Inversion of electromagnetic induction data using a novel wavelet-based and scale-dependent regularization term

Abstract

Summary The inversion of electromagnetic induction data to a conductivity profile is an ill-posed problem. Regularization improves the stability of the inversion and a smoothing constraint is typically used. However, the conductivity profiles are not always expected to be smooth. Here, we develop a new inversion scheme in which we transform the model to the wavelet space and impose a sparsity constraint. This sparsity constrained inversion scheme will minimize an objective function with a least-squares data misfit and a sparsity measure of the model in the wavelet domain. A model transform to the wavelet domain allows to investigate the temporal resolution (periodicities at different frequencies) and spatial resolution (location of the peaks) characteristics of the model, and penalizing small-scale coefficients effectively reduces the complexity of the model. The novel scale-dependent regularization term can be used to favour either blocky or smooth structures, as well as high-amplitude models in globally smooth structures in the inversion. Depending on the expected conductivity profile, a suitable wavelet basis function can be chosen. The scheme supports multiple types of regularization with the same algorithm and is thus flexible. Finally, we apply this new scheme on a frequency domain electromagnetic sounding (FDEM) dataset, but the scheme could equally apply to any other 1D geophysical method.

Flexible quasi-2D inversion of time-domain AEM data, using a wavelet-based complexity measure

“…In this section, we extend the work of Deleersnyder et al (2021) into two dimensions. The 2D scaledependent wavelet-based regularization term will be described in a step-by-step manner.…”

“…This is the foundation of the sparsifying nature of the wavelet transform. A more detailed example is found in Deleersnyder et al (2021) or in standard works about wavelet theory, such as Mallat (1999).…”

“…Deterministic regularization techniques impose constraints on the model parameters. It is generally a minimum-structure inversion, meaning that unrealistic electrical conductivities are filtered out and in Deleersnyder et al (2021) for 1D inversion. Additionally, we consider a complexity measure per orientation.…”

An overview of multimethod imaging approaches in environmental geophysics

“…The choice of the additional regularization parameters is not straightforward and often arbitrary, but may be estimated objectively from time-lapse data (Nguyen et al, 2016). With a similar goal, but a completely different approach, Deleersnyder et al (2021) present a regularization method, which transforms the model into the wavelet domain. This novel scale-dependent regularization term can favor both blocky and smooth models, as well as high-amplitude features embedded in an otherwise smooth model domain.…”

Assessing and Improving the Robustness of Bayesian Evidential Learning in One Dimension for Inverting Time-Domain Electromagnetic Data: Introducing a New Threshold Procedure

“…Hansen et al [53] studied the effect of using approximate forward models on the inversion of GPR cross-hole travel time data and demonstrated that the modelling error could be more than one order of magnitude larger than the measurement error, leading to unwanted artifacts in the realizations from the posterior probability. For EM methods, studies have demonstrated the negative effect of using fast approximations of the forward model on the accuracy of the inversion [54][55][56]. In particular, for TDEM methods, using accurate models is computationally too expensive to be attractive for stochastic inversion of large data sets, as are those obtained from surveys with airborne or tTEM methods.…”