浅谈反射地震走时层析中的正则化

Regularization in traveltime tomography for reflection seismic data

反射地震走时层析本质上是一个病态问题,而正则化是改善问题病态程度的有效手段.反射地震走时层析最终可归结为线性方程组的求解,本文讨论了在线性方程组求解过程中正则化的作用和方式.正则化的作用有:(1)用超定分量约束欠定分量和零空间分量;(2)用先验信息约束欠定分量和零空间分量;(3)对射线的不均匀覆盖进行阻尼;(4)对数据的不准确性进行阻尼.正则化的加入方式有:(1)加法型(将正则化矩阵补在层析矩阵后面,包括导数型正则化和零阶正则化,一阶导数型正则化对应最平坦解,二阶导数型正则化对应最光滑解,零阶正则化对应紧约束解);(2)乘法型(将正则化矩阵与层析矩阵相乘,主要包括阻尼型正则化).并利用简单的模型对正则化的效果进行了试验,发现经各种正则化约束后,与未加任何正则化约束得到的速度模型比较,尽管恢复的异常体的幅度不如后者大,但得到的速度剖面要平滑得多,更利于后续的射线追踪正演和层析反演.

In essence, the traveltime tomography in reflection seismic belongs to morbid problem,and regularization is an effective means to modify the morbid degree of problem. In this paper, the functions and the fashions of regularization are discussed when traveltime tomography is reduced to the solution of linear equation system, the functions of regularization include： constraint components which are underdetermined or lie in null space using overdetermined components and all kinds of known information; treat the uneven coverage of rays and the inaccurate datas by damping. The fashions of adding the regularization in the solution of tomography equation system include： addtion style, meaning padding the regularization equation system after the tomography equation system, derivative regularization and tight constraint regulariztion belong to this style; multiplication style, damping regularization belongs to this style. Furthermore the function of reguralization is tested for several regularization fashions using model, and the conclusion is that the model is smoother after regularization constraint, which is favorable for the following ray-tracing modeling and tomography inversion.