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 ...

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Small-loop frequency-domain electromagnetic (FDEM) devices measure a secondary magnetic field caused by the application of a stronger primary magnetic field. Both the in-phase and quadrature component of the secondary field commonly suffer from systematic measurement errors, which would result in a non-zero response in free space. The inphase response is typically strongly correlated to subsurface magnetic susceptibility. Considering common applications on weakly to moderately susceptible grounds, the in-phase component of the secondary field is usually weaker than the quadrature component, making it relatively more prone to systematic errors. Incorporating coil-specific offset parameters in a probabilistic inversion framework, we show how systematic errors in FDEM measurements can be estimated jointly with electrical conductivity and magnetic susceptibility. Including FDEM measurements from more than one height, the offset estimate becomes closer to the true offset, allowing an improved inversion result for the subsurface magnetic susceptibility.

Often, multiple geophysical measurements are sensitive to the same subsurface parameters. In this case, joint inversions are mostly preferred over two (or more) separate inversions of the geophysical data sets due to the expected reduction of the non-uniqueness in the joint inverse solution. This reduction can be quantified using Bayesian inversions. However, standard Markov chain Monte Carlo (MCMC) approaches are computationally expensive for most geophysical inverse problems. We present the Kalman ensemble generator (KEG) method as an efficient alternative to the standard MCMC inversion approaches. As proof of concept, we provide two synthetic studies of joint inversion of frequency domain electromagnetic (FDEM) and direct current (DC) resistivity data for a parameter model with vertical variation in electrical conductivity. For both studies, joint results show a considerable improvement for the joint framework over the separate inversions. This improvement consists of (1) an uncertainty reduction in the posterior probability density function and (2) an ensemble mean that is closer to the synthetic true electrical conductivities. Finally, we apply the KEG joint inversion to FDEM and DC resistivity field data. Joint field data inversions improve in the same way seen for the synthetic studies.

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