Inverse modeling of large-scale spatially variable parameters fields at fine resolution can be reduced to estimating the projections on dominant principal components of the underlying parameter fields based on principal component analysis of the spatial covariance. For unknown or biased prior structural parameters of spatial covariance models, an iterative procedure consisting of two successive steps is usually implemented, i.e., estimation of spatial covariance followed by estimation of the spatially variable parameter fields conditional on the spatial covariance and observations. In this study, we develop an iterative, computationally efficient method to update dominant principal components for nonlinear inverse problems of large-scale spatial fields that adaptively corrects the bias from the initially defined prior spatial covariance. Our algorithm involves two-layer iterations: the inner iteration is to obtain the best estimates of projections on given retained principal components, and the outer iteration implements an efficient rank-one updating to correct the retained principal components using the posterior covariance associated with the best estimates of the projections. Numerical experiments show that inversion results can be significantly improved for large-scale inverse problems with biased structural parameters for spatial covariance. The experiment results show that the iterative correction is essentially to match the distribution patterns of the spatially correlated parameter field with its most dominant principal components. We also investigate the performance of the developed method under different biased covariance model initialization, including model type bias, variance bias and correlation length bias. The correction cannot fundamentally change the smoothness defined by the covariance model type, but can still describe major distribution patterns including anisotropy. Biased variance can be corrected and yields similar best estimates and variance maps, and biased correlation length can be corrected within an applicable range.

可以将高分辨率的大规模空间可变参数字段的逆模型简化为基于空间协方差的主成分分析来估计基础参数字段的主要主成分上的投影。对于未知或有偏差的空间协方差模型的先前结构参数，通常要执行由两个连续步骤组成的迭代过程，即，先估计空间协方差，然后再估计以空间协方差和观测为条件的空间可变参数字段。在这项研究中，我们开发了一种迭代的，计算效率高的方法来更新大规模空间场的非线性逆问题的主要主成分，该方法可以自适应地校正最初定义的先前空间协方差的偏差。我们的算法涉及两层迭代：内部迭代用于获取给定保留主成分上投影的最佳估计，而外部迭代使用与最佳相关的后协方差实现有效的秩一更新以校正保留主成分。预测的估算值。数值实验表明，对于空间协方差有偏的结构参数的大规模反问题，反演结果可以得到显着改善。实验结果表明，迭代校正实质上是将空间相关参数字段的分布模式与其最主要的主成分进行匹配。我们还研究了在不同的偏差协方差模型初始化（包括模型类型偏差，方差偏差和相关长度偏差。校正不能从根本上改变由协方差模型类型定义的平滑度，但仍可以描述包括各向异性的主要分布模式。可以校正偏差方差，并产生相似的最佳估计值和方差图，并且可以在适用范围内校正有偏差的相关长度。