纯奇异一维声反射问题
The purely singular 1-D acoustic reflection problem
摘要：
本文分析了由一系列均匀水平层组成的三维半空间的反射率剖面（模型）和边界处的平面波脉冲响应（数据）之间的非线性对应关系。这种对应关系在地球物理成像中非常重要，半个多世纪以来，人们从各种角度对其进行了研究。本文的主要贡献是在有限时间确定性方法的背景下推导出一些新的东西：（i）模型中数据的精确有限（非渐近）公式，（ii）相应的精确逆算法，以及（iii）固有非线性的精确表征。关于（iii），对于一般模型，对应关系被表征为一对映射，其中一个是局部线性的，另一个是局部多项式的。两者都是由一个局部组合不变量，一个整数矩阵。关于（ii），基本的逆算法被修改，以允许错误的幅度数据，利用逆问题的超确定性，以恢复精确的模型，即使在数据严重失真的情况下。最后用数值例子说明了结果。
英文摘要：
This paper analyzes the nonlinear correspondence between the reflectivity profile (model) and the plane wave impulse response at the boundary (data) for a three-dimensional half space consisting of a sequence of homogeneous horizontal layers. This correspondence is of importance in geophysical imaging, where it has been studied for more than half a century from a variety of perspectives. The main contribution of the present paper is to derive something new in the context of a time-limited deterministic approach: (i) an exact finite (non-asymptotic) formula for the data in terms of the model, (ii) a corresponding exact inverse algorithm, and (iii) a precise characterization of the inherent nonlinearity. Regarding (iii), for generic models the correspondence is characterized as a pair of maps, one of which is locally linear, and the other of which is locally polynomial. Both are determined by a local combinatorial invariant, an integer matrix. Concerning (ii), the basic inverse algorithm is modified to allow for erroneous amplitude data, taking advantage of the overdeterminacy of the inverse problem to recover the exact model even in cases where the data is badly distorted. The results are illustrated with numerical examples.
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