位场垂向导数的稳定算法

A STABLE ALGORITHM FOR CALCULATING THE VERTICAL DERIVATIVES OF POTENTIAL FIELD

垂向导数广泛应用于位场数据处理和解释中,常规的空间域求导公式多以台劳级数为基础,这不可避免会引入截断误差,且不便于求取高阶导数。另一种应用较多的波数域算法,其求导算子在高波数成份产生振荡失真,很不稳定,在实际应用中很难取得理想结果。这里提出了一种计算位场数据垂向导数的稳定算法,在波数域求垂向积分,在空间域计算二个二阶水平方向导数,借助拉普拉斯方程,求取位场异常的各阶垂向导数。算法只在求垂向积分时引入了一次付里叶变换,其它计算全部在空间域中进行,从而有效地避免了吉布斯效应的重复累加传递,有效降低了常规波数域求导算子在高波数成份产生的振荡失真。通过理论模型对比,证明了这里所提出的算法较常规的波数域方法具有良好的稳定性,特别是在计算高阶垂向导数时有明显优势,能有效地提高与垂向导数有关的位场定量解释方法的准确性。

The vertical derivatives are widely used in processing and interpretation of potential field data,the conventional space domain formulas of calculating vertical derivatives are mostly based on Taylor series which do inevitably bring the truncation error,and are inconvenient to get higher-order derivatives.Another common approach to calculate vertical derivatives is to use derivative operator in wavenumber domain.Unfortunately,it produces high wavenumber oscillation and is very unstable,therefore,it is hard to obtain desirable results by using this approach in practice.This paper presents a stable algorithm to calculate any order vertical derivatives of potential field by means of Laplace equation that involves the computation of vertical integral in wavenumber domain and two second horizontal derivatives in space-domain.It is only once that Fourier transform was introduced in calculating the vertical integral,and the other steps were processed in space domain.So the accumulation and pass of Gibbs effect was avoided and the high wavenumber oscillation produced by conventional wavenumber domain derivative operator was depressed efficiently.It is illustrated by a composite model that the algorithm presented here is more stable than the common wavenumber domain method,especially,it has obviously advantage in calculating higher-order vertical derivatives,and it can improve the accuracy of quantitative interpretation of potential field data.based-vertical derivatives method.