General formalism for the efficient calculation of the Hessian matrix of EM data misfit and Hessian-vector products based upon adjoint sources approach

3-D electromagnetic (EM) studies of the Earth have advanced significantly over the past decade. Despite a certain success of the 3-D EM inversions of real data sets, the quantitative assessment of the recovered models is still a challenging problem. It is known that one can gain valuable information about model uncertainties from the analysis of Hessian matrix. However, even with modern computational capabilities the calculation of the Hessian matrix based on numerical differentiation is extremely time consuming. Much more efficient way to compute the Hessian matrix is provided by an ‘adjoint sources' methodology. The computation of Hessian matrix (and Hessian-vector products) using adjoint formulation is now well-established approach, especially in seismic inverse modelling. As for EM inverse modelling we did not find in the literature a description of the approach, which would allow EM 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 Hessian matrix using adjoint sources approach. We also show how this technique can be implemented to calculate multiple Hessian-vector products very efficiently. The formalism is general in the sense that it allows to work with responses that arise in EM problem set-ups either with natural- or controlled-source excitations. The formalism 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 two EM techniques: magnetotellurics and controlled-source sounding with vertical magnetic dipole as a sourc