Estimation of the depth of magnetization from 3-D elastic net sparse inversion model of geomagnetic data

SUMMARY Inversion of geomagnetic anomaly data poses an ill-posed problem, and extremal models such as equivalent source layers or point-source distributions can explain observations to the same degree as volumetric magnetization distributions. However, the spectral characteristics of magnetic anomalies provide fundamental constraints for magnetic source-depth estimation. Specifically, the maximum detectable depth of crustal magnetic sources is dictated by the longest wavelengths present in the field, which correspond to the low-wavenumber bands of the spectrum. This relationship is often analysed through the log power spectrum versus wave number plot, using the slopes of the linear segment for depth estimation. Methods aiming at reconstructing the depth to the bottom of magnetization from spectral field characteristics are commonly referred to as spectral methods. However, these methods are based on assumptions about the statistical properties of the source distribution and are prone to misinterpretations. Here, we apply sparsity-constrained 3-D inversion of magnetic data using an elastic net regularization to recover the susceptibility distribution and the bottom of magnetization. We claim that the elastic net ($\ell _2\ell _1$ norm) regularization, when properly tuned to balance the solution’s smoothness with sparsity, stabilizes the inversion, avoiding extremal magnetization distributions and generating a geologically plausible source depth distribution that is consistent with the expected source distribution. The $\ell _1$ norm brings sparsity and high resolution, while the $\ell _2$ norm brings inversion stability and structural continuity to the final model. From the recovered 3-D elastic net sparse inversion model, we extract the depths of all the deepest non-zero susceptibility values and suggest this to be an alternative estimate to the base of magnetization. Moreover, we suggest that the resulting 3-D model has a value in itself and may aid geological interpretation.