Probabilistic 1-D Inversion of Frequency-Domain Electromagnetic Data Using a Kalman Ensemble Generator

Abstract

Frequency-domain electromagnetic (FDEM) data of the subsurface are determined by electrical conductivity and magnetic susceptibility. We apply a Kalman Ensemble generator (KEG) to one-dimensional probabilistic multi-layer inversion of FDEM data to derive conductivity and susceptibility simultaneously. The KEG provides an efficient alternative to an exhaustive Bayesian framework for FDEM inversion, including a measure for the uncertainty of the inversion result. Additionally, the method provides a measure for the depth below which the measurement is insensitive to the parameters of the subsurface. This so-called depth of investigation is derived from ensemble covariances. A synthetic and a field data example reveal how the KEG approach can be applied to FDEM data and how FDEM calibration data and prior beliefs can be combined in the inversion procedure. For the field data set, many inversions for one-dimensional subsurface models are performed at neighbouring measurement locations. Assuming identical prior models for these inversions, we save computational time by re-using the initial KEG ensemble across all measurement locations.[7] describe how including the IP component can avoid systematic underestimation of EC when a significant IP shift is recorded.The aforementioned publications apply classical, deterministic inversion approaches yielding a single model parameter realisation [11]. We propose a pragmatic approach to probabilistic FDEM inversion by applying the ensemble-based KEG. The KEG method can be seen as a trade-off between two extreme approaches to inverse problems: (1) using a deterministic inversion that quickly finds a single model satisfying the data, and (2) computing a large number of possible parameter models in search techniques aiming to be exhaustive (e.g., Markov-chain Monte-Carlo methods [12]).Ensemble-based inversions have been proven to be efficient and robust [6], but at the same time comprehensive enough to characterize the uncertainty of the result. In the context of Kalman methods, uncertainty is characterized by standard deviations (STD) of model parameters, which are a measure for the parameter spread of the models that match the field measurements. The KEG method uses an equation equal to the update step of the Ensemble Kalman Filter [13], but it performs updates exclusively in the model parameters [6].The novelty of our work lies in the application of the KEG to the inversion of FDEM data. We update prior EC and MS ensembles simultaneously, based on the measurements of the IP and OP component of the secondary electromagnetic field. Additionally, it is shown how correlations computed in the KEG can be used as a proxy for the measurement sensitivity. Using this sensitivity proxy, we determine a depth of negligible sensitivity, the so-called depth of investigation of a particular measurement setup. The KEG provides an interesting inversion method for geophysical data, especially when moving towards large and multi-dimensional forward models.The application of the KEG is demonstrated ...

1D geological imaging of the subsurface from geophysical data with Bayesian Evidential Learning

“…Nevertheless, stochastic methods are often linked to high computation costs, since they require a large number of forward model runs to sample the posterior distribution in the prior model space, explaining why they are still often limited to scientific or numerical studies (e.g., Irving and Singha, 2010;Trainor-Guitton and Hoversten, 2011). Approximation of the posterior can be obtained (under some assumptions) through ensemble-based inversion (e.g., Bobe et al, 2019). Even more recently, machine learning also emerged as a potential contender to replace the inversion process, typically yielding to deterministic-like results due to the nature of neural networks and such (e.g., Laloy et al, 2019Laloy et al, , 2017Yang and Ma, 2019).…”

Efficient Probabilistic Joint Inversion of Direct Current Resistivity and Small-Loop Electromagnetic Data

“…The KEG was only recently adopted for the inversion of geophysical data. Bobe et al [12] applied it to the inversion of frequency-domain electromagnetic data to the probabilistic estimation of electrical conductivity and magnetic susceptibility. As for other probabilistic methods, a joint framework is readily available while using the KEG.…”

“…Highly-parametrized models result in computationally expensive forward model runs that limit the applicability of MCMC methods [11]. Only recently, efficient approximations of the posterior distribution have been proposed for geophysical problems, while using innovative methods, such as the Kalman ensemble generator (KEG, [12]) or Bayesian Evidential Learning [13]. Those techniques rely on a smaller number of forward model runs and are, thus, less expensive.…”

“…The objective of this paper is to apply the KEG for joint probabilistic inversion of DC resistivity and small-loop EM data. previously applied to the inversion of EM [12], yet, to the best of our knowledge, this is the first time that (1) the method is used for geophysical joint inversion and (2) that these types of data are jointly inverted in a probabilistic way. We present two examples of simulated geophysical measurements for synthetic one-dimensional subsurface models and one field data example showing vertical contrasts in electrical conductivity.…”

Sensitivity and depth of investigation from Monte Carlo ensemble statistics

“…A thorough compilation, featuring popular applications of Markov chain Monte Carlo methods (Mosegaard and Tarantola, 1995), can be found in Sambridge and Mosegaard (2002). Although, since the publication of their review, several new MC methods have been introduced, for example trans‐dimensional, multi‐chain or approximate Bayesian methods (Malinverno, 2002; Sambridge et al ., 2006; Socco and Boiero, 2008; Vrugt et al ., 2009; Bobe et al ., 2019).…”