method provides better results and allows a lower degrading of the signal-to-noise ratio than the standard Fourier method.