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.