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

The inversion of electromagnetic induction data to a conductivity profile is an ill-posed problem. Regularization improves the stability of the inversion and, based on Occam's razor principle, 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 in the wavelet domain has both temporal as spatial resolution, and penalizing small-scale coefficients effectively reduces the complexity of the model. Depending on the expected conductivity profile, an optimal wavelet basis function can be chosen. The scheme is thus adaptive. 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

Determining the optimal focusing parameter in sparse promoting inversions of EMI surveys

If the magnetic field caused by a magnetic dipole is measured, the electrical conductivity of the subsurface can be determined by solving the inverse problem. For this problem a form of regularisation is required as the forward model is badly conditioned. Commonly, Tikhonov regularisation is used which adds the $\ell_2$-norm of the model parameters to the objective function. As a result, a smooth conductivity profile is preferred and these types of inversions are very stable. However, it can cause problems when the true profile has discontinuities causing oscillations in the obtained model parameters. To circumvent this problem, $\ell_0$-approximating norms can be used to allow discontinuous model parameters. Two of these norms are considered in this paper, the Minimum Gradient Support and the Cauchy norm. However, both norms contain a parameter which transforms the function from the $\ell_2$- to the $\ell_0$-norm. To find the optimal value of this parameter, a new method is suggested. It is based on the $L$-curve method and finds a good balance between a continuous and discontinuous profile. The method is tested on synthetic data and is able to produce a conductivity profile similar to the true profile. Furthermore, the strategy is applied to newly acquired real-life measurements and the obtained profiles are in agreement with the results of other surveys at the same location. Finally, despite the fact that the Cauchy norm is only occasionally used to the best of our knowledge, we find that it performs at least as good as the Minimum Gradient Support norm

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

In frequency domain Electromagnetic Induction (EMI) surveys, an image of the electrical conductivity of the subsurface is obtained non-invasively. The electrical conductivity can be related to important subsurface properties such as the porosity, saturation or water conductivity via Archie’s law. The advantage of geophysical EMI surveys is its cost-effectiveness because it is a noncontacting method, one can easily walk with the device or mount in on a vehicle or a helicopter (AEM). The process of finding the conductivity profile from the collected field data is an ill-posed inverse problem. Regularization improves the stability of the inversion and, based on Occam’s razor principle, a smoothing constraint is typically used with a very large number of thin layers. However, the conductivity profiles are not always expected to be smooth. Another alternative is to use a predefined number of layers and to invert for their conductivity and thickness. This can yield sharp contrasts in conductivity. In practice however, the real underground might be either blocky or smooth, or somewhere in between. Those standard constraints are thus not always appropriate. We develop a new minimum-structure inversion scheme in which we transform the model into the wavelet space and impose a sparsity constraint. This sparsity constrained inversion scheme minimizes an objective function with a least-squares data misfit and a sparsity measure of the model in the wavelet domain. With a solid understanding of wavelet theory, a novel and intuitive model misfit term was developed, allowing for both smooth and blocky models, depending on the chosen wavelet basis. A model in the wavelet domain has both temporal (i.e. low and high frequencies) and spatial resolution, and penalizing small-scale coefficients effectively reduces the complexity of the model. Comparing the novel scale-dependent wavelet-based regularization scheme with wavelet-based regularization with no scale-dependence, revealed significantly better results (Figure A and B) w.r.t. the true model. Comparing with standard Tikhonov regularization (Figure C and D) shows that our scheme can recover high amplitude anomalies in combination with globally smooth profiles. Furthermore, the adaptive nature of the inversion method (due to the choice of wavelet) allows for high flexibility because the shape of the wavelet can be exploited to generate multiplerepresentations (smooth, blocky or intermediate) of the inverse model

Dataset: Multidimensional surrogate modelling for Airborne TDEM data

DataHF contains the synthetic data for a SkyTEM 304 TDEM system, flying at 40 m height for a two-layered model with an interface at an angle, described in parameters.csv, via 3D simulations. DataLF contains the 1D data (without angle) with a 1D analytical forward model. Look out for the published PhD dissertation "Improving Airborne Time-Domain Electromagnetic Imaging with Applications to Groundwater Salinity Mapping" for a description of the dataset in Chapter 5.Look out for the published PhD dissertation for a description of the dataset

AEM appraisal tool for multidimensionality issues

This code was used to create an appraisal tool which detects multidimensionality issues in Airborne EM inversion in which a 1D forward model was used. The results are published in Remote Sensing Deleersnyder, W., Dudal, D., & Hermans, T. (2022). Novel Airborne EM Image Appraisal Tool for Imperfect Forward Modeling. Remote Sensing, 14(22), 5757. Abstract: Full 3D inversion of time-domain Airborne ElectroMagnetic (AEM) data requires specialists’ expertise and a tremendous amount of computational resources, not readily available to everyone. Consequently, quasi-2D/3D inversion methods are prevailing, using a much faster but approximate (1D) forward model. We propose an appraisal tool that indicates zones in the inversion model that are not in agreement with the multidimensional data and therefore, should not be interpreted quantitatively. The image appraisal relies on multidimensional forward modeling to compute a so-called normalized gradient. Large values in that gradient indicate model parameters that do not fit the true multidimensionality of the observed data well and should not be interpreted quantitatively. An alternative approach is proposed to account for imperfect forward modeling, such that the appraisal tool is computationally inexpensive. The method is demonstrated on an AEM survey in a salinization context, revealing possible problematic zones in the estimated fresh–saltwater interface

Scale-dependent wavelet-based regularization scheme for geophysical 1D inversion

First release of our WABI-code! Scale-dependent wavelet-based regularization scheme for geophysical 1D inversion This flexible inversion scheme allows to easily obtain blocky, smooth and intermediate inversion models. Different inversion models are obtained by simply changing the wavelet basis. db1: blocky inversion models db2-db4: sharper inversion models db5+: smoother inversion models Daubechies (db) wavelets are ideal (see Deleersnyder et al, 2021), however, other wavelets can also be used. Simply run pywt.wavelist() to list the available options. The shape of the wavelet basis function (e.g., look here) is an indication of the type of minimum-structure the regularization method will promote. How to cite The method: Deleersnyder, W., Maveau, B., Hermans, T., & Dudal, D. (2021). Inversion of electromagnetic induction data using a novel wavelet-based and scale-dependent regularization term. Geophysical Journal International, 226(3), 1715-1729. DOI: https://doi.org/10.1093/gji/ggab182 Open Access version on ResearchGate The code: Wouter Deleersnyder, & Robin Thibaut. (2022). WouterDls/1D-wavelet-based-inversion: Wavelet Based Inversion (0.1.0). Zenodo. https://doi.org/10.5281/zenodo.655269

Novel Airborne EM Image Appraisal Tool for Imperfect Forward Modeling

Full 3D inversion of time-domain Airborne ElectroMagnetic (AEM) data requires specialists’ expertise and a tremendous amount of computational resources, not readily available to everyone. Consequently, quasi-2D/3D inversion methods are prevailing, using a much faster but approximate (1D) forward model. We propose an appraisal tool that indicates zones in the inversion model that are not in agreement with the multidimensional data and therefore, should not be interpreted quantitatively. The image appraisal relies on multidimensional forward modeling to compute a so-called normalized gradient. Large values in that gradient indicate model parameters that do not fit the true multidimensionality of the observed data well and should not be interpreted quantitatively. An alternative approach is proposed to account for imperfect forward modeling, such that the appraisal tool is computationally inexpensive. The method is demonstrated on an AEM survey in a salinization context, revealing possible problematic zones in the estimated fresh–saltwater interface.</jats:p

Improving Flanders’ salinization map : applying a scale-dependent wavelet-based regularization scheme on time-domain electromagnetic inversion

The salinization map of the region of Flanders, Belgium shows the depth of the interface between fresh and salt groundwater in the coastal and polder area. It serves as an exploratory tool to examine the potential of groundwater projects that improve freshwater availability in the shallow subsurface. Flanders environment agency published an updated map in 2019, based on airborne time-domain electromagnetic induction data. The result of the inversion is, however, overly smooth, which potentially conceals interesting features. Via an inverse problem, the electromagnetic induction data can be mapped onto a conductivity profile, which serves as a proxy for salinity via petrophysical laws. The inverse problem is ill-posed and regularization improves the stability of the inversion. Based on Occam’s razor principle, a smoothing constraint is typically used with a very large number of thin layers. However, the salinity profiles in the Belgian coastal plains are sometimes sharp, impeding the correct estimation of the fresh-saltwater interface. An alternative is to use a predefined number of layers and to invert for their conductivity and thickness. This can yield sharp contrasts in conductivity. In practice, however, the real underground might be either blocky or smooth, or somewhere in between. Those standard constraints are thus not always appropriate. With a novel wavelet-based inversion scheme, the original data can be re-interpreted in a flexible fashion. In simple terms, a wavelet function can be seen as a building block and a simple model is one that can be built with few building blocks of various sizes. Our proposed inversion scheme adds a regularization term that limits the number of building blocks in the wavelet-domain to make sure only the necessary complexity is retrieved. The scheme is tuned by only one additional parameter (which determines the wavelet basis function) and is able to recover blocky, intermediate and smooth structures. It is also capable to recover high amplitude anomalies in combination with globally smooth profiles, a common problem for smooth inversion, and an essential feature to accurately predict the salinity. We first demonstrated this alternative inversion scheme on 1D data and now extend it to two dimensions. We apply it to the data of the salinization map to validate our inversion model with ground-truth data collected in boreholes. Our approach yields an improved estimation of the fresh-saltwater interface and can thus be used to update the salinity map

Multidimensional forward modelling of EM induction data within a salinization context : is it worth the extra cost?

In (time-domain) Electromagnetic Induction (EMI) surveys, an image of the electrical conductivity of the subsurface is obtained non-invasively. The electrical conductivity serves as a proxy for salinity via petrophysical laws. The advantage of geophysical EMI surveys is their cost-effectiveness because it is a non-contacting method, one can easily walk with the device or mount it on a vehicle or a helicopter (AEM). An accurate interpretation of the data is computationally expensive as it requires a full 3D simulation of the induced electric currents embedded within an iterative and ill-posed inverse problem. Therefore, this forward model is usually approximated with an 1D forward model which only considers horizontal layers, for which fast analytical forward models exist. Quasi-2/3D inversion allows for lateral variation in the subsurface models, but uses those 1D forward models to generate the data. The final inversion model usually fits the (potentially intrinsic 2/3D) data well up to noise level. But what with the discrepancy between the 1D and 2D data? The biased modelling error, introduced by using a 1D forward model in a 3D problem, is difficult to estimate. Does the inversion model that fits the data via 1D model also fit the data via a 3D model? This question has already been addressed in the literature about fault detection, but in a saltwater intrusion context, the lateral variation is expected to be much smoother. And the question remains to what extent multidimensional modelling is crucial. The time-domain AEM field data from the salinization map of the region of Flanders, Belgium is used as the case study (Flanders environment agency published the map in 2019). A specific flight line is selected for which validation data is available that shows a 2D (lateral) variation. Both results from the quasi-2D and stitched inversion with a traditional smoothing regularization is presented. An accurate 3D forward modelling is performed on both inversion models via the SimPEG package. The results of the simulations are compared with the actual field data and help us to answer the question of whether multidimensional modelling is crucial in geophysical inversion at the AEM scales and a saltwater intrusion context