Sharp spatially constrained inversion with applications to transient electromagnetic data

Abstract

Time-domain electromagnetic data are conveniently inverted by using smoothly varying 1D models with fixed vertical discretization. The vertical smoothness of the obtained models stems from the application of Occam-type regularization constraints, which are meant to address the ill-posedness of the problem. An important side effect of such regularization, however, is that horizontal layer boundaries can no longer be accurately reproduced as the model is required to be smooth. This issue can be overcome by inverting for fewer layers with variable thicknesses; nevertheless, to decide on a particular and constant number of layers for the parameterization of a large survey inversion can be equally problematic. Here, we present a focusing regularization technique to obtain the best of both methodologies. The new focusing approach allows for accurate reconstruction of resistivity distributions using a fixed vertical discretization while preserving the capability to reproduce horizontal boundaries. The formulation is flexible and can be coupled with traditional lateral/spatial smoothness constraints in order to resolve interfaces in stratified soils with no additional hypothesis about the number of layers. The method relies on minimizing the number of layers of non-vanishing resistivity gradient, instead of minimizing the norm of the model variation itself. This approach ensures that the results are consistent with the measured data while favouring, at the same time, the retrieval of horizontal abrupt changes. In addition, the focusing regularization can also be applied in the horizontal direction in order to promote the reconstruction of lateral boundaries such as faults. We present the theoretical framework of our regularization methodology and illustrate its capabilities by means of both synthetic and field data sets. We further demonstrate how the concept has been integrated in our existing spatially constrained inversion formalism and show its application to large-scale time-domain electromagnetic data inversions

Voxel inversion of airborne electromagnetic data for improved groundwater model construction and prediction accuracy

“…Figure 1 provides a workflow for the method. First, the gathered airborne electromagnetic (AEM) data from the survey area are inverted with smooth or sharp horizontal and vertical constraints (Vignoli et al, 2015). This is done by using a recently developed voxel inversion scheme which decouples the geophysical model from the position of the acquired data .…”

“…The reason for this is found in its minimum-structure L2 norm inversion formalism (Constable et al, 1987;Menke, 2012). Following the notation used by Vignoli et al (2015), this can be expressed as…”

“…The second regularization method is the minimum gradient support (Portniaguine and Zhdanov, 1999;Blaschek et al, 2008;Vignoli et al, 2015), which allows for large sharp vertical and horizontal model transitions. The minimum gradient support regularization seeks to minimize the spatial variations vertically and laterally by penalizing the vertical and horizontal model gradients through the stabilizer expressed as (Vignoli et al, 2015) …”

“…If, for example, smooth regularization is used to estimate resistivity models in a sharply layered system, it will produce a blurred resistivity distribution from which one should be careful with inferring the spatial distribution of hydraulic conductivity to be used in a groundwater model. In this case, it would be better to use minimum gradient support regularization (Portniaguine and Zhdanov, 1999;Blaschek et al, 2008;Vignoli et al, 2015) for the geophysical inversion because the estimated resistivity distribution will tend to consist of fewer, more sharply defined layer boundaries (vertically and horizontally). However, it is often ignored that geophysical data can be inverted using alternative regularization schemes, and to test whether the alternative geophysical models affect the predictive capability of a groundwater model.…”

Calibrating electromagnetic induction conductivities with time-domain reflectometry measurements

“…Besides the classical regularization matrices based on the discretization of the first and second derivatives, in all the cases characterized by sharp interfaces, we tested a nonlinear regularization stabilizer promoting the reconstruction of blocky features and thus to improve the spatial resolution of EMI inversion results. (Zhdanov et al, 2006;Ley-Cooper et al, 2015;Vignoli et al, 2015Vignoli et al, , 2017. The advantage of this relatively new regularization is that, when appropriate prior knowledge about the medium to reconstruct is available, it can mitigate the smearing and over-smoothing effects of the more standard inversion strategies.…”