Analysis of the Impact of Model Nonlinearities in Inverse Problem Solving
Analysis of the Impact of Model Nonlinearities in Inverse Problem Solving
复制标题
模型非线性对反问题求解的影响分析
DOI：
10.1175/2008jas2534.1
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发表时间：
2008
影响因子：
3.1
通讯作者：
D. Posselt
中科院分区：
文献类型：
--
作者：
T. Vukicevic;D. Posselt
在这项研究中，非线性模型属性和反问题的解决方案之间的关系进行了分析，使用数值技术的基础上制定的反问题理论Mosegaard和Tarantola。根据这一理论，反问题和解决方案是通过卷积和概率密度函数（PDF），代表从联合多维空间中的模型，观测和先验知识获得的随机信息的连接定义。该理论提供了一个明确的分析的非线性模型函数，连同信息的不确定性模型，观测，先验知识，通过建设的联合概率密度，从边际解函数，然后可以评估。从理论推导出的数值分析技术通过在模型和观测相空间中的离散网格上的函数映射和从已知参数分布进行的Monte Carlo采样的组合来计算离散形式的分量PDF。数值分析技术的有效性证明通过其应用到两个著名的简化模型的大气物理：阻尼振荡和洛伦茨的三分量模型的干细胞对流。本研究的主要发现包括以下几个方面：（一）在反问题中使用非单调的正向模型会产生多峰后验概率密度函数，其实现取决于观测的信息内容以及观测和模型的不确定性。(ii)观察值随时间、空间或两者的累积效应可能使最终后验PDF单峰，即使使用非单调前向模型也是如此。(iii)对于给定的控制参数空间中的自由度，在非单调非线性模型的情况下，比单调非线性或线性前向模型需要更多的独立观测来约束解。(iv)一个非线性单调的前向模型会产生一个偏单峰后验概率密度函数，这意味着一个适定的最大似然逆问题。(v)模型误差的存在大大增加了非单调非线性模型在后验概率密度函数中捕获多个模式的可能性。(vi)在非线性前向模型的情况下，使用高斯近似的先验更新具有类似的影响，增加模型误差，这表明有可能产生一个有偏的平均中心估计，即使当观察和模型是无偏的。
In this study, the relationship between nonlinear model properties and inverse problem solutions is analyzed using a numerical technique based on the inverse problem theory formulated by Mosegaard and Tarantola. According to this theory, the inverse problem and solution are defined via convolution and conjunction of probability density functions (PDFs) that represent stochastic information obtained from the model, observations, and prior knowledge in a joint multidimensional space. This theory provides an explicit analysis of the nonlinear model function, together with information about uncertainties in the model, observations, and prior knowledge through construction of the joint probability density, from which marginal solution functions can then be evaluated. The numerical analysis technique derived from the theory computes the component PDFs in discretized form via a combination of function mapping on a discrete grid in the model and observation phase space and Monte Carlo sampling from known parametric distributions. The efficacy of the numerical analysis technique is demonstrated through its application to two well-known simplified models of atmospheric physics: damped oscillations and Lorenz’s three-component model of dry cellular convection. The major findings of this study include the following: (i) Use of a nonmonotonic forward model in the inverse problem gives rise to the potential for a multimodal posterior PDF, the realization of which depends on the information content of the observations and on observation and model uncertainties. (ii) The cumulative effect of observations over time, space, or both could render the final posterior PDF unimodal, even with the nonmonotonic forward model. (iii) A greater number of independent observations are needed to constrain the solution in the case of a nonmonotonic nonlinear model than for a monotonic nonlinear or linear forward model for a given number of degrees of freedom in control parameter space. (iv) A nonlinear monotonic forward model gives rise to a skewed unimodal posterior PDF, implying a well-posed maximum likelihood inverse problem. (v) The presence of model error greatly increases the possibility of capturing multiple modes in the posterior PDF with the nonmonotonic nonlinear model. (vi) In the case of a nonlinear forward model, use of a Gaussian approximation for the prior update has a similar effect to an increase in model error, which indicates there is the potential to produce a biased mean central estimate even when observations and model are unbiased.