General formalism for the efficient calculation of derivatives of EM frequency-domain responses and derivatives of the misfit

Electromagnetic (EM) studies of the Earth have advanced significantly over the past few years. This progress was driven, in particular, by new developments in the methods of 3-D inversion of EM data. Due to the large scale of the 3-D EM inverse problems, iterative gradient-type methods have mostly been employed. In these methods one has to calculate multiple times the gradient of the penalty function—a sum of misfit and regularization terms—with respect to the model parameters. However, even with modern computational capabilities the straightforward calculation of the misfit gradients based on numerical differentiation is extremely time consuming. Much more efficient and elegant way to calculate the gradient of the misfit is provided by the so-called ‘adjoint' approach. This is now widely used in many 3-D numerical schemes for inverting EM data of different types and origin. It allows the calculation of the misfit gradient for the price of only a few additional forward calculations. In spite of its popularity we did not find in the literature any general description of the approach, which would allow researchers to apply this methodology in a straightforward manner to their scenario of interest. In the paper, we present formalism for the efficient calculation of the derivatives of EM frequency-domain responses and the derivatives of the misfit with respect to variations of 3-D isotropic/anisotropic conductivity. The approach is rather general; it works with single-site responses, multisite responses and responses that include spatial derivatives of EM field. The formalism also allows for various types of parametrization of the 3-D conductivity distribution. Using this methodology one can readily obtain appropriate formulae for the specific sounding methods. To illustrate the concept we provide such formulae for a number of EM techniques: geomagnetic depth sounding (GDS), conventional and generalized magnetotellurics, the magnetovariational method, horizontal gradient sounding (HGS) and a method that combines HGS with GDS. We also show how the developed formalism can be adapted for the inversion of multisite responses—horizontal magnetic and electric tensor