大地测量与地球物理中病态性问题的正则化迭代解法

Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics

大地测量与地球物理中需要求解的大规模超定线性方程组常具有病态性,在使用共轭梯度法求解时必须克服病态性的危害影响,本文利用正则化思想改进了共轭梯度解法。首先通过构造干扰源向量,推导了与法方程同解且病态性大为减弱的新的解算方程,然后用共轭梯度迭代法对新方程求解,最后通过航空重力向下延拓等数值试验验证了新解法的有效性,并且将其与最小二乘法、共轭梯度法、Tikhonov等方法比较,结果表明新方法的精度较高。

In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often ill-conditioned.When the conjugate gradient method is used,their ill-conditioned effects to the solutions must be overcome.By regularization ideas,the conjugate gradient method is improved.Firstly,by constructing the interference source vector,a new equation is derived with ill-condition diminished greatly,which has the same solution to the original normal equation.Then the new equation is solved by conjugate gradient method.Finally,the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface.In the numerical experiments,the new method is compared with LS,CG and Tikhonov methods,and its accuracy is the highest.