Two New Methods Based on Implicit Expressions and Corresponding Predictor-Correctors for Gravity Anomaly Downward Continuation and Their Comparison

Downward continuation is a key technique for processing and interpreting gravity anomalies, as it has a major role in reducing values to horizontal planes and identifying small and shallow sources. However, it can be unstable and inaccurate, particularly when continuation depth increases. While the Milne and Adams-Bashforth methods based on numerical solutions of the mean-value theorem have partly addressed these problems, more accurate and realistic methods need to be presented to enhance results. To address these challenges, we present two new methods, Milne-Simpson and Adams-Bashforth-Moulton, based on implicit expressions and their predictor-correctors. We test the validity of the presented methods by applying them to synthetic models and real data, and we obtain stability, accuracy, and large depth (eight times depth intervals) downward continuation. To facilitate wider applications, we use calculated vertical derivatives (of the first order) by the integrated second vertical derivatives (ISVD) method to replace theoretical ones from forward calculations and real ones from observations, obtaining reasonable downward continuations. To further understand the effect of introduced calculation factors, we also compare previous and presented methods under different conditions, such as with purely theoretical gravity anomalies and their vertical derivatives at different heights from forward calculations, calculated gravity anomalies and their vertical derivatives at non-measurement heights above the observation by upward continuation, calculated vertical derivatives of gravity anomalies by the ISVD method at the measurement height, and noise. While the previous Adams-Bashforth method sometimes outperforms the newly presented methods, new methods of the Milne-Simpson predictor-corrector and Adams-Bashforth-Moulton predictor-corrector generally present better downward continuation results compared to previous methods.