How to Determine Spatial Resolution for an Inverse Problem
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摘要: 反演问题的时空间分辨率或称时空分辨长度是评估模型精细程度的重要参数,决定了该模型应用的范围和价值,但是分辨长度估算却是比反演更复杂和麻烦的数学问题。除了层析成像中广泛利用理论模型恢复试验定性提取空间分辨长度外,通过求解分辨率矩阵可定量获得分辨长度。通过矩阵操作给出的分辨率矩阵包括三类:直接分辨率矩阵、正则化分辨率矩阵和混合分辨率矩阵。这三类矩阵包含了反演本身不同侧面的信息,因此在一个反演应用中,同时提供这三类分辨率矩阵可更全面地评估反演模型分辨率分布。最近An(2012)提出了从大量随机理论模型及其解中统计出分辨率矩阵的方法。这种分辨率矩阵是从模拟真实反演实验的输入和输出模型中通过反演得到的,因此这种分辨率矩阵更能反映整个反演所涉及到的更多因素和过程;同时由于这种分辨率矩阵计算过程无需进行矩阵操作且不依赖于具体正演和反演方法,因此可以被应用于更普遍的反演问题。实际应用证明统计分辨率分析方法适用于对二维和三维层析成像反演模型进行分辨率分析。Abstract: The information of solution's spatial resolution is important for model appraisal in an inversion, however, the computation to determine a spatial resolution is nontrivial and often more difficult than to solve an inverse problem. Visual inspection of the restoration of a synthetic structure widely applied in tomographic studies can give indicative information on spatial resolution distribution, however, resolution matrix estimation can give quantitative information of spatial resolution length for a general inverse problem. Resolution matrices obtained by matrix operation may be divided into three classes: direct resolution matrix, regularized resolution matrix and hybrid resolution matrix. Each matrix can give part of the information on the inversion, and then the simultaneous implementation of all three resolution matrices in a single study can potentially provide a complete understanding on the resolution length information. An (2012) proposed a new class of resolution matrices generated by a simple one-parameter nonlinear inversion performed based on limited pairs of random synthetic models and their inverse solutions. The estimates were directly retrieved from synthetic models and their inverse solutions, and then it can include the information on the whole inversion procedure; The independence on the degree of inverse skill used and the absence of a requirement for matrix operations indicated that this approach is particularly suitable for very large linear/linearized inverse problems. Inversion examples even for 3D inversion problem demonstrated that reasonable resolution lengths can be determined from statistic spatial resolution calculation.
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Keywords:
- resolution length /
- resolution matrix /
- inverse problem /
- model appraisal
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