基于微分中值定理数值解的重力场向下延拓方法研究

Downward continuation is a high-resolution technique, which meets the requirement of high accuracy of the gravity data processing and interpretation. It is the key to present precise and detailed underground density distribution. However, inaccuracy, instability and small continuation depth seriously restrict its application. The numerical solutions of the differential mean-value theorem with the characteristics of accuracy, stability and large step-size can solve these problems of the traditional downward continuation methods. But the numerical solution method is different from the traditional downward continuation methods, and its theoretical research and analysis are still insufficient. This project will improve the downward continuation theory based on the numerical solutions of the differential mean-value theorem to ensure the accuracy of downward continuation. We develop new methods based on this theory and do downward continuation to different gravity data with existing methods. This optimizes the selection of various methods to make the results more stable. We quantitatively analyze the truncation error and spectral characteristics of the numerical solutions, and we further evaluate the advantages of this method in solving the downward continuation problems of accuracy, stability and depth. And we realize the application of this method on the condition of the calculated vertical gradient. This project will eventually establish a set of downward continuation theoretical methods based on numerical solutions of the differential mean-value theorem to solve the downward continuation problems. This will provide technical support for the study of the precise and detailed structure of the Earth's interior.

高分辨向下延拓技术可满足重力数据处理与解释的高精度需求，是精细刻画地下密度分布的关键。然而传统方法存在不准确、不稳定、延拓深度小等问题，严重制约了向下延拓应用效果。具有准确、稳定和步长大等特点的微分中值定理数值解可以改善这些问题，这一方法不同于传统向下延拓方法，目前对其理论研究和分析仍显不足。本项目将完善基于微分中值定理数值解的向下延拓理论，确保向下延拓的准确性；开发基于该理论的新方法，综合已有方法对不同类型重力数据进行向下延拓，优化方法的选择，提高结果的稳定性；定量分析数值解的截断误差、频谱特征等，评价这种方法在改进向下延拓的稳定性、延拓深度等方面的优势；实现该方法在换算垂向梯度条件下的应用。本项目最终将形成一套改善向下延拓现有问题的基于微分中值定理数值解的重力场向下延拓理论方法，有望为研究地球内部精细结构提供技术支撑。

针对矿产资源勘探、匹配导航等领域的实际问题，提高重力场数据处理和解释结果的精度和可靠性，具有重要应用价值和意义。向下延拓能够突出局部、浅部的异常体信息，进而提高数据处理和解释的分辨能力，在重力场数据处理和解释中起到十分关键的作用。但是传统方法存在不准确、不稳定、延拓深度小等问题，制约了向下延拓的实际应用。为了解决上述问题，本项目提出了多种准确、稳定、大深度且抗噪声能力强的向下延拓新方法。为了确保研究的准确性，本项目完善基于微分中值定理数值解的本项目向下延拓理论；提出基于微分中值定理数值解的Adams-Bashforth-Moulton“预估矫正”、Milne-Simpson“预估矫正”、Runge-Kutta等向下延拓新方法；为了提高结果的稳定性和延拓深度，本项目综合已有和新提出的方法，对不同类型重力数据向下延拓，定量分析数值解的截断误差、数据拟合误差等，评价不同方法在改进稳定性、延拓深度等方面的优势，选择优化的向下延拓方案；利用ISVD方法换算垂向梯度，实现本项目新提出的方法在有限条件下的应用。本项目最终将形成一套改善向下延拓现有问题的基于微分中值定理数值解的重力场向下延拓理论与方法，有望为研究地球内部精细结构提供技术支撑。