The Anatomy of Inverse Problems
The Anatomy of Inverse Problems
复制标题
反问题的剖析
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
R. Snieder
中科院分区：
文献类型：
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作者：
J. Scales;R. Snieder
地球物理学的一个主要任务是对地球内部进行定量描述。因此，反问题是地球物理研究和工业应用的一个重要领域。图1显示了有多少文本呈现了逆问题。地球模型是数学空间的一个元素，包含了所有允许的地球属性（或至少与给定实验相关的属性）的参数化;这个空间被称为模型空间。问题的物理性质决定了哪些数据d对应于给定的模型m。计算给定模型的模型响应（合成“数据”）的问题称为正演问题。相应的数据驻留在称为数据空间的数学空间中。在许多应用程序中，人们记录数据，目标是找到相应的模型。该任务称为逆问题，如图1所示。
图1.
逆问题的传统观点是：找到预测测量值的模型。
不幸的是，图1是错误的。原因很简单。一般来说，人们所寻求的模型是具有无限多个自由度的空间变量的连续函数。例如，地球中的三维速度结构具有无限多个自由度。另一方面，数据空间总是有限维的，因为任何真实的实验只能导致有限数量的测量。变量的简单计数表明，从数据到模型的映射不可能是唯一的;或者等价地，模型空间中必须存在对数据没有影响的元素。即使对于涉及理想化的无噪声测量的问题，这种唯一性的缺乏也是显而易见的。当真实的的不确定性.
A major task of geophysics is to make quantitative statements about the interior of the earth. For this reason, inverse problems are an important area of geophysical research and industrial application. Figure 1 shows how many texts present inverse problems. The earth model is an element of a mathematical space that contains all allowable parameterizations of the earth’s properties (or at least those properties relevant to a given experiment); this space is referred to as model space . The physics of the problem determines which data d correspond to a given model m . The problem of computing the model response (synthetic “data”) given a model is called the forward problem . The corresponding data reside in a mathematical space that is called data space . In many applications, one records the data, and the goal is to find the corresponding model. The task is called the inverse problem , as shown in Figure 1.
FIG. 1.
The conventional view of inverse problems: find the model that predicts the measurements.
Unfortunately, Figure 1 is wrong. There is a simple reason for this. In general the model that one seeks is a continuous function of the space variables with infinitely many degrees of freedom. For example, the 3-D velocity structure in the earth has infinitely many degrees of freedom. On the other hand, the data space is always of finite dimension because any real experiment can only result in a finite number of measurements. A simple count of variables shows that the mapping from the data to a model cannot be unique; or equivalently, there must be elements of the model space that have no influence on the data. This lack of uniqueness is apparent even for problems involving idealized, noise-free measurements. The problem only becomes worse when the uncertainties of real …