Numerical Solutions of the Mean‐Value Theorem: New Methods for Downward Continuation of Potential Fields

Downward continuation can enhance small‐scale sources and improve resolution. Nevertheless, the common methods have disadvantages in obtaining optimal results because of divergence and instability. We derive the mean‐value theorem for potential fields, which could be the theoretical basis of some data processing and interpretation. Based on numerical solutions of the mean‐value theorem, we present the convergent and stable downward continuation methods by using the first‐order vertical derivatives and their upward continuation. By applying one of our methods to both the synthetic and real cases, we show that our method is stable, convergent and accurate. Meanwhile, compared with the fast Fourier transform Taylor series method and the integrated second vertical derivative Taylor series method, our process has very little boundary effect and is still stable in noise. We find that the characters of the fading anomalies emerge properly in our downward continuation with respect to the original fields at the lower heights. Plain Language Summary A class of novel methods for downward continuation of potential fields are addressed. Because they are new methods and the results are stable, convergent and accurate, and have very little boundary, it is required for rapid publication. The mean‐value theorem for potential fields is presented and it shows the vertical variation of potential field simply. The new scientific advance allows convergent and accurate downward continuation because of the different numerical solutions of it. The results improve the anomaly's resolution and enhance the overlapping character. The mean‐value theorem and its numerical solutions of potential fields may be the new theoretical basis of the free‐air correction and downward continuation. Key Points We derive the mean‐value theorem for potential fields, the new theoretical basis of the free‐air correction, and downward continuation We present several formulae for downward continuation based on different numerical solutions of the mean‐value theorem for potential fields One of the proposed methods has been tested on both synthetic and real cases, and the downward continuation is stable and highly accurate