几种梯度场计算的数值算法及其地学应用

Several gradient vector field numerical algorithms and their applications in geosciences

梯度作为标量场在空间上最重要的微观变化特征,能较好地反映其在空间上的变化趋势。本文介绍了中心差分算子、相邻梯度因子和Sobel梯度算子三种梯度数值算法。以点电荷在三维空间中产生的电势场构造了一标量场,通过理论公式求解了其梯度值,并将理论梯度值与三种数值算法的结果进行了对比与统计分析,验证了三种梯度数值算法的有效性和适用性。结果表明,排除场源的影响后,Sobel梯度算子相较其它两种数值算法更逼近于理论梯度值。最后,利用Sobel梯度算子求取了某铜矿区可控源音频大地电磁法(CSAMT)三维视电阻率场的梯度场,用梯度场刻画了视电阻率在空间上的变化趋势,提高了对三维地下空间的电性分布规律和地质情况的认识。

Gradient is the most important micro-variation characteristics of scalar field in three-dimensional space,which can reflect its trends better.In this paper,the central-difference operator,adjacent gradient operator and Sobel gradient operator were introduced in detail.In addition,the 3D theoretical gradient vector field of electric potential scalar field produced by apoint charge was computed according to its physical formula,and the results calculated by three gradient numerical algorithms was statistically compared with theoretical gradient value to verify their precisions and validities.The experimental results showed that Sobel operator has higher precision in the space excluding the field source than the other two numerical algorithms.At last,the gradient vector field of apparent resistivity surveyed by the controlled source audio-frequency magnetotelluric（CSAMT）method was calculated by Sobel operator to characterize the spatial changes of apparent resistivity,which improves the cognition of the electric property spatial distribution and geological situation in the 3D subsurface space.