中文摘要：

假定位场向下延拓的系数矩阵为正定的条件下,直接采用共轭梯度法求解位场向下延拓的第一类Fredholm型积分方程。理论模型试验表明：该方法收敛速度快,但抑噪能力较差。与位场向下延拓的积分迭代法相比,该方法收敛速度较快、发散也较快、适用性较差。为获得稳定近似解,将此不适定问题实施正则化过程,转化为最小二乘求极小值问题,再采用共轭梯度法迭代求解,实现抑噪能力较强的位场向下延拓的CGNR法。理论模型检验表明：位场向下延拓的CGNR法抑制噪声能力较强,并且相比同样具有抑制噪声能力较强的最小二乘最速下降法,CGNR法收敛速度很快,具有明显的计算优势。

英文摘要：

The first kind of Fredholm integral equation in the downward continuation was solved by adopting conjugate gradient method with the assumption that the coefficient matrix is positive definite. The theoretical model test shows that downward continuation has a very fast convergence rate but suggests poor ability to compress the noise. Compared with integral iteration method, conjugate gradient method indicates a faster convergence rate, as well as divergence rate, and is not so applicable. In order to acquire stable solutions the regularization of the operator of the ill-posed problem was carried out and the problem was transformed to solve the minimum value of the least squares estimations. After using the conjugate gradient method to solve the least squares problem, a method was implemented for downward continuation based on CGNR method that has a better ability to suppress noise. Model test demonstrates that this method has strong ability in restraining noise. This method convergence fast and has obvious computation advantage by a comparison with least square steepest descent method.