Journal of Earth System Science ( IF 1.3 ) Pub Date : 2021-09-04 , DOI: 10.1007/s12040-021-01664-5 M Zulima Fernández-Muñiz 1 , J L G Pallero 1, 2 , Juan L Fernández-Martínez 1 , Tapan Mukerji 3
Abstract
In this paper, we present a modified version of the Pereyra–Rosen algorithm to the solution of ill-conditioned linear and nonlinear inverse problems arising in gravimetry. We perform a sensitivity analysis of the solution to the two main tuning parameters of this algorithm, comparing its solution with LSQR and the truncated SVD. First, we show the application to a general-purpose linear system whose system matrix has been created by conditional simulation and whose solution is known and to a synthetic 1D gravimetric problem for two different geological set-ups (smooth Gaussian and blocky geophysical anomalies) in the noise-free and noisy cases. The Pereyra–Rosen algorithm provides very good results using a reduced number of column vectors of the system matrix. We finally show the fast inversion of a real gravity profile in the Atacama Desert (north Chile). The algorithm is well suited for the solution of large-scale ill-conditioned linear problems as the ones encountered in the fine discretization of continuous linear inverse problems in several dimensions and in the iterative linearization of nonlinear inverse problems in several fields of geosciences.
Research Highlights
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Implementation of a modified Pereyra–Rosen algorithm and its application to find the solution of linear ill-conditioned systems that arise in integral equations, in different geophysical linear inverse problems, and in the linearization of nonlinear inverse problems.
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Comparation of the modified Pereyra–Rosen algorithm with LSQR and SVD algorithms for different synthetic problems with and without noise in data.
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Sensitivity analysis of the Pereyra–Rosen algorithm, depending to the values of the ORTP and SUPER parameters in order to obtain good solutions.
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Application of Pereyra–Rosen algorithm, LSQR and the truncated SVD algorithms methods to a nonlinear inverse problem in gravity inversion with real data from the Atacama Desert, observing similar results for all of them.
中文翻译：
通过 Pereyra-Rosen 算法的修改版本快速反演重力剖面
摘要
在本文中，我们提出了 Pereyra-Rosen 算法的改进版本，用于解决重力测量中出现的病态线性和非线性逆问题。我们对该算法的两个主要调整参数的解进行敏感性分析，将其解与 LSQR 和截断的 SVD 进行比较。首先，我们展示了对通用线性系统的应用，该系统的系统矩阵是通过条件模拟创建的，其解是已知的，以及对两种不同地质设置（平滑高斯和块状地球物理异常）的合成一维重力问题的应用。无噪音和嘈杂的情况。Pereyra-Rosen 算法使用减少的系统矩阵列向量提供了非常好的结果。我们最终展示了阿塔卡马沙漠（智利北部）真实重力剖面的快速反演。该算法非常适合求解大规模病态线性问题，如多维连续线性逆问题的精细离散化和地球科学多个领域非线性逆问题的迭代线性化中遇到的问题。
研究亮点
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改进的 Pereyra-Rosen 算法的实现及其在积分方程、不同地球物理线性逆问题和非线性逆问题的线性化中出现的线性病态系统的求解的应用。
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比较改进的 Pereyra-Rosen 算法与 LSQR 和 SVD 算法对于不同的合成问题，数据中有无噪声。
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Pereyra-Rosen 算法的敏感性分析，取决于 ORTP 和 SUPER 参数的值，以获得良好的解决方案。
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将 Pereyra-Rosen 算法、LSQR 和截断 SVD 算法方法应用于重力反演中的非线性反演问题，使用来自阿塔卡马沙漠的真实数据，观察到所有这些的相似结果。
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