For many geophysical measurements, such as direct current or electromagnetic induction methods, information fades away with depth. This has to be taken into account when interpreting models estimated from such measurements. For that reason, a measurement sensitivity analysis and determining the depth of investigation are standard steps during geophysical data processing. In deterministic gradient-based inversion, the most used sensitivity measure, the differential sensitivity, is readily available since these inversions require the computation of Jacobian matrices. In contrast, differential sensitivity may not be readily available in Monte Carlo inversion methods, since these methods do not necessarily include a linearization of the forward problem. Instead, a prior ensemble is used to simulate an ensemble of forward responses. Then, the prior ensemble is updated according to Bayesian inference. We propose to use the covariance between the prior ensemble and the forward response ensemble for constructing sensitivity measures. In Monte Carlo approaches, the estimation of this covariance does not require additional computations of the forward model. Normalizing this covariance by the variance of the prior ensemble, one obtains a simplied regression coefficient. We investigate differences between this simplified regression coefficient and differential sensitivity using simple forward models. For linear forward models, the simplied regression coefficient is equal to differential sensitivity, except for the influences of the sampling error and of the correlation structure of the prior distribution. In the non-linear case, the behaviour of the simplified regression coefficient as sensitivity measure is analysed for a simple non-linear forward model and a frequency-domain electromagnetic forward model. [...]

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蒙特卡罗集合统计的灵敏度和调查深度

对于许多地球物理测量，例如直流或电磁感应方法，信息会随着深度而逐渐消失。在解释根据此类测量估计的模型时，必须考虑到这一点。因此，测量灵敏度分析和确定调查深度是地球物理数据处理过程中的标准步骤。在确定性基于梯度的反演中，最常用的灵敏度度量，即微分灵敏度，很容易获得，因为这些反演需要计算雅可比矩阵。相比之下，蒙特卡罗反演方法中可能不容易获得微分灵敏度，因为这些方法不一定包括前向问题的线性化。相反，先验集成用于模拟前向响应的集成。然后，根据贝叶斯推理更新先验集成。我们建议使用先验集成和前向响应集成之间的协方差来构建灵敏度度量。在蒙特卡罗方法中，这种协方差的估计不需要对前向模型进行额外的计算。通过先验集合的方差对这种协方差进行归一化，可以得到一个简化的回归系数。我们使用简单的前向模型研究了这个简化的回归系数和差异敏感性之间的差异。对于线性前向模型，除抽样误差和先验分布的相关结构的影响外，简化回归系数等于微分灵敏度。在非线性情况下，针对简单的非线性正演模型和频域电磁正演模型，分析了作为灵敏度度量的简化回归系数的行为。[...]