Sensitivity and depth of investigation from Monte Carlo ensemble statistics

Abstract

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 simplified regression coefficient. We investigate differences between this simplified regression coefficient and differential sensitivity using simple forward models. For linear forward models, the simplified 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. Differential sensitivity and the simplified regression coefficient are similar for prior intervals on which the forward model response is approximately linear. Differences between the two sensitivity measures increase with the degree of non‐linearity in the prior range. Additionally, we investigate the correlation between prior ensemble and forward response ensemble as sensitivity measure. Correlation yields a normalized version of the simplified regression coefficient. We propose to use this correlation and the simplified regression coefficient for determining depth of investigation in Monte Carlo inversions.