GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 for defning the model parameters, e.g. the Cole-Cole ones, on several model meshes, for instance one for each inversion parameter. In this way, it is possible to defne the spectral parameters, like the tme constant and the frequency exponent in the Cole-Cole model, on meshes coarser than the resistvity and chargeability ones, vertcally and/or horizontally, with a signifcant improvement in parameter resoluton. For each dataset of the inversion process, a distnct forward mesh is defned. The interpolaton from the model parameters defned on the model mesh nodes into the values at the subdivisions of the i th forward mesh is expressed through a matrix multplicaton, in which the matrix holds the weights of the interpolaton, which depends only on the distances between model mesh nodes and the subdivisions of the i th forward mesh: (1) In EEMverter 1D, 2D and 3D forward & Jacobian computatons have been implemented. In partcular, Transient EM data are modelled in 1D following Efersø et al. (1999); in 3D the forward soluton is carried out in frequency domain, with the fnite element method, both with tetrahedral elements or with the octree approach, similarly to what has been done with the tme-stepping tme-domain approach in Zhang et al. (2021) and Xiao et al. (2022a). The fnite element approach is used also for frequency-domain galvanic computatons in 2D (Fiandaca et al., 2013) and 3D (Madsen et al., 2020). The transformaton to tme-domain is obtained through a fast Hankel transformaton (as in Efersø et al., 1999) for both the forward response and the Jacobian. The Jacobian of the model space is computed summing the contributons of all forward meshes up (Christensen et al., 2017; Madsen et al., 2020, Zhang et al., 2021), using the domain decompositon with a forward mesh for each sounding in 3D EM computatons (Cox et al., 2010; Zhang et al., 2021): (2) The total Jacobian is used for computng the inversion model in a Levenberg-Marquardt linearized approach as follows: (3) M m i F i m i = f i ( M ) = F i ∙ M J M J M = ∑ i J m i ∙ F i T M n +1, j = M n , j + [ J M , j T C d −1 J M , i + R T C R , j −1 R j + λ I ] −1 ∙ [ J M , j T C d −1 ∙ ( d − f n , j ) + R T C R , j −1 R j ∙ M n , j ]