GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 Where , and . Finally, the tme-domain IP decay is computed as the convoluton of the impulse response with the current waveform between the electrodes, solved as proposed by Fiterman and Anderson (1986) for piecewise linear current waveforms. EEMverter is implemented in such a way that the inversion parameters are defned on the nodes of the model mesh and migrated to the forward mesh through interpolaton (that can be chosen and selected by the user). The spatal decoupling between model and forward meshes allows for defning the model parameters, e.g., the Cole-Cole (Cole and Cole 1941; Pelton et al. 1978) ones, on several model meshes, one for each inversion parameter, for example. For each dataset of the inversion process, a distnct forward mesh is defned. The interpolaton from model parameters M into the values m i is expressed through a matrix multplicaton: (6) in which the matrix holds the weights of the interpolaton that depends only on the distances between model mesh nodes and the subdivisions of the i th forward mesh (Fiandaca et al., 2024). As for the forward response, the Jacobian matrix is computed in the frequency domain and then transformed into the tme domain. The tme-domain Jacobian in the i th forward mesh is computed as: (7) where the matrix holds the Hankel coefcients, the matrix implements the efects of current waveform, gate integraton and flters and the frequency-domain Jacobian is calculated in 1- D through fnite diference and in 2-D/3-D using the adjoint method and the chain rule as in Fiandaca et al. (2013) and Madsen et al. (2020), thus allowing to use any parameterizaton of the IP phenomenon in the inversion: where is the Jacobian of the i th forward mesh with respect to the complex conductvity and is the partal derivatve of the complex conductvity versus the model parameters (Fiandaca et al., 2024). The Levenberg-Marquardt linearized approach is used for computng the inversion model: (8) r = t 2 π λ = ω 2 π f 1 ( λ ) = 1 λ f ( λ 2 π ) i ( t ) m i = f i ( M ) = F i ∙ M F i J m i , TD = A ∙ T ∙ J m i , FD J m i , FD J m i , FD = J σ* , i ∙ ∂ σ * ∂ m i (7) J σ* , i σ* ∂ σ * ∂ m i M n +1, j = M n , j + [ J T M , j C −1 d J M , i + R T C −1 R , R j + λI ] −1 ∙ [ J T M , j C −1 d ∙ ( d − f n , j ) + R T C −1 R , j R j ∙ M n , j ]

RkJQdWJsaXNoZXIy MjQ4NzI=