标题： 一种使用样本扩展的半解析但有偏的不确定性评估方法，用于非线性旅行时间层析成像 显示英文标题

标题： A Semi-analytic but Biased Uncertainty Assessment Method using Sample Extensions, Analysed for Nonlinear Travel Time Tomography

摘要： 许多地球物理问题可以表述为反问题，这些反问题从观测数据中估计一组参数值。 在贝叶斯框架下，这些问题的解通过所谓的后验概率分布函数（pdf）以概率方式描述。 为了获得稳健的推断结果，通常需要抽取数百万个模型参数值样本并进行模拟；这是一个计算成本高昂的过程。 我们研究了样本扩展的概念，作为一种在解决完全非线性反问题时提高效率的方法。 一个样本的扩展被定义为一组模型或参数值，其正向函数值可以直接从一个已经计算过正向函数的样本中获得，从而避免了额外的正向函数计算。 在一个用于地震旅行时间层析成像的第一到达旅行时间计算的特定情况下，我们将样本扩展应用于获得参数空间中具有非零超体积的连续区域，其中所有位置的正向函数值仅需一次正向模拟即可确定。 我们设计了一种确定性采样技术，通过求解一个优化问题来识别最有信息量的扩展。 在一个涉及单个旅行时间数据的示例层析成像中，我们找到了51个最优样本，并利用它们构建了一个贝叶斯后验pdf的解析近似。 此外，我们提出了一种基于扩展的算法，用于实际层析成像场景，并将其应用于一个合成的二维示例。 本研究突出了使该方法效率低下的两个基本问题：(1) 扩展的超体积有限，(2) 为了简化解析计算而忽略了参数相关性。 解决这些问题将为未来的研究提供可能的方向。 显示英文摘要

摘要： Many geophysical problems can be cast as inverse problems that estimate a set of parameter values from observed data. Within a Bayesian framework, solutions to such problems are described probabilistically by the so-called posterior probability distribution functions (pdf's). To obtain robust inference results often requires millions of model parameter value samples to be drawn, and their simulation to be performed; this is a computationally expensive procedure. We investigate the concept of sample extensions as a means to improve efficiency when solving fully nonlinear inverse problems. A sample's extension is defined as the set of models or parameter values whose forward function values are directly accessible from a sample for which the forward function has already been evaluated, obviating the need for additional forward function evaluations. In a specific case of first-arrival travel time calculations used in seismic travel time tomography, we apply sample extensions to obtain a continuous region with non-zero hypervolume within parameter space, across all of which the forward function values are known given only a single forward simulation. We devise a deterministic sampling technique that identifies the most informative extensions by solving an optimisation problem. In an illustrative tomographic example that involves a single travel time datum, we find 51 optimal samples, and use them to construct an analytic approximation to the Bayesian posterior pdf. Additionally, we propose an extensions-based algorithm for real-world tomography scenarios and apply it to a synthetic 2D example. This study highlights two fundamental problems that make the method inefficient: (1) limited hypervolumes of extensions and (2) neglecting parameter correlations to simplify analytic calculations. Finding solutions to these problems defines possible directions for future research.