Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model
Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model
复制标题
高维变分反演的改进分析误差协方差矩阵：使用 3D 大气传输模型进行源估计的应用
DOI：
10.1002/qj.2495
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发表时间：
2015
影响因子：
8.9
通讯作者：
Dylan B. A. Jones
中科院分区：
文献类型：
--
作者：
N. Bousserez;D. Henze;A. Perkins;K. W. Bowman;Meemong Lee;Junjie Liu;Feng Deng;Dylan B. A. Jones
变分方法广泛用于解决地球物理反问题。尽管基于梯度的最小化算法可用于高维问题（维度 >106），但它们不提供最优解决方案中的误差估计。在本研究中，我们假设合理的线性模型，评估了几种近似分析误差协方差矩阵的数值方法的性能。使用 CO2 柱的综合遥感观测对 CO2 通量估计问题进行评估。考虑低维实验，以便将分析误差近似与全秩有限差分逆 Hessian 估计进行比较，然后进行实际的高维应用。两种随机方法，蒙特卡罗模拟和基于成本函数随机梯度的方法，产生的分析误差方差相对误差<10%。对于基于梯度的随机化，由于采样噪声导致的长距离误差相关性明显不那么明显，这在并行实现时也特别有吸引力。还测试了使用 Broyden–Fletcher–Goldfarb–Shanno (BFGS) 算法对逆 Hessian 矩阵的确定性评估。虽然现有的 BFGS 预处理技术产生的误差方差近似值很差（相对误差 >120%），但一种新的预处理器可以有效地累积逆 Hessian 对角线上的信息，从而显着改善结果（相对误差 <50%）。此外，使用相同的梯度和向量对执行 BFGS 算法的多个循环可以增强其性能（相对误差 <30%），并且对于获得收敛是必要的。利用这些发现，我们提出了一种 BFGS 混合方法，该方法使用来自一些（3-5）蒙特卡罗模拟的信息将新的预处理器与多个 BFGS 循环相结合。其性能与低维情况下的随机近似相当，而高维实验则获得了良好的可扩展性。这些新的 BFGS 方法的潜在应用范围从表征高维反问题的信息内容到提高当前最小化算法的收敛速度。
Variational methods are widely used to solve geophysical inverse problems. Although gradient‐based minimization algorithms are available for high‐dimensional problems (dimension >106), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis‐error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO2 flux estimation problem using synthetic remote‐sensing observations of CO2 columns. A low‐dimensional experiment is considered in order to compare the analysis error approximations to a full‐rank finite‐difference inverse Hessian estimate, followed by a realistic high‐dimensional application. Two stochastic approaches, a Monte‐Carlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error <10%. The long‐distance error correlations due to sampling noise are significantly less pronounced for the gradient‐based randomization, which is also particularly attractive when implemented in parallel. Deterministic evaluations of the inverse Hessian using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm are also tested. While existing BFGS preconditioning techniques yield poor approximations of the error variances (relative error >120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error <50%). Furthermore, performing several cycles of the BFGS algorithm using the same gradient and vector pairs enhances its performance (relative error <30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a BFGS hybrid approach which combines the new preconditioner with several BFGS cycles using information from a few (3–5) Monte‐Carlo simulations. Its performance is comparable to the stochastic approximations for the low‐dimensional case, while good scalability is obtained for the high‐dimensional experiment. Potential applications of these new BFGS methods range from characterizing the information content of high‐dimensional inverse problems to improving the convergence rate of current minimization algorithms.