解非线性反演问题的新策略

A new strategy of solving non-linear inverse problem

1. 中国新星石油公司华东石油局第六物探大队,南京 210009;2. 中国新星石油公司石油物探研究所,南京 210014

The Sixth Brigade of Geophysical Prospecting, East China 210009

本文提出了一种解非线性反演问题的新策略。它首先利用贝叶斯反演理论中的MAP方法建立非线性反演问题的线性化迭代格式, 然后运用模拟退火算法中有关全局收敛的思想来指导迭代的收敛过程, 同时还在迭代中根据迭代解估计的收敛性态对解估计加入随机扰动来进一步保证迭代解估计的全局收敛性。这种新的非线性反演策略不仅能克服常规线性化迭代反演方法（如非线性阻尼最小二乘法）要求初始解估计好, 否则易陷入局部最优解估计“陷阱”的缺点, 而且较模拟退火算法又具有计算量小的优点, 是一种比较好的解非线性反演问题的策略。尤其适用于先验知识较少的非线性反演问题, 具有很强的普适性。与常规的非线性阻尼最小二乘法比较, 本文提出的反演策略深入讨论了线性化误差对迭代解估计的作用。并在此基础上, 不仅对阻尼因子在迭代中的取值提出了一种新方案, 而且对迭代解估计在迭代中的走向和局部最优解估计提出了一些新认识。

This paper presents a new strategy to solve non-linear inverse problem. Firstly the MAPmethod in the Bayes theory is used to construct a linearized iterative algorithm. The idea of global convergence in simulated annealing algorithm is applied to guide the convergence process of iteration. Meanwhile the global convergence of iterative solution estimate is further guaranteed by adding random purturbation in solution estimate according to its convergence situation. In conventional linearized iterative inversion methoods, a good initial solution estimate is needed, or else the iteration may easily fall into a pitof locally optimal solution. The new strategy overcomes this drawback, and spends less computationalamount than the simulated annealing algorithm. It is adaptive specially to the non-linear inverse problemwith little priori information.The effect of linearized error on iterative solution estimate is discussed deeply through comparing theinversion strategy with the non-linear damped least-square methed. On the basis of this, the authars putforward viewpoints about the selection of damping factor, the trend of iterative solution estimate, andthe locally optimal solution estimation.