Abstract

A model of the medium, which makes it possible to use information more rationally for solving inverse problems in comparison with the well-known models of a layered and quasi-layered medium is introduced. A two-dimensional medium in which the fields are described by the Helmholtz equation is studied. A linearized statement of the problem of reconstructing the parameters of the medium, the inverse problem for the Helmholtz equation, is considered. The conditions for the uniqueness of the detection of layers are established. Examples of the ambiguity of the solution of the inverse problem according to information that initially even seemed redundant for the unique recovery of the medium are given. Algorithms and calculations for determining the characteristics of powerful layers are presented. Methods of interpreting the information known for a finite set of frequencies are proposed. The natural assumption about the possibility of restoring the n-layer medium from information at n + 1 frequencies is verified. It turns out that it is not possible to determine n conductivities and 2n boundaries, i.e., n functions and 2n numbers, from n + 1 functions, even if these n + 1 functions are specified by a large number of parameters. It is found that the n-layer medium can be restored from information known for 2n frequencies.

中文翻译：

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使用中间模型远程确定功能强大的层的参数

摘要

介绍了一种介质模型，与众所周知的分层和准分层介质模型相比，该模型可以更合理地使用信息来解决逆问题。研究了用亥姆霍兹方程描述场的二维介质。考虑了重构介质参数问题的线性化说明，即亥姆霍兹方程的反问题。建立了唯一的层检测条件。给出了根据最初甚至看起来对于介质的唯一恢复似乎是多余的信息来解决反问题的模棱两可的示例。介绍了确定强大层特征的算法和计算。提出了解释对于有限频率集合已知的信息的方法。关于恢复可能性的自然假设验证了来自n + 1个频率的信息的n层介质。事实证明，这是不可能确定Ñ电导率和2 Ñ边界，即，Ñ功能和2 Ñ号码，从N + 1层的功能，即使这些N +是由大量的参数指定的1层的功能。发现可以从已知的2 n个频率的信息中恢复n层介质。