METHODS FOR CALCULATING FRECHET DERIVATIVES AND SENSITIVITIES FOR THE NONLINEAR INVERSE PROBLEM - A COMPARATIVE-STUDY
METHODS FOR CALCULATING FRECHET DERIVATIVES AND SENSITIVITIES FOR THE NONLINEAR INVERSE PROBLEM - A COMPARATIVE-STUDY
复制标题
DOI：
10.1111/j.1365-2478.1990.tb01859.x
复制
发表时间：
1990-07-01
影响因子：
2.6
通讯作者：
OLDENBURG, DW
中科院分区：
文献类型：
--
作者：
MCGILLIVRAY, PR;OLDENBURG, DW
在大多数非线性反问题的解决方案的一个基本步骤是建立一个拟议的模型的变化之间的关系，并在正演模拟数据的变化。一旦建立了这种关系，就有可能改进初始模型，以获得与观测数据的更好拟合。在线性化分析中，Fréchet导数是模型变化和数据变化之间的联系。在某些简单的情况下，可以导出弗雷歇导数的解析表达式。在本文中，我们提出了三种技术来实现这一点，并说明他们通过计算的Fréchet导数的ID电阻率问题。对于更复杂的问题，如果无法获得Fréchet导数的表达式，则需要对模型进行参数化，并数值求解数据对模型参数的灵敏度-偏导数。讨论了计算一阶灵敏度的标准摄动法，并与更有效的灵敏度方程和伴随方程法进行了比较。还提出了允许计算高阶、方向和目标函数灵敏度的扩展。最后，这些不同的技术的应用都说明了1D和2D电阻率问题。
AbstractA fundamental step in the solution of most non‐linear inverse problems is to establish a relationship between changes in a proposed model and resulting changes in the forward modelled data. Once this relationship has been established, it becomes possible to refine an initial model to obtain an improved fit to the observed data. In a linearized analysis, the Fréchet derivative is the connecting link between changes in the model and changes in the data. In some simple cases an analytic expression for the Fréchet derivative may be derived. In this paper we present three techniques to accomplish this and illustrate them by computing the Fréchet derivative for the ID resistivity problem. For more complicated problems, where it is not possible to obtain an expression for the Fréchet derivative, it is necessary to parameterize the model and solve numerically for the sensitivities ‐ partial derivatives of the data with respect to model parameters. The standard perturbation method for computing first‐order sensitivities is discussed and compared to the more efficient sensitivity‐equation and adjoint‐equation methods. Extensions to allow for the calculation of higher order, directional and objective function sensitivities are also presented. Finally, the application of these various techniques is illustrated for both the 1D and 2D resistivity problems.