We develop a three-dimensional inversion code to image the resistivity distribution of the subsurface from frequency-domain controlled-source electromagnetic data. Controlled-source electromagnetic investigations play an important role in many different geophysical prospecting applications. To evaluate controlled-source electromagnetic data collected with complex measurement setups, advanced three-dimensional modelling and inversion tools are required.

We adopt a preconditioned non-linear conjugate gradient algorithm to enable three-dimensional inversion of impedance tensor and vertical magnetic transfer function data produced by multiple sets of two independent active sources. Forward simulations are performed with a finite-element solver. Increased sensitivities at source locations can optionally be counteracted with a weighting function in the regularization term to reduce source-related anomalies in the resistivity model. We investigate the capabilities of the inversion code using one synthetic and one field example. The results demonstrate that we can produce reliable subsurface models, although data sets from single pairs of independent sources remain challenging.

We present a synthetic inversion study illustrating two approaches which help to deal with heterogeneous sensitivities in 3D frequency-domain controlled-source electromagnetic inverse problems.

Using edge-based vector finite-element approximations on tetrahedral meshes and a preconditioned non-linear conjugate gradient algorithm, we invert for impedance tensor elements generated by a set of two coincident perpendicularly oriented horizontal electric or horizontal magnetic dipole sources. Depending on the number and locations of sources and the choice of impedance tensor components used for inversion, the sensitivity patterns can differ significantly. Measurement setups with a small number of sources, but many receiver stations at the surface covering near-field, transition zone and far-field, are often deployed for land-based controlled-source electromagnetic measurements. Such a setup can result in accumulated sensitivities close to the sources and receivers, which implies strongest model updates in these regions and can mislead the inverse algorithm to a search direction, where no physically meaningful model can be produced nor the data are fitted.

In order to mitigate the influence of strong sensitivities near sources and receivers on the inversion process, we apply an efficient preconditioner and customisable weights in the model regularisation matrix. The preconditioner is updated with the Broyden-Fletcher-Goldfarb-Shanno algorithm using the diagonal of the approximate Hessian matrix as start preconditioner. The latter is computationally expensive to obtain, but aims at finding a favourable search direction for the inverse algorithm already in early iterations and distributing the model update more evenly in the domain. To account for the sensitivity loss with depth, we implemented a depth weighting functional in the model regularisation term. The approach is based on counteracting the exponential and geometrical decay of the electromagnetic fields with depth and distance from the sources. In practical, we increase the smoothing in the shallow part of the model close to the source locations, where no structure is expected. We present synthetic examples indicating that this approach is an efficient way of helping the inversion to converge, obtaining a reliable model and resolving structure at depth.