A new regularization method for calculating the vertical derivatives of the potential field

Vertical derivatives of the potential field are commonly used in processing and interpretation of potential field data. These derivatives can narrow the width of anomalies and locate source bodies more accurately. The higher the order of the derivatives used, the more pronounced the effect. But as derivative filters are a form of high-pass filters, noise in the data is enhanced similarly. Thus calculation of higher vertical derivatives of the potential field is unstable operations in the sense of mathematical physics definition and their incorrect evaluation can contribute to enlargement of instability of the whole method. Many approaches have been proposed to solve this problem. Recently, the ISVD (integrated second vertical derivative) algorithm which is based on the Laplace equation and combines the wavenumber domain method with the space domain method has become the main method for the calculation of the vertical derivatives of the potential field. One way to stabilize higher vertical derivative evaluation is the utilization of the Tikhonov regularization. The key of this method is the choice of the regularization parameter. Based on the regularization theory and the radially averaged power spectrum of the potential field, we propose a new regularization method for calculating the vertical derivatives of the potential field. First, we simply define a special wavenumber named the cutoff wavenumber to divide the potential field spectrum into a signal part and a noise part based on the radially averaged power spectrum of potential field data. Next, we use the conventional vertical derivative operator to process the signal and the Tikhonov regularization operator to suppress the noise. Then, the parameters of the improved operator are defined by the cutoff wavenumber which has an obvious physical sense. The new regularization method only needs determining an ad hoc cutoff wavenumber based on the analysis of the radially averaged power spectrum of the potential field. Moreover, the new regularized method can not only eliminate the influence of high-wavenumber noise but also avoid. the attenuation of the signal. Additionally, as the vertical derivatives can be obtained precisely, the stable downward continuation by the Taylor series method can also be achieved. The comparison analysis of two theoretical gravity models and real aeromagnetic data shows that the new method is easy to implement and has clear physical sense. Furthermore, the calculation results of the vertical derivatives are more stable and precise than the ISVD algorithm. Even the cutoff wavenumber which is determined by the radially averaged power spectrum is not precise enough, the proposed method still has the advantage of stability. Based on the computation of the stable vertical derivatives which are obtained by the proposed regularization method and Taylor series expansion of the field, a large depth and high precision downward continuation of potential field can be realized.