将Occam反演与lasso融合、过完备字典及各向同性总变分正则化相结合
Extending Occam's inversion with lasso fusion, overcomplete dictionaries, and isotropic total variation regularisation
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中文总结 AI 辅助
本文将Occam反演与lasso融合、过完备字典及各向同性总变分正则化结合，揭示三种l1正则化求解器的数学原理，为地球物理反演推荐了高效求解方法，以鼓励该领域采用l1正则化反演。
中文摘要 AI 辅助
Occam反演是一种用于非线性地球物理反演的稳健算法，可在观测噪声范围内提供最平滑的模型，从而避免对地质情况的过度解释。最初的Occam反演采用l2范数惩罚模型粗糙度，而l1范数可用于生成视觉上更清晰的模型。不过，采用l1正则化的地球物理反演已与统计学和成像领域的主流研究脱节，例如多维度下采用差分算子（即总变分）的l1正则化存在两种不同形式，其中仅一种对边缘方向具有不变性，这一区别在地球物理学中常被忽略；在一维场景下，该问题可简化为统计学中的融合lasso问题。在Occam框架中，针对l2数据范数与l1模型正则化的情况，研究表明通过合成或分析方式实现的lasso融合会得到相同的一维问题。本文将该框架扩展至多维度，并应用于字典中的小波变换等算子，揭示了三种常用l1正则化问题求解器背后的数学原理：坐标下降法、迭代重加权最小二乘法（IRLS）及分裂Bregman法。研究人员利用这些求解器求解一系列线性问题（一维回归、二维去模糊）和非线性问题（一维航空瞬变电磁法），并包含一个野外数据示例。研究建议一维问题采用坐标下降法，二维问题采用IRLS而非分裂Bregman法，IRLS所需的最小二乘求解次数最多可减少一个数量级。本文通过将l1正则化反演的各种形式呈现为熟悉的Occam反演，希望进一步鼓励地球物理学家采用l1正则化反演。
英文摘要
Occam's inversion is a robust algorithm to perform nonlinear geophysical inversion. It provides the smoothest model within observation noise, thereby discouraging geological overinterpretation. While Occam originally penalised l2 model roughness, l1 can be used to provide models that are visually sharp. However, l1 regularised geophysical inversion has diverged from the larger body of statistics and imaging literature. For example, l1 regularisation with a difference operator (i.e., total variation) in multiple dimensions has two distinct forms, only one of which is invariant to edge orientation -- a distinction often overlooked in geophysics. In one dimension this reduces to the fused lasso problem in statistics. Within the Occam framework, for l2 data norm and l1 model regularisation, we show that lasso fusion through either synthesis or analysis leads to the same 1D problem. We extend the framework to multiple dimensions and to operators such as wavelet transforms in a dictionary, unravelling the mathematics behind three commonly used solvers for l1 regularised problems. These are coordinate descent, iteratively reweighted least squares (IRLS), and split Bregman. We use them to solve a sequence of problems that are linear (1D regression, 2D deblurring) and nonlinear (1D airborne transient electromagnetics) including a field data example. We recommend coordinate descent for 1D problems, and IRLS over split Bregman for 2D problems, with IRLS requiring up to an order of magnitude fewer least squares solves. We hope this work will further encourage geophysicists to adopt l1 regularised inversion, by presenting its various forms as a familiar Occam's inversion.
