位场向下延拓的最小曲率方法

针对位场向下延拓的不适定性,我们将位场向下延拓视为向上延拓的反问题,提出以位场最小曲率作为约束条件来求解稳定的下延位场.我们将剖面位场向上延拓表达式用傅里叶矩阵的形式表示,以矩阵乘法形式给出延拓的表达式,同时向待反演的下延位场引入最小曲率约束,得到向下延拓的最小曲率解,并利用正交变换给出了更为简洁的频率域解.随后,利用Kronecker积将上述全部结果拓展至三维位场,给出了三维位场向下延拓的最小曲率解.此外,我们将位场数据的填充、扩充问题与向下延拓问题统筹考虑,提出一种新的向下延拓迭代格式,该算法面向实际资料处理需求、无须预扩充或填补数据.下延迭代时,对原始数据直接向下延拓,而空白部分利用上一次下延位场估计的上延值替代其空白值并对其向下延拓,直至获得最小曲率约束下稳定的向下延拓结果.同时,我们也讨论了利用改进L曲线和广义交叉验证(GCV)计算正则参数最优估计的问题.对理论模型和实际航空重力资料进行了向下延拓检验,处理结果表明位场向下延拓的最小曲率方法解能满足实际位场资料对向下延拓处理的需求,具有较高的下延精度.

Minimum curvature method for downward continuation of potential field data

Downward continuation calculates the potential field that is closer to the source of anomalies is a powerful but unstable tool in data processing. To obtain a reasonable downward continued result, we formulate the downward continuation process as an inverse problem of upward continuation, and introduce a discretized smoothing continuous curvature spline regularization approach to solve it. The proposed method, which is based on the discrete Fourier transform(DFT) matrix, allows robust smoothing of downward continued field data. In the study, we formulate the upward continuation of 2-D potential field data as matrix multiplication in the form d=Gm represented by DFT matrix. Then, the inverse problem is formulated as minimizing a total objective function consisting of total squared curvature of downward continued field data and data misfit. Because the number of data is often large, calculating the matrix equation is a major computational load of the downward continuation. We greatly simplify and solve it by means of the DFT. Then we extend the method to 3-D potential field by using the Kronecker product, and obtain the minimum curvature solution of downward continued field data in the frequency domain. As the results of downward continuation are strongly influenced by the smoothing parameter, automatic choice of the amount of regularization parameter is carried out using the method of L-curve and generalized cross validation(GCV). In addition, we consider the problems on data missing and grid expansion in the downward continuation of potential field. An iterative robust scheme of downward continuation is then proposed to deal with data expansion or replacing dummy values with interpolated values. To test the performance of the method, we employ the synthetic and real airborne gravity data for downward continuation. Data processing results on both the synthetic and real data confirm that the performance of the proposed technique can satisfy the need of real data processing for downward continuation of the potential field.