Two new methods for gravity anomaly downward continuation based on implicit expressions of numerical solutions of mean-value theorem and their comparison

In practical problems concerned with an exploration of geological mineral resources, to enhance the resolution for its geological interpretation, downward continuation of the gravity anomaly is usually performed, as downward continuation can highlight local and shallow gravitational sources related to the ore body, which plays a very important role in the following processing and interpretation of gravity data. However, downward continuation is an ill-posed issue and has been a research topic for gravity exploration.

General classical methods for the downward continuation of gravity anomalies mainly include spatial-domain methods, of which, however, their convolution calculations are complicated; frequency-domain methods, the product calculations by Fourier transform from spatial-domain convolution, according to which not only do downward continuation factors have amplification effects, but also errors from the discretization and truncation of the Fourier transform cause oscillations in results. Improved methods, such as regularization filtering methods and generalized inverse methods, according to which although the stabilities of these downward continuations are improved, their downward continuation depths are not significant (generally no more than 5 times the measured interval); the integral iteration method, according to which stable results can be achieved for noise-free data and the depths of its downward continuation are large, but its number of iteration is giant, resulting in the reducing of computational efficiency and the accumulation of noises; Adams-Bashforth methods and Milne methods established by numerical solutions of the mean-value theorem, according to which they are easy to calculate and with greater depth of downward continuation (more than 15 times the measured interval). However, measured vertical derivatives are needed use to improve their accuracy.

As the coverage of measured vertical derivatives is low and their costs are high in real gravity explorations of geological mineral resources, which means it is not always possible to utilize measured vertical derivatives. To widen the real application for downward continuation methods of numerical solutions, instead of the measured vertical derivatives, we use the calculated ones by the ISVD (integrated second vertical derivative) method. At the same time, to improve the accuracy of the result using calculated derivatives, we present two new methods, Adams-Moulton and Milne-Simpson, based on implicit expressions of numerical solutions of the mean-value theorem for gravity anomaly downward continuation. These two methods have mathematical significance for improving the accuracy of numerical solutions. To demonstrate their effectiveness, we compare these four methods for downward continuation in the same degree including an Adams-Bashforth method, a Milne method, an Adams-Moulton method and a Milne-Simpson method by texting on the synthetic and real data of gravity exploration. The results show that the two implicit methods have higher accuracy, which has practical significance for the resolution improvement of gravity anomaly downward continuation in exploration interpretation.